Geophysics: Field Measurements (Experimental Methods in the Physical Sciences)

Methods of Experimen taI Physics VOLUME 24 GEOPHYSICS PART 6 Fleld Measurements METHODS OF EXPERIMfNTAL PHYSICS Robe...

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Methods of Experimen taI Physics VOLUME 24 GEOPHYSICS PART 6

Fleld Measurements

METHODS OF EXPERIMfNTAL PHYSICS Robert Celotta and Judah Levine, Editors-in-Chief

Founding Editors

L. MARTON C. MARTON

Volume 24

Geophysics PART B Field Measurements

Edited by

Charles G. Sammk Department of Geological Sciences University of Southern Cafifornia Los Angeles, California

Thomas L. Henyey Department of Geologlcai Sciences University of Southern California Los Angeles, Californla

ACADEMIC PRESS, INC. Hercourt Brace Jovanovich, Publishers

Orlando San Dlego New York Austln Boston London Sydney Tokyo Toronto

COPYRIGHT @ 1987 BY ACADEMIC PRESS. INC. ALL RIGHTS RESERVED. NO PART OFTHIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS. ELECTRONIC OR MECHANICAL. INCLUDING PHOTOCOPY. RECORDING. OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM. WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS, INC. Orlando. Florida 32887

Unired Kingdom Edition published by

ACADEMIC PRESS INC.

(LONDON) 24-28 Oval Road, London NWI 7DX

LTD

Library of Congress Cataloging in Publication Data (Revised for volume 24, Part B) Geophysics. (Methods of experimental physics; v. 24) Includes indexes. Contents: pt. A. Laboratory measurements - pt. B. Field measurements. 1 . Geophysics. I . Sammis, Charles G. 11. Henyey, Thomas L. (Thomas Louis), Date I l l . Title. IV. Series. QE501.G48 1987 551 86-17439 ISBN 0-12-475967-X (v. B : alk. paper)

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PRINTED IN THE UNITED STATES OF AMERICA 8 1 8 8 8 9 9 0

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CONTENTS PREFACE .

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10. Seismic Instrumentation TA-LIANGTENG 1. 2. 3. 4. 5. 6. 7. 8.

Introduction . . . . . . . Historical Development . . . . Nature of Seismic Ground Motions . Basic Types of Seismic Sensors . . Damping Devices and Transducers . Pendulum-Galvanometer Interaction Central Recording and Networking . Recent Developments in Seismographs References . . . . . . . .

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1 2 11 14 25 32 35 50 73

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77 87 97 110 120 123

11. Marine Acoustic Techniques F . N . SPIESS

. Introduction: The Ocean Acoustic Environment

1 2 3 4 5

. Echo Sounders . . . . . . . Bottom-Imaging Sonars . . . . . Acoustics for Position Determination . A System Example . . . . .

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12. Surface Measurement of the Earth’s Gravity Field JAMESH . WHITCOMB 1. Introduction . 2 . Instrumentation 3 . Applications . References . .

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CONTENTS

13. Satellite Measurement of the Earth’s Gravity Field WILLIAM M . KAULA

. Introduction and HistoricalSummary .

1 2 3 4

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. Geodetic Satellites . . . . . . . . . . . . . Satellite Orbit Dynamics . . . . . . . . . . . . DataAnalysis . . . . . . . . . . . . . .

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References .

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163 164 170 178 186

14 Experimental Methods in Continental Heat Flow

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DAVIDD BLACKWELL AND ROBERT E. SPAFFORD

. Introduction . . . . . . . Temperature . . . . . . . Thermal Conductivity . . . . Heatproduction . . . . . . Heat Flow Calculation . . . . Miscellaneous Techniques . .

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. FutureResearch References .

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189 192 206 213 213 217 219 221

15. Measurement of Oceanic Heat Flow R. P VON HERZEN

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. . 3. Thermal Conductivity 4. Heat Flow . . . .

1 Introduction . . . 2 Temperature Gradients

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16. Electrical Methods in Geophysical Prospecting STANLEY H . WARD

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1 Introduction . . . . . . . 2 Elementary ElectromagneticTheory

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3 Electrical Properties of Earth Materials.

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4 Basic Principles of Resistivity and Induced Polarization

Surveys

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5 MagnetotelluricMethod . . . . . . 6 Controlled-SourceElectromagneticMethods

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313 346 366

17. Measurement of In Situ Stress BEZALEL C HAIMSON

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. Introduction . . . . . . . . . . . . . .

1 2 3 4 5 6

. Surface Measurements . . . . . . . . . . . . Near-Surface Measurements . . . . . . . . . . . Deep Measurements . . . . . . . . . . . . . State of Stress in the Earth’s Crust . . . . . . . . . Future Research . . . . . . . . . . . . . References .

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377 379 383 393

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18. Continuous Measurement of Crustal Deformation DUNCAN CARRAGNEW

. Measurement . . . . . 2 . Quantities to be Measured . 3 . General Design Features . . 4 . Tiltmeters: Particular Designs 5 . Strainmeters . . . . .

1 Aims and Problems of Continous Deformation

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19. Geophysical Well Logging JAYTITTMAN

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1 Introduction . . . . . . . . . . . . . . 2. Geological and Petrophysical Interpretation of Logging

. . . . . . . . References . . . . . . . . . INDEX . . . . . . . . . . . . CONTENTS OF VOLUME24. PART A . . .

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Measurements .

3 The Physics of Logging Measurements

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PREFACE Geophysics uses many of the methods and techniques of physics to study the solid earth and planets. Geophysical data are collected both in the laboratory and in the field. Volume 24, Part A of this two-volume set discusses the laboratory techniques, and Volume 24, Part B discusses the field techniques used in geophysics. Field measurements provide the basic data set against which all geophysical models must be tested. Geophysical field methods are directed principally toward investigation of the earth’s interior-that portion of the earth that is inaccessible to geologic observation. The methodology involved in field research generally falls into one of three categories. In the first category, instruments are strategically deployed to passively detect the natural waves and potential fields of the earth. Properties of these fields are governed by the physics of the earth’s interior-its composition, rheology, and structure. Data from these measurements can then be “inverted” to recover or constrain the physics. Examples of these methods include the detection of wavefields generated by earthquakes and the measurement of gravitational and geomagnetic fields. In the second category, wave or potential fields are artificially generated at the earth’s surface and the resulting reflected or secondary fields are measured using carefully designed arrays of surface instruments. Examples include reflection seismology and electromagnetic methods. Finally, the third category involves the direct measurement of in situ properties of the earth, such as the state of crustal stress or the physical properties of rock into which deep borings have been made. Seismology is the most powerful method used for exploring the earth’s interior. In Chapter 10,T. Teng discusses instrumentation used to detect and record seismic wavefields. Instrumental techniques used to exploit the broad seismic bandwidth are described. In a related discussion in Chapter 11, F. Spiess reviews acoustic techniques used in the marine environment to characterize the sea floor. The emphasis is on high-resolution signal generators or transducers operating at frequencies above 1 kHz. We have not included a discussion of the seismic reflection method as carried out on land or at sea. This method uses signals generated in the 1 to 100-Hzbandwidth to map geologic structures in the continental and oceanic crust, and has been highly refined by the petroleum exploration industry. Excellent discussions of the experimental techniques and operational methods involved can be found in publications of the Society of Exploration Geophysics, and in texts such as “Exploration Seismology,” Volumes 1 and 2, by R. E. Sheriff and L. P. Geldhart. ix

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PREFACE

Knowledge of the distribution of mass in the earth, particularly in the outer heterogeneous layers, is fundamental to understanding the structure of the lithosphere and the geometry of convection currents that drive the lithospheric plates. In Chapters 12 and 13, J. Whitcomb and W. Kaula describe techniques for measuring the earth’s gravity field from the surface and from satellites, respectively. Surface measurements provide high resolution but spotty coverage of the earth as a whole. Satellites, on the other hand, obtain a synoptic view of the gravity field, uncovering areas that are otherwise inaccessible. Estimates of temperatures in the earth, particularly for the deep interior, vary widely. Much of the imprecision in our knowledge stems from an incomplete understanding of the distribution of heat sources. The nature of convection in the mantle is critically dependent on the internal temperature distribution. Surface heat flow measurements provide the only direct estimates of internal temperatures, particularly within the continental and oceanic lithospheres. Chapter 14, by D. Blackwell and R. Spafford, and Chapter 15, by R. Von Herzen, review methods for making heat flow measurements on the continents and in the oceans, respectively. Oceanic heat flow measurements have played a fundamental role in the development of the theory of sea floor spreading. Chapter 16, by S. Ward, reviews electrical methods in geophysical exploration. These methods, which include resistivity, induced-polarization, and magnetotelluric and electromagnetictechniques, have been used extensively for groundwater exploration, and by the mining and geothermal industries. Some, most notably magnetotelluric methods, are now being applied to deep exploration of the crust and mantle. We have not included a discussion of the magnetic method of exploration, a passive measurement technique based on perturbations of the earth’s largely dipolar magnetic field by rock masses displaying remanent and induced magnetization. Magnetic measurements are also important in paleomagnetism (see Chapter 9 in Volume 24, Part A) and in descriptions of spatial and temporal characteristics of the earth’s dipole and non-dipole fields. Discussions of magnetic methods can be found in a variety of texts, such as “Interpretation Theory in Applied Geophysics,” by F. S. Grant and G. F. West. The in situ state of stress is directly related to tectonic processes in the lithosphere-specificaIly earthquakes and the mechanics of faulting. In Chapter 17, B. Haimson reviews the methods for measuring stress in the crust. Perhaps the most important advance in in situ stress determinationin the past several years has been the refinement and general use of the borehole hydraulic fracturing technique. Tectonic stresses applied by one plate on another across lithospheric plate boundaries, by convection-induced basal tractions or by gravitational body forces, result in nonuniform strain fields with the plates. Faults represent important singularities in these

PREFACE

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strain fields. Chapter 18, by D. Agnew, summarizes the methods for continuous measurement of crustal deformation, many of which have been refined over the past 10 years in response to a national program in earthquake hazard reduction and fault zone monitoring. More than any other kind of instrument, strainmeters require the ultimate in long-term stability. Finally, Chapter 19, by J . Tittman, reviews the methods used in geophysical well logging. Because of the importance of these methods to hydrocarbon exploration and production, many specialized instruments and techniques, some proprietary, have been developed and refined by the petroleum industry. However, as more deep borings become available to the general scientific community, well logging methods will find their way into the arsenal of basic geophysical research tools for the characterization of rock physics. This area of geophysical instrumentation has the potential for significant scientific and economic profit.

THOMAS L. HENYEY CHARLES 0.SAMMIS


10. SEISMIC INSTRUMENTATION

Ta-Liang Teng Department of Geological Sciences University of Southern California Los Angeles, California 90089-0741

1. Introduction Elastic radiation emanating from a seismic source propagates as waves traveling over the surface and through the interior of the earth. An instrument registering seismic waves is a seismograph, and a record so registered is a seismogram. From seismograms comes our knowledge of the global distribution of earthquakes, of the internal structure of the earth, and of the nature of the seismic source process. In modern seismological observations, a number of seismographs form a network, which is the basic scientific tool (analogous to major telescopes in astronomy) that provides a continuing data base fundamental to the science of seismology. The quantities to be measured by seismological instruments are the time history of displacements and their derivatives (velocities, accelerations, and strains) at the surface, which give the boundary values from which the earth's internal constitution as well as the seismic source structures are deduced. The measurement of these seismic boundary values is itself a science called seismometry. It is the purpose of this chapter to give a comprehensive account of the seismic instrumentation that forms the backbone of seismometry. One of the characteristics of seismometry is that measurements are made over an enormous magnitude range (for displacements from lo-'' to 10" m and accelerations from lo-' to 10"g) and a broad frequency band to lo2 Hz). In some applications, a frequency range down to DC or up to lo4 Hz is required. Another salient feature of seismometry is that at the same time that measurements are being made, both the observer and the object are subject to disturbing ground motions. The measurement sought is the motion history of this observed object with respect to an inertial frame which does not exist on the earth at the time of the passage of seismic waves. Therefore, much of the effort in the development of seismometry has been devoted to establishing an adequate pseudostationary point for an inertial reference 1 METHODS OF EXPERIMENTAL PHYSICS Vol. 24, Part B

Copyright 0 1987 by Academic Press, Inc. All rights of reproduction in any form reserved.

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frame against which measurements may be made to a sufficient degree of accuracy. There are basically three devices that can be used for measuring seismic ground motions: (1) the pendulum sensor, which makes use of an inertial mass loosely coupled to the sensor housing ; (2) the strainmeter, which measures the difference between displacements at two distinct points ; and (3) the pressure sensor. A possible fourth device can be developed, at least in principle, using a gyroscope suspended in a balanced, frictionless gimbal system. Conservation of angular momentum of such a system offers the potential for measuring the rigid body rotation on top of displacement. Such devices are in common use in inertial navigation. If their state-of-the-art sensitivity and stability can be improved, they could potentially be adapted to seismological applications and could provide more complete descriptions of the earth’s ground motions. Although the sensor is the most crucial element, a complete seismic monitoring system today consists of five parts : the sensor, the signalconditioning device, the recorder, the timing device, and the seismic data telemetry link. To provide better areal coverage with an accurate common time base and to improve operational economy, seismic stations are interconnected through telemetry Iinks to form regional networks. Depending on the type of application, a multitude of designs for all these five parts are in current use. Detailed discussions of the designs and applications will be given in the following sections.

2. Historical Development Seismology is a young science. Yet the design and construction of earthquake-detecting devices can be traced to as early as A.D. 132, when a Chinese astronomer and mathematician, Chang Hang, invented a machine that would register the occurrence of an earthquake and give the approximate direction of the wave approach. It is quite an ingenious device, making use of mass inertia as the triggering mechanism. Once triggered, the instrument would cause the release of a copper ball held in the mouth of one of eight dragon heads arranged in eight equal azimuthal directions. The released ball would drop into the open mouth of one of eight waiting toads below, thus also giving the direction of wave approach. In the early 18th century, a French device consisting of a bowl of mercury with small holes drilled on the rim would indicate an earthquake and its direction by mercury overflowing through the holes during the passage of earthquake waves. The first European pendulum-based measuring devices that recorded the strengths of earthquakes came into use in the mid-1700s. A suspended mass, such as a

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SEISMIC INSTRUMENTATION

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simple pendulum, remains momentarily stationary as the earth quakes. A track record can be obtained on a thin sand layer or on smoked glass that gives the relative motion between the pendulum and the earth. These devices, called seismoscopes, have since been reinvented many times in different parts of the world. An obvious improvement is to have the instrument make a mark on a rotating drum to record the time of earthquake occurrence. An instrument that writes the earthquake motion as a time function is called a seismograph; it was formally developed in about 1880 in Japan by a group of British workers-Gray, Milne, and Ewing. A record giving the time function of the earth’s ground motions is called a seismogram. It is the study of seismograms that has established seismology as a branch of quantitative physical science. Since the first introduction of seismographs, the progress in seismic instrumentation has by and large followed the advances of technology. With the establishment of new and better seismic stations, the progress of observational seismology has been punctuated by milestone findings. Early examples include the 1889 first identification of a distant earthquake on a seismogram (an instrument in Potsdam, Germany, recorded an earthquake in Japan), the 1909 discovery of the base of the continental crust, now known as the Moho, the 1913 determination of the depth of the earth’s core, and the 1922 discovery of deep-focus earthquakes. The rate of new findings accelerates as more and better instruments are being deployed. T o follow the developmental history of seismic instrumentation, one must also take note of the worldwide deployment programs of seismograph networks-a special feature that makes seismology a truly international science. A seismograph system generally consists of four parts: the sensor, the signal-conditioning device, the clock, and the recorder. If a number of seismic stations are interconnected for central recording with a common time base, a fifth part-the telemetry links-will also be an integral component of the overall seismic network. Of the five parts that generally form a seismic network, the sensor is a component unique in seismic instrumentation. We will devote more discussion to its design principles and its evolution with time. Development of the other four components, which are common to many other instrument systems, followed closely the progress of technology. Taking the timing device, for instance, at the turn of the century, when the seismograph first came into common use, it was difficult to keep the daily drift rate of the mechanical clock of a seismograph system to within seconds. Synchronization of clocks among distance seismic stations was an impossible task. This difficulty was not alleviated until the 1940s, when electric clocks were incorporated in seismograph systems together with standard time radio receivers that allowed synchronization with the worldwide standard time broadcasting

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service that was then just beginning. This improvement permitted timekeeping in a seismograph station to be within a fraction of a second. As clocks with quartz oscillators (in a constant-temperature oven) came into common use in the 1960s, satellite broadcasting of standard time commenced in the late 1970s, and continuous synchronization became practical with modern electronic devices, the timing problem was essentially solved in seismograph operation with millisecond accuracy. Early seismographs made use of a rotating drum as a recorder. A sheet of smoked paper was wrapped around the drum. As the drum rotated, it also translated slowly, allowing a very fine helicoidal line to be scratched on the smoked paper by a pen attached to the inertial mass-the pendulum. Timing minute marks appeared as square pulses on the signal trace. A whole day’s record would register on one sheet of paper with a time scale of 30-60 mm to the minute. These smoked-paper recorders later evolved into other types that make use of an ink pen on ordinary paper or a light beam on photosensitive paper. The light beam device was able to increase the instrument magnification and remove the trace curvature on the recorded waveforms because of the short arm length of the recording pen. As technology progressed, the use of analog tape recorders together with multipen chart recorder playback greatly increased the recording capacity. In normal seismic network operation, the highest frequency t o be recorded is about 25-30 Hz. It was then possible to multiplex many signal channels (usually eight) on each recording tape track. A 1-inch-wide tape with 14 recording tracks, for example, has the capacity to record 112 seismic signal channels. This large capacity plus the advant of signal conditioning and telemetry electronics brought about in early 1960s the current mode of seismic network operation that combines central recording with data telemetry. These analog tape units require one long tape (7200 ft) per day continuously running at a low speed (15/16 ips). Normally, 99% of the tape content gives non-earthquake-related signals or background noise. Therefore, as digital computers come to common use, many seismic networks are converted to digital recordings. With a signal-discriminating device (usually software in the computer CPU, or a bank of microprocessors, one for each input signal channel) it is possible to record only seismic events, thus greatly compressing the tape contents. The digital tape of seismic events lends itself conveniently to downstream computer data processing pertinent to the seismic events. Although computerbased seismic recording is the current state of the art, drum recorders are still in common use, if sometimes only as a supplement. The visible helicoidal paper records, after all, provide a good means for quick diagnosis of network performance as well as an up-to-the-minute summary on seismicity. The signal-conditioning device was not part of the system when the seismography was first introduced, for then the differential motions between

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the pendulum and the recorder were magnified mechanically through level arm arrangements. The device became desirable in the 1900s, when electromagnetic transducers were introduced that converted penduludmotions into electric signals. These signals would have to be amplified and filtered before they could be applied to recorders. With the progress of electronics, the signal-conditioning device became increasingly complex and advanced to meet various applications. Up to the 1950s, a good deal of work was devoted to the design of stable long-period galvanometers that could be coupled with the electromagnetic seismographs. As low-noise, high-impedance preamplifiers became available, galvanometers were replaced by electronic amplifiers and filters, which not only eliminated the need for photographic recording as required by the galvanometer, but also gave high amplification and broadband performance. At the same time, high magnification permitted the seismometer to become much smaller. The low power consumption of these new signal-conditioning devices permitted their operation on DC power, and this has brought about unattended telemetered seismic stations. Telemetry is the latest entry of the seismograph system. The need for large areal coverage prompted a rapid increase in the number of seismic stations. For economical operation, telecommunication technology was first applied to seismic monitoring in the early 1960s. Through either telephone or radio transmission links, signals from seismometers, after preliminary conditioning, were used to modulate bands of carrier frequencies, then multiplexed and transmitted through the telemetry link. At the recording end (usually the observatory), the received signals were demultiplexed and further conditioned before being applied to recording devices. Today, a telemetered seismic network may consist of as many as several hundred seismic stations, with all signals centrally recorded on a common time base. Since field stations do not depend on AC power and are unattended, with infrequent maintenance visits, it is much easier to site stations in remote and strategic locations with a quiet background, resulting in networks of high sensitivity and good coverage. Lately, satellite telemetry is being tested on a small scale. The successful application plus a cost reduction in satellite transmission will lead to the establishment of a global digital seismic network. This would extend the current telemetered seismic networks from a regional to a global scale, significantly increase the network aperture. Finally, we will give a historical account of the development of the part unique to a seismograph-the pendulum. A simple pendulum has a natural period proportional to the square root of the pendulum length divided by gravitational acceleration. To sense short-period (e.g., with a pendulum natural period To = 1 sec) ground motions, a pendulum length of 25 cm will suffice. However, for picking up long-period surface waves (e.g., 6 = 20 sec), it is entirely impractical to erect a 100-m-high pendulum. This

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severe limitation prompted the search for a pendulum that could be operated at a long period with stability. The limitation was overcome with a garden gate-type horizontal pendulum, whose natural period was extended by tilting the hinge axis at a small angle 8 from the vertical, so that only a small fraction of the gravitational restoring force (g sin 8)was used. As 6 is made small, the natural period becomes long. A second design was the inverted pendulum shaped like a freestanding top-an unstable system. The inverted pendulum was stabilized by supporting springs attached to the instrument frame. With appropriate adjustments, the natural period of the inverted pendulum could be lengthened to seven to eight times that of simple pendulum of equivalent length. Besides the period problem, the sensitivity of these pendulum sensors was also a problem, because at the turn of the century instrument magnification could be achieved only by lengthening the mechanical level arm that held the recording pen. To increase the instrumental magnification, a larger pendulum mass was required to overcome the friction of the pen at the end of the long level arm. At the turn of the century, the German scientist Wiechert was building larger pendulums, with masses from 100 to 1000 kg. The heaviest one, which was built in 1906, weighed 17 tons and realized a maximum magnification of 2 x lo3. In the same year the Russian scientist Galitzin developed an electromagnetic seismograph, which employed a coil as the pendulum mass surrounded by a permanent magnet affixed to the instrument frame. The electric signal generated by the relative motion between the magnet and the coil was used to drive a galvanometer. Optical registration of the galvanometer deflection provided the needed instrumental magnification. The problem of instrumental magnifications was basically solved, and Galitzin further showed that a pendulum mass of 7 kg was adequate. However, the total system performance became much more complex due t o the coupling of the transducer circuit with the galvanometer circuit. While the garden-gate-type pendulum answered the problem of a longperiod horizontal sensor, the problem of constructing a long-period vertical sensor was not solved until LaCoste introduced the zero-length-spring seismograph in 1934. This spring required a special method of winding that imparted a residual compressional stress in its natural state and would have reduced the spring to zero length if it could collapse on itself. Theoretically, this spring could achieve an infinite period when operated in a vertical vibration mode; it has been used typically in vertical seismometers that have a natural period of about 30 sec. With progress in the development of seismometers, the size of the sensors has become smaller. This has made practical the installation of sensors inside small-diameter (4-6 in.) boreholes. Downhole installation of seismic sensors has significantly reduced background noise and increased sensitivity. The

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small sensor size also made possible portable seismometers, ocean bottom seismometers, and seismometers deployed on the moon. The recent development of force-balanced accelerometers (FBAs) has resulted in smaller sensors with broader response bandwidths. Velocity and displacement signals can be obtained by simply integrating the FBA output. Rapid developments in digital electronic technology and the universal availability of standard time broadcasting have made possible the miniaturization of yesterday’s seismological observatory into a briefcase-size package. The widespread availability of data telemetry and particularly, in the near future, satellite telemetry can potentially link together a seismic array of continental dimensions. Such large-aperture seismic arrays will be operated as inverted “telescopes” pointing into and imaging the earth’s interior. A new era of exploring the detailed properties of the earth’s interior is soon to begin. Parallel to instrumental developments are deployments of seismic networks. Although the first seismic network with some degree of worldwide distribution was established by John Milne in 1896, a network of real global distribution was not deployed until the early 1960s as part of the U.S. national effort to improve capability in detecting and identifying underground nuclear explosions. Known as the World-Wide Standardized Seismograph Network (WWSSN), it consists of stations which each have three-component longand short-period seismographs with uniform calibration, time synchronization, and means of data archiving and dissemination. The WWSSN today comprises 110 stations operating in 54 countries (Fig. l), and it has provided fundamental data for seismological research in the past two decades not only in the United States, but around the world. In 1973 the United States began the development and global deployment of 13 Seismic Research Observatories (SROs) that combined the new borehole seismometer with an advanced analog and digital recording system. The availability of high-quality digital data produced by the SROs has opened up exciting new directions and opportunities for seismological research utilizing the newly acquired digital computing power. The success of the SRO prompted the development in the late 1970s of a digital recorder that could be attached to thus upgraded existing WWSSN systems. Seventeen such recorders are being installed at WWSSN stations (termed DWWSSN). Moreover, an early version of five high-gain long-period (HGLP) seismographs installed in the late 1960s has been modified by using more advanced computer-controlled digital recording. These five stations are called Abbreviated Seismic Research Observatories (ASROs). Today, SRO, ASRO, and DWWSSN stations together form the Global Digital Seismic Network (GDSN) shown in Fig. 2, which provides the bulk of digital data for seismological research. Another modern development is the installation of special seismographs to record ultra-long-period surface waves and the free oscillation of the earth out t o a period approaching DC.

m

1 FIG. 1. Map showing the distribution of WWSSN stations.

10 Y

0 INSILLLED

LOYC-PERIOD (HGL?) SlhlIOYf ILLWID)

FIG.2. Map showing the distribution of GDSN stations.

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There are over 20 ultra-long-period observatories around the world. Instruments consist of strainmeters (quartz tube type or carbon fiber type) and gravimetric accelerometers. A detailed discussion of these ultra-longperiod instruments can be found in a recent review paper (Agnew, 1986). To improve the capability for detecting weak signals from nuclear explosions in the midst of microseisms, seismograph arrays have been installed by a number of countries. A notable example is the large-aperture seismic array (LASA) installed near Billings, Montana, in the mid-1960s by the Department of Defense. LASA consists of 525 linked seismometers grouped in 21 subarrays. In each subarray 25 seismometers are arranged in a hexagonal geometry. A schematic diagram is shown in Fig. 3. The array covers an area 200 km in diameter and is similar in operation to a radio telescope array except that it points downward. The advantages of array operation include timing synchronization, microseism suppression, and identification of the direction of energy approach. However, the array operation is very expensive, especially for data processing and archiving. During the past 20 years there have been about 20 arrays in operation ;today only about 5 large seismic arrays are still active. N

FIG.3. Schematic diagram of the LASA array. [Reprinted with permission from Aki, K . , and Richardson, P. (1980). “Methods o f Quantitative Seismology.” W. H. Freeman and Company, San Francisco, California. Copyright 0 1980.1


11

To monitor regional seismic activity for earthquake hazard studies, a large number of telemetered short-period microearthquake networks have been constructed. This activity began in the early 1960s, when data telemetry (both radio and telephone) became economically feasible. These networks are typically composed of single (vertical) component short-period band (1 -20 Hz) sensors with their outputs telemetered to a central recording facility, where timing is supplied. Some large networks may consist of several hundred stations covering areas of hundreds of thousands of square kilometres ; the smaller networks may have only a dozen stations over an area of 100 km’. High gain is of primary concern for these networks rather than high fidelity of reproduction of actual ground motions. The principal objective of these networks is to locate the hypocenters and determine the magnitudes and the fault plane solutions. In California alone, there are probably more than 500 such stations. To avoid excessive cultural noise in areas where stations must be set up to provide uniform coverage, downhole installations are not uncommon. For coastal areas, ocean bottom installations sometime become desirable to pinpoint offshore events. Finally, there is another type of seismic station for the recording of nearfield strong ground accelerations. These instruments are usually referred to as strong motion accelerographs ;they are inactive until triggered into motion by a preset level of ground acceleration (usually 1070 of a g). These accelerographs can also be linked to form an array, but most of them are operated independently with internal relative timing. The data output is invaluable for engineers engaging in earthquake-resistant designs.

3. Nature of Seismic Ground Motions A seismic disturbance (earthquake or explosion) excites elastic waves, which propagate in the form of body waves and surface waves. For a very large earthquake, these propagating waves interfere with each other to form standing waves known as the earth’s free oscillations. From the analysis of these waves, we have derived information which constitutes our present basic understanding of the earth’s interior as well as the nature of the seismic source. Observations of seismic ground motions include the waveforms of particle acceleration, velocity, displacement, and strain (or the spatial derivative of the displacement field). A special feature of seismic observations is the signal’s broad range in both amplitude and frequency. These broad ranges typically span six orders of magnitude. This large variation not only is a result of the great difference in energy content between earthquakes of different magnitudes, but also strongly depends on the epicentral distance. The most prominent signals recorded by a standard long-period seismograph

12

TA-LIANG TENG

for a distant shallow earthquake are surface waves with a period of about 20 sec. Shorter-period surface waves tend to suffer from scattering due to shallow heterogeneities, and longer-period surface waves more easily lose energy into the asthenosphere. Nevertheless, surface waves give rise to signals of much larger amplitude in a seismogram than do body waves, mainly because body waves suffer stronger amplitude drop-off due to spreading of the wavefront. A small (Ms 3) earthquake will show a surface-wave amplitude of 100 mp (10-5 cm) at an epicentral distance A = 20" and 10 m p at A = SO", whereas the largest earthquake will show an amplitude of several centimeters at A = 20" and several millimeters at A = 80". Only to record surface waves over these distance and magnitude ranges requires an instrumental dynamic range of at least 120 dB. For observation very close t o the epicenter, an additional 40dB in instrumental dynamic range would be necessary for an adequate recording of the displacement field, which can reach a maximum amplitude of several meters in close-in range for very large earthquakes. Body waves usually have shorter periods ;their wave amplitude drops off faster with distance due to both a stronger wavefront spreading factor and a heavier attenuation effect, which increases rapidly with the wave frequency. At a large distance, the displacement amplitude of body waves can be one to two orders of magnitude smaller than that of surface waves. In the absence of background noises, an ideal seismic instrument for recording all waveforms over the entire distance range would require a dynamic range of nearly 200 dB. This stringent requirement imposes great difficulty on various aspects of the seismic instrument design, from the sensor, the amplifier, and the telemetry electronics to the recording device. In terms of acceleration, the corresponding required sensitivity is about lo-'' g , and the corresponding strain is about This resolution is very difficult to achieve, and the always present seismic background noise prevents the detection of these small signals. In fact, the ambient seismic noise level on the earth's surface is generally several orders of magnitude higher than this minimum resolution and is both frequency- and site-dependent. Sources of the seismic noise include meteorological effects, ocean wave motions, industrial activities, and traffic. The effect of the ocean wave motions is particularly pronounced and persistent. An analysis of the noise power spectral density shows a generalized picture such as that in Fig. 4 with two noise peaks at about 0.07 and 0.14 Hz. The latter is much stronger, and both can be traced to an origin related to ocean wave motions. The level of the ambient noise is also site-dependent ; it can differ by almost two orders of magnitude between a quiet and a noisy site. The main noise peak at 0.14 Hz cm for a noisy site and has an equivalent displacement of about cm for a quiet site. This lessens somewhat the above stringent requirement on the dynamic range. It would therefore be sufficient for the instrument to

-

FIG.4. Generalized power spectrum density of seismic background noises (after Melton, 1976). Earth noise is plotted as squared acceleration per millihertz. The Queen Creek spectrum is shown with two branches, the lower branch (thin line) representing the noise after substraction of the measured instrumental noise included in the upper branch. Elsewhere along this curve the instrumental noise is a negligible portion of the energy represented. The Camp Elliott curve represents data from a Southern California site about 20 km from the West Coast. Thermal acceleration energy for several assumed seismometers is shown by the horizontal solid lines, and a proposed frequency plot of an earthquake with a surface wave magnitude of 3 at 60" epicentral distance is included to show the relation between earthquake and noise energy.

14

TA-LIANG TENG

resolve a ground displacement of cm at that frequency. However, the Ievel of ambient noise tends to drop away from these two noise peaks. An ambient noise minimum occurs at about 30-40 sec. A nominal resolution of lo-' cm for long period is desirable for this period band. For the short-period band that is commonly dominated by microearthquake spectral energy over the frequency range from 1 t o 30 Hz, a desirable resolution is about cm. From low8to lo2cm, the amplitude of seismic signals still covers 10 orders of magnitudes or 200 dB in dynamic range. But for different applications of earthquake recording, it is general practice to design different instruments to cover different bands. To avoid the noise peaks at 0.07 and 0.14 Hz, seismic instruments are typically divided into two classes : long-period instruments with their amplitude response peaking at 30 sec, and short-period instruments with their amplitude response peaking at about 10 Hz. Each class of instrument is generally operated at the maximum gain permitted by the ambient noise of the site, and a system dynamic range of about 120 dB is quite adequate. Even this reduced 120-dB dynamic range has been achievable only with the advent of digital computers. For the recording of very long-period earthquake motions such as the free oscillations of the earth as a consequence of large earthquakes, ultra-long-period instruments are designed. They require a strain sensitivity of about lo-" or a corresponding 10-9g in acceleration. However, the amplitude of the earth's free oscillations typically spans four to five orders of magnitude, and a recording device of 80 to 100 dB is adequate.

4. Basic Types of Seismic Sensors A distinct feature of the measurement of seismic ground motions is the absence of a stationary reference frame on which the sensing device can be placed. In other words, the observer moves with the object during an earthquake, which makes careful measurement of the trajectory of motion of the object difficult. An effective and practical method for overcoming this difficulty is t o make use of an inertial mass that is loosely coupled t o the moving frame (the earth) through certain pendulum arrangements. Therefore, a discussion of seismic sensors is basically an analysis of the motion of a damped pendulum. Since seismic ground motions are a vector field, this analysis should involve the equation of motion of both vertical and horizontal pendulums. Besides pendulum seismometers, other sensors have also been developed that measure the derivatives of the displacement field such as strain and pressure. Strainmeters, or strain seismometers, measure the spatial derivatives of the displacement. Quartz tube strainmeters, carbon fiber strainmeters, and various tiltmeters all belong to this category, which


15

makes no use of the inertial property of a pendulum. Their response to very low frequency (down to DC) motions makes them very useful in the study of the earth’s free oscillations. Another category consists of hydrophones, which measure the pressure field (in fluid) induced by seismic ground motions. Their sensing elements are typically piezoelectric pressure transducers. Principal applications of hydrophones are for seismic exploration work in water-covered areas. Some downhole seismometers use hydrophones as sensors to relax the requirement of locking the downhole package against the hole. There is one last method that, at least theoretically, can be used to measure the vector field of seismic ground motions. Based on the conservation of angular momentum, it is possible to use a gyroscope as a stationary point against which the rigid-body rotation of a point on the earth can be measured. Gyroscope-based systems are available for navigation applications as well as for downhole orientation in drilling. The state-of-the-art sensitivity and stability are not yet good enough for measurement of seismic ground motions. As mentioned before, a pendulum can provide an inertial mass for the needed stationary reference frame that is loosely coupled to the earth. Ground motions are thus measured with respect to this simulated reference frame. To adequately record seismic ground motions, an important and continuing effort has been the design and operation of stable long-period pendulums with minimum cross-axis coupling and parasitic vibrations. Since the ground motions are a vector field, both vertical and horizontal pendulums are used. A brief recapitulation of the analysis of pendulum motion is given here together with a discussion on period-lengthening measures commonly employed for stable long-period pendulums. 4.1. Vertical Pendulums

The simplest way to create a stationary reference frame for vertical motions is a vertical spring with a suspended mass M (Fig. 5a). Let LObe the length of the unstressed spring and L the length of the spring under stress. Then the equilibrium condition requires Mg = k(L - Lo)

where g is the gravitational acceleration and k the spring constant. The equation of motion for the rectilinear vertical motion of the mass M is M d 2 y / d t 2= - k [ ( L

or

+ y ) - Lo] + Mg = - [Mg/(L - Lo)]y

(1)

16

TA-LIANG TENG

Earth

Earth

A

0

Mg (a 1

(b)

FIG. 5 . (a) Simple vertical pendulum consisting of a mass and a spring. (b) Improved suspension for period lengthening and cross-axis coupling reduction.

where o1is the angular frequency of the simple harmonic motion 0 1

= 277fl =

J g / ( L - Lo)

(3)

Thus the pendulum period is =

~ z J ( L- Lo)/g

(4)

Clearly, with constant g the pendulum period depends on L , which physically has a practical limit. A natural period longer than 1 sec is difficult to realize. Also, this simple suspension lends itself easily to a nonlinear effect caused by cross-axis coupling ; that is, horizontal ground motions can contaminate the rectilinear motions in the vertical direction. A method of lengthening the pendulum period and reducing the degree of cross-axis coupling can be achieved by a new suspension shown in Fig. 5b. Here the equilibrium condition requires : MRog = ka(L - LO)

(5)

where ROis the distance from the center of gravity of the pendulum t o the turning axis, and a is the connecting point of the spring and boom to the turning axis. With a small angular motion 8, the equation of motion of this system is j l d28/dt2 5:

- ka[(L + ad) - Lo] + MRog cos 8

= -ka[(L

+ ae) - Lo] + MRog

where jl is the moment of inertia of A4 about 0.

(6)

10.

17


Putting Eq. (5) into Eq. (6), we get

+ ka28 = 0

j1 d28/dt2

(7)

or

Thus the angular frequency of the pendulum is 0 1

=

2nf1 =

a

(9)

and the corresponding period is

7i = 2 7 r d j Z 2 Neglecting the mass of the boom, j~ = MR;, giving =

TI

2

n

m = 2 n 4 L - Lo)/gJRo/a

(1 1)

Comparing Eq. (1 1) with Eq. (4),one finds that this new suspension can realize a lengthening of the pendulum period by a factor of (Ro/a)’”. For RO = 4a, the original period can be lengthened by a factor of 2 . However, mechanically, Ro cannot be much larger than a, which also imposes a practical limit to this approach. An improved approach is to place the point B, which joins the spring with the boom, below the line connecting the mass M and the pivot 0 as shown in Fig. 6 . For this system the equilibrium condition is MRog = kr(L - lo)

(12)

Here, again, Lo is the length of the unstressed spring, L the length of the spring as the mass Mis at the equilibrium condition, and r the distance from the pivot point 0 to the spring. A small deflection of the boom by an angle t9 is described by the equation of motion: jl

d28/dt2 = - MRog cos 8

+ k(L’ -

LO)^'

(13)

The primed values correspond to those for the disturbed state. Combining Eqs. (12) and (13) gives

+

j1 d26/dt2= - k(L - L O ) ~ C6O S k(L’ - LO)^'

(14) Setting up rectangular coordinates centered at 0, the point of spring suspension A is (x, y ) ; the point B is (XI, y l ) , which reduces to (a, 0) at the equilibrium position. Expanding r and r’ in Taylor series for small 8 and keeping terms up to 02, we have

+ kax[l - L o / L + Loay2/xL3]6 + (tka2yLo(a- X)[X(X - a) + y 2 ] / L 5 ) 8=2 0

j~ d 2 8 / d t 2

(15)

18

TA-LIANG TENG

Y

X

FIG.6. Further improvement on suspension for a linear vertical pendulum.

The term with 8’ in Eq. (15) causes a departure of the pendulum from the simple harmonic motion. Since O2 is always positive, during the oscillatory motion, the mass will spend more time below the equilibrium position than above it. However, the O2 term will vanish if any one of the following three conditions exists. (1) x = a. This reduces the pendulum suspension to the one shown in Fig. 5b. Equation (15) collapses back to Eq. (7) and the pendulum period given in Eq. (10) results. Thus the elimination of the e2term by setting x = a does not bring about further improvement in period lengthening. (2) LO = 0. This is the condition of a spring of zero initial length. When a spring is stretched, the applied load f is proportional to the elongation ( L - LO),where L is the actual length and LOis known as the initial length. Theoretically, LO = 0 implies that this spring will collapse to zero length if the size of the coils vanishes such that a residual tension can no longer be supported by the coils. For LO = 0, Eq. (15) reduces to j l d2B/dt2

+ kaxe = 0

(16)

or d28/dt2+ with 0 1

= 2Xfi =

=

0

(17)


19

and

TI

=

2ndjzZ

Here, as x = 0, that is, placing the supporting point A on they axis (Fig. 6), the pendulum period 2i approaches infinity. At equilibrium, the angle AOB = n / 2 and the equilibrium condition gives

MRog = kLr = kay

(20)

y = MRog/ka

(21)

yielding

Putting Eq. (21) into Eq. (19), we have

TI

= 27r-x

=

2nd10tan(cu

+ p)/g

(22)

where 10 is the equivalent pendulum length of the physical pendulum and angles CY and pare shown in Fig. 7. It is interesting to note that by maintaining the condition (21), one can adjust the pendulum period by changing the position o f x .Figure 8 illustrates a number of possible spring suspensions that result in varying degrees of pendulum period lengthening possibilities as dictated by the geometry. (3) x(x - a)2 + y2 = 0. For fixed Q , this is an equation of a circle centered at ( a / 2 , 0 )with radius a / 2 . This condition gives the locus of the spring suspension point A, which must fall on the above circle. The geometry is shown in Fig. 9. This, again, reduces Eq. (15) to Eq. (16) and leads to the Y

FIG. 7. Angular geometry of suspension for a linear vertical pendulum.

20

TA-LIANG TENG

FIG. 8 . Various spring suspensions resulting in varying degrees of pendulum period lengthening for a linear vertical pendulum. Y

0

FIG,9. Suspension that gives a linear vertical pendulum only at the equilibrium condition.

same result as for the case of a zero-initial-length spring. However, this result can be achieved only when the pendulum is at the equilibrium position; the effect of the O2 term will again appear during dynamic vibrations. Comparing the above three conditions, the case LO = 0 (zero-initial-length spring) is the only one that can both remove the nonharmonic oscillation and

10.


21

effectively extend the period of the pendulum. Since the sensitivity of a pendulum to acceleration at low frequency is proportional to the square of the pendulum period, finding a stable long-period vertical pendulum was the most important problem in instrumental seismology for many years. This problem was not solved until LaCoste and Romberg (1942) invented the zero-initial-length spring in 1935. LaCoste’s spring can theoretically achieve an infinite period,* but in actual applications, long-term stability can be achieved only at a period up to about 30 sec. The zero-initial-length spring must be wound with a twist applied to the wire as it is coiled; this plus the stringent requirement on the material properties makes its production nontrivial. Press et a/. (1948) were the first to introduce the zero-initial-length spring in the long-period vertical siesmometer design, which later became the backbone of the World-Wide Standardized Seismograph Network (WWSSN). A common problem associated with the vertical pendulum is the temperature dependence of the spring constant k, which, in turn, causes the pendulum to drift from its equilibrium position. If 4 is the pendulum drift angle, the drift period T ( 4 )can be approximated by TI(+)= Tldcos 4 - B sin 4

where B = 4n2/0/(7i2g). Thus, an upward drift of the pendulum (4 > 0) tends to reduce the pendulum period, and a downward drift increases it. Therefore, it is common practice to choose alloys with low temperature coefficients and control the temperature inside the instrument housing in order to stabilize the long-period vertical pendulum. In the area of period lengthening, much recent research has been done using electronic feedback and compensation circuits. These measures are generally referred to as the force-balanced approach and have achieved varying degrees of success ;some have realized a pendulum period up to several hundred seconds and a sensor response close to DC. A discussion of the force-balanced sensors will be given in a later section. 4.2. Horizontal Pendulums

The simplest horizontal pendulum consists of a mass M suspended by a string from a point 0 on the frame (Fig. 10a). The pendulum can couple to the frame through a leaf spring (Fig. lob). Again, if the moment of inertia of the pendulum about the point 0 isjl and the distance between 0 and the * In practice, this is unattainable because of inexact spring length and hinge positions, finite restoring force of the hinge, temperature dependence of k, variation of gravity, and other disturbances. The maximum period maintained in a routine observation was 80 sec, by Francis Lehner of the Seismological Laboratory of the California Institute of Technology.

22

TA-LIANG TENG

Earth

1

Earth

F I G . 10. (a) Simple horizontal pendulum. (b) Simple horizontal pendulum with a leaf spring hinge suspension. (c) Equivalent pendulum length of a physical pendulum.

center of gravity of the pendulum is R o , a small deflection B of the pendulum will excite a simple harmonic motion described by j , d2B/dt2= - M R o g sin B

(24)

For a small deflection, sin 8 = 0 , we have d28/dt2 + ( M R o g / j , ) 8= 0

(25)

This leads to the same equation as Eq. ( 2 ) , giving w1 = 2nf, =

(26)

and T, =-n2

=

2nJl/g

(27)

where 10

= jl/MRo

is the equivalent pendulum length as described in Fig. 1Oc. It can be shown that 10 > R o . For a spring-coupled pendulum (Fig. lob), the right-hand side of Eq. (24) has an additional term - CB to account for the elastic restoring force, with C ( > O ) being the equivalent spring constant. The natural frequency for the spring-coupled pendulum is 01

=

27rfi = Jg/Io

+ C/jl

(28)

10.

23


0'

I

i

I

( C )

FIG.1 1 . (a) Garden gate suspension of a horizontal pendulum. (b) Modification of the garden gate suspension resulting in tensional hinge points. ( c ) Period lengthening of a garden gate suspension.

Therefore, a spring-coupled pendulum will result with an increased natural frequency or reduced pendulum period. From Eq. (27), one finds that a 20-sec pendulum period requires a suspension length of almost 100 m. Thus, a simple pendulum cannot be a practical device for long-period horizontal seismic sensors. However, a garden gate-type suspension (Fig. 1la) can provide stable long-period pendulums for a horizontal seismometer. The pendulum make use of a fraction of the restoring force (Mg sin i), where i is the angle between the support axis 0-0' and the vertical. For a small deflection 8, sin 8 = 9. We have the equation of motion

+

j , d28/dt2 (MRog sin i)B = 0

(29)

The natural frequency of the pendulum is WI

=

2nf

=

J ~ M Rsin~i/j, = & G Z &

(30)

24

TA-LIANG TENG

or the pendulum period T,

=

2nJ10/(gsin i)

TI becomes large for small i, and approaches infinity as i approaches zero. Extending the support axis 0-0’,it will intersect at a point 0’with a vertical line passing through the center of the pendulum mass M(Fig. 1lc). One finds that the equivalent pendulum length 16 is

I6

= lo/sin

i.

As i becomes small, 16 becomes long. For i = 1 minute, I6 is about 3600 times longer than l o , which corresponds to a 60-fold pendulum period lengthening. A minor modification in suspension (Fig. l l b ) results in tension at both supporting points 0 and 0’,thus reducing the friction at pivots such as may exist in the case shown in Fig. l l a . This type of long-period horizontal pendulum is commonly used in present-day seismometers, especially those of the WWSSN. The maximum stable period for a standard instrument is about 30 sec. A longer operating period is again limited by the nonnegligible temperature coefficients of materials used in the pendulum construction, as well as the constancy to which the small tilt angle i can be held. It may be noted in passing that inverted pendulums were used in designing horizontal seismometers that are still being operated in a small number of older seismic stations around the world. The mass M is supported by a leaf spring (Fig. 12a) or by a rod and coupled to the pivot 0 through a leaf spring (Fig. 12b). Since the system is unstable, two springs are needed to balance it.

Eorth

Eorth

(a)

(b)

FIG. 12. (a) Inverted pendulum. (b) Inverted pendulum with a leaf spring hinge support.

10.

25


If the spring constants of both springs are ko, then for a small deflection 8, sin B = 8 and the equation of motion is j l d2B/dt2 = (MRog - 2ka2)8

where a is the length between 0 and A. The pendulum period is

& = 2 d l o / g ( D - 1) where D = 2ka2/MRog. For D 5 1, the inverted pendulum loses its stability. For D -+ 1+, TI 03 ; this can never be achieved, however. In actual applications, the inverted pendulum is far less stable even though it can realize a period seven to eight times longer than that of a simple pendulum. -+

5. Damping Devices and Transducers To provide meaningful ground motion data, the pendulum seismometer must give an output bearing a definite relationship to the input disturbance. For an undamped pendulum, it will oscillate freely and indefinitely when excited. For impulsive earthquake wave arrivals, the output signal will be an envelope of varying amplitude containing primarily oscillations at the free period of the pendulum, thus completely obscuring the real information and rendering the recording useless. To suppress the undesirable oscillation, damping is introduced. Either viscous damping or electromagnetic damping can be used. The latter is much more effective and easily adjustable, and it is not subject to the undesirable temperature dependence of a viscous fluid. Therefore, electromagnetic damping is in common use in almost all seismometer designs. A damping force is proportional to the angular velocity of the pendulum and in a direction opposite to the pendulum motion. Damping of instruments with velocity transducers is almost always accomplished by energy loss in the resistive elements of the output circuit. This may be a galvanometer circuit, the input resistance of an amplifier, or simply a resistive shunt. 5.1. Motion of a Damped Pendulum

Let B I be a damping factor such that the damping moment is given by - B I 8. Incorporating this damping term in Eq. (8) or Eq. (17) gives

B where

~ E = I

+Z

+ o:e = o

E , ~

(32)

B l / j l and c1 is called the damping coefficient. Solutions of

26

TA-LIANG TENG

Eq. (32) with initial conditions

8 = B0

and

.

.

8 = O0

at t = 0

can take one of three forms: (1)

0 1

> El:

with

tan 4 = Bovl/(B0

+ E~ e0)

This is a damped oscillation with period Ti = 2n/v1 = Z/(1 - D?)’”. As in Eq. (10) or Eq. (19), Z = 2n/01 is the natural period of the undamped pendulum, and D 1= E I / W ~is commonly referred to as the damping constant. Since 01 > e ~DI , < 1 gives the case of underdamped oscillation with TI’> Tl . The behavior of the pendulum is shown in Fig. 13. The damping ratio u is defined as = I ykl/lyk+ll= eTel/vl = eTDI/(1--Df)l” (2) o1= E1(or D1 = 1):

8

=

[(I

+ mlt)e0 + &t]e-‘lf.

This marks the inception of a nonperiodic motion, and DI = 1 is referred to as the critical damping constant. (3) 01 < E1(or DI > 1):

8 = e-’I‘(Cl sinh v1 t

+ C2 cosh vl t )

where 71 = ( E f

-

2 1/2

01)

c1= (eo + E1eo)/vl

c2= eo

This also gives a nonperiodic motion.

FIG. 13. Motions of a damped pendulum.

10.

27


It therefore can be concluded that (a) as D I= 0, the pendulum performs an undamped oscillation, (b) as D I< 1, the pendulum performs an underdamped oscillations, (c) as D I = 1, the pendulum gives a critically damped nonperiodic motion, and (d) D I > 1, an overdamped oscillation results. 5.2. Forced Oscillation of a Damped Pendulum

Assuming that the motion is restricted to the x direction, we denote the motion of the earth in the inertial reference frame as u(t). Then in view of Eq. (32), the equation of forced oscillation is

g + 2E4 + u:[ = -ii small motion, and AI is the effective pendulum

(33)

where [ = lo0 for length. Equation (33) shows that a linear combination of [ ( t )and its time derivatives can reproduce the acceleration ii of the earth’s motion. If the earth’s motion has a characteristic frequency u,then (1) For u ol, the first term on the left-hand side of Eq. (33) dominates, and ( becomes nearly equal to - ii. Thus the damped pendulum essentially records the earth’s displacement. (2) For w 4 U I ,the last term on the left-hand side of Eq. (33) dominates, and u:[ approximates - ii. Thus the damped pendulum essentially records the earth’s acceleration.

The response of a damped pendulum to an arbitrary forcing function u ( f ) can be obtained by directly solving Eq. (33). However, the frequency response X ( w ) ,or the system response, of a damped pendulum can easily be obtained by considering a sinusoidal input u ( t ) = exp( - i d ) ; then the response [ ( t ) = X(u)exp( - i u t ) with

X(U)= ( - u2)/(u2 + 2i.m

- 0:)

If we define the amplitude response IX(u)I and the phase delay $(u)by X(W) = IX(0)Iexp[i4(w)l

then

IX(U)( =

UZ/[U2 -

Uh2 + 4 E 2u2 I 1/2

and

4(0)

= - tan-’[2Eo/(u2 - a:)] +n

Figure 14 gives the amplitude response and phase delay over a range of damping constant D 1= 8/01 and frequency band. For u 0 1 , IX(u)I 4 1 and 4(u) + n ;the pendulum records the ground displacement correctly but with a reversed sign. For u 4 0 1 , there is no phase delay, but the pendulum

*

28

TA-LIANG TENG

2

1

0

3

w,/w

-n

1

I

I

0

2

w1lw

FIG. 14. Amplitude response IX(w)l and phase delay +(w) of a pendulum seismometer. [Reprinted with permission from Aki, K., and Richards, P. (1980). “Methods of Quantitative Seismology.” W . H. Freeman and Company, San Francisco, California. Copyright 0 1980.1

has little or no sensitivity to ground displacement with a period much longer than the pendulum period. However, the pendulum becomes a good accelerometer. The response [(t) for an arbitrary ground displacement u ( t ) can be obtained by convolution :

:l

[ ( t )=

u(t - t)x(r)dr

where x ( t ) is the inverse Fourier transform of X(o).

5.3. Types of Transducers Early seismometers are arranged for direct registration of the relative motion between the pendulum mass and the frame by either mechanical or

10.


29

optical means. In the former method, a system of levers magnifies the motion of a very heavy mass and applies it to a pen or stylus in contact with the recording surface. In order to develop enough force to overcome the stylus friction and lever inertia, the pendulum mass may weigh several tons. Very few of these mechanical magnifying, direct-writing instruments are now in use. For optical magnifying systems, a mirror is usually coupled to the suspended system, and a light beam focused through the mirror onto a sheet of photosensitive paper wrapped over a revolving drum exposes a data trace which becomes visible when the sheet is processed. Since no work is done in using the light beam, the suspended mass may be much smaller. Modern seismometers make use of transducers that convert the pendulum motion into an electrical signal which can conveniently be manipulated (amplified, filtered, etc.) and applied to various recording media. By and large, there are two types of transducers : electromotive and parametric. The former include electromagnetic transducers, piezomagnetic transducers, and piezoelectric transducers. All of them can convert the mechanical motion directly into electromotive force. Parametric transducers instead make use of the effect of mechanical motion on an element of an external circuit which indirectly modifies its current. Capacitive transducers, inductive transducers, and resistive transducers belong to this category. Electromagnetic transducers are most commonly used in seismometer design ; they are called velocity transducers as their output is directly proportional to the relative velocity between the pendulum mass and the frame. Displacement or acceleration signals can be simply obtained by integrating or differentiating circuits, respectively. Piezoelectric or piezomagnetic transducers have their output proportional to the acceleration and therefore are acceleration transducers. Since the piezo effect makes no use of the inertia, these transducers can be made very small and are particularly useful for high-frequency work. They are also made into the sensing elements of hydrophones. However, their sensitivity is relatively low. Capacitive transducers make use of the hyperbolic relationship of the gap between plates and the resulting capacitance. As the gap is small, the relationship between the displacement gap and the capacitance approaches linearity and gives rise to highly sensitive displacement transducers. These transducers are commonly used for long-period work. Various modifications of circuits are used in designs for ultra-long-period strain and tilt measurements. 5.4. Electromagnetic Transducer and Resistive Damping

The relative motion of a pendulum and the seismometer frame is most commonly measured by the electromagnetic transducer. Therefore the observed output voltage is a measure of the velocity. It can be either a moving

30

TA-LIANG TENG

Eanh

FIG. 15. Schematic system configuration of an electromagnetic transducer. [Reprinted with permission from Aki, K . , and Richards, P. (1980). “Methods of Quantitative Seismology.” W . H . Freeman and Company, San Francisco, California. Copyright 0 1980.

coil system in which a coil is attached to the pendulum mass and moves through a magnetic field, or a moving magnet system in which the pendulum mass is made into a magnet moving through a fixed coil. To generate sufficient output, the coil is made of a long conductor that has a finite resistance Ro. An adjustable shunt resistor R is commonly connected across the output terminals for damping. Following Aki and Richards (1980), Fig. 15 shows a schematic system configuration. Let 1 be the length of the core conductor within the magnetic field of flux density B , and assume that the directions of coil movement and magnetic field are mutually perpendicular. Then the force F necessary to drive the coil motion is

F = IlB where l i s the current generated in thecoil. The mechanical power produced is

10.


31

The power must dissipate through the resistive elements of the circuit R + Ro;this gives

VZ = ZIBt

or

V = IBt

Writing G for IB, we find that V = G t and F = GI. Therefore G is called the electromotive constant and has units of volts per centimeter per second. We can solve for Z and F : Z = G 100. Again, FFT techniques are most efficient at these high degrees (Goad, 1987).

3.3. Forced Perturbations Precessions of the perigee and node at rates on the order of one revolution per 3 months can be taken as observed. These arise from the term lmpq = 2010 in Eq. (27),used in Eq. ( 5 ) with Eq. (15) for the Poisson brackets [a, el, [a, I],and [a,I] (K39): d a / d t = 3nNzo(1

- 5 cos2I)(ae/a)2C2~/4(1 - e2)2

dWdt = 3nN20(cosZ)(a,/a)zC2~/2(1 - e2)2

(30) Higher even zonal harmonics (I even, m = 0) also make a contribution a factor of smaller to these secular motions. Sinusoidal oscillations of about 3-month period arise from zonal harmonics of odd degree due to the motion of the perigee with respect to latitude. These effects are generated

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WILLIAM M. KAULA

chiefly by terms in Eq. (27) with I - 2p = - q = f 1 . The most prominent effects generated by tesseral harmonics, m # 0, have rates of about m cycles per day arising from terms with I - 2p + q = 0 in Eq. (27). To calculate the amplitudes of these perturbations, normally it suffices to take (I, e, Z constant and n, w , A4 secularly changing on the right of the equations of motion, Eq. (5). For example, the rate of change of the inclination obtained by using Eq. (27) in Eq. (5) is (dI1dt)lmpq

=

iW - 2p)[I, 01 + mIZ, a l 1 R l m p q

(31)

The assumption of secular change then leads to an integral with respect to time, using Eq. (15), of (K40): AZ/mpq

= ((1 - ~P)[Z,0 1

+ m[Z,o l l R / m p q / V / / m p q = [(I - 2p) cos I - m ] ( ( ~ J a ) ' A , m f i m p G / p q x e~p(iy/lm,,)n/V//mpq(~(l - e2)1'2 sin I

(32)

where the rate in the denominator is tj/mpq

= ( I - 2p)ci,

+ ( I - 2p + q)ni + m(hz - 4)

(33)

Perturbations by all the harmonics of all the Kepler elements can be calculated in a similar manner (K40). In cases of very small eccentricity, numerical difficulties can be avoided by using in place of the elements e, o the variables P = ecoso

Q = esino

(34)

Substitution of AP = Ae cos o - e A o sin o and so forth leads to forms in which terms that have divisors going to zero have zero numerators. As a consequence of the linear perturbations described above, a particular term in the spherical harmonic expansion of the gravity field can be considered to generate a spectrum of variations in a nearly circular satellite orbit which have frequencies that are different combinations of the four rates &, 0,h, 6 : one generated by the central term of the earth's gravity, two by the oblateness, and one by the earth's rotation. For a term with I even, there will be an m cyclejday term corresponding t o p = 112, plus I terms with rates of about I - 2p cycles per orbit. Terms dependent on q = f 1 will exist, but have high rates. For a term with I odd, there will be no m cyclelday term for q = 0, but there can be significant terms of q = i 1 ;that is, skew symmetry in the field will force an eccentricity. Since the effects of interest are of order or less, in an analytic calculation of the orbit it is necessary to take into account the nonlinear perturbations by the oblateness CZO , since it is of order In the decade 1957-1967, an extraordinary number of papers were published on this

13.

SATELLITE GRAVIMETRY

177

problem. The options which fostered this abundance included : (1) definition of intermediary : there are several refinements to the secularly precessing Kepler ellipse taken empirically here ; (2) independent variable :longitude of intermediary and so forth, rather than time; (3) choice of instantaneous elements: Kepler, Delaunay, Hill, and so forth; (4) means of carrying out nonlinear interactions : simple Taylor expansion, Von Zeipel transformation, Lie transformation, numerical iteration. Probably the most effective theories are those which use an intermediary entailing a linear C20 departure from the simple Kepler ellipse, employ Hill elements or P,Q [Eq. (34)] as the final variables (to avoid difficulty with zero eccentricity), and perform the nonlinear interactions by applying Lie transformations to the Poisson brackets (Aksnes, 1970; Kinoshita, 1977). The principal defect of these algorithms for gravity field determination is thought to be interaction between C20 and tesseral harmonics (Lambeck and Coleman, 1983). 3.4. Resonance

The procedure outlined above can break down in cases where Glmpq = 0. In general, for any orbit with v revolutions per day, there will exist harmonics generating perturbations with rates less than a half-cycle per day, arising from m = v, 2v, 3v, ... .Since v is necessarily less than 17 [a good enough rule is 17(ae/~)~’~] and 1 1 m,these normally will be rather small terms and the effects can be treated as small divisor terms rather than true resonances, since i///mpq goes through full cycles, 0 to 27r. Some systematic attempts have been made to select sets of satellites of 16, 15, 14, 13, ... cycles per day and analyze them forterms of m = 16,32, 15,30, ...order, but they havenot beenmajor contributions to the overall determination of the field. On the other hand, in comprehensive analyses of the field, higher-degreeterms for the orders m corresponding to low rates for satellites in the set used will often be included. One case of true resonance which does often occur arises fromgeostationary satellites placed in orbit for the purpose of communication, meteorology, or other surveillance. Since these have d u e of 6.6 from the rule of thumb given above, they are sensitive mainly to the I, m = 2,2 harmonic, corresponding to an ellipticity of the equator. The most useful quantity based on observations is an acceleration along track arising from changes in the energy: that is, in the semimajor axis. From Eqs. (5), (lo), (15), and (27) (KSl),

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WILLIAM M. KAULA

A pendulumlike solution of Eq. (39, resulting in a libration about 1 - A22 = d 2 with a period proportionate to 1/Jj12can be made (K50-53), but information about the gravity field is actually obtained by comparing observed accelerations to Eq. (35) extended to include higher-degree terms (Wagner, 1983). 3.5. Miscellaneous Effects

The most comprehensive analyses (Lerch et al. , 1974; Gaposchkin, 1980) take into account (1) atmospheric drag, using an idealized atmospheric model (Jacchia, 1965);(2) radiation pressure from the sun ;(3) lunar and solar direct attractions; and (4)oceanic and solid planet tides. All these effects cause smaller accelerations on a close geodetic satellite than the variations of the gravitational field. However, they build up to larger displacements because they are nonconservative or long-periodic. Hence it is necessary to take them into account in classical satellite geodesy, in which the distribution of observations is nonuniform. 3.6. Numerical Integration

The major determinations of the gravitational field now all utilize numerical integration of the orbits and the partial derivatives with respect to parameters, since the termination of the effort at the Smithsonian Astrophysical Observatory in 1981.Examples of numerical integration systems are the Jet Propulsion Laboratory (JPL) Development Ephemeris (Devine, 1967) and the NASA/GSFC GEODYN (Martin et al., 1972). In the GEODYN system, position is obtained by a second-order Cowell predictor-corrector scheme, while velocity is obtained by a first-order Adams-Moulton predictor-corrector. These programs are quite reliable but lavish in use of computer time.

4. Data Analysis 4.1. General Considerations

In classical satellite geodesy-category 1 of Section I-fairly elaborate corrections of the observations are needed because of the variations in direction and rate of the earth’s rotation, atmospheric refraction, and aberration (Kaula, 1966a, K82-K86; Lerch et a/., 1974). Like the miscellaneous perturbations of the orbit, they are significant mainly because of the nonuniform distribution of observations. This nonuniformity affects the determination of the gravity field because the spectrum of effects given in Eq. (33) is generated by the same rotation that affects the observability by

13. SATELLITEGRAVIMETRY

179

a particular tracking station. A satellite is observable when its earth-referred latitude and longitude $ and 1 are close to those of the tracking station. But from the spherical triangle formed by the orbit, the equator, and the meridian for 1,

o

+ f = sin-’(sin $/sin I)

CI - e = I - cot-1( i dtan’~csc’ 4 - sec2I )

(36) Hence any one station will see the satellite only near two values of the angle vfmpq :one corresponding to a northward pass and the other to a southward pass. For this reason, determinations of the gravity field from systems of category 1-ground station to satellite tracking-must be large-scale leastsquares computations, solving simultaneously for station locations and orbital constants of integration with the spherical harmonic coefficients of the gravity field. It is a different matter, however, with the newer systems, where the measurement is made on board the spacecraft and hence is not so limited by intervisibility constraints. Radar altimetry is by far the simplest and most direct method, despite the nonidentity of the mean sea level with the geoid, since the measurement is linearly proportional to the quantity of interest :the potential. Although there is some art in the removal of orbital error by use of track crossings and so forth, altimetry comes closest of all systemsto direct mapping. Satellite-to-satelliterange rate, necessary to get the gravity to comparable resolution over the land, is more complicated and indirect because the range rate is affected by both the angular rate (which is only energy dependent) and the position (which is both energy and angular momentum dependent). Hence spherical harmonics of the same order rn and parity (l odd or even) have overlapping spectra, entailing simultaneous solution for each such “string. ” Therefore alternative means of representation are being tried (Kaula, 1983). 4.2. Observation Equation Formation

To use an observation to correct orbit-affecting parameters, the effect of the parameters on the observation must be computable, as well as the associated partial derivative. Thus in Eq. (1) for range rate, the station location and rotation rate of the earth must be taken into account (Kaula, 1966a, K63):

r = [Rxqq - Rxuu]‘[Rxqq - (tJRxu/tJ19)ue]/r

(37)

where u is the earth-fixed rectangular coordinate of the tracking station and R,, is R3(- 6). In differentiating Eq. (37) (or the analogous expression for

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WILLIAM M. KAULA

range or direction), the principal cautions which must be exercised concern the facts that (1) q and q depend on the eccentricity e through the eccentric anomaly E as well as directly, and ( 2 ) the instantaneous angular elements a, w , M depend on the constants of integration a0 , e o ,10through their effects on the precession rates, Eq. (30). Radar altimetry does not require such complicated differencing. However, it is instructive to derive the dependence of an altitude on a spherical harmonic through the radial coordinate as well as the geoid height. Using r = a(1 - e cos E), from Eq. (8), and allowing for the dependence through a, e, and M gives Ahm =A m n

C fimp(ae/a)’[2(1 - 2p)/@imp0 + Gip(-i)/eCi/(-i)mpo P

- Glpl/e@(i+ i)mpo] exp(iy/rmpo)- UePIm exp(imA)

(38)

Satellite-to-satellite range rate turns out to be a good deal more complicated because of the dependence of the range rate on the sum of two radial displacements as well as the difference between the two angular rates. For two satellites of angular separation 6M, and ( I - 2 p ) 6M 4 1,starting from the cosine law for the range,

R = (r? + rf - 2rlt-z cos 6M)”’

(39)

one obtains (Kaula, 1983): ARrmp = A 1 m n ~ a ( ~ e / ~ ) ’ F l m , 6 ~ [ 2 G 2p / p i+( l 1- / 2 ) / 3 ( 1 + l ) m p 0

- 2G/p(-i)(I - 2p - 1/2)/@(/-1)mp0 - 3(1- 2 ~ ) ~ / @ / m p oexp(iy/rmpo) li (40)

4.3. Techniques of Solution

4.3.1.Classical Satellite Geodesy. As mentioned, the overlapping spectra of different gravity coefficients together with the nonuniformity of tracking distribution and the stroboscopic effect, Eq. (36), necessitate the use of several satellites of varied specifications in solutions from classical ground-to-satellite data. The most elaborate solution (Lerch et al. , 1979, 1982) has 7954 unknowns : 592 spherical harmonic coefficients of the gravity field, 3 coordinates each for 146 tracking stations; 6 constants of integration each for 699 orbital arcs; and, for 2 . 5 years of Lageos, 3 components of earth orientation for 5-day intervals. But since each of the 1,300,000 observation equations has only one set of six constants of integration (plus, for Lageos, three earth orientation parameters), the normal equations for each arc can be partitioned so that only the parameters common with other arcs must be carried over for combination with other arcs. This “partitioned normals”

13.


181

technique (Kaula, 1966a, K105) thus reduces the maximum normal matrix which must be inverted to that corresponding to the parameters common to all arcs. Since most of the 146 tracking sites are associated with only a subset of the orbital arcs, a second partitioning can be performed to remove them from the final solution. These procedures make it most convenient computationally to retain separately the normal equations for the gravitational harmonic coefficients and station coordinates. Another technique employed to reduce instability of the inversion is to use zero apriori estimates of the coefficients with standard deviations according to a rule such as lO-’/I2. The inverse square of this quality is added to the diagonal of the normal equations before the final inversion. In these massive solutions, the errors of the observations are necessarily assumed to be uncorrelated. This assumption is obviously untrue, not so much because of systematic errors in the instrumentation as because the force model is unavoidably incomplete and the observations are nonuniformly distributed. Consequently, the determination of the weight factors which must be applied to the normal equations from different satellites plus the a priori weight described in the previous paragraph is a rather empirical matter. Figure 3 is a global map of the most recent solution by Lerch et al. (1982). 4.3.2. Altimetry. The great quantity of data generated by the radar altimetry necessitates further improvisation. The normal procedure is t o determine the orbits by using gravity fields from previous solutions, calculate the radial coordinates of the observations from these observations, and then obtain the geoid height by subtraction of the measured altitude (plus reference ellipsoid and tidal effects). This procedure gives good determinations of the short-wavelength (less than 2000 km) variations in the geoid, which then are affected mainly by ocean dynamics and instrumental effects. However, there remain longer-wavelength warps in the geoid of a few meters arising from errors in the orbits, which, in turn, are caused mainly by errors in the gravity field. For a map of the geoid over an ocean, the long wavelengths are largely removed by requiring consistency of the geoid heights at track crossings. In effect, the error varies smoothly enough that it can be replaced by a linear trend. The most detailed global compilation of oceanic geoid heights is probably that by Rapp (1982) ;see Fig. 4 for an example. Another global map at smaller scale is by Marsh and Martin (1982). Since then, the emphasis has been on exploiting the full resolution of the Seasat altimetry-about 20 kmfor regional imagery, using digital processing techniques developed for other remote-sensing data. Haxby et al. (1983) subtract out a 12th degree gravity field from the geoid, apply a two-dimensional fast Fourier transform, and use a flat earth approximation to obtain gravity anomalies which are represented on a multicolored map with slant illumination. Sandwell (1984)

FIG.3. Geoid heights referred to a mean ellipsoid (flattening 1/298.257), calculated from spherical harmonic coefficients to degree 36 based on a combination of satellite and surface data: solution GEM 10B of Lerch et al. (1981).

13.


183

LONGITUDE

FIG. 4. Detailed altimetry geoid; heights are in meters referred to a mean earth ellipsoid (flattening V298.257). From orbital radii determined at NASA Goddard Space Flight Center (Marsh and Williamson, 1980) minus Seasat altitudes and tidal heights (Schwiderski and Szeto, 1979), with crossover adjustments at Ohio State University (Rowlands, 1981). One of 53 maps in Rapp (1982).

takes along-track differences of altitudes and then uses an autocorrelation function to interpolate among passes. The resulting map of geoid height slopes highlights submarine features such as seamounts and fracture zones remarkably; see Fig. 5 . 4.3.3. Combination of Classical Data and Altimetry. To improve the long-wavelength features of the gravity field from radar altimetry, ideally observation equations based on Eq. (39) would be added. These would then be used to yield improved orbit constants of integration, as well as spherical harmonic coefficients of the gravity field. In practice, this use has not yet been made of the altimetry. Instead area means are formed, either of the geoid heights themselves (Gaposchkin, 1980; Lerch et al., 1981 ;Rapp, 1981) or of the gravity anomalies calculated therefrom (Reigber et al. , 1982). These area means are then subjected to harmonic analysis to obtain estimates of the spherical harmonic coefficients to combine with estimates based on

184

WILLIAM M. KAULA

180

200

820

240

260

aio

FIG.5 . Geoid gradient map of the southeast Pacific in the area of the Eltanin Fracture Zone and the Louisville Ridge. A gray-tone image scale is produced by sampling the geoid gradient at 0.2" intervals and assigning a dot density to each value. (From Sandwell, 1984.)

classical satellite geodesy. While there is some consistency in that the radar altimeter satellite orbits are calculated with the field determined by the classical technique, the radar altitudes have no effect on the orbital constants of integration. 4.3.4. Evaluation and Combination with Surface Data. Lambeck and Coleman (1983) give a detailed critique of the solutions by Gaposchkin (1977, 1980), Lerch et al. (1979, 1981), and collaborators Balmino et al. (1978) and Reigber et al. (1982). In each of these citation pairs, the earlier work is a purely classical solution, while the later incorporates altimetry. Lambeck and Coleman find discrepancies among these solutions of 3 m in geoid height, with a maximum of 10 m. These differences are greater than the purported accuracy of the solutions ; they also incorporate a large long-wavelength component from terms such as I, m = 3,3. Reasons can be found to criticize all the solutions. The Gaposchkin (1977, 1980) work may have insufficient variety of orbits and be affected by neglect of CZO interaction with tesserals in the analytic theory. The Lerch et al. (1979, 1981) work apparently does not use the secular perturbations to infer zonal harmonics and is obscure as to how the older data are weighted relative to the new. All the solutions are nonrigorous as to how the altimetry and classical data are combined, as discussed in the previous section. It therefore seems desirable to (1) add data which are sensitive only to the long-wave variations and to test the purely classical solutions against both (2) surface gravimetry and (3) geosynchronous satellite accelerations. Step (1) is carried out by Lerch et al. (1982), (2) by Lerch et al. (1983), and (3) by Wagner (1983).

13.


185

The incorporation of the Lageos-2 data did produce significant changes in harmonics of low degree, I I 4. The test against surface gravimetry entails comparison of two estimates for the mean anomaly Ag of a surface block (Kaula, 1966b):

AgT

=

AgL

+ Sg + ET

(41)

where the subscripts S and T connote satellite and terrestrial, respectively, AgL is the true value of the anomaly through the maximum degree L incorporated in the satellite solution, Sg is the true value of contributions from degrees I > L , and E S and ET are errors. Since all four of the values on the right of Eq. (41) should be independent, we can obtain from a sufficiently large sample an estimate

where the angle brackets indicate averages over the sample. The results for are 3.0 to 3.2 mgal testing against various sets incorporating altimetry, and 4.8 mgal against a set not incorporating altimetry. These values are, respectively, 2.1 and 3.3 times the internally predicted uncertainty of the satellite solution. The higher figure for the comparison to gravimetry may reflect long-wavelength error in the surface data. The geosynchronous satellite accelerations are now measured to five significant figures and hence reflect perceptible contributions from harmonics up to degree 6.The total discrepancy from the Lerch et al. (1982) solution is 0.10%, less than one-fourth that of any other solution (Wagner, 1983). Combination solutions using gravimetry on the land and altimetry at sea, carried to 1' x 1 area means and spherical harmonic coefficients to degree 180, have now been done by Rapp (1981) and Lerch et al. (1981). In these combinations the area means from altimetry are treated in the same manner as the gravimetry. As discussed above, this method does not extract the maximum advantage from the altimetry to improve the satellite solutions. However, the results for 1' x 1 area means are probably good within 5 mgal for 90% of the earth's surface. The remaining 10% is land and ice-covered areas of forbidding access and hence will require satellite-to-satellite range rate to survey. ES

Acknowledgment This work is supported by NASA grant NAG 5-317.

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WILLIAM M. KAULA

References K. Aksnes, Astron. J. 75, 1066 (1970). R. R. Allan, Pfanet. Space Sci. 15, 1829 (1967). American Geophysical Union, “National Geodetic Satellite Program,” NASA Spec. Publ. No. 365, 2 vols. Washington, D.C., 1977. American Geophysical Union, J. Geophys. Res. 84, No. B8 (1979). American Geophysical Union, J. Geophys. Res. 87, No. C5 (1982). APL : Johns Hopkins University Applied Physics Laboratory and Guidance & Control Laboratory, Stanford University, J. Spacecr. Rockets 11, 637 (1974). G. Balmino, C. Reigber, and B. Moynot, Ann. Geophys. 34, 55 (1978). D. Brouwer and G. M. Clemence, “Methods of Celestial Mechanics.” Academic Press, New York, 1961. 0. L. Colombo, EOS, Trans. Am. Geophys. Union 64, 680 (1983). C. J. Devine, “JPL Development Ephemeris Number 19,” Jet Propul. Lab. Tech. Rep. 321181. Pasadena, California, 1967. E. M. Gaposchkin, Philos. Trans. R. SOC. London, Ser. A 284, 515 (1977). E. M. Gaposchkin, J. Geophys. Res. 85, 7221 (1980). C. C. Goad, Manuscr. Geodaetica 12, 11 (1987). H. Goldstein, “Classical Mechanics,” 2nd Ed. Addison-Wesley, Reading, Massachusetts, 1980. W. F. Haxby, G. D. Karner, J. L. LaBrecque, and J. K. Weissel, EOS, Trans. Am. Geophys. Union 64, 995 (1983). 1. R. Izsak, J. Geophys. Res. 70, 2621 (1965). L. G. Jacchia, The earth’s gravitational potential as derived from satellites 1957.1 and 1957.2. Smithson. Astrophys. Obs. Spec. Rep. No. 19 (1958). L. G. Jacchia, Static diffusion models of the upper atmosphere with empirical temperature profiles. Smithson. Inst. Astrophys. Obs. Spec. Rep. No. 170 (1965). B. Jeffreys, Geophys. J. R. Astron. SOC. 10, 141 (1965). W. D. Kahn, S. M. Klosko, and W. T. Weils, J. Geophys. Res. 87, 2904 (1982). W. M. Kaula, “Theory of Satellite Geodesy.” Blaisdell, Waltham, Massachusetts, 1966a. W. M. Kaula, J. Geophys. Res. 71, 5303 (1966b). W. M. Kaula, J. Geophys. Res. 88, 8345 (1983). D. G. King-Hele, Science 192, 1293 (1976). H. Kinoshita, Third-order solution of an artificial satellite theory. Harvard Cent. Astrophys. Prepr. Ser. No. 594 (1977). K. Lambeck and R. Coleman, Geophys. J. R. Astron. SOC. 74, 25 (1983). F. J. Lerch, C. A. Wagner, J. A. Richardson, and J. E. Brownd, “Goddard Earth Models (5 and 6),” Goddard Space Flight Cent., Greenbelt, Maryland, 1974. F. J. Lerch, S . M. Klosko, R. E. Laubscher, and C. A. Wagner, J. Geophys. Res. 84, 3897 (1979).

F. J. Lerch, B. H. Putney, C. A. Wagner, and S. M. Klosko, Mar. Geod. 5, 145 (1981). F. J. Lerch, S. M. Klosko, and G. B. Patel, Geophys. Res. Let. 9, 1263 (1982). F. J. Lerch, S. J . Klosko, and G. B. Patel, EOS, Trans. Am. Geophys. Union 64,673 (1983). J. G. Marsh and T. V. Martin, J. Geophys. Res. 87, 3269 (1982). J. G. Marsh and R. G. Williamson, J. Astonaut. Sci. 28, 345 (1980). T. V. Martin, C. C. Goad, M. M. Chin, and N. C. Mullins, “GEODYN,” Wolf Res. Dev. Co., Riverdale, Maryland, 1972. Massachusetts Institute of Technology (MIT), The terrestrial environment : Solid earth and ocean physics. NASA [Contract. Rep.] CR NASA-CR-1579 (1970). R. H. Merson and D. G. King-Hele, Nature (London) 182, 640 (1958). P. M. Muller and W. L. Sjogren, Science 161, 680 (1967).

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H. J. Paik, BUN. Geod. 55, 370 (1981). R. H. Rapp, “The Earth’s Gravity Field to Degree and Order 180 Using Seasat Altimeter Data, Terrestrial Gravity Data, and Other Data,” Dep. Geod. Sci. Surv. Rep. No. 322. Ohio State Univ., Columbus, 1981. R. H. Rapp, “A Global Atlas of Sea Surface Heights Based on the Adjusted Seasat Altimeter Data,” Dep. Geod. Sci. Surv. Rep. No. 333. Ohio State Univ., Columbus, 1982. C. Reigber, G. Balmino, B. Moynot, and H. Muller, Annu. NASA Geodyn. Program Conf., 4th, 1982. V. S . Reinhardt, F. 0. Vonbun, and .I. P. Tuerneaure, Proc. IEEE Symp. Position, Location, Navig., 1982. D. Rowlands, “The Adjustment of Seasat Altimeter Data on a Global Basis for Geoid and Sea Surface Height Determination,” Dep. Geod. Sci. Surv. Rep. No. 325. Ohio State Univ., Columbus, 1981. D. T. Sandwell, J. Geophys. Res. 89, 1089 (1984). E. W. Schwiderski and L. T. Szeto, “NSWC Ocean and Geocentric Tide Tapes and Tide Computation Program,” U.S. Nav. Weapons Cent. Rep. Dahlgren, Virginia, 1979. P. T. Taylor, T. Keating, W. D. Kahn, R. A. Langel, D. E. Smith, and C. C. Schnetzler, EOS. Trans. Am. Geophys. Union 64, 609 (1983). F. Tisserand, “Mecanique Celeste,” Gauthier-Villars, Paris, 1889. C. A. Wagner, J. Geophys. Res. 88, 5083 (1983).


14. EXPERIMENTAL METHODS IN CONTINENTAL HEAT FLOW

David D.Blackwell Department of Geological Sciences Southern Methodist University Dallas. Texas 72275

Robert E. Spafford Department of Geological Sciences Southern Methodist University Dallas. Texas 72275

1 . Introduction Continental heat flow deals with the measurement, reduction, and interpretation of the conductive flux of heat through the earth’s surface by using a set of techniques which are appropriate for a continental setting. The quantities which are measured are temperature as a function of depth in a drill hole and the thermal conductivity of representative materials cut by the hole. Ancillary properties such as radioactive heat production, thermal diffusivity, heat capacity, and density may be determined as well. A number of corrections may be necessary so that the results of these measurements can be interpreted as representative of the deeper crust. These corrections include temperature and fluid content corrections to thermal conductivity if in situ conditions differ from the conditions under which thermal conductivity measurements were made, topographic corrections to account for the perturbing effects of nearby terrain, corrections for lateral surface temperature variations near the measurement site, corrections for time-dependent surface temperature variation, corrections for the effects of inhomogeneous thermal conductivity structure in the vicinity of the measurement site, and evaluation of the effects of subsurface fluid flow on ground temperatures. In a typical heat flow measurement on land, temperatures are measured at discrete intervals (e.g., 1-5 m) in a drill hole that is more than 100 m deep. Shallower holes often suffer from surface-induced thermal perturbations for which corrections are inaccurately or poorly known. The difference in temperature over each interval (the geothermal gradient) is then determined. 189 METHODS OF EXPERIMENTAL PHYSICS Vo1. 24, Part B

Copyright Q 1987 by Academic Press, Inc. All rights of reproduction in any form reserved.

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DAVID D . BLACKWELL AND ROBERT E. SPAFFORD

Samples of the rock encountered in the drill holes are obtained in the form of either cuttings or core and returned to the laboratory, where the thermal conductivity is measured. The heat flow is calculated as the product of the thermal conductivity and the geothermal gradient. Appropriate corrections described above are made to the value(s) obtained. Data are then interpreted in terms of the internal temperature distribution and the nature of the heat sources involved. Heat flow measurements are made for a number of purposes and the techniques vary depending on the intended use. Because the internal processes of the earth are thermally driven, heat flow measurements are typically used to study tectonics. Regional heat flow values contain information on the thermal structure of the lithosphere and some aspects of the geochemistry of the crust. Locally, conductive heat flow may be effected by ground water movement, particularly in geothermal systems. In recent years, heat flow techniques have been recognized as the most cost-effective geophysical techniques for geothermal exploration, and many measurements have been made for such purposes. Because many high-temperature geothermal systems are associated with cooling magma chambers, developmentmotivated heat flow studies have become important sources of scientific data for the study of these features. The exploration of geothermal systems and the scientificinterest in deep drilling to study the thermal field associated with magma chambers (Luth and Hardee, 1980) have generated a need for instruments that can function at elevated temperatures and pressures and in hostile chemical environments. Thus much recent instrumental development has focused on equipment that can function under extreme conditions. In this chapter, the equipment for measurement of temperature, thermal conductivity, and heat production from radioactive decay is discussed. Equipment associated with well testing and reservoir evaluation and/or determination of various water flow parameters will not be described. These methods are generally regarded as a part of reservoir engineering and/or hydrology and are not discussed here. The chapter is divided into three sections. The first section deals with temperature measuring equipment, the second with thermal conductivity measuring equipment, and the third with heat production measuring equipment. Beck (1965) has described the experimental techniques in use prior to 1965; the discussion here deals with methods in use since that time. A brief sketch of the history of heat flow studies is given by Bullard (1965). 1.1. Requirements for Heat Flow Determination

There has been some discussion in the literature of the requirements, in terms of hole depth, number of temperature measurements, and number of thermal conductivity samples, for a “reliable” regional heat flow

14.

EXPERIMENTAL METHODS IN CONTINENTAL HEAT FLOW

1.91

measurement. Chapman et al. (1984) discussed some specific examples where heat flow values calculated from parts of holes below 190m, without corrections, gave consistent values within f 10% of the regional value. Jessop (1983, p. 70) concluded that a hole depth of 320m, at least 20 temperature measurements, and at least 36 thermal conductivity measurements are “reasonable but arbitrary criteria for a realiable heat flow.” Drury and Lewis (1983) and Shearer and Reiter (1981) argued that hole depths of 600m or deeper are required. On the other hand, Balckwell et al. (1980) argued that in many areas reliable heat flow values may be obtained in 100-m holes, particularly if appropriate corrections for terrain and microclimatic effects are made. When heat flow and geothermal gradient measurements are used for exploration and evaluation of geothermal systems, holes as shallow as 1-20m may be used. In such cases the perturbations due to near-surface variability in thermal parameters may be only a small percentage of the elevated heat flow and can be essentially neglected. There is no doubt that in the Arctic recent climatic changes cause major disturbances to the geothermal gradient to depths of 100-200 m (Lanchenbruch and Marshall, 1969; Cermak, 1971). In temperate regions these climatic warmings of 1-3°C within the past 100-200 years do not seem to have occurred, and heat flow typically remains constant with depth in 100-600-m holes in impermeable rocks. Thus the main factors that perturb temperatures from the onedimensional heat conduction setting are terrain, microclimatic effects, ground water flow, and inhomogenous thermal conductivity. Water flow effects (not related to the drill hole, which is assumed to be grouted; see Section 2.2) are not a simple function of depth and no single depth can be used as a guideline for all geologic terrains. The depth to which temperature is affected by ground water flow may range from a few meters to several kilometers or more. The larger depth of perturbation occurs in regional flow systems such as the Madison and Dakota aquifers in the Great Plains (Majorwicz and Jessop, 1981; Gosnold, 1985; Back et al., 1983) and in geothermal systems, where fluid may circulate to depths of 5-10 km. In some cases fluid flow effects can be predicted, but in many cases they cannot. Thus depths required for a reliable regional heat flow determination, whether the holes are 100, 190, 320, or more than 600 m deep, cannot be established except with reference to specific geologic terrains and their particular tectonic, physiographic, and hydrologic settings. Many corrections used in heat flow analyses are summarized by Jaeger (1965). The most commonly required corrections are for topographic effects and/or microclimatic effects. Correction techniques for these effects have been discussed by Henry and Pollack (1985), Blackwell et al. (1980), Lachenbruch (1968, 1969), and Birch (1950), among others. In northern

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latitudes, corrections for Pleistocene climatic effects similar to those discussed by Jessop (1971) are often made. However, in most other areas such a correction is unusual, so users of heat flow data should be careful of the consistency of the correction types when data sets are being compared.

2. Temperature 2.1. Introduction

Temperature is a fundamental property, and there are many different ways in which temperature measurements can and have been made in the earth. A comprehensive summary of various general temperature measurement techniques is given by Brickwedde (1962). One group of techniques used in heat flow is referred to as wireline techniques (either electric or nonelectric). In these techniques, a sensor or instrument is attached to the end of a wireline cable. The only purpose of the cable is to position and retrieve the package. The temperature recording device is self-contained. The two most common nonelectric temperature measurement instruments of this type are mercury maximum-reading thermometers and Amarada (or Kuster) clock-driven temperature recorders. Remote electrical instruments are being developed. A second class of techniques uses a wireline cable with one or more conductors as an electrical connection from surface recording devices to the downhole sensor assembly. This class of techniques is by far the most commonly used. The main drawback is that electrical cables are expensive, may fail by mechanical or thermal breakdown of the insulation, and must be connected to the sensor by a cable head, which is another mechanical and electrical weak point. 2.2. Hole Preparation

The first requirement for a heat flow value is the measurement of in situ rock temperatures at different depths so that the geothermal gradient can be calculated. Typically a hole in the ground is required, preferably vertical and of small diameter. Small-diameter holes minimize potential temperature instabilities resulting from cellular convection (Diment, 1967 ; Gretener, 1967; Sammel, 1968). Measurements may be made in mine workings or tunnels, but good data may be difficult to obtain because of the effects of ventilation, dewatering, and sampling difficulties. In discussing the continental heat flow data collected as of 1975, Jessop (1983) found 1310 values measured in vertical drill holes and 389 values measured in mines, tunnels, or lakes and shallow seas. The hole must be at thermal equilibrium when the temperature measurements are made so that drilling effects do not affect

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the values. Thermal recovery times range from a few days for 100-150-m airrotary drilled holes to several months for deep mud-drilled hydrocarbon exploration holes (see, e.g., Lachenbruch and Brewer, 1959). The requirement for thermal equilibrium is the major factor limiting widespread use of temperature logs in hydrocarbon exploration. To guarantee an accurate measurement of the rock temperature the fluid in the drill hole must be static. Experience has shown that, unless proved to the contrary, many drill holes 50-100 m deep or even deeper intersect and connect fractures or stratigraphic zones containing water with different piezometric levels. The short circuit represented by an open hole will induce flow between thezones. This flow is often fast enough (Sorey, 1971 ;Mansure and Reiter, 1979) that the water temperatures no longer accurately reflect temperatures in the adjacent rock (Birch, 1947, 1966). To guarantee that the fluid is static the holes must be grouted around sealed tubing installed in the hole from top to bottom. If the tube is water-filled, temperature measurements can be made rapidly even in the part of the drill hole above the static water table in the surrounding rock. The grouting material may be cement, a chemical grout, drilling mud, or cuttings. Without the tubing and grouting, many holes, even in basement terrain, may not be usable for heat flow studies. More details of hole drilling and preparation techniques are discussed by Moses and Sass (1979). 2.3. Wireline Temperature Measurement Techniques

The techniques discussed in this section are designed to be used with no electrical connection to the surface. Thus cable and cable-head integrity problems are of no consequence. The oldest temperature device still used in the measurement of temperature in the earth is the mercury maximumreading thermometer. An extensive program of thermal measurements was begun in the 1920s by C. E. Van Ostrand of the U.S. Geological Survey (van Ostrand, 1926, 1951). Most of these data were not readily available in raw form until Gaffanti and Nathanson (1981) published a report detailing this data set. In addition, “maximum” temperature measurements are made in almost every hydrocarbon exploration well drilled in the United States because logging companies typically run maximum-reading thermometers just above their tool to obtain an estimate of the temperature for calculation of fluid resistivity and so forth. In fact, isotherm and gradient maps of North America have beenpublishedon the basisofthesedata(A.A.P.G.-U.S.G.S., 1976a, b). There is no way to monitor the temperatures continuously with depth or time and no way to find reversals in temperature with depth, because only the maximum temperature actually reached by the package is recorded.

194

DAVID D. BLACKWELL AND ROBERT E. SPAFFORD

Thus, from a practical point of view, these measurements are rudimentary and the amount of information returned is minimal. Such problems as difficulties with shakedown of the thermometer are usually minor if proper care is taken. The thermometers should be run in pressure-tight packages, however, as the effect of pressure on exposed bulbs can be significant. Another nonelectrical wireline technique, which returns more information, is the Amarada-type recorder. In this temperature instrument, the pressure in a fluid-filled Bourdon tube is recorded by the scratches of a stylus on a smoked bronze plate. The other axis is advanced by a mechanical (spring-wound) clock motor. On retrieval, the plate is put into a reader, and the temperatures (as a function of time) are read from the scratches on the smoked bronze, based on a calibration table supplied with the instrument. The same instrument can also record borehole pressure if the Bourdon tube is not pressure-sealed. The upper temperature limit of this device is in excess of 30O0C,although for the higher temperatures it is usually necessary to modify case seals and to use special oil for the clock (Major and Whitten, 1980). With this instrument, temperatures can be obtained at several discrete points, with time versus wireline depth noted at the surface, and changes in temperature with time at a given point can be obtained when the instrument is held at a constant depth. Temperature reversals can be handled, which is not possible with the maximum-reading thermometers. Quasi-continuous logs can be obtained if the tool is run slowly enough. The long time constant for response of the instrument and the low resolution (on the order of 0.5OC) put a limitation on the usefulness of the tool for comparison of gradients in different lithologies and for investigations of the small-scale thermal features in a hole. Development of low-power microcircuitry makes possible a different type of tool. A dewared (insulated) or heat-sinked tool with internal measurement and recording electronics could be designed and built. Correlation of time with a surface record of the time-depth history of the tool would allow a temperature-depth curve to be reconstructed once the tool was returned to the surface. Such a tool would give a certain degree of high-temperature capability in excess of electrical cable limits. Similar instruments have been designed by well logging companies to operate during drilling in a downhole environment. 2.4. Electrical Wireline Techniques

The electrical wireline (cable) apparatus for temperature measurement can be divided into three subsystems: the sensor, the cable, and the data acquisition system. In geothermal wells, the factor limiting high-temperature applications is the relatively low-temperature breakdown of insulation resistance of commonly available well logging cables and cable head assemblies.

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EXPERIMENTAL METHODS IN CONTINENTAL HEAT F L O W

195

The sensor most commonly used for temperature measurements in heat flow studies is a thermistor. A thermistor is a sensor composed of semiconductor material which has a steep negative temperature versus resistance curve (Robertson et al., 1966). The sensor is connected to the surface via electrical conductors, and an intermittent or continuous recording of resistance versus depth is obtained. Temperature is obtained from a calibration curve determined by laboratory comparison of the sensor to National Bureau of Standards temperature calibrations. These types of instruments have been used by the heat flow community for many years (Beck, 1965;Roy et al., 1968). Initially, the surface instrument was a Wheatstone bridge ; more recently, accurate and inexpensive digital voltmeters have become available. In order to obtain 0.01"C resolution, a 4i-digit meter is necessary. A 5t-digit meter will allow a resolution of 0.001 "C. Typically, a four-lead cable is used so that the effect of cable resistance variations can easily be compensated. If a cable with fewer leads is used the effect of temperature on the conductor resistance must be considered. One technique is to use a sensor with a high enough resistance that cable temperature-induced resistance changes are insignificant. Both armored and unarmored cables may be used, Silver graphite on coin silver slip rings are usually used to connect the cable conductors to the surface measurement system. Thermistors can be used at maximum temperatures which are presently limited only by cable insulation breakdown. Temperatures over 300°C have been measured with thermistor devices in EE-2 of the Dry Hot Rock project at Los Alamos, New Mexico (Cremer, 1981). The cable used had a construction of MP-35 alloy armor wires surrounding individual conductors insulated with layers of wrapped and sintered TFE Teflon. This construction is state-of-the-art and very expensive. The cable lasted for a period of several hours at temperatures as high as 317°C. A more practical construction is stainless steel armor over PFA Teflon-insulated conductors, rated to 260"C, at about 20% of the cost of the TFE Teflon cable. We have fitted portable systems (cable 600+ m long, 25 kg total weight) with FEP Teflon-insulated four-conductor cables, rated to 20O-22O0C, because FEP can be extruded in very small thicknesses. Measurement of temperatures with a minimum precision of f 0.013"C over the temperature range 0-200°C is possible with this equipment, using thermistors with high resistances at the ice point and multiranging digital voltmeters (DVMs). We have used these types of portable systems extensively and routinely in geothermal studies since 1977. A typical system is shown in Fig. 1. Response times of these types of sensors mounted in thin-wall stainless steel hypodermic tubing in water-filled holes are a few seconds. In air, 10-20 minutes may be required for a measurement. To speed measurements in air,

196


FIG. 1. Portable high-temperature logging equipment. Reel is fitted with 900m of fourconductor FEP Teflon-insulated cable with stainless steel braid jacket. Depths come from measurehead. Slip rings are silver graphite on coin silver. Meter is a 4f-digit digital multimeter which, when used with a thermistor with an ice-point resistance of about 400 k n , allows from 0 to 200°C. Reel can be driven temperature measurements with a resolution of ~ 0 . 0 1 3 " C with a 12-V motor if desired. The total system weight is 25 kg.

several measurements are made at different times at each depth and the equilibrium temperatures calculated by extrapolation. An objection sometimes raised against thermistor sensors is their supposed lack of stability. This was a problem only in the early stages of thermistor manufacturing, however, and for the past 20 years thermistors have proved to be extremely stable if used at temperatures between 0 and 100°C. For example, we still have a probe which was originally fabricated in 1963. Its ice point resistance has changed less than 0.001"C in the past 18 years (Roy et al., 1968, probe 5K-396-2).

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Another type of sensor which can be used is a platinum device. The advantage of platinum as a temperature sensor is that the metal has a more nearly linear temperature-resistance response (in fact, it is exactly parabolic) than does the thermistor material. The typical coefficient of change of resistance with temperature is lower than for a thermistor, however, and to obtain the same temperature resolution, at least an order of magnitude more accurate resistance measurements must be made. A third device which has come into recent use in logging is the semiconductor integrated circuit AD590, manufactured by Analog Devices, and second sourced by Intersil. This device passes a current which is proportional to absolute temperature (1 mA/"C). Its advantage is that it can be used with a two-wire cable (or a one-conductor armored cable with a sheath return); thus, a very long cable can be put onto a relatively small reel. This device has proved practical for deep-well logging. The temperature resolution and time constant appear to be about the same as for thermistors, but self-heating limits its usefulness for measurements in air. The major limitation at present is temperature. The sensors are rated for 150"C, but we have used selected units to log wells to 175°C. Exposure to temperatures over 150°C may cause calibration drifts of several tenths of a degree, requiring recalibration. A second difficulty is that to resolve *O.Ool"C,very high insulation leakages must be maintained (thousands of megohms). This requirement puts strains on cable head construction and cable integrity. A final type of sensor, which has not been used very often in well logging, is a thermocouple. The disadvantage of a thermocouple is that the resolution may be poor; the advantage is that very high temperatures can be measured. In 40-m-deep holes, Hardee (1980) has used thermocouple equipment to measure temperatures over 10oO°C, and phototype equipment is being tested for use in holes as deep as 1.2-1.5 km. Muecke et af. (1974) used a thermocouple device to measure temperatures to 200°C in a geothermal well in the Azores. Thus such instruments could be used for very hot geothermal wells, where other continuous-reading devices fail because of electrical leakage at high temperatures in currently available cable construction. The cable usually used with this type of device is a two-conductor construction with MgO insulation in a stainless steel tube. Recently, cable with this construction with four conductors surrounded by MgO insulation inside a stainless steel tube has become available. The cable is manufactured by BICC Pyrotenax Limited and is rated to temperatures of 800°C. This type of cable could be used with a platinum sensor to obtain precise temperature measurements to very high temperatures. Lengths to 10 km are available. The longevity and suitability of this cable in logging situations are being evaluated (J. Dunn, 1983, personal communication). Probe constructions are quite variable and have changed with time as new

198


materials and designs have become available. Thin-walled stainless steel needle tubing is usually used as a sensor holder so that the sensor is protected from direct contact with the downhole environment. Typical probe constructions have been described by Beck (1965), Simmons (1965), Roy et af. (1968), Costain (1970), and Reiter et al. (1980), among others. Typical time constants of probe designs in use are a few seconds in stirred water. All the sensors described above can be monitored by electronic data acquisition systems so that a continuous (samples taken as often as desired) temperature log can be obtained. This continuous recording is very useful in holes with layered geology so that thin interbeds can be delineated, and in geothermal wells, where variations in gradient and temperature associated with individual fracture zones can be identified. Custom digital recording systems can be built for hardware costs of less than $1500, using one of the low-cost microcomputers now available (see Fig. 2). In the commercial well logging industry, the most common method for transmitting downhole measurements to the surface is frequency modulation. With this technique an analog value of voltage representing the temperature response of a sensor (for example) is converted to a frequency which represents the desired information. In this case electrical leakage in the cable and connecting components, which causes analog signal amplitude change and thus causes problems for voltage or current measuring devices, does no damage to the data represented by signal frequency. Unfortunately, the signal-conditioningelectronics which must be placed downhole to achieve this temperature-to-frequency conversion are necessarily complex, subject to drift with age at modest temperatures, and subject to extreme inaccuracy and outright failure at temperatures exceeding about 200°C.The resolution of commercial tools is typically 0 3 ° C or worse, although this resolution is determined largely by the tophole data reduction technique, and calibration errors are often several degrees Celsius or more. The major reason for this low quality is related to use of the data, however. In hydrocarbon exploration and production, temperatures are used in a qualitative fashion and no serious attempt is made to make high-accuracy, high-resolution measurements, particularly at temperatures approaching those seen in geothermal exploration. Bristow and Conaway (1984) have described a tool with downhole temperature-to-frequency conversion that has a precision of a few tenths of a millidegree and so is suitable for heat flow applications. Remote wireline techniques are the rule in oceanographic heat flow studies, where rather large, self-contained packages are lowered to great depths. The temperatures are measured electricallyversus time and recorded, or transmitted by acoustic signals directly back to the surface. Temperature sensors are typically thermistors. Development of such techniques for use downhole during the drilling process for continuous temperature (and other

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FIG. 2. Low-cost digital-recording temperature-logging system. The system includes a 5i-digit DVM,5 K RAM microcomputer with built in 40-column LED display and printer, two cassette tape recorders, and digital depth encoder. The system was used to make the temperature logs shown in Figs. 3 and 6.

200


physical property) logging during drilling is being carried out by the major logging companies. In most cases the effect of temperature on nuclear phenomena in boreholes is considered noise to the signal that interacts with the rock. Ross eta/. (1982) turned this approach around. They investigated the possibility of using temperature effects on thermal neutrons to measure formation temperatures. They did not carry their investigation past laboratory testing, however. 2.5. Temperature Precision and Resolution

With thermistor sensors it is quite feasible to make temperature measurements with a resolution of 0.0001"C to *O.OOl"C in field conditions (Roy et a/., 1968; Sass et a/., 1968). Typical accuracies are much less and are at best &O.O2"C to typical values of *O.O5"C or worse. If the gradient in a hole is calculated with data from a single run of the same probe, the gradient accuracy is determined by the temperature precision, however. Since gradients are typically calculated over depth intervals of several meters from measurements made with the same probe, instrumental errors are generally negligible. Amajor question is, of course, to what accuracy do measured temperatures reflect in siter rock temperatures? In some holes detailed logs with closely spaced reading intervals can resolve very fine-scale variations in lithology. Examples of these fine-scale variations have been shown by Roy et al. (1968), Conaway and Beck (1977a, b), Reiter et al. (1980), and Blackwell and Steele (1987), among others. The best examples are from holes which have been properly grouted, have reached equilibrium, and have low gradients. If the recording is made while instruments are being lowered, the response of the instrument must be taken into account. Costain (1976), Conaway (1977), and Nielsen and Balling (1984) have discussed data analysis techniques based on temperatures obtained with a temperature logging tool moving at quite high velocities (above 5-10 m/min). Examples of precision temperature logs are shown in Fig. 3. Three logging runs in an abandoned water well on the Southern Methodist University (SMU) campus are shown. Two runs were made with a thermistor sensor and a four-conductor armored well logging cable lowered at 4 m/min and recording temperatures to 0.001"C every 1 m. The third temperature log was made with an AD590 sensor on a one-conductor armored cable with the same logging speed but recording temperatures every 0.5 m. The ability of a gradient log to outline thin units with small variations in thermal conductivity is clearly demonstrated. The gradient logs are much better at lithologic resolution below the water table than the natural gamma-ray log because many limestones and sandstones are "dirty" (include significant amounts of potassium in clay).

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GAMMA

0

700E

800

b

1{ 1 TRlNlTV

1 SMU 0

50

SAND

W A ~ E R WELL

TRUCW I

1000

CPS

TRUCK 2

100

50 100 150

FIG. 3. Temperature gradient and total gamma count logs for a well on the Southern Methodist University campus. The section includes Upper and Lower Cretaceous rocks. Formation names are shown. Sand units are shown by dot pattern and shale units are shown by dash pattern. The remainder of the units are composed of marl or limestone. Temperatures were digitized at 1- or 0.5-m intervals to a precision of O.OO1°C and are not shown above the water table. Truck 1 logs were made with a conventional thermistor probe. Truck 2 log was made with a semiconductor sensor.

The ultimate limit to the resolution attainable and desirable is not related to instrumentation, however, but to the drill hole itself and the use planned for the data. If only a rough idea of gradient is necessary, rather crude resolution is adequate. For example, average gradients which are the same as averages obtainable with more precise and accurate equipment were obtained by Van Ostrand, using only the maximum-reading thermometers. However, the interpretation of such data is difficult because a detailed correlation cannot be made between the various effects in and around the borehole and variations in gradient. On the other hand, if 0.0001"Ccould be resolved, measurements with f 1% error of a 1O0C/km gradient could be made over a 1-m depth interval. This resolution would obviously allow very detailed analysis. Such logs could be used for detailed lithologic correlation, relative thermal conductivity determination between lithologies, determination

202


of fluid entry and exit points for very small flow rates, as well as gradient calculation for heat flow. In most holes, the limit of temperature resolution is determined by convection in the hole rather than the instrument response. Since temperature increases with depth, the density of the water decreases with depth and thus the water column is typically unstable. Three people have discussed in detail the effect of this phenomenon on temperature measurements in wells (Diment, 1967; Gretener, 1967 ;Sammel, 1968). Each of these authors discussed different aspects of temperature stability in a drill hole. Diment (1967) also showed that essentially the same temperatures (within 0.05"C) are obtained in air and in water in the same hole. This result verifies that temperature measurements in air are as useful as temperature measurements in water for determining the rock temperatures. All three authors showed that for most gradients, convection cells will exist in the water column, and each showed plots of the amplitude of typical temperature variations at a single depth as a function of gradient, hole size, or fluid type. Sammel (1968) calculated the critical gradients for instability in the water column and showed them as a function of temperature, hole diameter, and type of fluid in the hole. For example, at 20°C a gradient of more than 35"C/km would be required for instability in a 5-cm-diameter well. On the other hand, if the well diameter is 15 cm, convection would be expected if the gradient exceeds only 8"C/km. Thus under most typical conditions there is convection in a well bore. Experience shows that the most stable temperatures are observed in holes in which the temperatures decrease with depth, in which the temperatures are below the maximum-density temperature of water (about 4"C), or in which there is water flow within the hole. In almost all other water-filled holes, some degree of temperature instability is observed. Empirical observations show that for hole diameters of 45cm or less, temperatures ranging from 10 to 50"C, and geothermal gradients up to lOO"C/km, typical point temperature variations approach a maximum of only about 10% of the temperature change associated with the geothermal gradient over a 5-m depth interval. For example, Urban et al. (1978) showed a comparison of the amplitude of temperature oscillation at two depths in a geothermal well in the Imperial Valley of southern California (see Fig. 4). Temperature oscillations in the 20-cm-diameter cased well ranged from a maximum of *O.I"C for a temperature gradient of 315"C/km (0.3"C/m) to an oscillation of k0.03"C for a temperature gradient of 87"C/km (0.09"C/m). The data of Urban et al. (1978) seem to reinforce the observation that convection-induced oscillations are not enough in most situations to significantly affect a gradient measured over an interval of several meters.

14.

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TIME CMINUTESI 0

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FIG.4. Temperature oscillations as a function of time at two depths in Mesa 31-1 in the Imperial Valley, southern California. The gradient is 315"C/km at 98 m and 87'C/krn at 122 m (Urban et a / . , 1978).

However, Sammel (1968) has shown that if the temperature gradient is 1000"C/km or greater, the measured fluid temperatures may not be related to the rock temperatures. He showed an example where the gradients between 1 and 3 m were completely homogenized by convection in a water-filled pipe in the winter due t o the very high gradient caused by cooling of the surface (see Fig. 5a). With Richard Bowen we inadvertently repeated this experiment and verified Samuel's result. We also determined that in the same hole, at the same time, the air column in the annulus around the tubing was stable and rock temperatures could be obtained even though convection with an aspect ratio of 60 : 1 was occurring inside the 5-cm light oil-filled tubing in the 15-cm-diameter hole (see Fig. 5b). Recent observations with a continuous recording system (1 -m interval)

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FIG.5 . (a) Temperatures measured in adjacent shallow wells showing convection in waterfilled well and thermal stability in glycerol-filled well (Sammel, 1968). (b) Temperatures measured in a 15-cm-diameter well near Vale, Oregon. The solid line connects points measured in the air annulus around a 5-cm-diameter plastic pipe filled with light oil. The dashed line connects points measured in the water-filled pipe.

~

205

14. EXPERIMENTAL METHODS IN CONTINENTAL HEAT FLOW GRADIENT 'CIKM

CALIPER INCHES

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FIG.6. Caliper, temperature gradient, and lithology logs for geothermal test well OMF-7A (Blackwell el a/., 1982). Hole is uncased below 400111 and 18.4cm in diameter but had 5-cm tubing installed to total depth. Two gradient logs made with a single-conductor armored-cable AD590 sensor system are shown (one was made on July I and the other on July 15, 1981). The key to the summary geologic log is as follows: diagonal lines, andesitic volcanic rocks; caret pattern, Cenozoic microquartz diorite porphyry intrusives ; horizontal lines, Columbia River basalt ; pebble pattern, volcanoclastic rocks ; wavy pattern, greenstones.

206


with a resolution of 0.001"C have shown an interesting, previously unrecognized effect. In a well in the Cascade Range (OMF-7A) completed by placing 5-cm tubing in an open hole (18 cm diameter) the detailed gradient log correlates in places with the caliper (or hole size) log (see Fig. 6). Apparently convection related to hole size variations disturbs the gradient on a scale of 5-10m, so that the gradient no longer corresponds to lithology. Further investigation of the question of temperature stability in drill holes as a function of rugosity is obviously needed. Because of the convective effects in many holes a temperature log based on a single run with an instrument having a resolution of 0.0001"C or better has little advantage over one based on an instrument with 0.001"C resolution. The requirement for small-diameter grouted observation tubing for optimum temperature measuring conditions is illustrated by the analysis of these borehole convection effects. An interesting conclusion based on the analysis of convenction in drill holes is that air-filled holes may be stable at higher geothermal gradients than are water-filled holes. In some situations, better data may be obtained in the portion of the well above the water table than in the portion below the water table. In all shallow exploration holes, whether geothermal test holes or existing water wells or the like, the air as well as the water column portion of the hole should be logged. The effect of casing (steel or plastic) and a cement annulus on temperature is not significant and similar temperatures are obtained in cased and uncased holes if no intrahole fluid flow exists. The lack of a casing effect can be demonstrated theoretically by considering the effect of a needle of one thermal conductivity embedded in a medium of a different thermal conductivity. Jaeger (1965) presents the solution for an elliptical cylinder in an infinite medium for heat flow parallel to the long axis. As the long ellipse axis is increased in relation to the two smaller axes, the temperatures inside the ellipse approach those outside it except near the end of the ellipse.

3. Thermal Conductivity The second quantity that must be known to calculate heat flow is thermal conductivity. Thermal conductivity is the property which describes the ability of a material to transmit heat. A property often used in thermal analyses but seldom measured is the thermal diffusivity. Drury et al. (1984) have discussed a modified Angstrom technique for the measurement of thermal diffusivity, and additional references to thermal diffusivity measurement techniques may be found in that reference. The thermal conductivity of rock depends on many factors and thus is very difficult to estimate to a useful accuracy from tables based on rock type.

14. EXPERIMENTAL METHODS IN CONTINENTAL HEAT FLOW

207

For most practical applications it is necessary to make measurements of thermal conductivity on sample materials actually encountered in the drill hole from which temperatures were obtained. There are a number of different techniques for thermal conductivity measurement ; however, two main techniques are in common use at present by the geothermal community. These are the divided-bar and needle probe techniques. It is likely the future will see changes in types of techniques and improvements in existing techniques because thermal conductivity measurements represent one of the more labor-intensive parts of heat flow determinations, yet they are vital for accurate measurements. 3.1. Divided Bar

The divided bar is the most commonly used technique. It can be used to obtain thermal conductivity of core samples or of cuttings samples of isotropic materials. The basic instrumentation was described by Birch (1950). A variant similar to that in use in many laboratories today was designed by Robert Roy, and a diagram of that equipment is shown in Fig. 7 (Goss and Combs, 1976). Goss and Combs (1976) also give the equations for calculating thermal conductivity with this device. This type of device has been used to measure thermal conductivity from permafrost (King, 1976) to geothermal (Sibbett et a)., 1979) conditions and with variable pore fluid conditions (Somerton, 1975). Basically, the temperature difference across a known standard material (usually quartz and/or silica glass) is compared to the temperature difference across the unknown sample, using a second interim reference material (lexan or epoxy in this form of the apparatus). A temperature drop is maintained across the stack by heaters or by constanttemperature water-circulating baths. To obtain reliable measurements, the core samples should be water-saturated by vacuum/high-pressure techniques and loaded to at least 10 MPa during measurements. Otherwise, the effect of microcracks at low pressures is significant (Walsh and Decker, 1966), and systematically low thermal conductivities may be obtained. One problem with the apparatus as shown in Fig. 7 is bonding the lexan to the copper blocks above and below it so that a reproducible stack thermal resistance is maintained. In our apparatus we have substituted epoxy for the lexan. Epoxy bonds well to copper and has about the same thermal conductivity as lexan. Both lexan and epoxy have a thermal conductivity about onetenth that of rock (0.3 W m-l K-'), so the thermal resistance of the stack can be adjusted for comparison to that of a typical rock sample with about onetenth of the rock sample length. The short stack has lower heat loss (or gain) than the longer stack required if a material of high thermal conductivity is used (Beck, 1965).

208

DAVID D. BLACKWELL.AND ROBERT E. SPAFFORD

HYDRAULIC C Y L INDER I

_____

(

+&&-----

SAMPLE

I HOT

TH

LEXAN

F O I Y RUBBER

COLD

"LAST ICS

RUIOER

-

.

-

.

.

- .

..

__

.

.

.

.

. I

FIG. 7. Idealized diagram of the divided-bar apparatus (Goss and Combs, 1976, after R . F. Roy).

14.


209

In general, an attempt is made to obtain thermal conductivity values at as close to in situ conditions of temperature as possible. Alternatively, temperature corrections can be applied. Measurements of the effect of temperature on thermal conductivity as well as general compilations of measurements have been summarized by Clark (1966) and Roy et al. (1981). The divided-bar technique, unlike the other techniques, can be used for anisotropic rocks. Core samples can be prepared in different directions so that the tensor components of thermal conductivity can be measured. Anisotropy effects are large for rocks such as shales and argillites composed of layered silicates. The divided-bar technique can also be used for thermal conductivity measurements on cutting samples of isotropic materials. It is suitable only for isotropic materials because there is no way to orient the cuttings fragments in their in situ direction (Blackwell and Steele, 1987). This technique has been described by Sass et al. (1971). A mixture of cuttings and water is put in a plastic sleeve and the whole container measured in the divided bar as if it were a normal core sample. Then the grain thermal conductivity of the cuttings material is calculated. This conductivity can be used, with an estimate of the in situ porosity, to calculate an in situ thermal conductivity, typically by use of the geometric mean mixing equation:

Ki, = K$KJ'-"

(1)

where Ki, is the in situ thermal conductivity, Kb the measured bulk thermal conductivity, K , the thermal conductivity of water (0.59 W m-' K-' at 20°C), and 4 the fractional porosity. Lack of knowledge of the in situ porosity is probably the factor limiting the accuracy of this method. Porosity values usually must be obtained from log data, core samples, or estimates based on knowledge of the rock type, although Morgan (1975) has described a technique for directly measuring the porosity of cutting samples. The divided-bar technique is relative, so the intra- and interlaboratory precision depends on the precision of the thermal conductivity determination of the standard materials used and the similarity of materials obtained by different laboratories at different times. In 1975 a comparison was made of four laboratories measuring thermal conductivity by the divided-bar technique (three in the United States and one in Canada). No instructions were given other than to measure the thermal conductivity. The results for eight samples are shown in Table I. The results indicate a worst-case error of I4% and errors of the mean of several samples of < 1% (D. S. Chapman, 1980 personal communication). The accuracy of the measurement depends on the accuracy of the silica glass and natural quartz thermal conductivity values taken for calbration. By convention, so that interlaboratory values are comparable, quartz and silica glass thermal conductivity values tabulated by

210


TABLE I. Interlaboratory Comparison of Divided Bar Thermal Conductivity Measurements' Vo deviation from mean

Sample number

Mean thermal conductivity (W m-' K-')

Lab 1

Lab 2

Lab 3

Lab 4

1 2 3 4 5 6 7 8

1.75 1.91 2.36 2.78 2.78 2.91 3.44 3.75

+0.3

+ 5.4

+2.5 0.0 1.7 1.4 +0.3 - 4.2 0.9

- 3.8

+ 0.3 + 0.9

- 1.8

Mean deviation from mean (Vo) R M S deviation from mean (Yo)

+ + 0.4

- 6.0 +0.4 + 2.8 - 1.2 1.1 -3.4 6.0 2.5

1.9

3.7

+ +

- 2.2 +4.1

+ 0.3

- 2.7 - 0.5

+ + +

-0.4 - 1.0 - 1.8 - 1.5

+0.3 3.5

- 0.8 1.4

a Results tabulated by David S. Chapman, Department of Geological Sciences, University of Utah, based on comparison made in 1975.

Ratcliff (1959) are used. Estimated accuracies for cuttings measured on the divided bar or by the needle probe (samples) are lower, with typical values of f 10% being given (Sass et al., 1971).

3.2.Needle Probe A second technique, most commonly used in oceanographic heat flow measurements, is the needle probe method. The geophysical applications of this method were first discussed by Von Herzen and Maxwell (1959; Von Herzen et al., 1962). With this instrument, a hypodermic needle containing a linear heater and a thermistor or thermocouple is implanted in the material whose thermal conductivity is being determined. The heater is turned on, and the change in temperature with time is recorded. The equation for calculating thermal conductivity is

T = (Q/4nK)ln(t)

+C

(2)

where Q is heat per unit length per unit time, K is conductivity, and C is a constant. Thus the quantities measured are temperature and time and the thermal conductivity can be calculated directly. An illustration of the equipment and some typical data are shown by Von Herzen and Maxwell (1959). The difficulty in applying this technique to rocks is that it is not easy to drill a 60-mm-deep, 0.9-mm-diameter (or equivalent) hole in a granite or a basalt. Furthermore, there is a problem with microcrack effects at 0.1 MPa pressure. Thus this technique is best used with cuttings, ocean or lake bottom sediments, or soft sediments such as might be encountered in valleys in the Basin

14.


21 1

and Range Province. In oceanic situations, measurements of thermal conductivity are often made in situ, with heaters and thermistors contained in probes which are driven into the bottom sediments (see also Sass et al., 1981). The technique is suitable only for isotropic materials.

3.3.Other Techniques There are many other techniques for measuring thermal conductivity, using both steady-state and transient approaches. A device called a QTM and using a transient method has been marketed by Showa Denko K.K., a Japanese company. Temperature is measured as a function of time, related to heat applied to the surface of the material. This technique has been used quite effectively for many different kinds of samples. Burch and Langseth (1981) have compared thermal conductivities of samples from Deep Sea Drilling Project (DSDP) holes measured by the needle probe and QTM techniques. Sass et al. (1984a) have discussed a detailed comparison of the divided-bar and QTM techniques. They find comparable results in most cases. Sample preparation is less tedious for the QTM technique than for the divided bar technique and minor surface roughness does not seem to be a problem. Large samples (30 mm x 60 m x 100 mm) are required, however, and saturation and microcrack effects are important. The materials studied must be isotropic. Following a suggestion by Vacquier (1989, Sass et al. (1984b) described a measurement technique using a device consisting of a conventional needle probe embedded in the flat surface of a half-cylinder of insulation material. The cylinder is placed on a flat rock surface and the needle probe operated in a conventional manner. This approach approximates the QTM technique except that a conventional needle probe with its recording setup is used. Data reduction is similar to that with the needle probe, and a comparison of results obtained with the QTM and half-space-needle probe techniques to dividedbar measurements shows similar accuracies. This technique shows promise as a useful supplement to the divided-bar technique for measurement of the thermal conductivity of consolidated isotropic materials. Other experimental techniques have been described in the literature but for various reasons are not commonly used. Jaeger and Sass (1964) discussed a line source technique requiring a saw cut in a rock sample. Hardee (1971) discussed an interesting technique in which a thermal penetrator was air dropped into a freshly emplaced Mount St. Helens pumice flow. Temperatures in the penetrator after emplacement were interpreted in terms of radial heat transfer to give the in situ thermal conductivity and thermal diffusivity of the pumice. Attempts have been made to determine thermal conductivity in situ in hard rocks by sealing off a section of the hole and then heating that section and

212


monitoring the temperature rise during heating or the temperature drop after a heat pulse has been applied (Beck, 1965; Beck et al., 1971). After much experimentation, the general conclusion is that uncertainty about thermal contact and effects of possible water convection when dealing with an actual rough, fractured hole wall are so great that useful results cannot always be obtained. At present we know of no one still pursuing this line of investigation. 3.4. Indirect Measurements Because of the difficulties of measurement and sampling, particularly in deep petroleum exploration wells, an indirect thermal conductivity measurement technique using commonly available well logs would be very useful. The typical approach to indirect thermal conductivity measurement is to correlate thermal conductivity with sonic velocity, density, electrical resistivity, and other properties which can be obtained from well logs. Correlation of thermal conductivity with velocity has been discussed for Imperial Valley sandstones by Goss and Combs (1976). Relationships between thermal conductivity and velocity, density, and porosity have been discussed by Merkel et al. (1976) for Cretaceous rocks in central Texas. Williams (1981) and Blackwell et al. (1982) have discussed the correlation of thermal conductivity with velocity for the Columbia Plateau and Snake River Plain basalts, Snake River Plain rhyolites, and Cascade Range andesites. In addition, Steele et al. (1981) discussed an empirical correlation between thermal conductivity and gamma-ray activity or sonic velocity for shales and limestonesin Kansas. Houbolt and Wells (1980) calculated thermal resistivity (the inverse of thermal conductivity) from seismic reflection travel times. Temperature gradient logs themselves give relative values of thermal conductivity in a hole if the heat flow remains constant throughout the depth range of the hole. All those techniques have recently been summarized by Blackwell and Steele (1987). An advantage of an indirect approach is that thermal conductivity values can be calculated as a continuous function of depth. Thus local areas of anomalous heat flow in the hole can be identified. In addition, the thermal conductivity obtained may be less biased than one calculated from measurements on core samples because the core samples may not be representative of the rocks encountered, particularly if the rock is highly jointed or fractured. For example, most of thermal conductivity values for shales reported in the literature appear to be in error on the high side by 50-100% because of sampling and anisotropy problems (Steele et al., 1981; Blackwell and Steele, 1987). The results from indirect techniques can be improved by combining them with a few core or cutting conductivity measurements from each hole studied.

14.

EXPERIMENTAL METHODS IN CONTINENTAL. HEAT FLOW

213

4. Heat Production There has been much discussion in the literature of various sources of heat and their effect on surface heat flow measurements. A major effect on regional heat flow is the generation of heat by the decay of uranium, thorium, and potassium in the outermost layers of the earth. In the continental crust, decay of the radioactive elements adds significantly to the surface heat flow (Birch, 1954). Identification of the linear heat flow-heat production relationship (Birch el al., 1968) has allowed quantification of this contribution. A typical technique for measuring the content of radioactive elements in a rock is the gamma-ray spectrum pulse-height analysis technique (Wollenberg and Smith, 1964). Both NaI(T1) and Ge(Li) detectors are used (Lewis, 1974). At present, most well logging companies can run a “heat production log.” Three-channel “spectra logs” can be obtained that separately measure the amounts of potassium, uranium, and thorium present in rocks cut by a well. From this type of log the heat production can be calculated. Descriptions of equipment and/or results of spectra-logging are given by West and Laughlin 1976), Lovborg et al. (1980), and Smith et af. (1983). Some authors have suggested that local generation of heat by chemical reactions may be important in the initiation or maintenance of some geothermal systems. For example, Parry el al. (1980) suggested that at Roosevelt Hot Springs, Utah, significant heat might be generated by chemical alteration reactions. Similarly, oxidation of sulfides in ore bodies might generate significant amounts of heat (Lovering and Morris, 1965 ; Edmiston, 1971). Such effects have usually been shown to be quite small in actual cases, however, and it is unlikely that they are significant in any but the rarest cases. The heat production from chemical reactions can be estimated, given the nature of the reaction and the rate at which it takes place. The biggest problem with this mechanism is having enough of the reactant, and having it react fast enough, for thousands of years to generate geothermally significant amounts of heat.

5. Heat Flow Calculation 5.1. Calculation of Average Conductivity

The effect of the water table on temperatures, gradients, and thermal conductivity is not generally understood. It might be supposed from Eq. (1) that, if the rocks above the water table are porous but not permeable, a significant change in gradient would occur at the water table (air having a much lower thermal conductivity than water). This effect might be important because in many areas water tables are several tens or even hundreds of

214


meters below the surface. Thus if reliable measurements could be made in air, it might be possible to use much shallower holes than would be required if reliable measurements could be made only below the water table. Of course, if the rocks are highly permeable and convective heat transfer is important, use of shallow holes above the water table would not be possible. We have found that, in general, in rocks with low porosity and low permeability there are no detectable differences in the mean gradient above and below the static water table. If water table effects exist, they are on the average less than the resolution of the gradient measurement (1- 1O%, depending on the particular drill hole). In rocks which are porous, differences seem to depend on the permeability and lithology of the rock. A sample temperature-depth curve from Vale, Oregon, is shown in Fig. 8. This hole was drilled in impermeable but porous siltstone. The thermal conductivity contrast between the siltstone (30% porosity) with air in the pores and that with water in the pores is 300%. On the other hand, it is obvious from the temperature-depth curve that the gradient above the water table is only 5-10070 higher than the gradient below the water table. Van Wijk (1966) has discussed in detail the effect of percentage of saturation on thermal conductivity. His conclusions are that at moderate fractional saturation, heat transport by the vapor is so effective that the thermal conductivity of the partially saturated material is approximately the same as that of the saturated material. Because the capillary force is strong for clays and siltstones, the fractional saturation at various depths

olo.o

TEMPERATURE, "C

,.

20.0 I "

'

"

M.0

30.0

"

VALE, OREGON 19S145E- 2 6 8 0 7/21/72

a a w

-

t r --

WATER TABLE

9100

I

I - -

.o W

0

200

-

l

t

,

l

,

l

,

,

FIG.8. Temperature-depth data for a hole in the Vale-Cow Hollow geothermal anomaly, Oregon. The hole was drilled in Pliocene Chalk Butte siltstone. The hole was not cased or grouted and measurements were made in air or water as indicated.

14.

EXPERIMENTAL METHODS IN CONTINENTAL HEAT FLOW O/o

215

WATER BY VOL.

Yo SATURATION (1.30

%)

FIG.9. Thermal conductivity of Pliocene Chalk Butte siltstone as a function of saturation. The in situ porosity is 30%. Calculated curve is from Van Wijk (1966). Data points are in situ measurements between the surface and 2 m by the needle probe technique. (Data from D. D. Blackwell and C. A. Brott, unpublished study.)

above the water table will be greater than for sands and gravels. Thus the thermal conductivity above the water table depends on the degree of saturation, which depends on other parameters, and so the amount of water-table effect will depend on the nature of the rock. Using an expression for the effect of partial saturation for siltstone, the thermal conductivity as a function of saturation was calculated for the rock type in which the Vale hole was drilled (see Fig. 9). Also shown in Fig. 9 are thermal conductivity measurements made as a function of depth below the surface by the needle probe technique. A 2-m hole was drilled with an auger and the measurements were made in the hole by sticking the needle into the walls of the hole. Based on these observations, the thermal conductivity is approximately 90% of the saturated thermal conductivity at a depth of less than 2 m. Hence, the agreement of the gradients above and below the water table is to be expected, and we may conclude that, in this case, it is appropriate to use the saturated thermal conductivity both above and below the water table. On the other hand, for a well-drained rock such as sand or gravel, there may be a significant thermal conductivity contrast above and below the water table, although it is not clear whether the observed effects are due to the degree of saturation or are due partially to water flow in these very permeable, as well as porous, materials.

216


In the calculation of heat flow the mean harmonic thermal conductivity K H is usually used. The equation for KH is 1 -=

Kn

CiI/Ki n

i = 1,2,3 ,..., n

(3)

The harmonic thermal conductivity is considered most appropriate because it is assumed that the variations in thermal conductivity in the earth are in horizontal layers and thus the proper mean value is the mean of the thermal resistance. The harmonic average tends to weight a value which is much higher than the average somewhat less than the arithmetic average does, but in most cases the harmonic average thermal conductivity and the arithmetic average thermal conductivity are very similar. A weighted mean harmonic thermal conductivity, calculated with representative thicknesses of the layer represented by each sample, should be used when a hole penetrates a layered medium. An equation for calculating the average thermal conductivity in a dipping anisotropic material such as shale, schist, or argillite is discussed by Hyndman and Sass (1966). 5.2. Heat Flow Calculation

Once the thermal conductivity has been measured and the geothermal gradient calculated, the heat flow in some interval can be calculated as a simple product of the geothermal gradient times the mean thermal conductivity : Qz = K(dT/dz)

(4)

There are various ways to calculate the heat flow values. The equations have been summarized by Hyndman and Sass (1966, p. 590). Sums are used rather than integrals because the actual measurements of gradient and thermal conductivity are not made at infinitely close spacing. If there is no systematic variation of geothermal gradient with depth, the heat flow may be calculated as the product of a least-squares fit to the temperature-depth data [so that the error of the slope (gradient) can be calculated] and the mean harmonic thermal conductivity or mean resistance over the length of the hole :

Q z = KH(dT/dZ)l.q. (5) If there is a systematic change in gradient with depth, the calculation of gradient and thermal conductivity must be over intervals rather than over the entire drill hole. The technique discussed in association with Eq. ( 5 ) can be used for each interval, or the heat flow can be calculated according to

14.


217

where Azi is a particular depth interval and Ki the appropriate thermal conductivity value for that interval. The heat flow can then be obtained from a least-squares calculation of the slope of a plot of ((2) versus temperature T(z).This calculation method is referred to as the resistance integral technique (Jaeger, 1965). It is particularly useful in layered media with temperature measurements that are widely spaced with respect to the lithologic variations. The error associated with the measurement can be calculated by propagation-of-error techniques from the errors calculated for the thermal conductivity and gradient if Eq. ( 5 ) is used. If Eq. (6) is used, the error can be calculated directly during the least-squares slope calculation. Calculation of the heat flow as a function of depth in a drill hole in which geothermal gradient varies with depth allows an evaluation of the assumption of conductive heat flow in that particular drill hole. If geothermal gradient and thermal conductivity variations cancel each other out and the heat flow remains constant after the various types of corrections have been applied, then the assumption of conductive heat flow is satisfied (at least locally). On the other hand, if there are variations in geothermal gradient which are not related to variations in thermal conductivity, then there is some problem, and previously unnoticed water flow, microclimatic effects, and so forth may be disturbing the geothermal gradient.

6. Miscellaneous Techniques While the discussion above has covered typical heat flow measurements, in a few cases other techniques may be employed. Techniques used in lakes are usually similar to oceanic techniques, in some cases modified for the shallower water (Haenel, 1970; Morgan et al., 1977). The use of very shallow holes in some situations is discussed in the next section. Sass et al. (1981) have described an unusual technique suitable for heat flow determination in unconsolidated to semiconsolidated clay units typical, for example, of valleys in the Basin and Range Province. Drilling is temporarily suspended and a scaled-up needle probe about 2 m long is forced hydraulically through the drill pipe and (a hole in) the drill bit about 1.5 m into the sediments ahead of the drill bit. Measurements of temperature versus time are made, and then electrical current is applied to a heater in the probe and needle probe-type thermal conductivity values are obtained. With their apparatus two gradient and three thermal conductivity measurements are obtained. Temperature measurements made in tubing installed after the hole had been drilled past the point of in situ measurement were generally within +O.O5"C of the temperatures measured with the needle probe. The main

218


advantage of the large-scale needle technique is that the holes do not have to be completed by installing tubing and by grouting because a heat flow measurement is made during drilling. In these geologic settings the holes often cave in when the drill pipe is removed from the hole and before tubing is installed, so conventional techniques cannot be applied. 6.1. Heat Flow in Shallow Holes

Some investigators have suggested that heat flow anomalies can be identified in holes between 1 and 3 m deep (see, e.g., Poley and Van Stevenick, 1970). The reason for the 1-m depth limitation is that this is below most of the diurnal effect. In holes 1-3 deep, the primary effect on the temperature is due to the annual cycle, which ranges from 2 to 10°C in amplitude. A background gradient would generate a temperature difference of 0.04"C (20"C/km) to perhaps 1.O"C (SOO"C/km) over a 2-m depth interval. Consequently, absolute measurement of geothermal gradient and heat flow in holes of this depth is not attempted, and exploration data in this depth range are used merely to locate temperature anomalies. Of course, these techniques cannot work at all unless the anomaly comes conductively or convectively to within a few meters of the earth's surface. In these kinds of applications noise effects are associated with temperature variations at shallow depths other than those related to geothermal anomalies. Such factors as microclimatic setting (albedo, roughness, mean wind speed, slope orientation, elevation, and vegetation cover), thermal conductivity variations in time and space, and the effects of rainfall and very shallow ground water movements must be considered. Temperature measurements must be made as nearly simultaneously as possible to avoid the effects of drift associated with the annual temperature cycle. Corrections are not simple or established for all of these effects; thus the typical noise associated with this technique is of the order of several degrees Celsius. The equipment used in these investigations is typical temperature-depth equipment capable of measuring accurate temperatures. Example studies have been presented by Bowen et al. (1977), Olmsted (1977), LeSchack et al. (1979), LeSchack and Lewis (1983), Lange et al. (1982), and Zielinski and Bruchhauser (1983). In general, these techniques are suitable for use only in geothermal systems and would not be used or considered if regional heat flow was the quantity to be measured. One unusual type of instrument suggested for use in the near-surface setting is a thermopile device which measures heat flow directly. This device might be installed at the surface and covered by insulation to effectively increase its apparent depth, or it might be emplaced in a drill hole or in a unventilated mine working to measure heat flow directly. Such a device has

14. EXPERIMENTAL METHODS IN CONTINENTALHEAT FLOW

219

been described by Poppendiek et al. (1982). At present, the accuracy of such a technique is undocumented. The next depth range usually discussed in the measurement of geothermal gradient or heat flow is 10-30 m. At these depths, annual effects are minor and, if heat transfer surface disturbances are small, significant heat flow anomalies can be identified. The factors that must be taken into account when trying to measure heat flow in holes of these depths have been discussed by Lovering and Goode (1963) and Lovering and Morris (1965). The conductive effects of annual temperature cycles at depth have been discussed for two- and three-conductivity-layer models by Lachenbruch (1 959) and Van Wijk (1966). Lachenbruch et al. (1976) described an example of the use of this technique in the Long Valley geothermal system, California, where data from holes in the depth range 10-30 m indicated the nature of the ground water flow system that was outlined by deeper drill holes (100-300 m). More detailed interpretations of microclimatic effects and of the effect of shallow ground water motions must be considered in analyzing these data. Even in the application to geothermal systems where heat flow is high, it is generally concluded that these shallow techniques, while possibly useful, are not cost effective. Thus most exploration uses deeper holes and more conventional heat flow measurement techniques.

7. Future Research A major influence on heat flow techniques in the past 10 years has been the development of these techniques as an exploration tool for geothermal systems. Making measurements in the geothermal environment of high temperatures, dynamic fluid motions, and complex chemical environments has become important. Plans and proposals for scientific drilling into the deep parts of geothermal systems (and the shallow parts of magmatic systems) will continue pressure for instrument development. Well logging tools developed for such applications in the Hot Dry Rock research program are described by Dennis et al. (1985). Accurate high-resolution temperature measurements can be made to 320°C at present, with a potential of up to 800°C at depths of several kilometers on the horizon. Pressure measurements can be made to similar depths and temperatures by using open gas-filled stainless steel needle-tubing with the pressure sensors at the surface. Thus practical equipment for making important physical measurements up to magmatic temperatures appears possible. Temperature and heat flow data have become increasingly important in hydrocarbon exploration as well (Gretener, 198 1 ;Blackwell, 1986). Exploration evaluation of sedimentary basins increasingly includes estimates of present and paleotemperatures and heat flow and their influence on the

220


thermal metamorphism (maturation) of organic matter. In addition, migration of hydrocarbons may be related to regional or local fluid flow, which in turn influences the thermal regime. In spite of the need for thermal data, the state of knowledge of the thermal characteristics of sedimentary basins is primitive and much research should be done (Blackwell and Steele, 1987). Continued interest in using temperatures in hydrocarbon exploration will eventually lead to development of techniques for obtaining useful thermal data from more of the thousands of deep exploration holes that are drilled annually. Measurement of temperatures at or just ahead of the bit during drilling and more accurate interpretation of drilling thermal effects should be investigated and developed. The requirement for thermal conductivity measurements specific to a drill hole with temperature measurements is a major factor limiting the number of heat flow determinations made. Development of an accurate, efficient, and convenient way to measure thermal conductivity in situ or on large masses of rock has the potential to revolutionize continental heat flow studies. In spite of much effort, however, this objective remains elusive. Perhaps the need to have this sort of information for proper evaluation of the thermal regimes of sedimentary basins will stimulate research into this important topic. Precision and resolution of temperature recording in wells also will be improved in order to extract the maximum amount of information about the relationship of gradient to lithology, fluid flow, and heat flow variations. Inexpensive digital recording systems will become more and more powerful and will allow more detailed logs to be made and more information to be extracted from the recorded data. Various data processing techniques will be developed for temperature logs. Combined studies (and synthesis of data collected at different times) of subsurface fluid flow and heat flow (and temperature) will become common in many geologic terrains. These types of studies will benefit hydrologists, who will learn more about very slow medium- and large-scale fluid flow in regional aquifers, geothermal systems, and sedimentary basins. At the same time, heat flow researchers will learn how to better estimate the potential errors from subsurface fluid flow and learn whether or not heat flow values in certain geologic and tectonic settings are accurate estimates of regional heat flow and can be used for crustal and lithospheric studies. Finally, major changes will occur in the way we model heat flow processes. Most techniques in use now are oriented toward current thermal conditions. Studies of temperature and heat flow history will become more common in the future. Thermal and time information obtained from apatite and zircon fission-track annealing studies, K-Ar diffusion, clay diagenesis, organic metamorphism, conodont color changes, and many other approaches not

14.


221

presently used will become available to allow tracking, within limits, of the thermal history of geologic terrains. These studies will provide an added dimension to what is currently considered heat flow research. Acknowledgments This work was supported in part by Department of Energy grant DE-AC03-81ER 10973 and National Science Foundation grant EAR-8420339, which are gratefully acknowledged.

References American Association of Petroleum Geologists, U.S. Geological Survey, Subsurface temperature map of North America. U.S. Geol. Surv. Map, 1 : 5,OOO,OOO (1976a). American Association of Petroleum Geologists, U.S. Geological Survey, Geothermal gradient map of North America. U.S. Geol. Surv. Map, 1 :5,000,000 (1976b). Back, W., B. B. Hanshaw, L. N. Plummer, P. H. Rahn, C. T. Rightmire, and M. Rubin, Process and rate of dedolomitization: Mass transfer and I4C dating in a regional carbonate aquifer. Geol. SOC.Am. Bull. 94, 1415-1429 (1983). Beck, A. E., Techniques of measuring heat flow on land. In “Terrestrial Heat Flow” (W. H. K. Lee, ed.), Monogr. No. 8, pp. 24-57. Am. Geophys. Union, Washington, D.C., 1965. Beck, A. E., F. M. Anglin, and J. H. Sass, Analysis of heat flow data-in situ thermal conductivity measurements. Can. J. Earth Sci. 8, 1-19 (1971). Birch, F., Temperature and heat flow in a well near Colorado Springs. Am. J. Sci. 245,733-753 (1947). Birch, F., Flow of heat in theFront Range, Colorado. Geol. SOC.Am. Bull. 6 , 567-630 (1950). Birch, F., Heat from radioactivity. In “Nuclear Geology” (H. Faul, ed.), pp. 148-174. Wiley, New York, 1954. Birch, F., Earth heat flow measurements in the last decade. In “Advances in Earth Science” (P. M. Hurley, ed.), pp. 403-430. MIT Press, Cambridge, Massachusetts, 1966. Birch, F., R. F. Roy, and E. R. Decker, Heat flow and thermal history in New England and New York. In “Studies of Appalachian Geology: Northern and Maritime” (E. Zen, W. S. White, J. B. Hadley, and J. B. Thompson, Jr., eds.), pp. 437-451. Wiley (lnterscience), New York, 1968. Blackwell, D. D., Use of heat flowhemperature measurements, including shallow measurements, in hydrocarbon exploration. In “Unconventional Methods in Exploration for Petroleum and Natural Gas, IV” (M. J. Davidson, ed.), pp. 321-350. Southern Methodist Univ. Press, Dallas, Texas, 1987. Blackwell, D. D., and J. L. Steele, Thermal conductivity of sedimentary rock-measurement and significance. In “Thermal History of Sedimentary Basins-Methods and Case Histories” (N. Naeser and T. McCulloh, eds.). Springer-Verlag, New York, 1987. Blackwell, D. D., J. L. Steele, and C. A. Brott, The terrain effect on terrestrial heat flow. J . Geophys. Res. 85, 4757-4772 (1980). Blackwell, D. D., C. F. Murphey, and J. L. Steele, Heat flow and geophysical log analysis for OMF-7A geothermal test well, Mt. Hood, Oregon. In “Geology and Geothermal Resources of the Mount Hood Area, Oregon” (G. R. Priest and B. F. Vogt, eds.), Spec. Pap.-Oreg. Dep. Geol. Miner. Ind. No. 14, 47-56 (1982). Bowen, R. G., D. D. Blackwell, and D. A. Hull, Geothermal exploration studies in Oregon. Misc. Pap.-Oreg. Dep. Geol. Miner. Ind. No. 19 (1 977). Brickwedde, F. G., ed., “Temperature, Its Measurement and Control in Science and Industry: Basic Concepts, Standards, and Methods,” Vol. 3, Part 1. Reinhold, New York, 1962.

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Bristow, Q., and J. G. Conaway, Temperature gradient measurements in boreholes using low noise high resolution digital techniques. Curr. Res., Part E, Geol. Surv. Can., Pap. 84-1B, 101-108 (1984).

Bullard, E. C., Historical introduction to terrestrial heat flow. In “Terrestrial Heat Flow” (W. H. K. Lee, ed.), pp. 1-6. Monogr. No. 2, Am. Geophys. Union, Washington, D.C., 1965.

Burch, T. K., and M. G. Lanseth, Heat-flow determination in three DSDP boreholes near the Japan Trench. J. Geophys. Res. 86,9411-9419 (1981). Cermak, V., Underground temperature and interval climatic temperatures of the past millennium. Paleocol., Paleoclimatol., Paleoecol. 10, 1-19 (1971). Chapman, D. S., J. Howell, and J. H. Sass, A note on drillhole depths required for reliable heat flow determinations. Tectonophysics 103, 11-18 (1984). Clark, S. P., Jr., Thermal conductivity. In “Handbook of Physical Constants” (S. P. Clark, Jr., ed.), Mem.-Geol. SOC.Am. No. 97, 459-482 (1966). Conaway, J. G., Deconvolution of temerature gradient logs. Geophysics 42, 823-838 (1977). Conaway, J. G., and A. E. Beck, Continuous logging of temperature gradients. Tectonophysics

41, 1-7 (1977a). Conaway, J. G., and A. E. Beck, Fine-scale correlation between temperature gradient logs and lithology. Geophysics 42, 1401-1410 (1977b). Costain, J. K., Probe response and continuous temperature measurements. J. Geophys. Res.

75, 3969-3975 (1970). Cremer, G. M., ed., Hot Dry Rock geothermal energy development program, Annual report, Fiscal year 1981. USDOE Rep. LA-885J-HDR, UC-669 (1981). Dennis, B. R., S. P. Koczan, and E. L. Stephani, High-temperature borehole instrumentation. U.S. Dep. Energy Rep. LA-10558-HDR (1985). Diment, W. H. Thermal regime of a large diameter borehole: Instability of the water column and comparison of air- and water-filled conditions. Geophysics 32, 720-726 (1967). Drury, M. J., and T. J. Lewis, Water movement within Lac DuBonnet batholith as revealed by detailed thermal studies of three closely spaced holes. Tectonophysics 75, 337-351 (1983).

Drury, M. J., V. S. Allen, and A. M. Jessop, The measurement of thermal diffusivity of rock cores. Tectonophysics 103, 321-333 (1984). Edmiston, R. C., Thermal gradients and sulfide oxidation in the Silver Bell mining district, Pima County, Arizona. M.S. Thesis, Univ. of Arizona, Tucson, 1971. Gaffanti, M. D., and M. Nathanson, Temperature-depth data for selected deep drill holes in the United States obtained using maximum thermometers. Geol. Surv. Open-File Rep. (U.S.) NO. 81-555 (1981). Gosnold, W. D., Jr., Heat flow and groundwater flow in the Great Plains of the United States. J. Geodyn. 4, 247-264 (1985). Goss, R., and J . Combs, Thermal conductivity measurement and prediction from geophysical well log parameters with borehole application. Proc. U.N.Symp. Dev. Use Geotherm. Resour., Znd, Sun Francisco, 1975 pp. 1019-1027 (1976). Gretener, P. E., On the thermal instability of large diameter wells-an observational report. Geophysics 32, 727-738 (1967). Gretener, P. E., Geothermics: Using temperature in hydrocarbon exploration. Educ. Course Note Ser., 7, Am. Assoc. Petrol. Geol. (1981). Haenel, V. R., A new method for the determination of the heat flow in lakes. 2.Geophys. 36, 725-742 (1970).

Hardee, H. C., Solidification in Kilauea Iki lava lake. J. Volcano/ Geotherm. Res. 7 , 21 1-233 (1980).

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Hardee, H. C., Thermal property measurements in a fresh pumic flow at Mt. St. Helens. Geophys. Res. Lett. 8, 210-212 (1981). Henry, S. G., and H. N. Pollack, Heat flow in the presence of topography: Numerical analysis of data ensembles. Geophysics 50, 1335-1341 (1985). Houbolt, J. J. H. C., and P. R. A. Wells, Estimation of heat flow in oil wells based on a relation between heat conductivity and sound velocity. Geol. Mijnbouw 59, 215-224 (1980). Hyndman, R. D., and J. H. Sass, Geothermal measurements at Mt. Isa, Queensland. J. Geophys. Res. 71, 587-603 (1966). Jaeger, J. C., Application of the theory of heat conduction to geothermal measurements. In “Terrestrial Heat Flow” (W. H. K. Lee, ed.), Monogr. No. 2, pp. 7-23.Am. Geophys. Union, Washington, D. C., 1965. Jaeger, J. C., and J. H. Sass, A line source method for measuring the thermal conductivity and diffusivity of cylindrical specimens of rock and other poor conductors. Br. J. Appl. Phys. 15, 1-8 (1964). Jessop, A. M., Distribution of glacial perturbation of heat flow in Canada. Can. J. Earfh Sci. 8, 162-170 (1971). Jessop, A. M., The essential ingredients of a continental heat flow determination. Zentralbl. Geol. Palaeontol. 1, 70-79 (1983). King, M. S., Thermal conductivity measurements on saturated rocks at permafrost temperatures. Can. J. Earth Sci. 16, 73-79 (1979). Lachenbruch, A.H., Periodic heat flow in a stratified medium with application to permafrost problems. Geol. Surv. Bull. (US.)No. 1083-A(1959). Lachenbruch, A. H., Rapid estimation of the topographic disturbances to superficial thermal gradients. Rev. Geophys. Space Phys. 6, 365-400 (1968). Lachenbruch, A. H., The effect of two-dimensional topography on superficial thermal gradients. Geol. Surv. Bull. (U.S.)No. 1203-E(1969). Lachenbruch, A. H., and M. C. Brewer, Dissipation of the temperature effect in drilling a well in arctic Alaska. Geol. Surv. Bull (U.S.) No. 1083-G,73-109 (1959). Lachenbruch, A. H., and B. V. Marshall, Heat flow in the Arctic. Arctic 22, 300-311 (1 969). Lachenbruch, A. H., M. L. Sorey, R. E. Lewis, and J . H. Sass, The near-surface hydrothermal regime of Long Valley caldera. J. Geophys. Res 81, 763-768 (1976). Lange, A. L., H. D. Pilkington, and J. Deymonaz, Comparative studies of geothermal surveys in 3-meter and temperature gradient holes. Geotherm. Resour. Counc. Trans. 6, 133-136 (1982). LeSchack, L. A., and J. E. Lewis, Geothermal prospecting with “shallo-temp” surveys. Geophysics 48,975-996 (1983). LeSchack, L. A., J. E. Lewis, D. C. Chang, R. I. Lewellen, and N. W. O’Hara, Rapid reconnaissance of geothermal prospects using shallow temperature surveys. USDOE Rep. for Contract EG-77-C-01-4021 (1979). Lewis, T. J., Heat production measurement in rocks using a gamma-ray spectrometer with a solid state detector. Can. J. Earth Sci. 10, 1494-1507 (1974). Lovborg, L., P. Nyegaard, E. M. Christiansen, and B. L. Nielsen, Borehole logging for uranium by gamma-ray spectrometry. Geophysics 45, 1077-1090 (1980). Lovering, T. S . , and H. D. Goode, Measuring geothermal gradients in drill holes less than 60 feet deep, East Tintic district, Utah. Geol. Surv. Bull. (U.S.) No. 1172 (1963). Lovering, T. S., and H. T. Morris, Underground temperatures and heat flow in the East Tintic district, Utah. Geol. Surv. Prof, Pap. (U.S.) No. 504-F (1965). Luth, W. C.,and H. C. Hardee, Comparative assessment of five potential sites for hydrothermal-magma systems: Summary. USDOE Rep. TIC-11303(1980).

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Major, B. A., and C. L. Witten, Upgrading Amarada-type survey clocks for high-temperature geothermal service. USDOE Rep. SAND80-0046 (1980). Majorowicz, J. A., and A. M. Jessop, Regional heat flow patterns in the western Canadian sedimentary basin. Tectonophysics 74, 209-238 (1981). Mansure, A. J., and M. Reiter, A vertical groundwater movement correction for heat flow. J. Geophys. Rex 84, 3490-3496 (1979). Merkel, R. H., L. M. Maccary, and R. S. Chico, Computer techniques applied to formation evaluation. SPWLA-Log Analyst pp. 3-10 (1976). Morgan, P., Porosity determinations and the thermal conductivity of rock fragments with application to heat flow on Cyprus. Earth Planet. Sci. Lett. 26, 253-262 (1975). Morgan, P., D. D. Blackwell, R. E. Spafford, and R. B. Smith, Heat flow measurements in Yellowstone Lake and the thermal structure of the Yellowstone caldera. J. Geophys. Res. 82, 3719-3732 (1977). Moses, T. H., Jr., and J. H. Sass, Drilling techniques presently in use by the geothermal studies project, US. Geological Survey. Geol. Surv. Open-File Rep. (U.S.) No. 79-763 (1979). Muecke, G. K.,J. M. Ade-Hall, F. Aumento, A. MacDonald, P. H. Reynolds, R. D. Hyndman, J. Quintino, V. Opdyke, and W. Lowrie, Deep drilling in an active geothermal area in the Azores. Nature (London) 252, 281-285 (1974). Nielsen, S. B., and N. Balling, Accuracy and resolution in continuous temperature logging. Tectonophysics 103, 1-10 (1984). Olmsted, F. H., Use of temperature surveys at a depth of 1 meter in geothermal exploration in Nevada. Geol. Surv. Prof. Pap. (U.S.) No. 1044-B (1977). Parry, W. T., J . M. Ballantyne, and N. L. Bryant, Hydrothermal alteration enthalpy and heat flow in the Roosevelt Hot Springs thermal area, Utah. J. Geophys. Res. 85, 2559-2566 (1980).

Poley, J. P., and J. Van Stevenick, Delineation of shallow salt domes and surface faults by temperature measurements at a depth of approximately 2 meters. Geophys. Prospect. 6 , 666-700 (1970).

Poppendiek, H. F., D. J. Connelly, and A. J. Sellers, Development of downhole geothermal heat flux and thermal conductivity transducers. Geotherm. Resour. Counc. Trans. 6 , 161-164 (1982).

Ratcliffe, E. H., Thermal conductivities of fused and crystalline quartz. Br. J. Appl. Phys. 10, 22-25 (1959).

Reiter, M., A. J. Mansure, and B. K. Peterson, Precision continuous temperature logging and comparison with other types of logs. Geophysics 45, 1857-1868 (1980). Robertson, E. C., R. Raspet, S. H. Swartz, and M. E. Lillard, Properties of thermistors used in geothermal investigations. Geol. Surv. Bull. (U.S.) No. 1203-B (1966). Ross, E. W., N. Vagelatos, J. M. Dickerson, and V. Nguyen, Nuclear logging and geothermal log interpretation : Formation temperature sonde evaluation. USDOE Rep. LA-9159-MS (1982).

Roy, R. F., E. R. Decker, D. D. Blackwell, and F. Birch, Heat flow in the United States. J. Geophys. Res. 73, 5207-5221 (1968). Roy, R. F., A. E. Beck, and Y. S. Touloukian, Thermophysical properties of rocks. In “Physical Properties of Rocks and Minerals” (Y.S. Touloukian and C. Y. Ho, eds.), McGraw-Hill/CINDAS Data Series, Vol. 11-2, pp. 405-502. McGraw-Hill, New York, 1981.

Sammel, E. A., Convective flow and its effect on temperature logging in small-diameter wells. Geophysics 33, 1004-1012 (1968). Sass, J. H., R. J. Munroe, and A. H. Lachenbruch, Measurement of geothermal flux through poorly consolidated sediments. Earth Planet. Sci. Lett. 4, 293-298 (1968).

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Sass, J. H., A. H . Lachenbruch, and R. J. Munroe, Thermal conductivity of rocks from measurements on fragments and its application to heat flow determinations. J. Geophys. Res. 76, 3391-3401 (1971). Sass, J . H., J. P. Kennelly, Jr., W. E. Wendt, T. H. Moses, Jr., and J . P. Ziagos, In-situ determination of heat flow in unconsolidated sediments. Geophysics 46, 76-83 (1981). Sass, J. H., C. Stone, and R. J. Munroe, Thermal conductivity determinations on solid rock-a comparison between a steady-state divided-bar apparatus and a commercial transient linesource device. J . Volcanol. Geotherm. Res. 20, 145-153 (1984a). Sass, J. H., J. P. Kennelly, E. D. Smith, and W. E. Wendt, Laboratory line-source methods for the measurement of thermal conductivity of rocks near room temperature. Geophys. No. 84-91 (1984b). Surv. Open-File Rep. (U.S.) Shearer, C., and M. Reiter, Terrestrial heat flow in Arizona. J. Geophys. Res. 87, 6249-6260 (198 1).

Sibbitt, W. L., J. G. Dodson, and J. W. Tester, Thermal conductivity of crystalline rocks associated with energy extraction from Hot Dry Rock geothermal systems. J. Geophys. Res. 84, 1117-1124 (1979). Simmons, M. G., Continuous temperature logging equipment. J. Geophys. Res. 70, 1349-1352 (1965).

Smith, H. D., Jr., C. A. Robbins, D. M . Arnold, L. L. Gadekan, and J. G. Deaton, A multifunction compensated spectral natural gamma ray logging system. SOC. Petrol. Eng., Annu. Meet. Pap. No. 12050 (1983). Somerton, W. H., Thermal properties of partially liquid saturated rocks at elevated temperatures and pressures. Am. Petrol. Inst. Res. Proj. Rep. No. 155 (1975). Sorey, M. C., Measurement of vertical groundwater velocity from temperature profiles in wells. Water Resour. Res. 76, 3391-3401 (1971). Steele, J. L., D. D. Blackwell, and D. W. Steeples, Heat flow determinations in Kansas and their implications for mid-continent heat flow patterns. EOS, Trans. Am. Geophys. Union 62, 392 (1981).

Urban, T. C., W. H. Diment, and M. Nathenson, East Mesa geothermal anomaly, California: Significance of temperatures in a deep drill hole near equilibrium. Trans. Geotherm. Resour. Counc. 2. 667-670 (1978). Vacquier, V., The measurement of thermal conductivity of solids with a transient linear heat source on the plane surface of a poorly conducting body. Earth Planet. Sci. Lett. 74, 275-279 (1985).

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15. MEASUREMENT OF OCEANIC HEAT FLOW

R. P. Von Herzen Woods Hole Oceanographic Institution Woods Hole, Massachusetts 02543

1. Introduction Oceanic heat flow measurements have had a significant role in the development of the concept of seafloor spreading and plate tectonics and particularly in quantitative thermal models of the evolution of oceanic plates (e.g., Parsons and Sclater, 1977). They have also provided the crucial evidence for the widespread phenomenon of hydrothermal circulation in ocean crust (Lister, 1972). Either of these processes causes surface heat flow values to vary by two orders of magnitude or more between the youngest and oldest parts of the spreading plates and between upwelling and downwelling parts of hydrothermal circulation cells. However, it is necessary to measure accurate heat flow values (&I5070 or better) for thermal modeling of important tectonic processes such as (1) the evolution of oceanic plates from the heat flow versus age relationship, (2) the reheating of the plates by hot spots (Von Herzen et a/., 1982; Detrick et al., 1986), and (3) the relative importance of conductive vs. advective heat transfer in sediments (Anderson et al., 1979). Depending on the problem, appropriate measurement instrumentation and sometimes careful corrections must be used for the highest accuracy. Most oceanic heat flow measurements, like their land counterparts, have been made with simplifying assumptions. First, with a few exceptions, it is generally assumed that heat is transferred vertically (in one dimension) by lattice conduction from the earth’s interior, at least over the regime in which measurements are made. Second, steady-state boundary conditions are assumed, unless the measurements or other data demonstrate otherwise. Under these conditions, heat flow values are determined by the product of two measured quantities : the vertical temperature gradient and the thermal conductivity of material over which the gradient is measured. The ubiquity of hydrothermal circulation in much of the permeable rock of the seafloor would seem to suggest that the general assumption of pure 221 METHODS OF EXPERIMENTAL PHYSICS Vol. 24. Part E


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lattice conduction may be inappropriate. However, since oceanic measurements usually require at least a few meters of surficial sediments penetrable by a temperature-gradient probe, but relatively impermeable to fluid flow, lattice conduction is generally the most important heat transfer mechanism over the measurement interval. The scales of hydrothermal circulation and the depth of most oceanic basement rock topography below the sediments are usually much greater than the measurement interval ;these factors cause the horizontal component of heat flux to be small or negligible compared to the vertical. The steady-state boundary conditions are verified by the general constancy of near-bottom ocean temperature and the uniformity of heat flow with depth below the seafloor. Indeed, the very close approximation to steady boundary conditions in the deep sea considerably simplifies the measurement of heat flow in that environment in comparison to most continental measurements. Improvements of instrumentation for oceanic heat flow measurements, as well as for data acquisition, have depended on technological advances as much as a desire for increased scientificunderstanding of the nature of ocean basins. The idea of obtaining oceanic measurements was probably first conceived by Sir Edward Bullard (1954), the “father” of both oceanic and continental heat flow measurements. Accurate oceanic measurements were not realized until 1950 (Revelle and Maxwell, 1952), primarily because of technological limitations. Indeed, the early measurements comprise one of the first successful applications of electronics to deep-sea scientific research. Although much of the instrumentation and techniques used now seem obvious. their development was not always straightforward and was sometimes tedious (Shor, 1984). The gradual accumulation of heat flow data in the 1950s was accompanied by a general increase in marine geophysical knowledge. A rapid expansion of marine geophysical activity occurred in the early 1960s, culminating in the unifying hypothesis of plate tectonics. The methods of early measurements were constrained to a large extent by the nature of the environment and available measurement platforms. Instrumentation was required to operate remotely and in the cold, highpressure environment of the deep sea. These factors stimulated the development of high-strength containers with watertight sealing. Because the initial instrumentation was developed before transistors generally became available, low power consumption and/or high energy density batteries were as important as instrumentation stability. Reduction of physical size has always had a high priority, both to reduce the requirements of high-pressure containment of physical instrumentation and to minimize the thermal response time of sensors. The fact that most of the seafloor is covered by soft mud with a constant boundary temperature led directly to the simple design of a thermal gradient probe which penetrates the bottom under its own

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weight. Thus many measurements could be made relatively rapidly in the deep sea, even though remote from the surface vessel. The constraints of the platform from which measurements are made, the research vessel, dictated relatively simple and reliable instrumentation. Such instrumentation was required to function accurately over many months at sea, with rough handling and often on small vessels without air conditioning. Long deep-sea cables and large but accurately controllable winches had to be developed to handle reliably the heavy, yet delicate, equipment to great depths (110 km) without tangling or breaking. The development of deep-sea drilling in the 1970s led to miniature temperature-measuring instrumentation which could be lowered inside the drill pipe to penetrate to greater depths beneath the seafloor. Modern geothermal instrumentation employs digital recording to enhance accuracy and broaden dynamic range and to ease the task of data reduction for many measurements. All of these technological developments were stimulated by the great increase in marine geophysical exploration following World War 11. Instrumentation has also benefited from the by-products of technology developed as a result of space exploration, especially the requirements for miniaturization and low power. A summary of the techniques used in the 1960s and the worldwide distribution of marine geothermal data was presented by Langseth and Von Herzen (1971). Heat flow measurements over the earth, both continental and marine, now number about lo4. As of about a decade ago, the data were compiled and summarized by Jessop et al. (1976), and recent marine geothermal investigations were discussed by Von Herzen (1984).

2. Temperature Gradients 2.1. Sensors

To measure temperatures in both the ocean and seafloor, it is desirable to use a sensor which responds accurately and rapidly to temperature, but is insensitive to other environmental factors such as pressure and electrical conductivity. During the early development of oceanic heat flow methods in the 195Os, the choice of sensors was not straightforward and led initially to several different types. An obvious possibility was the thermocouple, a small rugged device which had been applied in industry for many years. Thermocouples mounted in a probe were used by Bullard (1954) in the first Atlantic Ocean heat flow measurements. Thermocouples have typical temperature coefficients of about 100pV/"C, which serve quite adequately for the usual industrial applications requiring broad range and low precision (a few degrees kelvin). However, for the oceanic heat flow requirements of narrow

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temperature range and high precision (0.001-0.01 K), the use of thermocouples presented formidable measurement problems : instrumentation was required to measure automatically and remotely differences of a few microvolts or less. Furthermore, special care was needed with all electrical connections made to the thermocouples before amplification to ensure that only the temperature at the sensor, and not that at some other connections, was being measured. In fact, the “zero offset” of the thermocouples used in the instrumentation of Bullard (1954) was determined by periodically reversing the connections between the sensors and the measuring circuitry. Another method for increasing the sensitivity of a small and inexpensive sensor is simply to employ several in a series (additive) configuration. This “brute force” method has not generally been used in oceanic heat flow investigations because of the requirements for additional space and complexity of assembly ;however, such techniques have been applied in industry to measure the relatively large heat flow through walls and other man-made structures. An appropriate choice of sensor for marine heat flow investigations has been the thermistor. This device, made of mixtures of sintered metallic oxides, is now widely available as a result of industrial applications. It is a sensor with a large temperature coefficient of electrical resistance, typically about 5%/K, so it can readily be used in resistance bridge circuits. In the earliest applications for oceanic heat flow measurements (e.g., Von Herzen et al., 1962), available thermistors were relatively large and had questionable stability, the former condition requiring innovative probe design and the latter frequent recalibration. Now thermistors are available in a wide variety of sizes and shapes and have adequate stability for most applications, especially when covered with a glass envelope, a common manufacturing technique. Although resistance stability over time and with environmental cycling is not a property that is easily documented, this writer has found that glass-coated thermistor beads with a resistance of 5 to 10 kilohms at 0°C typically do not drift significantly (a few ohms) over periods of several years or more, even when subjected to temperature cycling (0-100°C). However, relatively large drifts have been found in very small beads with similar resistances used in thermal conductivity needle probes (see below). Apparently, the magnitude of drift is a function of manufacturing techniques, physical size, and configuration, as well as the materials used. Thermistor resistance can also be affected by hydrostatic pressure, so probes for deep-sea use are usually designed with the thermistor contained in a pressure-protected enclosure (e.g., cylindrical probe). This is not usually a severe additional requirement, since isolation of the electrical connections from seawater and protection from mechanical abuse are necessary in any case.

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Almost all equipment designed for oceanic heat flow measurements now employs thermistors as sensor. Their small size, low cost, wide availability, high sensitivity, and good stability have made them the optimum choice. They are usually available commercially with resistances which have a tolerance of about 10-20%, a range which is too broad for most applications (see below). Some manufacturers now provide thermistor units with matched resistance-temperature characteristics, although not always in the desired size or physical configuration. Thermistors can usually be selected by the manufacturer, with a correspondingly increase in costs for each unit, or by the investigator in the laboratory with relatively simple equipment (e.g., an ice bath). 2.2. Gradient Probe Design

2.2.1. Bullard Probe. The first gradient probes designed for geothermal measurements (Bullard, 1954) were in the shape of long, thin cylinders and are now commonly referred to as Bullard probes. Initially they were about 1 in. (2.5 cm) or more in diameter, with lengths ranging between 2 and 5 m; later designs had reduced diameters to minimize the thermal equilibration time constant (Von Herzen and Uyeda, 1963; Lister, 1970), which is operationally important for marine investigations. The usual construction is a thick-walled cylinder, into which two or more thermal sensors are placed at intervals along the length and in good thermal contact with the wall, and the ends sealed to seawater and hydrostatic pressure. The probe is rigidly attached at its top end to a weight stand containing a recorder and pressure case (Fig. 1). The entire assembly provides the mass to drive the probe into the bottom sediments. The probe assembly is lowered to within a few tens of meters above the ocean floor, where typically the water column is isothermal and the system calibration can be checked (see below). The probe is then allowed to fall freely or is lowered rapidly into the soft sediments on the seafloor. Immediately after the Bullard probe has penetrated the bottom, the temperature distribution along its length is different from that of the surrounding sediments because of its previous thermal equilibration with the water above and the generation of frictional heat during penetration. If the probe is very long compared to its diameter, which is usually the case, it dissipates heat almost entirely in the radial direction into the sediments. For a probe that is a good thermal conductor compared to the sediments, as is almost always the case with a steel probe, the theory of approach to equilibrium has been well established (Bullard, 1954). To obtain accurate estimates of sediment temperature gradients it is usually necessary to leave the probe undisturbed for at least several thermal time constants, 7 , of the

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k-1

METER-!

FIG. 1. Cylindrical (Bullard) gradient probe and attached recorder pressure case used at Scripps Institutionof Oceanographyin the 1960s. F and fi show locations of thermistor sensors in probe. [Reprinted with permission from Von Herzen, R. P., and ffyeda, S.(1963). Heat flow through the eastern Pacific Ocean floor. J. Geophys. Res. 68, 4219-4250. Copyright by the American Geophysical Union.]

probe/sediment system ( 7 = a 2 / a , where a is the probe radius and CY the thermal diffusivity of sediments) ; this time is 10-40 min, depending primarily on the probe diameter. Leaving the probe entirely undisturbed for such times with the other end of the hoisting wire attached to a drifting ship is not always convenient or possible, which has motivated modifications to the Bullard probe designs. A special modification of the Bullard probe developed for measurements in shallow waters is described by Matsubara el a!. (1982). The probe has a single thermistor sensor which can be moved along its length to sample the temperature distribution in great detail, measuring precisely the effects of bottom-water temperature fluctuations in the sediments. The effects of these temporal variations can be separated from the steady-state gradient by analysis. The measurements and control of the instrumentation are made through a conducting cable to the probe from an anchored platform (ship). One complete measurement to high accuracy may require several hours after penetration of the probe. It is desirable to use a probe length of about 10 m to remove accurately the annual temperature fluctuations on continental margins or other shallow marine regions. 2.2.2. Outrigged Probe. The outrigged probe (Fig. 2) is a useful and successful modification of the cylindrical Bullard probe. Individual small probes (I 1/8 in. diameter) are attached to a primary strength member and separated from it in the radial direction to minimize thermal disturbance. A number of such probes can be attached to the primary strength member, such

15.

233


CORE PIPE

-k

10 N ICHES d-

--I

k I / 4 INCH

S . S TUBING KNURLED BRASS

SLEEVE

S S . PLUG PLASTICJ CEMENT

LTHERMISTOR BEAD

FIG. 2. Small-diameter outrigged probe construction and mounting developed at Lamont Geological Observatory in the 1960s. [Reprinted with permission from Gerard, R., Langseth, M. G . , and Ewing, M. (1962). Thermalgradientmeasurementsinthewater and bottom sediment of the western Atlantic. J. Geophys. Res. 67,785-803. Copyright by the American Geophysical Union.]

as a core barrel, with electrical connections leading to a recorder contained in the driving weight above. The small thermal time constant of these probes (T 5-10sec) means that thermal equilibrium with the sediments can be nearly achieved in a time of 5 min or so in the bottom, providing a substantial advantage in operations from a drifting surface ship. Also, for times which exceed 107, the approach to equilibrium temperatures is closely proportional to the inverse of time (Bullard, 1954; Hyndman et al., 1979), which considerably simplifies the extrapolation to obtain in situ temperatures. 2.2.3. “Violin-bow’’ Probe. This most recent probe design (Hyndman et al., 1979)combines some features of both the Bullard probe and the outrigged probe described above. Its essential elements are a substantial strength member (cylinder) to which a relatively slim Bullard probe (1/4-5/16 in. diameter) containing an array of thermistors is attached. The probe attachment at both ends to the strength member gives a rigid construction to a slim probe with a relatively small thermal time constant (Fig. 3). Watertight integrity is ensured by a single rigid connection to the recording case. The small probe diameter also allows in situ thermal conductivity to be measured when used in the “pogo” mode (see below). One drawback is the requirement to contain many thermistors and a heater wire in a long tube of relatively small diameter, which makes construction somewhat tedious.

-

234

R. P. VON HERZEN

FIG.3. Photograph of violin-bow heat flow probe. (From Hyndman er al., 1979.)

15.


235

2.3. Recording Instrumentation

The design of recording instrumentation for the measurement of temperature gradients in the seafloor is strongly influenced by the environment. The uniformity of water temperatures near the seafloor at most deep-sea sites provides a convenient reference calibration of zero gradient for each measurement. Hence any instabilities in electronic circuitry or cable/plug connections can usually be minimized by comparing relative temperatures measured in the bottom with those measured while the instrumentation is suspended in near-bottom waters, both before and after each penetration. The high ambient hydrostatic pressure in the deep sea requires either specialized instrument design to function within that environment or protection of the instrumentation within a pressure case ;the latter design has been followed by most investigators. The refinement and standardization of manufacture of O-ring seals and bulkhead electrical feed-through connections have made this approach relatively straightforward and robust. Finally, the requirement for remote recording means that instrumentation is usually battery-powered ; this limitation on available energy affects the design of in situ thermal conductivity measuring equipment (see below) and determines the duration of multipenetration (pogo) operations. To measure typical temperature gradients of 0.05 to O.l"C/m in the seafloor with several sensors spaced at 1 m or less requires a precision in temperature measurement of at least several thousandths of a degree. Higher precision, at least 0.001 "C, is needed to accurately extrapolate the transient frictional heating pulse at each sensor due to bottom penetration. A common recording technique is to switch thermistor sensor inputs sequentially in a common resistive bridge circuit (Fig. 4). A variation on this method described by Hyndman et al. (1979) employs a separate bridge for each thermistor, switching each bridge output in sequence to the amplification circuitry. The latter allows the use of relatively high impedance solid-state switches, since the amplifier input can have very high impedance (megohms), although at the expense of additional bridge components. The sensor bridge network can be excited by either alternating or direct current. Alternating current methods provide for somewhat simpler amplification and null detection circuitry, although considerable care must be taken to minimize any capacitive and inductive effects associated with long sensor lead wires. In both cases, the power dissipated at the sensor must be sufficiently low to avoid any significant effects of self-heating. Thermistor resistances are typically several kilohms or more, with bridge outputs of about 10 mV/"C. Therefore voltage amplification of several hundred is required to achieve signal levels which can be digitized or recorded in analog form.

236

R. P. VON HERZEN

R

8

1.08

\

j.

9Q Q

1.04

1.0'

0

1

2

3

4

5

6

7

TEMPERATURE OC FIG. 4. Typical Wheatstone bridge used in temperature gradient measurement circuitry, showing enhanced linearity of bridge output with temperature compared to nominal thermistor resistance linearity. The thermistor nonlinearity with temperature compensates to a large extent for the bridge nonlinearity with resistance. Re is adjustable resistance to match the particular thermistor resistance&, several of which with closely matched resistance-temperature relationships are sequentially switched in the bridge circuit.

Although the analog recording schemes of earlier instrumentation (Von Herzen et al., 1962; Von Herzen and Uyeda, 1963 ; Langseth, 1965) usually provided adequate resolution, most modern instrumentation systems employ digital techniques. The primary reasons for this trend are the broad operating range of ambient temperatures and the usefulness of automatic data reduction when measurements are numerous (multiprobing). The instrumentation described by Hyndman et al. (1979) provides 12-bit digital resolution recorded on a digital printer within the instrument. A digital resolution of 16 bits (Von Herzen et al. , 1982) provides adequate resolution and range for all ambient ocean temperatures ( - 2-35"C), and the total data from long stations (- 1 to 4 megabits) can be contained on standard magnetic tape cassettes.

15.

237


It is convenient and cost-effective to provide a real-time acoustic link between the instrumentation and the research vessel, so that the status of the instrumentation can be monitored continuously. The digital data can be transmitted as a two-frequency serial code, as in the Hyndman et al. (1979) instrumentation, although the data link is sensitive to weather and sea conditions at the surface. The pulse-positioned telemetered ping (12 kHz) system used by Von Herzen et al. (1982) is recorded directly on a precision sweep recorder aboard the ship in analog form (Fig. 5 ) and is relatively '

(

1

1

'

LlGURlAN

I

HF 16.8

FIG.5 . Shipboard recording of the acoustically telemetered signal from a typical heat flow measurement. The recording is made with a stylus swept from approximately the bottom to the top of the record each 0.5 sec, with the paper (time) advancing from right to left. A mark is made each time an acoustic ping is received. Each variable is recorded over 2 sec in a preset, repeatable sequence as a reference (REF) pulse and a record pulse representing its relative resistance or temperature. The figure illustrates, in sequence, several minutes before penetration (PEN),penetration with thermal decay of frictional heating, in situ thermal conductivity (K) heating, and several minutes after pullout. Recorded traces include five thermistors (TW, T1,..., T4), two calibration resistors (LCAL, HCAL), tilt, in sifu K off/on indication, and pressure (P).Note the less prominent reflection (duplication) of traces when the instrumentation is above the bottom, providing monitoring of instrument height above bottom.

23 8

R. P. VON HERZEN

unaffected by normal ambient sea conditions. However, the range of data displayed aboard ship is limited and must be preselected, which is not usually an inconvenience for monitoring purposes. A recent trend in instrumentation is to incorporate microprocessor control for automatic scaling as well as other recording flexibility (Hutchison, 1983 ; Hsu et al., 1983 ; Wright and Fang, 1984). The use of microprocessor technology simplifies the electronic design and hardware but increasesthe task of software design, depending on the formatting and amount of processing desired in the acquisition instrumentation. For some very detailed investigations in topographically variable regions (i.e., near ridge crests) it is useful to record relative hydrostatic pressure to high resolution (- 1 decibar) and thus to obtain accurate relative depths during bottom penetrations. This allows locations of measurement to be more accurately determined if good topographic maps are available, especially if the instrumentation is not directly beneath the vessel, as is usually the situation during multiprobing stations (Green et al., 1981). Pressure recording also provides accurate near-bottom vertical profiles of water temperature during the lowering and hoisting of the instrumentation between penetrations (Galson and Von Herzen, 1981). 2.4. Data Acquisition Techniques

The data acquisition methods for determining seafloor temperature gradients have evolved with changes in instrumentation. Most essential for all measurements is the research vessel, which provides the platform for deploying the instrumentation as well as the mobility to occupy stations over the entire seafloor. Also important are the deep-sea wire cable and winch required to lower and raise the equipment to and from the seafloor. Ocean depths range from 0 to 10 km, with the usual range in the deep sea between 2.5 and 6 km. A few measurements have been obtained with instrumentation which falls freely to the bottom and returns after dropping ballast, but the occasional loss of instrumentation and small bottom penetration have discouraged extensive use of this technique. The early Bullard probes required long times ( 1/2 hr) in the bottom without disturbance to allow dissipation of the frictional heat of penetration. As it is difficult to maintain the surface ship directly over the instrument in the bottom, the probe was frequently bent on pullout. Therefore after a single penetration the probe was recovered aboard the vessel to be straightened or replaced before the next measurement. Retrieval of the instrumentation is also required for gradients measured by outrigged probes on piston coring operations, although the core barrel is frequently able to achieve greater penetration (up to 20 m) than a Bullard probe. To increase the rate of acquiring measurements, particularly for stations

-

15.

239


in closely spaced clusters, the pogo probe technique was developed. It is now used commonly with the violin-bow and outrigged probe sensors. In both cases a large strength member, usually a long cylinder (up to 6 m), with a heavy weight on top (up to 25001b), is used to penetrate the bottom. The equipment must remain undisturbed in the bottom for only a relatively short time (5-15min, depending on whether in situ thermal conductivity is measured) because of the small thermal time constant of the sensors. Thus the probe is not usually bent on pulling out, and additional penetrations can be made as the vessel moves slowly (- 1 knot) over the bottom. In this way as many as 20 or 30 penetrations can be made on one lowering over a small region (Fig. 6), with battery capacity being the primary factor controlling the number of penetrations. One operational problem with pogo-probing, especially in deep water, is that the cable continues to move laterally through the water for a considerable time ( - 1/2 hr or more) even after the vessel is stopped to prepare for a penetration. This can result in premature disturbance to instrumentation unless slack in the cable is maintained by releasing more wire at the vessel. 33'46N

'

I

PC-j3

'

8

I

I

5.

3356

I

-

I

6730

6720

6710

HEAT FLOW 8tTE 7

4 2'

a 22

10

Scole

23

A

24

5

0 I

33-25 61

W

20km

I

10 mmi.

I

I

FIG. 6. Example of small survey with locations of pogo-probe measurements in the NW Atlantic. [Reprinted with permission from Detrick, R. S., Von Herzen, R. P., Parsons, B., Sandwell, D., and Dougherty, M. (1986). Heat flow observations on the Bermuda Rise and thermal models of mid-plate swells. J. Geophys. Res. 91,3701-3723.Copyright by the American Geophysical Union.] Various symbols are locations of measurements on different pogo-probe stations (lowerings). Dotted lines are survey tracks, with contours of acoustic basement traveltime delay after bottom reflection. Water depth about 5.1 km.

240

R. P . VON HERZEN

However, this release may result in a considerable amount of cable being laid on the seafloor, with the possibility of entanglement and with large lateral forces usually applied to the instrumentation during pullout from the bottom. One technique for minimizing these effects involves raising the instrumentation 1 km or more above the bottom during the drift or tow between penetrations, then lowering it again rapidly after the vessel has stopped for the next penetration. This practically ensures a laterally stationary, nearly vertical cable above the instrumentation during the time of penetration, for only a small amount of additional time (- 10 min) required to lower the instrumentation down to the seafloor. Two technical improvements have contributed to the success of the pogoprobing technique. First, the acoustic telemeter allows real-time monitoring of the status and performance of remote instrumentation, as well as determining the height of the instrumentation above the seafloor (Fig. 5 ) . Especially in deep water and/or rough weather, when the weight of the hoisting cable far exceeds the instrumentation weight and cable tension at the surface undergoes large excursions as a result of vessel motion, the monitoring of penetrations by acoustic methods is essential. The required data rates are low, equivalent to 10-20 digital bits/sec, and signal-to-noise ratios in the 5-20-kHz band are usually good. Electrically conducting cables may be useful for high data rates and/or acoustically noisy environments, although such cables constructed to withstand the rigorous oceanographic environment are relatively expensive (- $100,000). Second, the development of relatively torque-free wire allows modest amounts of additional cable to be released during penetrations to decouple the instrumentation from vessel motion, without undue danger of cable kinking. Another solution to the latter problem is to place swivels between the cable and the instrumentation which allow release of some of the cable twist resulting from loading. Another recent use of marine geothermal instrumentation with highprecision temperature recording is in the search for hydrothermal vents associated with oceanic rifts (Corliss et al., 1979; RISE, 1980). The same sensors used for geothermal gradient measurements can be deployed in a vertical array both above and below the recording instrumentation (Fig. 7). The amplitude and restricted lateral extent of temperature anomalies associated with hydrothermal venting usually distinguish them from the temperature structure which results from near-bottom ocean dynamics (Fig. 8). To establish precise locations of any vents which are detected, it is usually necessary to employ a bottom-mounted acoustic transponder network (Spiess and Mudie, 1970). With bottom transponders carefully located relative to each other by a surface vessel, it is usually possible to achieve navigational precision of a few meters within a region several kilometers in lateral extent which includes the transponders. In addition to the original


24 1

F

Hydrographic Cable

+Lower

Thermistor

+Weight

FIG. 7. Schematic of configuration used for lateral profiling of near-bottom water temperatures. (From Williams et al., 1974.)

discovery and mapping of the warm-water vents near the Galapagos Islands with this technique (Williams et al., 1974), hydrothermal venting is being studied on other areas of the midocean rift system with similar methods, and such phenomena continue to be of intense scientific interest. 2.5. Deep-sea Drilling

Techniques for measuring vertical temperature gradients during deep-sea drilling operations have somewhat different requirements than the shallowpenetration equipment used from oceanographic vessels. The sensor(s) and recording instrumentation must be physically compact so that they can pass through the drill pipe. They must also be able to withstand larger shocks and accelerations associated with rapid passage through the drill string. The

R. P. VON HERZEN I

I

I

I

I

I

U +0.05 'C

M

L -0D5.C

I

I

00

1.0

I

I

I

21) 3.0 40 DISTANCE ALLWG TRACK fKM1

I

50

FIG.8. Lateral temperature profiles across GalPpagos spreadingcenter with instrumentation shown in Fig. 7. Upper (U), middle (M), and lower (L) thermistor traces are arbitrarily separated by 0.05"C. Average lateral velocity about 1.3 kmlhr. Hydrothermalventing is shown by sharp spikes near 3 and 5 km distances. Broad anomaly over hill at 1 km is caused by interaction of ocean currents with topography. (From Williams et al., 1974.)

sensor probe must be of sufficient strength to penetrate the relatively more indurated sediments at greater depths, up to 300 m, below the seafloor, yet maintain as small a thermal time constant as possible. Resolution of temperatures to ~ 0 . 0 1 " Cis generally sufficient, since the vertical interval of measurements at a drill site is usually tens or hundreds of meters, rather than the interval of 1 m or less for most oceanographic probes. The instrumentation developed for the earliest phase of ocean drilling was described by Von Herzen and Maxwell (1964). Physical size was minimized by use of a standard seven-conductor logging cable to provide both electrical power to the downhole package and real-time signals back to the surface vessel (temperature-modulated frequencies). The desire to coordinate downhole temperature measurements with coring operations during the main initial phase of drilling led to development of remote recording (Fig. 9), using custom-made magnetic drum or miniature cassette recorders. Results from the initial several years of use of such instrumentation are summarized by Erickson et al. (1975). Difficulties were encountered with the relatively large

15.

.. :.


:;I *

LOCKED LATCH (Optional)

*.

DOWN- HOLE TEMPERATURE . RECORDING INSTRUMENT

..

.:.‘.I

243

I

:. .. . ....

CORE CATCHERS

accelerations and shocks experienced by the instrumentation during drilling operations, which caused physical damage and loss of data. A significant improvement was made with the development of all solid-state instrumentation (Yokota et al., 1980),whereby the remote recording no longer depended on a precision-tuned mechanical apparatus. All the successful measurements with the instrumentation described above were obtained by separate lowerings of the instrumentation only for the purpose of temperature measurements. Thus additional drilling vessel time ( l i to 3 hr, depending on depth) was required for each downhole measurement attempt. A relatively recent development (Koehler and Von Herzen,

244

R. P. VON HERZEN

FIG.10. Miniature downhole temperature recorder and battery pack, which are inserted into special core cutter shoe (background) for temperature measurements in sediments during hydraulic piston coring operations of the Deep Sea Drilling Project. (From Koehler and Von Herzen, 1986.)

15. MEASUREMENT OF

OCEANIC HEAT FLOW

245

1986)of subminiature instrumentation incorporated in the wall of the core barrel (Fig. 10)allows temperature measurements to be made simultaneously with hydraulic piston coring, at little or no cost of additional rig time. This much-reduced rig time allows more detailed temperature-depth data to be obtained at sites where coring is the primary objective, providing more accurate gradient determinations and the data needed to determine, for example, whether conductive heat flow is uniform with depth beneath the seafloor. Some initial results for Deep Sea Drilling Project (DSDP) Leg 86 are described by Horai and Von Herzen (1985). Temperatures measured in boreholes soon after they have been drilled can be used to estimate equilibrium geothermal gradients. If sufficient time is available after the drilling disturbance, at least a factor of 5 to 10times longer than the duration of drilling the hole, the temperature distribution of most deep holes closely reflects the equilibrium gradient. Indeed, most heat flow measurements on land are determined from gradients measured in such holes. For oceanic boreholes, it is rare that time is available to establish equilibrium temperature gradients before measurement, and another method originally derived by Bullard (1947)can be used. Bullard showed that, to a good approximation, the approach of a borehole to equilibrium is dependent on three parameters : To, the magnitude of the disturbance; tl(z),the total time of the disturbance (circulation time), a function of depth in the hole; and tz , the time since the end of circulation. The temperature measured at any depth in the hole, Tm, is related to the equilibrium temperature, Z, by

Tm

=

Te +

Toln(1

+ fJt2)

(1) Therefore equilibrium temperatures at any depth in the borehole can be estimated by measuring the dissipation of the temperature disturbance at known times after cessation of drilling (circulation). In the oil industry, a graph of Tmversus ln(1 + f l / t 2 ) is known as a Horner plot (Dowdle and Cobb, 1975). The technique has been used successfully to estimate deep gradients at site 504B in the Pacific Ocean (Fig. 11). A more elaborate method, based on heat exchange between both downgoing and return circulation with the borehole walls, was used for deep-sea holes by Burch and Langseth (1981).Accurate determination of equilibrium temperatures depends on careful monitoring of volume and temperature of the circulating fluids. Hyndman etal. (1984,1987)have reviewed geothermal measurements made over most of the Deep Sea Drilling Project. In general, there are no large or systematicdiscrepanciesbetween heat flow values deduced from measurements in these drill holes and those obtained with oceanographic probes nearby. A few measurements show nonlinearities in temperature gradients, which are best explained by vertical flow of seawater in the hole as a result of ambient pore water pressures in the crustal rocks being different from hydrostatic.

246

R. P. VON HERZEN

5 0 4 - B D O W N H O L E TEMPERATURE M E A S U R E M E N l S

HOLE

0,--3473 0'

-

.

n\

0

\

0

0

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0

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m SEAFLOOR-

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3 50

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L E G 70 G - 0 TEMPERATURE LOGS MEASURED 3 D E C ' 7 9 , 4 3 DAYS AFTER LEG 6 9 D R I L L I N G EXTRAPOLATION OF LOGS (BULLARO. 1947) 5 0 4 - C SEDIMENT TEMPERATURES

\

0

-

100

A'

\ \

100

4\

370 _ O-

3 7 4 8 m EASEMENT-

\

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-

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-

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Y

E X T R A P O L A T I O N OF 0.1+4'C/m ESTIMATED

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0 1 0 U

\

[7

100

E

EASEMENT

GRADIENT

[L

\

100

3901

0 0

2 U w

um

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m I

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00 420(

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T E M P E R A T U R E ('CC) FIG.11. Borehole temperatures determined during Legs 69 and 70 of the Deep Sea Drilling Project at Site 504. Note good agreement of sedimentlbasement interface temperature determined by extrapolation of sediment temperatures from above and by gradient determined from temperature logs below. Low temperatures in the upper 400 m were caused by flow of seawater down the drilled hole. (From Becker el al., 1983.)

15.

247


2.6. Estimates from Gas Hydrate Reflectors

A somewhat different method for estimating temperature gradients which does not involve temperature measurements in sediments is available for some marine tectonic provinces, especially portions of the continental margins. It has been recognized that hydrocarbon gases can combine with water to form a gas hydrate molecule at ambient conditions close to the seafloor and that these substances may have anomalous seismic velocities in comparison to the sediments in which they form (Stoll et al., 1971). Where they occur, the gas hydrates may therefore be detected by reflections of seismic waves caused by the acoustic impedance contrast, and indeed such reflectors have been inferred for many regions of continental margins (Shipley et al., 1979). With a knowledge of the temperature of the seafloor where such reflectors can be identified, the phase diagram of the particular gas (usually methane) hydrate can be used to estimate the temperature gradient to the depth of the reflecting horizon. Such reflectors may cross sediment bedding horizons but are usually subparallel to the seafloor, so they are frequently designated as “bottom-simulating reflectors (BSRs).” The depth to a BSR in the Nankai Trough near Japan was used by Yamano et al. (1982) to estimate variations of temperature gradients and heat flow in this region, Relatively good agreement between estimates by this method and measurements with oceanographic probe techniques were obtained by Cande et al. (1986) in the region of the Chile ridgekrench collision. The accuracy of the method depends on data for in situ sediment physical properties (velocity, density, thermal conductivity), parameters which are usually not sufficiently well controlled to result in heat flow values as accurate as those obtained by the usual oceanographic probe methods. If these parameters are relatively well determined, accuracies of 10% seem possible; otherwise, *20% may be more reasonable, although this uncertainty may be reduced significantly if one or more accurate heat flow measurements by other techniques are available to calibrate the region. This method has the advantage of permitting spatially continuous estimates of heat flux in a region as determined from the BSR, compared to discrete measurements at a specific location with all other methods.

*

2.7. Calculation of Gradients

The methods used to calculate gradients and their uncertainties depend on the nature of the data. If measured temperatures versus depth fit a linear relationship, and thermal conductivity versus depth is relatively uniform, the procedures described by Von Herzen et al. (1982) are appropriate. Relative temperature-depth data and their uncertainties are determined by extrapolation of the transient heating decay at each thermistor probe and from the

248

R. P. VON HERZEN

relative temperatures measured before and after penetration with the probes suspended in near-bottom water of uniform temperature. The extrapolation of the transient decay of cylindrical probes is treated with the conduction theory developed by Bullard (1954). Gradients and their uncertainties can then be determined with the usual least-squares fitting routines, modified by weighting of the temperature-depth data according to their uncertainties. When the data are sufficient to show that gradients change systematically with depth, because thermal conductivity also varies with depth or heat flow is not uniform (see below), somewhat different computational procedures must be used. If variations in gradient and thermal conductivity are inversely related, such that heat flow remains uniform with depth, each depth interval defined by temperature-depth data should be analyzed (e.g., Von Herzen and Anderson, 1972). If heat flow appears to vary systematically with depth as a result of transient environmental situations (water temperature fluctuations, rapid sedimentation or erosion, etc.), it is usually best to estimate a correction for such effects before calculating equilibrium gradients. The vertical heat flux can also vary with depth due to (1) refraction of heat by irregular spatial distributions of thermally dissimilar media, (2) the presence of heat sources (radioactivity, chemical reactions), and/or (3) the vertical advection of pore fluids (see Section 4.2).

3. Thermal Conductivity 3.1. Measurements on Core Samples

Until recently, the most common method for determining thermal conductivity ( K ) for the purpose of estimating heat flow was by direct measurements on cores of material recovered at or near the location of gradient measurements. Timely conductivity measurements were emphasized to avoid the possibility that changes of material (e.g., dehydration) might occur between core recovery and the measurements. Under such conditions, the transient needle probe technique (Von Herzen and Maxwell, 1959) became a convenient method for determining conductivity aboard the vessel. The equipment required has a simple physical and electrical design (Fig. 12). The needle probe technique employs an electrical current supplied to a wire of uniform linear resistance inside a length of hypodermic needle tubing, which is emplaced in the sediment. When a constant current I is initiated in the resistance wire, the needle probe simulates a transient line source of heat with small radial dimensions. A small thermistor bead, also inside the needle tubing, is used to monitor the rate of temperature increase. Under these conditions, the temperature T at the needle rises logarithmically with time

15.

249


NEEDLE PROBE

EXPANDED CROSS SECTION HEATER WIRE

I

+

1-

#I8 GAUGE HYPODERMIC

POLYURETHANE

. 4 crl

1 I

THERMISTOR I N NEEDLE COMPUlER

PLUG CH4RT RECORDER

HEhlER WIREI N NEEDLE

CONSTANT CURRENT SOURCE

FIG.12. Needle probe apparatus for measurement of thermal conductivity (after Von Herzen and Maxwell, 1959). Up to five needle probes can be deployed simultaneouslyfor measurements with recently developed equipment.

t as heat is dissipated into the surrounding uniform core material:

T = (Q/47rK)In t

+C

(2)

where Q is the power input (I’R) per unit length dissipated by the needle and C is a constant that depends on the radial thermal resistance of the needle probe, thermal contact resistance, and so forth. Thus, by determining temperature at the needle for at least two known times after initiation of heating, K is evaluated from Eq.(2). This simple functional relationship for the needle probe is valid after 10 to 20 time constants r = a 2 / a of the probe in the sediments, which is typically about 10 sec for the 18- or 20-gage needles usually employed. At very long times the relationship breaks down when a significant portion of the heat flux conducted radially away from the needle reaches the boundaries of the core sample. For the 29 to 3 in. diameter typical

250

R. P. VON HERZEN

of sediment cores, this time is usually about 5 to 10 min and is readily detected by deviation from a linear T versus In t relationship at longer times. The needle probe is usually inserted into a sediment core normal to its long axis by making a hole through the plastic core liner. The conductivity determined by a needle probe with this orientation is a combination of the vertical and horizontal components of K, whereas for heat flow measurements only the vertical component is desired. However, most surficial ocean sediments have no detectable anisotropy in K over the scale of the measurement (= 10 cm), as determined by comparative measurements made by the author with the needle probe aligned parallel to the core axis. Equipment for measuring thermal conductivity has been constructed to include as many as five individual needle probes recording simultaneously. With such instrumentation it becomes desirable to include automatic data recording and reduction. The needle probe measurements can be controlled by a programmable sequencer which samples a voltage analog of temperature of the needle probes periodically in a repeating sequence. It is convenient to record the digitized data on a magnetic tape cassette in the same format used with the gradient measuring instrumentation and/or to transfer the data directly to the memory of a minicomputer or via a cassette tape reader (Fig. 13). The data are reduced to thermal conductivity with appropriate software arranged to fit all usable data for each needle probe (usually 15-20 temperature measurements) in a least-squares sense to Eq. (2) and to eliminate any constant temperature change which might be caused by ambient temperature drift of the core. Thermal conductivity standards, consisting of either ground silica glass saturated with water (Goldberg et al., 1980) or molded plastic Lab ashore and/or shipboard

Data logger/ instrument

\

Digital

--Conductivity instrument FIG. 13. Schematic of digital data reduction equipment for heat flow measurements.

15.


25 1

cylinders with conductivity values in the range of ocean sediments, are frequently measured to ensure proper performance of the system. They indicate a repeatability of measurement usually within about f3'-70 of their mean K value, although the systematic error in measurement of conductivity by the needle probe method is probably less. A modification of the usual needle probe method using an initial pulse of heat and monitoring the temperature decay (Lister, 1979) is being developed by E. Davis (1983 personal communication). This modification uses less power than the usual steady heating method for comparable measurement sensitivity, a substantial benefit for battery-powered in situ measurements (see below). The divided-bar apparatus was developed to measure conductivity of indurated rocks penetrated by boreholes, but it has also been used for ocean sediments (Ratcliffe, 1960). It is based on the steady-state measurement of temperatures across a sample when a calibrated amount of heat is steadily applied. The method requires a rather tedious sample preparation, and measurement times are relatively long (- 112 hr). It has been largely superseded by the more rapid transient methods. A modification of the transient needle probe method is the quick thermal conductivity meter (QTM), a commerciallydeveloped instrument. It consists of a flat plate in which a planar heat source and temperature sensor are embedded. Measurements are made by placing the plate on a flat surface of the sample and energizing with electrical current, as with the needle probe. Measurements can be made in as little as 2 min. Comparisons with the needle probe method (Horai, 1981) show systematic differences up to 20%. One source of the discrepancy for sediments may be evaporation of water on the flat sediment surface, causing the conductivity value to be higher. The method appears most useful for indurated samples on which a relatively flat surface can be machined (lapped), but in which drilling a small hole for needle probe measurements would be difficult. Carvalho et al. (1980) and Becker et al. (1983) have used similar apparatus for measurements on hard core samples.

3.2.In Situ Measurements With the increasing availability of inexpensive solid-state electronics (microprocessors) for controlling remote instrumentation, in situ conductivity measurements have become feasible. Such measurements remove the additional uncertainties due to disturbances caused by coring or to the ambient environment (e.g. pressure, temperature) being different between seafloor and laboratory. The first in situ measurements, by Sclater et al. (1969), employed a scaled-up version of the needle probe method. The

252

R. P. VON HERZEN Thermistor

9

/

line heater in probe

FIG.14. Photograph of outriggedthermistor probe with supporting fin and cables to connect to recording instrument. Resistance wire inside tubing to provide heat for in sifu conductivity measurement extends over length of probe between support fins.

equipment was designed to measure the conductivity of surficial sediments extending only several decimeters beneath the seafloor. A scaled-up version developed for deeper in situ K measurements in lakes is described by Christoffel and Calhaem (1969). A recent development of the in situ method with outrigged probes (Jemsek et al., 1985) is also a larger version of the needle probe technique, using a line heater located within the same probe as the thermistors used for gradient determinations (Fig. 14). After an appropriate time interval following bottom penetrations (- 5-6 min), to allow frictional heating to dissipate and the gradient to be measured, steady heating (approximately 2 W/m) is applied to the line heaters for 10min. For the relatively small-diameter (1/8in.) outrigged probes, with a thermal time constant of 5-8sec in sediment, the logarithmic asymptote is closely approached about 2 min after initiation of heater power. Data reduction is the same as that used for laboratory needle probe measurements (see above). Heater initiation is controlled by a preprogrammed microprocessor in the gradient recording instrument, sensing bottom penetration with an accelerometer or a pressure gauge input. All heater circuits of the outrigged probes (up to a total of seven) are electrically connected in series to the same constant-current supply to ensure uniform and constant heating to each probe. Up to seven conductivity values are determined thereby with each gradient measurement, requiring an


253

undisturbed time in bottom of about 15 min for each combined gradient and in situ conductivity determination. A somewhat different technique is employed for in situ measurements with the violin-bow probe (Davis et al., 1984). A pulsed current.method, first described by Lister (1979), is applied to a resistance wire over the total length (several meters) of this sensor probe. Many thermistor sensors for conductivity measurements are spaced at intervals of a few centimeters between the sensors used for gradient measurements and electrically connected so as to obtain the mean thermal resistivity between the gradient sensors. The sensor probe is relatively thin (1 cm diameter), but larger than outrigged probes, so the measurement of gradient and in situ conductivity requires about 30min undisturbed in the bottom (Davis et al., 1984, Fig. 4). The increased thermal time constant of the larger probe (1- 1.5 min) requires more complex data reduction for both gradients and in situ conductivity. The total energy in the heating pulse must be calibrated to an accuracy equal to or better than that desired for the in situ K determinations. The advantages are that the pulse-heating technique requires substantially less energy than the continuous-heating needle probe method, and the appropriate mean thermal resistivity is determined with many sensors distributed along the probe. Hutchison (1983) reports measurements with equipment of similar design constructed at Cambridge University. One peculiarity noted by Hutchison, not reported by Davis et al., is an occasional thermal decay which does not fit the theory. Hutchison suggests that disturbed material occasionally may be entrained adjacent to the long probe wall; this phenomenon deserves further investigation. In general, in situ conductivity measurements appear to compare favorably with those made on core material aboard ship or in the laboratory. Davis et ai. obtained better than 2% agreement between average values measured on cores and in situ measurements at four sites in the North Atlantic. Apparently the coring process does not greatly alter the thermal properties of seafloor sediments. The in situ methods determine thermal conductivity at exactly the same site as the gradient measurement, which should particularly improve the accuracy of heat flow estimates where conductivity may vary rapidly laterally or with depth. Detrick et al. (1986) report data obtained on the Bermuda Rise for which in situ conductivity measurements significantly improved mean heat flux estimates, apparently because of local variability in sediment conductivity. 3.3. Other Estimates

In some situations where it is not feasible or possible to conduct direct measurements of conductivity on the relevant core material or by in situ measurements, it becomes desirable to estimate conductivity from other

254

R. P. VON HERZEN

measured parameters. For most surficial ocean floor sediments with high water contents (>20% by weight), Bullard et al. (1956)showed that conductivity is closely related to water content, irrespectiveof sediment type. Bullard and Day (1961)further quantified the relationship, showing that thermal resistivity R (the inverse of conductivity) can be related to water content w by

*

*

R = l/K = (161 14) + (6.51 0.30)~ (3) where R is in degrees Celsius centimeters seconds per calorie and w in weight percent. Therefore, it appeared that K (or R) could be estimated to about 10% from values of w. However, a systematicallydifferent formula was obtained by Lachenbruch and Marshall (1966)from many measurements on Arctic Ocean cores: R = 95.2 + 7.41~

(4)

which, for a given water content, gives conductivity values 10-20070 higher than calculated from Eq. (3). The most likely explanation appeared to be a significantly higher grain conductivity of the Arctic Ocean cores, perhaps as a result of .more abundant carbonate minerals and quartz in the solid fraction ; experimental verification of such minerals was not sought. An alternative estimate of water content, or porosity, of desiccated sediments derives from measurements of chlorine, a constituent of seawater left with the solid residue after evaporation of pore waters (Ratcliffe, 1960; Gerard et a/. , 1962). With the assumption that the chemical constituents of seawater are relatively uniform, a measure of the amount of chlorine leads to a seawater content calculated to within several percent. presumably present as pore fluid in the sediment. It is assumed that the original pore fluid does not migrate before or during the desiccation process. For many deep boreholes, especially those drilled for commercial exploration or production, core recovery may be quite incomplete. However, when well logging is carried out, porosity estimates can be derived from the nuclear and electrical logs. Combined with conductivity measurements on drill cuttings and lithology estimates derived from the same cuttings, an estimate of mean conductivity can be made (King and Simmons, 1972). The cuttings must be separated according to the major lithologies present before conductivity measurements are made, a tedious and time-consuming task.

4. Heat Flow 4.1. Uncertainty Estimates

The steady-state heat flow is calculated as the product of a temperature gradient and an appropriate thermal conductivity. The method for computing gradients differs according to the nature of the data, as does the


255

method used for calculation of heat flow and its uncertainties. If one gradient estimate G is available, combined with either a measured or assumed mean conductivity value, the calculation is relatively straightforward. With G i AG and K f AK as the best estimates of gradient and thermal conductivity, respectively, the heat flow Q and its uncertainty AQ are (Von Herzen et al., 1982) Q = GK

A Q = G A K + KAG

(5) (6)

For cases where gradients and thermal conductivity vary inversely with depth such that heat flow remains constant, heat flow should be calculated for each depth interval defined by the data (see, e.g., Von Herzen and Anderson, 1972). The heat flow mean and its uncertainty will then be determined by a statistical combination of all intervals, weighted by the uncertainties associated with each interval. Where heat flow is thought to be uniform over a region with many measurements, a straightforward statistical estimate is appropriate. However, where the data quality may vary widely with each station (penetration), it may be more appropriate to combine all temperature-depth data into a single gradient estimate (Langseth et al. , 1980; Hutchison ei al. , 1981). The latter procedure, combined with the mean thermal conductivity of the region, sometimes gives tighter confidence limits on the heat flow estimate than the statistical combination of all individual heat flow values. 4.2. Environmental Disturbances and Their Evaluation

Although the deep ocean basins offer advantages of environmental stability compared to continents, other factors may affect heat flow measurements significantly. The relatively limited vertical range of most oceanic gradient measurements makes them susceptible to even small disturbances associated with the bottom interface and makes nonuniform heat flow with depth difficult or impossible to detect if it occurs over a scale large compared with the probe length. 4.2.1. Bottom Water Temperature Variability. For most of the deep ocean, temperature fluctuations of bottom waters are sufficiently small to be negligible for the purpose of determining reasonably accurate values of the geothermal flux. They do become important for many shallower ( 5 1 km) regions, however, and for that reason accurate measurements on shallow continental margins with oceanic probes are relatively rare. Some deep basins are subject to periodic temperature fluctuations due to sinking of bottom water-for instance, the Denmark Straits in the North Atlantic (Lachenbruch and Marshall, 1968 ;Sclater and Crowe, 1979). The northwest Atlantic Basin

256

R. P. VON HERZEN

bottom-water temperature appears to vary as a result of lateral motion, or oscillations, of the interface between near-bottom water masses (Galson and Von Herzen, 1981;Davis et al., 1984). Other deep ocean basins do not appear as variable as that of the northwest Atlantic, although that conclusion may be due to lack of data elsewhere. If the bottom-water temperature history can be estimated, its effect on temperature gradients can be calculated from conduction theory. For example, the temperature T and surface gradient go resulting from a step function of temperature A T applied at time t = 0 on the surface of a semiinfinite medium of uniform thermal diffusivity a are given as (Bullard et al., 1956)

T = ATerf~[Z/(4at)’/~] go = AT/(Rat)‘/’

(7) (8)

where Z is depth and erfc is the complementary error function (Carslaw and Jaeger, 1959). A linear change of boundary temperature starting at t = 0 with a rate of change such that the temperature difference is A Tat time t gives T = AT4i2 erfc[Z/(4at)’’’] go = ~ A T / ( T ~ / ~

(9) (10)

where i t erfc is the second integral of the complementary error function. For a harmonic boundary temperature of the form To = A Tsin of,where o = 2nf (f= frequency), the same parameters are T = ATexp[-Z(o/2a)’’’] go = - (w/2a)’’2 AT(cos o t

sin[wt - Z(o/2a)”’]

+ sin at)

(1 1) (12)

Indeed, if an arbitrary boundary temperature variation is described by its Fourier components, these parameters can be calculated by superposition of the solutions for the individual Fourier components to any degree of approximation, since the Laplace differential equation of heat conduction is linear and homogeneous. 4.2.2. Refraction. If the seafloor has laterally inhomogeneous thermal properties, heat flow refraction will occur and steady-state isotherms will not be horizontal. Depending on the magnitude and geometry of the inhomogeneities, heat flow will thus vary laterally over the seafloor. The effects of surface topography on surface heat flow were discussed in a thorough paper by Lachenbruch (1968). Significant anomalies occur near changes in slope of the bottom topography, although these would be attenuated by a sediment cover, a necessary condition for most heat flow measurements. Some analytical solutions to simple (but not necessarily realistic) geometries of

15. MEASUREMENT OF

OCEANIC HEAT FLOW

257

buried topography were developed by Von Herzen and Uyeda (1963) and Lachenbruch and Marshall (1966). Numerical methods are most useful for arbitrary geometries. Finite-difference techniques developed by Sclater et af. (I 970) were applied to heat flow measurements over and near salt domes on the continental margin by Von Herzen et al. (1972) and to heat flow measurements in lakes by Von Herzen et d. (1974). Finite-element methods were developed and applied to similar problems by Lee et al. (1980). Although it is not normally a significant problem for marine measurements, the “warm rim” effect can be important for corrections to measurements in small, temperate lakes. The effect was first described by Johnson and Likens (1967) and is a result of the thermal perturbation caused by a lake with a bottom-water temperature significantly different from that of its surroundings. Its effects can also be incorporated in the numerical methods mentioned above. 4.2.3. Sedimentation. A systematic effect on heat flow measurements in some regions is caused by sedimentation. If the sedimentation rate is sufficiently high, a significant part of the heat will be absorbed by the initially cold sediments. An analytical solution to a simple model of this process, including radioactive heat generation, was obtained by Von Herzen and Uyeda (1963). The model required the same thermal properties throughout and did not include the effects of sediment compaction. Birch et al. (1968) obtained a somewhat different formulation which gave similar results. The magnitude of the effect depends on the duration as well as the rate of sedimentation, and for sediment thicknesses observed over most of the deep sea floor (a few hundred meters) the reduction was found to be small ( c 10%) for sedimentation rates 5 10 cm/103 years. However, the effects on heat flow were found to be important for some regions near continents and in marginal seas, where sediment thickness and sedimentation rates may be much higher. A possibly more realistic numerical model has been developed by Hutchison (1985). Although still a one-dimensional model, it takes into account a nonuniform vertical distribution of thermal conductivity, variable sedimentation rates over time, and effects of compaction. In general, calculations show that the previous simpler models may have overestimated the effects on heat flow. Compaction of sediments with associated upward expulsion of pore waters compensates to some extent for the decrease in heat flux resulting from the transient heating of the sediments being deposited. In small basins with anisotropic permeability (i.e., shales) and boundaries which are hydrological barriers, models with lateral migration of pore waters and expulsion near the edges may be appropriate (Bethke, 1985). 4.2.4.Conductive versus Advective Heat Transfer. Most of the discussion thus far has focused on conductive heat flow measurements and the perturbations thereto. An important process in young ( 550 Ma) seafloor is

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R. P. VON HERZEN

the advective transfer of heat by hydrothermal circulation (see, e.g., Lister, 1972; Anderson et al., 1977). Detailed investigations of this process near actively spreading ridges, where it is most intense, show highly variable heat flow which tends to be ordered in linear oscillating patterns aligned along the axis of spreading, with wavelengths of several kilometers (Green et al., 1981 ; Becker and Von Herzen, 1983b). In such regions the conductive mean heat flux, although high, is generally significantly less than that expected, based on quantitative models of the evolution of oceanic plates (Parsons and Sclater, 1977). The differences between theory and observation are interpreted as indicating the existence of cellular pore water convection driven by cooling of the relatively young ocean crust. The lower than expected conductive heat flux is taken as evidence that much of the heat is removed by advective exchange of the pore waters with cold seawater. Indeed, high-temperature venting of pore waters at the ridge axis has now been observed directly at some localities (Williams et al., 1974; Corliss et al., 1979; RISE; 1980). Obviously, in such regions the total flux cannot be determined by conductive heat flow measurements alone, no matter how numerous. It is best estimated from theoretical models or from mass flux models derived from geochemical constraints (Wolery and Sleep, 1976; Jenkins et al., 1978) or from other measurements to estimate the advective heat flux. Significant advection of heat at midocean ridges has now been directly confirmed at many locations (see, e.g., Corliss et al., 1979; RISE, 1980), as well as theoretically predicted from models of hydrothermal circulation in porous media (Fehn et al., 1983). Measurement of the total advected flux is hampered by the uneven distribution of hydrothermal venting along the ridge axis, as well as the possible ephemeral nature of individual vent sites. Estimates of the thermal output of an individual vent were made by MacDonald et al. (1980) from temperature and flow velocity measurements at a vent orifice. This approach does not account for the possibly significant contribution to the thermal budget from the lower-temperature diffuse flow. Measurements of temperature and flow velocity in the thermal plume above a vent (Little et al., 1987) have the potential to determine both the hightemperature vent flow and diffuse flow around a vent. On a larger scale, measurements of the bottom-water temperature structure above the midocean ridge axis (Crane el al., 1985) have the potential to provide an estimate of the overall advected flux, although such estimates may be seriously contaminated by dynamics of the deep ocean waters. The uncertainties of these estimates increase with the scale of the region considered, ranging from perhaps 50% for individual vents to one or two orders of magnitude for a ridge segment. As the crust ages and becomes covered with sediment, the conducted heat

15.


259

flux tends to approach the theoretical value. The critical thickness of uniform sediment cover necessary to suppress significant advective exchange of pore water with seawater is unknown but appears to be at least several tens of meters (Williams et al., 1979). Experimental determination of this thickness is difficult because even a single basement outcrop apparently can function as an efficient conduit for fluid flow and thereby affect the conductive heat flux in the sediment for many kilometers surrounding it (Sclater et af., 1976; Anderson et af.,1977). However, theoretical considerations suggest a similar thickness (Anderson and Skilbeck, 1981). It is also not generally known whether hydrothermal circulation continues beneath a uniform sediment cover, although detailed heat flow and pore water chemical measurements suggest that it does beneath sediments more than 250 m thick on the south flank of the Costa Rica rift (Langseth eta/., 1986). It appears that the fluid permeability of the underlying basement rocks may be substantially lowered as chemical precipitation clogs the fractured pathways soon after establishment of such a sediment cover. Vertical flow of pore water through sediments has also been indicated by nonlinear temperature profiles (Anderson et af., 1979; Becker and Von Herzen, 1983a; Geller et af., 1983) and chemical gradients (Sayles and Jenkins, 1982 ;Bender, 1983). For steady-state upward flow, the temperature gradient decreases exponentially with depth, depending on the flow rates. Rates greater than lo-’ cm/sec produce significant nonlinearity of gradients within the uppermost few meters of sediments. Both the heat flux and the fluid flow rate can be determined from the curvature in this gradient, although accurate values depend on carefully resolved temperatures versus depth in the sediment as well as sediment thermal properties. Acknowledgments I am grateful for the support provided largely by the National Science Foundation, most recently through grant OCE85-16298,for my participation in part of the developments in marine heat flow measurements discussed in this chapter. Hardly any of those would have been possible without the encouragement and assistance of many colleagues, research associates, and students. Partial support for preparation of this chapter was provided by the Woods Hole Oceanographic Institution. I am indebted to T. Henyey, M. Langseth, and J. Heirtzler for comments on early versions of the manuscript and to R. Hyndman for providing Fig. 3. This is Contribution No. 6372 of the Woods Hole Oceanographic Institution.

References Anderson, R. N., and J. N. Skilbeck, Oceanic heat eow. In “The Oceanic Lithosphere: The Sea” (C. Emiliani, ed.), Vol. 7, pp. 489-523. Wiley, New York, 1981. Anderson, R. N., M. C.Langseth, and J. G. Sclater, The mechanisms of heat transfer through the floor of the Indian Ocean. J. Geophys. Res. 82, 3391-3409 (1977).

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Anderson, R. N., M. A. Hobart, and M. G. Langseth, Geothermal convection through oceanic crust and sediments in the Indian Ocean. Science 204, 828-832 (1979). Becker, K., and R. P. Von Herzen, Heat transfer through the sediments of the mounds hydrothermal area, Galapagos spreading center at 86”W. J. Geophys. Res. 88, 995-1008 (1983a). Becker, K., and R. P. Von Herzen, Heat flow on the western flank of the East Pacific Rise at 21”N. J. Geophys. Res. 88, 1057-1066 (1983b). Becker, K., M. G. Langseth, and R. P. Von Herzen, Deep crustal geothermal measurements, Hole 504B, DSDP Legs 69 and 70. Initial Rep. Deep Sea Drill. Proj. 69, 223-235 (1983). Bender, M. L., Pore water chemistry of the mounds hydrothermal field, Galapagos spreading center: Results from Glomar Challenger piston coring. J. Geophys. Res. 88, 1049-1056 (1983). Bethke, C. M., A numerical model of compaction-driven groundwater flow and heat transfer and its application to the paleohydrology of intracratonic sedimentary basins. J. Geophys. Res. 90, 6817-6828 (1985). Birch, F., R. F. Roy, and E. R. Decker, Heat flow and thermal history in New England and New York. In “Studies of Appalachian Geology: Northern and Maritime” (E-an Zen, W. S . White, J. B. Haddley, and J. B. Thompson, Jr., eds.), pp. 437-451. Wiley (Interscience), New York, 1968. Bullard, E. C., The time necessary for a borehole to attain temperature equilibrium. Mon. Not. R. Astron. Soc., Geophys. Suppl. 5,127-130 (1947). Bullard, E. C., The flow of heat through the floor of the Atlantic Ocean. Proc. R. SOC.London Ser. A 222,408-429 (1954). Bullard, E. C., and A. Day, The flow of heat through the floor of the Atlantic Ocean. Geophys. J. 4,282-292 (1961). Bullard, E. C., A. E. Maxwell, and R. Revelle, Heat flow through the deep sea floor. Adv. Geophys. 3, 153-181 (1956). Burch, T. K., and M. G. Langseth, Heat-flow determination in three DSPD boreholes near the Japan trench. J. Geophys. Res. 86,941 1-9419 (1981). Cande, S. C., R. B. Leslie, J. C. Parra, and M. Hobart, Interaction between the Chile Ridge and Chile Trench: Geophysical and geothermal evidence. J. Geophys. Res. 92, 495-520 (1987). Carslaw, H. S., and J. C. Jaeger, “Conduction of Heat in Solids,” 2nd Ed. Oxford Univ. Press, London and New York, 1959. Carvalho, H. da S., S. Purwoko, M. Thamrin, and V. Vacquier, Terrestrial heat flow in the Tertiary basin of central Sumatra. Tectonophysics 69, 163-188 (1980). Christoffel, D. A., and I. M. Calhaem, A geothermal heat flow probe for in-situ measurement of both temperature gradient and thermal conductivity. J. Phys. E 2, 457-465 (1969). Corliss, J . B., era/.,Submarine thermal springs on the Galapagos Rift. Science203, 1073-1082 ( 1979). Davis, E. E., C. R. B. Lister, and J. G. Sclater, Toward determining the thermal state of old ocean lithosphere: Heat flow measurements from the Blake-Bahama Outer Ridge, N.W. Atlantic. Geophys. J. R. Astron. SOC.78, 507-545 (1984). Detrick, R. S., R. P. Von Herzen, B. Parsons, D. Sandwell, and M. Dougherty, Heat flow observations on the Bermuda Rise and thermal models of mid-plate swells. J. Geophys. Res. 91, 3701-3723 (1986). Dowdle, W. L., and W. M. Cobb, Static formation temperature from well logs-an empirical method. J. Pet. Technol. 27, 1326-1330 (1975). Erickson, A., R. P. Von Herzen, J. G. Sclater, R. W. Girdler, B. V. Marshall, and R. Hyndman, Geothermal measurements in deep sea drill holes. J. Geophys. Res. 80,2515-2528 (1975).


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Fehn, U., K. E. Green, R. P. Von Herzen, and L. M. Cathles, Numerical models for the hydrothermal field at the Galapagos spreading center. J. Geophys. Res. 88, 1033-1048 (1983). Galson, D. A., and R. P. Von Herzen, A heat flow survey on anomaly MO south of the Bermuda Rise. Earth Planet. Sci. Lett. 53, 296-306 (1981). Geller, C. A., J. K. Weissel, and R. N. Anderson, Heat transfer and intraplate deformation in the central lndian Ocean. J. Geophys. Res. 88, 1018-1032 (1983). Gerard, R., M. G. Langseth, and M. Ewing, Thermal gradient measurements in the water and bottom sediment of the western Atlantic. J. Geophys. Res. 67, 785-803 (1962). Goldberg, D., R. P. Von Herzen, and J. Sclater, Thermal conductivity measurement of fused silica glass. Tech. Rep. Woods Hole Oceanogr. Inst. WHOI-80-34 (1980). Green, K. E., R. P. Von Herzen, and D. L. Williams, The Galapagos spreading center at 86"W: A detailed geothermal field study. J. Geophys. Res. 86, 979-986 (1981). Horai, K., Thermal conductivity of sediments and igneous rocks recovered during DSDP Leg 60. Initial Rep. Deep Sea Drill. Proj. 60, 807-834 (1981). Horai, K., and R. P. Von Herzen, Measurement of heat flow on Leg 86 of the Deep Sea Drilling Project. Initial Rep. Deep Sea Drill. Proj. 86, 759-177 (1985). Hsu, M., T. L. Henyey, D. V. Manov, and Y. K. Li, A digital heat flow probe. EOS, Trans. Am. Geophys. Union 64, 837 (1983). Abstr. Hutchison, I., Heat flow studies of the Gulf of Oman and western Mediterranean. Ph.D. Thesis, Darwin Coll., Univ. of Cambridge, 1983. Hutchison, I., The effects of sedimentation and compaction on oceanic heat flow. Geophys. J. R. Aston. Soc. 82, 439-459 (1985). Hutchison, I., K. E. Louden, R. S. White, and R. P. Von Herzen, Heat flow and age of the Gulf of Oman. Earth Planet. Sci. Left. 56, 252-262 (1981). Hyndman, R. D., E. E. Davis, and J. A. Wright, The measurement of marine geothermal heat flow by a multipenetration probe with digital acoustic telemetry and in-situ thermal conductivity. Mar. Geophys. Res. 4, 181-205 (1979). Hyndman, R. D., M. G. Langseth, and R. P. Von Herzen, A review of Deep Sea Drilling Project geothermal measurements through Leg 71. Initial Rep. Deep Sea Drill. Proj. 78, 813-823 (1984). Hyndman, R. D., M. G. Langseth, and R. P. Von Herzen, Deep-sea drilling project geothermal measurements: A review. Submitted to Rev. Geophys., 1987. Jemsek, J., R. Von Herzen, and P. Andrew, In-situ measurement of thermal conductivity using the continuous-heating line source method and WHO1 outrigged probe. Tech. Rep. Woods Hole Oceanogr. Inst. WHOI-85-28 (1985). Jenkins, W. J., J. M. Edmond, and J. B. Corliss, Excess 'He and 4He in Galapagos submarine hydrothermal water. Nature (London) 272, 156-158 (1978). Jessop, A. M., M. A. Hobart, and J. G. Sclater, The world heat flow data collection-1975. Geotherm. Ser., Dep. Energy, Mines Resour., Can. No. 5 (1976). Johnson, N. M., and G. E. Likens, Steady-state thermal gradient of the sediments of a meromictic lake. J. Geophys. Res. 72, 3049-3052 (1967). King, W., and G. Simmons, Heat flow near Orlando, Florida and Uvalde, Texas, determined from well cuttings. Inr. J. Georherm. Res. 1, 133-139 (1972). Koehler, R., and R. P. Von Herzen, A miniature deep sea temperature data recorder: Design, construction, and use. Tech. Rep. Woods Hole Oceanogr. Inst. WHOI-86-3 (1986). Lachenbruch, A. H., Rapid estimation of the topographic disturbance to superficial thermal gradients, Rev. Geophys. 6 , 365-400 (1968). Lachenbruch, A. H., and B. V. Marshall, Heat flow through the Arctic Ocean floor: The Canada Basin-Alpha Rise boundary. J. Geophys. Res. 71, 1223-1248 (1966).

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Lachenbruch, A. H., and B. V. Marshall, Heat flow and water temperature fluctuations in the Denmark Strait. J. Geophys. Res. 73, 5829-5842 (1968). Langseth, M. G., Techniques of measuring heat flow through the ocean floor. In “Terrestrial Heat Flow” (W. H. K. Lee. ed.), Monogr. No. 8, pp. 58-77. Am. Geophys. Union, Washington, D.C., 1965. Langseth, M. G., M. H. Hobart, and K. Horai, Heat flow in the Bering Sea. J. Geophys. Res. 85, 3740-3750 (1980). Langseth, M. G., M. Mottl, M. Hobart, and A. Fisher, Hydrothermal circulation in the vicinity of the DSDP 501/504 sites on the south flank of the Costa Rica rift. EOS67, 1222 (1986). Langseth, M. G., and R. P. Von Herzen, Heat flow through the floor of the world oceans. In “The Sea” (A. Maxwell, ed.), Vol. 4, Part I, pp. 299-352. Wiley, New York, 1971. Lee, T.-C., A. J. Rudman, and A. Sjoreen, Applications of finite element analysis to terrestrial heat flow. Occas. Pap. Indiana Geol. Surv. 29 (1980). Lister, C. R. B., Measurement of in situ sediment conductivity by means of a Bullard-type probe. Geophys. J. R. Astron. SOC. 19, 521-532 (1970). Lister, C. R. B., On the thermal balance of a mid-ocean ridge. Geophys. J. R. Asrron. SOC. 26, 515-535 (1972). Lister, C. R. B., The pulse-probe method of conductivity measurement. Geophys. J. R. Astron. SOC.57, 451-461 (1979). Little, S . A., K. D. Stolzenbach, and R. P. Von Herzen, Measurement of plume flow from a hydrothermal vent field. J. Geophys. Res. 92, 2587-2596 (1987). MacDonald, K. C., K. Becker, F. N. Spiess, and R. D. Ballard, Hydrothermal heat flux of the “black smoker” vents on the East Pacific Rise. Earth Planet. Sci. Lett. 48, 1-7 (1980). Matsubara, Y., H. Kinoshita, S. Uyeda, and A. Thienprasert, Development of a new system for shallow sea heat flow measurement and its test application in the Gulf of Thailand. Tectonophysics 83, 13-31 (1982). Parsons, B., and J. G. Sclater, An analysis of the variation of ocean floor bathymetry and heat flow with age. J. Geophys. Res. 82, 803-827 (1977). Ratcliffe, E. H., The thermal conductivities of oceansediments. J. Geophys. Res. 65,1535-1541 (1 960). Revelle, R., and A. E. Maxwell, Heat flow through the floor of the eastern North Pacific Ocean. Nature 170, 199-202 (1952). RISE Project Group, East Pacific Rise: Hot springs and geophysical experiments. Science 207, 1421- 1433 (1 980). Sayles, F. L., and W. J. Jenkins, Advection of pore fluids through sediments in the equatorial east Pacific. Science 217, 245-248 (1982). Sclater, J. G., and J. Crowe, A heat flowsurveyat anomaly 13on theReykjanesridge: Acritical test of the relation between heat flow and age. J. Geophys. Res. 87, 1593-1602 (1979). Sclater, J. G., C. E.Corry, and V. Vacquier, In-situ measurement of the thermal conductivity of ocean floor sediments. J. Geophys. Res. 74, 1070-1081 (1969). Sclater, J. G., E. J. W. Jones, and S. P. Miller, The relationship of heat flow, bottom topography and basement relief in Peake and Freen deeps, Northeast Atlantic. Tectonophysics 10, 283-300 (1970). Sclater, J. G., J. Crowe, and R. N. Anderson, On the reliability of oceanic heat flow averages. J. Geophys. Res. 81, 2997-3006 (1976). Shipley, T. H., M. H. Houston, R. T. Buffler, F. J. Shaub, K. J. McMillan, J. W. Ladd, and J . L. Worzell, Seismic evidence for widespread possible gas hydrate horizons on continental slopes and rises. Am. Assoc. Pet. Geol. Bull. 63, 2204-2213 (1979). Shor, E. N., E. C. Bullard’s first heat-probe. EOS, Trans. Am. Geophys. Union 65, 73-74 (1984).

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Spiess, F. N., and J. D. Mudie, Small scale topographic and magnetic features. In “The Sea” (A. Maxwell, ed.), Vol. 4, pp. 205-250. Wiley (Interscience), New York, 1970. Stoll, R. D., J. Ewing, and G. M. Bryan, Anomalous wave velocities in sediments containing gas hydrates. J . Geophys. Res. 76, 2090-2094 (1971). Von Herzen, R. P., Oceanic heat flow data. In “Geophysics of the Solid Earth, the Moon and the Planets” (K. Fuchs and H. Soffel, eds.), Landolt-Bornstein New Series, Group 5, Vol. 2, pp. 207-241. Springer-Verlag, Berlin and New York, 1984. Von Herzen, R. P., and R. Anderson, Implications of heat flow and bottom water temperature in the eastern equatorial Pacific. Geophys. J. R. Astron. SOC. 26, 427-459 (1972). Von Herzen, R. P., and A. E. Maxwell, The measurement of thermal conductivity of deep-sea sediments by a needle-probe method. J. Geophys. Res. 64, 1557-1563 (1959). Von Herzen, R. P., and A. E. Maxwell, Measurements of heat flow at the preliminary Mohole site off Mexico. J. Geophys. Res. 69, 741-748 (1964). Von Herzen, R. P., and S. Uyeda, Heat flow through the eastern Pacific Ocean floor. J. Geophys. Res. 68, 4219-4250 (1963). Von Herzen, R. P., A. E. Maxwell, and J. M. Snodgrass, Measurement of heat flow through the ocean floor. Temp. :Its Meas. Control Sci. Ind. 3, 769-777 (1962). Von Herzen, R. P., H. Hoskins, and T. van Andel, Geophysical studies in the Angola diapir field. Geol. SOC.Am. Bull. 83, 1901-1910 (1972). Von Herzen, R. P., P. Finckh, and K. J. Hsu,Heat-flow measurements in Swiss lakes. J . Geophys. 40, 141-172 (1974). Von Herzen, R. P., R. S. Detrick, S. T. Crough, D. Epp, and U. Fehn, Thermal origin of the Hawaiian swell: Heat-flow evidence and thermal models. J. Geophys. Res. 87,671 1-6723 (1982).

Williams, D. L., R. P. Von Herzen, J. G. Sclater, and R. N. Anderson, The Galapagos spreading center : Lithospheric cooling and hydrothermal circulation. Geophys. J. R. Astron. SOC. 38, 587-608 (1974).

Williams, D. L., K. Green, T. van Andel, R. P. Von Herzen, J. R. Dymond, and K. Crane, The hydrothermal mounds of the Galapagos Rift: Observations with DSRV Alvin and detailed heat flow studies. J. Geophys. Res. 84, 7467-7484 (1979). Wolery, T. J., and N. H. Sleep, Hydrothermal circulation and geochemical flux at mid-ocean ridges. J. Geol. 84, 249-275 (1976). Wright, J. A., and C. L. Fang, A microprocessor instrument for real-time marine heat flow measurement. EOS, Trans. Am. Geophys. Union 65, 1120 (1984). Abstr. Yamano, M., S. Uyeda, Y.Aoki, and T. H. Shipley, Estimates of heat flow derived from gas hydrates. Geology 10, 339-343 (1982). Yokota, T., H. Kinoshita, and S . Uyeda, New DSDP (Deep Sea Drilling Project) downhole temperature probe utilizing IC RAM (memory) elements. Tokyo Duigaku Jishin Kenkyusho Iho 55, 75-88 (1980).


16. ELECTRICAL METHODS IN GEOPHYSICAL PROSPECTING*

Stanley H. Ward Earth Science Laboratory University of Utah Research Institute Salt Lake City, Utah 84108 and Department of Geology and Geophysics University of Utah Salt Lake City, Utah 84112

1. Introduction This chapter will treat the resistivity, induced-polarization, magnetotelluric, audio-frequency magnetotelluric, and controlled-source electromagnetic methods. With the exception of the magnetotelluric method, all of these methods are used in mining exploration. Resistivity, induced-polarization, and magnetotelluric methods are used in a minor way in oil and gas exploration. All of the methods have been used in geothermal exploration, while resistivity, controlled-source electromagnetics, and audio-frequency magnetotellurics are employed periodically in coal basin studies. Resistivity, magnetotellurics, and controlled-source electromagnetic methods are used in deep exploration of the earth’s crust and mantle. Resistivity, inducedpolarization, and controlled-source electromagnetic methods are used in ground water exploration, while only resistivity is used routinely in applications in geotechnical engineering. Representative applications of each method will be given subsequently. The objectives of this chapter are to provide for each method (a) an overview, (b) an outline of important applications, (c) a summary of important references, (d) a summary of the theoretical and physical bases, (e) a description of typical modern field equipment, ( f ) a summary of data processhg, (g) a summary of interpretation procedures, and (h) a summary of the problems encountered with the method in its various applications. *The final revised manuscript for this chapter was received in December, 1985. 265 METHODS OF EXPERIMENTAL PHYSICS Vol. 24. Part B


266

STANLEY H. WARD

Each of the methods demands a knowledge of electromagnetic theory. Such theory as is necessary to understand each method will be presented, commencing with elementary electromagnetic theory at the outset. Within the discussion of each method, additional electromagnetic theory will appear. Each method also depends on contrasts in electrical properties of the earth media. Since these properties have some unusual features vis-his more homogeneous materials such as metals, it is necessary to discuss these properties at some length prior to entering into discussions of the five methods mentioned earlier.

2. Elementary Electromagnetic Theory 2.1. Introduction

I now present theory sufficient only for the purpose of this presentation. Throughout, mks units will be used and time dependence eiotwill be invoked. 2.2. Maxwell’s Equations

An electromagnetic field may be defined as the domain of the four vector functions E, ByD, and H,where E is the electric field intensity in volts per meter, B the magnetic induction in webers per square meter, D the dielectric displacement in coulombs per square meter, and H the magnetic field intensity in amperes per meter. The experimental evidence of Amptre and Faraday leads to the two fundamental Maxwell equations described in the time domain : V x E

+ aB/at

=0

(Faraday’s Law)

(1)

V x H

- aD/at

=J

(Ampere’s Law)

(2)

and

in which J is the electric current density and aD/at the displacement current density, both in amperes per square meter. It should be stressed that these are empirical equations which seem to govern all electromagnetic phenomena. Taking the divergence of Eqs. (1) and (2), I obtain

v-a~/at=o

and

-v-aD/at=v.J

(3)

because the divergence of a curl is zero. Provided the vector functions B and D are piecewise continuous and possess continuous first and second

16.

ELECTRICAL METHODS IN GEOPHYSICAL PROSPECTING

derivatives, then the operators V and

a/&

a

-(V.B)

=

at

267

may be interchanged to yield

0

(4)

and

a

--(V*D) at

= V.J

Equation (4) yields the third Maxwell equation

V-B=O if at any time B was zero. Equation ( 5 ) yields the fourth Maxwell equation

V*D=Pe

(7)

if at any time D was zero, provided that an equation of continuity V *J

+ &/at

=0

(8)

is applied. Equation (8) is a statement of the conservation of charge in the vicinity of a point. For homogeneous earth materials of conductivity s/m or greater, free charge pe dissipates in less than 10-6s. Thus for geophysical prospecting, in which frequencies less than lo5 Hz are employed, ape/at = 0 and we may write

V * D= 0

(9)

Equation (9) does not apply to inhomogeneous regions; at the interface between two different media a surface charge accumulates. 2.3. The Constitutive Relations

The first two Maxwell equations, Eqs. (1) and (2), are uncoupled differential equations describing the experimental behavior of the five vector functions E, B, H, D, and J. These two equations are coupled only through the frequency-domain constitutive relations

D = E(w, E, r, t , T , P,...) E

(10)

T, P,...) H

(1 1)

B

= p(w, Byr, t,

-

and

J = b(w, E, r, t , T, P,...) E in which the tensors 8, J, and b describe, respectively, the dielectric permittivity, magnetic permeability, and electric conductivity as functions of

268

STANLEY H. WARD

angular frequency o,electric field strength E or magnetic induction B, position r, time t , temperature T , and pressure P . Each of these three tensors is complex in the general case, permitting the phases of D and E, of H and B, and of J and E to be different. In most elementary electromagnetic earth problems the following assumptions are made to simplify analysis. 1. All media are linear, isotropic, homogeneous, and have electrical properties which are independent of time, temperature, or pressure. 2. The magnetic permeability p is assumed to be that of free space, i.e., p =po.

Comments on these assumptions follow. 1. Anisotropic media are included in some simple electromagnetic boundary-value problems and aid in interpretation of data obtained with plane wave sources. 2. Inhomogeneous media entering into electromagnetic boundary-value problems are treated as one-dimensionally inhomogeneous (plane-layered), two-dimensionally inhomogeneous (infinite cylinders of arbitrary cross section), or three-dimensionally inhomogeneous. 3. In shallow prospecting the effect of pressure is small and is customarily ignored. 4. The time dependence of electrical conductivity due to varying moisture content in surface soils is usually ignored, although not correctly so.

For the purpose of subsequent discussion, the following three constitutive relations will suffice :

D=

[E’(O)-

J = [a’(o)

i ~ ” ( o ) ] E= EE

(13)

+ ia”(w)]E = aE

B=pH

(15)

in which dielectric permittivity E and electrical conductivity a are complex functions of angular frequency while magnetic permeability p is independent of frequency and is real. 2.4. Fourier Transformation of Maxwell’s Equations

We wish to effect Fourier transformation of the Maxwell equations, given in the time domain by

V x E

+ dB/&

=

0

and

V

X

H

- dD/dt = J

(16)

16.


269

The vector field quantities E, B, H, D, and J are each functions of position r and time 1. A typical field quantity transforms according to

.

' 1 &

Pm

E(r, t ) = -

-~

E(r, o)e'"'do

--OD

in which E(r, t ) is described in the time ( 1 ) domain whereas E(r, o)is described in the frequency (0) domain. The curl of Eq. (18) is V x E(r, I ) = V

. I

Pm

-

X

6 -"

E(r, o)eiW'd o

which for E(r, o)piecewise continuous, with continuous first and second derivatives, becomes

3

rm

V x E(r, w)eiw'd o

V x E(r, t ) = 7T

-m

The quantity aB/dt transforms according to

1

1

=6

R

"

iwB(r, o)e'"' d o

--o

Thus the first Maxwell equation (1 6)becomes under Fourier transformation V x E(r, w)eiW'd o

+-

iwB(r, w)e'"' dw = 0 (22)

The second Maxwell equation similarly transforms according to V x H(r, o)e'"'dw = -

J(r, o)+ ioD(r, w)e'"' do (23)

Insofar as Eqs. (22) and (23) apply to arbitrary functions, E(r, o)and H(r, o),provided they satisfy the existence conditions for Fourier transformation, these equations must apply to each element of the integral. Thus we find that V x E(r,o)

+ ioB(r,w) = 0

(24)

and V x H(r, o)- iwD(r, w ) = J(r, w )

(25)

270

STANLEY H. WARD

These are the frequency-domain versions of the first two Maxwell equations. If we now substitute, in Eqs. (24) and (25), the constitutive relations of Eqs. (13)-( 1 3 , we obtain, after dropping the functional dependences, V xE

+ ipwH = 0

(26)

and V x

H - (a + iew)E = 0

These are now two coupled differential equations. In Eq. (27), the term J = aE is conduction current density and the term aD/at = $oE is displacement current density, so V x H must represent total current density. It is customary to make the following shorthand identifications (Harrington, 1961) :

i = ipo

(impedivity )

(28)

9 = a + i&w

(admittivity)

(29)

and

so that Eqs. (26) and (27) may be rewritten VXEXLH=O

and VXH-PE=O 2.5. The Wave Equations

2.5.1. Wave Equations in the Time and Frequency Domains. If we take the curl (i.e., V X ) of Eqs. (1) and (2) we obtain From (1) V x (V x E)

);(

+Vx

=0

From (2) and

V x ( V x H ) - V X

The constitutive relations in the time domain

D=EE

B=pH

J=aE

(32)

r:>

- = V X J

16.


27 1

may now be substituted in Eq. (32) to yield

V xV xH

and

-Vx

[:t

-(EE)

1

=

V x (aE) (33)

If p , E , and a are constant in space and time in a homogeneous region, then Eq. (33) reduces to

aH at

V x V x E + p V x -= 0 V xV xH

and

aE

- EV x = aV x E at

(34)

Provided the vector functions H and E are piecewise continuous and have continuous first and second derivatives, the operators V x and a/at may be interchanged so that Eq. (34) becomes 0

a x V x E + p (V x H)= 0 and

V xVxH

a

-E(V x E) = aV x E at

(35)

The quantities V x H and V x E are, of course, given in Eqs. (1) and (2), so Eq. (35) is readily converted to a2E at

aE at

VXVXE+~ET+~O-=O and

V xV xH

aH + pe-a2H +pa=0 at2 at

(36)

The vector identity in Cartesian coordinates

V X V X A E V V - A - V2A

(37)

permits us to expand the first term in each of Eqs. (36). Taking cognizance of the fact that V - E= 0 and V . H = 0, from Eqs. (6) and (9), for homogeneous earth regions, then Eq. (36) becomes a2E aE - pa- = 0 V2E - pue7 at at

and

a2H aH V2H - p~~ -p a x =0

272

STANLEY H . WARD

These are wave equations for the electric and magnetic fields, stated in the time domain. A one-dimensional Fourier transformation of Eq. (38) leads to

V2E + ( p m 2 - ipao)E = 0

and

V2H + ( p & 0 2 - ipao)H = 0 (39)

and

V2H + k2H = 0

or

VZE+ k2E = 0

(40)

in which

k2 = p&w2 - ipoo = -29

(41)

Equations (40) are the wave equations in the frequency domain or, more commonly, the Helmholtz equations in E and H. In Eqs. (39), p&o24 paw for earth materials at frequencies less than lo5Hz; that is, displacement currents are less than conduction currents. Thus Eqs. (38) and (39) may be rewritten as

aE V2E - p a - = 0 at

and

aH V2H - pa=0 at

V2E - igaoE = 0

and

V2H - ipaoH = 0

and (43)

where, under this circumstance,

k=-

(44)

Either Eqs. (42) or Eqs. (43) represent a diffusion equation. Their onedimensional versions are

a2E

aE at

and

ipawE = 0

and

--pu--=O aZz

a%

_.-

az2

a2H a22

-

aH at = O

a2H

- - ipowH

az2

=0

(45) (46)

2.5.2. Solutions of the Wave Equations. Equations (46) are secondorder linear differential equations with solutions E = + E o e - i ( k z - w t ) + -Eoei(kz+w0= +E + -E (47)

and

H=

+Hoe-i(kZ-Wt)

+ -Hoei(kZ+Of)

= +H

+ -H

(48)

16.


273

Since k is complex, it is written

k=a-ip

(49)

in which a andp are both real. The quantities a and pare given, in general, by a = u[(pUE/%)(Jl

+ a2/UE2C02+ 1)]1’*

(50)

p

+ Q2(E2C02 - 1 ) p

(51)

and = O[(p&/2)(Jl

When conduction currents dominate over displacement currents (tan 6 % l), as is customary in electrical prospecting, we find that a and fi are identical real quantities defined by

a=p=-

(52)

Then the positive solutions of Eqs. (47) and (48) may be written E=+ ~ ~ ~ - i ~ ~ z ~ - P z ~ i d

(53)

and

H=

+ ~ ~ ~ - i ~ ~ z ~ - L k ~ i w t

(54)

From Eqs. (53) and (54) we may draw the following conclusions: 1. Since p is real e-(32becomes smaller as z becomes larger. Hence it represents attenuation. An electromagnetic wave will be reduced in amplitude by a factor of l/e at a distance within a medium described by the depth of penetration d, where

d= 2. with 3. with 4.

e-iaz= cos(az)

z. eior= cos(wt)

=5 0 3 m

- isin(az) states

+ isin(ot)

(55)

that the wave varies sinusoidally (56)

states that the wave varies sinusoidally t. (57) Over a plane fixed in space, E and H vary with time as in Fig. la. 5 . If the wave propagates in the z direction, E and H will vary sinusoidally with z as in Fig. lb. How do we know that the wave propagates? The arguments follow.

1. Wax = Way = 0 states that Ex and Hx are of constant magnitudes over a plane as below. This is a uniform plane wave as in Fig. lc. 2. Planes at different distances along z will have their own magnitudes. At (1) and (3) in Fig. lb, Ex and Hy are maximum. At (2) in the same figure, Ex and Hyare zero.

274

STANLEY H. WARD

"I

FIG. 1 . Uniform plane waves : (a) electric and magnetic fields vary sinusoidally with time, (b) electric and magnetic fields vary sinusoidally with distance, and (c) 'electric and magnetic vector amplitudes and phases are uniform over a plane normal to the direction of propagation.

3. In any plane, the Evector (or theHvector) will exhibit the same phase; i.e., at each point over the plane, E (or H)will reach its positive peaks, zero crossing, or negative peak at identical times. Any such plane is therefore referred to as a plane of constant phase. 4. The peaks of the E or Hfield which occur at t = 0, z = 0, will occur downstream at t = t l , z = z1. That is, a plane of constant phase will propagate in the z direction if we set a/az = Way = 0 as appropriate to a uniform plane wave. 5 . A plane of constant phase is described by E=

+E~~-~(CLZ-O = ~-iC )

(58)

where C is the phase, i.e., description of the amplitude of the sine wave, as a function of z and 1. If

crz - o t =

c

(59)

16.

ELECTRICAL METHODS IN GEOPHYSICAI. PROSPECTING

275

dz/dt = o / a = ?&= phase velocity (positive)

(60)

then

Similarly, for

E= dz/dt = - w / a =

-bei(uz+Wt)'=iC

V p V p h

= phase velocity (negative)

(61) (62)

This explains why we used +Eoand -Eo.

2.6.Boundary Conditions Electromagnetic problems arising in the physics of the solid earth generally deal with the resultant current, field intensity, or potential in response to an impressed or primary field. The primary field gives rise to a secondary distribution of charges and currents and, hence, to a secondary field. The resultant field is the sum of the primary and secondary fields. Each of the fields must satisfy Maxwell's equations, or equations derived therefrom, plus appropriate conditions to be applied at boundaries between the homogeneous regions involved in the problem, e.g., at the air-earth interface. The problems we meet most frequently, therefore, are referred to as boundaryvalue problems. Boundary conditions are readily derived from the integral forms of Maxwell's equations (e.g., Stratton, 1941, p. 34). We shall merely state them here.

Normal B. The normal component Bn of B is continuous across an interface separating medium 1 from medium 2. This is written Bni

= Bnz

(63)

NormalD. The normal component Dn of D is discontinuous at an interface due to the accumulation of a surface charge density p s , i.e., Dn1

- Dnz

= ps

(64)

Tangential E.The tangential component Et of E is continuous across an interface, i.e.,

4, = Et,

(65)

Tangential H . The tangential component Ht of H is continuous across an interface, i.e., Ht, = Htz

(66)

276

STANLEY H. WARD

Current density J. The normal component Jn of J is continuous across an interface, i.e., Jnl

(67)

= Jnz

Strictly speaking, this result applies only to direct current, but it is totally satisfactory for earth materials up to 10’ Hz in which displacement currents may be neglected. Potentials. The static potentials V and U defined by E=-VV

(68)

n = -vu

(69)

are continuous across an interface, i.e.,

J4=&

(70)

u1 = u2

(71)

We note from the above that of the quantities considered, only one is discontinuous across an interface. It is essential to explore the nature of this discontinuity at an interface separating media of different conductivities. Equation (64) may be rewritten as En1

- En2 = ~ s / e o

and Eq. (67) as En1

-

( 0 d ~ d E n z=

0

(73)

Combining Eqs. (72) and (73) results in ps =

-

EO[(~I

~ ~ Y o l l E n z

(74)

which informs us that a surface charge ps occurs at the boundary between media of different conductivities 01 and 0 2 . Although the surface charge faradlm, its electrical field E is density is small because EO = 8.854 x not necessarily small, as we can deduce from

in which ds is an element of the surface over which the charge occurs. In the case of two adjacent plane boundaries separating a region of 0 2 from a background of 01 the charge accumulation is as depicted in Fig. 2 for the case 02

> 01.

16.


277

ai>ai FIG. 2. Charges associated with boundaries representing discontinuities in electrical conductivities.

2.7. Wave Impedances A uniform plane wave is defined as a plane wave in which the field intensities are independent of the coordinates in each equiphase plane. This condition is expressed as

a

a

- [ E , HI = - [E,HI ax aY

=0

for a plane wave propagated in the positive z direction. The frequencydomain components of Maxwell’s equations, given by Eqs. (30) and (31), with the constraint of Eq. (76), become aEY- 2H. az

aEx - - - -2Hy az

HE = 0

(77)

Thus in a uniform plane wave E and H are contained in planes perpendicular to the direction of propagation; the wave is said to be transverse electromagnetic (TEM). The four transverse components of Eqs. (77) and (78) may be considered as the superposition of two independent pairs [Ex,H,] and [E,, Hx].As shown earlier, solutions to Eq. (46) are linear combinations of eikzand e-ikzas follows : Ex = + E x e - i ( k Z - d ) + - E x e i ( k Z + w f ) (79)

E, =

+ ~ ~ ~ - i ( k z -+ w t -) ~ ~ ~ i ( k z + w t )

(80)

H, =

+ ~ , ~ - i ( k z - w t+ ) -Hxei(kz+wr)

(81)

Hy =

+Hye-i(kz-at)

(82)

+

- ~ ~ ~ i ( k z + w t )

The superscript plus denotes a wave traveling in the positive z direction and the superscript minus a wave traveling in the negative z direction. In general,

278

STANLEY H. WARD

'Ex,-Ex , ..., -Hy are complex constants and combine to form the complex vector amplitudes. Hence we may write E = + E e - i ( k Z - w t ) + -Eei(kz+ot) (83)

H

=

+He-i(kZ-of)

+

-Hei(kZ+ot)

(84)

Not all eight amplitude given in Eqs. (79)-(82) are independent, according to Eqs. (77) and (78). If we now substitute Eqs. (79) and (82) in the second of Eqs. (77) we obtain - ik+E, - i(kz- O t ) + ik-Exei(kZ+@f) = - t + ~ ~ ~ - i ( k z -o t?) - ~ ~ ~ i ( k z + ~ f ) (85)

The coefficients of the exponentials eikz and e-ikz must vanish independently, and hence we obtain

Similarly, we find

+Hx = (- k/mp)+Ey

(88)

-Hx = (k/wp)-Ey

(89)

Ratios of components of E to components of H have the dimensions of impedance (volts per meter divided by ampere-turns per meter) and are called wave impedances Z, where c

In an infinite medium, the ratios of the field components are determined by the frequency and the constants of the medium, and the wave impedance then - of the medium. Given that Z = and becomes the intrinsic impedance k = G, we may obtain t = ikZ and JJ = ik/Z. In a dielectric the wave impedance becomes Z=@i

which for the free space is

zo = djzG

- 377

n

(92)

In a conductor the wave impedance becomes

z=-

n

(93)

16.


279

I

II

zi- 1 zi

1

I

FIG.3. An n-layered earth on which a uniform plane wave is normally incident. E, and ff,, electric and magnetic fields ; k vector wave number; ui and hi, conductivities and thicknesses of layer i.

2.8. The Plane-Wave Impedance of an n-Layered Isotropic Earth 2.8.1. Normal Incidence. We wish to consider normal incidence of a uniform plane wave on an n-layered isotropic model of the earth. The plane wave is propagated in the positive z , or downward, direction. The earth model is illustrated in Fig. 3. In any layer we may write the electric and magnetic fields in terms of an outgoing wave and a reflected wave. For normal incidence of a uniform plane wave on a plane-layered, nonpermeable isotropic earth, the fields in the ith layer are Eyc. (+E. I e-iki(z-zO + - E i e i k i ( Z - Z i ) lei"' (94)

where ki is the wave number in the ith layer, POthe permeability of free space, o the angular frequency, Zi the vertical distance to the bottom of the ith

280

STANLEY H. WARD

layer, z any vertical distance within a layer at which the field is measured ; 'Ei the amplitude of the positive-traveling electric wave in the ith layer, -Ei the amplitude of the negative-traveling electric wave in the ith layer, and eiot the harmonic description of the wave. At this juncture, for convenience, we note that selection of E, only and Hx only assumes that the electric vector is normal to the plane of incidence. Over the plane z = Zi we find

Eyi = 'Ei

+ -Ei

and Hxj = (-Ei - 'Ei)(l/Zi) in which Zj = wpo/ki

is the intrinsic impedance of the ith layer. Equations (96) and (97) yield +Ei = &Eyi - ZiHxi)

-Ej = i(Eyi

+ ZiHi)

At z = zi-1, continuity of tangential E and H demands that

Hxi = Hxci-1) Eyi = Ey(i-1) Therefore we may write Ey(i-1) = +E.l e- i k i ( z i - l - Z i ) Hx(i-1)

=-

+

-Eieiki(Zi-i-Zi)

(1/ z i ) (l e+ -iki(zi~ . L - Z i ) - -Eieiki(Zi-l-Zi) )

Now if we let Zi

- zj-1

= hi

and substitute Eqs. (99) and (100) in Eqs. (103) and (104), we find

Ey(i-1) = Eyi cosh(ikihi) - ZiHxi sinh(ikih;)

(106)

Hx(i-1) = H,i cosh(ikihi) - (1/Zi)Eyi sinh(ikihi)

(107)

where use has been made of the identities coshx = (e" + e-")/2

(108)

sinhx = (e" - e-X)/2

( 109)

16.


28 1

Equations (106) and (107) may be written in matrix form as

[

Ey(i- 1)

Hx(i-lJ

=

cosh(ikihi) - Zi sinh(ikihi) l/Zi sinh(ikihi) COsh(ikihi)I

[-

'

[21

(1 10)

and symbolically by

The matrix i7 is referred to as the transfer matrix of the ith layer. For n layers we can find a succession of Z from Zi through Tn each transfer matrix permitting us to write the fields in one layer in terms of the fields in the next layer. Thus, we can readily find the matrix relationship between the fields in the (i - 1)th layer and those in the infinite medium terminating the nth layer.

The product Z E of n matrices is itself a matrix S, so Eq. (112) may be written

in which

Then the impedance Zj-1 looking into the n-layered medium from the surface of the ith layer, is

where the substitution & + I = - E y ( n + l ) / H x ( n + l ) has been made. For an earth model consisting of one layer overlying an infinite half-space (this model is usually referred to as a two-layered earth)the impedance may

282

STANLEY H. WARD

be found by making the substitutions for a0 from Eq. (110), i.e., a11

a12 (YZI

= cosh(ik1hl)

- ZI sinh(ik1h l ) = - (1/Z1) sinh(ik1hl) =

Thus we obtain from Eq. (1 16) the impedance

21 = -Eyi/H.i which is defined in terms of the electric and magnetic fields measured at the surface of the earth as

21 = z1

Z2 + 21 tanh(iklh1) Z1 + ZZtanh(iklh1)

Expression (122) is the impedance at the surface of the two-layered earth. For an n-layered earth model, we start with the impedance at the top of the first layer above the basal half-space. By analogy with Eq. (122) this will be

2,

=

n + l + 2,tanh(iknhn) z,ZZn + Z,+l tanh(ik,h,)

(123)

Once this impedance is computed we may use it as the terminating impedance of an equivalent homogeneous half-space and write for the impedance at the top of the (n - 1)th layer 2 n - 1 = Zn-1

2,

+ Z,-1

Zn-l

tanh(ik,-lh,-l)

+ 2, tanh(ikn-lhn-l)

( 124)

and so on up to the surface, where

We have used the notation 2j to denote the impedance at the top of the ith layer and the notation Zi to denote the characteristic impedance of the ith layer. 2.8.2.Oblique Incidence. The previous development may be generalized to accommodate an arbitrary angle of incidence. As it turns out, the impedance contrast between the air and the earth is so large that regardless of the angle of incidence of a plane wave in the air, the wave in the earth will travel vertically. This is readily seen from Snell’s law

kl sin 0i = k2 sin 8r

(126)

16. ELECTRICAL METHODS IN GEOPHYSICAL PROSPECTING

283

in which k~ is the wave number of air, k2 the wave number of the earth, 8i the angle of incidence, and el the angle of transmission relative to the vertical. When k2 % kl in (126) then 8t = 0 and this always holds at an air-earth interface when displacement currents are neglected.

3. Electrical Properties of Earth Materials 3.1. Introduction

Bulk resistivities from the surface to in excess of 15 km depth in a normal crust are controlled by aqueous electrolytic conduction via pores, fractures, and faults. A slight increase in resistivity with depth in this region is the result of decreasing pore, fracture, and fault porosity due to increased hydrostatic load. Fractures and faults are known to remain open to depths in excess of 5 km due to departures from hydrostatic loading. From about 15 km to the Moho, mineral semiconduction dominates and the resistivity decreases downward. Semiconductionwill remain the dominant conductionmechanism in excess of 100 km into the normal upper mantle. 3.2. Aqueous Electrolyte Conduction 3.2.1. Normal Mode of Conduction. Conduction in near-surface rocks is largely electrolytic, taking place in pore spaces, along grain boundaries, in fractures, and in faults but negligibly through the silicate framework. The ions which conduct the current result from the dissociation of salts when the salts are dissolved in water. Since each ion is able to carry only a definite quantity of charge, the more ions that are available in a solution and the faster they travel, the greater the charge that can be carried. Hence, the solution with the larger number of ions will have the higher conductivity. Thus, in general, a rock which contains saline water within its pores will have a greater conductivity when the salinity of the water is high than when it is low ; salinity is a major factor in determining the resistivity of a rock. An increase in temperature lowers the viscosity of water, with the result that ions in the water become more mobile. The increased mobility of the ions results in an observed resistivity decrease with increase in temperature according to pI = p d [ l

+ 4 - 1811

(127)

in which a is the temperature coefficient of resistivity (usually given as about O.O25/"C), t the ambient temperature, pt the resistivity at this temperature, and p18 is the resistivity at 18°C.

284

STANLEY H. WARD

Archie’s law,

F = p,/pw = 4-m

(128)

usually is satisfied for aqueous electrolytic conduction. In Eq. (128), F i s the formation factor, prthe resistivity of the rock, pwthe resistivity of the saturating electrolyte, 4 the porosity, and m the cementation factor, which varies between 1.0 and 3 ; m = 2 is the value usually taken for sandstone while m = 1 satisfies conduction in rocks in which fracture porosity dominates. 3.2.2. Effect of Clays on Rock Resistivity. A clay particle acts as a separate conducting path in addition to the electrolyte path. The resistance of this added path is low. The origin of this abnormally high clay mineral conductivity lies in the double layer of exchange cations, as shown in Fig. 4. The cations are required to balance the charge due to substitution within the crystal lattice and to broken bonds (Grim, 1953). The finite size of the cations prevents the formation of a single layer. Rather, a double layer is formed, consisting of a fixed layer immediately adjacent to the clay surface and a diffuse layer which drops off in density exponentially with distance from the fixed layer. The diffuse layer, in contrast to the fixed layer, is free to move under the influence of an applied electric field. The cations of the diffuse layer add to the normal ion concentration and thus increase the density of charge carriers. The net result is increased surface conductivity. Although clay minerals exhibit this property to a high degree because of their large ion exchange capacity, all minerals exhibit it to some extent. All rocks containing clay minerals have an abnormally high conductivity for this reason.

- - -

CLAY

@

----

-

-

PARTICLE

ABSORBED CATIONS

t NORMAL CATIONS

- NORMAL ANIONS FIG.4. Schematic representationof ions adsorbed on clay particle. (After Ward and Fraser, 1967.)

16.


285

The effect of disseminated clay or shale on rock resistivities becomes increasingly important as the conductance through the pores diminishes. In geothermal and mining environments, hydrothermal alteration converts feldspars to kaolinite, montmorillonite, and other clay minerals, especially in silicic rocks. In basic rocks, chlorite and serpentine may also be produced. All of these alteration products exhibit high surficial conductivity. As the concentration of the electrolyte increases, the relative contribution of the electrolyte conduction path to the clay conduction path increases, as may be seen from Or

= (a,

+ as)/F

(129)

in which ar, a e ,and asrepresent the observed conductivities of the rock, the electrolyte, and the clay surface path. Ward and Sill (1976) demonstrate that as 30e for altered rocks at Roosevelt Hot Springs, Utah, despite the presence of an electrolyte containing 7000ppm total dissolved solids.

-

3.3. Induced Polarization

3.3.1.Introduction. Pyrite and clay minerals often are found as alteration products in geothermal and mining areas. Hence the induced electrical polarization mechanisms of electrode polarization and membrane polarization might be expected to occur there. 3.3.2. Electrode Polarization. Whenever there is a change in the mode of current conduction, e.g., from ionic to metallic, energy is required to cause the current to flow across the interface. This energy barrier can be considered to constitute an electrical impedance. The surfaces of most solids have a very small net attraction for either cations or anions, as mentioned earlier for clay minerals. Immediately adjacent to the outermost solid layer there is an adsorbed layer of essentially fixed ions, one or a few molecular layers in thickness (Fig. 5a). These are not truly exchangeable and, hence, constitute the fixed layer. Adjacent to the fixed layer of adsorbed ions there is a group of relatively mobile ions, of the same or opposite charge, known as the diffuse layer. The anomalous number of ions in this zone decreases exponentially from the fixed layer outward to the normal ion concentration of the liquid. (The normal balanced distribution of anions and cations has been deleted from Fig. 5 for clarity.) The particular distribution of ions shown is only one of several possible distributions, but it is the most common. The electrical potential across the double layer has also been plotted in Fig. Sb; the potential drop across the diffuse layer is known as the zeta potential (2). While the fixed layer is relatively stable, the diffuse layer thickness is a function of temperature, ion concentration in the normal electrolyte, valence

286

STANLEY H. WARD

=- + ) +

o - _x

+

+

+

t

+

(a)

+ I +

+I

--I=

+

o=t;+ w =

t +

+

= + 1 +

t

+ J

I

-h

'FIXED

LAYER

4

I-

(b)

DlSTA N C E FIG. 5. (a) Hypothetical anomalous ion distribution near a solid-liquid interface; (b) corresponding potential distribution. (After Ward and Fraser, 1967.)

of the ions, and dielectric constant of the medium. Most of the anomalous charge is contained within a plane distance d from the surface (Grahame, 1947) :

d=

(EO KekT/2ne2~2)"2

( 130)

where n is the normal ion concentration of the electrolyte, 1) the valence of the normal ions, e the elementary charge, Ke the dielectric constant of the medium, k Boltzmann's constant, and T temperature. The thickness is, therefore, governed by the balance between the attraction of unlike charges at the solid surface and the thermal redistribution of ions. Obviously, increasing n, the salinity, or u, the valence, decreases the thickness of the diffuse layer. Returning now to polarization at electrodes, there are two paths by which current may be carried across an interface between an electrolyte and a metal (Fig. 6). These are called the faradaic and nonfaradaic paths. Current passage in the faradaic path is the result of an electrochemical reaction such as the oxidation or reduction of some ion and may involve diffusion of the ions toward or away from the interface. The charge is carried physically across the interface by an electron transfer. In the nonfaradaic case, charged particles do not cross the interface; rather, current is carried by the charging and discharging of the double layer. The nonfaradaic component, thus, may be represented by a simple capacitance insofar as the variation of its impedance with frequency is concerned.

16.

ELECTRICALMETHODS IN GEOPHYSICAL. PROSPECTING REACTION RE S I STAN C E

287

WARBURG IMPEDANCE

F A R A D A I C PATH

NO N- FAR A D A l C PATH 11 I DOUBLE LAYER CAPACITANCE

FIG. 6. Circuit analog of interfacial impedance. (After Ward and Fraser, 1967.)

In the faradaic path, the impedance associated with the electron transfer is represented by the reaction resistance. The ion diffusion process is not representable in so simple a fashion and, in fact, may not be adequately represented by any combination of fixed capacitors and resistors. It is customarily referred to as the Warburg impedance W and its magnitude varies inversely with the square root of the electrical frequency. The interfacial impedance of many metal-electrolyte interfaces may be described roughly as follows. Above 1000Hz most of the electric current is carried across the interface by the nonfaradaic path ; hence, the interfacial impedance varies with frequency as approximately f-’.As the frequency is lowered, more and more current is carried via the faradaic path, so the lowfrequency impedance varies with frequency in the range f - ” 2 to fo, depending on the magnitude of the impedance ratio W / R . The discussion above applies to an ideal electrode in a pure electrolyte. The concepts, however, are important in understanding the processes occurring when current is passed through a rock. Any rock sample is dirty from the viewpoint of the physical chemist, since the electrodes (semiconducting mineral grains) and electrolytes (pore solutions) are anything but pure. Nevertheless, perhaps we are justified in using equivalent circuits based on pure systems since a phenomenological explanation for rock behavior results. With this caution, one might suggest the equivalence of the elementary rock system of Fig. 7a with the equivalent circuit of Fig. 7b, where W is the Warburg impedance [= k(l - i ) / f ” * ;k is a constant], CFthe doublelayer capacitance, CCHthe chemical capacitance, R the reaction resistance, R’ the resistance representing a higher-order reaction, Ri the resistance of the ionic path, and Rm the resistance of metallic vein path or particle. In noting these circuit elements, it must be appreciated that one chemical reaction at the interface may lead to a chain of subsequent reactions involving electrons, ions, and molecules of all reaction products present. At each point

288

STANLEY H. WARD

(a)

--

>IONIC

PATHS

LM E T A LL I C P A R T I CL E R, and R, R i and R,

Ri and R,

FIG. 7. (a) Simplified representation of mineralized rock ; (b) corresponding equivalent circuit; (c) equivalent circuit of all mineralized rocks. (After Ward and Fraser, 1967.)

of the reaction chain, the accumulation of the reaction product represents a capacitance CCHto the electrode. Escape of the product is achieved either by diffusion, represented by a Warburg impedance W ,or by a reaction, represented by a resistor R. The product of this reaction in turn follows a similar circuit behavior, which we have omitted for simplicity, except to lump all such products as R'. Although the circuits of Figs. 7a and 7b satisfy the expected physical/ chemical processes in mineralized rock, they are too complicated for practical use. Thus, the simple circuit of Fig. 8a is used to predict induced polarization, of both electrode and membrane type, in a rock. The frequency- and timedomain responses of the circuit of Fig. 8a are shown in Fig. 8b and 8c, respectively. This is the Cole-Cole model of relaxation used by Pelton et al. (1978a). 3.3.3. Membrane Polarization. In rocks containing a few percent clays distributed throughout the rock matrix, membrane polarization is important. Membrane polarization arises chiefly in porous rocks in which clay particles partially block ionic solution paths (Fig. 9a). The diffuse cloud of cations (double layer) in the vicinity of a clay surface is characteristic of clayelectrolyte systems. On application of an electrical potential, positive charge carriers easily pass through the cationic cloud but negative charge carriers accumulate (Fig. 9b) ; an ion-selective membrane, therefore, exists. Consequently, a surplus of both cations and anions occurs at one end of the membrane zone, while a deficiency occurs at the other end. This is because the number of positive charges cannot deviate significantly from the number

16.


R1

289

..

( c ; " i b EXCITING C U R R E N T

t-

VDC' R z I

v

,v ,RIR2 1

0

Rl+R2

FIG. 8. Simplified analog circuit model of rock. (a) Elementary circuit; (b) frequency response of elementary circuit-sine wave excitation ; (c) transient response of elementary circuit-square wave excitation. (After Ward and Sill, 1983.)

of negative charges at any one point in space, or large electric fields would result. These ion concentration gradients oppose the flow of current, and the overall mobility of ions is reduced by this process. This reduction in mobility is most effective for potential variations which are slow (e.g., 0.1 Hz) with respect to the time of diffusion of ions between adjacent membrane zones. For potential variations which are fast (e.g., 1000 Hz)with respect to the diffusion time, the mobility of ions is not substantially reduced. Hence, the conductivityof a membrane system increases as electricalfrequency increases.

3.4.Semiconduction The intrinsic conductivity of a solid at temperature T is computed from rs

= lel(nepe

+ nhph)

(131)

where n e and n h are the electron and hole equilibrium concentrations, ,ueand the mobilities of electrons and holes, respectively, and e the elemental charge.

,uh

290

STANLEY H. WARD NORMAL ELECTROLYTE CHARGE CARRIERS

NEGATIVE CHARGE ZONE OF ION DEFICIENCY ZONE OF ION UGH

(b)

+

FIG.9. Depiction of ions in a pore space forming an ion concentrationbarrier which creates membrane polarization: (a) pore path before application of an electric potential; (b) pore path after application of a potential. (After Ward and Fraser, 1967.)

Kinetic theory leads us to expect a temperature dependence of the form

e-E’kTfor the concentration of electrons in the conduction band of a solid. Assuming a relatively small variation of mobility with temperature, we are then led (Kittel, 1953) to predict a conductivity dependence of the form (i

= (ioe -Eg/2kT

(132)

in which Eg is the gap energy, 00includes the mobility function and, in this form, is the conductivity as T -+ 00, and kis Boltzmann’s constant. Thermal, electrical, or optical excitation of electrons across the band of forbidden energy renders the solid conducting. Impurities and imperfections in the material produce extrinsic conductivity. Above some temperature, impurities may be unimportant, so we define the temperature range above extrinsic conductivity as the intrinsic range in which the previous mechanism is operative. However, below the intrinsic range, certain types of impurities and imperfections markedly alter the electrical properties of a semiconductor. Extrinsic semiconduction arises by thermal excitation of electrons (occupying intermediate energy levels in the forbidden gap produced by impurities in solid solution) into the unoccupied conduction band, or by the excitation of electrons from the occupied valence band into unoccupied impurity levels. Ionic conduction in a solid occurs as a result of mobile ions moving

16. ELECTRICAL METHODS IN GEOPHYSICAL PROSPECTING TABLEI.

29 1

Values of uo and E as the Temperature Ranges of Importance for the Extrinsic Electronic, Instrinsic Electronic, and Ionic Mechanisms' Range of importance

Type of semiconduction

uo Wrn)

E (eV)

("C)

Extrinsic Intrinsic Ionic

10-6

10-3

1.o 3.3

10-3

3 .O

600 600 to 1100 1100

'Semiconduction follows the formula u = uoe-"T conduction mechanism.

but uo and E are different for each

through the crystal lattice as a result of defects in it. The simplest imperfection is a missing atom or lattice vacancy (Schottky defect). Diffusion of the vacancy through the lattice constitutes transport of charge. The conduction mechanism above 1100°C is recognized as ionic because, when an iron electrode is used in contact with a magnesium orthosilicate, iron diffuses into the silicate, replacing the magnesium. Table I illustrates the temperature ranges important for extrinsic, intrinsic, and ionic conduction. 3.5. Melt Conduction A silica magma chamber can be expected to exhibit a resistivity two to three orders of magnitude lower than its solid rock host, as demonstrated by the experiments of Lebedev and Khitarov (1964). Duba and Heard (1980) measured resistivity on buffered olivene and Rai and Manghnani (1978) measured electrical conductivity of basalts to 1550°C ; the latter measurements establish that mafic rocks can also demonstrate low resistivities. Resistivities of order 1 fi m are to be expected in either silicic or basic melts due to ionic conduction. For partial melts, the melt phase serves as an interconnection of low resistivity in a residual crystal matrix of resistivity two or more orders greater and determines the bulk resistivity (Shankland and Waff, 1977). An Archie's law dependence is hence expected.

4. Basic Principles of Resistivity and Induced Polarization Surveys 4.1. Introduction

Electrical resistivity surveys are used routinely in geothermal, base metal mining, coal, and ground water applications (Zohdy, 1964; Al'pin ei al., 1966; Keller and Frischknecht, 1966; Kunetz, 1966; Van Nostrand and

292

STANLEY H. WARD

Cook, 1966; Bhattacharya and Patra, 1968; Keller, 1969; Meidav and Furgerson, 1972; Parasnis, 1973; Telford et al., 1976; Verma et al., 1982; Ward and Sill, 1982). They are used much less routinely in oil and gas and deep crustal exploration (Keller, 1968; Eadie, 1981 ; Ward, 1983a, b). Resistivity surveys are capable of mapping overburden depth, stratigraphy, faults, fractures, rock units, conductive ore deposits, thermal brines and associated hydrothermal alteration, and variations in the deep conductivity of the crust, and may be capable of direct detection of oil and gas. The induced-polarization (IP) method was developed for detecting small concentrations of disseminated mineralization in base metal exploration (Seigel, 1949; Hallof, 1957; Marshall and Madden, 1959; Wait, 1959; Van Voorhis et al., 1973 ;Wynn and Zonge, 1975 ;Sumner, 1976,1979; Angoran and Madden 1977; Pelton et al., 1978a; Hohmann and Ward, 1981 ; Ward and Sill, 1982). Subsequently it has been used experimentally in geothermal exploration (Ward and Sill, 1982). Resistivity and induced-polarization surveys are performed in boreholes and at the earth’s surface. In the interest of uniformity throughout this chapter I will limit my discussion to surveys performed at the earth’s surface. Dyck (1975) reviewed electrical borehole methods. 4.2. Basic Principles

As Hohmann and Ward (1981) indicate, the resistivity and inducedpolarization methods involve measurement of an impedance, with subsequent interpretation in terms of the subsurface electrical properties and, in turn, the subsurface geology. An impedance is the ratio of the response (i.e. output) to the excitation (i.e. input). In the resistivity and IP methods, the input is a current injected into the ground between two electrodes and the output is a voltage measured between two other electrodes. In frequency-domain impedance measurements, the input current is a sine wave with frequencyfand period T = l/J The output is also a sine wave, as shown in Fig. 10; its amplitude A and phase @ depend on electrical properties of the earth. In general, the output is delayed by CP x T O R seconds relative to the transmitted waveform. Often it is convenient to decompose the output wave into in-phase (real) and quadrature (imaginary) components, as shown in Fig. 10. If we denote their peak amplitudes as R and Z, respectively, then the amplitude and phase of the output waveform are given by

A = R2 + Z2

(133)

@ = arctan(l/R)

(134)

and

16.


293

I c

FIG.10. Transmitted and received waveforms in the frequency domain. (After Hohmann and Ward, 1981.)

Impedance can also be measured in the time domain, in which case the current is periodically turned on and off. As shown in Fig. 11, the output is the voltage measured at various times when the transmitter current is off. Note that the input again is periodic, because measurements must be made for each of several periods and then added together, or stacked, to eliminate

294

STANLEY H. WARD

TRANSMITTER CURRENT T (perlod)

V

RECEIVED SIGNAL

*

time

FIG. 1 1 . Transmitted and received waveforms in the time domain. (After Hohmann and Ward, 1981.)

noise. Time- and frequency-domain measurements are directly related through the Fourier transform and, in that sense, are equivalent. However, in practice, each domain has certain advantages and disadvantages. There are three basic modes of operation for any electrical method: sounding, profiling? and sounding-profiling. In sounding, the transmitterreceiver separation is changed, or the frequency is changed, and the results are interpreted in terms of a layered earth. Because the earth is usually not layered, we believe that sounding has only modest application. In profiling? the transmitter or receiver or both are moved along the earth’s surface to detect lateral anomalies. The most useful method is a combination of sounding and profiling, which delineates both lateral and vertical variations. The resistivity and induced-polarization methods are based on the response of earth materials to the flow of current at low frequencies. The dc resistivity method is based on potential theory, which requires direct current, but noise and measurement problems quickly lead to the use of alternating currents of low frequency, so the resistivity method now employs ac exclusively. The IP method, on the other hand, requires the use of alternating current because it is based on changes in resistivity as a function of frequency. As the frequency increases, in some critical frequency range determined by

16.


295

the resistivity of the materials and the scale size of the measurement, electromagnetic coupling between transmitting and receiving circuits violates potential theory, and electromagnetic theory is required. Measurements are made with a four-electrode array consisting of two current and two potential electrodes. Resistivity data are always recorded along with IP data to aid in interpretation. For a homogeneous earth, the resistivity is given by

(135)

p = KAV/I

where Z is the current, A V the measured potential difference, and K a geometric factor that depends on the electrode configuration. When the ground is not homogeneous, the voltage and current data are reduced in the same fashion, but the resistivity is called the apparent resistivity. It is the resistivity of a homogeneous earth that would produce the same measurement. The potential due to a single electrode on a three-layer earth is given by (Sunde, 1949)

where

k123 = (1 u123 =

+~ ~ 2 3 e - ~ ~ 1 )

-

(el

- pZkZS)/(pl

-t p2k23)

(137) (138)

k 4 ) = (1 - ~23e-~"2)/(1 + ~23e-"~*)

(139)

and U23 =

(p2

- pl)/(p2

-t p1)

(140)

Jo(1r) is the Bessel function of the first kind of order zero, r the distance from the current electrode at which the potential is measured, and 1 a Hankel transform variable. With two current and two potential electrodes in use, as is customary, the potential difference between the two potential electrodes is measured as A V = (6- b) - (K -

h)

(141)

where the first term is the potential difference due to the positive current electrode and the second term the potential difference due to the negative current electrode. An apparent resistivity is then readily derived as pa

'I

-K-=-

-

211

~mki23(A)[J0(1rl)- Jo(Ar2) 0

- JO(Ar3)

-k

Jo(Ar.)]dA

(142)

296

STANLEY H. WARD

RESISTIVITY AND IP ARRAYS ARRAY

K

GEOMETRY

Pa v s a

WENNER c1 SCHLUMBERGER

P1

P2

7rn(n+l)a

tQ1

DIPOLE-DIPOLE

p1

Pz

p1

PI

na

c,

c z

SOUNDING

pa vs (nt 1/z)a SOUNDING

B

MN

c1

LEiE

c2

L-@K-l

A

POLE-DIPOLE

DISPLAY

2xn(n+1)a

Pavsn

SOUNDINGPROFILING

rn(n+l)(n+2)a

Pavsn

SOUNDINGPROFILING

FIG.12. The common arrays used in resistivity and induced-polarization surveys.

Forward solutions involve evaluation of the last integral as the ri are changed systematically. To illustrate, when conducting field soundings with the Schlumberger array, the current electrodes of Fig. 12 are expanded about six times per decade of distance, starting with a current electrode separation of a few meters, until their separation reaches 1 km or more. The potential electrodes are left fixed at, say, m apart until the voltage becomes too small, which occurs as the current electrodes get farther apart. Then the potential electrodes are expanded to, say, 5 m, and the current electrode expansion continues. A plot of Pa versus half the current electrode separation (AB/2) is made as in Fig. 13. This curve may be compared with catalogs of curves based on Eq. (142). Equation (136) is one member of a Hankel transform pair, the other member of which is

Thus all the information about the earth that is present in the kernel klz3(A) is available on effecting the Hankel transform of Eq. (143). Attempts to use this technique were made by Slichter (1933) and Vozoff (1958). Unfortunately, the technique seldom works because it demands a range of r from zero to infinity, which is never available in practice. Today, a least-squares fit is performed between observed values of pa versus AB/2 and values calculated from Eq. (142). Severalinversion methods are available for performing this operation and they will be referenced later.

16.


297

X:= 1.00 10'3(7.6%>

I LOW -.

h

P

BEST HlGHi LOW BEST 24.0

126.0 35.9 4.3 4.82ll.S 115.0 123.0? 7.0% 9.0 2000.0

--

1

3

5

SOIL SAND AND SILT

1

A

A

10

5.41

-

132.0

.-

62.1

.8 2.5

6.0 __

2.0 14.0 65.5f4.4% 684 *FIXED VALUE5

3 5 l o o LEGEND CLAY

HIGH1

1- f

3 5 1 0 0 0 A812 SANDSTONE

GRAVEL (AQUIFER)

FIG.13. Six-layer interpretation of Schlumberger sounding. The table gives best estimates for layer thicknesses (meters) and resistivities (ohm-meters) as well as low and high estimates corresponding to 1 standard deviation departure in log parameter space. The section at the bottom compares estimated resistivity and depth with geological information from a well. (After Rijo e t a / . , 1977.)

When performing combined sounding and profiling, that is, when searching for both lateral and vertical variations in resistivity, the dipole-dipole array of Fig. 12 is most commonly used. Referring to Fig. 14, the transmitting dipole is established between stations 1 and 2; that is, electrodes are placed at 1 and 2 and are connected to a source of low-frequency current. Frequencies in the range 0.03-3 Hz are used most commonly. The receiving dipole is first connected to electrodes at stations 3 and 4, and the current I,

298

1

STANLEY H. WARD

2

3

4

Plot value of

5

6

PFE for electrodes

7

at 2-3, 6-7

FIG.14. Method of plotting data in a pseudosection for the dipole-dipole method. Numbers on profile are electrode positions. Current Zis entered via a transmitting dipole between stations 5 and 6.Value of resistivity, or of induced-polarization parameter, is plotted at intersection of lines drawn at 45' from center of each dipole. x dipole length; n separation, which assumes values ranging from 1 through 6 for every location of the transmitting dipole.

voltage V, and geometric factor Kare entered into Eq. (135) to compute an apparent resistivity. This resistivity is plotted at the intersection of lines drawn at 45" as in Fig. 14. Then the receiving dipole is moved to stations 4-5, 5-6, 6-7,7-8, and 8-9. This expansion of the array provides information mostly on the vertical variation of resistivity. Then the transmitting dipole is moved to stations 2-3 and the expansion process repeated. With, say, 10 or 20 transmitter locations established along a traverse line, a whole field of apparent resistivity data points will appear as in Fig. 14. These data are contoured to produce what is known as a pseudosection. It is not a true representation of the distribution of resistivities in the subsurface. Theoretical pseudosections are computed iterativeIy until one is found which reasonably matches the observed one. Figure 15 contains an observed pseudosection, a computed one, and a two-dimensional model on which the computed pseudosection is based. Three-dimensional earths may also be modeled, as will be discussed subsequently. For induced-polarization surveys, both the amplitude pa of apparent resistivity and the phase shift 4 between the transmitted current and the

16.

299

ELECTRICAL METHODS IN GEOPHYSICAL PROSPECTING FIELD DATA

-5

-4

-3

-2

-I

0

I

2

3

4

5

3

4

5

I

(a)

COMPUTED RESULTS

-5

-3

-4

-2

-I

0

I

2

I

(b)

\,

0

202'

4

%oo\o'

56

63

52

60

56

5

OU

133'

53 00

58 51/

9O

I

I

1

FIG. IS. (a) Observed pseudosection from dipole-dipole field survey; (b) computed pseudosection using two-dimensional finite-element algorithm ; (c) model of the subsurface used in producing the computed pseudosection. (After Hohmann, 1982.)

received voltage are measured in the frequency domain ; the dipole-dipole array is usually used and the lpal and r#~ values are plotted in pseudosection as in Fig. 16. Two-and three-dimensional modeling of [paland 4 are then performed.

300

STANLEY H. WARD

FIG.16. Induced-polarization response from deep sulfide mineralization beneath resistive overburden-Kennecott, Safford, Arizona, porphyry copper deposit. (From Hohmann and Ward, 1981.).

4.3. Data Acquisition

Table I1 lists the features of a microprocessor-based resistivity and induced-polarizationreceiver engineered by the Earth Science Laboratory of the University of Utah Research Institute. Its features facilitate coherent detection, which is necessary for enhancing signal-to-noise ratio and for

16.


30 1

TABLE11. Typical Features of IP Receiver" Frequency domain 0.001 to 2000 Hz in 1,2,3.3,5 steps Sequence Automatic gain ranging Automatic S.P. buckout Sample at M PTS per cycle M = 5 1 2 , f l 10 Hz Decreasing to 4 for f = 2000 Hz Stack 2" cycles n = 0 to 6 , f s 0.33 Hz n = 4 to 6, 0.5 S f < 10 Hz Increase to n = 10, 20 5 f -C 2000 Hz Compute (paland extrapolate phase Compute running std. dev. for f d 0.33 Hz Full phase and amplitude calibration

" These are Geotronics DR-1 preliminary specifications.

recognizing and removing electromagnetic coupling. The high-pass filtering before stacking significantly improves signal-to-noise ratio, as SanFilipo and Hohmann (1982) have established. The automatic gain ranging and selfpotential buckout features speed observations, as does the use of dual channels. A transmitter of equally modern design does not exist but has been outlined by engineers of the Earth Science Laboratory in Table 111. The electrodes used for resistivity and IP surveys require special consideration. The current electrodes must be of low impedance so that a modest voltage source of order 1OOOV may drive 5-20A of current. Usually, aluminum foil of dimensions 0.3 by 1.O m is placed in a pit dug to about 0.3 m or more. Earth is cast over the foil and about 1-5 gal of salt water (NaCl)

TABLE111. Typical Features of Microprocessor-based Transmitter Computer control monitors all power circuits Computer controls load up Monitors input power Abrupt load change shutdown Displays all operating parameters Analyzes system faults Makes operational logs Programmable waveform Facilitates remote control

302

STANLEY H. WARD

is applied before and after emplacing the aluminum foil. The salt solution effectively increases the area of the electrode, especially if the electrode is not used for 24 hours after its emplacement. One hopes to obtain impedances of a pair of electrodes of order 100 hz if current of the order of 10 A are sought, as is usual. Several electrodes in parallel, placed 1-2m apart, will lower impedances where required. Potential electrodes, on the other hand, are not required to be of low impedance. Rather, they must be of low noise. Nonpolarizing Cu-CuSO4 electrodes are usually employed (Sumner, 1976). Dipole lengths used in dipole-dipole surveys range from 30m to 1 km. Current from the transmitter will range from 1 A to 20 A, depending on the application, but also depending on how low the impedance of the transmitting electrodes can be made. Receiving dipole wires are typically 18 or 20 AWG, while transmitting dipole wires are typically 8 to 12 AWG. 4.4. Data Processing

The induced-polarization parameters measured depend on whether the system makes use of a time-domain or frequency-domain waveform (Figs. 10 and 11). For time-domain measurements, the maximum value of the voltage during the on cycle, along with the current, can be used to calculate the apparent resistivity. The transient during the off cycle contains the basic information on induced polarization in the time domain. This transient is specified by its normalized value just after the current is turned off and by the form and rate of decay. For frequency-domain measurements, the basic data are the magnitude and phase of the measured voltage as functions of frequency, from which the amplitude and phase of the apparent resistivity are calculated. Older analog time-domain receivers integrate one or several intervals under the decay curve, at sampling times ranging from about 0.05 to 2.0s after current shutoff. When the integrated voltage is normalized by the primary voltage VO and the integration time At, the unit of the measurement is given as millivolts per volt and is called the chargeability M. Another definition of chargeability, the Newmont standard, does not normalize by the integration time; the units are millivolt seconds per volt or milliseconds. Since the equivalent integration time of the Newmont standard is 1 s, normalization by the integration time does not change the numerical value of the chargeability. The Newmont standard is often written as M331 , which refers to a standard pulsed square wave of 3 s on, 3 s off, and an integration time of 1 s. Often measurements are made with different pulse lengths and integration times, which are then reduced to an equivalent M331 by using various model-dependent normalization factors (Sumner, 1976).

16.


303

Analog frequency-domain receivers often use two to five frequencies, and many have no current waveform reference, so phase information is lost. The basic data are then the magnitudes of the apparent resistivitypl and pz at two frequenciesf1 and fz , which can be used to calculate the percent frequency effect (PFE), ~

PFE = lOO(pi

- pz)/pi

where pi is the resistivity at the lower frequency. Modern digital receivers sample the waveform at discrete points in time and store the samples as numbers in the computer memory. Manipulation of the data stored in memory is under program control and, in principle, either time- or frequency-domain processing can be done. To increase the ratio of signal to noise, multiple cycles are stored and averaged, or stacked, in the memory. Phase information is obtained by using a pair of very accurate synchronized oscillators at the receiver and transmitter or by using a cable link between the receiver and transmitter. For the Newmont standard of chargeability, time-domain and frequencydomain IP units are related by 7M = 7Cg (mrad) = 1 PFEIdecade of frequency

(145)

Normally, IP effects produce a positive percent frequency effect, a phase lag (negative phase angle), and a secondary decay voltage with the same sign as the primary (M positive) ; by convention these are referred to as positive IP effects. Negative IP response (positive phase angle) can be caused by geometric effects with normally polarizable materials and by inductive coupling. Precise measurements are required in I P surveys; even a large IP response of 20 mrad is a phase shift of only 3".

4.5,Arrays The most common arrays used in resistivity surveys are the Wenner, Schlurnberger, dipole-dipole , pole-dipole, and bipole-dipole arrays. If induced-polarization surveys are to be conducted, either the pole-dipole or the dipole-dipole array is used in order to minimize electromagnetic coupling. The bipole-dipole array was used extensively after the success that Risk et af. (1970) experienced with it at the Broadlands geothermal region in New Zealand. It has been used much less in recent years because the apparent resistivity contour plans obtained with it are complicated, difficult to interpret, and vary significantly with bipole orientation and position. Because of these problems, I will not discuss it further, but refer the reader to articles by Dey and Morrison (1977), Hohmann and Jiracek (1979), and Frangos and Ward (1980) for evaluations. The remaining four arrays are

304

STANLEY H. WARD

TABLEIV. Basis for Selecting p,/IP Arrays Time or frequency domain Decades of spectrum Signal-to-noise ratio Lateral and vertical resolution Depth of exploration Lateral effects Electromagnetic coupling

illustrated in Fig. 12. Of these, the Wenner array has largely been replaced by the Schlumberger array because the latter is least affected by near-surface inhomogeneities beneath the array (Kunetz, 1966). The dipole-dipole array has largely replaced the pole-dipole array in conductive environments (e.g., geothermal) because it exhibits less electromagnetic coupling. Table IV lists seven factors to consider when selecting an array for resistivity or induced-polarization surveys. Time-domain and frequencydomain operations are equivalent but equipment convenience may dictate one or the other. In either domain, one would prefer three decades of spectrum from about 0.1 to 100 Hz for IP surveys to permit determination of the polarization spectrum. Table V provides an evaluation of the last five factors of Table IV. Where 1 is entered in a box it indicates the preferred array ;where 3 is entered it indicates the least desired array, for that particular factor. Signal-to-noise ratio is superior for the Schlumberger array because the transmitting and receiving electrode pairs are nested. For the same reason, electromagneticcoupling is greatest, i.e., worst, for the Schlumberger array. Dipole-dipole techniques are always superior to other techniques for lateral resolution of two adjacent steeply dipping bodies. Vertical resolution of adjacent beds in a horizontally layered sequence depends on the range and density of measurements laterally ; the Schlumberger array is worst in this regard (Oldenburg, 1978). The depths of exploration of resistivity arrays are given by Roy and Apparao (1971)for Schlumberger as 0.125L and for dipole-dipole as 0.195L, where L is the maximum separation between extreme electrodes (AB for TABLEV. Resistivity Array Evaluation

Schlumberger Pole-dipole Dipole-dipole

Surface effects

S/N ratio

1 2 3

1 2 3

Lateral Vertical Depth of Lateral resolution resolution exploration effects 3 2 1

3

2 I

EM coupling

3 2

3 2

3 2

1

1

1

16.


305

Schlumberger). Thus Schlumberger uses 1.6 times the maximum electrode separation of the dipole-dipole method for the same depth of exploration ; this makes it more susceptible to the effects of inhomogeneities offset from the sounding, i.e., lateral effects. In view of the evaluation of Table V, it should not be surprising to find that (a) usually the Schlumberger array is used at many scattered sites within a large region where estimates of the thicknesses and resistivities of assumed horizontal layers are required, while (b) the dipole-dipole array is used on a regular grid of lines where the earth is assumed to be two- and threedimensionally inhomogeneous.

4.6.Interpretation Hohmann (1982) provided a current review of numerical modeling for all electrical geophysical methods. Forward and inverse techniques of interpreting resistivity and inducedpolarization data over one-dimensional (I-D) earths, i.e. plane-layered, are readily available. Representative references are La Compagnie Generale de Geophysique (1955, 1963), Mooney and Wetzel(1956), Zohdy (1965, 1975), Al’pin et al. (1966), Kunetz (1966), Koefoed (1968), Ghosh (1971), Inman et af. (1973), Inman (1975), Petrick et af. (1977), Rijo et af. (1977), Van Zijl (1977)’ Oldenburg (1978), and Coen and Yu (1981). While inversion of data for a two-dimensional (2-D) earth has been attempted (see, e.g., Pelton et al., 1978b; Tripp et al., 1984), forward modeling of resistivity data is customarily used in interpreting dipole-dipole resistivity and IP data. Pertinent references include Coggon (1971, 1973), Ward et af. (1973), Lee (1975)’ Snyder (1976), Rijo (1977), and Fox et af, (1980). Petrick et af. (1981) published a three-dimensional (3-D) inversion scheme for interpreting resistivity data. Based on the concept of a centers, the scheme provides information on the locations of conductive bodies. Three-dimensional forward solutions have been presented by Dieter et af. (1969), Hohmann (1975), Lee (1975), Dey and Morrison (1979), Lee et al. (1981), Pridmore et al. (1981), and Petrick (1983). The two- and three-dimensional forward interpretations are based on finite-difference, finite-element, transmission surface, integral-equation, or hybrid finite-elementlintegral-equationformulations. Hohmann (1982) has provided a review of all such methods. Quoting from Hohmann, “Differential equation (finite element and finite difference) and integral equation methods have been used. Differential equation (DE) solutions are easiest to implement, and they result in large banded matrices. Becausethe entire earth is

306

STANLEY H. WARD

TABLEVI. Problems with p,/IP Surveys Natural field noise Cultural noise Effect of overburden Effect of other geologic noise Effect of topography Resolution, lateral and vertical Electromagnetic coupling

modeled on a grid, DE methods are preferable for complex geology. Integral equation (IE) formulations involve more difficult mathematics, but the unknown fields only need to be found in anomalous regions. Thus IE solutions are less expensive for calculating the response of one or a few small bodies and hence are most useful for evaluating field techniques, for designing surveys, and for generating interpretation catalogues. Much recent research on 3D modeling has focused on hybrid methods, which attempt to combine the advantages of DE and IE solutions.” 4.7. Problems with Resistivity and Induced-PolarizationSurveys 4.7.1. Introduction. As with any geophysical method, applications of the resistivity and IP methods encounter problems which can be only partly overcome. Table VI lists the problems encountered when applying resistivity and/or induced-polarization surveys. Each of these problems will be addressed briefly in the following. 4.7.2. Natural Field Noise. Natural electric and magnetic fields below 1 Hz are due mainly to the interaction of fields and particles from the sun with the earth’s magnetic field; their magnitude depends on solar activity. Above 1 Hz they are primarily due to worldwide thunderstorms. As Fig. 17 shows, their amplitudeincreases rapidly with decreasing frequency below 1 Hz, which effectively prevents measurements below about 0.03 Hz. Since electromagnetic coupling is too high above 1 Hz, IP measurements with large arrays are limited to the range 0.03-1 Hz.Even in that range, coherent detection and digital high-pass filtering are required to make accurate measurements because of the natural field noise. Stacking, that is, adding successive transients, is necessary to reduce noise in time-domain measurements, but noise rejection is not as good as for coherent detection in the frequency domain. Commonly, the range of frequencies is extended to 100 Hz or higher in order to obtain spectra of complex conductivity, as will be described subsequently. 4.7.3. Cultural Noise. Table VII lists the sources of cultural noise. Grounded structures such as fences, power lines, and pipelines redistribute

16.

307


-

-

-

W

wJ

-

CAVITY

0.5

w

A I

0.0001

I

0.01

I

I

1.o

FREQUENCY

I00

I0 K

(Ha:)

FIG. 17. Generalized spectrum of natural magnetic fields. (After Campbell, 1967.)

current from a grounded wire source so that part of the current flows through the cultural feature. Spurious resistivity and induced-polarization anomalies arise as a result. In a definitive analysis of the problem, Nelson (1977) found that the only certain means of eliminating such spurious responses is to keep IP transmitting and receiving lines away from grounded structures. However, he did a commendable job in computing the response of a grounded structure for comparison with the resistivity phase measured over the structure (Fig. 18). Cultural features also can introduce noise into measurements by providing a path for various interfering signals. Of course, strong noise voltages are present in the vicinity of power lines, requiring filtering TABLE VII. Cultural Noise Passive Fences Pipelines Power lines Telephone lines Rails Active Power lines Telephone lines Electrified rails

308

STANLEY H. WARD

N 5

3

4

4 ,

1

2

3 3 1 1 5 .I 10 7 3 -1 6 0 - 3 1 0 -6 5 1 10 5

0 5

1 7

1

2 1

0

2

2

11

0

1 3

4

4

1 1

11 3

3

2

3

z

s s

3

87

4

3

-1 3 I1

2

7 4 1 5

0

11

500'dipoler

F ~ L TEST D N 5

4 3

3

t

2

3 3

3

2

o 8

1 7

2 1

3 3

4

5 5

3

3

0 9 3 8 0 3 3 - 1 1 0 3 3 1 0 . 1 3 3 -2 12 3 3 4 11 -1 3 -4 13 3 3 3 4 13 -3 1 5 3 3 3 3 4 1 4

3

3 4 1

COMPUTED MODEL IP

electrodes

N 5

4

3

2

2

3

4

5 5

FIG. 18. Phase lag in milliradiansdue to a power line and computed model using the grounded impedance measured on one of the power poles. The computed model half-space parameters were 50 O-m and 3 mrad. The grounding impedance were 100 O-m at 160 mrad, with 1 1 grounds in the calculation. One of the grounds is 5 m from thecenter IP electrode. (After Nelson, 1977.)

at the front end of the receiver. Furthermore, pipelines often carry electrical current for cathodic protection, and this current is a source of noise. 4.7.4. Overburden and Other Geologic Noise. Conductive overburden, generally in the form of porous alluvium or weathered bedrock, prevents current from penetrating to the more resistive bedrock. Hence detection of bedrock features is less certain than when overburden is absent. When the overburden is of irregular resistivity, as illustrated in Fig. 19, the geologic noise produced by the near-surface features readily obscures the anomaly due to the target in the bedrock. Anomalies due to geological heterogeneities of no geothermal significance can also obscure, or partly obscure, the anomaly due to a geothermal system. 4.7.5. Topography. Much geothermal exploration is done in mountainous terrains, where topography can produce spurious resistivity anomalies. Fox et af. (1980) systematically analyzed the effects of topography for the dipole-dipole array, using a two-dimensional numerical solution. Figure 20, for example, shows the apparent resistivity anomaly

16.

309


/ , /

Pa

I

I

I

I

FAULT

I

,

150

150

67

150

1

,

150 375 327 313 150 150 386 336

150 150 150 150 150 394 150 150 150 150 150 150 I

1

150

150

150

69

I

I

5015i50\4;;

150 150

I

I

I

I

I

,

I

,

344

I

I

+

FAULT SULFIDES 99

FAULT +

FIG.19. Resistivity pseudosections over an earth model consisting of a contact between rock types, a massive sulfide body at the contact, and an irregular overburden. (After Pridmore et al., 1981.)

310

STANLEY H. WARD

-

1

I

EARTH

p = I00

APPARENT RESISTIVITY -5

I

-4

-3

-q

-I

0

I

2

3

4

5

FIG.20. Apparent resistivityanomaly due to a two-dimensionalvalley with 30" slopes. (After Fox er ol., 1980.)

produced by a valley with 30" slopes. The pseudosection is characterized by a central zone of low'apparent resistivity flanked by zones of high apparent resistivity. The low is most pronounced when the transmitting and receiving dipoles are on extreme opposite sides of the valley. This example shows that a valley can produce a large, spurious resistivity low which could easily be misinterpreted as evidence for a buried conductor. Similarly, a hill can produce an apparent resistivity high. Because induced polarization is a normalized measurement, current focusing and dispersion produced by an irregular terrain surface do not significantly affect IP data. Thus if the earth were homogeneous and polarizable, irregular terrain would produce no significant spurious response. However, second-order topographic effects in IP surveys are introduced by variations in distances between surface electrodes and a polarizable body relative to a flat earth. In general, topographic effects are important where slope angles are 10" or more for slope lengths of one dipole or more. The solution to the

16.


311

problem is to include the topographic surface in numerical models used for interpretation. 4.7.6. Resolution, Lateral and Vertical. To facilitate resolution of the resistivities and thicknesses of horizontally layered media, a wide range and high spatial density of electrode separations are required. Even so, the principle of equivalence (Kunetz, 1966) indicates that substantial ambiguity exists in determining layer thicknesses and resistivities. Resistivity techniques usually provide information on resistivity-thickness products for resistive layers and conductivity-thickness products for conductive layers. The problem of vertical resolution is illustrated in Fig. 21. Superposition of resistivity or induced-polarization responses from two or more bodies frequently leads to misinterpretation. Figure 21 shows how the responses of two prisms superpose as they are moved closer together. Each prism is conductive (p2/p1 = 0.2), has dimensions of 1 width x 4 depth extent x 5 length, and occurs at depth 1 . These units are normalized by the dipole length. This case dramatically illustrates the need for sophisticated interpretation of resistivity and IP anomalies: a pseudosection should not be construed as a cross section of the earth. Drilling would be unsuccessful if a hole spotted over the IP high in the pseudosection in the two cases where the bodies are separated. Bull’s-eye pseudosection anomalies such as these often are caused by superposition. When the bodies join, their responses merge into that for a single wide body, as shown in the lower pseudosection of Fig. 21. 4.7.7. Electromagnetic Coupling. The resistivity and induced-polarization methods typically use dc formulation which requires that the transmitting and receiving wires be coupled only resistively. However, when ac is used, which is customary, electromagnetic coupling between the transmitting and receiving wires also occurs. This is readily seen in the expression for mutual coupling between a pair of grounded wires (Sunde, 1949):

in which Q(r) = 1/2nor

( 147)

is the resistive coupling term and

is the electomagnetic coupling term. In these relations r is the distance between the electrodes a, byA, and B which terminate the wires, 0 the angle

312

STANLEY H. WARD

I

15

15

-3

I

!I

I

FIG.21. Resolution of adjacent bodies. Induced-polarizationresponses due to two prisms are superposed; width, la; depth extent, 4a; length, Sa; depth, la; p2/,71,0.2. Dipole length is a. Anomaly contours are in EZ(To), which is the fraction of the intrinsic polarization of 100 given to the bodies. Therefore, EZ(To) can represent PFE, Mor 6.(After Hohmann and Ward, 1981 .)

between the wires, CT the conductivity of the half-spqce on which the wires are situated, k = (-iapw)’” the wave number of the half-space, p the permeability of the half-space, and w the angular frequency. The electromagnetic coupling between the wires increases with the frequency, the lengths ab and-AB, the separation between ab and AB, and the conductivity of the half-space. Electromagnetic coupling is particularly important in the induced-polarization method, where one is attempting to

16. ELECTRICAL METHODS

-IOJ

.01

I

1

IN GEOPHYSICAL PROSPECTING

I

I

fHz

'O

I

100

313

A

1000

FIG. 22. Phase spectra for various dipoles and spacings from an IP survey in conductive terrain, Northern Territory, Australia. (Data by Phoenix Geophysics Ltd.)

measure resistivity as a slowly varying function of frequency, the latter due to electrochemical reactions in the subsurface. As can be seen from the formulation above, electromagnetic coupling is also frequency-dependent and it can totally obscure the IP effects. Figure 22 illustrates how electromagnetic coupling increases with frequency. Extrapolation of the resistivity phase to zero frequency will eliminate the electromagnetic coupling and leave only the IP effect. Hence an IP survey should use several decades of spectrum to permit this phase extrapolation.

5. Magnetotelluric Method 5.1. Introduction

The magnetotelluric (MT) method has been used in geothermal, hydrocarbon, and crust/mantle exploration for about 30 years; it relies on measurement of three orthogonal components of natural magnetic fields and two horizontal orthogonal components of natural electric fields in the frequency band to 10 Hz (Tikhonov, 1950; Cagniard, 1953). The audiomagnetotelluric (AMT) method has been used in mining and geothermal

314

STANLEY H. WARD

exploration for about 10 years; it relies on measurements of the same components of magnetic and electric fields, but in the frequency range 10 to lo4 Hz (Strangway et af.,1973). While the two methods use different sets of equipment and rely on fields from fundamentally different sources, they are essentially the same method and will be so treated in this chapter to the extent possible. The abbreviation MT/AMT will be used throughout to refer to the combined method. A representative set of early references on the magnetotelluric method includes Tikhonov (1950), Cagniard (1953), Cantwell (1960), Bostick and Smith (1962), Wait (1962), Swift (1967), Sims et af. (1971), and Vozoff (1972). The paper by Vozoff (1972) has become the standard reference for a reasonably current description of the magnetotelluric method, especially as applied to oil and gas exploration. Gamble et af. (1979a, b) describe the use of a remote reference for eliminating bias errors in resistivity estimates obtained with MT data. Papers describingits application in geothermal areas include Hermance et af. (1975), Hermance and Pedersen (1977), Stanley et af. (1977), Goldstein et al. (1978, 1982), Morrison et al. (1979), Dupis et al. (1980), Gamble et af. (1980), Musmann et al. (1980), Ngoc (1980), Wannamaker et af. (1980, 1983), Aiken and Ander (1981), Berktold (1982), Berktold and Kemmerle (1982), Goldstein et al. (1982), Hutton et af.(1982), Martinez et al. (1982), Stanley (1982), and Wannamaker et al. (1983), among others. Pertinent references on the audiomagnetotelluric method include Keller (1970), Strangway and Vozoff (1970), Strangway et al. (1973), Dupis et af. (1974), Dupis and Iliceto (1974), Keller and Rapolla (1974), Hoover and Long (1975), Hoover et af. (1976, 1978), Isherwood and Mabey (1978), Jackson and O’Donnell(1980), Long and Kauffman (1980), and others. The article by Strangway et al. (1973) is usually taken as the point of departure for literature surveys of the AMT method.

5.2. Basis of t h e MT/AMT Method 5.2.1. Basic Principles. 5.2.1.1. SOURCESOF FIELDS.The MT/AMT method uses the earth’s natural electric and magnetic fields to infer the electrical resistivity of the subsurface. Figure 17 contains a generalized spectrum of natural magnetic field amplitude taken from Campbell (1967). There is, of course, a corresponding electric field spectrum, related through Maxwell’s equations. Fields above about 1 Hz are due to worldwide thunderstorms, the principal centers being in equatorial South America, Africa, and the southwest Pacific. Because the ionosphere is a plasma, i.e., a highly conducting medium, the energy propagates in a waveguide mode in the earth-ionosphere cavity. The resonances shown in Fig. 17 are due to constructive interference.

16.


315

Below 1 Hz the fields, called micropulsations, are mainly due to the interaction of the solar wind with the earth's magnetic field and ionosphere. As Fig. 17 shows, the amplitude of the electromagnetic field increases with decreasing frequency below 0.1 Hz. Important references on natural electromagnetic fields are Bleil(1964),Matsushita and Campbell (1967), and Jacobs (1970). These natural fields represent noise for controlled-source electromagnetic methods (CSEM), but they are the source fields for MT. Since low frequencies are needed for deep penetration, it is easy to see from Fig. 17 why MT has been used so extensively for crustal studies and deep exploration: the source fields increase at low frequencies for MT while the same fields constitute noise for CSEM, and hence noise increases as frequency is lowered in CSEM. Furthermore, CSEM sources undergo a strong geometric decay, which plane waves do not. Audiomagnetotelluric, which is simply MT in the audio frequency range 10 to 104 Hz, has the advantage that data can be collected much faster, but, of course, depth of exploration is less than for lower frequencies. Unfortunately, low source fields have hindered the application of AMT, especially in regions remote from the equator. One remedy has been to use an artificial source, usually a grounded wire carrying current, at a large distance from the survey area. This technique is called controlled-source audiomagnetotellurics (CSAMT). 5.2.1.2. FORMULATION FOR A ONE-DIMENSIONAL EARTH. The basic formulation for the MT/AMT method applied to a homogeneous earth is given in Eq. (151). Orthogonal electric and magnetic field pairs, [Ex.H,] or [Ey,H,], are measured at the surface of the earth. These quantities are simply related to the electromagnetic impedance Z of a plane wave. When displacement currents are neglected, which is justifiable for earth materials at the frequencies employed in MT/AMT surveys, the impedance may be computed from

Z = a p o / k = Ex/Hy = - Ey/Hx

(149)

Under these conditions Eq. (149) can be rewritten as

z

= o p o / G j i Z i i = J ; G p = =eiTl4

(150)

The impedance phase is 45", with Ex leading Hy by this amount. The resistivity of the half-space is then given as p = (l/aflo)(z(z = o.2TIEx/Hy12

(151)

where Ex is in millivolts per kilometer, Hy in nanoteslas, and the period T in seconds.

316

STANLEY H. WARD

air

FIG.23. Typical model, apparent resistivity, and impedance phase for a layered (I-D) earth. (After Ward and Wannamaker, 1983.)

When the earth is layered, as in Fig. 23, the plane wave impedance is given by the recursive formula developed earlier : ZI

ZZ+ Z1 tanh(iklh1)

=

z 1

=

~ n - 1

ZI + 2%tanh(ik1h l )

through Zn-1

2 + Zn-ltanh(ikn-lh,-l) Zn-1

+ Zntanh(ik,-Ihn-l)

(153)

in which Zi = Opo/ki is the intrinsic impedance of ith medium, the impedance at the top of the ith layer, and ki and hi are the wave number and thickness, respectively, of the ith layer. From Eq. (152) one can compute the impedance phase d, and the apparent resistivity pa via 2 1

= IZlle'+

(154)

Figure 23b shows schematically the appearance of the pa vs. f and d, vs. f curves for the three-layer earth depicted in Fig. 23a. Boehl et al. (1977) show that one can predict the phase from the apparent resistivity approximately by d, = 45"

+ 45"alnp,/alnw

(156)

16.


317

TE MODE E,, Y

TM MODE E,

z

Rotate to minimize Z,,

Zyy

FIG. 24. Model, modes of excitation, and formulation for a 2-D earth. (After Ward and Wannamaker, 1983.)

which may be observed in principle in Fig. 23b. Equation (156) is based on an assumption that the resistivity and impedance phase are related through a Hilbert transform for a 1-D earth (Kunetz, 1972). Our observations suggest, but do not prove, that the Hilbert transform relationship usually is obeyed by 2-D and 3-D earths (Wannamaker et al., 1982). FOR A Two DIMENSIONAL EARTH. 5.2.1.3. FORMULATION 5.2.1.3.1. Modes of Excitation. For a two-dimensional earth, i.e., one in which the resistivity in the strike direction differs from the resistivity in the orthogonal direction, the electric field in either of these principal directions may be related to the magnetic fields in both directions. Then a tensor relationship between electric and magnetic fields must be used, as illustrated in Fig. 24. The mode of excitation in which the electric vector is oriented parallel to strike, Ell, is referred to as the transverse electric (TE) mode, whereas when the electric vector is perpendicular to strike, E L , the transverse magnetic (TM) mode is excited. Usually both modes are excited simultaneously. The electric and magnetic fields for TE and TM mode excitation of a conductive 2-D body are shown schematically in Fig. 25. For the TE mode, where the electric field is parallel to the body, the anomalous normalized

P, FIG. 25. Illustrative behavior of electric and magnetic fields over a 2-D body in a homogeneous half-space. (After Ward and Wannamaker, 1983.)

318

STANLEY H. WARD

electric field E,”,/Egvaries from its normal value well off to the side of the body to low values over the body. The corresponding TE mode magnetic field H!‘/H$ reverses over the body, while H$/H$ is negative outside the body and positive over the body, as appropriate for a line source of current along the axis of the body. These secondary induced fields become vanishingly small as frequencies approach zero (Wannamaker et al., 1982). For the TM mode one observes in Fig. 25 that the normalized anomalous electric field E$/E: is positive outside the body and negative over it. This characteristic of the TM mode is indicative of dipolar fields, does not vanish as frequency falls, and requires some explanation. 5.2.1.3.2. Surface Chargeand Current Channeling. The explanation for this dipolar behavior lies in the existence of a surface charge density p , ,which we established in Section 2.6. For MT, the E-field response is predominantly dipolar, although higherorder multipoles may be important at higher frequencies (Stratton, 1941, pp. 563-573). The electric field at C in Fig. 26 due to this polarization is in the direction of the external electric field En,, while the electric field at D due to the polarization is in the opposite direction to En,. On addition of the incident field, this gives the appearance of currents in the external medium being deflected into the more conducting medium. This phenomenon is referred to as current channeling. If the two-dimensional body of Fig. 26 was more resistive than its surroundings, i.e., p2 > P I , then the currents would be deflected away from the body. Current channeling as opposed to local induction of eddy currents is illustrated i n Fig. 27. The two effects are superimposed when an electromagnetic field impinges on an earth in which a conductive inhomogeneity exists. Figure 28 contains plots of apparent resistivity and impedance phase versus frequency for points A and B of Fig. 25. The apparent resistivity and

I-

+I

*D FIG. 26. Illustration of surface charges at boundaries, due to an electric field excitation. (After Ward and Wannaker, 1983.)

16.


319

CURRENT CHANNELING

-L /

P2

P1

Dl INDUCTION

FIG.27. Illustration of concepts of current gathering and local induction.

“1‘ Pa PI

FIG.28. Typical behavior of TE and TM mode apparent resistivities and impedance phases at two points near a 2-Dbody in a homogeneous half-space. (After Ward and Wannamaker, 1983.)

320

STANLEY H. WARD

impedance phase for both points at the highest frequencies will be that for a homogeneous half-space of resistivity p1 since the skin depth

61 = 2/=

=

5 0 3 m

(157)

in the half-space is so small that little energy reaches the 2-D body. On the other hand, at the lowest frequencies the 2-D body will be transparent to the downward-traveling electromagnetic wave, since the TE mode does not involve surface charges and current gathering, so once again the apparent resistivity and impedance phase at both A and B will be that for a homogeneous half-space of resistivity p1 (Wannamaker et al., 1982). Between the low- and high-frequency extremes, the TE mode apparent resistivity, ~ T ,E drops below p1 at A and B since the 2-D body is of resistivity lower than pl and its effect is observed. The behavior of the impedance phase &E is then somewhat predictable from m~ if Eq. (156) is loosely applied. On the other hand, ~ T Mat point A starts at p1 at the highest frequency, where the waves have not penetrated to the 2-D body, but continues to decrease with decreasing frequency until it becomes asymptotic at some value dictated by the current channeling effect. Note that A is located in a region where the total electric field is lower than the incident field, as for point D of Fig. 26. Off to the side of the 2-D body, as at B of Fig. 25, the electric field due to the polarization charges adds to the incident field. Hence, the apparent resistivity, calculated from an expression of the form

0.2T1Ey/Hx12 (158) will increase with decreasing frequency until a low-frequency asymptote has been reached. Once again +TM roughly follows the gradient of ~ T M versus frequency. 5.2.1.3.3. The Impedance Tensor. We have seen that there are two basic modes of excitation, TE (Ell) and TM (EL), as illustrated in Fig. 24. In practice, we do not know the strike or x direction a priori, so our field data are taken in rotated directions which may be at any angle to x and y . Hence we need some means of rotating field data into TE and TM modes. If x is the strike direction, we write ~ T = M

ZTE= EX/Hy = Z ,

(159)

ZTM= - Ey/Hx = Z y x When the fields are aligned parallel and perpendicular to strike, the impedance tensor given in Fig. 24 becomes

i.e., Z,, = Zyy = 0.

16.

fi


32 1

W

x'

x

0 LA c ri

Y' O

L

Y

PLAN VIEW

1

-W

Ftc. 29. Plan view of measuring axes (x, y ) and symmetry axes (x', y ' ) for an MT survey over a 2-D body in a homogeneous half-space. (After Ward and Wannamaker, 1983.)

Now let us turn to the field case where the electrical strike direction is unknown. Then measurements are made in the rotated xy coordinate system of Fig. 29. The x'y' coordinate system is aligned with the strike, so we have

E: = ZTEH; E; = - ZTMHi

(161) (162)

In the xy coordinate system rotated by angle 8,

+ E; sin 8 E,, = -E:sinB + E;cosB Hx = H: cos 8 + H;sin 8 Hy = - H i sin 8 + H; cos 8 Ex = E: cos 8

(1 66)

Substituting Eqs. (161) and (162) in Eq. (163), we obtain Ex = ZTEH; cos 8

- ZTMH: sin 8

(167)

We can write for the reverse coordinate transformation,

H: = Hx cos 8 - Hysin 8

(168)

+ Hy cos 8

(169)

H; = Hx sin 8

322

STANLEY H. WARD

When Eqs. (168) and (169) are substituted in Eq. (167) there results

Ex = Z T E ( Hsin ~ 8 + Hycos 8) cos 8 - Z T M ( Hcos ~ 8 - Nysin 8) sin 8 = H ~ ( Z TE ZTM)sin 8 cos 8

+ H'(ZTE cos28 + ZTMsin28)

where use has been made of the trigonometric identities 2 sin 8 cos 8 = sin 28

2 s i n ' ~= 1 - ~ 0 ~ 2 8 2 C O S ~e = 1

+ cos 20

Hence, if we write in the rotated coordinate system

Ex = ZxTHx

+ Zxy Hy

then by comparing Eq. (170) with Eq. (171) we get Z, = ~ ( Z T-E ZTM) sin 28 and

z,

=

WTE + ZTM)+ )(zTE - ZTM)cos 28

(173)

Similarly,

Zyx = - +(ZTE+ ZTM)+ &ZTE - ZTM)cos 28

(174)

Zyy= ~ ( Z T M ZTE) sin 28

(175)

and

The important conclusion to be drawn from Eqs. (172)-(175) is that the impedance elements obtained in the field coordinate system are complicated combinations of TE and TM mode impedance elements. From Eqs. (172) and (175) we find that zxx =

-zyy

(176)

16.


323

while from Eqs. (172) and (173)

- z,,

= ZTM+ ZTE

( 177)

Both Eqs. (176) and (177) are indepedent of the angle 0. If the earth is 2-D, then one measures orthogonal electric and magnetic fields in the arbitrary field coordinate system and calculates the impedances ZA, Z& , Zjx, and Z;,. From these data one attempts to find a direction in which Zxx= Z,, = 0 and the resulting 2, = ZTEand Z,, = - ZTM. The direction at which this occurs, for clockwise rotation, are given by Vozoff (1972) as tan(400) =

(ZA

- Zj,)(Z& + ZjJ + (ZA + Zj,)(Z& Iz:, - ZjY(2- \Z&+ zj#

- Zjd

(178)

Such principal directions occur every go", so the strike direction cannot be distinguished from the dip axis by using the impedance alone. This ambiguity is removed by using the tipper T, defined as

where

Hz = AH,

+ BH,

( 180)

For the TM ( E l ) mode, no Hzresults, so the direction x which results in A decreasing to zero is the strike direction. 5.2.1.4. FORMULATION FOR A THREE-DIMENSIONAL EARTH. Figure 30 illustrates that for an equidimensional3-D object, mode identification is no longer possible, but for an elongate 3-D object it is possible. All components of the secondary field are induced by any orientation of the incident field. Furthermore, current channeling takes place for any orientation of the incident field. x'

x

PLAN VIEW

E, FIG.30. Plan view of measuring axes (x, y ) and symmetry axes (X', y ' ) for an MT survey over a 3-Dbody in a homogeneous half-space. (After Ward and Wannamaker, 1983.)

324

STANLEY H. WARD

Sims and Bostick (1969) showed that the usual impedance tensor of Fig. 24 is valid for 3-D models. In Fig. 30 we illustrate the 3-D body, the measuring axes xy, and the symmetry axes x’y’. The electric and magnetic fields are related not by Eqs. (161) and (162), but by the following equations:

E: = ZiXHi

E;

=

Z;,H;

+ Z& H; + Z;:,H;

(181) (182)

in which the impedance elements Zij are functions of 8, as are the fields. The electric field in the x direction is

E, = E: cos 8 + E; sin 8

(183)

When Eqs. (181) and (182) are substituted in Eq. (183), there results

+ (Z,!,H: + ZjYH;)sin 8 = (ZX cos 8 + Z;, sin 8)H: + (Z& cos 8 + Z;, sin 8)H;

Ex = (ZiXH:+ Z&H;) cos 8

(184)

Then we substitutelliand Hi according to Eqs. (168) and (169), respectively, to obtain

+ Z;, sin @(H,cos 8 - H, sin 8) + (Z& cos 8 + ZiY sin B)(H, sin 8 + Hy cos 8)

Ex = (2% cos 8

( 185)

When rearranged, Eq. (185) yields

E,

+ Z;, sin’ 8 + (Z;, + Z&) sin B cos 8]Hx + [Z& cos28 - Z;, sin28 + (ZiY - ZiX)sin 8 cos 8]Hy

= [ZiXcos2 8

(186)

In the field coordinate system xy we expect to observe that Ex = ZXxH,+ Z,Hy

(187)

so we identify Z,, and Zyy as follows:

zX,= ziX60s’ 8 + z;, sin’ 8 + (z&+ z;,)sin 8 cos e 2 , = Z& cos’ B - ZiXsin’ 8

+ (ZiY - 22,)

sin 8 cos 8

(188) (1 89)

By using the same trigonometric identities as for the 2-D case, Eqs. (1 88) and (1 89) reduce to Zxx= 21 + 2 , = 24

cos 28 + 2 3 sin 28

(190)

+ Z3 cos 28 - ZZ sin 28

(191)

2 2

16.


325

Similarly, we find

Zyx =

- Z4 + Z3 cos 26

-

Zyy = Z1 - Z2 cos 26 -

2 2

sin 26

(192)

2 3 sin 26

(193) In Eqs. (190)-(193) we have shortened the notation by using the following definitions

+ Z;,)/2 2 3 = (Z& + Z;J2 z1

= (2%

2 2

= (Z& - Z;,)/2

(194)

24

=

(Z& - Z,',)/2

(195)

From Eqs. (190) and (193) we observe that

+ 2yy)/2

(ZXX

=

z1

=

z 4

( 196)

while from Eqs. (191) and (192) (2, - 2,)/2

(197) Comparison of Eqs. (194) and (196) and Eqs. (195) and (197) shows that Z1 and 2 4 are invariant under rotation. In the 2-D case, Eq. (176), we found that Z I = 0. Hence the skewness S has been introduced as a measure of threedimensionality. S is defined by

s = 1z11/1z41

= JZ:, -

2)!yl/p&- &I,!

(198) If S is large, three-dimensionality is indicated. If S is small, it is not easy to deduce whether the earth is 2-D or 3-D. It is clear from Eq. (196) that the elements Zxxand Zyyof the impedance tensor do not become zero in the presence of a 3-D body except along any axis of symmetry. However, for a 3-D body, principal axes generally may be defined where Z,, and Zyy are minimized. Hence it is customary to estimate an approximate strike direction 8 0 and to estimate the principal impedances Z , = ZTEand Zy, = ZTMin such principal directions. Several methods have been used to find the angle 60 between the measuring axes and the principal axes. For example, one can maximize 1 2 , 1 2 + JZyx)2, minimize lZxx)2+ maximize I z , ~ or \ Z ~ , I , minimize 1 ~ or lzYy\, ~ ~ maximize 1 1 2 , + zYx), and so on. Each procedure will give the strike direction if the earth is twodimensional. When the earth is three-dimensional these methods do not give the same results. The most common method used is that of maximizing the absolute value of the sum of the off-diagonal elements, i.e., maximizing lZ, + ZYJ. This is done analytically (Swift, 1967; Sims and Bostick, 1969). As for the case of a 2-D structure, principal directions of the impedance occur every 90". Defining TE and TM modes requires that this 90" ambiguity be removed. This can be accomplished precisely for a 2-D structure by using tipper strike, since Hz is correlated with the horizontal magnetic field perpendicular to the strike. A unique tipper strike can be defined for 3-D

Iz~~~~,

326

STANLEY H. WARD

bodies as well, with the principal impedance closest to this strike being assigned to the TE mode. Principal apparent resistivities are p& = O . ~ T ) Z & ( O O ) ) ~(TE mode)

(199)

O.~T)Z,L(OO)(~(TM mode)

(200)

=

with impedance phase derived directly from Z& and Z,!x. Tipper strike has the additional advantage that it is relatively insensitive to near-surface geological noise (Wannamaker et al., 1980, 1982). However, this procedure does not necessarily allow 2-D algorithms to be applied routinely to principal apparent resistivities and impedance phases gathered over 3-D structures. The basic behavior of apparent resistivities pxu and pyx and impedance phases and +yx at points A and B over the 3-D body in a half-space, where the x and y coordinates of the MT quantities pertain to those drawn in Fig. 31, has been plotted in Fig. 32. At point A, note that pxu and pyx fall as frequency falls until they become asymptotic to a low-frequency limit somewhere below p1. Regardless of the orientation of the inducing electric field, some boundary polarization charge will exist. This charge creates qualitatively a dipolar electric field anomaly over the body, which resembles that of the TM mode of a 2-D body. At low frequencies current gathering is by far the dominant factor in determining both pxr and pyx. The character of the impedance phases +xuand 4yxat point A is complementary to that of pxuand pyx,although departures from the 2-D responses and +yx at all but the highest frequencies again occur. For the 3-D body, have values exceeding 45". Eventually, at low frequencies, 4- and 4yxwill become asymptotic to 45", but will never drop below it. Note that qualitatively the apparent resistivities and impedance phases obey Hilbert transform

Y

p2

FIG.31. Plan map showing location of measuring points A and B over a 3-D body in a homogeneous half-space. (After Ward and Wannamaker, 1983.)

16.


A

327

t

I I f f

FIG. 32. Typical apparent rehistivity and impedance phase curves for points A and B of Fig. 31. (After Ward and Wannamaker, 1983.)

relations as observed over 2-D structures. Again, I have no proof of this relation for 3-D bodies, but it has been my experience with 3-D computer simulations and the vast majority of field observations that in general this transform relation seems to hold. In the lower part of Fig. 32, the sounding curves correspond to point B outside the 3-D prism. Apparent resistivity pv and impedance phase c$xu here resemble those at point A, except that their variations are relatively subdued.

5.3.Data Acquisition Vozoff (1972) provides a useful overview of MT data acquisition. Sternberg et al. (1982) present an updated and more detailed description of the subject. Figure 33 shows a schematic representation of the disposition of E and H sensors for MT soundings. Because of the steeply varying nature of the spectrum of Fig. 17, data are collected in a number of bands from loe4 to 103Hz. Figure 34 displays typical data from the midband extending nominally from 0.05 to 5 Hz. The E fields are detected between orthogonal sets of nonpolarizing electrodes. The electrodes are connected by 50-300-m wires to electric field preamplifiers in the recording truck. While Vozoff (1972) advocated use of large distances (2600 m) between electrodes, Wannamaker (1983) advocates electrode separations as short as possible, consistent with adequate signal. Modern E field preamplifiers are of sufficiently low internal noise that

328

STANLEY H. WARD

E t

FIG.33. Magnetotelluric sensor deployment in the field. (After Sternberg et al., 1982.)

shorter spacings are possible. One seeks to avoid placing electrodes of a pair on opposite sides of a surficial resistivity change; the shorter the wire, the less likely the electrodes will be on opposite sides. We will refer to this matter again later. The electrodes are either Cd-CdCl2 , Pb-PbClt , or Cu-CuSO4 nonpolarizing type. The former are thought to have slightly lower noise, i.e., chemical drift, but CdClz is highly toxic. The H fields are detected with induction coils or Squids (cryogenic magnetometers). Most modern MT surveys employ two complete MT stations so that the E or H fields from one may be used as a remote reference for the other. Gamble et al. (1979a,b) demonstrated that bias in estimates of impedances derived from MT measurements may be reduced by use of a remote reference. Stodt (1983) made a comprehensive review of bias and tandom errors in MT surveys and demonstrated that bias removal can be effected, under certain conditions, without resort to remote reference. The use of a remote reference is, nevertheless to be preferred. Figure 35 shows the system used for MT research by Conoco, Inc. According to Sternberg et al. (1982): The system consists of two data acquisition (or DA) vehicles, each being equipped with the necessary electronicsto record three components

16.


329

SCALE IN Seconds

FIG.34. High-quality MT data low-pass filtered at 5 Hz. Five field components are illustrated.

Data Acquisition (DA) System

Data Acquisition (DA) System

330

STANLEY H. WARD

of the magnetic field with either squid magnetometers or induction coils along with four electric field sensors to record two pairs of orthogonal electric fields. The extra satellite electric field sensor is typically deployed at a distance of a few hundred to a few thousand feet from the main-base electric field sensor. The two DA vehicles may be separated by any distance but are generally 1 to 10miles apart. Time synchronization between the two vehicles is obtained by signals from WWVB. A signal analysis (SA) truck is located in the vicinity of both DA trucks and is used to process the MT data. Magnetic tapes are transported from the DA trucks to the SA truck. We have found that the use of telemetry links to transmit the data from one vehicle to another can severely restrict one’s flexibility in choosing site locations. Furthermore, transportation of the tapes usually leads to an insignificant delay in comparison with the site occupation time, particularly for a research system. The SA truck is capable of performing all of the MT processing, including remote reference processing, modeling and generation of final resistivity cross sections. Figure 36 presents a block diagram overview of a system designed by the Earth Science Laboratory of the University of Utah Research Institute to cope with the wide dynamic range of signal levels encountered in MT/AMT (Stodt, 1983). Output from the electric field sensors is fed into a signal conditioning and line drive box located at the sensors. Line drivers send preamplified and conditioned signals to the recording truck, approximately MEASUREMENT SITE -+-w

I N S T R U M E N T TRUCK

\

lOOm--d

f FIG. 36. Earth Science Laboratory MT system overview. (After Stodt, 1983.)

16.

CA 1I B RO AR 110N


331

L F E A T U R E6

* PROGRAMMABLE

G A I N - O d b , 21.6 d b

OPTIONS

* HIGH

PASS

F E A T U RE6

*PROGRAMMABLE, GAIN -0db,10.3 d b , 3 l d b D I F F E R E N T I A L OUTPUT

* NOTCH * * LOW N O I S E 6 0 H z , 180 H r *RFI FILTER * TRANSIENT SUPPRESSION * L O W T H E R M A L DRIFT

FIG. 37. The field and coil H field receivers of the Earth Science Laboratory MT system. (After Stodt, 1983.)

100 m from the sensors. At the recording truck, the incoming signals are presented to a line receiver and fed to four-pole low-pass filters with programmable cutoffs for antialiasing protection. Optional programmable gain and high-pass filter stages are also available at this point. The signal on each channel is then fed to a separate sample and hold amplifier, controlled by a programmable time base. Voltages are then digitized with a minimum of 12 bits of resolution and stored for processing. Figure 37 presents an expanded block diagram of the electric field and coil receivers of Fig. 36. The electric field measurements are processed in the following stages. They are presented to differential preamplifiers with radio frequency interference (RFI) filter and transient suppression, then to optional high-pass and 60-1 80-Hz notch filters, and finally to programmable gain differential output amplifiers with programmable offset. The coil magnetic field signals are presented to a differential preamplifier with RFI filter and transient suppression, and then to optional high-pass and 60180-Hz notch filters. Line drivers send the conditioned signals to the recording truck. The Squid magnetic field signals are sent directly to the truck. Gain in the system is introduced as early as possible to avoid contaminating the measurements wfth instrument noise. The purpose of the optional highpass and notch filters and the programmable offset in the electric field channels is to tailor the signal so the gain can be turned up without incurring saturations by energy at frequencies which are not of interest. Because of the steeply varying nature of the spectra as a function of frequency and the nonstationary character of MT signals, dynamic range in the analog-to-digital (A/D) conversion is a problem which requires very careful consideration. Usually 14-16-bit AID conversion is required for each

332

STANLEY H. WARD

of several bands over the range 10-3-100 Hz of frequencies to be recorded. A preemphasis filter is also used to whiten the spectrum and thus reduce the dynamic range requirements.

5.4.Data Processing Stodt (1983) presents a particularly clean approach to MT data processing, from which I quote in part : Magnetotelluric (MT) data are obtained as sets of simultaneous measurements of orthogonal electric (Ex,Ey) and magnetic field (Hx,Hy ,Hz)components at a given site on the earth’s surface. The data sets are Fourier transformed and used to calculate complex transfer functions which relate the field components to each other in the frequency domain at the air-earth interface. When the usual assumptions concerning the plane-wave nature of the source fields are satisfied (see, e.g., Madden and Nelson, 1964; Swift, 1967), the signal components (subscript s) of the measure fields are related to each other in the following manner : ESi = ZixHsxZiyHsy

i = x or y

(201)

and Hsz =

GxHsx + &Hsy

(202)

The tensor impedances, Zjx and Ziy ,and the tipper functions, Tu and Tv , are functions of frequency and conductivity structure. Equations (201) and (202) can be written in the general form Osi = GixZsx

+ GjyIsy

i = X , y, or z

(203)

where, from the viewpoint of linear system theory, Gix and Gi, are transfer functions of a dual input, single output linear system through which the inputs, Isxand Zsy, are related deterministically to the output, Osi. The goal of MT is to deduce the conductivity structure of the earth from the frequency dependent behavior of the impedance and tipper functions. Generally, MT field measurements consist of signal components of variable amplitude, contaminated by noise. Noise can be defined in general terms as any components of the processed field measurements which do not satisfy the plane-wave impedance relationships given by equations (201) and (202). This general definition includes systematic errors in addition to additive random noise components. Systematic errors are caused by deviations from the assumed model, e.g., errors

16.


333

due to sources which are not plane waves, cultural noise, and analogue or digital processing errors from instrument drift, aliasing, or truncation effects. It is important to distinguish between systematic errors and random noise when developing estimation procedures and error analysis for the impedance and tipper functions. Impedances and tippers are usually calculated as unweighted leastsquares estimates. We distinguish between conventional and remote reference impedance and tipper estimates. Conventional estimates are calculated entirely from field measurements obtained at a single base site (see, e.g., Sims et al., 1971). Two of the horizontal field measurements are used as references with equations (201) or (202) to compute the estimates, In contrast, remote reference estimates (Goubau et al., 1978; Gamble et al., 1979a,b) are computed by introducing two reference fields which are measured at a separate location. This is done to avoid correlations between the noises in the base and reference field measurements which introduce bias errors into the estimates. Details of the derivation of the MT transfer function calculations are given in a number of references, including Swift (1967), Sims and Bostick (1969), Word et al. (1970), and Vozoff (1972). A brief summary of the salient points is presented here. The impedances Zxx,Z,, and Zyy are complex and, as noted earlier, are given by

Ex = ZxxHx+ Z,Hy Ey = ZyxHx

+ ZyyHy

(204) (205)

The problem is to solve for the Zij. Since there are more observed field quantities than unknowns, this information can be used by multiplying Eq. (204) and (205) by the complex conjugate of each of the fields,

(ExA*( = Zxx(HxA*)

+ ZV(HyA*)

+ Zxy(HyB*) (EyA*> = Zyx(HxA*) + Zyy(HyA*) (EyB*> = Zyx(HxB*) + zyy(HyB*) (ExB*) = Zxx(HxB*)

(206) (207)

(208) (209)

where A* and B* are the complex conjugates of any two of H x ,Hy ,E x ,and Ey and the angle brackets denote frequency band averages. This yields more possible equations that unknowns. Since each of the solutions to these equations responds differently to noise on any one of the

334

STANLEY H. WARD

field components it is customary to discard some of the solutions and average others to obtain the best estimate. Solving for the Zjj (see, e.g., Vozoff, 1972) gives zxx =

(ExA* )(HyB*) - (ExB* )(HyA*) (HxA*)(HyB*) - (HxB* XHyA * )

(2 10)

z,

=

(ExA*)(HxB*) - (ExB*)(HxA*) (HyA* >(HxB*)- (HyB* >(HxA*)

(211)

zyx

(EyA* )(HyB* ) - (EyB* )(&A* ) = (HxA*)(HyB*)- (HxB*)(HyA*)

(2 12)

(EyA*)(HxB*) - (ByB*)(HxA*) - (HyA*)(HxB*) - (HyB*)(HxA*)

(213)

z yy

where A* and B* are the complex conjugates of any two of H x ,Hy , E x, and Ey .Any quantity such as (E,A*) is the cross-power of Ey and A* calculated from

's

wi+A0/2

<EyA*)(od=

wl-A0/2

EyA* d o

(214)

These are the Zij estimated in the conventional way, For remote reference estimates of the Zu Eqs. (204) and (205) are multiplied by the magnetic fields (Hxr and Hyr) at a distant site. Then

-- --

(Ey H,*,HxH$ H.ff.xH$) zyy= (HxH.ZHyHy*r -- Ey - HxHfiHy H.Z)

(218)

where the overbar denotes an average over a frequency window as well as over all data sets. Equations (215)-(218) involve only cross-powers between the base and the remote stations. If the noise at the base station is not correlated with the noise at the remote reference station and if a sufficient number of data sets are averaged, these impedance estimates will be unbiased by noise. Furthermore, since Eqs. (204) and (205) were multiplied in turn by a single reference field,

16.


335

the values of the impedance elements are independent of the magnitudes and phases of the reference fields and of the resistivity structure at the reference site. Further details on the remote reference processing method are given in Goubau et a/. (1978), Gamble et a/. (1979a, b), and Clarke et al. (1983). 5.5. Data Interpretation

For 1-D earths, inversion is used to obtain the thickness and resistivity of each layer (Wu, 1968; Nabetani and Rankin, 1969; Patrick and Bostick, 1969; Jupp and Vozoff, 1975; Patella, 1976; Bostick, 1977; Petrick et a/., 10

&I

= 300

A

= 100 = 39

= 19

10

/

-

9

-

4

* 7/3 * 3/2

= 1/9

10-

= 1/19 I

10-

\

1/20

= 0

FIG.38. Two-layercurves for the magnetotelluric apparent resistivity. (After Patella, 1976.)

336

STANLEY H. WARD

1977; Oldenburg, 1979; Larsen, 1981; Parker and Whaler, 1981; and others). From inversion one expects to obtain estimates of the parameters of the earth model plus one or more estimates of uncertainties in the parameter estimates. This method has almost entirely replaced the former curvematching system in which a field curve was matched with one or more members of a catalog of curves computed from forward modeling of a layered earth. Catalogs of forward-computed curves appear in Cagniard (1953), Yungul(1961), Srivastava (1967), and Patella (1976). In recent years it has become increasingly apparent that the earth is seldom plane-layered, so numerical algorithms for two- and three-dimensional earths are becoming OBSERVED APPARENT

W

RESISTIVITY

E

STAT ION

OBSERVED IMPEDANCE PHASE I

W

B-B'

1

I

STATION

I

O

I

1 1

8-B' I

I

E

FIG.39. Observed apparent resistivity and impedance phase pseudosections for profile B-B'. Contours of pyxare in ohm-meters while those of & are in degrees. (After Wannamaker et af., 1980.)

16.

ELECTRICAL METHODS IN GEOPHYSICALPROSPECTING

337

necessary in the interpretation of MT/AMT data (Ranganayaki and Madden, 1980; Wannamaker et al., 1980, 1982). Figure 38 shows apparent resistivity curves over a two-layer earth. Two-dimensional models of the earth which have been reported in the literature include the vertical contact (Neves, 1957 ; Patrick and Bostick, 1969; Vozoff, 1972; Morrison et al., 1979), a vertical dipping dike (Vozoff, 1972), a 2-D prism (Patrick and Bostick, 1969), a deep valley fill (Ward et a/. , 1973), and the general two-dimensional earth (Pascoe and Jones, 1972 ; Rijo, 1977; Stodt, 1978; Morrison et al., 1979; Wannamaker et al., 1980; Nutter, 1981; Jiracek et al., 1982). The MT effects of two-dimensional topographic features have been studied by Ku et al. (1973) and Ngoc (1980). MODELED APPARENT RESISTIVITY

W

E

STATION

MODELED IMPEDANCE PHASE

m

F W

5-6'

L

E

.

mr

m

o

e*

8 0

Lez:s,-:kr

STATION

5-5'

f

p

2 E

FIG.40. Computed apparent resistivityand impedancephase pseudosectionsfor model finiteelement section for profile B-B'. (After Wannamaker eta/., 1980.)

338

STANLEY H. WARD

I Lrn

1400

-

0

5-rn

3000

FIG,41. Best 2-D TM finite-element section fitting the observations for profile 9-9' of Fig. 24. Values for individual media are in ohm-meters. Vertical exaggeration is 6 :I . (After Wannamaker et al., 1980.)

All of these algorithms compute the MT/AMT responses for both TE and TM modes of excitation, Two-dimensional MT inversion has been discussed by Jupp and Vozoff (1977). Observed apparent resistivity and impedance phase TM mode pseudosections are shown in Fig. 39 (Wannamaker et af., 1980). Modeled apparent resistivity and impedance phase are shown in Fig. 40; a 2-D finite-element algorithm was used in the computation. The resulting model of the subsurface is shown in Fig. 41. Means for calculating the MT responses of 3-D earths have been reported by Jones and Vozoff (1978), Ting and Hohmann (1981), Wannamaker and Hohmann (1 982), Wannamaker et al. (1 982), Wannamaker (1983), Park et af. (1983), and others. Most MT data have been interpreted by using 1-D earth models at each site along a profile of stations. The resulting interpretation is a 2-D cross section of the earth (see, e.g., Stanley et al., 1977). Wannamaker et af. (1980, 1982) demonstrate that this approach can produce grossly misleading earth models. For 3-D environments with strong preferred orientations, 2-D TM mode modeling is preferred; TE algorithms are of limited use due to current gathering (Wannamaker et al., 1982). Otherwise, full 3-D interpretation is required. Wannamaker etal. (1982), Newman etal. (1983), and Wannamaker (1983) also demonstrate the importance of layering in which 3-D bodies are situated. 5.6. Problems with the MT/AMT Method 5.6.1. Overview. A number of problems make it difficult to acquire MT data of high quality. If high-quality data are gathered, however, a new set


339

of problems arises in interpreting the data. These problems are sketched in the next few paragraphs. 5.6.2.Source Dimensions. In the formulation presented above it is assumed that the MT fields are propagated as plane waves. This assumption was the source of much controversy in the early days of MT, but Madden and Nelson (1 964)showed that the field is usually a plane wave at frequencies greater than Hz in mid-latitudes. At frequencies below 1 Hz, the primary concern appears to be whether the fields due to equatorial and auroral electrojet ring currents in the E layer of the ionosphere can be treated as planar. Hermance and Peltier (1970)and Peltier and Hermance (1971)studied the effects of such ring currents. They concluded that in conductive environments, the plane wave assumption is valid in the frequency range to 1 Hz. However, significant errors can occur at frequencies less than lo-’Hz in areas where high resistivities are encountered if measurements are made within 500 km of the position vertically beneath the electrojet. At frequencies above 1 Hz, the proximity of lightning discharges becomes important. Bannister (1969)studied the fields radiated from a vertical electric dipole over a homogeneous earth and concluded that the plane wave assumption is valid for distances greater than seven skin depths from the source. Dmitriev and Berdichevsky (1979)showed that the plane wave formulation is applicable to a layered earth even if the fields vary linearly in the horizontal plane. However, if the plane wave assumption is not valid, the extra field components associated with nonplanar waves will be processed so as to produce bias in MT estimates (Stodt, 1983). 5.6.3.Random Noise. Random noise may arise in (a) the electrodes for E field measurement via chemical disequilibrium, (b) movement of the E field wires in the earth’s magnetic field when wind agitates them, (c) movement of the H field sensors in the earth’s magnetic field due to wind or seismic activity, (d) microphonics in the H field sensors due to any motion, (e) thermal noise in the E and H field preamplifiers, ( f ) quantization noise in A/D converters, (8) nonlinear behavior of the total recording system, (h) sporadic departure from plane wave propagation, and (i) sporadic cultural noise due to power lines, telephone lines, rail electrification, pipeline corrosion protection, radio interference, and the power sources in the recording instrumentation. Obviously, one attempts to minimize these noise sources prior to recording and processing. Ultimately, the processing system must be designed to minimize, evaluate, and place statistical limits on errors introduced into MT transfer functions by random noise. 5.6.4. Systematic Noise. Most of the noise sources described above are also capable of introducing systematic noise into estimates of the MT transfer

340

STANLEY H. WARD

functions. As Stodt (1983) points out, the systematic noise must be treated independently of the random noise in any statistical evaluation of noise in MT data. Systematic noise leads to biased estimates of the MT transfer functions. To attempt to eliminate this problem, the use of a remote reference has become common practice (Gamble et al., 1979a,b). Stodt (1983) demonstrates that to some extent this bias can be removed from conventional data, i.e., data recorded without a remote reference. Nevertheless, use of a remote reference is recommended. 5.6.5. Geological Noise due to Overburden. In areas where there is an irregular conductive overburden, current channeling into a patch of deeper or more conductive overburden will produce anomalies even to the lowest frequencies. Unless these anomalies are interpreted via 2-D or 3-D modeling, they can be mistaken for deep-seated features. Wannamaker (1983) illustrates these effects. 5.6.6. Resolution. In MT surveys, resolution of layers in a flatly dipping layered structure is usually of more concern than resolution of adjacent steeply dipping bodies. For example, we rely on active source systems for delineating fractures and faults in the shallower parts of geothermal systems, while we rely on MT for detecting the more diffuse heat sources at depth. Madden (197 1) demonstrated the principle of equivalence in MT soundings. For simplicity, I will analyze the MT response of a two-layer earth, although the analysis is readily extended to more layers. If layer 1 is electromagnetically thin-i.e., wavelengths in the layer are much greater than the thickness of the layer (kl hl e 1)-then tanh(ik1hl) = i k l h ~and Eq. (153) becomes

21 = z2

1 + ik2h1 1 + Z2crlhl

where the definition of the intrinsic impedance has been used. If layer 1 is conductive, cr1 s 02, then Eq. (219) reduces to Z2/(1 + Z 2 ~ l h l ) (220) The effect of a thin conductive layer on 21 arises only from its conductivitythickness product ; neither cr1 nor hl can be resolved independently. On the other hand, if layer 1 is resistive, 01 e a2, then Eq. (219) reduces to 2 1

21 = Z2(l

+ k2hl)

(221)

Thus the effect of a thin resistive layer on 21 arises solely from its thickness. These conclusions can be shown to apply to any layer within an arbitrarily layered sequence. In the general case, a buried layer 1 appears thin if krhl 4 1 throughout the frequency range for which EM waves are able to penetrate from the surface to the buried layer.

16.


34 1

5.6.7. Topography. The effect of topography on the results of an MT survey may be significant. Figure 42 illustrates distortion of telluric currents at low frequencies. Anomalous secondary electric and magnetic fields result. The TE and TM mode apparent resistivities due to a ridge of two different resistivities at two different frequencies are illustrated in Fig. 43 (Ku et al., 1973; Ngoc, 1980). A valley causes the opposite effect on the electric field. Also, as is the case for buried structures, the TM response of 2-D topography becomes asymptotic to a nonzero low-frequency limit where boundary charges on the surface of the earth act as sources for current variations below. Note also that the TM mode topographic effects are much stronger than those of the TE mode, especially near corners of the structure. The magnitude of the responses in this case indicates that one should give careful consideration to the effects of topography if the breaks in slope in the survey area are as steep and abrupt as those studied here. 5.6.8. Current Channeling. Current channeling is merely part of the response of a 2-D or 3-D earth. One-dimensional earth models do not include it at all, while 2-D earth models include it only for the TM mode. Thus current channeling is a problem only if it ignored in the modeling process. Ting and Hohmann (1981) provide a particularly good illustration of the effects of current channeling, and we repeat it here. Theoretical results for a three-layer model are compared with those for horizontal 3-D square slabs in place of the middle layer. The l-D model consists of an anomalous layer with resistivity 5 bZ-m. For comparison with 3-D models, we replace the infinite anomalous layer by a finite square slab having different lateral extents. The apparent resistivity is calculated over the center of the slabs and plotted as a function of frequency. The comparison is shown in Fig. 44 for square slabs 400, 800, 1200, and 1600 m on a side. The 3-D results should be resonably accurate, based on convergence checks and comparisons with 2-D TM models. The largest 3-D slab shown is 1600m. The 3-D results appear to be converging to the 1-D curve, but the convergence is very slow at the lower frequencies. This illustrates an important point : because surface charges at

\

f=O.IHz

- --

- - --

f=

2 Hz

800

16000

7 00

14000

600

l2000

500

4 00

8000

300

6000

200

4000

100

2000

0

0

FIG.43. TM mode (EL) and TE mode (Ell) anomalies over a 2-Dridge. (After Ngoc, 1980.)

its boundaries are important, a 3-D slab must be very large for l-D interpretation to apply. If l-D inversion is applied to the results obtained for the largest slab, the results will be erroneous. 5.6.9.Depth of Exploration and Detectability. Depth of exploration is often stated to be one skin depth 6,where

a=-

(222)

This simplification is misleading, because noisy data or surface geological noise can obscure the responses of deep bodies. However, with care in both data acquisition and data interpretation, depths of exploration Well in excess of 100 km can be achieved for infinite interfaces. For 2-D or 3-D bodies, depth of exploration can be considerably less. Newman et al. (1983) explored the possibility of detecting deep magma

16.


I50

-

1009000 70 -

343

SLA0

-

A

400m z 400m

0 D

BOOm r B O O m l2OOm x I 2 0 0 m

0

1600m r 1 6 0 0 m

60-

c 5040-

v)

W

30-

10

0.0I

0.1

10

I

100

FREO. ( H z 1 MODEL

h, = 2 0 0 m

p,

h, = 3 )

p,

KX)fL-m

:

=

ear'h

loon-m

FIG.44. Apparent resistivity 1-D curve and computed data points for four different equidirnensional 3-D slabs. (After Ting and Hohmann, 1981.)

chambers with MT. If the magma chamber is electrically connected to a highly conducting half-space, it probably will not be detected. On the other hand, if the basal half-space is resistive or if the earth is not layered, the magma chamber is more readily detected. Apparent resistivity and impedance phase sounding curves directly over the 3-D model of Fig. 45 appear in Fig. 46 and are compared to curves centered over a 2-D structure of identical cross section and to curves representing the response of just the 1-D layered sequence in the absence of the inhomogeneity (Newman et a/., 1983). Note that departures from the purely layered response by both modes of the 3-D body signature are very subdued ; such a response would have a low probability of being recognized, particularly in light of the frequent existence of near-surface geological noise which obscures target MT anomalies (Wannamaker et al., 1982). The 2-D TM and corresponding 3-D results agree very closely, as expected, indicating that increasing the strike length of this structure will not facilitate its

344

STANLEY H. WARD

I

f C

C

4

J5

0

lOtm

SCALE

CROSS

SECTION

JUVENILE CHAMBER ( v i r t u a l l y undetectable)

FIG.45. Three-dimensional model of magma chamber in a layered earth. (After Newman et al., 1983.)

detection with the apparent resistivity pyxand impedance phase &. On the other hand, the 2-D TE response in Fig. 46 is quite strong, dwarfing the corresponding anomalies in pxuand & caused by the 3-D structure. This discrepancy is interpreted as being due to current gathering in the 3-D body, whereby secondary currents induced about the 3-D structure are essentially short-circuited into deeper, less resistive media of the layered host and are inhibited by the material of 4000 Q-m from reaching the surface to produce an anomaly (Newman et al., 1983). To verify this interpretation, the 3-D body of Fig. 45 was removed from its layered host and simulated within a uniform half-space of 400 Q-m

16.


3D -1-0

--P..

-2

0

-1

r2

11

log f (Hz)

- 1-0 1

15

15 65 F.

0

2

55 45

35 25

l5

L L

-3

-2

0

-1

11

*2

15 -3

0

-1

-2

+2

*1

log f

(Hz) log f (HA FIG.46. Apparent resistivity and impedance phase curves over the magma changer of Fig. 45, layered earth. (After Newman et at., 1983.)

30

2D

'4

-.....- PP,." I I -3

-2

-1

0

+1

*2

-3

log f (Hr) 75

5 .

* ;

5

-2

-1

0

+1

*2

log f (HA

-

L

35

251 1 -3

-2

-1

log

0

11

*2

f (Hz)

FIG.47. Apparent resistivity and impedance phase curves over the magma chamber of Fig. 45, homogeneous half-space. (After Newman et al., 1983.)

346

STANLEY H. WARD

(see Fig. 47). In this case, the 2-D TM anomaly as well as the anomalies in both ply, & and pyx,&x over the 3-D body are much stronger than the anomalies in Fig. 46. (Anomalies here mean departure from the 1-D response.) This high sensitivity of the response of a 3-D body to its layered host underscores the importance of simulations using an algorithm handling 3-D bodies in arbitrarily layered earths.

6. Controlled-Source Electromagnetic Methods 6.1. Introduction

Controlled-source electromagnetic methods (CSEM) are applied in mining, ground water, geothermal, sedimentary basin, and deep crusthpper mantle exploration. Surveys are performed in boreholes, on the earth’s surface, and from aircraft flying 35-135 m above the earth’s surface. In the interest of uniformity throughout this chapter. I will limit my discussion to surveys performed at the earth’s surface. Dyck (1975) reviewed electrical borehole methods, Becker (1979) reviewed airborne electromagnetic methods, while Ward (1979,1982) and Hohmann and Ward (1981) reviewed ground electromagnetics, all as applied to mineral prospecting. Ward (1983a, b) reviewed ground electromagnetic methods applied to deep crustal and geothermal exploration. Numerous references are given in each of these seven review articles. 6.2. Basic Principles

6.2.1. Geoelectric Sections. Figure 48 portrays a generalized model of the earth, in mining exploration, in which a massive sulfide body is the object of search by means of the electromagnetic (EM) method. Unfortunately, all the other members of the geoelectric section of Fig. 48 will also contribute to the secondary magnetic field detected at the receiver. The ratio of secondary to primary magnetic fields is recorded as the ratio AE/E of secondary to primary voltages in the receiver. Thus the objective of the electromagnetic method in mining exploration is to detect and evaluate each element of the geoelectric section so that the resistivity environment surrounding the ore can be assessed. In this fashion, for example, one can hope to distinguish overburden response from the response of the massive sulfide body in deeply weathered terrains. To achieve this objective, the following inequality must hold :

el = a l p l w t l L Q ez = 0 2 p 2 w t 2 ~

(223) where o,p, w ,and t are the conductivity, magnetic permeability, angular frequency, and thickness for the overburden (subscript 1) and the ore body

16.


3 47

R E C E I V I N G COIL

T R A N S M I T T I N G COIL

3 tWEATHERED HOST ROCK

-‘-OVERB

MASSIVE SULFIDE DISSEMINATED SULFIDE-(

Fro. 48. Generalized geoelectric section appropriate to the electromagnetic exploration of massive sulfides. (After Hohmann and Ward, 1981.)

(subscript 2). The parameter L is the separation between the transmitting and receiving coils. The 8i are induction numbers, which control the responses of overburden and ore body. Any electromagnetic boundary-value problem will involve one or more induction numbers, as can be seen by dimensional analysis of the wave equation

This equation indicates that a dimensionless quantity kL remains invariant under transformation from one scale to the next, i.e.,

The induction number 8 = kL is thus similar to the Reynolds number in fluid flow. Hence it is a fundamental quantity governing electromagnetic phenomena. It is expressed by kz in Eq. (47) and by kihj in Eq. (123). If the inequality of Eq. (223) holds, then the contribution of the ore body can be distinguished clearly from the contribution of thepverburden in AE, provided all other induction numbers are much less that 8i. As one might

348

STANLEY H. WARD

expect, the various Bi are not always far apart, with the result that separation of the contributions of the various elements of the geoelectric section to A E is not clear-cut. In addition, interactions between the elements of the geoelectric section can and do take place; an example is current channeling into a highly conductive medium from a less conductive one. The electromagnetic problem in mining geophysics can be described as a search for procedures to separate the geological signal arising in a massive sulfide body from the geological noise arising from the other elements of the geoelectric section. The procedures must be sought with the realization that each geological noise source may shift the phase, alter the amplitude, and change the spatial spectrum of each component of the secondary fields scattered by the massive sulfide deposit. To solve this problem, it is necessary to (1) obtain precise data over several decades of freqency, (2) avoid spatial aliasing of data, (3) select an optimum transmitter-receiver configuration, and (4)use three-dimensional models to simulate the real earth. Compromises between complete solutions and economical or practical solutions are to be expected within this framework (Hohmann and Ward, 1981; Ward, 1982). Figure 49 shows a generalized model of a convective hydrothermal system being explored by an electromagnetic method. Except for the subsurface target, it is similar in most respects to the mining problems of Fig. 48. However, the resistivity of the geothermal reservoir is usually not as low as

OUTLINE OF

FIG.49. Generalized geoelectric section appropriate to the electromagnetic delineation of a convective hydrothermal system.

16.

349

ELECTRICAL METHODS IN GEOPHYSICAL PROSPECTINO

the resistivity of the massive sulfide, the target in mining exploration. Further, only the faults and fractures constitute the geothermal reservoir. Thus detection of thin planar features, or an aggregate of them, becomes the goal of applying electromagnetic methods in geothermal environments. There are, however, geothermal resources which are stratigraphically controlled, so that electromagnetic delineation of roughly horizontal porous strata then becomes the objective of electromagnetic surveys. Caprock, overburden, and other lateral and vertical variations of resistivity of no consequence to the geothermal reservoir may obscure its detection and delineation (Keller, 1970; Ward, 1983a,b). For the grossly inhomogeneous resistivity environments that are normally encountered in mining and geothermal exploration, continuous sounding and profiling are usually done. Sounding is carried out by varying frequency, while profiling is carried out by laterally moving the receiver relative to a fixed transmitter or moving both transmitter and receiver in unison. More will be said about this later. In deep crustal sedimentary basin and ground water studies, the application of active electromagnetic methods is directed primarily toward sounding, that is, estimation of layer resistivities and thicknesses in an essentially horizontal sequence of layers (Zohdy, 1964, 1965, 1975 ; Keller, 1970; Ward, 1983a, b). 6.2.2. Response of a Sphere. In mining applications, the object of search with the electromagnetic method is most often a lenslike massive sulfide body with conductivity of the order of 1 S/m buried in a resistive host of about S/m. An approximation to this situation is a conducting sphere in a vacuum. Hohmann and Ward (1981) presented the time and frequencydomain responses of a sphere of radius R and conductivity 7. They stated that : The time domain electromagnetic response can be approximated by a single exponential decay :

h ( t ) = e-f’T where the time constant

7

(226)

is given by 5

=

0p0 R 2 / z 2

(227)

H/m is the magnetic permeability of free in which po = 1.26 x space, The equivalent frequency domain electromagnetic response is

Larger time constants, then, correspond to larger (OR’)products.

350

STANLEY H. WARD

0.4

I\ \

- -1

T I M E DOMAIN

\

\ (Poor Conductor)

"0

2

OO

94

-

8

6

4

Ti m e (mrecl

I

187

A

281

374

I

I

468

I

562

Frequency (Hz)

FIG. SO. Time- and frequency-domain responses for good and poor conductors. (After Hohmann and Ward, 1981.)

In Fig. 50, we compare frequency- and time-domain electromagnetic responses of a good conductor (T = 3.2ms) and a poor conductor ( 5 = 0.64ms). For two spheres of radii 50 and 100m, these time constants correspond to conductivities of 10 and 0.5 S/m, respectively. In the time domain, the poor conductor is characterizedby a more rapid


35 1

decay, while in the frequency domain the peak quadrature response and maximum slope of the in-phase response occur at a higher frequency for the poor conductor. Figure 50 is useful for gaining insight into eddy current induction in confined conductors. In the time domain, after the transmitting current is terminated, the eddy currents will be confined initially to the surface of the conductor. As a result of ohmic losses, the initial induced currents will begin to dissipate. “The region inside the conductor will see a decreasing magnetic field and thus eddy currents will start flowing through it. This process is repeated in time at successive interior points and can best be described as an inward diffusion” (Nabighian, 1982). The currents will decay with a time constant as noted above. In the frequency domain, eddy currents due to highfrequency excitation will be confined to the surface, while eddy currents due to low-frequency excitation will appear throughout the body. As it turns out, the response of any body will be similar to that of a sphere, so Fig. 50 is useful in obtaining a basic understanding of any earth induction process. 6.2.3.Response of a Half-Space. A finite source of electromagnetic waves, such as a loop of wire, will radiate a field of complex form. This field can always be decomposed into a spectrum of plane waves, although complex angles of incidence must be invoked (Clemmow, 1966). In Section 5, on the magnetotelluric method, I noted that if displacement currents are negligible, then because of the large conductivity contrast between air and earth, electromagnetic waves of any angle of incidence will traverse vertically in a homogeneous or horizontally layered earth. The electric vectors will then be confined to the horizontal. Thus, the spectrum of plane waves due to a finite source will also give rise only to horizontal electric fields. A loop source of wire, when placed above the earth’s surface, will induce horizontal eddy currents regardless of its orientation. This result has been established as a well-known theorem (see, e.g., Weaver, 1970). Nabighian (1979) showed plots of the current density in the ground induced by a rectangular loop situated on the surface of the earth. It is reproduced here as Fig. 51. Nabighian described the downward and outward traveling current pattern as “smoke rings”. Further, he showed that the combined effect of all induced currents in the ground can be approximated by the effect of a single current filament of the same shape as the transmitting loop and moving downward with velocity u = 2-/

while increasing its horizontal dimensions in proportion to

(229)

e.

352

STANLEY H. WARD

H I

t / P 10.4

Conrours X loe9 A / m *

1.6

I

Contours

X

10.’~

A/m2

FIG. 51. Current density in the “smoke ring” of induced current around a step-varying vertical magnetic dipole on a half-space. (After Nabighian, 1979.)

6.3. Data Acquisition 6.3.1. Time Domain, Frequency Domain, and Decades of Spectrum. Table VIII itemizes factors to consider in selecting controlled-source electromagnetic systems. Figures 10 and 52 illustrate the waveforms typically used with frequency-domain (FEM) and time-domain (TEM) electromagnetic TABLE VIII. Basis for Selecting ControlledSource Electromagnetic Systems TEM or FEM Decades of spectrum Signal-to-noise ratio Lateral and vertical resolution Source configuration Transmitter coil size Depth of exploration Current channeling Effects of topography

16.

f

ELECTRICAL. METHODS IN GEOPHYSICAL PROSPECTING

353

TRANSMITTER C U R R E N T (AND PRIMARY MAGNETIC FIELD)

SECONDARY (TARGET) MAGNETIC F I E L D

C U R R E N T AND

FIG.52. Typical time domain transmitted and received waveforms. (After McNeil, 1980.)

systems. Time-domain systems are much in vogue today in shallow crustal exploration, especially mining and sedimentary basin exploration, because measurements over a broad spectrum may be made in a short period of time, whereas FEM systems require much more field time. Further, at present TEM has an inherently higher sensitivity than FEM because TEM measurements are made in the absence of the primary field. On the other hand, power is concentrated in a narrow bandwidth with FEM and spread over a broad bandwidth with TEM. A higher signal-to-noise ratio results with FEM. This can be countered in TEM if a square wave of low duty cycle is used; it provides a short on-time and a long off-time with high instantaneous power. Finally, most commercial state-of-the-art TEM systems operate over two decades of spectrum, whereas the few state-of-the-art FEM systems operate over four decades of spectrum. If exploration to great depths is desired, measurements must be made at long times after cutoff of the current in the transmitter.

354

STANLEY H. WARD

TABLE IX. Evaluation of Frequency-Domain (FD) and Time-Domain (TD) Electromagnetic Methods ~~~~

FD TD

FD TD

~

Spectrum

Bandwidth

Transmitter-receiver separations

Alignment errors

Four decades Two decades

Narrow Broad

Large Small

Problem No problem

Timeheading

Signal

Noise

Instantaneous power

Large Small

Model-specific Model-specific

Low High

Low High

For deep crustal exploration, one might expect FEM to be favored over TEM because of the higher signal-to-noise ratio expected with narrowband (FEM) than with broadband (TEM) systems. While SanFilipo and Hohmann (1982) confirm this idea, further study of the problem is needed, especially since the signals received from the subsurface targets are model-specific. Table IX summarizes the known relative advantages of FEM and TEM. Apart from these, one should note that a single coil can be used as both transmitter and receiver in TEM. This has not yet been exploited in geothermal or deep crustal exploration, but it is a feature of SIROTEM (McCracken and Buselli, 1978), which has been used in mining and sedimentary basin exploration. Further, alignment errors between transmitter (Z) and receiver (Rx)are unimportant in TEM because only of secondary (scattered) fields are measured, whereas in FEM measurements are always made of primary (source) and secondary fields combined. One can depart from the basic waveforms of Fig. 10 and 52 to achieve specific objectives. Duncan et al. (1980) and others reported the use of a pseudorandom binary sequence (PRBS) but have yet to demonstrate that this modestly broadband system enjoys advantages of FEM narrowband systems which also employ cross-correlation to extract signal from noise. Several other waveforms have been used in mining exploration (see, e.g., Barringer, 1962; Lamontagne, 1975; Won, 1980). 6.3.2.Source Configurations. Figure 53 portrays five basic transmitting source configurations used in mining exploration. For detailed discussions of the many variants on these basic source types, see Grant and West (1965), Ward (1967), and Telford et al. (1976). The two-loop array (Fig. 53a) is moved in-line or broadside across the expected strike of the structure. In the frequency domain, real and imaginary parts of secondary magnetic fields are recorded as a percentage of the

16.


Two-loop

355

Large source loop

Single -loop Ax

Tx 8 Rx

Section

Plan

FIG.53. Five basic source types used in electromagnetic exploration : (a) coplanar horizontal, coplanar vertical, or coaxial loop pairs; (b) large rectangular source loop to which a single horizontal or vertical receiving coil is referenced; (c) single loop which is used sequentially as transmitter and then as receiver in the time domain or whose impedance is measured in the frequency domain; (d) Grounded wire source to which electric and magnetic field components are referenced ; (e) vertical transmitting loop, tilt angle and ellipticity measured by receiver.

primary field. The phase reference is hard-wired from the transmitter to the receiver, by which means the primary field is also canceled. In the time domain, the secondary transient is simply recorded and stacked at the receiver. Measurements are made every 25 or 50 m along the traverse with the transmitter and receiver separated by 100-200 m. The large source loop portrayed in Fig. 53b ranges in dimensions from 200 x 400m to 500 x 1OOOm. Measurements are made in the frequency domain of the field strength and phase of one to three magnetic field components or of field strength ratio and phase difference with a pair of horizontal coplanar or vertical coaxial coils. For the former, synchronized crystal clocks at the receiver and transmitter provide a phase reference for coherent detection. Measurements of one to three components are also made in the time domain, for which crystal clocks provide a time reference for stacking. Traverses of the receiving coils are made outside the loop on lines perpendicular to a long side of the loop, and hence nominally perpendicular to the geologic strike. In the time domain, measurements are also made inside the loop when one wishes to minimize current channeling (B. Spies, 1982,

356

STANLEY H. WARD

personal communication). Typical reading intervals are 25-50 m. If two receiving coils are used, they are separated by about 50 m. Lajoie and West (1976) demonstrated that the size of the large source loop can be matched to the size of the target in order to achieve an optimum response. They concluded that the source dimensions should be of the order of the target dimensions if the target is a three-dimensional body. A scaled-up version of this system suitable for deep crustal studies is represented by the experiments reported by Connerney et al. (1980), in which the electrical resistivity of the crust was determined to more than 30 km. In the time domain it is possible to use a single loop, first as a transmitter and then as a receiver (Fig. 53c). Fast switching of the loop from the transmitter to the receiver facilitates this approach. The loop is moved along a traverse normal to the geologic strike between measurements, with receiving stations being occupied every 50 or 100 m. The loop typically range from 50 to 100 m to the side. The method is also used in shallow sedimentary basin applications. The fourth transmitting source, shown in Fig. 53d, is a grounded bipole. As used in the controlled source audiomagnetotelluric method, the bipole is typically 1-2 km in length. Readings are made over the frequency range 10 Hz to 10 kHz of components of electric and magnetic fields parallel and perpendicular to the bipole and also of the vertical magnetic field. If measurements are made three to five skin depths away from the source, a plane wave formulation can be used in the interpretation. Skin depths are calculated for the most resistive medium. The bipole is oriented parallel to the strike to excite the TE mode, whereas it is oriented perpendicular to the strike to excite the TM mode. The method is being used increasingly for applications in mining, geothermal, and sedimentary basins ; the measurement station intervals depend on the scale of the problem (Goldstein and Strangway, 1975; Sandberg and Hohmann, 1980; Bartel and Wayland, 1981). The grounded bipole studies of Sternberg (1979) and of Duncan et al. (1980) can be considered related to CSAMT, but their measurements were made closer to the grounded bipole than is typical of CSAMT. The fifth transmitting source, shown in Fig. 53e, is a coil whose plane is vertical, i.e., a horizontal magnetic dipole. The receiver is located in the plane of the coil and rotated about an axis joining transmitter and receiver. The tilt of the major axis and the ellipticity of the ellipse of magnetic field polarization are measured. In typical use, in which it is called a rotating vertical loop, the transmitter is fixed at a central point on a survey grid and the receiver is moved in increments along adjacent lines as in Fig. 53f. The plane of the transmitting loop is rotated so as to contain the receiver, In this fashion, the transmitted primary field is always horizontal at the receiver, regardless of elevation differences between transmitter and receiver. Thus

16.


357

any inclination of the total field at the receiver is an indication of a secondary field. With inductive CSEM, there has been no systematic and objective comparison of the wide range of transmitting and receiving coil pairs which may be used in exploration for various targets. The reasons for this are the size of the task and the lack of availability of all of the computational methods required to make such a comparison. For deep crustal exploration the selection of systems becomes considerably simplified-for example, compared to the options for mining exploration. To start with, the source is most likely either a grounded bipole or a square coil. Once one has gone to the considerable logistical trouble of laying out the source, it makes the most sense to collect all three orthogonal components of magnetic field and the two horizontal orthogonal components of electric field. However, costs, logistics, and objectives may dictate otherwise. A large single loop, serving first as transmitter and then as receiver, has not been used for deep crustal studies, as noted earlier. An objective comparative analysis of the advantages and limitations of each of these sources, and of MT/AMT in various one-, two-, and three-dimensional terrains, is a highly desirable objective for future analysis. 6.3.3. Time and Frequency bands. For mining exploration, the typical frequency band used is 100-3000 Hz, but this is extended in some systems to 5-50,000Hz. In the time domain, samples of the induced transient are typically made over the range 0.5-50 ms, but in some systems this is extended to 0.1-100 ms. These time and frequency bands are also used in coal and sedimentary basin exploration. For geothermal exploration, where greater depth of exploration is desired, frequencies as low as 0.1 Hz have been used (Goldstein et al., 1982). In deep crustal exploration, frequencies as low as 0.05 Hz have been used (Connerney et al., 1980) in purely inductive systems. 6.3.4.Electronic Equipment. The receiver and transmitter described for use with the resistivity and induced-polarization methods (Tables I1 and 111) would serve for electromagnetic systems operating over the frequency range to 2000 Hz. A commercially available time-domain electromagnetic system of modern design, the Geonics EM-37, has the following characteristics. The transmitting loop varies in dimensions from 40 x 70x11 to a maximum of 300 x 600 m. The current pulses have a repetition rate of 3 or 30 Hz and a maximum amplitude of 20 A. The transient is sampled at 20 time channels. Successive operation at 30 and 3 Hz effectively allows readings at 30 time channels from 80 ps to 80 ms. The output from each channel can be read from a digital LED display or can be recorded digitally with the addition of an extra box. The transmitter can be synchronized to the receiver via dualprecision crystal clocks, hardwire, or radio link (Nabighian, 1982).

358

STANLEY H. WARD

6.4. Data Processing

Modern digital receivers sample the waveform at discrete times and store the samples as numbers in the computer memory. Notch filtering at 60 and 180 Hz reduces noise due to power lines. Stacking of signals in the time domain and correlation detection in the frequency domain are routinely used. 6.5. Interpretation 6.5.1. One-DimensionalEarth Models. The review article by Hohmann (1982) covers the state-of-the-art numerical methods used in the interpretation of electromagnetic data. The half-space and horizontally layered earth are one-dimensional models. The frequency-domain response of a vertical magnetic dipole of moment m over a layered earth is given by

H,(P, X) = 41c and

Hr(P, X) = -

r 0

R(1, P, X)Jo(1r)A2 d1

Im

471 0

R(1, P, X)Jl(1r)A2 d1

(230)

(231)

where Jo(1r) and Jl(1r) are Bessel functions of the first kind of orders 0 and 1 , respectively, 1 is the Hankel transform variable, and R(1, P, X) is the kernel of the integral and is a function of the model parameters. The unknown model parameters are the components of the n-dimensional vector P, and hence are the layer conductivities aj(i = 1 to n) and the layer thicknesses hi(i = 1 to n - 1). Therefore, there are m components of P,where m = 2n - 1. The system parameters forming the vector X are the sourcereceiver separation rand the angular frequency of the transmitter o.Glenn el a/. (1973) define R(1, P, X)

+ 1 = Z d ( Z 1 + 2,)

(232)

where

2, = z,

22

+ 21tanh(iklh1)

21+

22

tanh(ik1h)

(233)

that is, the plane wave impedance at the top of an n-layered structure as given by Eq. (125). The time-domain response of a layered earth is obtained by taking the inverse Laplace transform of Eq. (230) or (231). Forward application of equations such as (230) and (23 1) is rapidly being replaced by inverse solutions in whichR(A, P,X) is estimated by least-squares

16.


1

0

.

’

“”“’I

~ . . . “ ‘ I

’.’.-

OBSERVATION MODEL

- 3 LAYER

-

359

- 25

-

-

+-5-lo* 10’

-

-

. I

10’

lo3

*

I

...... I

lo4

-

. ...lo5

F R E OUE NCY Hr

R E S I S T I V I T Y Qm

3

(b) FIG.54. (a) Horizontal magnetic dipole sounding for a receiver 30.5 m south of transmitter (after Ward et a/., 1976); (b) deduced vertical resistivity profile.

360

STANLEY H. WARD

inversion methods. Pertinent references include Glenn et al. (1973), Ward et al. (1974, 1976), Glenn and Ward (1976), La Brecque (1984), and Fullagar and Oldenburg (1984). Figure 54a, from Ward et al. (1976), shows ellipticity and tilt angle observations relative to a horizontal magnetic dipole 30.5 m distant from the receiver. The solid line is an inversion computed for a three-layered earth. The model is shown in Fig. 54b. 6.5.2.Two-dimensional Earth Models. Hohmann (1982), in his review of numerical models applied to electrical methods, gives a number of useful references, including Coggon (1971), Hohmann (197 l), Jones and Price (1971), Parry and Ward (1971), Swift (1971), Stoyer and Greenfield (1976), Lee (1978), Brewitt-Taylor and Weaver (1 979), and Lee and Morrison (1 980). Plane-wave and line-source excitation of a two-dimensional earth is readily treated. However, tested and reliable algorithms for a 2-D body in the presence of a 3-D source are currently questionable. No inverse techniques have been attempted. Figure 55 shows a comparison of finite-element computations by Coggon (1971) and integral-equation computations for the vertical magnetic field over a 2-D conductor excited by a line source. The

140 I50

160

- HOHMANN o

COGGON

-1-170

FIG.55. Integral-equation(solid lines) and finite-element (circles) computations of amplitude and phase of normalized vertical magnetic field over a 3-D body near a line source. (After Coggon, 1971), and Hohmann, 1971.)

16.


361

agreement is excellent, confirming the validity of both solutions. Checks such as these are essential when using numerical methods. 6.5.3.Three-Dimensional Earth Models. Once again I refer to Hohmann’s (1982) review article, which cites articles by Lines and Jones (1973), Hohmann (1975), Wiedelt (1975), Lajoie and West (1976), Lee et al. (1981), Pridmore et al. (1981), and Tripp (1982). More recently W. A. SanFilipo (personal communication) seems to have found at least two approaches to solving this class of problems. 6.6. Problems with Inductive CSEM Methods 6.6.1. Natural Field Noise. Figure 17 illustrates a typical natural magnetic field spectrum. Its field strength characteristics are a low near 3 Hz, a rapid increase with decrease in frequency below 3 Hz, an interim peak just below 100Hz,and a trough near 2000Hz followed by a rise at higher frequencies. Shallow CSEM for mining, geothermal, and sedimentary basin studies utilizes the frequency range from 1 to lo5 Hz. As noted earlier, in the range 1to lo5 Hz natural field noise arises in atmospheric discharges, especially those associated with lightning, The major worldwide centers of lightning storms are in Indonesia, Central America and northern South America, and north-central Africa. These thunderstorm centers shift north in northern hemisphere summer and south in southern hemisphere summer. The energy from these major lightning discharge centers propagates in a waveguide bounded by the ionosphere and the earth’s surface. At any point on the earth’s surface the measured noise includes this waveguide-propagated energy plus atmospheric discharges from nearby sources. The resulting spectrum of noise exhibits a wide dynamic range and a very transient type of individual peak. Schemes to deal with such a difficult noise source include truncation of the highest peaks by limiters and data point removal in digital systems designed to accept data points only within a prescribed range of amplitudes. Narrowband CSEM systems effect signalto-noise enhancement and are used as frequency-domain systems. Broadband CSEM systems can only use stacking and data point removal. 6.6.2. Cultural Noise. As Table VII indicates, cultural developments create active and passive noise. Circuits completed through fences, pipelines, power lines, telephone lines, rails, and other conductive cultural structures produce anomalies largely unrelated to subsurface geology. These sources of noise can, in rare instances, be reduced by removing the offending structure but, by and large, they can be avoided only by placing transmitters and receivers well away from them. This is not always possible in areas of concentrated industrialization, and hence important geological problems simply cannot be attacked in such areas.

362

STANLEY H. WARD

Some of these cultural developments also serve as sources for narrow- or broadband electric and magnetic field noise, especially power lines, telephone lines, and electrified rails, as Table VII indicates. The problem is compounded by the fact that these active sources of cultural noise induce eddy currents in the passive cultural noise sources such as fences and pipelines. Notch filters centered at 60Hz (50Hz) and 180Hz (150Hz) are characteristically used in CSEM systems to eliminate power line noise. 6.6.3.Geological Noise due to Overburden. Overburden can be described variously as unconsolidated sediments, weathered rock, or both. It may be resistive or conductive. Weathered rock is invariably conductive because the geological process of weathering leads to (a) increased porosity, (b) increased presence of clay minerals with their surficial electrical conductions, and (c) increased concentrations of ions in the pore waters of the weathered rocks. In dry climatic environments, evaporation strongly increases the concentration of ions in the pore waters, on the average. Dilution of these ions takes place during the rainy season in dry or wet environments. A worldwide study of these factors indicates that the shallow resistivity profile is closely related to local climatology, glaciation, and tectonic style. The depth of overburden related to weathering seldom exceeds 100 m but can easily reach 2 km for valley fill in the Great Basin of the United States. Sedimentary rocks in such areas as the Gulf Coast oil-producing region of Texas and Louisiana commonly exhibit resistivities as low as 1-10 h2-m due to interstitial brines (Pirson, 1963). This also is true of some geothermal areas such as the Imperial Valley in California (Meidav and Furgerson, 1972) and of deep valley fill containing evaporites in Nevada and Utah (Ward and Sill, 1976). Values as low as 0.1 0-m have been recorded in some of these areas. In a broader context than has been used here so far, these conductive sediments also can be treated as overburden if one is attempting to study the electrical properties of the deep crust. If a surficial conductive horizon, i.e., overburden of any sort, overlies a substratum to be studied with CSEM, the percentage of current entering the substratum becomes of utmost importance. Figure 56 illustrates the fraction of current f(a) confined to the overburden as a function of Q = 2Sp/L, where S is the conductivitythickness product of the overburden, p the resistivity of the basement, and L the distance between the current electrodes (Edwards and Howell, 1976). To ensure that, say, 80% of the current persists below the overburden, a! must be about 0.3. If the overburden is 1 km thick with resistivity 10 h2-m and the resistivity of the bedrock is lo3h2-mynot unreasonable numbers in areas of thick conductive overburden, then the distance between current electrodes must be lo00 km to ensure that 80% of the current flows in the bedrock. This is an incredible requirement which clearly demonstrates the difficulty of

16. ELECTRICALMETHODS IN GEOPHYSICAL PROSPECTING

0.03

0.I

I

I

L

I

I

0.3

1.0

3.0

10.0

30.0

363

+

a FIG.56. Fraction of currentf(a) confined to the overburden as a function of a. (See text.) (After Edwards and Howell, 1976.)

electrically detecting geological structure beneath conductive overburden with bipolar electric sources. The analysis has been made for dc and may not apply strictly for ac. When horizontal coil sources are used, induction in the overburden is strictly controlled by 61 , the induction number of the overburden. This induction number can be decreased by lowering the angular frequency. For example, given an overburden 1 km thick with 10 Q-m resistivity excited by an inductive coil source with measurements made at 3 x 10-*Hz at a distance of 10 km from the source, the value of 61is 0.2, well below values resulting in significant overburden response (Wait, 1955). For MT the situation is probably better because the skin depth for the same earth model given by

6=5 0 3 m

(234)

is 15 km, implying that the l-km-thick overburden of resistivity 10 0-m is virtually transparent to plane electromagnetic waves of frequency 3 x Hz. From the discussion above it might be concluded that the order of preference for sources to be used in regions of high surficial conductivity is (a) plane waves, (b) inductive coils, and (c) grounded bipoles. This is, of course, only a partial analysis of the problem, but it does show the nature of the difficulties of probing the crust in regions of thick conductive overburden or sediments. Geometric decay of fields from the three source types and attenuation of electromagnetic waves from each source type are other factors to consider (Hohmann and Ward, 1981). 6.6.4. Resolution and the Effect of Geological Noise. To facilitate vertical resolution, i.e. , resolution of the resistivities and thicknesses of

364

STANLEY H. WARD

horizontally layered media, an inductive electromagnetic system would have to sample at three or four frequencies per decade. At least four decades of spectrum are required if one wishes to explore both shallow and deep layers. If lateral inhomogeneities are superimposed on the layering, there must also be an adequate spatial density of receiving stations over a distance sufficiently large to permit delineation of all inhomogeneities of interest. A broad spectrum is necessary if the Bj of the inhomogeneities cannot be predicted in advance. Data are then best plotted as contours of field quantities in frequency-distance space. This will be illustrated in the discussion of the CSAMT method below. Roving two-loop sources, described below, provide the best lateral resolution. However, for deep crustal electromagnetic exploration it is necessary to use a fixed transmitter, either a loop or a grounded bipole, so it is also necessary to accept the lateral resolution achievable with these sources. It is well known (Madden, 1971) that inductive techniques, passive or active, usually provide information on conductivity-thickness products of conductive layers, whereas they usually provide only thickness information on resistive layers. In contrast, resistivity techniques usually provide information on resistivity-thickness products for resistive layers and conductivity-thickness products for conductive layers. Vertical resolution of resistive and conductive layers is well illustrated via inversion (see, e.g., Fullagar and Oldenburg, 1984). Joint inversion of inductive and resistive data sets can markedly improve the resolution (Petrick et al., 1977). 6.6.5. Topography. Three effects of topography on measurements made with an inductive electromagnetic method are elevation errors, alignment errors, and current channeling in ridges. Variations in elevation of the receiver relative to the transmitter will produce elevation errors in electric or magnetic fields along a traverse, relative to the fields that would be observed over a flat surface. These can be severe for short separations between transmitter and receiver, which might occur in shallow geothermal exploration. If topographic relief is large, one seeks to ensure that a square coil or bipole source is horizontal and that measurements are made of horizontal and vertical magnetic and horizontal electric fields. Alternatively, the plane of the transmitting coil must contain the axis or the plane of the receiving coil, and orthogonal magnetic field components are measured relative to this axis or plane. If either of these alternatives is ignored, alignment errors will result. If, for example, a transmitter is located below and adjacent to a ridge, induced currents will occur in the ridge at the higher frequencies and will contribute a source of noise which may obscure subsurface features. 6.6.6. Current Channeling. Current channeling was discussed previously for MTIAMT. It is most pronounced for MT where plane waves are involved. It becomes of increasing importance at lower frequencies


365

TAEZE X. Relative Importance of Current Channeling in Various Electromagnetic Methods" Most

Least

MT AMT AFMAG VLF CSAMT UTEM (Univ. Toronto) TURAM EMP (Newmont) GEOPROBE Vertcal Rotaining Loop PEM (Crone) MAX MIN I1 Shootback SIROTEM

Detecting contact follows the same order.

(Wannamaker et a)., 1982). It diminishes as the size of the transmitting source decreases. If the receiver is restricted to regions close to the transmitter, the Oi referred to earlier may be so small that the system may not observe the current channeling. For a one-, two-, or three-dimensional source, current channeling may occur along one or all axes of the body, depending on the direction of propagation of the plane wavelets associated with the source. Finally currents can be channeled from regions exceptionally remote from the body, especially for plane wave excitation. Table X shows an ordering of passive and active electromagnetic systems according to their intiation of current channeling. The order is subjective, but probably correct in principle. Current channeling will lead to enhanced detection of three-dimensional inhomogeneities in some circumstances. However, unless it is taken into account, estimation of the parameters of a three-dimensional body may be grossly in error (see, e.g., Ward et al., 1974). 6.6.7. Depth of Exploration. Aruleof thumb used by most geophysicists is that the depth of exploration is 0.3 to 1.0 times the separation between transmitting and receiving coils. Depth of exploration is controlled mostly by target response, geological noise, and separation between transmitter and receiver (Hohmann and Ward, 1981). Acknowledgments I am grateful to my colleagues G. W. Hohmann, W. R. Sill, and P. E. Wannamaker for collaborating on several reports and manuscripts from which I have drawn most of the material for this chapter. Joan Pingree typed the manuscript and Doris Cullen and Sandra Bromley supervised preparation of the illustrations. I am grateful to the Earth Science Laboratory of the University of Utah Research Institute for funding the typing and drafting.

366

STANLEY H. WARD

References Aiken, C. L. V., and Ander, M. E. (1981). A regional strategy for geothermal exploration with emphasis on gravity and magnetotellurics. J. Volcanol. Geotherm. Res. 9, 1-27. Al’pin, L. M., Berdichevskii, M. N., Vedrintsev, G. A., and Zagarmistr, A. M. (1966). “Dipole Methods for Measuring Earth Conductivity” (G. V. Keller, trans].). Consultants Bureau, New York. Angoran, Y., and Madden, T. R. (1977). Induced polarization: A preliminary study of its chemical basis. Geophysics 42, 788-803. Bannister, P. R. (1969). Source distance dependence of the surface-impedance conductivity measurement technique. Geophysics 34, 785-788. Barringer, A. R. (1962). The INPUT electrical pulse prospecting system. Min. Congr. J. 48, 49-52.

Bartel, L. C., and Wayland, J. R. (1981). Results from using theCSAMTgeophysica1 techniques to map oil recovery processes. SOC.Pet. Eng. AIME, Prepr. SPE No. 10230. Becker, A. (1979). Airborne electromagnetic methods. In “Geophysics and Geochemistry in the Search for Metallic Ores” (P. J. Hood, ed.), Geol. Surv. Can., Econ. Geol. Rep. No. 31, 33-43.

Berktold. A. (1982). Electromagnetic studies in geothermal regions. Proc. Workshop Electromagn. Induct. Earth Moon, 6th. IAGA, Phys.. Univ. Victoria, B.C. Berktold, A., and Kemmerle, K. (1982). Distribution of Electrical Conductivity in the Urach Geothermal Area, a Magnetotelluric and Geomagnetic Depth Sounding Investigation :The Urach Geothermal Project,” pp. 289-300. Schweizerbart, Stuttgart. Bhattacharya, P. K., and Patra, H. P. (1968). Direct current geoelectric sounding. In “Methods in Geochemistry and Geophysics,” Elsevier, New York. Bleil, D. F., ed. (1964). “Natural Electromagnetic Phenomena Below 30 kc/s.” Plenum, New York. Boehl, J. E., Bostick, F. X., Jr., and Smith, H. W. (1977). “An Application of the Hilbert Transform to the Magnetotelluric Method.” Rep. Electr. Geophys. Res. Lab., Univ. of Texas, Austin. Bostick, F. X., Jr. (1977). A simple almost exact method of MT analysis. In “Report on Workshop on Electrical Methods in Geothermal Exploration” (S. H. Ward, ed.), U.S.G.S. Contract 14-08-0001-G-359, pp. 174-183. Dep. Geol. Geophys., Univ. of Utah, Salt Lake City. Bostick, F. X., Jr., and Smith, H. W. (1962). Investigation of large-scale inhomogeneities in the earth by the magnetotelluric method. Proc. IRE 50, 2339-2346. Brewitt-Taylor, C. R., and Weaver, J. T. (1979). On the finite difference solution of twodimensional induction problems. Geophys. J. R. Astron. SOC.47, 375-396. Cagniard, L. (1953). Basic theory of the magnetotelluric method of geophysical prospecting. Geophysics 18, 605-635. Campbell, W. H. (1967). Geomagnetic pulsations. In “Physics of Geomagnetic Phenomena” (S. Matsushita and W. H. Campbell, eds.), pp. 822-890. Academic Press, New York. Cantwell, T. (1960). Detection and analysis of low-frequency magnetotelluric signals. Ph.D. Thesis, MIT. Clarke, J., Gamble, T. D., Goubau, W. M., Koch, R. H., and Miracky, R. F. (1983). Remotereference magnetotellurics: Equipment and procedures. Geophys. Prospect. 31. 149-170. Clemmow, P. C. (1966). “The Plane Wave Spectrum Representative of Electromagnetic Fields. Pergamon, New York. Coen, S., and Yu, M. W. H. (1981). The inverse problem of the direct current conductivity profile of a layered earth. Geophysics 46, 1702-1713.

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Coggon, J . H. (1971). Electromagnetic and electrical modeling by the finite element method. Geophysics 36, 132-155. Coggon, J. H. (1973). A comparison of I P electrode arrays. Geophysics 38, 737-761. Connerney, J. E. P., Nekut, A., and Kuckes, A. F. (1980). Deep crustal electrical conductivity in the Adirondacks. J. Geophys. Res. 85, 2603-2614. Dey, A., and Morrison, H. F. (1977). An analysis of the bipole-dipole method of resistivity surveying. Geothermics 6, 47-81. Dey, A., and Morrison, H. F. (1979). Resistivity modeling for arbitrarily shaped threedimensional structures. Geophysics 34, 615-632. Dieter, K., Paterson, N. R., and Grant, F. S. (1969). IP and resistivity type curves for threedimensional bodies. Geophysics 34, 615-632. Dmitriev, V. I., and Berdichevsky, M. N. (1979). The fundamental model of magnetotelluric sounding. Proc. IEEE 67, 1034-1044. Duba, A., and Heard, H. C. (1980). Effect of hydration on the electrical conductivity of olivene. EOS, Trans. Am. Geophys. Union 61,404. Duncan, P. M., Hwang, A., Edwards, R. N., Bailey, R. C.,and Garland, G. D. (1980). The development and application of a wide-band electromagnetic sounding system using a pseudo-noise source. Geophysics 45, 1276-1296. Dupis, A., and Iliceto, V. (1974). An example of rapid magnetotelluric investigation of faulted structures : The Carboli area (Lardarello, Italy). Boll. Geofis. Teor. Appl. 16, 125-136. Dupis, A., Iliceto, V., and Norinelli, A. (1974). First magnetotelluric measurements on Lardarello site. Boll. Geofis. Teor. Appl. 16, 137-152. Dupis. A., Marie, P., and Petian, G . (1980). Magnetotelluric prospecting of the Mont Dore area. In “Advances in European Geothermal Research,” pp. 935-943. Reidel Publ., Dordrecht, Netherlands. Dyck, A. V. (1975). Electrical borehole methods applied to mineral prospecting. In “Borehole Geophysics Applied to Metallic Mineral Prospecting: A Review” (A. V. Dyck, ed.), Geol. Surv. Can., Pup. 75-31, 13-28. Eadie, T. (1981). “Detection of Hydrocarbon Accumulations by Surface Electrical Methods: A Feasibility Study,” Rep. No. 15. Dept. Phys., Univ. of Toronto. Edwards, R. N., and Howell, E. C. (1976). A field test of the magnetometric resistivity (MMR) method. Geophysics 41, 1170-1 183. Fox, R. C., Hohmann, G. W., Killpack, T. J., and Rijo, L. (1980). Topographic effects in resistivity and induced polarization surveys. Geophysics 45, 75-93. Frangos, W., and Ward, S. H. (1980). “Bipole-dipoleSurveyat Roosevelt Hot Springs Thermal Area, Beaver County, Utah,” Rep. DOE/ID/12079-15. Res. Inst., Earth Sci. Lab., Univ. of Utah, Salt Lake City. Fullagar, P. K., and Oldenburg, D. W. (1984). Inversion of horizontal loop electromagnetic soundings. Geophysics 49, 150-164. Gamble, T. D., Goubau, W. M., and Clarke, J. (1979a). Magnetotellurics with a remote reference. Geophysics 44,53-68. Gamble, T. D., Goubau. W. M., and Clarke, J. (1979b). Error analysis for remote reference magnetotellurics. Geophysics 44,959-968. Gamble, F. D., Goubau, W. M., Goldstein, N. E., and Clarke, J. (1980). Referenced magnetotellurics at Cerro Prieto. Geothermics 9, 49-63. Ghosh, D. P. (1971). The application of linear filter theory to the direct interpretation of geoelectrical resistivity sounding measurements. Geophys. Prospect. 19, 192-217. Glenn, W. E., and Ward, S. H. (1976). Statistical evaluation of electrical sounding methods. Part I. Experiment design. Geophysics 41, 1207-1221.

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17. MEASUREMENT OF IN SlTU STRESS

Bezalel C. Haimson Rock Mechanics Program Department of Metallurgical and Mineral Engineering University of Wisconsin Madison. Wisconsin 53706

1. Introduction Knowledge of the in situ state of stress is fundamental to improving our understanding of tectonic processes, assisting in studies of earthquakes (mechanics, prediction, and control), and estimating rock reaction to changes in boundary conditions caused by human activities (drilling, excavation, reservoir impoundment, fracturing, etc.). The difficulty in assessing the state of stress stems from our inability to measure stress directly. Rather, the measurement is indirect and based on the response of the rock to a disturbance of the stress field. This response, be it a strain, a deformation, or a pressure, is recorded and then related back to the initial state of stress. Thus, the mechanical properties of the rock play a crucial role in the accurate determination of in situ stress. Under ideal conditions of elasticity, homogeneity, linearity, and isotropy, indirect measurements can yield reliable stress values. However, where the mechanical characterization of rock is not straightforward it may affect the interpretation of stress measurements. The many methods suggested over the years for measuring in situ stress and the few that have actually been used in the field with different degrees of success suggest that the ideal technique has yet to be devised. The existing methods do provide reliable estimates of the in situ stress in most cases, but may fail under unusual conditions of stress magnitudes and directions, rock anistropy, inhomogeneity, temperature, state of natural fractures, depth of measurement, and others. Jaeger and Cook' define the state of stress at a point as a tensor quantity, the unique determination of which requires knowledge of six independent components. In terms of principal planes (three mutually perpendicular planes which are unique at any given point in that no shear stresses exist along them) the six components are simply the three principal stresses acting across these planes and their respective directions. 377 METHODS OF EXPERIMENTAL PHYSICS Vol. 24, Part B


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BEZALEL C. HAIMSON

It is generally accepted that the state of stress in situ is the result of the superposition of a number of stress fields. The most obvious stress field is that due to gravitational forces. The vertical component of stress av resulting from this field is simply equal to the weight of the overlying rock yD,where y is the rock weight density and D the depth of the point of interest below the surface. The horizontal components of stress resulting from gravitational forces would be expected to be uniform (Le., OH = Oh, where DH and ah are the largest and least horizontal stresses, respectively) and not to exceed the value of av. In fact, in the event of lateral confinement restricting any horizontal strain the magnitude of the horizontal stress is given by vov/(l - v) (where v is the Poisson ratio for the rock), which is generally no more than 0.50~.If no lateral strain restriction exists, the horizontal gravitational stresses are zero. To earth scientists a most important stress field is that due to tectonic plate boundary forces. This field generally acts in the horizontal plane and is directional, often creating a distinct difference between OH and O h with the stress orientations related to tectonic loads. Because of the freedom of plate deformation in the vertical direction, resulting from the proximity of the free surface of the earth, ov is not affected significantly by this field. A third major stress field is that due to cooling or heating of the rock. This thermal stress can result from the proximity of a magma chamber, from diurnal and seasonal temperature changes, or from other natural or manmade sources. An important stress field, but quite different in nature, is the one generally termed residual stress or locked-in stress. This is the measurable stress prevailing in a rock block even when all the boundary forces have been reduced to zero. Residual stresses of 100 MPa that have apparently lasted since early Precambrian time have been recorded.* Normally, residual stresses are more important in near-surface measurements, while at greater depths they become only a fraction of the total measured stress. Methods of in situ rock stress measurement were first developed in the 1950s and 1960s in response to a need to quantify stability evaluations of underground engineering structures such as mine openings reaching great depths (2000m in Idaho, close to 4000 m in South Africa and India) and caverns for hydroelectric power plants of unprecedented size (typically 25 x 40 x 100m). Thus, instruments were devised mainly for the determination of stress conditions in the vicinity of excavation^.^-^ The importance of stress measurements to the understanding of geological phenomena such as seismic slip along a fault plane was demonstrated when hydraulic fracturing, the only method capable of measuring stress at great depths, was employed in the Rangely, Colorado, experiment. 6 p 7 Strong agreement between in situ stress measurement results and focal mechanism

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MEASUREMENT OF IN SITU STRESS

379

solutions for earthquakes and their possible relation to the driving mechanism of plate tectonics were subsequently reported.*-” In the following section the major methods of in situ stress measurement are described. They have been arranged in three groups: surface, nearsurface, and deep methods. A summary of our present knowledge of crustal stresses is also presented.

2. Surface Measurements Surface measurements of in situ stress are those conducted directly on the free surface of the rock or within a depth of less that 1 m. The free surface can be a relatively flat rock outcrop or a quarry floor at a point well removed from any topographic effects. Surface measurements, although based mainly on two established methods, have only recently been used for regional in situ stress determination. In one method the measuring device is a strain ro~ette,’~’ l4 and in the other it is a flat jack.” The advantage of both of these methods is the relative ease of measurement. The tests can be conducted quickly and inexpensively. However, surface measurements require that the rock outcrop be unweathered and not affected by nearby fractures or other discontinuities or by topographic relief. Components due to temperature changes and residual stresses make sorting out the tectonic portion of the measured stress difficult. In addition, no information is gained on the variation of stress magnitudes with depth. 2.1. Strain Rosette

The stress relief method employing strain gages was probably among the first techniques ever used to measure rock stresses. It was used, for example, in a tunnel below the Hoover Dam in 1932 and in the Soviet Union in 1935.5 More recently the method has been used for determining surface regional horizontal in situ stre~ses.’~’ l4 The method consists of bonding a strain gage rosette (consisting of three independent strain gages, Fig. 1) to a prepared surface of a rock mass, such as an unweathered outcrop or a quarry floor. After allowing sufficient time for strain gage curing, an initial reading is taken in each gage. A slot around the rosette is then cut or drilled to a depth sufficient to completely release the in situ stresses from the cored-over rock piece. This is followed by a final reading of the strain in the rosette gages. If electrical leads can be maintained in contact with the power supply and the recording equipment during the slot cutting, continuous plotting of the strains can be carried out, allowing better verification of gage condition and of the final stress-relieved strain. A temperature-compensating gage attached to an undisturbed portion of the rock surface is incorporated in the circuit for more reliable strain readings.

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BEZALEL C. HAIMSON

CIRCULAR

(b)

FIG. 1 . The strain rosette and its application to rock surface measurements. (a) Plan view; (b) profile of strain rosette cemented to rock. Also shown is the slot resulting from overcoring carried out after initial strain readings are taken.

The angles between the three rosette gages are set in advance. The unknown values are the two horizontal principal stresses and their directions. The three strain readings ( E i , where i = a, b, c) yield uniquely the largest horizontal (EH) and least horizontal (&h) principal strains and the direction Oj between the known orientation of the ith gage and that of EH . All 0,'s are interrelated. The three relationships are

2&i =

+ &h) + (&H - &h) cos 20i

(2.1) The resultant E H , &h, and Bi can be used to calculate the principal stresses OH and o h if the stress-strain relationship for the rock is known. In the case of isotropic linear elasticity this relationship requires knowledge of the modulus of deformation (E) and the Poisson ratio (v). The principal stresses are thus obtained from (&H

The directions of QH and Oh are the same as those of EH and &h, respectively.

17.

MEASUREMENT OF IN SZTU STRESS

38 1

Handin16reports on the use of strain rosette measurements to determine the state of stress at the surface in the Rangely oil field. Acomparison of these tests with other overcoring methods of stress measurement was reported by de la Cruz and Raleigh." Engelder and Sbar13used strain rosettes to estimate the regional surface stress field in the Potsdam sandstone of northern New York just west of Lake Champlain. They found the reproducibility of both the principal strain orientations and magnitudes to be less than desirable, although the mean orientations were quite consistent. The measurements were hampered by the near-surface presence of a major component of residual strain, which was difficult to ascertain accurately and thus made the isolation of the tectonic component rather difficult. Schaefer and Kiel14 used strain rosettes at some two dozen locations around Iceland in an attempt to establish the surface state of stress. Their results with respect to orientation show considerable consistency, although they are not always in agreement with deep measurements.18 2.2. Flat Jack A flat jack consists of two thin metal plates welded together around their edges except for an orifice through which a hydraulic fluid can be injected into the space between the plates. Following Tincelin," flat-jack testing procedure involves first the installation of a pair of markers into the horizontal rock surface. These markers are separated by a distance of approximately 20 cm in the direction along which the value of the horizontal normal stress is to be determined. A high-precision ( f 1 pm) extensometer is mounted between the markers and the initial distance is determined. Sometimes two or three pairs of markers are installed in parallel to obtain several independent readings (Fig. 2). The extensometers are then removed

FIG.2. The flat-jack method of surface stress measurements.

3 82

BEZALEL C . W S O N

and a vertical slot is cut into the rock face, halfway between the markers in a direction perpendicular to the marker alignment (Fig. 2). The slot, typically about 0.5-1 .O m2in areal extent, can be made by drilling overlapped vertical holes or by cutting with a diamond saw. In the former case the shape of the slot is square; in the latter case it is semicircular (Fig. 2). The slot creates a free surface perpendicular to which the normal stress is reduced to zero. The deformation accompanying this stress relief is reflected in the relative displacement between the installed markers. To determine this displacement, the extensometer is remounted and the new distance between the markers is recorded. The difference between the two readings (before and after slot cutting) equals the displacement. The flat jack, which is made to fit the cut slot in shape and size, is then inserted into the slot and cemented in place. With the extensometer(s) mounted, the flat jack is pressurized while the displacement between the markers is monitored. When the displacement decreases to zero (i.e., the extensometer reading equals that before the slot cutting) pumping is stopped. The pressure in the flat jack is recorded and is assumed to be equal to the in situ, preslotting, horizontal normal stress acting in the direction of the marker pair(s). The accuracy of this assumption is still a matter of debate and corrections have been suggested.’ To define the principal horizontal in situ stresses at the surface, values of normal stresses in three different directions must be determined. This is accomplished by conducting three separate flat-jack tests in slots cut at different known azimuths. In practice, several additional redundant slots may be tested to improve the reliability of the results. If the angles between the three flat-jack slots are carefully measured and the rock is assumed to be isotropic and linear elastic, the relationship between the normal horizontal stress Oi as measured by the ith flat jack and the horizontal principal stresses is given by

where i = 1, 2, 3 and 8i is the angle between the ith marker-pair known direction and that of the largest horizontal compressive stress OH (Fig. 2). The unknowns, OH, O h , and 0, can be determined uniquely by solving the three equations representing the three flat-jack readings. Thus, the complete state of stress at the surface (both magnitudes and directions) is established. Successful application of the flat-jack method for surface in situ stress measurements requires that the rock tested have a bed thickness exceeding the depth of the slot and that surface weathering be low or nonexistent. Hence, fresh surfaces of quarry floors are often preferred to natural outcrops. Thin bedding planes, fractured rock, the presence of nearby joints, and rocks rich in clay do not lend themselves to this method.”

17. MEASUREMENT OF IN SZTU STRESS

383

In cases where the normal stress is tensile, the distance between the markers increases when slotting is carried out and cannot be canceled by the inflation of the flat jack. Hence, a tensile stress cannot be measured directly by the flat-jack method. The technique is also bound to fail if the regional horizontal stresses are too high. In this case the rock deformation following stress relief could cause excessive closure of the slot and prevent the flat jack from fitting in. The flat-jack method has been used in many underground civil and mining excavations since the early 1950s for the purpose of determining the stresses in the walls of tunnels and caverns.’992oThe only known crustal in situ stress measurement program using the flat jack has been conducted in France.”

3. Near-Surface Measurements Near-surface measurements of in situ stress are those conducted in boreholes at depths of 0-50 m. The primary technique used is “overtoring,” which encompassesdrilling a test hole, instrumenting it, and then coring over the instrument to relieve the stresses. This explains why overcoring methods of stress measurement are also called “stress relief” methods. The stress relief instrument typically utilizes a strain gage device which, on stress relief, enables the measurement of changes in test hole diameter (“borehole deformation gage”), changes in strain on the flattened bottom of the test hole (“doorstopper”), or changes in strain on the wall of the test hole (“triaxial cell” in its different varieties). The strain changes can be transformed into values of the in situ stress components if the elastic properties of the tested rock are known. In the following, each of the major overcoring techniques is described separately. 3.1 . Borehole Deformation Gage

The borehole deformation gage is a cylindrical borehole probe consisting of a central core and an outer stainless steel protective casing. Affixed to the core are six beryllium-copper cantilevers symmetricallydistributed about the center and axially aligned. This arrangement creates three diametrically opposed cantilever pairs at 120” from each other. Each cantilever is equipped with two strain gages mounted near the fixed end, one on the top and one on the bottom surface. At the free end of each cantilever a carbide button of adjustable length is attached (Fig. 3). The six buttons protrude through the protective casing and make solid contact with the test hole wall as the tool is inserted. The button length adjustment enables the operator to ensure that contact is made. Thus all the cantilevers are bent by the test hole wall towards the hole axis and the strain gages sense the amount of bending

3 84

BEZALEL C . HAIMSON CABLE CLAMP ,ORIENTATION

PLACEMENT

PIN

/

GAGE, CASE

\

END SEALING GROMMET

/

CABLE CONNECTOR

G A G E BODY

CARBIDE BUTTON

\

STRAIN GAGES

FIG. 3. Cross section of the borehole deformation gage. (After Hooker and Bickel.’’)

in each of the six arms. A multiconductor cable connected to the instrument core transmits dc current to the stain gages and in return feeds back to a strain indicator on the surface voltage changes resulting from strain gage length fluctuations. The placement end of the borehole deformation gage is equipped with two orientation pins which engage matching hooks in the installing rods. Thus, a rigid connection is formed between the operator on the surface and the instrument in the hole which allows orientation of the borehole deformation gage. The gage was developed by the US. Bureau of Mines4and the testing procedure has been described in detail by Hooker and Bickel.21

DRILLING

INSTALLATION

OVERCORING

FIG. 4. Major steps in borehole deformation gage measurement procedure.

17. MEASUREMENT

385

OF IN SITU STRESS

The testing procedure (Fig. 4) involves first drilling the overcore-size hole (100-150 mm in diameter) to some 20-40 cm above the predetermined testing depth. Then a concentric 38-mm-diametertest hole is drilled into the bottom of the overcore hole for at least 50 cm. The calibrated borehole deformation gage is now installed at the required depth, oriented in a known direction, and the initial diameter readings taken. The installing rods are removed, and the conductor cable is threaded through the overcore diamond bit, drilling rods, and water swivel and is reconnected to the surface power supply and strain indicator (or other recording device). The gage is now ready for overcoring. The overcored hole must be perfectly concentric with the test hole containing the gage. During the overcoring operation, which continues to about 10-20cm beyond the gage nose, the changes in the test hole diameters as sensed by the gage cantilevers are recorded continuously (Fig. 5). The overcoring procedure produces a thick-walled rock cylinder detached from the rock surroundings and, hence, free of the prevailing in situ stress. The difference between the initial and final diameter readings for each cantilever pair corresponds to the total diametral deformation in the respective direction resulting from the state of stress at the point of measurement. The overcored rock cylinder is removed from the hole and used for the OVERCORE DISTANCE (cm) 0

5 I

m-

‘0 €

0

s

-0.5

E

,

cU f

-1.0

BWp

-1.5

20

25

30

DIA. I A DIA. 2 DIA. 3 0

0.5

U

-

10

&

-

k; -

W

-J

0

I W K

2

-2.0

PLANE O F -MEASUREMENT -2.5

-

FIG. 5. Continuous diametral deformation readings taken along three directions as the borehole deformation gage is overcored. The plane of measurement is that of the six carbide buttons (see Figs. 3 and 4). (After Fischer.”)

386

BEZALEL C . HAIMSON

determination of elastic properties such as modulus of deformation and Poisson ratio. This is accomplished by inserting the overcore into a biaxial cell and subjecting it to known confining pressures while reading the corresponding diametral deformations. Calculation of the in situ stress tensor is based on the theory of elasticity and the Kirsch solution.’ Assuming that the rock is homogeneous, continuous, linear, and isotropic and that the test hole is vertical and parallel to a principal stress direction (z), the relationship between the ith diametral change (Adi) and the principal stresses is Adi = (~/A??)[(OH + Oh)

+ 2(OH - Oh)(l

-

V2) COS2 Oj

- VtYz]

(3.1)

where i = 1,2,3 ;d is the initial test hole diameter, and Bi is the angle between the known direction of the ith cantilever pair and that of the largest horizontal compressivestress OH. The set of three equations representing the three cantilever pair readings yield unique values for O H , Oh, and 91, since the angles between the cantilevers are known and the value of O, can usually be estimated (typically assumed to equal the weight of the overlying rock). The case of anisotropic rock has been treated by Amadei.22 The borehole deformation gage is a relatively inexpensive rugged tool which can be used repeatedly in dry or wet holes. It has been found to work well in rock which will maintain an intact overcored cylinder. If the overcored rock fractures during drilling the test fails, since the diametral readings are bound to be erroneous. Thus, densely fractured or bedded rock and areas of high stresses (which may induce core disking) are not suitable for this method. Testing depths cannot be too great and have been limited to about 50 m because of the difficulty of concentric overcoring and of orienting the tool as the distance increases.23 Like all the overcoring techniques, the borehole deformation gage has been developed for use both from the surface of the earth and in underground mining or civil excavations. Most of the in situ stress measurements by this method have been confined to North America, and a recent summary of results in the United States has been published in Zoback and Zoback.”

3.2. Doorstopper The doorstopper strain cell is designed to determine the in situ state of stress in rock by measuring the strain changes on the flattened end of a borehole on relieving the stress by o v e r c ~ r i n g .The ~ * ~name ~ “doorstopper” comes from the resemblance of the strain cell to a household doorstopper. It consists of a foil strain gage rosette whose top surface forms the bottom of a cylindrical rubber casting fitting a cylindrical acrylic shell. At the other

17. MEASUREMENT OF rN

srm STRESS

387

ORIENTATION

ACRYLIC

RUBBER CA

FIG.6. Doorstopper gage. (After Fischer.2’)

end of the cylinder a multipin connector enables the rosette and the temperature compensating gage (bonded to a rock chip which is suspended in the rubber casting) to be electrically connected to the surface power supply and strain indicator via a conductor cable (Fig. 6). The rubber cylinder protects the strain rosette during the gage installation and subsequent overcoring. The sequence of events in a typical doorstopper measurement is shown in Fig. 7. A test hole 60-70 mm in diameter is drilled to the depth of interest, the end of the hole is ground flat and smooth with a special diamond bit, and the surface is cleaned as required for strain gage application. A doorstopper strain cell mated to an installing tool is lowered to the hole bottom with steel connecting rods. There it is oriented in a predetermined direction and is pushed against the flat end, applying pressure for several minutes. The fastsetting adhesive previously spread over the strain rosette backing is now

DRILLING

INSTALLATION

GLUEING

OVERCORING

FIG.7. Major steps in doorstopper measurement procedure.

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BEZALEL C . HAIMSON

squeezed between doorstopper and hole end and allowed to cure. When the gage is firmly attached the initial strain rosette readings are taken. Typical rosette strain gage arrangements are at 45” or 60” from each other. The installing tool is removed and the test hole is deepened by using a coring diamond bit which overcores the doorstopper for a distance of about 10 cm and thus relieves the stresses at the hole end. In the original design the conductor cable was disconnected during overcoring and had to be reattached at this stage for the final set of strain rosette readings.24 However, the reconnection is awkward in deeper holes, and by missing the continuous recording a potential malfunction of any of the gages may go unnoticed. In more recent versions of the tool the cable remains connected throughout the drilling and the changes in strain gage readings are recorded c o n t i n u ~ u s l y . ~ ~ After completion of the final strain readings the overcored cylinder is removed, brought to the surface, and used for the determination of the elastic parameters. The hole is now ready for the next doorstopper test. Calculation of the principal stresses in the plane of the hole end involves two steps using well-established relationships. First, the principal hole-end strains and their directions are calculated from the three strain gage readings in a way identical to that described in Section 2.1 [Eq. (2.1)]. Then the holeend principal stresses are calculated from the elastic strain-stress relationships, which require accurate knowledge of the modulus of deformation and the Poisson ratio of the tested rock [Eq. (2.2)]. However, determination of the far-field state of stress from doorstopper overcoring is not as simple as in the case of the borehole deformation gage because no analytical solution exists to the relationship between the hole-end stresses and those prevailing in situ. Assuming that the vertical hole axis is aligned with the direction z, along which one of the principal stresses acts, the relationships between the hole-end principal horizontal stresses OH, and Oh, e and the in situ principal horizontal stresses OH and Oh are -BH,e

=acH

+ bah

-bh,e = b a H

+ agh

(3 4

where a and b are coefficients determined by numerical or photoelastic modeling. Over the years workers have obtained different values of a and b, depending on the approximation method used. Goodmanz6 prefers the values a = 1.30 and b = (0.085 + 0 . 1 5 ~- vz) as obtained in de la Cruz and Raleigh.” The resulting OH and Oh [Eqs. (3.2)] are the horizontal in situ stresses, and they act in the same directions as O H , e and O h . e , respectively. The doorstopper technique has a number of advantages over other stress relief methods. The major one is that its overcoring is performed with the same coring bit required for drilling the test hole. Thus, costs are lower, the concentricity requirement between overcore and gage is easily met, and tests can be performed at greater depths (approaching 100m or so). The

389


doorstopper is inexpensive and can be purchased commercially or produced in one's own lab~ratory.'~' 25 Like the borehole deformation gage, the doorstopper helps determine the stresses in a plane (that of the flat hole-end) and knowledge of the third principal stress is necessary. Doorstoppers can be used successfully in both soft and hard rocks. Major disadvantages of the method include the imprecise relationship between the hole-end stresses and those prevailing in situ and the difficulties of cleaning and preparing flat ends in deep holes and of bonding strain rosettes to rock under water. As with other overcoring techniques, the segment of hole tested should be solid and the overcore should remain unfractured for at least 10 cm below the doorstopper. This requirement, however, is considerably less stringent that in the case of the borehole deformation gage, in which the overcore involved is substantially larger. The doorstopper has been employed widely for determination of regional stresses in countries such as South Africa,2 Germany2' and Canada.28 Most of these measurements were conducted from deep underground mines.

3.3. Triaxial Strain Cell The triaxial strain cell is an instrument for measuring the complete state of stress from one test by the overcoring technique.24In that respect it departs considerably from the previously described overcoring methods, which are used to estimate the state of stress merely in a plane perpendicular to the test hole axis. The triaxial strain cell consists of a temperature-compensating (dummy) gage glued to a rock chip and three active strain gage rosettes arranged in a plane normal to the axis of the test hole, typically at 120" from each other (Fig. 8). In each rosette one of the strain gages is aligned with the direction of the hole axis, one is along the circular cross section of the borehole wall (i.e., at 90" to the first gage), and the third is at 45" to each GAGE ORIENTATION IN ROSETTES

Z FIG.8. Triaxial strain cell showing the location of the three strain rosettes. Also shown are the gage orientations within each rosette. (After Leeman.")

390

BEZALEL C . HAIMSON

of the other gages (Fig. 8). In principle, the triaxial strain cell technique consists of inserting the cell into a pilot hole at the required depth, cementing the three active strain gage rosettes to the hole wall, taking the initial gage readings, overcoring, and taking the final gage readings. From the recorded strain changes resulting from overcoring around the three rosettes the complete state of stress can be calculated if the elastic constants of the tested rock are known. The calculation of the stress tensor is rather tedious and involves several steps. The secondary principal strains in the plane of each strain rosette are determined first. This assumes that in each rosette the strain gages are sufficiently short and close to each other to render the surface to which they are bonded a plane. In each rosette the three strain readings yield uniquely the magnitudes and direction of the respective secondary principal strains as shown in Eq. (2.1). The secondary principal stresses can also be calculated if the elastic properties of the rock are known [Eq. (2.2)]. From these results the specific stresses Pie, Piz,and Pie in the ith rosette plane (i = 1,2, 3 ) can be determined. These are the tangential, axial, and shear stresses, respectively, in the borehole wall at the position occupied by the rosette. If the axis of the test hole is parallel to the z direction of an arbitrary set of Cartesian coordinates xyz (Fig. 8), the relationships between the in situ far-field stress components om,, (where m, n = x , y , z) along these axes and the stresses on the wall of the borehole Pie, Piz, Pie area

piz = - v[2(oxx- ayy)cos 28i

Pie

= -axzsin Bi

+ 2ayzcos 8i

+ 40,

sin 2OiI

+ aZz (3.3)

where i3i is the angle measured counterclockwisefrom the positivex direction in the xy plane to the ith rosette (Fig. 8). The set of Eqs. (3.3) has six unknowns representing the six components of the far-field stress tensor in the directions x , y, z. To obtain a solution, six independent equations are required. Thus, as a minimum two strain rosettes, each supplying values Pie, Piz,Pie for its position O i , should be sufficient to provide the six required equations. However, P& should theoretically have the same value in all strain rosettes. Hence, a third strain rosette is added to provide both the sixth independent strain reading and a redundancy necessary to improve the reliability of results. Once the six stress components in the x , y , z directions are determined they can be used to calculate the principal stress magnitudes and their directions. The first known triaxial strain cell was developed by Leeman.24It consists of a cylindrical brass body with three holes around its circumference, into which three rubber plugs are tightly fit. The holes are set at predetermined


391

relative orientations. A strain rosette is molded to the outside of each rubber plug and the leads are connected to the electrical connecting plug of the brass body. To lower the strain cell into the hole, it is connected mechanically and electrically to an installing tool. This tool facilitates the forcing out by means of compressed air of the three rosette plugs into contact with the borehole sidewalls. Thus, with the proper epoxy applied to the rosettes prior to insertion into the hole, the air pressure will assist the bonding of the gages to the rock. The tool is oriented so that the precise direction of each gage is known. The instrument is typically used in 76-mm boreholes. The hole is drilled to the desired depth and the end is ground flat to ensure concentric 38-mm hole drilling for a distance of 50 cm. The triaxial strain cell attached to an installing tool is inserted into the hole, and the strain rosettes are glued to the wall midway along the 38-mm hole. The initial readings are taken when the epoxy cement has cured. The installing tool is then removed and the slim hole is overcored with a thin diamond coring bit of 76 mm outside diameter. The cylindrical core containing the triaxial strain cell is removed from the hole and the stress-relieved strain readings taken. The difference in strain reading in each gage is expected to represent the strain condition resulting from the prevailing in situ stress and is used to calculate the three-dimensional stress tensor as explained above. The major steps of the testing procedure shown in Fig. 4 are followed when using triaxial strain cells. Two of the major disadvantages of the original Leeman24triaxial strain cell are that it requires good rock wall conditions for proper rosette bonding and it does not provide protection of electrical circuitry from drilling or ground water. A modified design of the triaxial strain cell makes use of a solid cylindrical plastic body in which the gages are encapsulated. Bonding is accomplished by filling the annulus between cell and rock with epoxy cement.29In this design the cell is less dependent on the borehole wall conditions and is completely watertight, allowing continuous recording of rosette strains during the process of overcoring. A different improvement in the original triaxial strain cell which renders it capable of being used at considerably greater depths (it has been tested down to 500m) was introduced by the Swedish Power Board.3o Their equipment consists of a long (1 .S m) and wide (60mm) tube-shaped probe which can be lowered on a wireline to the bottom of a 76-mm hole. A strainrosette carrier is attached to the lower end of the probe. The carrier consists of three plastic tongues, each carrying a strain rosette near the free bottom end. The three rosettes are submerged in a glue pot during the downhole trip. When the carrier enters the concentric 38-mm hole and the larger-diameter probe hits the bottom of the 76-mm hole a central cone is released from the probe, falls into the carrier, detaches the glue pot, and presses the tongues

392

BEZALEL C . HAIMSON

with the epoxy-soaked rosettes against the hole wall. The cone supplies the necessary pressure to facilitate good bonding between rosettes and rock while the epoxy cement is curing. During this time a magnetic compass located in the probe is used to establish the orientation of the strain gages. The special compass fluid is first heated electrically to allow correct positioning of the compass needle. It is then allowed to freeze (at temperatures below lS°C), fixing the needle position until it is read off when returned to the surface. When the cement has hardened sufficiently the initial strain readings are taken in all nine gages. Then the probe is hoisted after cutting off the gage wires and leaving the carrier with the three tongues glued to the pilot hole wall. After overcoring, the hollow core with the gages attached is lifted carefully from the hole, the gages are rewired, and the second set of readings is taken. A complete test at an average depth of 200 m can take about 5 hours if no complications arise. Figure 9 shows the stages of the measurement procedure. The Swedish Power Board method is the most sophisticated of all triaxial strain cell versions and has allowed measurements at depths never before attempted with overcoring techniques. However, two major disadvantages of this method are that the strains cannot be read continuously during overcoring and the second set of readings can be taken only when the

I

2

3

4

5

6

FIG.9. Major steps in measurement procedure with the Swedish Power Board triaxial strain cell: 1, a 76-mm hole and concentric 38-mm extension are drilled; 2, probe is lowered and strain gage carrier with glue pot is inserted into the small hole; 3, probe touches bottom of large hole, pushing glue pot down and freeing strain gage-carrying tongues; 4, after allowing for cement curing and initial gage readings, probe is hoisted, leaving gage carrier in small hole; 5 , overcoring is done with 76-mm bit; 6, overcore is hoisted and final gage readings are taken. (After Hiltscher et

d 3 0 )

17.


393

overcore has been brought up to the surface. The first disadvantage implies that a malfunction of one of the gages during drilling will go unnoticed and may bias the interpretation of the strain readings. The second implies that some tests that could otherwise have been successful may have to be discarded if the overcore breaks during its detachment and lifting to the surface. The extended time between the initial and final readings and the possible effects of temperature variations in the overcore can also distort the resulting strain readings. There is a substantial literature on successful tests with the different versions of the triaxial strain cell. However, these have mainly been associated with geotechnical investigations of underground mines or civil In one case the Swedish Power Board cell compared favorably with the measurements conducted by hydraulic fracturing in the same 400-m vertical hole. 31

4. Deep Measurements Deep measurements of in situ stress are those conducted in boreholes deeper than 100 m. The primary technique used is the hydraulic fracturing method, which in the past decade has become the most common means of estimating crustal stresses. However, other techniques are being developed which may revolutionize this field in the next decade. Four such techniques are described briefly. All four methods are already at a stage where they produce useful results. 4.1, Hydraulic Fracturing

Probably the greatest advance in the field of in situ stress determination was made in the past 15 years or so with the development and use of the hydraulic fracturing measurement technique (also called hydro fracturing or hydrofrac). Hydrofracturing is a borehole technique like most stress relief methods, but it does not require precision concentric drilling, tedious underwater strain gage gluing, or delicate overcoring. The downhole hydrofracturing equipment is rugged, has practically no moving parts, and in some versions has no electrical components. These advantages render hydrofracturing the only method commonly used for stress measurement at great depths. Test holes l00Om deep have become rather r ~ u t i n e , ~and ”~~ measurements to depths of 5000 m have already been conducted.34 The hydrofracturing stress measuring technique consists of sealing off a section of the test hole at the depth of interest by means of two inflatable rubber packers and pressurizing the section with a hydraulic fluid such as water. At some critical (also called breakdown) pressure, the rock at the

394

BEZALEL C . HAIMSON 0

2

loL 0

L

2

I

4

TIME (min)

FIG.10. Typical pressure-time and flow rate-time curves during a hydrofracturing test. (a) Two shut-in pressure levels are recorded ; one (P?)corresponds to vertical hydrofracture and the other (P?)is related to horizontal hydrofracture. (b) Only a vertical hydrofracture was induced. Both packer and test-zone pressures were recorded downhole.

borehole wall bursts and develops a tensile fracture. This fracture can be extended away from the hole by continued pumping. When the pump is shut off without venting the hydraulic line a shut-in pressure is recorded (Fig. 10). This is the pressure necessary to keep the fracture open. The pressurization cycle is repeated several times. The breakdown and shut-in pressures, which are carefully monitored during the test, can be related to the prevailing in situ stress magnitudes. Determination of hydrofracture orientation enables estimation of the principal stress directions. In this manner the complete state of in situ stress can be assessed, provided certain conditions on the rock mechanical characteristics are met. Theoretical expressions relating hydrofracturing pressures to in situ stresses have so far been developed only for linear elastic and isotropic rocks in which one of the principal stresses acts in a direction parallel to the axis of the test hole. The parallelism prerequisite between hole axis and principal stress direction is usually achieved by drilling vertical test holes, since the expectation is that one of the principal stresses is nearly vertical in most locations. Two distinct stress conditions exist. In one the vertical stress (av) is the least principal compressive stress; in the other the vertical stress is either the intermediate or the largest principal stress. A hydraulic fracture will follow the path of least resistance, that is, a direction perpendicular to that of the

17.


395

least principal stress. Laboratory and field results show that when inflatable rubber packers are used to seal off an unprefractured interval in massive rock, the initial hydrofracture is almost always vertical and perpendicular to the least horizontal principal stress (Oh), regardless of the magnitude of ov . If ov is the least principal stress, the hydrofracture may turn around and become horizontal as it travels away from the stress field imposed by the pressurized test hole. If ov is not the least principal stress, the induced hydrofracture will extend along its initial direction. The shut-in pressure Ps needed to keep the hydrofracture open when pumping is stopped is equal to or slightly larger than the in situ compressive stress perpendicular to the fracture plane. If ov is not the least principal compressive stress, Pswill stay constant provided there are no leakages past the packers or into preexisting joints or rock pores, and the relationship

Ps

= (Th

(4.1)

will determine the least horizontal principal stress. The vertical principal stress is calculated from the weight of the overlying rock :

uv = YD

(4.2)

where y is the rock weight gradient and D the depth. If oV is the least principal compressive stress, a vertical fracture may nonetheless initiate first, yielding a primary shut-in pressure (Ps'). Soon the fracture will become horizontal and a second shut-in pressure (Ps")will be recorded (Fig. 10). Clearly, P,"> P," and

In this situation, both the least horizontal principal stress and the vertical stress are directly determined from hydrofracturing pressures. In some hydraulic fracturing tests the pressure change after the cessation of pumping is gradual and shut-in pressures are not distinct enough to be measured accurately. This phenomenon is due primarily to leak-off from the straddled interval and fracture into the surrounding rock, leakage past the packers, or further fractures propagation after pumping stops. Various techniques based on different theoretical models are used by practitioners to deal with indistinct shut-in pressures.35 These models have not been developed rigorously to suggest one definitive approach to determining P,. Pumping at a very low flow rate in one or more of the pressurization cycles, yielding a pressure level that is just sufficient to overcome leaks and keep the induced hydrofracture open, can be used with confidence as representing Ps in these situations (Fig. 10).

3 96

BEZALEL C. HAIMSON

To estimate the value of the major horizontal principal stress (OH), the poroelastic relationship between the critical (breakdown) pressure (Pel) necessary to induce a vertical hydrofracture and the two horizontal principal stresses is given by36 Pcl

- PO = ( T + 3411 -

OH

- 2Po)/K

(4.5)

where compressive stresses are taken as positive, POis the initial pore pressure in the rock at the tested depth, T is the hydrofracturing tensile strength, and K is a poroelastic parameter which can be determined independently in the laboratory. The range of K is 1 < K 2. K = 2 when the Poisson ratio equals 0.5. In practice, the values of T and K can be derived from a plot of (Pel - PO) against (301, - OH - 2P0) based on laboratory-simulated hydrofracturing tests in which the principal stresses are known since they are the controlled variables. The trend emerging from such series of tests in five rock types37 in which no pore pressure was applied is presented in Fig. 11. The parameter K is not constant, but in the approximate range 0 < PCl - PO - T / K < 25 MPa, K is not significantly larger than 1.O. Beyond 25 MPa, the value of K increases and approaches the value 2.0 asymptotically. In the field tests conducted to date, it has always been assumed that rock is perfectly impermeable to the fracturing fluid, and the universal relationship used for calculating OH has been38 Pcl

= T

+ 3Oh - OH - PO

(4.6)

The complexity of determining the variable parameter K and of using it correctly in Eq. (4h)has caused practitioners to prefer the simple approximation given by Eq. (4.6).However, as measurements reach greater depths

30,- OH (MPo) FIG. 1 1 . Relationship between breakdown pressure ( P c ,representing either P,I or Pc2)and horizontal in situ stresses as obtained by averaging laboratorytest results for five hard rock types and normalizing for T = 0. In these tests PO was held at zero. (After Edl.”)

17. MEASUREMENT

OF IN SITU STRESS

397

and zones of high in situ stress, the use of Eq. (4.5) may become inevitable to prevent considerable errors in calculating OH. The value of T is a rock parameter that varies with type of loading, rate of loading, and size of specimen. Obtaining a correct value of the hydrofracturing tensile strength would require hydrofracturing an additional borehole of the same size as that of the field test in a rock identical to that tested but subjected to no far-field stresses. Since this is difficult to accomplish, simulated laboratory hydrofracturing tests in small specimens of extracted core have been conducted to approximate T. The uncertainty in the value resulting from borehole size difference has led to an alternative method of determining OH that does not require knowledge of T.39This method uses the pressure necessary to reopen a hydraulic fracture, Pc2, instead of that causing initiation of the fracture, PCl(Fig. 10). The basic assumption is that the hydraulic fracture closes tightly at the completion of the pressurization cycle and that it opens suddenly to accept fluid when the borehole is repressurized. Using Pc2, Eq. (4.6) becomes

Pc2 =

3Uh

- OH - PO

(4.7)

Owing to the various assumptions made in arriving at Eq. (4.7), the value of OH is necessarily only an estimate. However, comparisons made between hydrofracturing results and various overcoring techniques have repeatedly shown excellent correlation^.^' The direction of hydrofractures can be determined by a number of methods. The most common and reliable way is to use an impression packer, which is an inflatable sleeve covered with a sheet of very soft rubber. When forced against the wall of the borehole, the soft rubber takes an imprint of the rock face condition and maintains a clear picture of it long after the packer has been deflated and retrieved to the surface. Packer impressions are oriented by employing a borehole surveying instrument (magnetic or gyroscopic) or other techniques. Impressions of successful hydrofractures are consistent within the same area and determine the directions of Oh and OH. When ovis the smallest compressive stress, both a vertical and a horizontal hydrofracture will often be traced by the impression packer. Thus, the complete state of stress in the vicinity of the test hole is determined from the results of two operations, pressurization and impression. The equipment required to run a hydraulic fracturing test can be divided in four groups according to tasks : borehole sealing, pressurization, recording, and fracture delineation. Two inflatable rubber packers, spaced apart a distance equal to at least six hole diameters, are interconnected mechanically and hydraulically and form the “straddle packer” tool used for sealing off a test interval. The packers are available commerciallyin a variety of sizes so that hydrofracturing can be conducted in virtually any hole diameter. The

398

BEZALEL C. HAIMSON

straddle packer is lowered into position either on a string of high-pressure tubing or drill rods or by using a geophysical well-logging irel line.^^' Packer inflation and pressurization are carried out through the tubing or via a separate slim flexible hose. The pressure in the packers is maintained at about 2 MPa over that in the interval so that leaks past the packer are usually prevented. In commercial straddle packers a valve attached to the straddle tool, and controllable from the surface, seals the packers at the conclusion of their pressurization and opens the straddled interval to the same hydraulic line. In other tools two separate hydraulic lines connect the surface pumps to the test zone, one for hydrofracturing the interval and the other for packer inflation. Thus, each line is separately controllable throughout the tests. Pressure in the two hydraulic lines can be monitored during the test by using electronic pressure transducers at the surface and, preferably, in the sealed-off interval (Fig. 12). If downhole transducers are not installed a selfcontained pressure gagelrecorder package can be lowered into the test interval. The volume of fluid injected into the rock is also monitored through a flow meter at the surface. The outputs of the flow meter and the pressure

FIG. 12. Wireline hydrofracturing field testing setup. (After Haimson and Lee.42)

17. MEASUREMENT OF IN SITU STRESS

399

transducers are recorded permanently on strip chart recorders for immediate observation. More recent developments are improved analog recording on tape (through a data tape recorder) and digital recording and analysis by computer. Fracture orientation is obtained by use of an impression packer. This is a regular inflatable packer with an outer layer of semicured rubber. The packer is lowered to the depth at which the hydrofrac test was conducted and is pressurized to a level higher than the secondary breakdown pressure. This allows the soft rubber cover of the packer to penetrate the slightly open fracture and take an imprint of it. An orienting device, magnetic or gyroscopic, is used to provide the azimuth of the fracture strike (Fig. 13). Another device used to determine fracture orientation is the borehole televiewer, which is a sonic logging tool that takes an oriented acoustic picture of the borehole wall.43This tool is considerably faster than the impression packer, but it is sometimes not sufficiently sensitive to detect hydraulic fractures that have completely closed after the pressurization stage of the test. The borehole

FIG. 13. Wireline packer impression setup. (After Haimson and Lee.42)

400

BEZALEL C . HAIMSON

FIG. 14. Photograph of borehole sonic televiewer record before and after hydrofracturing. A subvertical hydrofracture cutting diametrically through the hole is clearly noticed in the picture at the right. The azimuth of the induced fracture can be read from its position on the borehole wall; major directions are marked at the bottom of picture. (Courtesy of Dr. H. Tsukahara.)

17.


40 1

televiewer provides additional information, such as the condition of the borehole wall and the existence of “breakouts,” but is significantly more expensive than the impression packer-orienting tool (Fig. 14). In recent tests both instruments have been made available at the test site. The televiewer is then used first, and only where it fails to discern hydrofractures is there a need to employ the impression packer. Starting with the historic test at Rangely, color ad^,^.' hydrofracturing stress measurements have been conducted to date on four continents.”’ 12*41*44.45 There have been measurements for the rational design of underground caverns,46 for the determination of regional crustal stresses,”’ 33 for earthquake prediction for induced seismicity studies,6i35B42 and for a number of other engineering and geological purposes. A complementary or alternative method to conventional hydrofracturing, which eliminatesthe need for PCz(or PCland T)in determining the horizontal in situ stresses, has recently been ~uggested.~’ This new approach makes the reasonable assumption that the vertical stress is a principal component and that each of the non-zero components of the in situ stress tensor varies linearly with depth. To calculate the stress tensor, a minimum of six (but preferably seven or more to reduce uncertainties) hydrofracturing tests are required over the depth range of interest, from which reliable P, and fracture orientation values should be obtained. These yield at least six values of normal stress (not necessarily a,)magnitudes and directions from which the principal stresses can be calculated in a manner similar to the strain rosette analysis. This method actually works best in rocks for which the induced hydrofractures are not always perpendicular to the direction of Oh. In many crystalline rocks, such as granites and quartzites, various weaknesses in the rock may in fact induce fractures which are not well aligned with the principal stresses. The suggested method could also be used to advantage in hydrofracturing tests conducted in inclined holes. In such cases one can at most obtain from conventional hydrofracturing interpretation the least horizontal stress (Oh) and its variation with depth.50Using the new technique the complete stress tensor could theoretically be obtained in inclined holes provided sufficient tests are conducted which do not all result in identical hydrofracture orientations. Finally, this method can also be employed in fractured rock. Since the shut-in pressure method does not require the creation of a new fracture, it can use the reopening and pressurization of closed natural fractures from which the correct Ps values are obtained for principal stress calculations. In general, the great potential of this method is in providing an independent check on stress values obtained conventionally, as well as in replacing common hydrofracturing when the latter cannot be used.

402

BEZALEL C . HAIMSON

4.2. Borehole Breakouts

Intervals with elongated cross sections (breakouts) have been observed repeatedly in boreholes and oil wells.s1i52 Leeman" interpreted the elongations as borehole sidewall fracturing resulting from excessive differential stresses in the plane normal to the hole axis. He noticed that the long axis of the cross section was aligned with the least compressive stress direction in that plane. His interpretation was independently supported and extended by Gough and Bells3to indicate that breakouts are the result of shear failure caused by stress concentration. In a vertical hole the nature of this stress concentration implies that the larger horizontal compression (OH)is normal to the azimuth of the elongated axis. Thus, if the direction of the breakouts can be established, the orientations of the horizontal principal stresses can also be determined. The most complete analytical model available to date for predicting the occurrence as well as the location, size, and shape of borehole breakouts is based on rock remaining linear elastic until a Mohr-Coulomb type of shear failure occurs.s4 In this plane strain model, the radial, tangential, and shear stresses in a horizontal cross-section are calculated at every point around the vertical borehole. They are then compared with the stresses required to cause shear failure, and a contour line can be drawn enclosing the zone of failed rock. This zone is assumed to represent the size and shape of the borehole breakout and can be characterized by the maximum breakout depth and the breakouts occur in the direction of Oh, and suggests that one can estimate the two principal horizontal stresses, CTH and Oh, by accurately determining the breakout depth and span. This model has the advantage of mathematical simplicity; however, it does not anticipate the existence of discontinuities, the potential for material yielding, the time-dependent characteristics of some rocks, and the possibility that breakouts are episodal. A comprehensiveseries of laboratory simulations of borehole breakouts in one rock type under conditions analogous to those conducted in the field has recently been c ~ m p l e t e dThese . ~ ~ tests confirmed that breakouts occur in two diametrically opposed zones along the borehole wall, in the direction of the least horizontal stress. It was also found that the total depth and the lateral span of breakouts appear to be directlyproportional to the state of horizontal insitu stress. Thus, a clear potential exists for also using breakouts to estimate stress magnitudes if the dimensions of the failed zone can be determined. The tool that has provided the means of detecting borehole elongation in oil wells is the modern four-arm high-resolution dipmeter, a wellbore logging device used to obtain subsurface structural information and hole deviation. The tool has four hydraulically actuated caliper arms spaced 90"apart. These arms measure the width of the hole and, where a tangible deviation from


403

circularity occurs, record the length and azimuth of the long and short axes. Although elongated boreholes can arise from a number of causes, such as anisotropic rock characteristics forcing the drilling bit to remove more material in one direction than in the other, it is believed that the most frequent borehole asymmetry is caused by preferential wall spalling. Surveys of oil company dipmeter records52s53 reveal that large areas of Alberta, northern Canada, Texas, Colorado, and other regions have uniformly oriented borehole breakouts, implyingthat they are under consistent regional stress directions. Although the four-arm dipmeter does not yield quantitative stress values, it does provide very important qualitative information regarding principal stress directions. The four-arm dipmeter has a number of limitations : it is made for use only in large-diameter holes, the arms are rather thick so that only wide breakouts will be detected, and the tool is available only through oil-well service companies. A recent development has been the discovery that the borehole sonic televiewer43can also detect borehole breakout^.'^ In the televiewer photographs, breakouts are discernible as regions of low reflectivity. The advantages of the televiewer are that the tool can be used in a hole of any size, starting from 75 mm in diameter, and that it provides a more precise means of determining the tangential and radial extent of the breakout. In the future the geometry of the breakouts may also be used estimate the magnitudes of horizontal stresses. 4.3. Holographic Interferometry

A method of measuring the in situ stress in deep boreholes by holographic interferometry has been de~eloped.’~ It consists of lowering an instrument into a borehole and using it to drill a small hole into the borehole wall. The stresses around the hole are relieved and the resulting displacement field is recorded on film as an interference hologram. The deduction of the state of stress from the local stress relief is similar to that in conventional threedimensional overcoring methods.24Holographic interferometry is an ultrasensitive method of recording small displacements. An interference hologram is obtained by recording holograms of the borehole surface before and after drilling of the small hole. Any deformation resulting from the introduction of the new hole will appear as a series of dark lines, or fringes, and may be related to a displacement value. The advantage of the hologram over strain gage devices used in overcoring techniques is that, instead of point measurements of displacements, it yields a complete picture of the displacement field of the borehole wall around the stress-relieving hole.’6 A holographic stress measurement has been carried out in a shale mine in Colorado, yielding stress components in good agreement with previous measurements by hydr~fracturing.’~ The method, however, is still developmental, and at

404

BEZALEL C. HAIMSON

present it can be used only in holes that are at least 30cm in diameter. A 15-cm-wide instrument is currently being designed. 4.4. The Differential Strain Curve Analysis and the Anelastic Strain Recovery Technique

The differential strain curve analysis is based on a laboratory experimental technique for measuring total microcrack volumes and orientations in rock cubes.57 Samples are prepared from oriented core and a minimum of six strain gages are bonded to their outside surfaces in predetermined directions. Each sample is jacketed and loaded hydrostatically to a pressure of about 200 MPa to eliminate any crack porosity. From the pressure-strain curve obtained for every gage the contribution of crack closure to the recorded strain can be established. The three principal crack strains and their directions can be determined using the six or more gages. Assuming that most of the microcracks in the sample are due to in situ stress relief occurring during the cutting of the core, and that in the undisturbed field conditions the number of open microcracks is negligible, the established crack strain tensor can be interpreted in terms of the in situ state of stress.58s59The method has been used on several occasions, including the Hot Dry Rock Geothermal Site at Los Alamos, New Mexico, and has yielded reasonable results. Its major limitations are the need for extracting oriented core, the requirement that all or almost all rock microcracks are closed in situ, and the necessity to measure rock elastic parameters in order to estimate the stresses. This method is still largely developmental, but may become valuable in deep stress measurements where high in situ stresses prevail and where the probability of microcrack closure is greatly enhanced. A similar method of estimating stresses from laboratory tests uses the phenomenon of anelastic strain recovery.60This method exploits the timedependent response of rock to in situ stress relief during core cutting. If precision strain gages are applied to samples prepared from fresh oriented core during the first few hours followingrecovery, and the continuing deformation measured, the observed strain recovery is proportional to the stress relieved in the direction of the measurement. The recovered strain tensor can be determined in a way similar to that described for the differential strain curve analysis. With appropriate assumptions regarding the creep law applicable to the tested rock, the virgin state of strain can be estimated, and from it the in situ stress tensor can be evaluated if material properties are known.

5. State of Stress in the Earth’s Crust As a result of in situ stress measurements conducted in the past several decades, some broad generalizations about crustal stress can already be

17.


405

made. Summarizing all the known measurements around the globe, it is apparent that the vertical stresses in the top 2.5 km of the crust scatter about a straight line representing the gravitational gradient for a rock density of 2.7 g/cm3." This finding confirms the suggestion that the vertical stress is not significantly affected by stress fields other than gravitational. As explained in Section 1, if the horizontal stresses were also a result of only gravitational forces, they would be expected to be uniform (i.e., OH = ah) and equal to or smaller than a v . In situ measurements, however, reveal that the principal stresses are typically unequal. At shallow depths at least one of the horizontal components is often larger than the vertical stress, whereas at greater depths horizontal stresses can be either smaller or greater than QV ,l 1 Horizontal stress orientations are generally consistent over large areas and are amenable to geologic analysis ;magnitudes of horizontal stresses cannot, generally, be predicted. However, it is clear that measured horizontal stress magnitudes increase with depth and that the relative magnitudes of all principal stresses are consistent within tectonic provinces.". I' One of the first compilations of horizontal stress data in North America was publishedin 1978.6l Amap based on hydrofracturing and overcoring tests showed that the maximum horizontal stress direction varied from northnortheast along the San Andreas fault, to northeast in the Basin and Range, to east-northeast in the Great Lakes area, eastern United States, and southern Ontario. A plot of the magnitudes of Oh and CJH versus depth indicated that in the top 5 km of the crust they increase linearly with depth but the rate of increase is lower than that of ov. A more complete stress map of the United States, published in 1980," included in situ stress measurements as well as stress indicators such as earthquake focal mechanisms and geological observations. McGarr6' used data from North America, southern Africa, and Australia to conclude that the maximum shear stress [(OH - av)/2 in compressional regimes, i.e., where OH > Oh > a ~ (av ; - ah)/2 in extensional regimes, i.e., where ov > OH > Oh] increases linearly with depth to at least 4 km. Generally, the maximum shear stress in hard rock (such as granite and quartzite) increases at twice the rate observed in soft rock, and the maximum shear stress in compressional regimes in considerably larger than under extensional conditions of stress. These results agree with the limiting values for lithospheric stress provided by laboratory tests of rock

6. Future Research Major advances in the in situ measurement of crustal stress have been accomplished in the past 20 years, with respect to both the development of techniques and the number of field tests conducted. The latter have contributed to an improved picture of the general stress regimes in some areas

406

BEZALEL C . HAIMSON

of the world, such as North western E u r ~ p e , ~Iceland,18 ’ southern Africa,2 and Japan.44 Much remains to be done toward improving the present techniques and developing new ones so that measurements become faster, less expensive, more reliable, and adaptable to difficult rock conditions. Probably the most common difficult rock condition is that in which the rock has at least one of the following characteristics: nonlinear elastic, anisotropic, inhomogeneous, anelastic. Establishing appropriate stressstrain relationships under these conditions is essential for correctly analyzing the field data. Anisotropy also requires a new approach to interpreting the directions of principal stresses from the known orientations of induced hydro fractures. Another difficult environmental condition is high rock temperature, which is encountered as test holes reach deeper into the crust (holes 10,OOO m deep are being planned) and as anomalously hot rocks are approached at relatively shallow depths (such as the Jemez plateau in New Mexico and the Salton Sea geothermal field in California). Most of the present stress measurement techniques require thorough redesign in order to function properly at temperatures of several hundred degrees. Some materials, such as that required to replace the rubber of inflatable packers, may not even exist at this time. In addition to technical difficulties at elevated temperatures, there is also the problem of interpreting the results, that is, sorting out tectonic stresses from thermal stresses. A different type of hostile environment to which stress measuring techniques should be adapted is ubiquitously fractured rock such as that encountered in the vicinity of active faults. To date, all of the in situ stress measurement methods require that rock be continuous or intact at least in the neighborhood of the test zone. Measurements in fractured rock require the development of new techniques, or at least substantial modifications of existing ones, and revision of inteipretation methods. Together with the research effort to improve existing techniques and develop new ones for the in situ measurement of stress in rock under nonideal conditions, there is also a need to investigate the potential for remote measurements. Attempts to use geophysical methods have been unsuccessful so far, but a greater effort to develop such methods may be necessary as we begin to study the deeper zones of the earth’s crust. References 1 . J . C. Jaeger and N. G . W. Cook, “Fundamentals of Rock Mechanics,” 3rd Ed., p. 593. Chapman & Hall, London, 1979. 2. N. C. Gay, Tectonophysics 29, 44 (1975). 3. N. Hast, Sver. Geol. Unders., Ser. C52, l(1958).

17. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.


407

L. Obert, Min. Eng. 14, 51 (1962). E. R. Leeman, J. S. Afr, Inst. Min. Metall. 65, 82 (1964). B. C. Haimson, Proc. U.S. Symp. Rock Mech., 14th. Penn. State Univ.. 1972 p. 689 (1973). C. B. Raleigh, J. H. Healy, and J. D. Bredehoeft, Science 191, 1230 (1976). M. L. Sbar and L. R. Sykes, Geol. Soc. Am. Bull. 84, 1861 (1973). G. Ranalli and T. E. Chandler, Geol. Rundsch. 64, 657 (1975). B. C. Haimson, in “The Earth’s Crust” (J. C. Heacock, ed.), Monogr. 20, p. 576. Am. Geophys. Union, Washington, D.C., 1977. A. McGarr and N. C. Gay, Annu. Rev. Earth Planet. Sci. 6 , 405 (1978). M. L. Zoback and M. D. Zoback, J. Geophys. Res. 85, 6113 (1980). J. T. Engelder and M. L. Sbar, J. Geophys. Res. 81, 3013 (1976). K. Schaefer and S . Kiel, Mestech. Briefe 15, 35 (1979). C. Froidevaux. C. Paguin, and M. Souriou, J. Geophys. Res. 85, 6342 (1980). J. Handin, Twelfth Quarterly Tech. Rep. to the U.S. Geol. Surv., Texas A&M, College Station, 1971. R. V. de la Cruz and C. B. Raleigh, Int. J. Rock. Mech. Min. Sci. 9, 625 (1972). B. C. Haimson and F. Rummel, J. Geophys. Res. 87, 6631 (1982). M. E. Tincelin. Ann. Inst. Tech. Butim. Truv. Publics 58, 972 (1952). M. Rocha, B. Lopes, and J. Da Silva, Proc. Int. Congr. Rock Mech.. Ist, Lisbon 2 , 57

(1966). 21. V. E. Hooker and D. L. Bickel, In5 Circ. U.S. Bur. Mines No. 8618, p. 32 (1974). 22. B. Amadei, Ph.D. Thesis, p. 472. Univ. of California, Berkeley, 1982. 23. B. C. Haimson and C. F. Lee, Proc. Can. Rock Mech. Symp., 13th, Toronto CIM Spec. Vol. 22, 42 (1980). 24. E. R. Leeman, Rock Mech. 3, 25 (1971). 25. D. J. Fischer, M.S. Thesis, p. 171. Univ. of Wisconsin, Madison, 1982. 26. R. E. Goodman, “Introduction to Rock Mechanics,” p. 478. Wiley, New York, 1980. 27. G. Greiner and J. H. Illies, Pure Appl. Geophys. 115, 11 (1977). 28. G. Herget, Proc. Can. Rock Mech. Symp., 13th, Toronto CIM Spec. Vol. 22, 42 (1980). 29. G . Worotnicki and R. J. Walton, Proc. I.S.R.M. Symp. Invest. Stress Rock, Sydney Suppl., p. 1 (1976). 30. R. Hiltscher, J. Martna, and L. Strindell, Proc. Int. Congr. Rock Mech., 4th. Montreux 2, 227 (1979). 31. B. C . Haimson, in “Hydraulic Fracturing Stress Measurements,” p. 107. Natl. Acad. Press, Washington, D.C., 1983. 32. M. D. Zoback and S . Hickman, J. Geophys. Res. 87, 6959 (1982). 33. B. C. Haimson and T. W. Doe, J. Geophys. Res. 88, 7355 (1983). 34. B. C. Haimson, J. Geophys. Res. 83, 5857 (1978). 35. M. D. Zoback and B. C. Haimson, Proc. Symp. RockMech., 23rd, Berkeley, Cali$ p. 143 (1982). 36. B. C. Haimson and C. Fairhurst, SOC.Pet. Eng. J. 7 , 310 (1967). 37. J. N. Edl, M.S. Thesis, p. 100. Univ. of Wisconsin, Madison, 1973. 38. M. K. Hubbert and D. 0 .Willis, Trans. Am. Inst. Min. Metall. Pet. Eng. 210, 153 (1957). 39. J. D. Bredehoeft, R. G . Wolff, W. S. Keys, and E. Shuter, Geol. SOC.Am. Bull. 87, 250 (1976). 40. B. C. Haimson, ASTMSpec. Tech. Publ. STP 554, 156 (1974). 41. F. Rummel, J. Baumgartner, and H. J. Alheid, in “Hydraulic Fracturing Stress Measurements,” p. 3. Natl. Acad. Press, Washington, D.C., 1983. 42. B. C. Haimson and M. Y. Lee, Proc. Symp. RockMech., 25th. Evanston, Ill. p. 194 (1984). 43. J. Zemanek, R. C . Caldwell, E. E. Glenn, Jr., and L. J. Norton, Geophysics 35,254 (1970).

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44. H. Tsukahara, in “Hydraulic Fracturing Stress Measurements,” p. 18. Natl. Acad. Press, Washington, D.C., 1983. 45. J . R. Enever and B. A. Wooltorton, in “Hydraulic Fracturing Stress Measurements,” p. 28. Natl. Acad. Press, Washington, D.C., 1983. 46. B. C. Haimson, in “Rock Mechanics: Caverns and Pressure Shafts” (W. Wittke, ed.), Vol. 1, p. 31. Balkema, Rotterdam, 1982. 47. Li Fangquan, Li Yan-mei, Wang En-fu, Zhai Quing-shan, Bi Shang-xu, Zhang-Jun, Lin-

48. 49. 50.

51. 52.

53. 54.

55. 56. 57. 58. 59.

60. 61. 62. 63. 64.

Peng, Wei Quing-yun, and Zhao Shi-guang, in “Hydraulic Fracturing Stress Measurements,” p. 130. Natl. Acad. Press, Washington, D.C., 1983. M. D. Zoback, H. Tsukahara, and A. Hickman, J. Geophys. Res. 85, 6157 (1980). F. H. Cornet and B. Valette, J. Geophys. Res. 89, 11527 (1984). B. C. Haimson, W. F. Bawden, and P. Baumgartner, EOS Trans. AGU 67, 1206 (1986). E. R. Leeman, J. S. Afr. Inst. Min. Metall. 65, 45 (1964). E. A. Babcock, Am. Assoc. Pet. Geol. Bull. 62, 1 1 1 1 (1978). D. I. Gough and J. S. Bell, J. Can. Earth Sci. 19, 1358 (1982). M. D. Zoback, D. Moos, L. Mastin, and R. N. Anderson, J. Geophys, Res. 90,5523 (1985). B. C. Haimson and C. Herrick, in “Rock Stress” (0.Stephansson, ed.), p. 271. CENTEK Pub., Sweden, 1986. J. D. Bass, D. Schmitt, and T. J . Ahrens, Geophys. J. R. Astr. SOC. 85, 13 (1986). G . S. Simmons, R. W. Siegfried, and M. Feves, J. Geophys. Res. 79, 4383 (1974). N. K. Ren and J. C. Roegiers, Proc. Intl. Conf. Intl. SOC.Rock Mech., 5th. F117 (1983). T. N. Dey and D. W. Brown, in “Rock Stress” (0. Stephansson, ed.), p. 351. CENTEK Pub., Sweden, 1986. L. W. Teufel, SPE/DOE, paper 13896, 467 (1985). B. C. Haimson, in “Sciences de la Terre et Mesures,” Memoire du B.R.G.M., No. 91, p. 163. Paris, 1978. A. McGarr, J. Geophys. Res. 85, 6231 (1980). W. F. Brace and D. L. Kohlstedt, J. Geophys. Res. 85, 6248 (1980). J. D. Byerlee, Pure Appl. Geophys. 116, 615 (1978).

18. CONTINUOUS MEASUREMENT OF CRUSTAL DEFORMATION

Duncan Carr Agnew Institute of Geophysics and Planetary Physics University of California. San Diego La Jolla, California 92093

1. Aims and Problems of Continuous Deformation Measurernent From the beginning of geology, many measurements have been made of deformation near the earth’s surface, at least partly in the hope that, given a sufficiently full description of the kinematics of the motions, it will be possible to say something about the underlying dynamics. Geological and geophysical mapping give data on deformation over durations of lo4 years or more, and for periods from 1 to 100 years repeated measurements with traditional geodetic techniques (triangulation, trilateration, and leveling) provide data for active seismic zones such as California’ and Japan ;’indeed, geodetic data form the basis of much of our understanding of the seismic cycle.3 But geodetic measurements are usually made too infrequently (repeat times of a year to decades) and in any case are too imprecise strain or tilt with recent improvements in strain giving perhaps lo-’) to give more than a fuzzy and fragmented picture of crustal deformation. The role of tiltmeters and strainmeters is to fill in this picture by making more sensitive measurements continuously. They not only refine our knowledge of gradual motions, but also measure other phenomena (such as tides, seismic waves, and possible earthquake precursors) that are otherwise undetectable. This chapter discusses instruments for making such continuous measurements, with the goal of providing background on available techniques and of giving some appreciation for the problems peculiar to this field, somewhat in the spirit of an earlier paper by G ~ u l t yA. ~fuller review has been published elsewhere.’ So far the level of success reached in this field has been less than might be hoped. Instruments for measuring earth tides and seismic waves accurately have been developed, but the detection of other phenomena remains elusive. In large part this is because the measurements not only must be of very high 409 METHODS OF EXPERIMENTAL PHYSICS Vol. 24, Part 6


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quality but also must be made under unfavorable conditions. For example, a typical tectonic strain rate (determined from geodesy) is E s-l (lo-' &/year), which is comparable to the stability of many material length standards-but the maintenance of a standard to better than this, and its comparison with the earth, must be done not in an air-conditioned laboratory, but in a damp tunnel, a water-filled borehole, or a desert soil. Because continuous deformation measurement is very difficult, the perfect instrument has not been built and may never be. The first step in design is therefore to determine the primary purpose of an instrument, deciding which criteria it must meet and which may be compromised. Then, knowing the strengths and weaknesses of particular techniques, their appropriateness to the application may be judged. I have attempted to give such evaluations of techniques throughout this chapter, but first describe the different types of crustal deformations and what is needed for each one.

2. Quantities to be Measured Crustal deformation measurements are usually separated into strain and tilt; but while very different instruments are used to measure these phenomena, both result from motion of the earth. In general6the motion of a continuum may be divided into three parts : 1. A rigid-body translation x. 2. A rigid-body rotation r = V x s (where s is the total displacement minus x). 3. A strain field E = t ( V s sV).

+

Strainmeters measure some part of E. The commonest type is the extensometer, which measures the change in length along a line; this is d~ E d~,, where do is the direction of the line. Dilatometers, or volumetric strainmeters, measure the volume change tr(E). Ideal tiltmeters would respond only to the second class of motions (the rotations) but in fact measure all three. They respond to strain because a line with original direction do can change direction not just because of a rotation r but also because of strain (for example, a uniaxial strain not directed along or perpendicular to do). They respond to translation because the direction of the vertical is the only one relative to which a rotation of do can usefully be measured; the vertical is found with an inertial sensor, which will respond to accelerations (and hence to the translation x) and to changes in the gravitational potential. The complete expression for the tilt vector is :5

- -

-

-

d ~ ) ( dE) ~ - dO(2o.do E) + Z0 x [&(r do) - r] - g-" v u 1 - 20(20 * v Ul) + x - to(2o a)]

= (20

(1)

18.

CONTINUOUS MEASUREMENT OF CRUSTAL DEFORMATION

41 1

where 20 is the direction and g the magnitude of the undisturbed gravity vector ; U Iis any change in the gravitational potential from its undisturbed state. For a propagating wave with frequency o and phase velocity c, the ratio of the second line to the first is of order o c / g , which is small for the phase velocities of elastic waves and periods greater than an hour. At these periods tiltmeters measure primarily deformation (the term in E and r), except for tides, where the terms in U Iare important. Near the surface of an elastic body the terms in E vanish if the direction do of the tiltmeter baseline is either normal to the surface or along a horizontal surface, conditions which often hold approximately. Since strain and tilt are spatial derivatives of displacement, both are dimensionless. (To take a simple example, on the surface of a half-space the strain in the x direction is a,s, and the tilt is a,s, .) It is, however, sometimes convenient to use the symbol E for strain, and if tilts are expressed in radians similar amounts of deformation are numerically equal. For historical reasons tilts are sometimes given in arc seconds; 1 arcsec = 4.848 prad. 2.1. Areas of Study and Design Criteria

The greatest difficulty in designing strainmeters and tiltmeters is that, to an extent unequaled in most other instrumentation, the coupling to the outside environment is as important as the actual transducer-and to a large extent this environment is beyond the designer’s control. Figure 1 makes this point diagrammatically (Shichi and Okada’ have produced a more elaborate version). The signal we are trying to measure passes through a series of stages before it is recorded. We would like each stage to faithfully reproduce its output, but there will inevitably be imperfections. A perfect reproducer must meet four criteria : 1. The output is a linear function of the input, not subject to hysteresis or to distortion at large values, and not affected by extraneous inputs (such as high accelerations). 2. The relation between input and output is invariant with time. 3. If the first two criteria are met the relation between output and input is given by the transfer function, a complex function of frequency. To interpret the output we must known this function for each stage: the instrument must be calibrated. 4. No noise is added to the signal being reproduced. To see the relative importance of these criteria, it is useful to describe which ones matter most for each phenomenon measured by tiltmeters and strainmeters; but first a little must be said on the final class of imperfections, namely noise. The amplitudes, both of deformations and of instrument noise,

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output

t Recorder (finite resolution 1 Electronics mecha n ica I

,

Coupling

t -----_ Noise

Ins rumen

Weather, soil mechanics

1 E a r t h (seismic waves, tides, tectonics 1 FIG.1 . Flowchart for crustal deformation measurement, showing the various sources of noise that can contaminate a tilt or strain record. Though such noise sources as soil motion are often not considered part of the instrument, in a complete design they must be.

vary with frequency. Figures 2 and 3 show the range of tilt and strain noise observed at reasonably good locations. Because the level varies so much with frequency, specifications of instrument noise (or of equivalent quantities such as sensitivity)are not useful unless a frequency specification (or at least a range) is included. The ideal description is a power spectrum, unfortunately not often provided. The first measurements of deformation were made to detect earth tides, a field that remains active.' The amplitude of these tides is about lo-' in strain and 3 to 4 times larger in tilt. Because the tides occupy a narrow frequency band, recording for enough time gives a good signal-to-noiselevel. If the noise is - 130 dB (relative to 1 cZ/Hz) then a year's record of a sine wave with amplitude lo-' will give a signal-to-noise ratio of 40dB, equivalent to a 1070 error in estimating the wave amplitude. As this precision is needed for many tidal studies, it is fortunate that with some care such a noise level can be reached for periods near 1 day. Calibrating an instrument this accurately is much more difficult, and for some types it is impossible because of unknowns in the coupling between the instrument and the earth. At present, progress in most tidal studies requires not lower noise but better calibration. The reverse is true for measurements of tectonic deformation. Even in

18.

41 3

CONTINUOUS MEASUREMENT OF CRUSTAL. DEFORMATION

-4 2

rad,

HZ

dB

-200 10-6

10-5

IO-~

10-2

i0-I

I

10

f(Hz) FIG.2. Range of tilt noise power spectra: the lower curve is the quietest recorded, and the upper the expected level at a noisier but still observationally reasonable site. The lower curve to 3 x Hz is from the long fluid tiltmeter at Piilon Flat Observatory (F. from 3 x Hz is from horizontal seismic noise at Wyatt, personal communication) and above 3 x to Hz is from a Kinemetrics tiltmeter at Piaon Queen Creek."' The upper curve from Flat4' and above 3 x lo-' Hz is based on horizontal seismic noise recorded near the Pacific coast. '08*' 0 9

active areas these changes (determined from geodesy) are usually very slow, typically strain or tilt per second. (The rates may be two orders of magnitude higher in volcanic regions or in areas undergoing subsidence because of ground water withdrawal.) The drifts of even the best current tiltmeters and strainmeters are of the order of tectonic rates. For there to be any hope of tracking tectonic motion continuously, the instrument noise level must be reduced for periods of days to years. Such reduction could safely sacrifice accurate calibration, for two reasons. Even with lower noise levels the signal-to-noise ratio will be poor and the signal measured only imprecisely ;the additional loss of information from an inaccurate calibration will then be small. Furthermore, we know so little about tectonic motion that even an inaccurate measurement would constitute a great advance in our knowledge. At the other end of the frequency scale are the deformations associated with earthquakes, which separate into two classes : quasi-static deformations

0

414

4

DUNCAN CARR AGNEW

-

€2

'H z

d0

- 280

10-7

10-6

10-5

10-4

10-3

10-2

io-'

1

10

f (Hz) FIG.3. Range of strain noise power spectra, as in Fig. 2. The lower curve up to Hz is from the NW laser strainmeter at Piiion Flat;'" from this frequency to 0.3Hz it is from measurements with a rod strainmeter at Queen Creek,"' and from 1 to 10Hz from laser strainmeter data in the Poorman Mine.'" The upper curve is from the laser strainmeter in Queensbury tunnel."'

and radiated elastic energy. Both of these are rapid and therefore require low noise only for frequencies from a few millihertz to a few tens of hertz. The quasi-static deformations (strain and tilt steps) fall o f p with distance R from the source as R - 3 . The best data therefore come from the nearsource region; but in this region the dominant signal will be from elastic waves, which cause large transient deformations and acceleration. Accurate recording of coseismic steps therefore demands extreme linearity in a strainmeter or tiltmeter ; the problem is greater for tiltmeters because the inertial terms in seismic tilt are much larger that the deformational ones. Since seismic wave amplitudes vary roughtly as R-2, radiated energy can be successfuIly recorded at greater distances than static effects. Low noise at high frequencies (> 1 mHz) is the main need in instruments used for this, together with high dynamic range (to record large earthquakes). Accurate calibration is also becoming important as the ability of seismologists to synthesize seismograms improves ; 1'To uncertainty is desirable for some applications.

18.


41 5

2.2. Economic Criteria

One final set of design considerations are reliability and cost. Because all the phenomena to be observed either occur unpredictably or must be recorded for long times, high reliability is always needed. In a field instrument attention to apparently trivial engineering details makes the difference between something that works and something that does not; the latter is useless no matter how well it meets all the other criteria. Low cost is obviously desirable, but has often been overemphasized. Other things being equal, an array of instruments is certainly more useful than just one; this, together with the limited funds usually available, has led to an emphasis on keeping down the capital cost of instruments and their modes of installation. However, because most instruments used to measure deformation are operated for years or decades, their capital cost is almost always much less than the cumulative operating cost. It is worth reducing the latter as long as reliability is not sacrificed, but much of it (such as the costs of data recording and analysis) is fixed. One partial exception to the relative size of capital and operating cost is volcano monitoring, where the lifetime of an instrument is only until the next eruption. Fortunately, the deformations associated with volcanoes are large and rapid, so inexpensive instruments” can be built that will measure them adequately. (Of course, if an instrument cannot measure the signals of interest, it does not matter how cheap it is.)

3. General Design Features The variety of different designs for tiltmeters and strainmeters, while a testimony to human ingenuity, often obscures the basic elements that they all have in common: 1. A stable reference of length for extensometers, volume for dilatometers, and the direction of the vertical for tiltmeters ; 2. Some method (usually a displacement transducer) of comparing this reference with the motion or deformation of the instrument frame; 3. An attachment between the instrument frame and the ground, forcing the frame to move only in response to “true” ground deformation. Of these, the last, while the most difficult, is often the least discussed because it is “outside” the instrument; but the biggest unsolved problems lie just here. This section discusses the attachment techniques developed for different field environments, after first describing some of the mechanical difficulties inherent in precise measurement and the displacement transducers most commonly used.

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DUNCAN CARR AGNEW

Because ground deformation is generally very small or less), precision mechanical design is crucial in instruments that measure it. This is as much an art as a science, to which the papers of Jones", l2 are a useful guide. Much of this art lies in reducing effects caused by unwanted properties of the materials used, of which the two most troublesome are thermal expansion and departures from elastic behavior. In terms of the discussion in Section 2.2, thermal expansion adds noise by coupling temperature changes into dimensional ones. A partial solution is to use materials with a low thermal expansion coefficient a, such as Invar (a = 6 x 1 0 - 7 ~ K - ' 0.04 , that of stainless steel) or fused quartz (a = 5 x E K-'). Large amounts of passive insulation will screen out rapid temperature changes; a thickness w of material with thermal diffusivity K d will attenuate temperature fluctuations of frequencyfby roughly exp(- w m ) . Burying the instrument m2 s-l, and so 1 m of it gives such screening; K d for earth is roughly attenuates daily changes (f= loF5Hz) by 50 dB, though 10 m attenuates annual changesby only 30 dB. If an instrument is both small and well insulated it will change temperature uniformly, in which case compensation methods (whereby expansions in opposite directions balance out) may be useful. With perfectly elastic behavior the strain (and hence displacement) depends linearly on the instantaneous value of the stress; departures from elasticity introduce poorly known nonlinearities and time dependences. Fused quartz is one of the most nearly elastic materials available. If the elastic constants vary with temperature, imperfect elasticity becomes a source of noise; for fused quartz the fractional change is a relatively large, + K - I . Another imperfection of materials is gradual changes in size, obviously the bane of instruments intended to measure tectonic deformation. These changes are largest when the material is under stress but even in unstressed material can be high, depending on prior machining, heat treatment, and so on. The most precise measurement^'^ show creep rates of to s-' in materials such as fused quartz, not much less than strain rates in tectonic areas. 3.1. Displacement Transducers

Good mechanical design ensures that the small displacements in an instrument are undistorted ; high-sensitivity transducers must then be used to convert the displacements into a signal (usually a voltage) large enough to be recorded easily. There is a vast engineering literature 14* l5 on transducers, and correspondingly much development, which has changed them from a difficult part of strainmeter and tiltmeter construction to one of the easiest. There are still trade-offs in choosing a transducer; this section will describe the advantages and problems of the three commonest types : optical, capacitive, and inductive.

18.

"


m

I

B

41 7

R

I

C

FIG.4. Types of displacementtransducer often used in strainmeters and tiltmeters, discussed in Section 3.1. The optical lever, in which a light beam reflects off a rotating mirror into a pair of detectors is shown in A. B is a schematic Michelson interferometer,with the movable mirror at the right and the detector at the top; C is a Fabry-Perot interferometer. D is a capacitive transducer, with an adjustable transformer as the voltage source. E is an LVDT (linear variable differential transformer).

The most obvious way to magnify motion is with a pointer; the optical lever (Fig. 4A) makes a massless pointer out of a light beam reflected from a rotating mirror. Used for direct photographic recording, this is an old technique; in more modern form, with photoelectric recording16 it can give noise levels as small as rad' Hz-' at around 1 Hz. Related arrangements, such as using a moving vane to shade a split ph~tocell,'~ are also possible. The principal advantage of all such methods is that they apply practically no force to the moving object. Another type of optical sensor is the interferometer," which uses the wave nature of light. Optical designers have invented numerous types," the commonest of which for measuring purposes is the Michelson (Fig. 4B). In this interferometer, a light beam is split by a partially reflective mirror; the two beams produced travel to separate reflectors and back to the beamsplitter, where they interfere. A detector monitors the light modulated by the interference. There are two special cases of interest : 1. If the light is monochromatic (wavelength A) and the distances to the two mirrors are such that the fields of both returning beams are in phase at the beamsplitter, the beams will interfere constructively and a detector looking at the beamsplitter will see light. If one of the mirrors moves away from (or toward) the beamsplitter by A/4, the two beams will interfere destructively and the detector will receive no light. Counting light-to-dark

418

DUNCAN CARR AGNEW

transitions (interference fringes) gives a record of the motion of the moving mirror; the direction of motion is found by altering one of the beams. Because monochromatic light is used all the fringes look exactly alike (the interference pattern varies sinusoidally) and if the working of the system is interrupted the motion during that time is uncertain by an arbitrary multiple of 112. 2. If the distance from the beamsplitter to both mirrors is the same, constructive interference will take place for all frequencies (or destructive, if a A/4 phase shift is introduced in one beam). In such a white-light interferometer there is a single, unique fringe, which occurs only for a particular geometry. These two types of Michelson interferometer serve very different purposes. In the white-light system, if one arm changes length the other must follow it to preserve interference. It is therefore most useful as a method of noncontact location, the position of one mirror being measured and used as a proxy for the other A Michelson using monochromatic light is itself a distance measurer, giving motions of one mirror (relative to the other) in terms of the wavelength of light. This is both a strength and a weakness : the wavelength of light is relatively large (- 600 nm) so that in its simplest form an interferometer is not as sensitive as other techniques, but since this wavelength is so well known (easily to four figures), interferometric measurements have an accuracy unequaled by any other method. Another interferometer used in deformation instruments (though less often) is the Fabry-Perot (Fig. 4C). In this, two mirrors (highly reflective but not completely so) form an optical cavity; a beam of monochromatic light enters at one end and is reflected back and forth between them. If the cavity length is an integral number of wavelengths it becomes resonant for that wavelength, constructive interference occurs, and all of the beam energy is transmitted ; departures from resonance decrease the amount transmitted very sharply. Changes in the length of a Fabry-Perot thus give fringes to be counted just as in a Michelson, but with potentially sharper peaks and thus greater sensitivity. All these interferometers measure changes not in physical but it optical path length, which is the integral of the index of refraction n over the optical path. Changes inn affect both arms of a white-light system or a nearly equalarm Michelson equally and therefore cancel, but in other designs are a major source of noise. For air n is approximately 1 + 2.8 x pa, where pa is the air density relative to that at standard temperature and pressure. The easiest way to reduce the effects of changes in n is to make po small by evacuating the system. Mechanical pumps easily give pa = (1 Pa Pressure) ;such pumps are very reliable but add substantially to the operating cost.

18.


41 9

Except for the white-light interferometer, all the designs described use monochromatic light and are practical only because such light is easily produced with a laser. It is not usually appreciated that without special precautions laser light, while very narrowband, is neither perfectly monochromatic nor fixed in frequency. The generating mechanism (stimulated emission) takes place over a relatively broad spectral line. The emitting medium is contained in an optical cavity (like the Fabry-Perot described above) and the wavelengths emitted are those for which the cavity resonates. Most lasers emit at several (closely spaced) wavelengths, but with proper design may be made to emit at only one (a single-mode laser). However, because this wavelength will vary with changes in the length of the laser cavity, it will not be constant. The wavelength emitted must therefore be stabilized; there are many ways2’ of doing this, but the technology, while advancing rapidly, has only just begun2’ to produce systems that can be purchased off the shelf. In principle, atomically stabilized lasers, which force the wavelength to coincide with a spectral feature, offer the best long-term stability. A good measure of this is the discrepancy between independently built systems ; for the iodine-stabilized laser22this is about 10- lo (fractional wavelength difference, which is equivalent to strain in an interferometer). At seismic periods, however, the stability of most laser systems is actually less than that of the earth.’ Lasers have several disadvantages : most users of deformation instruments are unfamiliar with them, and lasers are relatively unreliable; even those made for wide commercial use (which are neither stabilized nor single-mode) have warranty periods of only about 2 years. This is certain to improve, but at the moment laser interferometers are hard to keep running. Capacitance transducers’’ 12*23 are the most sensitive displacement transducers used in strainmeters and tiltmeters. There are many possible geometries ; the commonest and easiest to understand is three parallel plates, the center one moving and outer ones fixed (Fig. 4D).Putting equal and opposite voltages of magnitude V on the fixed plates generates a field between them that is uniform (except for fringing fields). The potential at a distance q from the center is then qV/d, where the separation of the fixed plates is 2d. A plate spacing of 1 mm and an imposed voltage of 10 V give a field of 5000 V/m, 0.25% of the breakdown gradient in air. Measuring the voltage on the center plate with a good amplifier properly used (see below) m2 Hz-l, which over a 1-Hz bandgives a displacement noise of 4 x width is an RMS displacement of 2 pm, less than an atomic diameter (this precision is meaningful when viewed as the average over a large object). Noise levels 40 dB lower have been reachedI2 with special techniques. In practice, ac rather than dc voltages are put on the fixed plates and detected on the center plate. The voltage on an object in a steady field cannot

420

DUNCAN CARR AGNEW

be measured precisely because the measurement drains charge from the object and so itself distorts the field. Even if a dc measurement were practical it would be undesirable because the voltage to be measured would vary only at the extremely low frequencies (in electronics terms) of the plate motionand all amplifiers become noisy at low frequencies. If the applied voltage is varied at a few kilohertz the voltage on the center place will also vary, with the plate motion causing an amplitude modulation. This slow modulation affects the output signal only near the frequency of the input sinusoid, so this signal can then be amplified with very little noise. Suitable circuitry, known as a phase-sensitive detector," then measures that part of the output in phase with the input, giving substantial protection from noise at other frequencies. Alternating-current excitation also means that the voltages put on the fixed plates may be generated with a transformer (Fig. 4D),which is a nearly ideal voltage source when properly designed. In particular, the ratio of the voltages produced on the two sides depends only on the ratio of the number of turns, independent of loads applied to the output (such as stray capacitances to ground). In some systems this ratio is left fixed (and equal to l), in which case the voltage on the center plate is ideally the qV/d given above, though because the source impedance of the center plate varies with position the actual output depends nonlinearly on q. If the voltage ratio R between the two sides is made adjustable by introducing taps at different points along the windings, the ideal output voltage is 0.5 Y [(R + l)q/d + R - 11. The actual output will again be nonlinearly dependent on q, except that if R is set to make the ideal output zero then the value of q/d is given by the ideal expression to an accuracy limited only by the fringing field effects, which may be made very small. Multitap transformers (called ratio transformers) can be made2' that will give R (and thus q / d ) to seven significant figures. A capacitive transducer used in combination with a ratio transformer can combine very high sensitivity with a dynamic range up to 140 dB, the linearity over most of this range 26 being very high (0.2% out to q = *0.9d). Capacitive transducers are not without problems. Systems of the type described are not commercially available but must be constructed by the individual experimenter. Typical parallel-plate designs have very small capacitance and hence behave as large impedances. Their gains therefore depend very much on extraneous factors such as the capacitance to ground of the cable from the center plate to the amplifier, though the nulling approach described above avoids this. More important, the electric field distribution will be changed by the presence of conductors or by dielectrics such as water; capacitance transducers can be used only in a protected environment. The simplest electromagnetic transducer moves a coil through a magnetic field to generate an emf; for periods of 100 s and less this design can have

18.


42 1

even lower displacement noise than a capacitive transducer, but because its output depends on velocity it is of little use at long periods. At such periods active excitation is needed ; the commonest example is the linear variable differential transformer, or LVDT (Fig. 4E). (Hugillz3 describes other possible designs.) In most LVDTs, two sensing coils are wound at each end of a primary coil driven by an ac voltage. A movable cylindrical ferromagnetic core extends beyond the primary into the secondary coils ; when this core is centered, the voltage induced in the secondaries is equal and opposite. Any motion away from the center produces an unbalance, and so long as the core extends through the primary coil the unbalance voltage varies nearly linearly with the displacement. Just as in a capacitive sensor, the output of the secondaries can be measured with a phase-sensitive detector. The outstanding advantage of the LVDT is that it is very rugged and resistant to extreme environments: the coils are inside a sealed case, and contamination between them and the core is not a problem unless the contaminant is magnetic. This has made LVDTs very popular for industrial use27and many different styles are commercially available. The sensitivity is normally given in terms of output volts per unit of displacement and input voltage (volts per volt meter); this is numerically equal to the inverse plate spacing d-’ in a capacitive transducer. The most sensitive commercial LVDTs are equivalent to d = 5 mm; special designs” have achieved the equivalent of d = 0.15 mm and a noise level of 6 x m2 Hz-’ at 1 Hz, comparable to the levels of capacitive transducers. The output impedance of the LVDT coils is also relatively low, so long cables can be used with less distortion of the signal. The actual performance of an LVDT depends on such things as eddy currents and hysteresis and is therefore very difficult to model mathematically, though simple descriptions are a~ailable.~’ These effects also keep LVDTs from being low-power devices and cause the output to depend in general on temperature and input frequency. Because of the complex structure it is harder to design for long-term stability with an LVDT than with a capacitance transducer. Tests of the stability and linearity of commercial LVDTs are badly needed.

3.2.Instrument Attachment With the improvement of tilt and strain transducers in the past two decades, it has become more obvious that no one has solved the problem of attaching the instrument to the ground. This attachment has often been neglected in the past, even though it is just as much a part of the design as what goes inside the instrument case. By its very nature the attachment problem does not admit of a universal solution: the particulars of local

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geology and hydrology will always enter in. In this section I summarize general problems and what solutions have been worked out. Many practices, perhaps inevitably in view of the crude state of the field, are only rules of thumb. The problems vary with the setting in which the instrument is installed, and so the discussion is organized around these. To get thermal stability early tiltmeters and strainmeters were installed in caves, mines, or tunnels; many still are. Such openings, if available, are relatively convenient to use, though usually damp ;but there must always be room for doubt, especially with artificial excavations, whether such openings are dimensionally stable. These doubts are greatest for deep openings, otherwise the most desirable because the most isolated from meteorological influences. The actual coupling29(for example, rockbolts) introduces further instability. In a large cavity it is also never entirely clear what a tiltmeter or strainmeter is measuring. A uniform strain field will be distorted near the free surface introduced by the cavity: the walls and floor of the cavity deform differently than they would if the cavity were filled with rock. The resulting differences in deformation, which affect tiltmeters and strainmeters in the cavity, are called3’ cavity effects, sometimes divided into strain-strain coupling (distortion of the strain) and strain-tilt coupling [tilts resulting from a strain, as in Eq. (l)]. Cavity effects can be very large (as for the case31132 of a strainmeter placed across a tunnel) but even if small will introduce significant uncertainty into problems for which precise calibration is essential, such as tidal measurernent~~~ or seismic recording. Finite-element modeling34can be used to reduce this uncertainty, but how well the model approximates reality must always be in doubt. Unless accuracy is unimportant, large underground cavities are best avoided. Primarily for reasons of cost, installations near the surface (in shallow pits or trenches) have also been popular : if one is planning either to build a very large instrument or to install many instruments, the costs of going deep underground will be prohibitive except in the rare case that “holes of opportunity” are available. Geodetic and seismic measurements made at the surface show that the surface layers respond to deeper deformations. Unfortunately, shallow materials also respond vigorously to meteorological inputs, or (in words that everyone in this field wishes they had thought of) “the change in strain comes mainly from the rain”.35 This has been thoroughly documented for h ~ m i d , semiarid,3g ~ ~ - ~ ~ and arid4’ climates. On steep slopes soil creep caused by weathering has long been known to geomorphologists :41 what has been surprising is the discovery that motions of a few tenths of a millimeter per year occur at depths of several meters in what can appear to be competent material. Comparisons of shallow tiltmeters 10 m apart4’ show that these shallow motions are not correlated over large distances, so their effect on an

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instrument of baselength I will vary as I-’. Even for I = 1000 my0.2 mm/yr of displacement at each end would give a deformation rate of s-l, comparable to tectonic rates. For the low-frequency noise of long-base instruments to be reduced below this level, the motion of each end must be referred to a depth at which the effects of weathering are small. This has been done for along-basestrainmeter by Wyatt4’ using an “optical anchor,” which is a Michelson interferometer with two nearly equal arms going 27 m deep into the ground at 60”angles and in opposite directions. Horizontal motion of the beamsplitter (attached to the end of the strainmeter) shortens one arm and lengthensthe other, yielding a signal that shows the near-surface motions. To “anchor” a long-base tiltmeter to depth a vertical strainmeter is needed at each end ;both interferometers and mechanicalstrainmeters have been used.44 Unless large signals are expected, there appears to be little point in putting short-base instruments at shallow depths ; even at seismic frequencies the noise is high because of pressure-induced tilts .45 There is also evidence (F. Wyatt , personal communication) that shallow short-base instruments give spurious offsets at times of ground acceleration over 0.05 m s-’. Of course, if a shallow installation is to be made it may be done well or badly. The most work on specific procedures has been done by M ~ r r i s s e ywho , ~ ~ finds that the best results come from surrounding the instrument with carefully tamped sand. The grains interlock to give a tight bond, but may be removed if needed to recover the instrument. The decreasing size of instruments (made possible by improved transducers) and the realization that the surface was too noisy for long-term measurements have led to a continuing development of strainmeters and tiltmeters for use in drilled boreholes, which are the cheapest way of getting to great depth. Boreholes suffer from cavity effects; for a vertical hole these are zero for a tiltmeter attached to the side of the hole or for vertical strain, but they are very large for strains measured across the hole (if the Poisson ratio v is 0.25, an empty hole multiplies a uniaxial strain by 2.83). How an instrument is to be coupled to the hole is an area of active study. Most tiltmeter installations4749 cement in a sealed stainless-steel casing within which the tiltmeter sits ;this allows the instrument to be recovered and keeps it dry. Most borehole strainmeter installations use expanding to bond the instrument directly to the wall rock. The relative merits of these procedures are still unclear. Both introduce cement (a material with poor stability) between the instrument and the earth, though in the unchanging environment of a borehole it should behave in a predictable way. Work on better cements is needed, as is study of what rock types, diagnostic logs, and installation methods give the best result. A borehole provides a stable environment but at the price of many disadvantages. A notable one is that if it is at all deep it is usually full of

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water (installations at the depth of the water table should be avoided in any case).52The biggest problem is inaccessibility. For tiltmeters, this means the added complexity of a remote-controlled leveling system, and for any instrument except a dilatometer some means of orienting the instrument. In shallow holes orientation can be done with alignment rods or by sighting down to the instrument ;in deep holes a magnetic compass or gyroscope has to be installed in the instrument or used beforehand to align a fixture. Finally, borehole installations are much more worrisome than other kinds : only in a borehole can momentary inattention cause the complete loss of an instrument.

4. Tiltmeters: Particular Designs The number of separate designs for tiltmeters is large, in part because no particular one has proved dramatically better than all the others. A useful division is between short-base instruments, which define the vertical with some sort of pendulum, and long-base instruments, which use an extended liquid surface. The short-base instruments are sometimes subdivided into those used in cavities and those used in boreholes, but this is not an especially fundamental division since with enough auxiliary engineering (not, admittedly, a trivial task) most small instruments can be made to work in a borehole. With the possible exception of the Michelson-Gale long-base tiltmeter described at the end of this section, the dynamics of all tiltmeters are well described by the indicator equation. If q is the displacement of the part of the tiltmeter that we want to measure and SZ is the component of the tilt vector Q along the sensitive axis, then this equation is where 00 is the natural frequency of the tiltmeter, c its damping relative to critical, and Kp a dimensionless constant that depends on the geometry of the system. At periods long compared to the period of the tiltmeter the sensitivity q/a varies as cut2. 4.1. Short-Base Tiltmeters

The most obvious design for an instrument to sense the direction of the vertical is a simple pendulum (Fig. 5A), but the low sensitivity of such an arrangement (for which Kp = 1) made it impractical until very sensitive displacement transducers were available. With capacitive transducers even short pendulums'2*53can achieve low enough noise to measure tides ; while the simplicity of the suspension (wires or strips) should give high transducer

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425

cc

c )

c )

L.

”

I

E

FIG. 5 . Tiltmeter designs, with the arrow beneath each sketch showing the direction of sensitivity. A-D are short-base instruments (Section 4.1). A is a simple pendulum and B a horizontal pendulum (shown in two views for clarity). C uses a mass suspended in a magnetic field, and D is a bubble tiltmeter with resistive sensing. E and F are long-base instruments (Section 4.2). E is the standard pot-and-tube design (with the level measured at both ends), and

F is the center-pressure system. G is the Michelson-Gale instrument.

stability, this has not been tested. A very high-quality vertical pendulum intended primarily for tidal rnea~urements~’”~ has been built by the Askania company (now part of Bodenseewerke). It uses a single pendulum hung so as to swing in two axes, the motion being detected capacitively (and in the newer models fed back electromagnetically). The shape of the vertical pendulum makes it excellent for borehole use, and the Askania instrument was designed especially for this, including an elaborate leveling system. This tiltmeter also includes an unusual calibration device, the so-called ballcalibrator, in which a small sphere is moved to and fro inside the pendulum, changing its center of mass and producing a precisely known apparent tilt.

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A somewhat similar (though apparently less elaborate) fedback simple pendulum tiltmeter has been developed in Japan, also for borehole use.49 In early tilt measurements the transducers available were poorer and greater instrument sensitivity was needed. The standard method of achieving this was (and is) the horizontal pendulum, in which the boom rotates in a nearly horizontal plane (Fig. 5B). If this plane is inclined at an angle i to the horizontal the frequency is O$ = wb/sin(i), where ovis the natural frequency of the boom hung as a vertical pendulum ;if we measure the displace: ment of the end of the boom Kp = 1. A serious practical problem with this design is that if sin(i) is small (to give high magnification) tilts perpendicular to the direction being measured will alter the sensitivity, making frequent calibration necessary. Small values of sin(i) also make construction difficult since the gravitational restoring force is small, and the suspension system must therefore be nearly frictionless and very stable. A quartz-fiber suspension, pioneered by Ishimoto and further developed into all-quartz instruments by Blum" and Verbaandert and M e l ~ h i o r ,reduces ~~ this problem considerably. Such instruments are relatively fragile, and so their use is restricted to large underground openings, with all the associated distortions. These instruments are therefore unsuitable for measuring tides ;since smaller horizontal pendulums with capacitive sensing give good tidal measurements in bore hole^^^ the quartz instruments should be viewed as obsolescent. Another way to get a long period and high sen~itivity'~ is to float a mass in a suitably shaped magnetic field, which can be done with a permanent magnet if the mass is made from the right material (Fig. 5C). Such a suspension is completely frictionless. The small mass of the suspended object means that optical position sensing must be used ;it also keeps the instrument from being as quiet as the ground-noise minimum shown in Fig. 2. Installed in a borehole, it has given good tidal results.57 A compact long-period sensor can also be built by measuring the position of a bubble trapped beneath a slightly curved surface; the long-period tilt sensitivity is just equal to the surface's radius of curvature. The position of the bubble can be sensed capa~itively~' but usually the surrounding liquid is conducting and a resistance bridge measures the motion of the bubble relative to fixed electrodes (Fig. 5D).Because bubble levels are relatively immune to vibration and high acceleration, they are made commercially for avionics use. An inexpensive tiltmeter intended for volcano monitoring" was designed around a level of this type. A higher-quality two-axis level5' built by the Autonetics division of Rockwell was used by kine metric^^^ as the sensor in their shallow borehole tiltmeter. In the Hughes bubble tiltmeterm the bubble was trapped beneath an optically flat plate, feedback being used to stabilize the bubble position.

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Bubbles trapped beneath a surface do not in general slide easily but cling to one spot61because of capillary effects. This behavior is unacceptable in a bubble tiltmeter ;eliminating it appears to involve using suitable fluids and a properly roughened surface and avoiding contamination of either. It is also difficult to get long-term reliability in the seals where electrodes enter the bubble chamber. Without a major effort the potential user of bubble levels is therefore constrained by what is made commercially, which may not be suitable for very precise work. A small tiltmeter used for tidal measurements must be calibrated to 1% in the range of tidal tilts, which means imposing a tilt of 0.1 prad known to 1 nrad. Because of the difficulty of making accurate small displacements, no way of doing this is generally available. Tilt tables may be used for larger tiltsa but are not calibrated in the range over which the measurement is made. The distensible support (“crapaudine”) of Verbaandert62 creates small displacements through the elastic deformation of a pressurized liquidfilled capsule ;the capsule is calibrated interferometrically for large displacements, linear elasticity being used to interpolate to smaller ones. Further work is needed in this area. 4.2. Long-Base Tiltmeters

The only way yet thought of to determine the vertical over a long baseline is to use the free surface of a liquid to define an equipotential. This technique brings with it a large sensitivity to t e m p e r a t ~ r because e ~ ~ of the high thermal coefficient of volume expansion of liquids (for water, one of the least sensitive, about 2 x 10-4K-’, or roughly that of most plastics). Because long-base tiltmeters must often be installed on or near the earth’s surface, the thermal sensitivity of a particular design is an important consideration. For the oldest and commonest arrangement-two end pots and a connecting tube with the level measured in each pot (Fig. 5E)-the temperature sensitivity is very high : if the liquid in one pot is heated the level in that pot rises while remaining unchanged in the other. Though a portable system has been built for field useu most pot-and-tube tiltmeters have been set up in tunnels and used to monitor crustal motion, the large majority in Japan.65 The natural frequency and damping of such a system are (to a good approximation) given by

where 1 is the base length and bt and b, are the diameters of the tube and pot ; m2s - l for water, 1.2 x lo-’ for mercury). If the level is measured at each end and differenced to get tilt,

x is the kinematic viscosity of the liquid

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DUNCAN CARR AGNEW

the tilt sensitivity is I - ’ ;this mode of measurement eliminates such commonmode effects as changes in the amount of liquid. In most pot-and-tube tiltmeters the levels are read manually, by moving a submerged micrometer point upward until it touches the surface. This is adequate for monitoring slow tectonic motion, but automatic measurements are needed to measure tides. Many Japanese tiltmeters7*66 use displacement transducers on floats ; though these systems work well for tidal measurements their long-term stability is unknown. A separate family of designs have mercury as the sensing liquid and measure the level with a capacitance sensor, with one fixed plate in each pot and the mercury serving as the center plate. The first such instrument was designed by BeniofP7 as a long-period seismometer;smaller mercury tiltmeters using ratio-transformer sensors68have been built for tidal measurements. After thermal sensitivity the biggest problem with the pot-and-tube arrangement is that if the instrument is made very long (to reduce the effects of end-mount motions) the response is very sluggish. In an attempt to reduce both of these problems Horsfall and King69built a pot-and-tube instrument in which the signal is the differential pressure measured halfway between the pots (Fig. 5F). The pressure sensor greatly reduces flow and so speeds up the response. Thermal sensitivity is also reduced63 but not eliminated ; an instrument of this design installed in California44showed spurious tilts which correlated with air temperature. The best solution to the temperature problem remains that found by A. A. M i ~ h e l s o n ~in~ ~1914: ” leave the surface of the liquid unbroken from one end to the other. The only equilibrium position for this surface is then an equipotential ; temperature fluctuations can cause localized convection but not a static tilt of the whole surface. The original Michelson-Gale instrument was intended to measure earth tides and was successful, though very precise tidal measurements with such a tiltmeter are hindered by ignorance of the dynamics of slow motion in an open channel. Perhaps because of a belief that the channel must be both level and straight, this design was neglected for many years. In fact, the channel need only follow a contour line : a restriction, but not a severe one. The Michelson-Gale design has recently been revived for crustal deformation m e a ~ u r e m e n tand , ~ ~ two versions of it are being compared in a test in C a l i f ~ r n i aBoth . ~ ~ are 535 m long and have tubes buried about 1.5 m deep. One (built by a group from Lamont-Doherty) has two small tubes (8 cm diameter), one serving as an air return; the other (developed at the University of California at San Diego) has a single tube 15 cm in diameter, an arrangement less affected by settling. The Lamont instrument senses the water level with a laser Michelson interferometer, in which the varying depth of a submerged corner cube causes changes in the optical path length of one arm. Occasional micrometer readings supply the absolute

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readings unavailable from such a system. The UCSD instrument uses a mechanically fed back white-light interferometer to track the water surface, giving a continuous absolute reading at some cost in added complexity. Both instruments have vertical anchors at each end, without which the inherent stability of the tilt measurement would be useless because of vertical soil motion.

5. Strainmeters The tiltmeter designer has the advantage that the fundamental referencethe direction of the vertical-is external and invariable; it merely has to be followed. In a strainmeter the reference is part of the instrument, and it is convenient to divide strainmeters according to what the reference is : mechanical strainmeters, in which the reference is a solid material, hydraulic strainmeters, in which it is a volume of liquid, and laser strainmeters, in which it is the wavelength of light (itself sometimes referenced to a solid). 5.1. Mechanical Strainmeters

The most obvious length standard is a solid rod, and it was this, in the form of a 20-m iron pipe, that B e n i ~ f used f ~ ~ in the first successful strainmeter (Fig. 6A). As this instrument was intended for seismic recording, it used a velocity transducer, and this, together with the high thermal coefficient of expansion of the rod, made it unusable at longer periods. It served its original purpose very well, and the design has been imitated by most other strain seismometers. These have included an array of shallow instrument^^^ and a vertical strainmeter7’ used in experiments on noise reduction and phase identification, and an extremely sensitive i n ~ t a l l a t i o nfor ~ ~detecting small surface waves. To make strain measurements at longer periods B e n i ~ f later f ~ ~ built strainmeters with fused-quartz rods for length standards having both displacement and velocity transducers. Instruments of this general design have been for seismic, tidal, and secular strain measurements ; though the long-term stability of the length standard is probably too low for the last purpose, at higher frequencies these instruments can give good results. An attempts3 to design a variant suitable for shallow installation was not satisfactory. Making the precise calibrations necessary for tidal and seismics4 measurements has been difficult because the displacements to be measured are small. Interferometers may be ~ s e d ~to~give * a~coarse ’ ~ ~ calibration; ~ another approach77is to apply a known force t o the free end of the rod and compute the displacement from its elastic constants. Given sensitive enough displacement transducers, the length of the rod can be very short. The strainmeter can then be put in a borehole, and two

430

DUNCAN CARR AGNEW

U

-

FIG, 6. Strainmeter designs, with A-C being long-base instruments and D and E borehole designs. A is the Benioff rod strainmeterand B a constant-tensionwire strainmeter. C is a laser strainmeter using a Michelson interferometer. D is a Sacks-Evertson dilatometer. E is a miniature borehole extensometer (shown looking along the borehole).

workers87s88have done so. Both instruments use small rod strainmeters extending across the hole (Fig. 6E), with capacitive transducers measuring three components of horizontal strain. Many sources of noise (such as expansion of the bonding cement, barometric loading, and temperature changes) give apparent dilatational strain; with a record of all three components one can calculate the shear strain, which in principle is unaffected by any of these if the instrument is centered in the hole. In the stable environment of a deep borehole some of the constraints otherwise present can be relaxed; for example, the length standards of one instrument are stainless steel. An alternative to a solid rod as a length standard is a wire suspended in catenary, as formerly done8’ for geodetic baselines. Recent designs stem from the development by Sydenham” of a strainmeter in which the wire is kept under constant tension and motions of one end are recorded (Fig. 6B). Sufficient tension must be applied to keep the wire from sagging far, but not so much that it is o v e r ~ t r a i n e d . ~Wire ~ ’ ~ strainmeters are easy to transport

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and install, though the flexibility of the wire introduces resonances that keep them from being useful as seismometers. The earliest instrumentsg1*92 used Invar wire, which is easy to work with and readily available, but whose weight makes it unsuitable for lengths over 20 m. More recentlyg3carbon fiber, which is lighter and also has a small temperature coefficient (CY = 8 x K-’), has been used. The tension is usually provided by a counterweight acting through flexure pivotsg4with an inductive sensor to measure displacements ; adjusting the counterweight occasionally allows strains up to to be followed. Relative calibration of wire strainmeters can be made to a few percent by interconnecting several end transducers. 5.2. Hydraulic Strainmeters

Hydraulic amplification was first used in an (unsuccessful) extensometer in 190O9’ and was proposed by B e n i ~ f in f ~1935 ~ for a dilatometer, but does not seem to have been successfully applied before the work of Sacks and Evertson, which is most fully described in the latter’s thesis.96The present version of their instrument holds a volume of silicone oil between two concentric stainless-steel tubes closed at both ends (the sensing volume). A capillary at one end of the volume leads into a small metal bellows to which an inductive displacement transducer is connected (Fig. 6D).The space outside the bellows is partly filled with gas, so that as the sensing volume changes oil flows through the capillary and moves the bellows. The entire instrument is put in a borehole partly filled with expanding cement, which bonds the instrument to the borehole wall. The factor relating dilatation of the sensing volume to displacement of the bellows would ideally be -V,/Ab, where V, is the total volume inside the outer tube and Ab the area of the bellows; in practice the factor can be half this because of the compressibility of the oil and the spring constant of the bellows. Since the actual strain-todisplacement factor can be around 100 m, a high sensitivity is easier to get than in the borehole extensometers described above. This high sensitivity makes the Sacks-Evertson instruments particularly useful as seismometers, at least so long as a precise calibration is not needed ; the latter depends on the details of borehole-to-instrument coupling, which are no more easy to estimate than those of any other cavity effect. A field in which this instrument appears to be preeminent is the recording of coseismic strain. At very high frequencies (1 Hz and above) the capillary attenuates pressure changes in the sensing volume before they reach the bellows. Because this hydraulic filtering isolates the only moving parts from large rapid changes and because the instrument is intimately bonded to the rock over a large surface, it performs well under high accelerations that for most other strainmeters cause spurious A disadvantage of this

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DUNCAN CARR AGNEW

instrument is that it gives only dilatation; a recent Japanese design99 has attempted to measure all three components of horizontal strain by dividing the annulus of sensing liquid into 120" sectors and measuring the volume change of each separately.

5.3.Laser Strainmeters The fundamental nature of the wavelength of light makes it an attractive length standard for an extensometer, but building a completely optical strainmeter was nearly impossible before the invention of the laser and even now remains difficult. Though commercial measuring devices can be adapted86 for geophysical measurements, the best results have come from specially designed systems. Section 3.1 described the two interferometers found in laser strainmeters : the Michelson and Fabry-Perot. There are also two possible ways to measure changes in the interference pattern. The simplest is to supply the interferometer with stabilized light and then to count fringes ;this gives relatively low resolution. This procedure is especially useful with a Michelson interferometer since then the fringe pattern varies fairly uniformly (sinusoidally) with motions of one end mirror. The other procedure is to vary the wavelength of the laser supplying the interferometer (the "slave laser") to keep the fringe pattern fixed; this can be done by changing the length of the laser cavity. The wavelength from the slave laser is then I(1 + E), where E is the extension of the interferometer and I the laser wavelength for zero strain. A second reference laser is maintained at a fixed wavelength I and light from it and the slave laser shine together onto a photodetector. The detector responds to the intensity of the light (the square of the electric field), and this varies at a frequency twice that of the incoming light ( 1 0 2 9 H ~which , is averaged out) and at the beat frequency between the two beams, which is almost exactly c d I . Since frequencies can be measured very precisely this can be a very sensitive method of measuring strain. This technique may be used with either type of interferometer but is more suited to the Fabry-Perot, the sharpness of whose fringes makes it possible to lock the slave laser more tightly. The three successful designs of laser strainmeter were all built around 1970 and, as it happened, used three out of the four possibilities outlined above. The laser strainmeter built at UCSD used a Michelson interferometer (Fig. 6C)and fringe-counting recording.'00 To get the necessary resolution it had to be made very long (732 m) with one fringe corresponding to 0.108 nE. Since the instrument was designed to operate on the surface this length was not a big disadvantage and was in fact necessary to reduce noise from motions of the end monuments. The surface mounting complicates the mechanical

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design of the system ; for example, servo-controlled telescopic joints are needed at each end of the vacuum pipe to compensate for its expansion and contraction with temperature changes. The interferometer is illuminated by a single-mode laser whose wavelength is kept fixed by locking it to a small Fabry-Perot cavity in the manner described just above. In this case, however, the goal is not to follow variations in the cavity length but to assume that it, and thus the laser wavelength, stays constant. As the cavity is inside a wellcontrolled chamber, its length should change only very slowly. At seismic frequencies this system gives better stability than all but the best atomically stabilized lasers, but after correcting for motions of the end monuments, drifts in the material of the stabilizing cavity appear to be the main source of noise at periods of months to years (F. Wyatt, personal communication). The laser strainmeter built at Cambridge University"' also used a Michelson interferometer, in which the arm lengths were chosen to allow illumination by a multimode laser. The long arm was 54 m long, and since the instrument was in a tunnel, bellows could compensate for the vacuum pipe motion. Light from the slave laser was combined with that from an iodine-stabilized laser to produce a beat-frequency signal with a nominal resolution of 0.2p&,though the actual noise was some 40 times larger. In a beat-frequency system the dynamic range is limited by how much the slave laser can be adjusted, which for this instrument was ~ 9 p . z . The shortest but most sensitive laser strainmeter was a 30-m instrument built by the National Bureau of Standards"' in a mine near Boulder, Colorado. This used a slave laser locked to a Fabry-Perot interferometer, and beat-frequency measurement. Though the Fabry-Perot cavity had to be designed carefully to ensure that only one of its modes was excited, the sharpness of the fringes raised the gain of the servo loop controlling the slave laser and so decreased the noise. An important reason for the high quality of this instrument was the use of a methane-stabilized He-Ne laser as the reference laser ;this operates in the infrared, making alignment of the optics more difficult, but at frequencies above 1 mHz is much more stable than any other laser. The noise level at 1 Hz was - 254 dB (relative to 1 &' Hz-I). A short instrument, if it does not entail loss of sensitivity or undue noise from end-mount motion, is actually desirable because each fringe corresponds to a larger strain (0.1 1 pe for the NBS system) so that, after a break in operation, the data may be patched together with much less ambiguity. By their very nature laser strainmeters are very well calibrated ;the biggest uncertainty lies not in the displacement measurements but in defining the base length. They can also offer better long-term stability than any other strainmeter and with enough effort very high sensitivity. The effort needed is, however, very large ;indeed any laser strainmeter is a complicated apparatus, requiring special skills to build and both skill and effort to keep operating.

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6. Conclusions The construction of tiltmeters and strainmeters remains a field with many unsolved problems, some of them mentioned above. Because of the diversity of measurements needed, it is probably unrealistic to aim for a single perfect design; more likely, a range of designs will appear, each most suited for a particular applications (and budget). Some examples are the tiltmeter of Westphal et al. lo and the Cambridge wire trainm meter,^^ which cost little and will measure rapid signals quite well. For tidal tilt measurement, the Askania tiltmeter provides excellent results, though at high cost. In improving instruments it is very important that proper tests be made to demonstrate the levels achieved. This usually requires side-by-side tests of instruments at an actual field site. One reason for this is the importance of the instrument-to-ground coupling, which is not checked in laboratory tests; only field operation tests the complete system. But if a field test is to produce any useful results, it must be clear that both instruments should see the same signal, which is unquestionable only if the instruments are side by side. Despite the apparent redundancy, such tests are the only way to see how well an instrument works. Of course, the data must also be analysed correctly. Visual comparison, though a useful beginning, is seldom enough, since it will usually show only that the tides are about the same; a detailed cross-spectral analysis (or, for the tides, harmonic analysis) must be made to quantify discrepancies properly. Few such tests have been made. An early one1O3 showed that for frequencies near 1 day wire strainmeters were noisier than a laser strainmeter. An examination of shallow borehole tiltmeters4' showed no agreement except for tides and microseisms, an experience repeated in a later study5' which did show good agreement between the tides measured on a deeper borehole instrument and those on a 535-m Michelson-Gale tiltmeter. A recent test44 has shown possible coherence at periods of a few days between this long-base tiltmeter and another of similar design. At longer periods, an early comparison'04 showed general agreement in the secular trend of a Sacks-Evertson dilatometer and nearby quartz-rod extensometers. However, the best test of secular measurements comes not from comparisons between instruments but from comparisons with repeat geodetic measurements. Tests comparing tiltmeters and leveling have been made in Japan"" lo' with somewhat discouraging results :while long-base tiltmeters show fluctuations one-tenth of those on short-base instruments, leveling shows changes ten times smaller yet. The reliable measurement of long-term tilt and strain (or indeed of any tilts and strains in normal areas at frequencies other than tidal or seismic)remains an unsolved problem. It is clear that because of near-surface noise an

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instrument that will accomplish this must be very long or buried in a deep borehole. Making either type of instrument work well poses many problems (e.g., sensing liquid levels in long-base tiltmeters and choosing cement for borehole strainmeters) and which, if either, will succeed remains to be seen. References 1. J. C. Savage, Strain accumulation in western United States. Annu. Rev. Earth Planet Sci. 11, 11-43 (1983). 2. W. Thatcher and T. Matsuda, Quaternary and geodetically measured crustal movements in the Tokai district, central Honshu, Japan. J. Geophys. Res. 86, 9237-9247 (1981). 3. G. Mavko, Mechanics of motion on major faults. Annu. Rev. Earth Plunet. Sci. 9,81-111 (1 98 1). 4. N. R. Goulty, Strainmeters and tiltmeters in geophysics. Tectonophysics 34, 245-256 ( 1976). 5. D. C. Agnew, Strainmeters and tiltmeters. Rev. Geophys. 24, 579-624 (1986). 6. L.E. Malvern, “Introduction to the Mechanics of a Continuous Medium.” Prentice-Hall, Englewood Cliffs, New Jersey, 1969. 7. R. Shichi and Y. Okada, Strain measurement in the vault. J. Geod. SOC. Jpn. 25, 101-134 (1979). In Jpn. 8. T. F. Baker, Tidal deformations of the earth. Sci. Prog. (Odord) 69, 197-233 (1984). 9. Y. Okada, Surface deformation due to shear and tensile faults in a half-space. Bull. Seismol. SOC. Am. 75, 1135-1154 (1985). 10. J . A. Westphal, M. A. Carr, and W. F. Miller, Expendable bubble tiltmeter for geophysical monitoring. Rev. Sci. Instrum. 54, 415-418 (1983). 11. R. V. Jones, Some uses of elasticity in instrument design. J. Sci. Instrum. 39, 193-203 (1962). 12. R. V. Jones and J. C. S. Richards, The design and some applications of sensitive capacitance micrometers. J. Phys. E 6, 589-600 (1973). 13. J. W. Berthold, S. F. Jacobs, and M. A. Norton, Dimensional stability of fused silica, Invar, and several ultra-low thermal expansion materials. Metrologia 13, 9-16 (1977). 14. J. E. Roughton and W. S. Jones, Electromechanical transducers in hostile environments. IEE Rev. 126, 1029-1052 (1979). 15. J. D. Garratt, Survey of displacement transducers below 50mm. J. Phys. E 12, 563-573 (1979). 16. R. V. Jones, Some developments and applications of the optical lever. J. Sci. Instrum. .38, 37-45 (1961). 17. I. Simon, A. G. Emslie, P. Strong, and R. K. McConnell, Sensitive tiltmeter utilizing a diamagnetic suspension. Rev. Sci. Instrum. 39, 1666-1671 (1968). 18. J. Levine, Laser distance-measuring techniques. Annu. Rev. Earth Planet. Sci. 5,357-369 (1977). 19. M. Born and E. Wolf, “Principles of Optics,” 4th Ed. Pergamon, Oxford, 1970. 20. K. M. Baird and G. R. Hanes, Stabilization of wavelengths from gas lasers. Rep. Prog. PhyS. 37, 927-950 (1974). 21, H. P. Layer, A portable iodine-stabilized helium-neon laser, IEEE Trans. Instrum. Meas. IM-29, 358-361 (1980). 22. J.-M. Chartier, Results of international comparisons using methane-stabilized He-Ne lasers at 3.39 bm and iodine-stabilized He-Ne lasers at 633 nm. IEEE Trans. Instrum. Meas. 32, 81-83 (1983).

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23. A. L. Hugill, Displacement transducers based on reactive sensors in transformer ratio bridge circuits. J. Phys. E 15, 597-606 (1982). 24. P. Horowitzand W. Hill, “The Art of Electronics.” Cambridge Univ. Press, London and New York, 1980. 25. J. J. Hill and A. P. Miller, A seven-decade adjustable-ratio inductively-coupled voltage divider with 0.1 part per million accuracy. Proc. ZEE 109B, 157-162 (1962). 26. M. T. Gladwin and J . Wolfe, Linearity of capacitance displacement transducers. Rev. Sci. Instrum. 46, 1099-1100 (1975). 27. E. E. Herceg, “Handbook of Measurement and Control.” Schaevitz Eng., Pennsauken, New Jersey, 1976. 28. E. Wielandt and G. Streckeisen, The leaf-spring seismometer: Design and performance. Bull. Seismol. SOC. Am. 72, 2349-2367 (1982). 29. G. J. Jeffrey and P. H. Sydenham, Stability of strain-meter mounts. Geophys. J. R. Astron. SOC.33, 185-193 (1973). 30. J . C. Harrison, Cavity and topographic effects in tilt and strain measurement. J. Geophys. Res. 81, 319-328 (1976). 3 1. S. Takemoto, Effects of local inhomogeneities on tidal strain measurements. Bull. Disaster Prev. Res. Inst., Kyoto Univ. 31, 211-237 (1981). 32. J. Beavan, R. Bilham, D. Emter, and G. King, Observations of strain enhancement across a fissure. Veroeff. Dsch. Geodaet. Komm., Reihe B 231, 47-58 (1979). 33. T. F. Baker, Tidal tilt at Llanrwst, north Wales: Tidal loading and earth structure. Geophys. J. R. Astron. SOC. 62,269-290 (1980). 34. J. Berger and C. Beaumont, An analysis of tidal strains from the United States of America: 11. The inhomogeneous tide. Bull. Seismol. SOC.Am. 66, 1821-1846 (1976). 35. J . E. Wolfe, E. Berg, and G . H. Sutton, “The change in strain comes mainly from the rain”: Kipapa, Oahu. Bull. Seismol. SOC. Am. 71, 1625-1635 (1981). 36. K. Herbst, “Interpretation of Tilt Measurements in the Period Range Above That of the Tides,” Rep. AFGL-TR-79-0093 (NTIS AD-A074 525/7), Air ForceGeophys. Lab. (1979). 37. R. Edge, T. Baker, and G. Jeffries, Borehole tilt measurements: Aperiodic crustal tilt in an aseismic area. Tectonophysics 71, 97-109 (1981).

38. M. Kasahara, R. Shichi, and Y. Okada, On the cause of long-period crustal movement. Tectonophysics 97, 327-336 (1983). 39. N. R. Goulty, P. M. Davis, R. Gilman, and N. Motta, Meteorological noise in wire strainmeter data from Parkfield, California. Bull. Seismol. SOC. Am. 69, 1983-1988 (1979). 40. F. Wyatt, Displacement of surface monuments: Horizontal motion. J. Geophys. Res. 87, 979-989 (1982). 41. I. Saunders and A. Young, Rates of surface processes on slopes, slope retreat and denudation. Earth Surf. Processes Landforms 8,473-501 (1983). 42. F. Wyatt and J . Berger, Investigations of tilt measurements using shallow borehole tiltmeters. J. Geophys. Res. 85, 4351-4362 (1980). 43. F. Wyatt, K. Beckstrom, and J . Berger, The optical anchor-a geophysical strainmeter. Bull. Seismol. SOC.Am. 72, 1707-1715 (1982).

44. F. Wyatt, R. Bilham, J. Beavan, A. G. Sylvester, T. Owen, A. Harvey, C. Macdonald, D. D. Jackson, and D. C. Agnew, Comparing tiltmeters for crustal deformation measurements: A preliminary report. Geophys. Res. Lett. 11, 963-966 (1984). 45. J. Peterson, H. M. Butler, L. G. Holcomb, and C. R. Hutt, The Seismic Research Observatory. Bull. Seismol. SOC.Am. 76, 2049-2068 (1976). 46. S.-T. Morrissey, “A Study on the Adaptation of a Commerical Tiltmeter for Monitoring Earth Tilts in UnfavorableEnvironments,” Spec. Tech. Rep., Contract 14-08-0001-15848, U.S. Geol. Surv. (1977).

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47. 0. Rosenbach and H. Jacoby, First experience with the Askania borehole tiltmeter (earth

tide pendulum). In “Problems of Recent Crustal Movements” (Y. D. Bulanzhe, ed.), pp. 467-478. USSR Acad. Sci., Moscow, 1969. 48. J. C. Harrison and J. Levine, “A Measurement of Long-Term Tilt in Colorado and Wyoming,” Rep. AFGL-TR-81-0304 (NTIS AD108-865-17), Air Force Geophys. Lab. (1981). 49. H. Sato, H. Takahasi, E. Yamamoto, N. Fukuo, M. Uehara, and Y.Terasawa, Development of the crustal tilt observation method using borehole-type tilt meters. Jishin 33, 343-368 (1980). 50. A. M. Neville, “Properties of Concrete,” 3rd Ed. Pitman, London, 1981. 51. C. E. Kesler, Expansive cement concretes-present state of knowledge. In “ACI Manual of Concrete Practice,” pp. 223/1-223/28. Am. Concr. Inst., Detroit, Michigan, 1970. 52. K.Evans and F. Wyatt, Water table effects on the measurement of earth strain. Tectonophysics 108, 323-337 (1984). 53. R. V. Allen, A borehole tiltmeter for measurements at tidal sensitivity. Bull. Seismol. SOC. Am. 62, 815-821 (1972). 54. Anonymous, “Borehole Tiltmeter Gpb 10,” Tech. Bull. 19, Bodenseewerk Geosyst. ( 1979). 55. P. A. Blum, Contribution a I’ktude des variations de la verticale en un lieu, Ann. Geophys. 19, 215-243 (1962). 56. J. Verbaandert and P. Melchior, Les stations geophysiques souterraines et les pendules horizontaux de I’Observatoire Royal de Belgique. Monogr. Obs. R. Belg. 7 , 1-147 (1961). 57. F. Wyatt, G. Cabaniss, and D. Agnew, A comparison of tiltmeters at tidal frequencies. Geophys. Res. Lett. 9, 743-746 (1982). 58. Xie Liangyun, A capacitive bubble level and its application to the transit instrument. Chin. Astron. Astrophys. 7, 150-153 (1983). 59. G. L. Cooper and W. T. Schmars, Selected applications of a biaxial tiltmeter in the ground motion environment. AIAA Cod. Guidance, Control, Flight Mech. (1973). 60. J. C. Harrison, Tilt observations in the Poorman Mine near Boulder, Colorado. J. Geophys Res. 81, 329-336 (1976). 61. E. B. Dussan V. and R. T. Chow, On the ability of drops or bubbles to stick to nonhorizontal surfaces of solids. J. Fluid Mech. 137, 1-29 (1983). 62. J. Verbaandert, L’ktalonnage des pendules horizontaux. Boll. Geof. Teor. Appl. 4, 419-446 (1962). 63. J . Beavan and R. Bilham, Thermally induced errors in fluid tube tiltmeters. J. Geophys. Res. 82, 5699-5704 (1977). 64. J. P. Eaton, A portable water-tube tiltmeter. Bull. Seismol. SOC.Am. 49,301-316 (1959). 65. T. Hagiwara, Observation of changes in the inclination of the earth’s surface at Mt. Tsukuba. Tokyo Daigaku Jishin Kenkyusho Iho 25, 27-31 (1947). 66. M. Kato, Observations of crustal movements by newly-designed horizontal pendulum and

water-tube tiltmeters with electromagnetic transducers. Bull. Disaster Prev. Res. Inst., Kyoto Unive. 27, 155-171 (1977). 67. W. W. Gile, A mercury pendulum seismometer. Geophys. J. R. Astron. SOC.36, 153-165 ( 1974). 68. F. D. Stacey, J. M. W. Ryan, E. C. Little, and C. Croskell, Displacement and tilt transducers at 140 dB range. J. Phys. E 2, 945-949 (1969). 69. J. A. C. Horsfall and G. C. P. King, A new geophysical tiltmeter. Nature (London) 274, 675-676 (1978). 70. A. A. Michelson, Preliminary results of measurements of the rigidity of the earth. Asrrophys. J. 39, 105-138 (1914).

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71. A. A. Michelson and H. G. Gale, The rigidity of the earth. Astrophys. J. 50, 330-345 (1 9 19). 72. R. G. Bilham, R. Plumb, and J. Beavan, Design considerations in an ultra-stable, long

baseline tiltmeter-results from a laser tiltmeter. In “Terrestrial and Space Techniques in Earthquake Prediction Research” (A. Vogel, ed.), pp. 235-254. Vieweg, Wiesbaden, 1979. 73. H. Benioff, A linear strain seismograph. Bull. Seismol. SOC.Am. 25, 283-309 (1935). 74. R. C. Shopland, Shallow strain seismograph installation at the Wichita Mountains seismological observatory. Bull. Seismol. SOC.Am. 56, 337-360 (1966). 75. R. C. Shopland and R. H. Kirklin, Application of a vertical strain seismograph to the enhancement of P waves. Bull. Seismol. SOC. Am. 60, 105-124 (1970). 76. J. E. Fix and J. R. Sherwin, A high-sensitivity strain-inertial seismograph installation. Bull. Seismol. SOC. Am. 60, 1803-1822 (1970). 77. H. Benioff, Fused-quartz extensometer for secular, tidal, and seismic strains. Geol. SOC. Am. Bull. 70, 1019-1032 (1959). 78. M. Major, G. Sutton, J. Oliver, and R. Metsger, On elastic strain in the earth in the period range 5 seconds to 100 hours. Bull. Seismol. SOC.Am. 54, 295-346 (1964). 79. J. L. Blayney and R. Gilman, A portable strain meter with continuous interferometric calibration. Bull. Seismol. SOC.Am. 55, 955-970 (1965). 80. J. Dratler, Inexpensive linear displacement transducer using a low power lock-in amplifier. Rev. Sci. Instrum. 48, 327-335 (1977). 81. T. Tanaka, On an extensometer of variable capacitor type. Bull. Disaster Prev. Res. Inst., Kyoto Univ. 15, 49-59 (1966). 82. L. Latynina, E. Starkova, B. Podgornykh, and R. Karmalyeva, Deformations of the

83. 84. 85. 86. 87. 88. 89.

earths’s crust at the Kondara station of the Tadzhik Socialist Soviet Republic. Izv. Acad. Sci. USSR, Phys. Solid Earth (Engl. Transl.) pp. 184-189 (1968). M. Major, Strainmeters. In “ESSA Symposium on Earthquake Prediction,” pp. 69-71. U.S.Gov. Print. Off., Washington, D.C., 1966. T. Mikumo and K. Aki, Determination of local phase velocity by intercomparison of seismograms from strain and pendulum instruments. J. Geophys. Res. 69,721-731 (1964). G. Hade, M. Connor, and J. Kuo, Laser interferometer calibration system for extensometers. Bull. Seismol. SOC. Am. 58, 1379-1383 (1968). S. Takemoto, Laser interferometer systems for precise measurement of ground strains. Bull. Disaster Prev. Res. Inst., Kyoto Univ. 29, 65-81 (1979). Chi Shunliang, Preliminary experimental result of a capacitance-type borehole earth strain meter. Acta Seismol. Sin. 4, 98-103 (1982). In Chin. M. T. Gladwin, High precision multi-component borehole deformation monitoring. Rev. Sci. Instrum. 55, 2011-2016 (1984). G. Bomford, “Geodesy.” Oxford Univ. Press (Clarendon), London and New York,

1962. 90. P. H. Sydenham, A tensioned-wire strain seismometer. J. Phys. E 2 , 1095-1097 (1969). 91. R. 0 . Bilham and G. King, Strain-gauges for geophysics. Commun. Obs. R. Belg., Ser. A 13, 258-278 (1971). 92. V. B. Gerard, An Invar wire earth strain meter. J. Phys. E 4, 689-692 (1971). 93. E. Hauksson, J. Beavan, and R. Bilham, Improved carbon-fiber extensometers. EOS, Trans. Am. Geophys. Union 60, 936 (1979). 94. G. King and R. Bilham, A geophysical wire strainmeter. Bull Seismol. SOC. Am. 66, 2039-2047 (1976). 95. J. Dewey and P. Byerly, The early history of seismology (to 1900). Bull. Seismol. SOC.Am. 59, 183-227 (1969).

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96. D. W. Evertson, “Borehole Strainmeters for Seismology,” Rep. ARL-TR-77-62, Appl. Res. Lab., Univ. of Texas, Austin (1977). 97. 1. S. Sacks, S.Suyehiro, D. W. Evertson, and Y. Yamagishi, Sacks-Evertson strainmeter, its installation in Japan and some preliminary results concerning strain steps. Pap. Meteorol. Geophys. 22, 195-207 (1971). 98. A. McGarr, I. S. Sacks, A. T. Linde, S. M. Spottiswoode, and R. W. Green, Coseismic and other short-term strain changes recorded with Sacks-Evertson strainmeters in a deep mine, South Africa. Geophys. J. R. Astron. SOC.70, 717-740 (1982). 99. S. Sakata, S. Shimada, and S. Noguchi, Development of new-type three-component borehole strainmeters. Proc. Joint Panel Meet. U.N.J.R. Panel Earthquake Predict. Technol., 3rd (1982). 100. J. Berger and R. Lovberg, Earth strain measurements with a laser interferometer. Science 170, 296-303 (1970). 101. N. R. Goulty, G. C. P. King, and A. J. Wallard, Iodine stabilized laser strainmeter. Geophys. J. R. Astron. SOC.39, 269-282 (1974). 102. J. Levine and J. L. Hall, Design and operation of a methane absorption stabilized laser strainmeter. J. Geophys. Res. 77, 2595-2609 (1972). 103. J. Beaven and N. Goulty, Earth-strain observations made with the Cambridge laser strainmeter. Geophys. J. R. Astron. SOC. 48, 293-305 (1977). 104. I. S. Sacks, J. Snoke, Y. Yamagishi, and S. Suyehiro, Borehole strainmeters: Long-term stability and sensitivity to dilatancy. Year Book Carnegie Inst. Wash. 74,287-291 (1975). 105. K. Kasahara, Tiltmeter observation in complement with precise levellings. J. Geod. SOC. Jpn. 19, 93-99 (1973). In Jpn. 106. T. Sato, K. Tachibana, and H. Ishii, Observation of crustal movements at the Akita geophysical observatory ( 5 ) . J. Geod. SOC.Jpn. 25, 277-288 (1979). In Jpn. 107. J. E. Fix, Ambient earth motion in the period range from 0.1 to 2560 sec. Bull. Seismol. SOC.Am. 62, 1753-1760 (1972). 108. B. Block and J. Dratler, A review of tidal, earth normal mode and seismic data obtained with quartz torsion accelerometers. Geophys. J. R. Astron. SOC. 31, 239-269 (1972). 109. R. A. Haubrich and H. M. Iyer, A digital seismograph system for measuring earth noise. Bull. Seismol. SOC. Am. 52, 87-93 (1962). 1LO. D. C. Agnew, Strain tides at Pirlon Flat: Analysis and interpretation. Ph.D. Thesis, Univ. of California, San Diego, 1979. 11 1. J. E. Fix and J. R. Sherwin, “Development of LP Wave Discrimination Capability Using LP Strain Instruments,” Rep. TR-72-3 (NTIS AD-748-232), Teledyne-Geotech (1972). 112. J. Berger and J. Levine. The spectrum of earth strain from lo-* to I d Hz. J. Geophys. Res. 79, 1210-1214 (1974).


19. GEOPHYSICAL WELL LOGGING

Jay Tittman Technical Consulting Services Danbury, Connecticut 06810

1. Introduction 1.l.Background

Geophysical well logging consists, for the most part, of lowering instrument packages into holes in the earth in order to measure physical parameters that characterize the formations. These measurements are presented versus depth and are referred to as a log, i.e., a record of geological and/or petrophysical information. The holes may be wells, intended for the production of fluids, or boreholes drilled exclusively for the purpose of exploration. The instrument package includes a sonde, or probe, containing the sensors which perform the measurements. An electronicscartridge connected to the sonde controls the sensors, providespower at appropriatelevels, receives and processes sensor output signals, often performs data reduction, and may include the downhole modem for a digital telemetry system. The sonde and electronicscartridgejointly are referred to as the logging tool, or merely as the tool. The logging tool is suspended on the end of a cable that is usually multiconductor (4 - 7 wires), although monoconductor cables are in conventional use also. (Coaxial cables and fiber-optic cables are in development.) The cable permits the downward flow of electrical power (in rare cases downhole battery packs are used) and the upward flow of electrical signals from the tool. Surrounding the bundle of insulated conductors are usually two steel wire-wraps, one inside the other, wound in opposite senses. This armor provides abrasion protection for the inner conductors and tensile strength for the cable. At the surface the cable is spooled on a powered winch-drum carried on a speciallydesigned truck or portable loggingunit for offshore use. Collector rings on the drum permit electrical connectionsto be made to the cable’s inner conductors. The uphole end of the cable communicateswith a computer-based control and data acquisition system. This is programmed for on-line data processing so that while the logging tool is drawn up the borehole by the winch a continuously recordedlog is made of one or more physical parametersversus 44 1 METHODS OF EXPERIMENTAL PHYSICS Vol. 24, Part B


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depth. The uphole portion of the system, in addition to its data-processing and recording functions, often has the capability to receive telemetered information about the state of the tool downhole. It can also send down commandswhich alter that state, often functioningas an element in a digital feedback loop. Figure 1 illustrates a typical logging setup at the well site. Originally,well logging was applied principally to infer the nature of fluids fillingthe pores of sedimentaryrocks, i.e., to determinethe saturationsof oil, water, and gas. Today, the spectrum of available logging measurements is sufficiently broad to permit, with varying degrees of accuracy, in-situ determination of rock density, porosity, major-element constituents,clay types, presence of fractures and their orientation, formation structural and stratigraphic dips, permeability to fluid flow,the nature of naturally occuring

Tension Gauge

Control and

Winch

Cable

FIG. 1. Typical logging setup. [From R. Desbrandes, “Diagraphies dam les Sondages.”

Editions Technip, Paris, 1982.1

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radioactiveelements, etc. The range of physical phenomena exploited is very wide. It includes electrical current flow, low- and high-frequency electromagnetic wave propagation, neutron moderation, thermal-neutron absorption, natural and artificial radioactivity, gamma-ray spectra, Compton scattering, photoelectric absorption, body and surface waves in the sonic frequency range, subsonic seismic waves, and nuclear magnetic resonance. Many conventional logging measurements developed for hydrocarbon exploration have been extended to other applications such as uranium, potash, copper, and iron ores, coal quality, coal ash content, and the location of potable water. 1.2. Logging versus Coring

Another way to learn about the earth’s subsurface is to extract cores for laboratory analysis. Logging and coring in some respects are competitive techniques but are in some ways complementary. Coring conventionally uses a hollow-barrel drill collar to which is affixed an annular, diamond impregnated drill. As the core hole is drilled, a continuous cylinder of rock is fed up into the core barrel where it is seized and brought to the surface. It is usually practical to retrieve cores in lengths up to only about 30 -60 ft (9 - 18 m) at a time. The whole drill string, consisting of lengths of pipe screwed together, must be pulled from the well each time a section of core is to be extracted. For wells of even moderate depth the process of pulling the drill string and reinserting it with an empty core barrel is time consuming and costly. In comparison, logging measurements can be made over the completedepth ofthe well and, ordinarily, the drillstring need be pulled out only once to permit the tools to enter the hole. Since modern loggingtools are designed to be combinable, it is usually possible to make all the measurements desired in only one or two “trips” into the well. When the core barrel is emptied at the surface some sections of core are frequently missing. This occurs most often with materials that disintegrate upon exposure to water in the borehole, such as shales containing the clay mineral montmorilionite, and with easily friable rocks, such as poorly cemented sandstones. Not only does the lost core leave a knowledge gap but it may also permit movement of the retrieved core within the core barrel. This slippage interferes with accurate determination of the depth of the retrieved core. Although logging measurements are sometimes incorrect because of tool failure, calibration drift, or other limitations (washout of borehole, beds thinner than the vertical resolution of the sonde, etc.) the lost-information problem is substantiallymitigated and measurement depth is generallymore accurate. Coring samples roughly 10- 1O2 in.3/foot( 5 X lo2- 5 X 1O3 cm3/meter)of

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formation thickness, depending on the diameter of the core barrel. By comparison, logging averages over formation volumes ranging from 50 to lo4 in.’/foot (lo3- 5 X 1O5 cm3/meter), depending upon the type of measurement performed. Since many formation properties vary substantiallywithin these volumes, coring can suffer more from statistical sampling problems. Conversely,because many logging measurements average over large vertical distances they are sometimes incapable of resolving parameter variations without the use of deconvolution techniques. The comparison of logs with cores requires that the properties of the cores be measured at least every 6 - 12 in. ( 15 - 30 cm), on the average, to provide statistical validity. Although coring has the obvious advantage of providing an actual sample for laboratory analysis, this sample may have been altered in the process of cutting and carrying it to the surface. As pressure is relieved the core can bleed pore fluids and suffer a change in the relative proportions of, for example, gas and liquid. Bulk properties such as elastic constants and mechanical strength of rocks change as overburden pressure is removed; fractures can open; clays can swell upon exposure to drilling mud, thus drastically changing permeability to fluid flow. The in-situ nature of logging measurements avoids most of these difficulties. Cores do, however, provide the material for laboratory measurementsthat in many cases cannot yet be matched satisfactorily by logging. High-precision mineralogical analysis, determination of low-concentration or trace elements in unaltered parts of the core, and measurement of caloric content of coals are examples. Nevertheless, there are synergisms between logging and coring. Some log measurements depend upon empirical calibration of the sondes. Although these calibrations often can be canied out in laboratory mock-up formations of known composition, it is frequently necessary to tie them to real earth formations, some of whose characterizations are well known from analysis of cores. An ancillary use for cores is in the interpretation of loggingmeasurements. Although individuallog measurements may be correct, the geological or petrophysical interpretation may be wrong if assumptions about some properties of the rock are wrong. For example, the recorded value of electrical resistivity may be accurate but the derived oil saturation will be incorrect if it is not known that conductive clays are present. Core analysis permits the laboratory determination of the types and concentrations of the clays or, what is often even more useful to log interpretation, their contribution to the rock’s resistivity. 1.3. The Logging Environment

Logging measurements and instrument design depart considerably from those encounteredin the laboratory. This arises fiom the peculiar and hostile

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environment in which loggingtakes place, The impact of this environment is evident also in corrections for environmentalperturbations and in log interpretation. Thus, we digress briefly to describe how wells are drilled and to review certain relevant properties of the environment. 1.3.1.Drilling the Well. Most wells are drilled with a special rotary bit positioned at the end of a long string of pipe. The pipe is turned by powerful engines at the surface. While drilling is in progress a liquid mud is pumped down the inside of the drill pipe and out through holes in the drill bit. The drilling mud, usually a specially prepared thixotropic medium, is forced back to the surface through the annular space between the drill pipe and the borehole wall. Several functions are served by the drilling mud. In addition to lubricating the bit and carrying cuttings to the surface, it provides a weighted column of liquid whose hydrostatic pressure is adjusted to exceed that of the pore fluids in the formations, thus preventing blowout. This last function has important consequencesfor logging because of the ways in which it alters the properties of the space under investigation by the sonde. The mud contains special weighting materials, usually clays, for adjusting the density, and chemicals chosen to provide a desired pH and thixotropy. The weighting material usually consists of “natural” clays (coming from formationsthrough which the well has been drilled), additive clays of special type, or barite (BaSO,). Barite has both high density and desirable properties related to suspension. Typical mud densitiesrange between about 1.1 and 2 g/cm3. Because a pressure drop is maintained across the borehole wall there is initially a flow of the mud’s liquid phase into permeable formations. This mudfiltrate displaces movable connate liquids in the pores, thus forcing them deeper into the formation and creating an altered zone around the borehole (Fig. 2). This is referred to as the invaded zone. As invasion proceeds, the particulates in the mud are filtered out on the borehole wall to form ajlter cuke, or mudcuke. Because of the platelike shape and chemical properties of the particulates the mudcake permeability to filtrate flow diminishes rapidly at first and then more slowly, until it approaches zero. In the equilibrium state the mudcake forms an impermeable layer that supports the pressure difference [in the vicinity of 50 psi (0.35 MPa)] between the mud column and the pore fluids. The mudcake thickness usually ranges from less than a millimeter to about 2 - 3 cm. Drilling methods other than rotary exist, some without the use of drilling mud. These can leave the borehole air-wed at the time of logging. Usually in this circumstance a thin, plasterlike layer of powdered rock and formation water is left on the borehole wall. Another alternativeapproach uses drilling muds that are oil-based or consist of an oil emulsion. These usually do not deeply invade the formation, because of oil -water- rock surface tension effects, and they do not leave significant mudcake.

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FIG.2. Schematic representation of a borehole illustrating the logging environment. Only beds permeable to mud filtrate exhibit mudcakes and invaded zones. Clays and shales are frequently ‘‘caved‘‘, or enlarged, as a result of sloughing off material made soft by exposure to drilling mud. Boreholes in hard formations (marl in this illustration) are usually circular and at bit size. [Adaptedfrom R. Desbrandes, “Thkorie et Interprktationdes Diagraphies.” &litions Technip, Paris, 1968.1

1.3.2. Downhole Conditions. In the case of mineral exploration, borehole diameters are usually in the range 1-4 in. (2.5-10 cm). The only purpose of the borehole is to acquire core and/or provide a means for logging-tool entry. Hydrocarbon exploration, which accounts for the overwhelming majority of logging activity worldwide, generally utilizes boreholes in the 6- to 10-inch (15- to 25-cm) range. In some cases, such as those in which very high liquid or gas production rates are expected, diameters can be as large as 12to 16in. (30-40 cm). Hole sizes in hydrocarbonexploration are larger than those in mineral exploration for three reasons: (1) Generally the

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wells are an order of magnitude deeper. (2) It is a primary intention that they will be used for production of hydrocarbons. (3) Adequate diameter must be provided in which to set casing pipe. Nearly all exploration logging is carried out with the tool immersed in the drilling mud.* Consequently, the properties of the mud and the mudcake influence significantly the accuracy with which formation-characteking parameters can be measured. For example, highly conductive mud surrounding the tool in the vicinity of an electrically resistive formation can short-circuit the currents used for probing the formation; gas bubbles in the mud can so seriously attenuate sound waves that measurement of formation sonic velocity becomes impossible; and the hydrogen content of the mud influencesthe moderation of fast neutrons used to probe the formation for hydrogenous fluids. An important class of measurements uses pad-type sondes which are mechanically forced against the borehole wall and which have only shallow depth of investigationinto the formation. Their measurements generally are affected by the thickness and nature of the mudcake. For such sidewallsonde measurements it may be necessary to know the mudcake resistivity, hydrogen content, or barite content, for example, so that appropriate corrections can be made. The mudcake properties that must be known or measured in situ depend upon the physics phenomenon used in the logging measurement. Further environmental complications arise from the diameter and shape of the hole’s cross section. The string of drill pipe typically reaches a mile (1.6 km) in length, and may be as long as 6 miles (- 10 km). Thus, its dynamical behavior under rotation resembles that of an elastic string as much as that of a rigid tube. When anisotropicstresses and fracture networks exist in the formations, the rotary motion ofthe drill string can erode the wall into an oval or egg-shaped cross section. Hence, logging tools that are not properly centralized and pad-type sondesthat are urged against the wall by a single eccentralizing spring or back-up arm will ride on the larger-axis, downhole side of the hole. Even when the hole is circular, diameter variations alter the influence of the mud on the measurement. Borehole size corrections must then be made on line or after recording. In formationsthat are soft or composed of poorly cemented granular materials the borehole may be eroded to a diameter larger than bit size by the action of mud flow and/or drillpipe rotation. This enlargement, or cave, (Fig. 2) occurs most frequently in beds containing certain kinds of clay or shale that have been

* This open-hole logging has as its goal the measurement of formation properties prior to cementing a casing pipe in the hole. Measurement through the casing and cement is referred ta as cased-hole logging.

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JAY TITTMAN

exposed to low-salinity drilling muds. Because these clays absorb fresh water in large quantities they soften, swell, and fall into the well, leaving hole enlargements that may vary either smoothly or irregularly with depth. The caves, filled with drilling mud, replace original formation material within the volume of space investigated by the sonde. Hence, the measured values of formation parameters, being volume averages, may be affected by the mud properties. Another important perturbation arises from the process of invasion described in Section 1.3.1.In its simplest description invasion involves replacing the connate water near the borehole wall by the mud filtrate. The invaded-zone annulus may be only several inches thick or extend to several feet, depending upon the nature of the formation and the drillingmud. Thus, logging measurements, which generally have limited depths of investigation, are affected by the properties of the mud filtrate rather than solely by those of the connate fluids. If oil is present initially it may be displaced also, although only partially. The fractionalvolume of pore space filled with the remaining oil is called residual oil saturation. This oil is held in place as a result of capillary forces in the interconnected rock pores. Hence, both the nature and the proportions of fluids in the invaded zone are different from those present in the virgin rock. Although the physics characterization of the invaded zone yielded by sonde measurements, e.g., electrical resistivity, sonic velocity, or neutron slowing-down length, may be correct, it may not be the one desired, that of the virgin formation. Electrical resisitivity, for example, is strongly affected by invasion since the conductivity and proportion of the connate water are significant determinants of the virgin-formation resistivity. At the other extreme, neutron slowing-down lengths are relatively unaffected by invasion because the hydrogen content of the mud filtrate is usually approximately the same as that of the oil and connate water. Similarly, direct logging measurements of elements in the rock matrix, provided by neutron-induced or natural gamma-ray spectrometry are unaffected by invasion. Pressure and temperature are the last downhole conditions to be considered. Temperaturenear the earth's surface increaseswith depth at an average rate of - 1"Fper hundred feet (- 2" C per hundred meters), although at many locations on the earth there exist significant departures from this mean geothermal gradient. Typically, bottomhole temperatures range up to roughly 212°F (lOOOC),although a significant number reach the 300°F350"F(150°C-175°C)level. From an instrumentation standpoint, the high temperatures encountered in medium-to-deep well logging impose severe requirements on downholeelectronics, sensors, and sonde materials. Special logging tools have been built for operation at ambient temperaturesof 500"F and higher for times long enough to secure a log in deep wells (5- 10 (260°C) hours).

19. GEOPHYSICAL. WELL

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449

Pressure imposes further constraints on the design of logging tools and logging measurements. Usually, downhole pressure is that of the hydrostatic head of drilling mud, 0.5- 1 psi/foot (6.6- 13 kPa/meter) of depth. Hence, most oil and gas logging tools are designed to function properly at pressures between 15 and 20 kpsi (100 and 150 MPa). Special equipment exists for operation at 25 kpsi (175 MPa) and above. 1.4. Standard Sonde Configurations

Logging measurements can be classified according to whether the sensed “field” is natural or artificially produced by a source in the sonde. The former case imposes more-or-less fundamental limitations on signal strength and on the volume of rock sampled. The latter obviously permits an increase in signal strength, when desired, by increasing the strength of the sonde source. In addition, by the use of special configurationsof sourcesand sensors it is possible to influence the location and volume of space sampled. This permits minimizing the perturbing influencesof variations in borehole diameter, mudcakes, and invaded zones. Thus, sondes can be classified as having sensors only, a source and a sensor, or multiple sources and sensors. 1.4.1. Natural Fields-Sensors Only. The first natural field used in logging was of electrochemical origin. Between two vertically separated points in the mud column there may exist a Spontaneous Potential (SP).’ The SP is created by a separation of charge (ions) resulting predominantly from two phenomena: (1) A liquid- liquidjunction, or difusion, potential is created across the boundary between two liquids with different ionic concentrations if the positive- and negative-ion mobilities are merent. This situation can occur, for example, at the boundary between mud filtrate and connate water. (2) A membrane potential can appear across a clay layer between two solutions of different ionic concentration. The membrane allows transport of positive ions and inhibits transport of negative ions, thereby creating macroscopic charge separation. These phenomena lead to the creation of battery-like cells which produce measureable (millivoltrange) potential variations in the mud column. Electrodes located on the exterior of an insulated sonde are used to sense these potential variations. The SP is used primarily for the delineation of sand- shale sequencesand for the determination of the resistivities of connate waters. In many ways the best understood field in logging is that of gravity,2 although it is one of the least used for formation evaluation. The reason for this is the conflict between the requirements of high accuracy and short measurement time in the borehole environment. Borehole gravimeters, mostly of the vibrating string type,3have been in use for a few decades. They must remain stationary in the hole for roughly 10 min in order to reach the

450

JAY TITTMAN

equilibrium needed to achieve the required accuracy. In addition to being very time consuming, this point-by-point measurement method increases the risk of having the tool stuck in the hole. In contrast, the gamma-ray field produced by natural radioactivity is probably the most universally used. Radioactive elements are present to a measurable degree in most sedimentary rocks. The most common sources are potassium-40, a lattice constituent in many mica, feldspar and clay minerals, and thorium- and uranium-serieselementsdeposited over geological time. These elements create a gamma-ray field which is sensed by detectors in the sonde. Modem refinementsin gamma-ray logging are being used with increasing frequency. Scintillation spectrometry separates the contributions from K, U-series, and Th-series gamma-rays. High-resolution spectrometry utilizes intrinsic-Ge detectors; these permit quantitative determination of U238 concentration, for e ~ a m p l e . ~ The naturally occurring temperature field is another one of interest. Whenever a sonde is lowered into a borehole a maximum-readingthermometer is attached so that bottomhole temperature can be ascertained. Resistance thermometers of short time constant (seconds) and high sensitivity (- 0.0 1"C)yield continuouslogs of the temperaturein the mud column. For many purposes, e.g., detection of gas entry, only departures from the geothermal gradient are of interest. In other cases temperatureanomaliessignal the presence of overpressuredzones or nearby large-scaleinhomogeneitiesin thermal difisivity. Formation pore pressure and its response to induced transients near the borehole wall yield important macroscopic characteristics of hydrocarbon reservoirs5: the pressure drive available for natural production of oil, and permeability to liquid flow. As in the case of gravity logging,these measurements are made discontinuously, level by level, in the well. The modem system of formation testing, as this is called, operates in the following manner.6 A pad is forced against the borehole wall, making a pressure-tight seal. Then, through the center of the pad is driven a hollow metal probe which removes the mudcake and slightly penetrates the wall, permitting hydraulic communication between the pores and an empty chamber located in the tool. Upon command from the surface an intervening valve is opened, pore fluid enters the probe and is captured in the chamber for later examination. The absolutepressure and the transient produced during the flow are sensed by gauges located in the device. Although most passive logging measurements, as these are often called, are relatively straightforward in principle, they present obstacles to both accuracy and precision. For example, the precision of natural radioactivity response and bed-boundary location are both limited by a trade-offbetween counting statistics and logging speed. More important, however, is the limi-

19.


45 1

tation imposed upon the kinds of formation characteristics that can be ascertained by logging natural fields. 1.4.2. Artificially Introduced Fields-Source-plussensor. An expanded range of formation characteristics can be evaluated by applying techniques that use fields established by sources in the sonde. These fields reflect physical properties of the formation and are sensed by detectors located in the sonde some distance above or below the source.* When neutrons or gamma rays are used to irradiate the formation, the field is sensed in the borehole by scintillation, solid-state, or gas-discharge detectors. Sonic fields are both produced and detected by piezoelectric or magnetostrictive transducers; high-frequency electromagnetic radiation is produced and detected by small antennas. The electrical resistivity of the formation is often measured by injecting currents from metallic source-electrodes and sensing potential differencesbetween other electrodesalso mounted on an insulated sonde. Another case is that of the electromagneticinduction field produced by a coil oscillating in the 10- to 100-kHz range. The field sensed by a coil some distance above or below the transmitter depends essentially on the conductivity of the formation. Two important properties of all logging measurements, but of special concern in the case of electromagnetic induction, are the shapes and volumes of formation over which averages are measured. This leads to a discussion of the concept of geometric factor, which then leads to definitions of depth of investigation and vertical resolution. 1.4.3. The Geometric Factor. Consider a vertically separated sourcedetector pair located on the axis of an infinitely long mud-filled borehole. If the detector is shielded from the direct influence of the source and the borehole is surrounded initially by infinite vacuum, the detector will register a small signal in response to the mud column. If we now surround the borehole with an isotropic homogeneous medium in the form of an annulus of thickness dr, the detector evidences an incremental response dR. As the thickness of the annulus increases, the response R(r) asymptotically approaches its “infinite-medium” value. Figure 3a pictures the experimental arrangement and Fig. 3a’ shows a schematic response function. The exact form of R(r)is determinedby the nature of the measurement, by the value of the parameter characterizingthe medium, and by design details of the sonde. The last of these includes features such as collimation of source and/or detector, detector energy sensitivity, source energy or frequency, and

* In logging methods using wave phenomena, e.g., sonic, induction, and high-frequency electromagnetic,the customaryterminology is fransmitferand receiver.In other methods, such as nuclear,the terms source and detectorare in common use. In this articlethese conventions are respected in those sections devoted to particularloggingtechniques,but detector,or sensor,and source are used when generic terms are appropriate.

452

JAY TITTMAN

z -

FIG.3. (a) Experimental configuration for defining the radial geometric factor G(r). (a') A representativeform for G(r) showing its monotonic increaseas the formation boundary moves out radially.The correspondingg(r) = dG(r)/dris also sketched. (b) Configurationfor defining the vertical geometric factor G(z). (b') Sonde response as the tool is moved vertically across a horizontal boundary. As for the radial case, g(z) = dG(z)/dz is also shown.

source -detector spacing. Usually R(r) is normalized to its infinite-medium value, G(r)= R(r)/R(m).Here G(r) is called the integral radial geometric factor. From the operational manner in which G(r)is constructed it is clear that, except for some special cases that will be noted later, it represents the fraction of the signal that is affected solely by the medium inside the cylindrical boundary at r. In nuclear radiation logging it is customaryto define the value of r for which G(r)= 0.9 as the depth of investigation; for resistivity measurements the value 0.5 is more commonly used. We can now define a differentialradial geometricfactor g(r) = dG/dr.In a certain sense g(r)can be interpretedas a radial weightingfunction indicating the relative contribution of an infinitely long formation annulus between r and r dr to the measurement provided by the detector. However, this interpretation ofg(r)is not rigorous in general. It imputes to each differential annulus a weight which is independent of the presence or absence of other annuli. Mathematically, this is the equivalent of assuming that g(r) is a

+

19.


45 3

function only of instrumental design constants and is independent of the values of the parameters characterizing the formation, which is usually not the case. For neutron and gamma-ray fields the meaning of G(r)is rigorous since any particle that traverses the cylindrical boundary at r is lost to the vacuum and cannot contribute to the detector response. However, g(r’), where r’ < r, does not include the interactions in dr’ of particles scattered back from beyond r‘, and the interpretation given above does not hold. This question is of special interest when electrical induction fields are used to probe the formation. For the induction case the above interpretation of g(r) is rigorously correct in the limits of either zero frequency or zero conductivity. However, it loses accuracy progressively as frequency or conductivity increases (Section 3.1.2.3). The vertical resolution of a source-detector pair can usually be examined in a manner similar to that for the radial geometric factor. As shown in Fig. 3b, surround the borehole with vacuum in the upper half-space and a semiinfinite medium in the lower. Then move the sonde vertically past the boundary. The normalized response G(z),recorded by the detector, is called the integral vertical geometricfactor. The vertical resolution of a sonde can then be taken as the distance between the values of z for which G(z)= 0.1 and 0.9. This definition is modified somewhat as it is applied to various sondes, but serves satisfactorily as a temporary paradigm. The dzferential vertical geometric factor, or vertical responsefunction, can be written as g(z)= dG/dz. The interpretations of G(z)and g(z) are similar to those for G(r)and g(r).The same comments concerningrigor that were made for g(r) apply to g(z).* The limitations mentioned above notwithstanding, the concepts of geometricfactor, depth of investigation,and vertical resolution have important heuristic value in comparing different sondes measuring the same formation parameter. Furthermore, they provide a basis for correcting for radial inhomogeneities such as mudcake and invasion, and for thin-bed effects. These notions will be extended and made more specific in later discussions of particular logging methods. Satisfactory 1.4.4. Systems with Multiple Detectors and/or So~rces.~ logging measurements could usually be made with a single source-detector pair if boreholes were perfectly circular and just large enough to contain the tool, if geological beds were always sufficiently thick, and if mudcakes and invaded zones did not exist. If these conditions prevailed, whatever volume of space was sampled would be a good representation of the formation. (We

* While this approach to the vertical geometric factor is useful for phenomena which persist in vucuo (neutrontransmission, electromagneticpropagation, etc.), it breaks down completely for those which do not. Thus, for example, it is clearly useless in sonic logging or downhole pressure testing. Appropriate methods for these and other logging techniques are discussed in Chapter 3.

454

JAY TITTMAN

ignore here any macroscopic heterogeneity intrinsic to the formation itself.) Instrument design parameters, e.g., source-detector spacing, could be chosen by criteria related solely to signal-to-noise ratio, dynamic range of response, etc. However, since all the borehole requirements mentioned above are rarely met simultaneously,many sondescontain more than one detector, and in some cases several source- detector pairs. The resulting additional measurements either cancel the undesirable effects or provide corrections, some of which are made automatically. There are two approaches in general use. One exploits symmetries in the measurement method in order to compensate for perturbations. The other combines measurements made with Merent geometric factors so as to produce an effective geometric factor that supresses the undesirable influence of certain regions of space in the vicinity of the sonde. These two methods are exemplified by sonic and induction sonde configurations, respectively. (a) Sonic. The elementary measurement of sound velocity in the formation is made by clocking the time between the firing of a pulsed transmitter and the detection of the first arrival of the sound wave at a receiver. The least-time path usually runs vertically through the formation near the borehole wall (Fig. 9). In addition to the transit through the formation,this path includestwo passes ofthe pulse through the mud between the centered sonde and the borehole wall, one radiating outward from the transmitter and the other refracted back to the receiver. The travel time to a second receiver, located farther from the transmitter than the first, contains the same mud transits. Thus, measurement of the difference between the times of arrival at the two detectorsautomatically removes the influence of paths through mud and mudcake. It remains to remove the effect on the transit time of a sudden change in borehole diameter, such as occurs at the boundary of a cave. This can be achieved to first order by averagingthe travel times in the upward and downward directions. Hence, another transmitter is placed symmetrically with the first, on the other side of the two receivers. The upward-going and downward-going inter-receiver transit times are separately measured before the sonde moves significantly, and are then averaged. Sonic sondes of this type are referred to as borehole compensated. (b) Induction. Typically, in an induction sonde the basic transmitterreceiver spacing is chosen to provide a depth of investigation of at least a few feet. This simultaneouslyproduces broad vertical resolution. Consequently, a highIy conductive bed at considerable distance above or below the sonde can unduly influence the conductivity value inferred from the receiver signal. Additionaltransmitter and receiver coils, suitably designed with respect to position and spacing, number of turns, and sense of the windings, are

.19. GEOPHYSICAL WELL LOGGING

455

placed on the sonde. Each resulting transmitter- receiver pair is characterizable by its own g(r, z).* When all the receiver outputs are combined, the resulting signal corresponds to that from a differential geometric factor which is the weighted mean ofthe individual g‘s: (g(r, 2)) = Xy-, wig,@,z), where n is the number of transmitter-receiver pairs and wi is the normalized weight for the ith pair. By judicious design (g(r, z)) can be tailored so that its values are drastically reduced at large distances above and below the sonde and at reasonably small radii. The latter diminishes the influence of the conductivity of media close to the sonde: borehole mud, mudcake, invaded zone, and mud-filled caves. Most sondes using neutron and gamma-ray sources have vertical resolutions in the range of 1 to 2 feet (30 to 60 cm), which is usually satisfactory. However, their depths of investigation are generally small, S 6 in. ( 15 cm), and their measurements would suffer serious borehole-size and mudcake effects if special design precautions were not taken. Formation-density tools, which utilize Compton scattering, compensate for mudcake effect by employing a single gamma-ray source and two detectors, one at “short” spacing and the other at “long.” These are mounted in a pad forced against the borehole wall and are shielded in the rear. The shortspacing measurement corresponds to a small depth of investigation, while the long spacing “sees” deeper. Each yields an apparent formation density that depends on both mudcake properties and true formation density. The sonde produces two measurements which are linearly independent functions of the sametwo variables.Hence, knowledgeof the countingrates from the two detectors permits determination of the correct formation density and, incidentally,an indication of the magnitude of the mudcake influence. Most neutron sondes contain at least two detectors. These permit the measurement of formation propertiesonly modestly affected by mudcake or borehole size. Deconvolutiontechniques have been under intensive development,principally for sharpeningvertical resolution. Induction measurements,because of their large-volumeaveraging, have been the principal beneficiaries of this

* Although operational definitions ofg(r)andgfz), individually, were given in Section 1.4.3, a more detailed treatment (Section 3.1.2.1) leads to a single g(r, z) appearing in a linear convolution expression for the measured conductivity:

where a,,, is the measured value of conductivity and n is the true formation conductivity. The permittedspatialvariation of aallows applicationto practicalcaseswhich include mud column, mudcake, invaded zone, caves, and bedding.An evenmore generaltreatment is given in Section 3.1.2.3.

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JAY TITTMAN

effort. Although crude on-line analog deconvolution techniques have been in use for a long time, the current availability of computers in the surface portion of logging systems has spurred the development of high-speed programs for this purpose. Because most other logging measurements have much finer vertical resolution than those for resistivity, deconvolution methods for them have been neither so badly needed nor so well developed. Nuclear methods for uranium and coal logging are exceptionsbecause many deposits of these minerals are thin and/or laminated. For these applications sondes have been designed with intrinsically sharper vertical resolution (even at the expense of depth of investigation) and special deconvolution computer programs are in use.* 1.5. Sonde Combinations

The earliest logs were recordings of a single formation parameter, electrical resistivity. During the intervening half century, downholemeasurements have proliferated to the point where it is no longer practical to make them one at a time. There are two principal reasons for this: (a) It is excessively time-consuming and (b) depth-matching of different runs is often difficult. (a) If a well is left without mud circulation for a long period of time, such

as would be required for many individual logging runs, clay-bearing formations can swell into the hole and block it, or poorly consolidated sands can fall into the hole and plug it. Either can prevent the tool from moving. Thus, it often becomes necessary to clean out the hole periodically between logging runs by lowering the drill pipe and recirculating the mud. This process, known as “tripping the well”, is very time consuming and expensive,as is the one-measurement-at-a-time logging procedure itself. (b) Depth matching of measurements made during different logging runs can be difficult because of the cable’s elasticity, although an auxiliary measurement, such as a natural gamma-ray log, common to each run usually permits a correct match. Because sonde depth is determined by the length of cable spooled off the drum at the surface, residual uncorrected cable stretch may introduce systematic error in the recorded depth. This is exacerbated by variable friction as the tool is drawn up the hole. Intermittent sticking and releasing of the tool produces a motion, called yo-yo, which is not easily sensed at the surface. Although cable-stretch and yo-yo corrections for depth matching can often be made by careful manual correlation among different logs, totally automated depth matching is not yet completely reliable. Over the past few decades there has been consistent progress in the development of combinable tools, making many measurements simultaneously, in order to avoid these problems. High-speed digital telemetry systems and

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LOGGING

low-power solid-state electronics have been key elements in the success of this effort. Also, the capability to digitally depth-shift data while logging has been an important element in implementing sonde combinations, since the measure points of individual sondes in the string can be many tens of feet apart. This permits several measurements, made at the same depth sequentially as the tool string is drawn up the hole, to be combined while logging. Thus, derived outputs of geological or petrophysical interest are available immediately.For example, by storing in the computer memory the response functions and environmental corrections for each sonde in the tool string, the four measurements indicated in Table I could be made to yield, on line, the derived quantities shown. The choice of sondesto be combined in a particulartool string depends on the problem to be solved and the synergy among the measurements. Consider the group of tools shown in Table I. They provide four physics measurements that arejointly functions of five petrophysical descriptors.Pore-liquid density, the undetermined parameter, can be established separately from produced water samples or the value unity can be used as a first approximation. The SP provides the water salinity value to combine with the conductivity measured by the induction sonde. (The connection between water resistivity and resistivity of the water-containing rock is described in Section

TABLE I. Typical Combination of Sondes and Their Derived Formation Descriptors Sonde name Induction

Measured physics parameter

Contributing petrophysical descriptors

Electrical conductivity

Porosity Water saturation Oil saturation

SP

SP

Water salinity

Neutron

Slowingdown length or migration length

Porosity Rock type

Petrophysical descriptors derived by combining measurements

I

Bulk density

Porosity Pore liquid density Rock matrix density

of porosity and oil saturation

i

, Gamma-gamma density

A function

Porosity and rock type

Porosity Oil saturation Rock type

45 8

JAY TITTMAN

2.1 .) The combination of neutron and density logs yields both porosity and rock type. Combiningknowledgeof porosity with the results of the induction and SP measurements then yields oil saturation. Not indicated in Table I is the ubiquitous natural radioactivitylog. Nearly every time any logging run is made a gamma-ray sondeis in the tool string. It usually contains a Dewar-flasked NaI scintillation detector which records the total gamma-ray counting rate above a preset low-energythreshold. This log provides a rough measure of formation shalinessand easy recognition of geological markers. If the well is to be put in production, the latter feature permits accurate positioning of the gun used for perforating the casing in front of the hydrocarbon reservoir. The longest string of combinable sondes in use today consists of gammaray/neutron/density/electromagneticpropagation/induction/poweredcaliper tools. Its length is nearly 100 ft (30 m).

References 1. J. R.Jordenand F. L. Campbell, “Well Logging 11-Electric and Acoustic Logging” (Sects.

6.1 and 6.2). H. L. Doherty Monograph Series, Vol. lo., SOC.Pet. Eng., Dallas, 1986. 2. A 1 18-entry bibliography (most with abstracts) on subsurface gravimetry can be found in S.L.Robbins, U.S.Geol. Surv. Open-File Report 80-170 (1980). 3. M. B. Dobrin, “Introduction to Geophysical Prospecting,” 3rd Ed., p. 399, McGraw-Hill, New York, 1976;L. G.Howell, K. 0.Heintz, and A. Barry,Geophysics 31,764 (1966); R. R. Goodell and C. H. Fay, Geophysics. 19,774 (1964). 4. L. H. Goldman and H. E. Marr, SOC.Pro$ Well Log Anal. Ann. Logging Symp. Trans., 20th, Tulsa, 2, Pap. GG (1979). 5. D. K. Sethi, W. C. Vercellino, and W. H. Fertl, Soc. Prof Well Log. Anal. Ann. Logging Symp. Trans., Zlst, Lafayette, La..Pap. CC (1980);G.Stewart and M. Wittman, Soc. Pet. Eng. Ann. Fall Tech. Conf, 54th, Las Vegas, SPE Pap. 8362 (1979); J. H. Moran and E. E. Finklea, J. Pet. Tech. 225,899 (1962). 6. Sethi et al(l980) see Ref. 5; A. L.Schultz, W. T. Bell, and H. J. Urbanosky, SOC.Pet. Eng. Ann. Fall Tech. Con., Houston, 49th, SPE Pap. 5035 (1974). 7. References for most of the systems mentioned in this section are provided in Chapters 2 and 3, where measurement methods and apparatus are described in more detail. 8. R. D. Wilson, D. C. Stromswold, M. L. Evans, M. Jain, and D. A. Close,SOC.Prof Well Log Anal. Ann. Logging Symp. Trans., ZOth, Tulsa, 2, Pap. FF (1979); J. G.Conaway and P. G. W e e n , Geophysics.43,1204 (1978);E. P. Howell, 0.J. Grant, Jr., and T. J. Crebs, SOC. Pet. Eng. Ann. Fall Tech. Conf, 53rd, Houston, SPE Pap. 7434 (1978);J. A. Czubek, Soc. Prof Well Log Anal. Ann. Logging Symp. Trans., 14th, Lafayette, La., Pap. W (1973); Acta Geophys. Pol. A, 9, 121 (1961).


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459

2. Geological and Petrophysical Interpretation of Logging Measurements Nearly all logging measurements characterize formations by the use of physics-type parameters, e.g., electrical resistivity, electron density, sound velocity, etc. Although of interest in their own right, most of these quantities are not the ones used by geologists,geophysicists,and petroleum engineersas their primary characterizations. We will refer to the latter as descriptors to simplify nomenclature and because some of them are intrinsically nonquantitative in nature. Descriptors include lithology, mineralogy, porosity, liquid saturations, formation dip, and permeability. Thus, it is necessary to derive descriptorsfrom measurementsof physics parameters. It is essentially this process that is known as log interpretation. Often the relations between these two modes of characterizationare neither simple nor even unique, and the interpretation process exploits any means available: empirical correlation, theoretical analysis of rock-structure models, information from core analysis,local field experience,etc. The structureofTable I already alludes to the translation from physics characterizationto descriptorcharacterization. Table I1 expands upon this by providing larger, but not exhaustive, lists of TABLE11. Measured Physics Parameters and Formation DescriptorsDerived from Combinationsof Them ~~

~

Physics parameters Electrical conductivity Dielectric permittivity Spontaneous potential Neutron slowing-downlength Neutron migration length Thermal-neutron macroscopic absorption cross section Electron density Gamma-ray photoelectric cross section Inelastic-neutron induced gamma-ray spectrum Thermal-neutron capture gamma-ray spectrum Natural gamma-ray spectrum Velocity of sound Nuclear magnetic resonance relaxation time Temperature Borehole pressure

Formation descriptors

Oil saturation Gas saturation Water saturation Irreduciblewater saturation Water salinity Porosity Petrography/lithology Mineralogy Shale/clay content clay type Permeability stratigraphic dip structural dip Coal content Ash concentration Sulfur concentration Uranium concentration Formation pore pressure

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JAY TITTMAN

directly measured physics-type quantities and the descriptor information derived from various groupings of the measurements. This chapter is a necessarily superiicial overview of basic interpretation methods since the subject is a large and complex one.' We treat only a limited number of measurement methods, selected for their widespread use and/or their special interest as examples of physics measurements in the borehole. It is assumed in most cases that the physics parameter has already been measured and corrected for perturbing environmental effects. Where necessary or u s e l l for understanding an interpretation method, a brief introduction to the physics of the logging measurement is given. The most important measurement techniques are described more fully in Chapter 3. This overview of interpretation also exposes some of the motivation behind the variety of logging methods in current practice and in development. 2.1 . Electrical Resistivity

The usefulness of electrical resistivity logging rests on the fact that rocks (with a few notable exceptions) and hydrocarbons are insulators, whereas connate waters are generally saline and, therefore, good conductors. Table 111 lists nominal resistivities for several materials of interest in logging sedimentary formations. Metallic conduction, although operative in metals prospecting, plays no significant role in most resistivity logging. Rather, the electrical conduction met here is electrolytic in nature. It is used mostly for the determination of water saturation S,, defined as the fraction of the rock pore space that is filled with water. For the simple case we will consider, both the rock matrix and any hydrocarbon present are assumed to have infinite resistivity. If the pore structure consisted of straight parallel tubes saturated with salty water, we would have

Ro =

%/+9

(2.1)

where Ro is the resistivity of the rock sample measured in the direction parallel to the tubes, R, is the water resistivity, and $I is the porosity, i.e., the volume fraction of the rock occupied by pores. Of course, the pore structure of real rocks is extremely complex, and the coefficient of R, is not simply l/$I. Observations on a wide variety of rocks leads to an empirical law of the form

%=F%

(2.2) where F is called theformation resistivityfactor, or simplyformation factor. Clearly, Pis determinedby both the porosity and the tortuosity that the pore structurepresents to electrical current flow. For& 5 1 m, Fis found to be


46 1

TABLE111. Electrical Resistivities of Earth Materials Resitivity

(nm at 18-20°C)

Material Pure Materials. Marble Mica Quartz 1I Quartz I Slate Sulfur Petroleum Distilled water Salt Water at 1 5 T b(kppm NaC1) 2 10 20 100 200 Typical Formationsc Clay/shale Saltwater sands

5 x 107- 109 1011- 1014 1 x 10’2 3 x 1014 1 - 2 x 106 loi4- 10” temperature unknown 2 X l O I 4 temperature unknown 0.5 X 104

3.4 0.72 0.38 0.09 0.06 2- 10 0.5- 10 5-10’

Oil sands

103

Compact limestone Dolomite Lignite

103 102

““Handbook of Chemistry and Physics,” 38th Ed., pp. 2237-2238. Chemical Rubber Publishing Co., Cleveland, 1956. R. Desbrandes, “Diagraphies dans les Sondages,” p. 124.Editions Technip, Paris,France, 1982. c R. Desbrandes, “ThBorie et Interprbtion des Diagraphies,” p. 8. Editions Technip, Paris,France, 1968.

approximately constant for rocks of the same intergranular or intercrystalline porosity. In general, the formation factor is empiricallyfound to take the form

F = a/+*, (2.3) where m is known as the cementation factor.2 The constants a and m fall roughly in the following ranges:

sandstones

carbonates

-0.6< a < - I 1.5 < m < -2.5

-

462

JAY TITTMAN

If a particular formation’slithology is constant across a field, a and m can be determined by the analysis of cores from one well and then used in the interpretation of logs in other wells in the same field. Usually R,is found Replacing F in from measurements on water samples or from the SP Eq. (2.2) gives

R,, = (a14”)Rw,

(2.4) a relation much closer to a description of reality than Eq. (2.1).When a is set equal to unity in Eq. (2.4), the latter is usually referred to as Archie’s law. Although discovered empirically,it can be derived from first principles, with a clear meaning assigned to the constant m.4 The condition of greatest interest occurs when the pores are partially water saturated and partially hydrocarbon saturated. From an examination of a large collection of samples Archie found that for granular rocks

Rt = (Fl~;)R,,

(2.5) where R,, the “true” resistivity ofa rock partially saturated by hydrocarbon,* replaces R,, ,and n generally falls in the range 1.7 2.2. Then, using Eq. (2.2) we have

- --

Rt = ( 1Is;)%, (2.6) which permits the evaluation of hydrocarbon saturation So = 1 - S,. Sometimes R,, can be measured in a deeper, completelywater-saturated part of the formation and then used in Eq. (2.6),with R, measured in the upper, hydrocarbon-bearingzone of the same bed. Alternatively, and more generally, we replace F i n Eq. (2.5) to get

Rt = [a/t#PS;]Rw. (2.7) Here we see the need for the additional,independent measurement of q6 and R,,,if we are to determine S,. If m = n, Eq. (2.7) becomes which has the same form as Eq. (2.4),with &replacing&, and q6SW replacing 4. This corresponds to treating the “infinitely” resistive hydrocarbon as if it were part ofthe rock matrix. To the extent that m and n are not exactly equal, the ideal equivalence between hydrocarbon and rock matrix breaks down.

* In most of the logging literatureR, refers to the resistivity of the virgin formationbeyond the invaded zone. Here, we will use it for the correct(true)value in the volume of formation probed by the sonde. The context will make clear whether this refers to the invaded zone or the virgin formation.

19. GEOPHYSICAL W E L L LOGGING

463

Presumably, this occurs not because their resistivities are unequal but because surface tension and capillary effects distribute the oil in such a fashion as to alter the tortuosity for electrical conduction. The factor l/S;createsan effective formation factor, as it were, which takes into account the altered tortuosity. It is often an acceptableapproximation to set a = 1,m = n = 2.& Then Eq. (2.8) takes the very simple form

4 = t 1/((6sw)21R,.

(2.9)

The approach outlined above for obtaining S, from resistivity measurements is the basis for attacking more complex problems not treated here.5 These include fractured systems and, of special importance, formatidns containing clay or shale, which exhibit surface conductivity effects arising from ion exchange phenomena. Since we have assumed that the true virgin formation resistivity4 was in hand before interpretationwas attempted,we have also ignored a practicalproblem of great importance: how to determine 4 from the log-measured resistivity, which is materially influenced by the resistivity of the invaded zone. This is treated in Section 3.1.1.5. 2.2. Neutron Moderation and Diffusion

We have seen that if & cannot be measured reliably, as is often the case, then Eq. (2.7) must be used to get S,, and an independent measurement o f 4 is required. An additional and quite separate need for the value of (6 arises in the determination of the total hydrocarbon content per unit volume of formation, +( 1 - S,). Neutrons are used effectively in the measurement of porosity because hydrogen strongly affects neutron moderation and because rock pores are nearly always filled with hydrogen-rich liquids, i.e., oil and/or water. (We ignore the case of hydrocarbon gas in this discussion.) However, just as we saw that resistivity is a function of as well as of S, ,leading to the need for an independent determination of 4, neutron moderation depends upon the mineral composition of the rock matrix as well as upon (6, leading to the need for still another measurementin order to establish the value of (6. This pattern is repeated throughout much of logging and is responsible, in part, for the large number of different logging measurementsshown in Table

+

11.

Consider a formation composed of a single mineralogicalmaterial, such as quartz (sandstone) or calcite (limestone), and uniform* porosity saturated with fresh water. The counting rate of, for instance, an epithermal-neutron detector located 1-2 ft (30-60 cm) above the fast neutron source in the

* Uniformity is adequatelyrealizedwhen the sizes of the rockgrainsand of the pores are small comparedto a neutron mean-free-path. In many cases this condition can be relaxed, however, to permit the characteristicdimensions to be small relative to the neutron slowing-downlength.

464

JAY TITTMAN

sonde, is the elementary measurement made. The formation’s moderating properties can be approximately characterized by the two independent parameters of simple diffusion theory$ D, the diffusion coefficient, and L,, the slowing-down, or diffusion, length. L,, the quantity which is more important for our needs, is proportional to the mean rectified distance a neutron travels in slowing down from some initial energy Eoto some final energy of interest E,, e.g., thermal energy. Clearly, L, characterizesa moderating medium only when Eo and Ef remain fixed. Both L, and D are strong functions of the hydrogen concentration and, generally to a lesser degree, of the rock matrix material. Thus, if the only important property varying were the porosity, we (2.10)

c

1

0

I

I

I

I

10

20

30

40

Porosity, %

FIG.4. (a) Porosity responses of a sidewall neutron sonde using a single epithermal detector. (b) The experimentalresponsesof (a)presented as a function of slowing-downlength calculated for each experimental point. [From H. Edmundson and L. L. Raymer, SPWZA 20th Ann. Logging Symp. Trans.,Tulsa,June 3-6, 1979, Vol. 1, paper 0.1

19.

.^ 10

-

8

t

Sandstone A Dolomite o Limestone @ Water (100%) 0

(4 14

-

12

-

465


/

10-

X

v)

0. 0

6 a-

I?

C 0)

.c C

6 -

a 0 4 -

2-

L, Slowing Down Length, cm RG.4. (Continued)

and a unique value for the porosity could be derived from the recorded counting rate. In practice, the response functionsf($) of neutron sondes are determined by measuring the counting rates in laboratory mock-up formations under a set of standard conditions, e.g., 8-in. (20-cm) borehole diameter, fresh water in the rock pores, and rock matrices of pure calcite, quartz, or dolomite. (Field logs are usually corrected, prior to interpretation, for departures from standard conditions.) For each rock type several standard formations of different and known porosity are constructed? Figure 4 presents the counting-rate responses of one type of single-detector, sidewall neutron sonde to porosity in the three most common rock-matrix materials.* Figure 4b is a transformation of Fig. 4a created by a Goertzel-Greulingcalculation of the formation L, for each experimental point? The small scatter of points indicates that such a single-parameter characterization of the formation is a useful approximation at porosities above about 10%.in the lower porosity range (larger L, values) the increasing scatter suggests a growing dependence on D. Figure 4b permits estimating sonde responses in rock matrices for which standard formationsdo not exist, but for which L, can be

466

SAY TITTMAh'

calculated. Similarly, responses in rock matrices consisting of any posited mixture of minerals can be synthesized, provided that the computed values of L, fall inside the range encompassed by the experimental data. In contrast with resistivity and several other types of logs, neutron logs are usually scaled directly in terms of the descriptor, porosity. Since the sonde response to a particular mineral or the nature of the rock matrix itselfmaybe unknown, the counting rate is often converted into limestone-equivalent porosity through relations of the type shown in Fig. 4a. This procedureplaces the burden of interpretation on the translation from limestone-equivalent porosity to true porosity ofthe actual matrix. Section2.3 showshow the need for knowledge of rock matrix, or lithology, is met by still another physical measurement. Another type of neutron sondeutilizes the ratio of counting ratesfrom two epithermal-neutrondetectorslocated at different spacingsfrom the source.lo This measurement is shown in Fig. 5 to depend, even better than that of Fig. 4, upon the single parameter L, for formation characterization. The large number of experimentalpoints representing different porosities, lithologies, formation-watersalinities,and even gas saturation, validates single-parameter characterization of formations for this type of logging measurement. In this discussion we have considered for the sake of brevity only sondes using epithermal-neutron detection. These exhibit, in a certain sense, the largest effect of liquid-filled porosity. In practice, thermal-neutron detection is more widely used because of the high counting rates available. The responses,however, are still dominated by the neutron-moderatingproperties of the environment. Thus, the principles outlined above remain operative. This subject is considered in more detail in Section 3.2. As in the discussion of resistivity, we have treated only an ideal formation, this one consisting of a pure-mineral rock matrix with fresh-water-fled porosity. In practice it is necessary to take into account the widespread presence of clay minerals, some of which contain substantial amounts of chemically bound water that would otherwise be interpreted as porosity. Further complications, which require additional measurements and/or more extended interpretation methods, include: water salinity, which reduces the hydrogen atom density in water and increasesL,;lO the occurrence of unexpected minerals in the rock matrix; the occurrence of hydrocarbon gas in the pores, which generally reduces the hydrogen atom density;' * water of hydration in certain minerals, e.g., gypsum; the somewhat different hydrogen concentrations in oil and water; the slightly different moderating effects of carbon (in oils) and oxygen (in water); and the presence of the mud-filled borehole. The additional measurementmost commonlycombined with the neutron log is that of bulk density. Its role in formation descriptionis described in the next section.

19. I 10

467

GEOPHYSICAL. WELL LOGGING

Q

-

8-Inch Water-Filled Borehole

a,

= 'OoYo

.-0

3 a:

1 -

SS LS DOL 0

w o o SW. GAS

4

0

0

5

10

15

20

25

3

Ls, Slowing-Down Length, cm

FIG. 5. Experimental responses of a neutron sonde using two epithermal detectors to a variety of laboratory formations. Slowing-down lengths were calculated for each formation condition. Legend: SS, sand; LS, limestone; DOL, dolomite; FW, 100%saturated with fresh water; SW,100%saturated with salt water; GAS, 37% porous sands with several different gas/water ratios. [Adapted from H. D. Scott, C. Flaum, and H. Sherman, Soc.Per. Eng. 57th Ann. Fall Tech. Con&Sept. 26-29, 1982, New Orleans, paper number SPE 11146. Copyright 1982 SPE-AIME.]

2.3. Gamma-Ray Scattering

Density logging was originally conceived to be a superior, stand-alone technique for the determination of porosity,12even though this method also requires knowledge of lithology for the interpretation. Today, however, the more-or-less standard interpretation procedure is to combine the density measurement with that of the neutron log to yield both porosity and lithology. In addition, density logs assist in the interpretation of borehole gravity surveys and both surface and borehole seismic surveys.13J4 The bulk density of an ideal formation consisting of uniformly distrib-

468

JAY TITTMAN

uted, fluid-filled pores in a rock matrix is simply PB = $Pf

+ (1 - $IPma.

(2.1 1)

Here, pf and pmaare the fluid and rock-matrix densities, respectively. Rearranging Eq. (2.1 1) to extract $ yields $=Pm-PB Pma

- Pf

(2.12)

Herep, is provided by the log;pfis often known from produced fluid samples or can be assumed, as an approximation, to be that of water; and pmacan be assumed or determined from analysis of cores from nearby wells. Several factors contribute to the usefulness of density logging for the determination of porosity: the accuracy with which p~ can be measured in situ (+ 0.02 g/cm3), the relatively narrow range of values encountered for pf,the constancy of pmoin a given type of rock, and the firm foundation of the mixing rule, Eq. (2.11). Table IV lists pmaand pf values for a variety of materials encountered in sedimentary formations. Under suitable conditions the log-measured value of p~ and the use of Eq. (2.12) alone solve the porosity problem. However, the more widely used technique for density-log interpretation rests on the recognition that density logs and neutron logs are independent measurements depending on two variables, lithology and porosity.* For this purpose, cross-plots of the form shown in Fig. 6 are used. Figure 6 refers to a two-detector neutron sonde somewhat similar to that described in conjunction with Fig. 5. It allows the determination of both lithology and true

* Of c o w , lithology is not a single, mathematically defined variable. For density logs, lithology is completely characterized by the single physics parameter p-. To the extent that a single parameter similarly characterizes the matrix for epithermal neutron logging, it is L,. Since the matrix L, is inversely proportional to pm, its independence might at 6rst glance be questioned. However, it is also a function of the independently variable chemical constituents of the rock matrix. Actually, for the neutron logging problem the situation is even more complicated if rigor is demanded. In this event no single characterizing parameter exists in the sense that if the parameter were known for the matrix and for the fluid, a mixing rule in the form of Eq.(2.1 l), linear (or even nonlinear) in the volume-fraction variable 4, could be written for the mixture. This results from the fact that at every energy the influence of each constituent on the moderation process depends upon the other constituents and upon the integrated effect of all of them in moderating neutrons down to that energy. The integral-form definition of L, used in Fermi age theory, for example, makes this clear. Section 3.2 treats this problem in more detail. Despite these observations, the cross-plot interpretation procedure described in the text is a useful device. This is because the pure lithologies encountered in the field are confined to a relatively discrete and well-known set, and because uniqueness of lithology prediction is not of paramount importance (in contrast with porosity prediction).


469

porosity when p, and limestone-equivalent porosity are known. If the formation is composed of a single-mineral rock matrix, the point determinedby a given value ofp, and a given value of limestone-equivalentporosity will fall on one of the lithology curves shown. This identifies the lithology and indicates the porosity simultaneously. If the rock matrix is composed of a mixture of any pair of the pure lithologies indicated in Fig. 6, the point will fall between the two appropriate curves. By connecting points of equal porosity on two lithology curves by a straight line, the porosity of the point in question is approximated. By linear interpolation along the isoporosity line, the proportions of the two lithologies are also approximated. Although the linearity assumption used in this mixed-lithologyinterpretation is rigorously correct only on the density scale, the method provides an excellent approximation to the porosity and to the proportions of the rock matrix minerals. Since more than two minerals may be present in the matrix, it is clear that the cross-plot cannot by itself yield unique answers. However, other logs, cores, or local knowledge ofthe formation help narrow the range of possible combinations of minerals present. The closeness and near parallelism of a set of isoporosity lines between different pairs of minerals permit reasonable estimation of porosity even when the lithology prediction may be considerably in error. The physical basis for density logging by gamma-ray scattering is the fact that the Compton cross section per electron is essentially independent of the atom in which the electron is bound. If we irradiate the formation with gamma rays of initial energy below the pair production threshold (1.02 MeV) and detect those returning to the sonde with energy well above the photoelectric absorption region (say, 150keV) the only interaction of consequence is Compton scattering. Thus, irrespective of the number of scatterings taking place in the formation, only the electron density (number per unit volume)* determines the counting rate. This electron density is just

n, = NA(Z/A)p,,

(2.13)

where NA is Avogadro’s number, Z the average atomic number, and A the average atomic weight of the formation. For most of the elements in sedimentary formations Z/A is closely 4; thus the only property of the formation to affect the counting rate is p,. By appropriate mass filtering or electronic energy discrimination at the detector output, logging measurements usually can achieve independence of the chemical composition of the formation for values of Z typically less than 20 (calcium). This range includes the most common elements in sedimentary formations: hydrogen, carbon, oxygen,

* The quantityp, defined in Table IV, footnote b, is also called electrondensityin the logging literature. The context should make clear which quantity is intended.

TABLEIV. Formation Parameters of Interest in Logging Density and Lithology by Compton Scattering and Photoelectric Absorption [Adapted from W.Bertozzi, D. V. Ellis,and J. S.Wahl, Geophysics 46,1439 (1981).]

Name

Formula

Photoelectric cross section ~ o l ~ u l a rper electron weight (7)O

Photoelectric effective atomic number

Bulk

Electron

density (g/cm3)

density

Z U T

PB

(g/cm3)

d:

):(

Macroscopic photoelectric

cross section U '

Elements

H C

0 Na Mg Al Si

S

cl K ca Ti Fe Sr

zr

Ba Minerals Anhydrite Barite Calcite Carnallite Celestite Corundum

1.008 12.011 16.000 22.991 24.32 26.98 28.09 32.066 35.457 39.100 40.08 47.90 55.85 87.63 9 1.22 137.36

0.00025 0.15898 0.44784 1.4093 1.9277 2.57 15 3.3579 5.4304 6.7549 10.081 12.126 17.089 31.181 122.24 147.03 493.72

136.146 233.366 100.09 277.88 183.696 101.96

5.055 266.8 5.084 4.089 55.13 1.552

1 6

8 11 12 13 14 16 17 19 20 22 26 38

2.700

2.602

2.070

2.066

2.960 4.500 2.710 1.61 3.960 3.970

2.957 4.01 1 2.708 1.645 3.708 3.894

40 56 15.69 47.2 15.71 14.79 30.4 11.30

1.984 0.999 1 Loo00 0.9566 0.9868 0.9637 0.9968 0.9979 0.9589 0.9719 0.9980 0.9 186 0.931 1 0.8673 0.8770 0.8154 0.9989 0.8913 0.999 1 1.0220 0.9363 0.9808

14.95 1070. 13.77 6.73 204. 6.04

Dolomite Gypsum Halite Hematite Ilmenite Magnesite Magnetite Marcasite Pyrite

Quartz Rutile Sylvite Zircon Liquids Water Salt water

oil

H20

184.42 172.18 58.45 159.70 151.75 84.33 231.55 119.98 119.98 60.09 79.90 74.557 183.31

3.142 3.420 4.169 21.48 16.63 0.829 22.08 16.97 16.97 1.806 10.08 8.510 69.10

13.74 14.07 15.30 23.45 2 1.87 9.49 23.65 2 1.96 21.96 11.78 19.02 18.13 32.45

2.870 2.320 2.165 5.240 4.70 3.037 5.180 4.870 5.OOO 2.654 4.260 1.984 4.560

2.864 2.372 2.074 4.987 4.46 3.025 4.922 4.708 4.834 2.650 4.052 1.916 4.279

0.9977 1.0222 0.9580 0.9518 0.9489 0.996 1 0.9501 0.9668 0.9668 0.9985 0.95 12 0.9657 0.9383

9.00 8.11 8.65 107. 74.2 2.5 1 109. 79.9 82.0 4.79 40.8 16.3 296.

18.016

0.358 0.807 0.119 0.125

7.52 9.42 5.53 5.6 1

1.Ooo 1.086 0.85od 0.850"

1.110 1.185 0.94gd 0.970"

1.1101 1.0918 1.1157 1.1407

0.40 0.96

( 120,000 ppm NaCI) m,.6 C H 2

Miscellaneous Berea sandstone Pecos sandstone Average shale' Anthracite Wal Bituminous Coal

C:H:O= 93:3:4 C:H:O= 82:5: 13

1.745 2.70 3.42 0.161

11.67 13.18 14.07 6.02

2.308 2.394 2.650" 1.7W

2.330 2.414 2.645d 1.749d

.9993e 1.ooo(r 0.998 1.0287

4.07 6.52 9.05 0.28

0.180

6.21

1.w

1.468d

1.0485

0.26

Since only relative values are required ( 7 ) is given as (z,a/10)3.6. = 2(Z/A)pB. U = (7 ) p e .See Eq.(2.19) for the utility of U.The units of U are arbitrary; see footnote u above. Variable; values shown are illustrative only. Value is for matrix only. 'Elemental compositiontaken &om F. J. Pettijohn, "sedimentary Rocks," p. 271. Harper, New York, 1949. bpe

O.lld

0.12d

472

JAY TITTMAN

Equivalent Limestone Porosity, O h

FIG.6. A typical cross-plot for the determination of lithology and liquid-filled porosity from measurementsof density and neutron sondes. Note the inverteddensity scale. In additionto the three principal lithologies, certain other minerals can often be identified by their characteristic locations on the cross-plot chart. [Adapted from “Log Interpretation Charts.” Schlumberger, Ridgefield, Connecticut, 1979.1

silicon, magnesium and calcium. Departures of Z/A from the value f, indicated in Table IVYare taken into account when necessary. As in the cases of resistivity and neutron moderation, we have here touched on only the most ideal density interpretation approach. Complications which occur in practice include the variation in water density with salinity, the occasional presence of high-Z minerals or gas, variable pmafor some lithologies, and shale peculiarities. In the actual logging measurement the problem which receives the greatest attention is that of sonde standoff from the borehole wall. This arises from the presence of intervening mudcake or caves of vertical extent less than the sonde pad length. Brief mention of this was made in Section 1.4.4; a more detailed discussion is found in Section 3.3.

19.


473

2.4. Gamma-Ray Photoelectric Absorption

It was noted in Section 2.3 that sondes for density logging are designed to exclude the portion of the gamma-ray spectrum affected by photoelectric absorption. This guarantees that essentially only Compton scattering can determine the detector counting rate and that the log yields p~ practically independent of chemical composition. However, the photoelectric portion of the spectrum, because of its high sensitivity to the effectiveatomic number of the formation, is exploited as a separate logging mea~urement.'~ This log responds strongly to formation lithology and only weakly to porosity variation. The 2-values of rock matrices are roughly two to four times larger than those of pore fluids (see Table IV). For an individual element of atomic number 2and gamma rays of energy E greater than that of the K-absorption edge, the photoelectric absorption cross section per electron has the approximate formI6 (2.14)

where Cis a proportionality constant. [In this discussion we use the photoelectric cross section per electron, rather than the physically more meaningful cross section per atom, to facilitate the calculation of averages (Eq. 2.15) and comparison with the Compton cross section: z(e1ectron) = z(atom)/Z.] For most elements encountered in logging sedimentary formations, the energy of the Kedge is sufficiently low that Eq. (2.14) applies. The K edge for calcium is at 4 keV, for example. Barium, sometimes abundant in mudcakes but only rarely in formations, is an important exception; its K edge is at 37 keV. This case is discussed in Sections 3.3.4and 3.4. Since the energydependence of 7 is nearly identical for all the elements of interest, it is the very strong Z-dependencethat determinesthe character of the low-energy part of the spectrum of multiply scattered gamma rays. To calculatethe value of ( 7 ) for a collection of elements, such as appears in a mineral or formation, it is necessary merely to weight the 7 for each element by the electron fraction contributed by that element. This leads to (2.15)

where Z,, A i , and mi are the atomic number, atomic weight, and mass fraction, respectively, of the ith element, and the sum is taken over all the elements present. Substitution of Eq. (2.14) into Eq. (2.15) makes explicit the strong dependence of (7) on even modest concentrations of high-Z

474

JAY TITTMAN

elements in the mixture. An approximation useful for making rapid estimates results from the facts that &/A, = 3 (see Table IV) and Zimi = 1: (2.16)

where the common energy-dependence has been absorbed into C’ (= C A ? - ~It~is.often useful to assign an efectiveatomic number,defined by analogy with Eq. (2.14), to the fonnation.I7 This is the (generally non-integer-valued) atomic number of a fictitiouselement having the same photoelectric cross section as the mixture:

The photoelectric absorption logging measurement (Section 3.4) is made in a manner that requires only one independent parameter, Z,, to uniquely characterize the formation. However, Eq. (2.17) shows Z,to have a complicated mixing rule. Thus, it is common practice to use (T) instead. (In this context most logging literature uses the symbol P, .) Another quantity U = ( z) n, ,or (7 ) p ,,which has the attractive feature of obeying a linear, volumeweighted mixing rule is also used.I8 Thus, for the formation material

u- Wf+(1 - w

m,

(2.18)

where the subscripts f and ma refer to fluid and rock matrix, as before. The log determination of Z,, (or (z)) adds another independent physicstype characterizingparameter to our list, and cross-plottingthis parameter againstp~ (or n,) immediatelysuggestsitself. Figure 7 makes clear the usefulness of this procedure. Several features of Fig. 7 are notable: (a) Zeffis much more dependent upon lithology than upon porosity. (b) The Dolomite curve falls between those for Limestone and Sandstone, in contrast to the sequence seen in Fig. 6. This helps eliminate some ambiguity in lithology determination. (c) The regions between the dry-matrix points and their respective “water-filled” curves provide the opportunity for estimating low-pressure gas content. For example, we can calculate isoporosity curves connecting equal porosity values on a “water-saturated” curve and a “gas-saturated” curve for the same lithology. The isoporosity curve can be scaled in units of gas saturation S,. (d) The Coal points fall in a distinctive region of the chart. Ash content estimatescan be made by noting where the representativepoint for a coal bed falls between the Coal point and the Sandstone curve, for example. Clearly, the additional parameter provided by the photoelectric-region mea-

19.

475


79 DN Limestone I 4 40%

61

=

5-

0

>

Typical Shale

4-

Dry Dolomite

Gypsum

-

14

3-

Aluminum

2-w--

40

30

20

0

'0

sandstone

1-

Coal 1.2

1.4 .

1.6

1.8

2.0

I

1

I

8

2.2

2.4

2.6

2.8

13

- 12 - 11 - 10 8 6 3.0

Electron Density, pe = 2pB, (gm/cm3) FIG.7. Cross-plot of the mean photoelectric absorption cross section per electron ( r ) ,and effective atomic number Z,, against electron density pe. (Note that pa = ps since Z/A = f for most cases of interest.) Points marked ''Dry'' represent 4O%-porous rocks containing only air in the pores. Thus, their ordinate values are the same as those of the corresponding rocks with zero porosity; densities, however, are reduced by 40%. characteristic locations on the cross-plot for several other minerals are shown.

surement can be used to solve problems of one more dimension than before: 3 minerals water-filled porosity, 2 minerals water-and-gas-filledporosity, etc. It provides another constraint on possible interpretations,thus helping to define the formation more accurately. The measurement itself is especially convenient since it can be carried out by a sonde which is simultaneously measuringp, .The scintillation-detector countingrate in an energy window located convenientlyin the band 50 - 100 keVI9is proportionalto the product of a function of Zeffand a hnction ofp, , i.e., j&Zeff)h(p,). The counting rate in the higher-energy window (above about 200 keV) used for the density measurement is proportional to&.+,). Thus, simple division of the low-energy counting rate by the high normalizes out the density dependence and yields a one-to-one relation with ZeS.This relation is established by measurement of sonde response in laboratory mock-up formations of accurately known density and Zefi19The "soft/ hard" window ratio depends strongly upon Zeff(closely as Z;g6) in the range 11 5 2 5 16, where most sedimentary formations fall.I8 This leads to a measurement of considerable sensitivity. Problems of both measurement and interpretation can arise from the occasional presence of even low concentrations of high-2 elements, e.g., uranium, barium, strontium, and zirconium. However, the principal obsta-

+

+

476

JAY TITTMAN

cle to reliable measurement is the presence of barium in barite-loaded mudcakes. Corrections can be applied when absorption of the low-energy gamma-ray flux entering the borehole is modest (Section 3.4). Unfortunately, as the barium mass per unit area of the mudcake increases, it eventually obliterates the low-energy portion of the spectrum and makes the measurement of formation Z,, impossible. 2.5. Velocity of Sound

The logging measurement of sound velocity serves the analysis and interpretation of seismic surveys made on the earth’s surface. Also, it is another means for determining formation porosity. These two applicationsare discussed sequentially below. Seismic surveys directly record wavetrains initiated by a vibration source at the surface and reflected from acoustic-impedancediscontinuities in the subsurface.20Since the received wavetrains are time functions, each surface geophone or hydrophone yields a two-way-transit-timemap of the subsurface, rather than a depth map. Although it is possible in principle to develop depth maps from seismic measurements alone, this is not done in practice because of inadequate knowledge of bed velocities. Other difliculties encountered in seismic interpretation include the following: (1) An intrinsic limitation in depth resolution is encountered because of the earth’s poor transmission of high frequencies (250 Hz).(2) The reception of multiply reflected waves (multiples) frequently produces ambiguous conclusions. (3) Some formations are not good seismic reflectors because the impedance mismatch between them and their neighbors is too small. Thus, for beds under seismic investigation it is valuable to have independent knowledge of sound velocities, depths, and acoustic impedances. The first two can be provided directly by drilling a borehole and recording a sonic log. Calcula, v is the sound velocity, is made tion of acoustic impedance p ~ v where possible by the addition of the density log. Conventional sonic logging actually measures the transit time between two receivers in the sonde,21as described briefly in Section 1.4.4 and more extensively in Section 3.6.4. Dividing by the inter-receiver spun yields the or slowness,usually reciprocal velocity, known as intewul-trunsit-time(ITT) expressed in microsecondsper foot or per meter. Since the acoustic impedance of each bed isjust Mt,where tis the ITT, density and sonic logs permit the construction of synthetic seismograms for comparison with actual ones.22This comparison helps resolve uncertainties and ambiguities in the analysis of the surface seismic survey, verifies reflection events, and relates seismic features to geological structures at accurately known depths. Synthetic seismograms sometimes suffer from errors in the measured

19.


477

values ofpe and t.These occur principallyin rugose boreholes. In some cases, even an accurate ITT itself may not be representative of the virgin formation through which the seismic wave passes. Formations containing clay can be softened near the borehole wall by the absorption of water fiom the drilling mud. Since the least-time path of the rehcted sound (Fig. 9) may lie in this altered zone, the logged value ofITT may be greater than that of the virgin formationthrough which the seismicwave travels.23Another sourceof error, cycle-skipping, can occur when the amplitude of the sonic wave arriving at the “far” receiver is too small to permit detection of the first arrival, i.e., the first half-~ycle.~~ In this case it may be the second or third cycle of the wave which is detected, leading to an erroneously large value oft. Problems arise, also, in comparing sonic log measurementswith seismic recordings because of the large difference in frequenciesused. Logging wavetrains are usually in the 10- to 30-kHz range, while seismic frequencies are in the 10- to 100-Hz band. To circumvent these problems another type of log is sometimes made. The vertical seismicprofile (VSP)is obtained by loweringgeophonesinto the well and recording, one depth at a time, wavetrains produced by seismic sources This technique has the advantage of producing what is on the ~urface.2~ essentiallya seismic record without the disabilitiesof two-way travel. Depths are accurately known, surface noise is eliminated, the impact of multiples is reduced, and distortion and frequency content are improved. To detect geological features at some distance fiom the well it is necessary to offset the seismic source horizontally from the well-head. Sonic logging measurement of porosity derives from the fact that in a given lithology the sonic ITT increases as the liquid-filled porosity increases.26When boreholes are rugose it is often possible to make more reliable porosity determinations from ITT measurements than from neutron or density logs because of the borehole compensation feature described in Section 1.4.4. Another porosity-related use of the sonic log is in distinguishing between total and efeclive porosity, and in estimatingclay content in pore fluids that contain clays in su~pension.~~ These clays affect the density log as if they were part of the rock matrix. Hence, to the extent that pchY pma,density-derived porosity is the volume fraction of the rock which is filled with pure liquid, excluding suspended particulates, and is called the effective porosity. On the other hand, suspended clay makes a very small alteration in the velocity of sound in the rock. Thus, the porosity derived from the sonic measurement is approximately the total porosity, i.e., the volume fraction of the rock filled with liquid including the suspended clay. The opposite situation exists when part of the porosity consists of isolated pores or vugs. Neutron and density logs respond to all the liquid present, while the sonic-derived porosity ignoresthe isolated portions. In the remain-

-

478

JAY TITTMAN

ing discussion of sonic log interpretation we will assume for simplicity that all pores are interconnected and filled with pure water unless it is indicated otherwise. The standard sonic log measurement of ITT may be directly inserted into the commonly used empirical time-average formula (2.19) t = &+ (1 - 9 b, where tf is the ITT for the fluid and tmafor the rock matrix material?* This expression has the attractive property of volume-weightedlinearity already noted with respect to Eq. (2.1 1) for density. However, it has neither the firm foundation nor the universal accuracy of the density equation. It applies rigorously only to sound waves travelling perpendicular to a sequence of parallel-boundarylayers of fluid and rock matrix, the thicknesses of which are sufficientlylarger than the wavelength and proportioned as 9 and 1 - 4, respectively. Nevertheless,it is found to be a good workingapproximation in many formations, particularly in well-compactedones. Another empirically based relation, derived from a larger data set, includes some nonlinearity in the function t &*.The likelihood that a formation will produce water generally increases with the distance of its


497

representativepoint from the straight line. The interpretation in shaly sands requires a more complicated procedure. (b) Permeability. The NMR determination of Sw,halso permits estimates of permeability k to be made. Several expressions of the form

k = A(P/S$,,)

(2.34)

have been proposed,6*where A, B, and Care empirical constants. Measurements on cores provide the information necessary to fix local values of the constants. Then log-derived estimates of Sw,irr and 4 are used to determine permeability in nearby wells. (c) Viscous oil saturation. If the oil has viscosity greater than about 3000 cp (3 Pa s), its free-precession signal is negligible by the end of the instrumental delay. Hence, the observed decay is that of the free water only (Fig. 14).If, in addition, Sw,irr< S,, then 4 - (bf = +So.The availabilityof even rough values for oil viscosity and Sw,h/Sw thus may permit the direct estimation of oil saturation. This procedure is helpful, for example, in evaluating heavy-oil reservoirs that are potential candidates for enhanced oil recovery (by steam flood or other means). (d) Residual oil saturation. The saturation of oil in the invaded zone can often be determined by an NMR log even when the oil is “light”, i.e., of low enough viscosity to contribute to the logged free-precession signal. Special preparations of paramagnetic ions are added to the mud so that they can be carried into the formation with the mud filtrate69.These agents remain in solution or suspension in the water phase but not in the oil. Thus, they drastically reduce T2of the free water in the pores, leaving Tj’affectedby the oil phase only. The derived FFI then represents only the oil, and it follows that (bf = $So. This technique can be made more sensitive and reliable by comparing NMR logs run before and after the paramagnetic ion injection.

References 1. “Well Logging and Interpretation Techniques,” Chapter 1. Dresser Atlas, Houston, 1982; “Log Interpretation Principles.” Schlumberger, New York, 1972; “Log Interpretation -Applications.” Schlumberger,New York, 1974. 2. G. E. Archie, Trans. Am. Znst. Min., Mettal. Pet. Eng. 146, 54 (1942). 3. “ b g Interpretation- Principles,” Chapter 13. Schlumberger, New York, 1972; “Well Logging and Interpretation Techniques,” Chapter 9,Dresser Atlas, Houston, 1982. 4. A. E. Bussian, SOC.Prof: Well Log Anal. Ann. Logging Symp. Trans., 23rd, Corpus Christi,

-

lYPap.E(1982);P.N.Sen,Geophysics46,1714(1981);P.N.Sen,C.Scala,andM.Cohen, Geophysics 46,78 1 (1981). 4a. However, see M. H. Dorfman, J. Pet Technol. 36,2195 (1984). 5. See, for example, A. E. Bussian, SOC.Prof: WellLogAnal. Ann. Logging Symp. Trans., 24th Calgary, 1, Pap. E (1 983);A. Poupon, C. Clavier,J. Dumanoir, R. Gaymard, and A. Misk, J. Pet. Technol. 22, 867 (1970); “Well Logging and Interpretation Techniques”, Chapter

498

JAY TITTMAN

25, Schlumberger, New York, 1972 contains a concise tabular summary of shaly sand interpretation techniques developed to 1977. 6. One-group diffusion theory is usually used to describe monoenergeticsystems,e.g., thermal neutrons. However, in hydrogenous media, such as are encountered in well logging,this approach is usehl even for slowing-down problems. For example, when the spatial distribution of epithermal neutrons about a point source of fast neutrons is to be described,the slowing-down length plays the role of the diffision length in the sense that 6L: ( r 2 ) .See J. R. Lamarsh, “Introduction to Nuclear Engineering,” 2nd Ed., Chapter 5 (especially pp. 213 @. Addison-Wesley, Reading, Massachusetts, 1983; L. S. Allen, C. W. Tittle, W. R. Mills, and R. L. Caldwell, Geophysics, 32, 60 (1967). For an introduction to neutron moderation and diffusion see S. Glasstone and M. C. Edlund, “The Elements of Nuclear Reactor Theory,” Chapters 5,6, and 14. Van Nostrand-Reinhold, Princeton, New Jersey, 1952; S. Glasstone and A. Sesonske, “Nuclear Reactor Engineering,” Chapter 3. Van Nostrand-Reinhold, Princeton, New Jersey, 1981. For a more extensive mathematical treatmentof neutron transportsee A. M. Weinberg and E. P. Wigner, “The Physical Theory of Neutron Chain Reactors,” Chapters 8 11. Univ. of Chicago Press, Chicago, 1958. 7. J. Tittman, H. Sherman,W. A. Nagel, and R. P. Alger, J. Pet. Technol.18,1351 (1966); W. B. Belknap, J. T. Dewan, C. V. Kirkpatrick,W. E. Mott, A. J. Pearson, and W. R. Rabson, “Drilling and Production Practice,” p. 289, Am. Pet. Inst., New York, 1959. 8. Reference 7 , Tittman et al. (1966). 9. H. Edmundson and L. L. Raymer, Soc. Prof: Well Log Anal. Ann. Logging Symp. Trans., Tulsa, 20th, 1, Pap. 0 (1979). 10. H. D. Scott, C. Flaum, and H. Sherman, Soc. Pet. Eng. Ann. Fall Tech. Conj, 57th, New Orleans SPE Pap. 11146 (1982). 11. J. J. Ullo, Soc.Pet. Eng. Ann. Fall Tech. Con$. 56fh,San Antonio, SPE Pap. 10295 (1981); F.Segesman and 0. Y. Liu, SOC.Prof: Well Log. Anal. Ann. Logging Symp. Trans.. 12th, Dallas, Pap. N (197 1). 12. J. Tittman and J. S. Wahl, Geophysics 30, 284 (1965); J. S. Wahl, J. Tittman, C. W. Johnstone, and R. P. Alger, J. Pet. Technol. 16, 14 1 1 (1 964). 13. Reference 2, Chapter 1. 14. See Section 2.5. 15. D. Ellis, C. Flaum, C. Roulet, E. Marienbach, and B. Seeman, Soc. Pet. Eng. Ann. Fall Tech. Con!,58th. San Francisco, SPE Pap. 12048 (1983); W.Bertozzi, D. V. Ellis,and J. S. Wahl, Geophysics 46,1439 (198 1); J. S. Gardner and J. L. Dumanoir, Soc.ProJ WellLog Anal. Ann. Logging Symp. Trans., 21st, Lafayefte, La.,Pap. N (1980). 16. J. H. Hubbell, US.Natl. Bur. Stand. Circ. 542 (1969). 17. For a slightly different approach see J. A. Czubek, in “Radioisotope Instruments in Industry and Geophysics,” Proc. Symp., Warsaw, 2,249,1965, LAEA, Vienna (1966). 18. Reference 15, Bertozzi et al. (1981) and Gardner and Dumanoir (1980). 19. Reference 15, Ellis et al. (1981). 20. J. A. Coffeen,“Seismic Exploration Fundamentals,” Petroleum hbl., Tulsa, 1978; M. B. Dobrin, “Introduction to Geophysical Prospecting,” 3rd Ed., pp. 25 356, McGraw- Hill, New York, 1976. 2 1. See for example,F. P. Kokesh, R. J. S c h w a , W. B. Wall, and R. L. Moms, J. Pet. Technol. 17,282(1965);M.P.Tixier,R.P.Alger,andC.A.Doh,J.Pet. Technol.l1,106(1959);A. A. Stripling, Trans. Am. Inst. Min.. Metall. Pet. Eng. 213,207 (1958). Pro$ Well LogAnal. Ann. LoggingSymp. Trans.,242h. 22. J. P. Castagnaand J. E. G a k ,SOC. Calgary. 2, Pap. NN (1983); D. G. Stone and H. B. Evans, SOC.Prof:Well Log Anal. Ann. Logging Symp. Trans., 21st, Lafiyette, La., Pap. KK (1980); D. H. Thomas, Log Anal. 19, 23 (1978); L. Dupal, J. Gartner, and B. Vivet, Soc. Proj WellLogAnal. Eur. LoggingSymp.

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-

-

19.


499

Trans., 5th, Paris, (1977);B. E. Ausburn, Soc. Prof: Well Log Anal. Ann. Logging Symp. Trans., lffth,Houston, Pap. F (1977);P. C. Wuenschel, Geophysics 25, 106 (1960);V. Baranov and G. Kunetz, Geophys. Prospect. 8,315 ( 1960); R. A. Peterson, W. R. Fillip pone, and F. B. Coker, Geophysics20,5 16 (1955). 23. Reference 22,Thomas (1978). 24. P. E. F. Goetz, L. Dupal, and J. Bowler, Aust. Pet. Explor. Assoc. J. 19, 131 (1979); Reference 22,Thomas (1978). 25. J. F.Lewkowicz, R. Reischman, and J. J. Walsh, SOC.Prof: Well Log Anal. Ann. Logging Symp., 24th, Calgary,2,Pap. MM (1983);“Well Evaluation Conference, South East Asia” (A. Winchester,ed.) p. 74.SchlumbergerTech. Serv.,Paris, 1981; K. D. Wyatt, Geophysics 46,880(1981);P. Kennett, R. L. Ireson, and P. J. Conn, Geophys.Prospect. 28,676 (1 980); E. I. Gal‘perin, “Vertical Siesmic Profiling,” Soc. Explor. Geophys., Tulsa, 1974. 26. L. L.Raymer, E. R. Hunt, and J. S.Gardner, SOC.Prof: WellLogAnal. Ann. LoggingSymp. Trans.,21st, Lafayette, La., Pap. P (1980);J. Geertsma and D. C. Smit, Geophysics26,169 (1961);M. R. J. Wyllie, A. R. Gregory, and L. W. Gardner, Geophysics23,459 (1958)and

21,41 (1956). 27. M. P. Tixier, R. L. Moms, and J. G. Connell, Soc. Prof: WellLogAnal.Ann. LoggingSymp. Trans., 9th, New Orleans, Pap. E (1968). 28. Reference 26,Wyllie et al., Geophysics 21,41 (1956). 29. Reference 26,Raymer et al. (1980). 30. K. Hartley, SOC.Prof: Well Log Anal. Ann. Logging Syrnp., 22nd, Mexico City, Pap. PP, (1981);W. G. Hicks and J. E. Beny, Geophysics21,739 (1956);A. M. Biot, J. Acoust. SOC. Am. 28,168,179 (1956);H. Brandt, J. Appl. Mech. 22,479 (1955);F. Gassmann, Geophysics 16, 673 (1951). 31. “Log Interpretation Charts,” p. 53. Dresser Atlas, Houston, 1983; “Log Interpretation Charts,” pp. 32,33.Schlumberger,New York, 1979. 32. R. E. Hoard, Soc.Prof: Well LogAnal. Ann. Logging Symp.., 24th. Calgary, 2, Pap. X X (1983);H. D. Brown, V. E. Grijalva, and L. L. Raymer, LogAnal. 12,27 (1971);R. L. Moms, D. R. Grine, and T. E. Arkfeld, SOC.Pet. Eng. Ann. FallMeet. 38th, New Orleans,

SPE Pap. 723 (1963). 33. S.K. Chang and A. H. Everhart, J. Pet. Technol. 35,1745 (1983);J . W. Minear and C. R. Fletcher, SOC.Prof: Well Log Anal. Ann. Logging Symp., 24rh, Calgary, 2,Pap. EE ( 1983); R. W.Siegfried and J. P. Castagna, SOC. Prof: Well Log Anal. Ann. Logging Symp. Trans., 23rd, Corpus Christi,Pap. I (1982);J. D. Ingram, C. F. Moms, E. E. MacKnight, and T. W. Parks, Ann. Int. SOC.Explor. Geophys.Meet. 51st, Los Angeles, Pap. S113 (1981) (Geophp sics, in press); J . Aron, J. Murray, and B. Seeman, SOC.Pet. Eng. Ann. Fall Tech. Conf:, 53rd. Houston, SPE Pap. 7446 (1978). 34. C. B. Officer, “Introduction to the Theory of Sound Transmission,” pp. 1 - 13,McGrawHill,New York, 1958;G. Joos, “Theoretical Physics,” pp. 169ff.,Blackie, London, 1934; A. E. H. Love, “The Mathematical Theory of Elasticity,” Vol. 1,p. 130,Cambridge Univ. Press, Cambridge, 1892 (Dover, New York, 1944). 35. M. B. Dobrin, “Introduction to Geophysical Prospecting,” 3rd Ed., p. 346;McGraw-Hill, New York, 1976;C. B. Stone, in “Developments in Petroleum Geology” (G. D. Hobson, ed.), Vol. 1, p. 275,Applied Science Publ., London, 1977;H. Ritch and J. T. Smith, SOC. Prof: Well Log Anal. Ann. Logging Symp. Trans., I7th, Denver, Pap. X (1976). 36. M. P. Tixier, G. W. Loveless, and R. A. Anderson, SOC.Pet. Eng. Ann. Fall Tech. Conf:, 48th. Las Vegas, SPE Pap. 4532 (1973);SOC.Pet. Eng. Ann. Fall Tech. Conf:, 47th. San Antonio, SPE Pap. 4135 ( 1972). 37. J. T Watson, SOC.Prof: WellLog Anal. Ann. Logging Symp. Trans., 24th. Calgary,Pap. FF (1983);P. Westaway, R. Hertzog, and R. E. Plasek, Soc. Pet. Eng. Ann. Fall Tech. Conf:,

500

JAY TITTMAN

55th, Dallas, SPE Pap. 946 1 ( 1980);R. C. Hertzogand R. E. Plasek, ZEEE Trans.Nucl. Sci, NS26,1558(1979); W.E.SchultzandH.D.Smith,Jr.,J. Pet. Technol.26,1103(1974);G. A. Lock and W. A. Hoyer. J. Pet. Technol.26,1044 (1974);J. Tittman and W. B. Nelligan, J. Pet. Technol.12,63 ( 1960).See. also a bound collectionof 10 papers issued under the title “Carbon/Oxygen Log,” Dresser Atlas,Houston, 198 1. 38. C. Flaum andG. Pine, Soc. Pro$ Well LogAnal. Ann. LoggingSymp. Trans.,22nd, Mexico City, l,Pap.H(198~);J.H.MoranandJ.Tittman,U.S.Patent3,521,064, July21, 1970. See also Reference 37, Westaway et al. ( 19801, Hertzogand Piask ( 1979),and Tittman and Nelligan (1 960). 39. See Fig. 13a in D. W. Oliver, E. Frost, and W. H. Fertel, SOC.Prof: Well Log Anal. Ann. Logging Symp. Trans., 22nd, Mexico City, Pap. TT ( 198 1) and Figs.1 and 10 in Reference 37, Hertzog and Plasek (1 979). Note that Fig. 1 1 is the calculated real C/O atom ratio. The “fan plots” in Oliver et al. (198 1) are based on sonde measurementsand show substantial zero offset even for the sandstone matrix. This results principally from spurious contributions to the carbon window. The absence of zero offset for sandstone in the sonde measurements shown in Fig. 10 of Hertzog and Plasek (1979) is a consequenceprincipally of the data-processing method used. 40. Reference 37, Westaway et al. (1980), Table 11. 41. F. J. Pettijohn, “Sedimentary Rocks,” p. 271. Harper, New York, 1949. 42. Reference 37, Hertzog and Plasek ( 1 979) and Tittman and Nelligan (1960);Reference 39, Oliver et al. (1981). 43. Reference 39, Oliver ez al. ( 1981). 44. Reference 37, Hertzog and Plasek (1979); Reference 38, Moran and Tittman (1970). 45. Reference 37, Hertzog and Plasek (1979). 46. Reference 37, Hertzog and Plasek (1979) and Reference 38, Moran and Tittman (1970). 47. M. Hassan, A. Hossin, and A. Combaz, Soc, Prof: Well Log Anal. Ann. Logging Symp. Trans., J 7th, Denver, Pap. H (1976). 48. See Table 1 in W. H. Fed, J. Pet. Technol.36,249 (1984);J. Suau and J. Spurlin,SOC.ProJ WellLogAnal.Ann. Logging Symp. Trans.,23rd, Corpus Christi, 1, Pap. G (1982);Table I1 in W. H. Fertl, Log Anal. 20,3 (1979). 49. See, for example, Reference 48, Suau and Spurlin (1982). 50. H. D. Smith, Jr., C. A. Robbins, D. M. Amold, L. L. Gadeken, and J. G. Deaton, SOC.Pet. Eng. Ann. Tech. Conf:,58th, San Francisco, SPE Pap. 12050 (1983);0. Serra,J. Baldwin, and J. Quirein, SOC.ProJ Well LogAnal. Ann. Logging Symp. Trans.,2Jst, Lafayetre, La., Pap. Q (1980). 5 1. Reference 15, Bertozzi d al. ( 198 1); H. Goldsteinand J. E. Wilkins, Jr., “Calculationsofthe Penetration of Gamma-Rays,’’ NYO-3075, USAEC, 1954; L. V. Spencer and U. Fano, Phys. Rev. 81,464L ( 195 1) and J. Res. Nut. Bur. Stand. 46,446 ( 195 1); P. R. Karr and J. C. Lamkin, Phys. Rev.76, 1843 (1949). 52. See, for example, G. F. Knoll, “Radiation Detection and Measurement,” pp. 328ff, Wiley, New York, 1979. 53. Reference 50, Sera et al. (1980). 54. J. C. SIater andN. H. Frank, “Electromagnetism,”p. 93. McGraw-Hill, New York, 1947;J. A. Stratton, “ElectromagneticTheory,” pp. 275-276. McGraw-Hill, New York, 1941;W. C. Chen and S. C. G. Gianzero, IEEE Trans. Geosci. Remote Sensing CE19,1(198 1); R. P. Wharton, G. A. Hazen, R. N. Rau, and D. L. Best, SOC.Pet. Eng. Ann. Fall Tech. Con$, 55th, Dallas, SPE Pap. 9267 (1980);R. Freedman and J. P. Vogiatzis, Geophysics 44,969 (1979). 55. R. N. Rau and R. P.Wharton, SOC.Pet. Eng. Ann. Fall Tech. ConJ,55th, Dallas, SPE Pap.

19.


50 1

9380(1980);Reference 54,Wharton et al. (1980);J. P. Poley, J. J. Nooteboom, and P. J. de Waal, Log Anal. 19,8 (1978). 56. Reference 54,Stratton (1941)and Wharton et al. (1980). 57. Reference 4,Sen (1981);Reference 55,Rau and Wharton (1980);T. J. Calvert, R. N. Rau, and L. E. Wells,SOC.Pet. Eng. Ann. CaliJ:Region. Meet. 47th. Bakersfreld,SPE Pap. 6542 (1977). 58. Reference 54,Wharton ef 01. (1980). 59. Reference 4,Sen (198I). 60. Reference 54,Wharton et al. (1981);Reference 57,Calvert et al. (1977). 6 1, Reference 54,Wharton et al. ( 1980). 62. P. T.Cox and W. F. Warren, SOC.Prof: WellLog Anal. Ann. Logging Symp., 24th, Calgary, 1, Pap. H (1983);B. Anderson and S . K. Chang, ibid., Pap. T; R. P. Mazzagatti, D. J. Dowling, J. C. Sims, A. E. Bussian, and R. S. Simpson, SOC.Pet. Eng. Ann. Tech. ConJ. 5 8 d San Francisco, SPE Pap. 12097(1983);G. S.Huchital, R. Hutin, Y. Thoraval, andB. 56th. SanAntonio, SPEPap. 10988 (1981);R. Clark, SOC.Pet. Eng. Ann. Fall Tech. Conf:, A. Meador and P. T. Cox, Soc. Pet. Eng. Ann. Fall Meet. SOth, Dallas, SPE Pap. 5504 (1975). 63. Reference 4,Sen (1981); Reference 55,Poley et al. (1978). 64. Reference 62,Cox and Warren (1983). 65. S. D.Sentuna and J. D. Robinson, Trans. SOC.Pet. Eng. AZME 249,237(1970);J. D. Loren and J. D. Robinson, ibid., p. 268;R. J. S. Brown, Nature 189,387 (1961). 66. R. C. Hemck, S. H. Couturie, and D. L. Best, SOC.Pet. Eng. Ann. Fall Tech.Conf:,54th, Las Vegas. SPE Pap. 8361 (1979). 67. C. Neuman and R. J. S. Brown, J. Pet. Technol. 34,2853 (1982);Reference 66,Hemck et al. (1979);J. D.Robinson, J. D. Loren,E. A. Vajnar, and D. E. Hartman, J. Pet. Technol. 26,226 (1974);A. Timur, SOC. Prof: Well Log Anal. Ann. Logging Symp. Trans., 13th. Tulsa, Pap. N (1972);A. Timur, J. Pet. Technol. 21,775 (1969);D.0.Seevers, SOC.Prof: Well Log Anal. Ann. Logging Symp. Trans., 7th. Tulsa, Pap. L (1966). 68. “Log Interpretation Charts,” p. 83.Schlumberger, New York, 1979;Reference 67,Timur (1969);A. Timur, Log Anal. 9 (S), 8 (1968). 69. C.H. Neuman, Sac. Pet. Eng.Ann. Fall Tech. Conf:,55th, Dallas, SPE Pap. 8844(1980);R. J. S.Brown and C. H. Neuman, SOC.Prof: WellLogAnal. Ann. Logging Symp. Trans.,21% Lafayette, La.,Pap. K (1980).

3. The Physics of Logging Measurements Most measurement techniques exploited in loggingare derived from laboratory methods. However, they have a character of their own that is dictated by the peculiarities of the logging environment and the particular petrophysical or geological objectives of the measurement. Chapter 2 described how the results of measurements based on physics phenomena are interpreted to provide geological descriptors of interest. In this chapter the most widely used measurements, those based on electrical, neutron, gamma-ray, and sonic phenomena are described in more detail.

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JAY TITTMAN

Some methods have received extensive development over the past halfcentury,’ and in general several different sonde embodiments exist for each kind of measurement. Usually the different embodiments or techniques are tailored to the solution of specific problems, e.g., accuracy at high resistivity values or measurement with deep investigation. In most instances these are variations, however inventive, on a small number of basic themes. This chapter concentrates on the measurement principles, often through a description of a particular embodiment. Some “variations” are discussed only briefly, described pictorially, or merely given literature reference. Most logging measurements require corrections, often quite large, before they yield the formation-characterizingphysics parameters that are desired. In general, this need arises because of the influence of mudcake, invasion, layering, temperature, borehole diameter, etc. on the raw measurement. It will be possible here only to indicate how the most important correctionsare made. Books of correction charts, usually referred to as departure curves, are published by logging companies for the use of log analysts. 3.1. Electrical Resistivity Methods

For our purposes it is convenient to divide conventional resistivity methods into two classes, those injecting currents into the formation by means of electrodesand those using coils for creating low-frequency electromagnetic induction fields that produce eddy currents in the formation. Present-day electrode methods are extensions of the original ones invented ~ induction by Conrad and Marcel Schlumberger in the late 1 9 2 0 ~The method, created by Henri-Georges Doll, first appeared about two decades later.3 State-of-the-artresistivity tools are intricate and complex devices, and to understand their characteristicsit is fruitful to trace their evolution. Thus, we will first describe some sondes that, although now obsolete, constituted important developmental steps. 3.1.1. Electrode Devices. Consider a point source of current embedded in an infinite homogeneous isotropic medium. The potential difference dV between two concentric spherical surfaces at radii r and r dr from the source is just

+

dV = (iR/4nr2)dry (3.1) where i is the current and R is the resistivity ofthe medium. Integrating from r to infinity and setting the potential there to zero gives V = iR/4nr. (3.2) Since most of the potential drop occurs close to the source, the medium at a


503

FIG.15. Infinite medium consisting of spherical layers concentric with a point source of current. Successive layers have outer radii a, b, c, . . . and resistivities R,,R2,R,, . . . , R,, . . . . The shell on which the potential is measured is at r < a.

sufficiently great distance negligibly influences the value of Vat small r. A very simple model allowing for variation in resistivity is that of a medium consisting of spherical layers, each with a different resistivity (Fig. 15). Integration of Eq. (3. l ) for this case yields V=”1[(1-i)+-(---)+-(---)+ R , r 47tr R , a

r

b

R , r R , b

r c

*

.].

(3.3)

Thus, for r e~ a, Vis determined essentially by R , unless the resistivity of one or more of the other layers is excessively greater than R I . Clearly, the more distant a layer is, the greater must be its resistivity if it is to influence V(r). 3.1.1.1. THENORMAL SONDE.To introduce the simplest practical electrode arrangement we consider the earth as a homogeneousmedium pierced by a borehole containing mud with the same resistivity as the earth. Electrodes can be suspended in the hole by negligibly thin insulated conductors. Sincethe electrodesare alwaysdeep enough to permit the surface of the earth to be considered at infinity, Eqs. (3.1) and (3.2) apply directly. Figure 16a illustrates this idealized situation. The surveying current is emitted by electrode A and returns to B “at” infinity, M is the potential-measuring elec-

504

JAY TITTMAN Current

Current Generator

Polentio-

A

-

-t-

-

Potentio-

Spacing

(a) Two-electrode circuit

(bJ Actual circuit

FIG. 16. Schematic diagram of the normal sonde. Current is emitted by electrode A and returns to B. Potential is measured between electrodesM and N. B and N are effectively at infinity. (a) Electrode arrangement illustrating the principle. (b) Actual arrangement used in field practice. [From “Interpretation Hand-Book for Resistivity Logs-Document 4.” Schlumberger, Houston, 1950.1

trode, and N is the reference at zero potential. A sonde using this arrangement of electrodes is referred to as a normal device. The generator at the surface provides the current and the potentiometer, or voltmeter, there measures the potential between M and N. The known vertical distance AM, the spacing, plays the role of r in Eq. (3.2). Thus, measurements of Vand i determine the value of R.In reality, of course, the earth is layered and the mud resistivityin general does not equal that of any of the layers. Thus, there are defined an apparent resistivity R,, and a sonde coeficient K N ,for the normal device:

R, = 4a(AM)(V/i)= KN(V/i).

(3.4) R,is just the resistivity of the infinite homogeneous isotropic medium that would produce the same ratio V/i as that observed at a particular depth. ExtractingR,,the true resistivity of a layered bed immediately surrounding A and M, from R, alone is in principle an indeterminate problem. Different combinations of R,?borehole diameter, invasion diameter, bed thickness, shoulder-bedresistivity, etc. can produce the same value of R,.Thus, determining R,generally requires departure curves for the normal device and additional measurements by other sondes.

19.


505

Figure 16b shows a closer approximation to the actual arrangement of the electrodes. The current-return electrode B (often the cable armor) is located in the hole, typically 15-30 ft (5- 10 m) above the A electrode. This is usually far enough away to have only a small influence on the potential measured at M. (B is placed downhole in order to implement the simultaneous measurement of the spontaneous potential [Section 1.4.11. Low-frequency [- 20 Hz] AC is used as the current source, instead of the DC tacitly implied in the foregoingdiscu~sion,~ as part of the same implementation.)If the source current is held constant, R, = V and the tool output can be calibrated directly in ohm meters. The short normal typically uses an AM spacing of from 5 to 20 in. (13 - 50 cm) and the long normal from about 20 to 84 in. (50-2 13 cm), depending upon the combination of vertical resolution and depth of investigation desired. In a homogeneous medium half the potential drop between Mand N occurs within a distance 2AM from A. [See Eq. (3.2).] Thus, the depth of investigation of the normal sonde is conventionally assigned to be UM.* This, of course, is only a loose approximationin heterogeneousmedia.5The measure point of the normal sonde is taken halhay between A and M because of the symmetryimposed by the reciprocity theorem, i.e., in crossing a bed boundary the same log results if the locations of electrodesA and Mare interchanged. 3.1.1.2. THELATERAL SONDE. Another electrode arrangement, which grew out of the concept of the simple normal sonde, is that of the lateral device. The lateral sonde of Fig. 17 places M and Nrelatively close together, with.4 at a distance. This permits the relatively clear marking ofthin resistive beds sandwiched between two more conductive ones, but produces highly asymmetric logs of R, as bed boundaries are crossed. A commonly used lateral spacingis about 19 ft (5.8 m).Sincethe distance MNis relatively small and is fixed for a particular logging run, the measured potential difference is closely proportional to the electric field strength at 0 (Fig. 17). Using Eq. (3.2) it is straightforward to show that for the lateral sonde

where KLis the lateral sonde coeficient. The values of R, yielded by the normal and lateral devices in a highly resistive bed are significantly different from R,,even when the bed is otherwise adequately thick. Furthermore, the lateral logs of R, are disturbingly * Note that this does not follow from the experimental procedure given in Section 1.4.3. Furthermore, for electrode devices of this general kind that procedure for defining vertical resolution breaks down completely.

506

JAY TITTMAN

Current Generator

Potentiometer

AMN Lateral Sonde

FIG.17. One form of lateral sonde. Local potential difference is measured between electrodesM and N, both of which are on the sonde, [From “Interpretation Hand-Book for Resistivity Logs-Document 4.” Schlumberger, Houston, 1950.1

asymmetricwithin the bed. Both effects arise from the fact that nearly all the electrical current flows in the relatively conductive mud column. Hence, the logs recorded as either sonde passes through the bed are more influenced by mud resistivity, borehole diameter, and distance to the nearest conductive shoulder bed than they are by the bed’s R,value.6 3.1.1.3. THELIMESTONE SONDE. Historically, this situation led to the development of still another electrode configuration, the limestone sonde’ illustrated in Fig. 18. Here, short circuits, insulated from the mud, join M with M’and N with N’. When MN ez AM the limestone sonde is just the symmetrical superposition of two lateral devices. The log is symmetric within the bed, although the center-bed reading ofR,is still far less than, and nearly independent of, R, in highly resistive beds, even if they are thick. Furthermore, if a conductive streak is present within a thick bed, the raw log shows symmetric reduction in R,. 3.1.1.4. THE MICROLOG.In order to better locate permeable streaks within compact highly resistive formations such as limestone, the microlog was developed.8Because of its fine vertical resolution and very shallow depth of investigation it has found extensive application also in locating permeable sands in sequences of thin sand and shale streaks. Figure 19a illustrates the electrode geometry and mechanics of the sonde, while Fig. 19b shows the electrical principle. The insulating rubber pad is forced against the borehole

19.

507

GEOPHYSICAL WELL LOGGING Current Generator

Potentio. meter

FIG. 18. The limestone sonde. When MN -K AM this is just the symmetric superposition of two lateral devices. [From “Introduction to Schlumberger Well LoggingDocument 8.” Schlumberger, Houston, 1958.1

wall so that the electrodes make electrical contact with the mudcake, if one exists, and with the formation if the mudcake is absent. Usually AM, is 1 in. (2.5 cm) and AM, is 2 in. (5 cm). Hence, two logs ofR, can be recorded, both with quite shallow investigation, but one sampling about twice as deeply as the other. Therefore, we can determine whether resistivity varies with horizontal distance from the pad. The presence of mudcake causes a separation between the two logs, i.e., R,(M,) # R,(M,), and this signals the location of a permeable formation. When the formation resistivity is much higher than that of the mud and the borehole is somewhat rugose, most of the surveying current passes through a conductive mud layer between the sonde and the wall. Under these conditions not much importance can be attached to the measured values of R, recorded or to the separation between them. 3.1.1.5. DEPARTURE CURVES.Since the principal use of resistivity measurements is in the determination of hydrocarbon saturation,it is usually the resistivity of the virginformation behind the invaded zone that is desired. (At this point it is fruitful to turn to conventional nomenclature and symbol usage, in which it is the virginformation’strue resistivity that is symbolized by R,. In the interest of simplicity, in Section 2.1 “true resistivity” was used differently.) Even when complete radial symmetry prevails, the resistivity measurement is affected by five variables in addition to R, : mud resistivity R, ,invaded-zoneresistivityR , ,borehole diameter d, invaded-zonediameter d i , and sonde spacing (AM or AO). The effect of mudcake is usually negligible for conventional sondes, i.e., other than the microlog and its descendents.The number of free variables is reduced by scaling all resistivities in units of R, and all distances in units of d, both of which are readily known. Here R, is determined by capturing a mud sample at the surface and

(b)

FIG.19. The microlog sonde. (a) Electrodeconfiguration and mechanical arrangement ofan early microlog sonde. More recent versions use hydraulically controlled linkages, similar to that in Fig. 23b, for forcing the pad against the wall. (b) Electrical circuit showing how potentials are yielding two values of apparent resistivity, both shallow but measured at electrodesM , and M2, with different depths of investigation. [From H. G. Doll, Pet. Trans.AZME 189, 155 (1950). Copyright 1950 SPE-AIME.]

19.


509

measuring its resistivity in a specially designed cell. Sometimesthe resistivity of the mud filtrate R,is desired;this is obtained by filteringthe mud sample through filter paper and measuring the filtrate in the resistivity cell. Surface measurements of R, or R, are corrected to their values at downhole temperature before R, is extracted. Borehole diameter is logged simultaneously by calipers located in the tool string. Estimates of invasion diameter can be made by comparing R, values from sondes having different depths of investigation or from local experience. Departurecurves, calculated for the case of point electrodeslocated axially in a cylindrical borehole piercing an invaded, but otherwiseinfinite homogeneous isotropic formation, provide the means for deriving R, from R, in thick beds. (In recent years computer modeling programs have been used extensivelyfor this purpose. In addition, some departure curvesare based on measurements in laboratory mock-up formations.) An example is shown in Fig. 20 for the normal sonde in a thick, invaded formation. Given particular values for R, and d from the logs, interpolation between curves of constant RJR, permits determination ofRtS8Departure curveshave been calculated also for the case of both invasion and finite bed thickne~s.~ In this regard it is instructive to outline the derivation for the potential field produced by a point current-source located on the axis of a borehole piercing an invaded formation of infinite thickness. The potential satisfies Laplace's equation, V2V = 0, everywhere except at the current electrode. Sufficiently close to the source the potential takes the form V=i R , $ 4 l r m ,

where r and z are the cylindrical coordinates of the field point, the z axis coincides with the borehole axis, and the origin is located at the current electrode. Boundary conditionsrequiring continuity of Vand of the normal component of current density are applied at both the borehole wall and the assumed cylindrical interface between the invaded zone and the virgin formation. Laplace's equation is solved by separation of variables, determination of particular solutions, and integration over the range of the separation constant (0 to ~0 in this case). The solution9can be cast in two forms, one convenient for numerical evaluation at large distances,

and the other at small distances,

510

JAY TITTMAN

look

Ra Apparent Resistivity

d Drill Hole Diameter

Rm Mud Resistivity RI Resistivity 01 Invaded Zone Rt Resiaivity of Formation

Di Invaded Zone Diameter

Beyond Invaded Zone I----CurvelorDi = d. LI = 0 -Curve for Di = 2d Li = di2

LI. Extent 01 Invaded Zone from Wall of Hole AM Spacing (normal Device)

-- Curve for Di = 56: LI = 26

.'" '

Curve for DI = 1Od Li = 41 2d

I I

I

FIG.20. Calculated departure curves for the normal sonde in an infinitely thick, invaded formation With RJR, = 2 1. [Adapted from "Review of Schlumberger Well Logging and Auxiliary Methods-Document 2." Schlumberger, Houston, 1958.1

In these equations and D represent different integrals over the separation constant 2. The variables in brackets are those appearing in the integrands and U symbolizes a collection of Bessel functions of the first and second kinds, the arguments of which are Ar, ld/2, and AdJ2. Similarly, W represents modijed Bessel functions of the first and second kinds. Through Eq. (3.4) these calculations relate Vto R , for the normal sonde if we set r = 0 and z = (AM). (The finite size of the electrodesand the presence of the insulated sonde mandrel can be ignored in most practical cases.) The departure curves shown in Fig. 20 are calculated in this way for one value of RxJRm. For invaded beds offhire thickness the problem is more difficult. An approximationusing the method of images has been successful in describing experimental results, provided that the resistivites of the shoulder beds are

19.


511

sufficientlysmallerthan R, of the object bed, the thicknessof the object bed is greater than the sonde spacing, and the effect of invasion is not too large.l0 For many years useful results were produced by a specially built resistor network that simulated assumed resistivitydistributionsin the vicinity ofthe sonde. More recently, numerical solutionsbased on finite-element modeling have had outstanding success, and, in general, digital computer modeling has replaced both analog simulation and analytical methods. 3.1.1.6. THEGUARDED ELECTRODE. The foregoing discussion of early electrode devices is primarily didactic. Although the present tense is used throughout, the sondes discussed are rarely, if ever, used today. However, their principles are the basis for modern, more complex electrode arrangements. The past thirty years have seen the extensive development of focused electrode devices. By appropriate design it is possible, within limits, to achieve simultaneously deep investigation and fine vertical resolution. This reduces the influence of borehole mud, invasion, and shoulder beds on the determination of R,. All these focused systems are related to the classical guard-ring technique of the electrical measurements laboratory. The earliest such logging device, the guarded electrode,I1is shown schematically in Fig. 2 1. Its principle is the basis for a whole family of sondes that followed. A description of its operation is quoted directly from Doll.” “The system . . . comprises one short central electrode M and two elongated short-circuited electrodes, G and G‘ symmetricallyplaced above and below M, and connected to Mby a low resistance shunt S.A current is fed to the electrodes, and that part i of the current which flows through the central electrode forms a sheet which is confined between two approximately horizontal surfaces. The current i is measured by means of a meter which is located at the surface and connected to both ends of the shunt. The potential AV of electrode M is also measured by means of another meter at the surface, and the ratio AV/i gives the value of the resistance offered to the current, which is proportional to the resistivityof the formation situated within the current sheet . . . Because of the presence of the shunt, the potential of the guard electrode is not rigorously equal to the potential of the central electrode, but is greater by an amount which depends on the resistance of the shunt and on the resistivity of the surrounding media. The shape of the beam of current emitted by electrode M is, therefore, affected to an extent which cannot easily be ascertained, and, consequently, the response of the system is not quite defined, particularly when the borehole is filled with low resistivity mud.”

3.1.1.7. THE LATEROLOGS. The defect mentioned at the end of the preceding quotation can be eliminated.” Instead of using a shunt to keep GG’ at approximately the same potential as M, an automatic control circuit governs a current fed separately to GG’ so as to null the potential difference between GG’ and M independently of RJR, and R,JR,. This arrangement, known as the 3-electrode laterolog or laterolog 3, is a conceptually intermediate step between the original guarded electrode and the manyelectrode devices that followed. Although the laterolog 3 provided an improved R, measurement, it was not used extensively in the field because its

512

JAY TITTMAN

-x

u

I --

FIG.21. The guarded-electrodesonde. The short-circuited guard electrodes G and G' are at nearly the same potential as M, which serves as both current-emitting and measure electrode. The electric field gradient thus established forces the survey current to fan out in a horizontal sheet between the dashed Lines and to penetrate deeply into the formation before curving upward to the current return. [From H. G. Doll, Pet. Trans.AIME 192,305 (1951). Copyright 1951 SPE-AIME.]

long metallic body interfered with SP and induction measurements made simultaneously. Substantially the same focusing effect is achieved by an arrangement of essentially point electrodes known as laterolog 7.' Figure 22 illustrates schematically both the electrode configuration and the distribution of current in a homogeneous medium. The system is symmetrical above and below A,, the survey-current electrode. Electrodes MI and Mz are short-circuited, as are M', to M i and A, to A,. The last pair inject an auxiliary,or bucking, current i, of the same polarity as the survey current io. A control signal representing the potential differencebetween M,Mz and M',MIis sent to the surface where it governs the current delivered to A , A z . The system is selfnulling in that the bucking current is continuouslyadjusted so as to maintain

19. GEOPHYSICAL

WELL LOGGING

513

FIG.22. Electrode configuration and current distribution for the Laterolog 7 sonde in an infinite homogeneous isotropic medium. [From “Introduction to Schlumberger Well Logging-Document 8.” Schlumberger, Houston, 1958.1

the condition V(M44,)- V(M{M;)= 0. The potential measurement can be made at any of the M-electrodes. It is converted to R, through multiplication by a sonde coefficient which depends only on the distances between the electrodes, as in the case of the normal and lateral sondes. Even when the mud column is taken into account, essentially no current from A, can flow vertically through the mud past the M-electrodes since V(M1M2) = V(MiM2.Thus, iofans out radially in a horizontal sheet, or slab, penetrating the formation deeply before curving upward toward the distant current return B (not shown in Fig. 22). The potential drop through which the surveying current flows is predominantly in the formation lying between O1 and 0,. Since the laterolog is usually used in salty muds, where R, < R,, the borehole mud has a very small influence on V(M),and thus on R, .Also, when the object bed is at least as thick as O,O,, the shoulder-bedresistivities have little effect if they do not contrast too greatly with that of the object bed. l2 Thus, ofthe usual environmentalperturbationsonly the invaded zone influences V ( M ) ,In general, if R,> R, and di < - 2d, nearly all the potential drop between Mand infinity occurs in the uninvaded part of the formation, and R, = R,.

514

JAY TITTMAN

To extend the usefulness of the latero3.1.1.8. THEDUALLATEROLOG. log measurement the dual laterolog was deve10ped.I~This tool provides simultaneouslytwo laterolog measurements with different depths of investigation but the same vertical resolution. In addition, the tool contains a multi-electrode device mounted on a pad which is forced against the borehole wall. This device, based on a prindpal known as sphericalfocusing,I4 is designed to have shallow investigation in the formation, yielding R,, but remaining reasonably free of mudcake effect. Thus, apparent resistivities with three different depths of investigation are measured simultaneously. These are entered into departure curves, created from experimental data and/or computer modeling of the sondes, to determine R, and di.I3 We discuss here only the dual laterolog portion of the tool. Figure 23a shows schematically the electrode array and the current paths.15The illustration is split only for convenience in representing the two sets of currents; the sonde itself is axially symmetric in construction. The left-hand portion refers to the deep laterolog (LM) and the right-hand to the shallow laterolog (LLs). Note that long electrodes A2 and A& in the fashion of the guardedelectrode sonde, have been added.* For operation in the L M mode these are short-circuited to AIA{ so as to force io deep into the formation before it returns to the surface. The auxiliary current i,, emitted by A1A{A2A;,is under feedback control, as in the laterolog 7. The shallow laterolog pictured on the right-hand side of Fig. 23a performs its measurement by using the same physical electrodes in a different configuration and operating at a different frequency. Here A2 and A; function as the common current return for both io and i,, the latter emitted by AIA{ only. (Because of the use of a different frequency there is, in effect, an open circuit betweenA,A{ andA,A; in the LLs mode.) As a result, the LLs survey current fans into the formation initially with the same vertical width as that for the LLd, about 2 ft (60 cm). However, since i, returns to the nearby electrodeA,A;, rather than to one at infinity, iodiverges after penetrating the formation a short distance. Thus, the potential measured at Mis determined essentiallyby the resistivity ofthe invaded zone, R,. In addition to yielding R,with good accuracy in high-resistivity formations, the dual laterolog operates over a much greater resistivity range than the earlier laterolog sondes. All the electrode sondes described to this point used constant io and detected variations in V(M),thus providing an output linear in resistivity. [Although they can alternatively be operated holding V ( M )constant and detectingthe variations in io,thus giving an output linear in conductivity, we have not discussed this mode.] The dual laterolog In this system the SP electrode is located on the cable at a considerable distance from the sonde proper and a trade-off is made between the improved LLd measurement and ability to run an induction log simultaneously.


LOGGING

515

FIG.23. (a) Schematicdiagram of electrodesand current distribution for the Dual Laterolog in an infinitehomogeneousisotropicmedium. Two separateresistivity measurementswith the same vertical resolution,but different depths of investigation,are made simultaneouslythrough use of different frequencies. The sondeis split here only for the purpose ofillustration.(b)Actual arrangement of electrodeson the sonde. The Rxopad providesa separate’‘sphericallyfocused” (shallow)resitivity measurementand is not part of the Dual Laterolog system. [FromJ. Suau, P. Grimaldi, A. Poupon,andP. Souhaite,Soc. Pet. Eng. 47thAnn. FaNMtg., Oct. 8 - 11,1972, San Antonio, paper number SPE 4018. Copyright 1972 SPE-AIME.]

achieves its increased dynamic range by operating at constant power, i.e., both i, and V(M)are varied according to a predetermined protocol, and individually measured, while the product is held constant. This permits the tool to cover a resistivity range of from 0.1 to 4 X lo4 i2 m. Figure 23b is a schematic diagram of the actual sonde. All electrodes except A2 and A: are metal rings of small vertical extent (a few centimeters). They are connected to insulated leads which run up through channels in the insulated sonde body and then through pressure seals into the electronics cartridge located above the sonde. Here A$ includes the metal body of the spherically focused R, system at the bottom. The hydraulically operated four-arm linkage which supportsthe R,,pad contains a caliper sensor, so the hole diameter is measured simultaneously with resistivity. This linkage serves also as a lower centralizer for the dual laterolog part of the system. In

516

JAY TITTMAN

order to achieve both the 2-ft (30-cm)vertical resolution and deep investigation simultaneously, the overall length of the sonde is required to be about 28 ft (8.4 m). 3.1.2. Coil Induction Devices. Electrode devices require direct contact with conductive mud in order to inject currents into the formation. When logging is performed in air-filled boreholes or holes filled with nonconductive oil-base mud this is not possible. Since induction logging measures resistivity through the use of eddy currents created in the formation, this obstacle is circumvented. Induction logging has proved to be such a successful technique that it is now extensively used also in holes filled with conductive mud. For electrode tools the normal sonde performs the pristine measurement; for induction logging it is the two-coil sonde, a longitudinally separated transmitter- receiver pair positioned on the borehole axis. We will study this basic system first through an approach known as geometricfactor t h e ~ r y . ~ (The origin of this name and its relation to the material presented in Section 1.4.3 will become apparent in the discussion that follows.) As is shown in Section 3.1.2.3, this theory is rigorously correct only in the limit of zero frequency or infinite resistivity. Despite this limitation, it serves as an excellent vehicle for exposing the fundamental ideas of induction logging. An outline of the more accurate and complete electromagnetic theory, which starts from Maxwell’s equations, is provided later. 3.1.2.1. GEOMETRIC-FACTOR THEORYOF THE TWO-COILSONDE. Consider a pair of coils wound on a mandrel of insulating material, as shown on the left side of Fig. 24. The transmitter coil is driven by a constant-amplitude current having time dependence of the form e-’”‘, where w is the angular frequency and i = fi. Faraday’s law predicts the establishment of electromotive forces (emf’s) with time dependence ioe-iwt in loops coaxial with the sonde, throughout space. In an elemental loop of unit cross-sectional area (Fig. 24) Ohm’s law then predicts an eddy current proportional to iowe- iwt ,where Q is the electrical conductivity in the loop. This eddy current produces its own magnetic flux which threads the receiver coil. Thus, it acts as a secondary source and contributes an induced voltage across the receiver terminals proportional to - aw2e-iur. We note that this contribution is linear in conductivity, increases rapidly with frequency, and is in phase (although of opposite sign) with the transmitter current.*

* The last feature is ofpractical importance because theemfproduced in the receiverby direct mutual coupling can be several orders of magnitude larger than the total signal produced by the formation. Fortunately, this “mutual” emf is in quadrature with the formation signal. In practice, its influence is eliminated by two expedients: (a) Various means are used for adding an approximatelyequal and opposite quadrature voltage to the receiver so as to “buck out” most of the mutual emf. (b) The receiver output is passed through a high-quality phase-sensitive detector which discriminates against any residual out-of-phase component.

19.

517


FIG.24. The basic two-coil induction sonde. Transmitter Tand receiver R, separated by a distance L, are wound on an insulatingmandrel. Tproducesan eddy cument in a loop of unit cross-sectional area in the formation.This in turn induces in R an emfwhich is proportional to the conductivityof the material in the loop. [Adapted from H. G. Doll, Pet. Trans. AIME 186, 148 (1949). Copyright 1949 SPE-AIME.]

To describe the spatialfeatures of the induction field we return to the transmitter current. The Biot-Savart law provides the spatial distributionof the magnetic field whose time derivative produces the emf around each unit loop. The resulting circular eddy current then creates at the receiver output terminals a voltage attributable solely to the unit loop in question. With the factor e+Orr suppressed, this is3 V =

x [A

+

r3

+

3

a(r, z). (3.8) 2 (r2 (L/2 (L/2 z)2)3/2 The symbols refer to Fig. 24: L is the transmitter - receiver spacing; A, and A, are the area-times-turns for transmitter and receiver, respectively; 1, is the magnitude of the transmitter current; and r, z are the cylindrical coordinates of the unit loop. The first bracket is the sonde constant K. For fixed L the secondbracket is a function of only rand z, i.e., of the position of the unit

+

518

JAY TITTMAN

loop. We refer to it as the diflerentialgeometric factorg(r, 2). The coefficient L/2 is inserted to effect the normalization .fZO SF g(r, z) dr d z = 1. The differentialgeometric factor isjust the relative weight given to the conductivity of each unit loop contributing to the receiver voltage. Since each loop contributes independently to the emf developed in the receiver, the total output voltage is just a linear convolution V= K

1

g(r, z)a(r, z) dr dz.

-0

(3.9)

This is a direct consequence of the tacit assumptions of geometric-factor theory that the induction field is established instantaneously, ohmic losses can be ignored, and the eddy current loops do not interact with one another. The electromagnetic-theory approach presented later does not yield this simple result, as has been alluded to in the footnoteof Section 1.4.4.Sincewe do not consider here dipping beds or slanted boreholes, rotational symmetry prevails and each unit loop contains material of only one conductivity. This absence of azimuthal variation is made explicit by the notation a(r, z). Because geometric-factor theory permits the direct computation of the sonde output for any given variation in a which is rotationally symmetric, it provides the means for computing departure curvesfor the effectsof invaded zones, shoulder beds, caves, etc. To examine separately the radial and vertical investigation characteristics of the two-coil sonde it is necessary merely to compute the two integrals g(r) = J-2, g(r, z) d z and g ( z ) = .f; g(r, z ) dr. We recognize these as just the differential radial and vertical geometric factors described in Section 1.4.3. The integral over z is expressible in terms of tabulated elliptic integrals, which allows computation of the function shown in Fig. 25a.3 The radial position of maximum weight is seen to lie at a distance of nearly L/2 from the sonde axis, but even at distances r > 3L the contributionsare non-negligible. The function g(z),representing the weights assigned to discs of infinite radial extent and unit thickness, takes the form

-

g(z) = 1/2L

for IzI < L/2,

g ( z ) = L/(8z2)

for IzI

> L/2.

This function (Fig. 25b) shows that the vertical resolution width is approximately L or 2L, depending upon the definition chosen. Because conductivity from one bed to another can vary by a factor as Iarge as lo4,g ( z )cannot be ignored even for values of z > 1OL,i.e., at distanceswhere g ( z )itself is very small but nonzero. Equation (3.9) shows that a highly condhtive bed, even at a considerabledistance above or below the sonde can make a contribution

-

519


1

0

2

4

3

5

3

2

1

. L O

L

-1

-2

-3

I

L

I

1

I

*

g(z) FIG. 25. Differential geometric factors for the two-coil induction sonde calculated using geometric-factor theory. (a) Radial and (b) vertical. (Adapted from H. G. Doll, Pet. Trans. AIME 186, 148 (1949). Copyright 1949 SPE-AJME.]

5 20

JAY TITTMAN

to the receiver signal as large as that of a much less conductive bed in the vicinity of the measure point 0. In an infinite homogeneous isotropic medium (a constant) Eq. (3.9) reduces to V/K = a. Thus, for the usual case of formations which are invaded and/or layered it is useful to define an apparent conductivity a, = VJK, at each depth, where V, is the measured receiver output voltage. When the sonde is in an uninvaded formation of sufficientthickness, a, = a, except for a small borehole effect. Synthetic logs illustrating the features described above have been calculated by geometric-factor theory for various conditions of invasion, bed thickness, and shoulder-bed c~nductivity.~ Also, departure curves permitting the evaluation of a, from a*,given the other required parameters, are a~ailable.~ 3.1.2.2. FOCUSED SONDES.The idea of adding auxiliary transmitters and/or receivers to the two-coil sonde in order to compensate for unwanted influences was briefly introduced in Section 1.4.4. Here we illustrate the principles of “focused” induction sondeswith a few introductory cases. Even though it is possible to create departure curves which permit the determination of a, from a, for invaded beds, for example,the additional inputs a,, and di are often not known with desired accuracy. The sensitivity of the 0,-estimate to variations in a,, and di can be substantially reduced by designingthe sonde so that g(r) is close to zero in the “near” zone. Similarly, the desirability of sharpening the vertical resolution of the two-coil sonde is made clear in Fig. 25b. The coupling between vertical resolution and depth of investigation of the two-coil sonde is fixed by the coil spacing L. To improve the sonde’s radial characteristics, consider an arrangement such as that shown in Fig. 26a. An auxiliary transmitter T’ is located halfway between the primary transmitter and receiver, T and R.It is wound in series opposition to T. When a possesses only radial variation, integration of Eq. (3.8) over z gives the voltage contributed to the receiver by an infinitely long, unit-thickness annulus at r as [Kg(r) K’g’(r)]o(r).The prime refers to the two-coil sub-sonde consisting of T’ and R.Ignoring common factors in the Ks and noting the winding sense of T’, we reduce the bracket to ATg(r)- 2A,g’(r). If both transmitter coils have the same area this expression simplifies further to N,g(r) - 2N,g’(r), where the Ns are the numbers of turns in the respective coils. Using the symbol ( g ( r ) ) for the effective differential radial geometric factor for the composite three-coil sonde, we have

+

where w and w‘ are the weights mentioned in Section 1.4.4. The factor

19.

52 1


A

f I

‘\

lbi

Lateral Distance From

LI

I

2L I

5U2

FIG.26. Radialfocusingby addition ofan auxiliary transmitter. (a) The auxiliarytransmitter TI is (in this particular embodiment) wound in series opposition to T and is located halfway between T and R. (T, R) comprises the primary sonde, (T, R) the auxiliary sonde, and (T, TI, R)the composite sonde. (b)With reference to Eq. (3.1 l), curve A is wg(r) for (T, R), curve B is w’g‘(r)for (T’, R)when the turns ratio NT*/NT= 0.164, and curve Cis ( g ( r ) )for the composite sonde (T, TI,R).Curve D is ( g ( r ) )when N , I N , is 0.125, a choice which forces (g(r))close to zero over a finiteradialdistance near the sonde. [Adapted from H. G. Doll, U.S. Patent 2,582,314, 1952.1

+

l/(NT- 2N,) makes w w’ = 1, thus ensuring the normalization I; ( g W ) dr = 1. Equation (3.10)suggestshow to suppress the influenceof the borehole and invaded zone. Set ( g ( r ) ) to zero at a particular radius r = a by choosing the turns ratio such that N,g(a) = 2NT8g’(a).For the choice N , = 0.164N,, Fig. 26b shows wg(r)and w’g’(r)as curves A and B, respectively.16Curve C is ( g ( r ) )for the corresponding composite sonde, i.e., C is the sum of curves A and B. The maximum of ( g ( r ) )for the three-coil sondeis a little deeper in the formation than that for the primary sonde. But what is more significant, the relative importance given to the region 0 < r < -(L/4) has been reduced appreciably. Of course, it is not necessary to locate T’ at L/2,as was done for simplicity in this example. Removing this constraint provides another free design parameter which can be chosen to advantage. Thus, it is possible to

.

522

JAY TITTMAN

-

choose a position for T’ and a value for NF such that ( g ( r ) ) 0 from the axis to some preassigned radius.16 Curve D in Fig. 26b is ( g ( r ) )for a composite sonde with T’ remaining at L/2, but with NT = 0. 125NT.(If T’ were located farther from R,this arrangement would be even more effective in suppressing the influence of borehole mud, caves, and invaded zone on oa.) Clearly, additional transmitter and/or receiver coils can be added. By choosingspacings, winding senses, and numbers of turns a variety of predictable radial characteristics can be produced.16 The argument leading to Eq. (3.10) is readily generalized for a sonde consisting of any number of series receivers and series transmitters: (3.1 1)

where

Here, N,, is the number of turns in the ith transmitter, NRl the number in the jth receiver, L, the distance between the ith transmitter andjth receiver, and gi,(r) is the differential radial geometric factor of the subsonde consisting of the (Ti,R,)pair. If the coil areas are not all equal, the Ns are merely replaced by the correspondingA,, and AR,. The two-coil sonde’s vertical response characteristic can readily be improved by reducing L. The price of this simple expedient is a reduction in the depth of investigation and retention ofthe poor ratio between the amplitudes of the wings of g ( z ) and the central lobe. (Note the abscissa scale of Fig. 25a and the ordinate scale of Fig. 25b.) However, significant improvement can be made in ( g ( z ) ) ,while retaining the desirable radial focusing discussed above, by the addition of both auxiliary transmitter and receiver coils with appropriately chosen winding senses, area-times-turns values, and spacings. This can be achieved by a coil arrangement such as that pictured in Fig. 27a.I6 (The coils T” and R” are merely trimmers which compensate for mutual inductance effects, but they slightly perturb the near-zone radial sensitivity of the system. Thus, they are included in the discussion.) Using an argument similar to that given above for amving at (g(r)),we determine ( g ( z ) )for the composite sonde consisting of all nine subsondes (T,R), (T’, R),(T”, R),(T, R’),etc. Figure 27b shows the resulting (g(z)) and, for comparison, g ( z ) for the primary two-coil sonde (T, R).Although the auxiliary focusing coils narrow the central lobe only slightly, the relative amplitude of the wings is reduced substantially.Consequently,the effectsof highly conductive beds above or below the sonde are greatly diminished. Although originally sonde specifications were determined largely by cutand-try, computer modeling programs now provide the means for rapidly

523


FIG.27. (a) Example of a six-coil sonde consisting of three transmitters and three receivers arranged so as to improve vertical resolution while not materially reducing depth of investigation. (b) ( g ( z ) ) for the same sonde (solid curve) and, for comparison, that of the primary twocoil sonde consisting of T and R. [Adapted from H. G. Doll,U.S.Patent 2, 582, 314, 1952.1

establishing (g(r,2 ) ) for coil systems of nearly any degree of complexity. In addition, deconvolution programs applied during or after logging essentially remove the remaining effects of the wings of (g(z)). In this regard the only significant remaining limitation on induction sondes is the vertical resolution width produced by the central lobe of ( g ( z ) )(Fig. 27b), usually 5.5 ft (1.7 m). In field practice the most commonlyused sondeis comprised ofthree pairs of transmitters and receivers, with a primary spacing of 40 in. (1 m).la It is usually referred to as the 6FF40. 3.1.2.3. ELECTROMAGNETIC THEORYAND SKIN EFFECT. Although geometric-factor theory is heuristically very useful, it fails to account for the attenuation and finite velocity of electromagneticfields propagating in conductive media. Both affect the magnitudes, phases, and spatial distribution

-

524

JAY TITTMAN

of the eddy currents which are at the heart of induction logging. One consequence is that some of the emf induced in the receiver is thrown into quadrature with the transmitter current; another is a reduction in the absolute magnitude of the receiver voltage. The amplitude reduction and phase shift are the observed manifestations of propagation, or skin, efects. The latter name arises by analogy with the limitation of very high-frequency currents to only a thin surface annulus, or “skin,” of a cylindrical metallic conductor. Although the physical phenomenon involved is the same in logging, skin depths in earth formations are generally in the 1- to 10-meter range at the conventional operating frequency of 20 kHz. In order to expose the origins and consequencesof skin effect, this Section briefly outlines the electromagnetic theory of induction logging based on Maxwell’s equations. The discussion is limited to infinite homogeneous isotropic media. Several classic papers present solutions to this problemlg; the approach of Moran and Kunz is outlined here. We start with the general form of Maxwell’s equationsmand the standard constitutive relations for linear and isotropic media. The time dependence of Since all the field quantities are created by the transmitter current is cia‘. this source, they have the same time dependence, and the time variable can be ignored. This reduces Maxwell’s equations to V X E - iopH = 0,

(3.12a)

V * E =q/E,

(3.12b)

J,,

(3.12c)

VXH-(a-im)E=

V.H=O,

(3.12d)

where q is charge density, J, the source current density in the transmitter, and the other symbols have their usual meaning. At the 20-kHz frequency used in induction logging OE < o (the opposite of the situation discussed in Section 2.7). Thus, Eq. (3.12~)reduces to V X H - OE = J, .

(3.12c’)

There are four important consequencesof the axial symmetry evident in the geometry of the problem and in the source current: (1) Only the azimuthal component of the electric field, E4, is nonzero (see Fig. 28). (2) All induced = 4 oE4). (3) No field component can be a (eddy) currents are circular (.I function of 4. (4) No charge can build up, so q = 0. From the last of these, V*E=O.

(3.12b’)

A vector potential A, defined by H = V X A, is introduced into Eqs. (3.12a, b’, c’, d). Then, setting to zero the arbitrary, additive zero-curl quantity by

19.

525


which E may differ from iwpA, it can be shown that the Helmholtz equation results: V 2 A +k 2 A = - J

(3.13)

s9

where k2 = iwpo. For a transmitter coil very small in both length and diameter (Fig. 28) the solution is shown to be A,

=

[

]

lra2NTIT p 4n $1

- ikr)eik’,

(3.14)

where a is the coil radius and A, is the only nonvanishing component of A. (Note that in Fig. 28 and throughout this section, p is used for the radial coordinate and r for the distance from the transmitter to the field point.) In this approximation the source is seen to be the equivalent of an oscillating magnetic dipole of moment nu2NT1,located at the origin in Fig. 28. Now, integration of

E+ = iwpA,

(3.15)

around a circuit permits calculation of the emf induced in the circuit. Thus,

Y

FIG.28. Geometry for the analysisof the electromagnetic field produced by a transmittercoil of negligible size. The circular induced current at the point Pin the surrounding medium has only an azimuthal component J+. [From J. H. Moran and K. S. Kunz, Geophysics 27, 829 ( 1962).]

526

JAY TITTMAN

for a receiver coil of NRturns and radius a located on the axis a distance L from the transmitter, the terminal voltage is V = 2adRA!?,(a, L) = 2naNRia/d,(,(a, L).

(3.16)

Insertion of the expression for A, from Eq. (3.14), withp = a and r = L, then yields

(3.17) For comparison with geometric-factortheory it is useful to separate Vinto its real and imaginary parts. [As in alternating-current-circuitnomenclature these are usually referred to as the resistive and reactive ( R and X)components, respectively.] Then expanding in powers of kL we get for the in-phase, or R, components

+

where 6=-=(1 i)/k is the skin depth and K = [ ( ~ ~ . a 2 ) 2 N ~ ~ RThe z ~ leading / 4 ~ ~ term ] ~ isjust the geometric-factor theory result, validatingthe assertion in Section 3.1.2.1 that geometric-factor theory is correct in the limit of zero frequency and/or infinite resistivity. The remaining terms produce a net reduction in VRand represent the skin effect on the R-component of the signal. Induction sondes are usually calibrated as if geometric-factor theory were correct?' Le., as if V, were truly linear in a. Thus, even for a homogeneous medium a, # a because of skin effect. Rather, from Eq. (3.18), to first order in L/S, O a = - KVR-,(l -

-g).

An approximate boosting correction is usually applied electronicallyto a,so as to yield on the recorded log a value closer to the true a." This homogeneous-medium multiplier, the equivalent of [I - $(L/d)]-',can be expanded to take into account, after logging, higher order terms if the magnitude of L/6 requires it. For the X-component of the receiver output the expansion produces

Vx=-[l2K wpL2

--(2 Lr+-(-y1 L 3

3

2

s

*

.

-3.

(3.20)

Here the leadingterm is independent of a and representsthe mutual emf (see footnote in Section 3.1.2.1). This same result appears from setting a = 0 in


LOGGING

521

the exact expression for V, Eq. (3.17). The remaining terms in Eq. (3.20) are a-dependent manifestations of the propagation phase shifts ignored in geometric-factor theory. Sondes which measure V, are capable of providing their own skin-effect correctionz3to VR while logging. With the mutual induction term in Eq. (3.20) bucked out, V,= -Ka&5/G) to first order. Except for sign, this is just the first-order skin effect indicated in Eq. (3.18). Thus, the X-signal is a fmt approximation to the "lost" skin-effect signal in VR *

For further discussion and additional consequences of the electromagnetic theory of induction logging the reader is referred to references 19 and 23. These present discussions of the correct spatial distribution of the eddycurrent amplitudesand phases, the influenceof the invaded zone, the effects ofbed boundaries, treatment of focused systems,effectsof finite coil size, etc. The general theory outlined above makes clear that the receiver voltage is not accurately described by the linear convolution ofEq. (3.9). Nevertheless, the concept of geometric factor retains much of its usefulness, and the R-component of the output voltage can be expressed as a nonlinearconvolution VR = K

Lm

g(p, z, 4 0 , z ) dP dz,

(3.21)

where g( p, z, a)is often referred to as the generalized,or propagated,geometWe ric factor. It can be shown that g ( p , z, a) = g(p, z) Re[( 1 - i/~r)e'k7.~' note that g( p, z, a) still maps a(p, z) into VR,although the mapping function itself depends on both the absolute value and spatial distribution of a.z4a Since g( p, z, a)is very complicated in the general case, some simplificationis made in practice. For example, g(p, z, a) can be calculated for the homogeneous case, assuming a spatially constant value for a, and then convolved with some assumed spatial variation in a(p, z)to produce an approximateor trial value for ap.ZsCalculation of both simple and generalized geometric factors has yielded highly instructive three-dimensional maps for a variety of sondes and environmental conditions.z6 Although numerical methods have been applied fruitfully to the induction logging problem, they have generally been limited by computation time to environments in which CJ can vary in either the radial or the vertical direction, i.e., either the invaded-zone problem or the layered-bed problem. Recently¶an accurate and efficient finite-element calculation has been successfully applied to the combination problem.27For further study and additional references on the generalized geometric factor and accurate computation of sonde response see references 24 and 27. The influence of conductive beds above or below the central lobe of g(z) led to early implementation of elementary vertical deconvolution. A tech-

528

JAY TITTMAN

nique using three-station analog memorization followed by analog linear deconvolution while logging was widely implemented.28 However, this method has limitations which are apparent from the foregoing discussion: (a) The wings of g ( z ) or g ( z , a) require far more stations than three for adequate deconvolution of even focused sondes. (b) The generalized geometric factor is not a constant function (even in an infinite homogeneous formation), but varies with 6.(c) The presence of invasion and/or shoulder beds can distort the actual tool response function still further. More recently, the ready availability of high-speed computers has permitted the development of a method for nonlinear deconvolution of the wings, which nearly removes these difficulties. Digital telemetry and truck-borne computers have, in turn, made this deconvolution possible while l0gging.2~This effects accurate vertical deconvolution of the wings and, by use of the X-~ignal,~~ a simultaneously improved correction for skin effect. Because of its use of the quadrature signal (with the mutual induction voltage bucked out) the method is called phasor deconvolution or phasor processing. Other computer-based deconvolution methods have been developed, but in their present state must be applied after the logging run. 3.1.2.4. INDUCTION/LATEROLOG COMPARISON AND THE PSEUDO-GEOMETRIC FACTOR.The notion of geometric factor was initially discussed in Section 1.4.3 by means of a set of thought experiments. As a consequence of the admittedly artificial environment used, we noted that the operational definitions given there were not relevant for certain types of logging measurements. An ideal spatial weighting function, applicable to any type of measurement, would be like the geometric factor described in Section 3.1.2.1. It would weight each unit volume of space independently with respect to some characterizing parameter of interest, e.g., conductivity, and permit the logging measurement to be expressed as a linear convolution. Unfortunately, nature does not fully cooperate, and it was shown (Section 3.1.2.3) that even for the induction measurement it is only in the limit as wa +0 that the ideal situation is realized. Departures from this ideal become even greater in some of the other (nonelectrical) logging methods. Despite these difficulties it remains important to define suitably a quantity which permits both comparison of different measurements with respect to depth of investigation and estimation of the effects of radial perturbations such as invasion. This capability helps in choosing the best type of measurement to use under a given set of circumstances, e.g., laterolog versus induction in the presence of invasion. Furthermore, it allows ensuring that when two or more measurements are melded in an interpretation they refer to the same volume of formation. For all these reasons it is useful to define the (Although a vertical pseudo-geometric (radial) pseudo-geometric factor can be defined in a manner similar to that given below, it is rarely


529

used.) We introducz it through the examples of induction and laterolog measurements in infinitely thick, invaded formations.From these examples it becomes clear how to create pseudo-geometric factors for other types of measurement. Consider a formation characterized by resistivities Rxoand R,,and allow the depth of invasion dito vary. Then, using R, for the apparent resistivity measured with a laterolog, we can write

R,(di) = Jddi)Rxo + [1 - Jddi)lRt, (3.22) where JLL is defined as the (integral) pseudo-geometric factor for the laterolog measurement under consideration. Rearranging terms yields Jm(di) = [RLL(di) - RtI/[Rxo - RJ. (3.23) Here RLL can be determined by analytical calculation, computer modeling, or experiment. Figure 29 illustrates for some cases ofinterest the trajectory of JLL as di increases. This defining procedure is similar to that used in Section 1.4.3. However, here the cylindrical boundary lies between two ordinarily realizable values of resistivity and the dependence of JLL on R, and Rt is made explicit. Fortunately, in many cases of interest J, depends only

0

40 di (inches)

-

80

120

FIG.29. Comparison of calculated pseudo-geometric factors for the deep laterolog (JLL) and 6FF40 induction(JIL) sondes. The JL calculationincludesskin effect. The values chosen for R, and RJR, nearly span the range found in sedimentary formationsof interest.R,is expressedin ohm-meters.[From P. Souhaite, A. Misk, and A. Poupon, SPWLA 16th Ann. Logging Symp. Trans., New Orleans, June 4-7, 1975, paper LL.]

530

JAY TITTMAN

weakly on R JR,, .Thus, for practical purposes the family of curves parameterized by R JR,, can be replaced by a single average curve out to rather large values of di.29 For the induction measurement we can write

, a = Gddi)axo + [ 1 - Gddi)lat, (3.24) where a, is the induction-derived apparent conductivity and G, is the integral radial geometric factor. If skin effect can be ignored, G,(di)= J$d2 g(r) dr and is independent of a,, and a,. (For focused sondes g(r) is merely replaced by ( g ( r ) ) . )If skin effect cannot be ignored, then a, can be cal~ulated'~ and cast in the form of Eq. (3.24) so as to define G,(di). In this a and a, To compare the induction and case G, is a function of , laterolog measurements it is necessary to express the radial investigation characteristics of both sondes as functions of the same variable, either a or R = l/a. Choosing R, we then can define J, by analogy with Eq. (3.22):

R,

= J,R,,

+ (1 - J d R , .

(3.25)

Eliminating R , between Eq. (3.24) and Eq. (3.25) then yields (3.26) Thus, even if geometric-factortheory appliesin the calculation of G, ,J, is a function of R JR, as well as of di, and J, is a pseudo-geometric factor with respect to resistivity for the induction sonde. Figure 29 shows Ju for the deep laterolog and J, for the 6FF40 induction sonde. The representative values chosen for R, and R,, nearly span the ranges found typically in field practice. We see that when R, < R,, the induction sonde has the deeper investigation. For the conditions leading to the lowest pair of curvesthe invaded zone has no effect, i.e., R , = R,, until it exceeds 40 in. (1 m); even then the effect grows slowly with increasing di. On the other hand, when R JR, = 10 the induction measurement (uppermost pair of curves) is markedly affected when di > 40 inches (1 m) and the L M measurement is preferred. Additional comparisons between the two sondes may be found in reference 29. Pseudo-geometric factors similar to those in Fig. 29 can be generated for measurements other than resistivity. See, for example, Section 3.2.6.

-

-

3.2. Neutron Methods

A brief introduction to neutron logging was given in Section 2.2 We now examine the physics of the method and the design and performance characteristicsof the most widely used sondes. The application of theory to neutron


53 1

LOGGING

logging is not as elegant, complete, or accurate as that for resistivity. This derives from a number of sources, including the difficulty in solving analytically the governing Boltzmann transport equation for borehole geometry and the complex energy- and angledependence of neutron interaction cross sections. Consequently,sonde design and determination of response characteristics have until recently been predominantly experimental, although they have been guided by theoretical models. Increasingly, computer codes using Monte Carlo, discrete-ordinate,and multigroupdiffusion methods are being used in conjunction with laboratory design experiment^.^' However, the essential physics of the measurements can be understood from even the simplified and approximate diffusion model described in the following sections. 3.2.1. Neutron Scattering. Fast neutrons interact with nuclei in three ways: absorption, or reaction, usually followed immediately by emission of protons or a-particles; elastic scattering, in which the neutron changes its direction and transfers part or all of its kinetic energy to kinetic energy of the recoiling nucleus; and inelastic scattering, in which kinetic energy is not conserved because the struck nucleus is left in an excited state. In conventional neutron logging, which uses sources with average energy 4.2 MeV, the most important interaction is elastic scattering. Although absorption of fast, intermediate, and epithermal neutrons does reduce the neutron population in the formation and can affect the spatial distribution of slow neutrons, its consequences are generally modest and are usually ignored in the theory of neutron sondes. In practice, inelasticscatteringis automatically taken into account by experimental determination of sonde responses in accurately known laboratory or field formations. Although inelastically scattered neutrons can suffer large energy losses, they constitute a relatively small fraction of the neutrons cascading to low energies, most of which reach low energies through repeated elastic collisions. The magnitude of the energy loss depends on the energy of the (usually) first excited state of the struck nucleus. Since elastic scattering is the most important interaction for neutron logging, we briefly state some of its relevant characteristics. In a collision the initial velocity of the nucleus is essentially zero relative to that of a fast neutron. Thus, the conservation of energy and linear momentum require that the neutron lose kinetic energy in scattering. The relation between the neutron energies before and after s ~ a t t e r i n g,~~ E’ and E, respectively, is

-

E/E’ =+[(I

+ r ) + ( I - r) cos 41,

(3.27)

where 4 is the neutron scattering angle in the center-of-mass coordinate system, r = [(A - l)/(A 1)12, andA is the ratio ofthe mass ofthe nucleus to that of the neutron, i.e., the atomic weight of the nucleus. Equation (3.27) exposes several features of interest: (1) The fractional energy loss depends

+

532

JAY TITTMAN

only on the mass ratio and scattering angle, and not the energy. (2) For

4 = 180’, correspondingto a head-on collision, E/E’ = r. Thus, 1 - r is the maximum fractional energy loss that can occur in scattering from a nucleus of atomic weight A. (3) For 4 = 0, a glancing collision, E = E’, and no energy loss occurs. (4) When A = 1, correspondingto collision with a hydroE = 0 and all the gen nucleus, E/E’ = (1 cos 4)/2.Hence, for 4 = 180”, neutron’s kinetic energy is transferred to the recoiling proton. Figure 30 shows, for several target nuclei of interest, the probability that a neutron will have a relative energy in d(E/E’) at E/E’ after scattering. Because the right-hand side of Eq. (3.27) is independent of energy it is convenient to measure the neutron energy in logarithmic units, u = ln(Eo/E), known as the neutron lethurgy, where Eo is a reference initial energy. Thus, each scattering adds an increment to u. It can be shown that

+

6050 -

CALCIUM

40 -

SILICON

30> k

OXYGEN

d 20m

CARBON

a m 0 a: a

w

>

F a

1

w

a

‘01 9

87-

65.

4-

HYDROGEN

-

-

--


533

the mean lethargy increase(logarithmicenergy decrement)per collision,{, is

t = 1 + [r/(1 - r)] In r,

(3.28)

where the average is taken over all scattering angles.32For hydrogen, { is indeterminate but L'Hospital's rule shows that lim t = 1. For all other -0

elements of consequence in logging the approximation

t = 2/(A + 3)

(3.29)

is accurate to within 1%. Sample values for {illustrate the unique position of hydrogen in slowing down neutrons: H( l), C(0.158), 0(0.12), Na(0.084), Mg(0.08l), A1(0.072), Si(0.070), S(0.06l), C1(0.056), K(0.050),Ca(0.049), Fe(0.035). The mean number of scatterings required to slow down to lethargy u from an initial value u = 0 is merely u/t. For example, in slowing down from the average energy of a typical logging source, 4.2 MeV, to an epithermal energy of 0.42 eV a neutron acquires a lethargy u = In lo7 = 16.1. Then the mean number of collisions for the elements cited above are: H( 17),C( 102),O(135), Na( 19l), etc. Although its exceptional {-value alone would make hydrogen special in neutron moderation, its influence is further enhanced by the size of its scattering cross section. Most of the other elements abundant in sedimentary formations have effective scattering cross sections in the range of severalbarns or less. (1 barn = 10-24cm2)In contrast, the proton's cross section is about 20 barns between los eV and 0.5 eV, i.e., over about 3 of the lethargy range of interest. If the moderating medium consists of a homogeneouscollection of dzfcerentelements, as is normally the case, the macroscopic (elastic) scattering cross section Zs= X i n,oat,where ni is the number of nuclei of the ith element per cubic centimeter and the summation is taken over all the elementspresent. Here 2,is the reciprocal of the scattering mean-free-path and has the dimensions of reciprocal length. For such a mixture (t)is the cross-section-weighted average taken over all the different kinds of nuclei present: ({) = X,(np&)/Z,. Neither { alone nor a, alone is a measure of moderating efficiency. In materials with small absorption this property is measured by the mean lethargy increase per unit length of travel of the neutron and is called the slowing-down-power, which we will designate with the symbol (. For a medium composed of an essentially monoisotopic element, e.g., carbon, ( = nos{ = Z,t, while for a mixture it is 2,({) = X i n,os,ti.Figure 3 1 illustrates the contribution to (-- made by each of the elements in a 15%porous, water-filled quartz sand. The energy dependence results from the energy variation in a, for each element. The extraordinary influence of hydrogen over nearly the whole slowing-down range used in logging is apparent, even

-

-

534

JAY TITTMAN

GLEAN SAND, POROSITY = 15%

(L

W

i

cn

10-3

.I

1

I

1

I

I

I.

10

10'

10'

10'

1

lo5

I

1

10'

10'

NEUTRON ENERGY IN ELECTRON VOLTS

FIG.3 I. Element contributions to the slowing-down power of a water-iilled 15%-porous quartz sand. [From J. Tittman, in "Fundamentals of Logging." University of Kansas, Lawrence, 1956.1

though there are nearly ten times more oxygen atoms present at the porosity indicated. 3.2.2. Neutron Transport and Diffusion. Neutron moderation ends when the neutrons arrive at thermal equilibrium with the medium. In this state the neutrons continue to diffuse, while maintaining a constant energy on the average. On each scattering collision an individual neutron may now gain or lose energy since the neutron's velocity may be less or greater than that of the struck nucleus. It follows that the energy distribution of the neutron population approximately follows the Maxwell- Boltzmann law. Eventually, each neutron suffers a collision in which it is absorbed, or captured, and the resulting compound nucleus is formed in an excited state. Usually de-excitation occurs almost instantaneously, with the emission of one or more capture gamma-rays characteristic of the isotope formed. The thermal behavior of neutrons is well described by classical diffusion theory with the addition of absorption. The epithermal neutron flux acts as a spatially distributed source for thermal neutrons, and the thermal diffusion which follows broadens the distribution still further. The complete description of neutron moderation and thermal diffusion in extended media derives from the Boltzmann transport equation.33This conservation equation expresses the equilibrium existing between the production and removal of neutrons in a differential volume element dz = dV dl du of a six-dimensionalphase space. Three dimensions come from the configuration-space coordinates, two from the velocity-direction unit vector i2 (corresponding to the polar and azimuthal angles), and one from

19.

535


the lethargy u. For our purposes the transport equation can be cast in the form

un * VN(r, n,u) + N(r, n,u)vZt(u) =

I,”

du’

lo,

dQ’N(r,

a’, u’)v’Z,(Q’ * n,u’

+

u)

+ S(r, u). (3.30)

Here N is the number of neutrons per unit volume in phase space, i.e., per unit volume at the configuration-spacepoint r, per unit lethargy at u, and per unit solid angle around the direction n.The neutron speed (at lethargy u) is v, and Zt is the total macroscopic interaction cross-section. Z,(n’ * R, u‘ -,u) is the macroscopic cross section for neutrons moving initially in a direction R’,with lethargy u’, to be scattered into a new direction with a new lethargy u. S(r, u) represents the rate at which neutrons are created per unit volume at r and per unit lethargy at u. (Isotropicemission is assumed, so there is no source dependence on n.) Now the meanings of the terms in Eq. (3.30) can be described. The first term accounts for the convective, or transport, loss rate of neutrons from a unit volume in configuration space, while the direction R and the lethargy u remain fixed. The second term represents the loss rate per unit phase-space volume, resulting from collisions of all kinds. Since a stationary state is being described, this loss rate must be balanced by an equal rate of increase on the right-hand side of the equation. Here S(r, u)isjust the rate at which neutrons are produced by sources per unit volume in phase space. [For our purposes S can be set equal to the Dirac delta function S(r, u), corresponding to a point source emitting monoenergetic neutrons. This is usually an adequate approximation for the encapsulatedamericium -beryllium mixture generally used in neutron sondes.] The integral term is the rate at which neutrons are scattered into unit elements at lethargy u and direction Q from all smaller lethargies u’ and all initial directions n’,* while remaining in the same configuration-space unit element at r. It is outside the scope of this article to explore the many approximation methods for solving Eq. (3.30).34Suffice it to say, there exists a set of conditions applying reasonably well to the logging problem that permits the transport equation to be approximated by a series of coupled diffision equations. These differential equations describe the neutron’s behavior within a finite lethargy interval called a one-velocitygroup. Each group is characterizedby a diffusion coefficient and macroscopic removal cross section. The latter is a This description applies only to the slowing-down phase. If Eq. (3.30) is to be applied to the thermal phase exactly, “upscattering”must be taken into account as well as “down-scattering”.

536

JAY TITTMAN

scattering cross section (absorption is assumed negligible) that determines the removal of neutrons from that group and their deposition in other groups of greater lethargy. (In general, the removal cross section is nearly the total cross section, since any interaction “removesy’the neutron from a sufficiently narrow group.) The rate at which neutrons are removed from the ith group, appropriately weighted, contributes to the source term in the equation for the (i j)th group, wherej = 1,2,3. . . . For the solution of some problems this multigroup method uses as many as several tens of groups or more. In these cases it yields the neutron energy distribution as well as the spatial distribution. For the logging problem it is convenient and usually adequate, in order to illustrate the underlying physics, to reduce the number of groups to only The group diffusion equations then take the form36

+

(3.3 1) and (3.32) where subscript 1 refers to the group comprising all the neutrons above thermal energy and subscript 2 to those at thermal energy. The symbols have the following meanings: J(r) is the neutron 5ux at r in group i, i.e., the number of neutrons per unit time crossingthe surface of an imaginarysphere of unit cross section at r;* S is the source of neutrons in number per unit volume and per unit time; 2,,is the macroscopic cross section for removal from the ith group; and Di is the diffusion coefficient for the ith group. In general, both 2,and D, may vary spatially. For the thermal group, absorption is the only removal mechanism, so Zr2= 2,. In using Eq. (3.3 1) to represent the epithermal 5ux produced by a point source of fast neutrons located at the origin, it takes the form

(3.33) Vzf,(r) - t.&(r)/W)I= 0 everywhere except at the source. The quantity is defined as the diffusion length L1.For the epithermal neutron group this is L,, the slowing-down length from source energy Eo to epithermal energy. (Since the symbols L, and L are commonly used for slowing-down length and thermal diffusion length, respectively, they will replace L , and Lzhereafter.) For a monoenergeticpoint source in an infinite homogeneous isotropic medium the solution to Eq. (3.33) is3’

a

A(r) = (Q/47W(e-rL*/r),

(3.34)

19.


537

where Q is the total number of neutrons per unit time issuing from the source. Within the framework of this diffusion approximation, and given a particular source energy, the moderator is characterized by the two indepenthe slowingdent parameters L,and DI.Since it can be shown that X,, = rYsa down power, the pairs (Dl, or (L, ,() can be used equallywell to characterize the moderating medium. [L, in Eq. (3.34) is calculated by integrating over the whole slowing-down energy range, as is noted in Eq. (3.36) below. However, neutron conservation requires that the determination of Dl(= (L:) use ( evaluated at epithermal energy.36]Single-detector epitherma1 neutron logging for porosity depends, then, on the strong effect of hydrogen on both L, and D,. Because source-detector spacing is usually chosen such that r/L, 3 the exponential factor dominates,and hydrogen’s influence on the detected flux is exerted mostly through L,. Although Eq. (3.34) is usually a good qualitative guide in assessing the importance of formation constituents in determining sonde response, it fails badly when r < L, since the predicted epithermal flux diverges as r + 0. The epithermal problem has been solved also in cylindrically symmetric borehole geometry without bedding,38but the solutions are cumbersome and will not be displayed here. The results, in the form of curves offi along the borehole axis, are similar to those based on Eq. (3.34) for a homogeneous medium, especially when r/L, > 1. Since many sondes are designed to detect thermal neutrons, we turn our attention to Eq. (3.32). In this case removal is the result of absorption, so Z, = Z the thermal-neutron macroscopic absorption cross section, and L = e2/2 a. For a point fast-neutron source in an infinite homogeneous isotropic medium, the source termf,Z;,, has the spatial distribution of the epithermal neutron flux given by Eq. (3.34). Then it can be shown that the solution to Eq. (3.32) is

r)

-

(3.35) The thermal neutron flux is a function of threeindependent parameters, one from the slowing-down phase and two from the thermal diffusion phase. For sedimentary formations of interest L, > L. Thus, for sufficiently large r (which may be < 1 m) the bracketed spatial factor in Eq. (3.35) is determined essentially by L,, whereas the magnitude depends upon the thermal parameters as well. This fact is the basis for the currently most widely used neutron sonde design, described in Section 3.2.3.2. LENGTH. Since L, plays the central role 3.2.2.1. THESLOWING-DOWN in the physics of neutron logging, a few remarks on its calculation and physical significanceare appropriate.A mathematical definition of L, stated

-

538

JAY TITTMAN

in terms of the variables we have been using is39 (3.36) Thus, a proper calculation of L, takes into account D/C at each lethargy (energy) as the neutrons slow down. Within the framework of multigroup theory Eq. (3.36) is replaced by Lf = Xi(Di/Ci)A u i ,.where Di and Ci are average values for the ith group. The reason for associating L, with L ,the slowing-down-group diffusion length, in the remarks following Eq. (3.33) now becomes evident. Equation (3.34) shows that L, is a measure of the width of the spatial distribution of epithermal neutrons. This point can be quantified by calculating the mean-square distance from the source, (r2) =

1

rzf,(r)4zr2dr/[h(r)4zr2

dr.

Substitution off,(r) from Eq. (3.34) yields directly (r 2 ) = 6L:. Thus we see that L, has a few (related) interpretations. It measures the rate of decline of the epithermal flux with distance from the source, the width of the flux distribution, and the rms distance a neutron travels in slowing down to lethargy u. Calculated values of L, as a function of fresh-water-filledporosity, for a logging-source of energy 4.2 MeV, are shown in Fig. 32. The steep initial slope of L, is responsible for the high sensitivity of neutron measurements at low porosities. The converse is true, of course, at the higher porosities. The influence of the rock matrix on L, is also to be noted. Not shown in Fig. 32 are the effects of several other petrophysical descriptors: (1) Water salinity increases L, by displacing H,O molecules, effectively reducing the H 2 0 density. The added Na and C1 nuclei have much higher atomic weight and, therefore, lower ( values. (2) Bound water in clays and hydrated minerals moderates neutrons as effectively as free water. Thus, it tends to lower L, values and make formations “look” more porous than they really are. (3) Gas, mostly methane, is usually of much lower density than oil or water. Hence, by displacing these liquids it reduces the hydrogen density, thereby increasingL, and making formations“look” less porous than they really are. 3.2.3.Single-Detector Sondes. The earliest neutron sondes were axially symmetric devices using encapsulated radium-beryllium sources.* * Except for the occasionallyused fission neutron emitter CT52,all encupsuiured,or chemical, logging sources consist of an intimate mixture of a strong a-emitterwith beryllium, which has an exceptionally low energy-threshold for the (a,n) reaction. Radium, polonium, plutonium and americium are the naturally radioactive a-emittersthat have usually been used. The last of these is now employed nearly universally because of its low gamma-rayemission,long half-life, relatively low toxicity, and freedom from governmental regulation.

19.

539


40.

35 -

v= Sandstone 0=

30

i

I

0

Limestone

*= Dolomite

v.

10

Fresh water Formations

20 30 40 Porosity (P.u.)

50

60

FIG.32. Calculated slowing-down lengths (L,)and migration lengths (L,) as a function of porosity in fresh-water-filled formations. Sandstone matrix was taken to be quartz (SiOJ, limestone to be calcite (CaCO,), and dolomite to be CaCO, MgCO,. Source energy is 4.2 MeV. [From H. D. Scott, C. Flaum, and H. Sherman,SOC. Pet. Eng. 57th Ann. Fall Tech. Con$ Sept. 26-29, 1982, New Orleans, paper number SPE 1 1 146. Copyright 1982 SPE-AIME.]

The detector, usually of either thermal neutrons or capture gamma-rays, with the latter type predominating,was 1 - 2 ft (30- 60 cm) above the source. Capture gamma-ray detection was employed to sense the thermal-neutron flux in the vicinity of the detector. This indirect method works because the gamma-ray production rate per unit volume isf,Z,v, where v is the average number of gamma-rays emitted per neutron capture. Sampling the thermal neutron flux in this roundabout way was practiced because of the relatively high counting rates resulting. However, both thermal-neutron and capture gamma-ray detection were eventually replaced because of both fundamental and practical objections. (1) At the fundamental level, the thermal-neutron flux near the detector is a function of three independent formation-characterizing parameters [Eq. (3.35)]. Each varies with porosity in its own way and, in addition, L is affected by the large thermal-neutron absorption of chlorine in salt water and of boron or gadolinium present in many shales. Thus, translating measured thermal counting rates into accurate porosity estimates is complicated and requires knowledge of the formation that is often unavailable. When capture gamma-ray detection is used, v enters as a fourth parameter, the value of which depends on the chemical composition of the formation. Furthermore, one or more additional parameters that

540

JAY TITTMAN

characterize gamma-ray transport contribute to the determination of the gamma-ray flux arriving at the detector. (2) At the practical level, both types of detection yield porosity estimates that are excessively sensitive to variations in borehole diameter, mud composition, mudcake thickness, tool position in the borehole, etc.40This sensitivity is primarily a consequence of flux detection in a single detector located too close to the source. Larger spacings were prohibited by the low counting rates they yielded. To circumvent some of these difficultiesa sidewall sonde utilizing epithermal detection was developed in the mid-1960s.4l Efficient detection of epithermal neutrons was made possible by the advent of high-pressure He3 proportional counters. Thermal neutron detection in these counters is prevented by wrapping them in an appropriately thick sheet of cadmium, an excellent thermal neutron absorber. The detection of the epithermal flux reduced the number of formation characterizing parameters to two [Eq. (3.34)]. The source and detector were mounted in a pad that was forced against the wall by a spring-loaded backup arm.The system was made directionally sensitive by shielding the detector over the sector of the sonde facing the borehole. Consequently, sensitivity to the borehole variables mentioned above was reduced significantly. By calibration in laboratory formations of accurately known properties, the modest influence of D, on the porosity estimate was made even smaller, leaving L, as the dominant parameter affecting the counting rate. These features are shown in Fig. 4. This sonde has been more or less superseded, because of its sensitivity to mudcake and borehole wall rugosity, by a twodetector system discussed in the next section. Nevertheless, it still finds use as a high-accuracy sonde for porosity determination where adverse borehole conditions are absent. 3.2.4. Two-Detector Systems. The basic idea behind two-detector systems using either thermal or epithermal detection is exposed by taking the ratiof(rn)/f(rf) from Eq. (3.34) or Eq. (3.35),42at two distances, r,(ear)and rf(ar)* 3.2.4.1. EPITHERMAL NEUTRON DETECTION. The ratio R , of the counting rates of two finite-size epithermal detectors should refle; the behavior ofA(rn)/A(rf),i.e., (3.37)

,

where Ar = rf - rn. Since r, and r, are design constants, R is a direct measure of the single parameter L,, with sensitivityl(l/R,)(dR,/dLJ1=(Ar/Li). This is greatest at small values of Ls,.i.e., at high porosity (see Fig. 5). However, as Fig. 32 shows, L8(c$)is relatwely insensitiveto variations in c$ in this region. As a consequence, the sensitivity of R,to porosity is actually greatest at the low-porosity end of the scale.

19.


541

The major obstacle to the use of two-detector epithermal neutron measurements is the low counting rates observed when the detectors are located far enough from the source to keep borehole effects small. Typical chemical sources contain 16 curies of Am and produce -4 X lo7 neutrons/s. Increasing the strength by a desirable factor of several makes the transportation shield heavier and source handling on the drilling floor cumbersome. One solution to this problem is found by examining an interesting feature of Eq. (3.33 . 4 3 3.2.4.2. THERMAL-NEUTRON DETECTION.In formationsof usual interest L, > L. Thus, at distances large enough that r/L, >> 1, e-'lL < eUrlL*, and Eq. (3.35) for the thermal neutron flux becomes

-

(3.38)

which has the same form as Eq. (3.34). Thus, by taking the ratio of the counting rates from two thermal-neutron detectors located at a suficient distance from the source we expect R2 earlL.,as for epithermal detection. Principally because thermal-neutron detectors are much more efficient than epithermal, counting rates are increased by roughly an order of magnitude. The homogeneous-medium argument that led to Eq. (3.38) cannot address questions related to borehole effects. However, solution of the problem in borehole geometry predicts that if both r,, and r, are sufficientlylarge, the influence of hole diameter on the estimate of porosity can be made quite ~mall.4~ This theoretical conclusion is confirmed experimentally. However, most thermal-neutron sondes in field use do not employ sufficiently large spacings (e.g., r, = 70 cm and rf = 90 cm) to achieve the borehole independence that is potentially available.43From this it also follows that the condition r/L, >> 1, required for the validity of Eq. (3.38), is not always adequately satisfied, at least for r,. In this case the measured ratio R, retains traces of L from Eq. ( 3 . 3 9 , and thermal absorption effects manifest themselves.44Attempts to relate measured ratios to the neutron migration length L, = have been moderately successful in accounting for these absorber effe~ts.4~ (Figure 32 presents calculated values for L,, as well as for L?, for pure matrices with fresh-water-filledporosity.) However, even when this approach is effective, the estimate of porosity from the measured value of R2 still implicitly involves thermal absorption, a complication that is absent when epithermal detection is used. At present the great majority of two-detector neutron sondes in the field use thermal-neutron detection. 3.2.5. The Dual Compensated-Neutron Sonde. In order to take advantage of the high counting rates produced by thermal-neutron detection and simultaneouslyavoid absorber effectsthrough epithermaldetection, a sonde

-

m,

542

JAY TITTMAN

combining both types of measurement has been developed.fi A schematic drawing of the source-detector configuration is shown in Fig. 33. Both measurements are placed on the log at the same depth by the use of memorization and depth-shifting in the uphole computer. Letting 4aland (bp, be the apparent porositiesderived from the epithermal in formaand thermal measurements, respectively,we expect that & = 4a2 tions that are free of thermal absorbers. In shaly formations, which usually > as a result of the inadequate thermal-detector contain absorbers, spacing discussed earlier. The important influence of saltwater is more complex. An increase in salinity of the borehole fluid or of the formation water, or both, always reduces d,, because of H 2 0 displacement. However, it can either increase or decrease 4az,depending on the location of the salinity increase and on the porosity. This occursbecause of the opposing influences of H,O displacement and increased Pa. These observations assume that 481 and 4, have already been corrected for the borehole effects resulting from the use of insufficiently large spacings. In an attempt to address these and other problems, the effectiveness of processing the counting rates from each detector individually, rather than the ratios, has been under study. The effects examined include those of borehole size, tool standoff from the wall, mudcake thickness, mud weight, salinity, temperature, gas, and rock matrix.47 3.2.6. Pseudo-Geometric Factor for Neutron Sondes. The pseudo-geometric factor introduced in Section 3.1.2.4 measures the ability of resistivity

THERMAL DETECTORS

EPITHERMAL DETECTORS

U

U

FIG.33. Schematic diagram of source and detector arrangementin a dual compensatedneutron sonde. The bow spring forces the sonde against the borehole wall. [From R. R. Davis,J. E.Hall, and Y .L. Boutemy, SOC.Pet. Eng. 56th Ann. Fall Tech. Con$. Oct.5-7, 1981, San Antonio, paper number SPE 10296. Copyright 1981 SPE-AIME.]

19.


543

tools to penetrate the invaded zone and respond to R,. In Ziquid-filZedporosity, however, there is usually little difference in neutron transport properties between the invading filtrate and the connate liquids. Consequently, with regard to the invasion process it usually suffices to make only salinity corrections (of modest size)to the neutron log in order to acquire good estimates of porosity. In gas-bearingformations, on the other hand, the problem is more acute. In these, the neutron transport properties of the invaded zone suffer a large change as the low-density gas is displaced by mud filtrate. Thus, most pseudo-geometric-factor studies of neutron sondes have been made with respect to water invasion into air-filled porosity. However, there are a few recent calculations of pseudo-geometric factors for invasion into C02-filled porosity and into partially water-saturatedporosity.48The results in the case of C02 are very little Werent from those for the void, whereas those for partial saturation are markedly different. Measurements of pseudo-geometric factors have been published for an epithermal sidewall sonde and for two-detector sondes of both the thermal and epithermal The experimental arrangement consisted of nested coaxial thin-wall tanks. The 1-in. (2.54 cm) or 2-in. (5 cm) annuli between them were filled with -35% porous quartz sand. The smallest, or “borehole”, tank was 8 in. (20 cm) in diameter and iilled with water. Sonde response measurements were made as successive annuli were saturated with fresh water, simulating step-profile invasion fronts with increasing depth of invasion di. Detector output at each invasion depth was converted into apparent porosity and the pseudo-geometric factor was calculated as (3.39) where c$,(O) is the apparent porosity with zero invasion depth and 4a(m) with infinite invasion. This equation is the neutron-sonde equivalent of Eqs. (3.23) and (3.26) for laterolog and induction sondes. Figure 34 presents the results for a single-epithermal-detectorsidewall sonde and for two-detector thermal and epithermal sondes. For the two-detector sondes pseudo-geometric factors are given for apparent porosities derived from each detector individually, as well as those derived from the ratio. The ratio-derived J’s indicate deeper investigation than those for either detector alone. Thus, ratio-taking partially suppresses nearby influencesin a manner similar to the combining of subsonde outputs for induction measurements (Section 3.1.2.2). The sonde using thermal-neutron detection shows deeper investigation than the epithermal. However, this is predominantly because of its larger spacing rather than because of the difference in detection energy. Equation (3.39) and Fig. 34 present what may be called “porosity-de-

Depth Saturated Inches From Borehole Wall

For App. Porosities From Count-Rate Ratio Near Detector Far Detector



19.


545

rived” J’s. Alternatively, a “counting-rate-derived”J or a “ratio-derived” J could have been defined instead, by replacing (6, with either a detector counting-rate or the ratio from a pair of detectors. Since the relations between each of these quantities and 4aare not strictly linear, the resulting J’s would be somewhat different. Reference 48 contains a collection of calculated curves of J versus di with respect to fresh-water invasion of gas-filled porosity. From the discussions of the preceding sections we expect the vertical resolution of a one-detector sonde to be roughly equal to the sourcedetector spacing. Similarly, for a two-detector sonde it should be roughly the distance between the two detectors. The latter surmise has recently been confirmed (with a few interesting exceptions) by a set of discrete-ordinates calc~lations.~~ Peculiarities,such as small overshootswhen the sonde passes from one bed to another, are shown to occur under some circumstances. These depend on the sequence in which the source and detectors pass the boundary, as well as on the direction ofthe porosity change, but not upon the neutron detection energy. 3.3. Density Measurement Using Gamma Rays

An introduction to the use of gamma-raysfor density loggingwas given in Section 2.3. In addition, the remarks introducing Section 3.2 apply almost verbatim, although three important differences are noteworthy: (1) The Compton cross section varies smoothly with energy, following the KleinNishina formula.51(2) The detected gamma-ray flux is a direct measure of bulk density, the property desired. (3) The mixing rule for density is intrinsically linear in the volume concentrations of the formation constituents. Because density logging utilizes transport phenomena, as neutron logging does, we will follow a format similar to that of Section 3.2. 3.3.1. Gamma-Ray Interactions. Gamma-rays interact with matter in three ways,S2each dominating in a particular energy region, as shown in Fig. 35. ( 1 ) Photoelectric absorption leads to the immediate ejection of an electron from the atom, and for most elements of logging interest is dominant only at energies < 50 keV. The electron is emitted with kinetic energy

-

FIG. 34. Pseudo-geometric factors for three neutron sondes in an air-filled 35%-porous quartz sand invaded to different depths by fresh water. (a) For an epithermal sidewall sonde, using the &derived Jof Eq. (3.39). (b) For a two-detector thermal sonde. The “near detector” curve is derived using apparent porosity values read from a counting-rate-versus-porosity calibration ofthe near detector,and similarly for the far detector.The “count-rateratio” curve is determined from ratio-derived apparent porosities. (c) For a twodetector epithennal sonde. Remarks are the same as for (b). [Adapted from H. Sherman and S. Locke, SPWLA 16th Ann. Logging Symp. Trans., New Orleans, June 4-7, 1975, Paper Q.]

546

JAY TITTMAN

Energy (MeV)

FIG.35. Gamma-ray mass-absorptioncoefficients at energies of interest in logging. [FromJ. Tittman and J. S. Wahl, Geophysics 30,284 (1965).]

Ee = Ey - Eb , where Ey is the gamma-ray energy and Eb the binding energy of the electron in the atom, usually the K-shell energy. (2) Compton scattering prevails for 0.1 MeV < Ev < 10 MeV. Since usually Ey &and the wavelength is sufficiently small, the gamma-ray is considered to scatter off individual free electrons. The scattering kinematics are determined by the relativistic conservation of energy and linear momentum. (3) Pair production, the creation of an electron-positron pair, is energetically possible only when Ey 2 2m0c2 = 1.02 MeV, where m ~ c is * the rest-mass energy of the electron and of the positron. The gamma ray disappears in the process, and the electron and positron share equally any excess energy, Ey - 2moc2.For elements of ordinary interest to us pair production dominates only when Ev > 10 MeV. Thus, this interaction plays no role of consequence in density logging. As has been noted in Section 2.3, at energies above the Kedge the photoelectric absorption cross section per electron increases with atomic number approximately as Z3.6, whereas the Compton cross section is independent of 2. The pair-production cross section is approximately linear in 2. Thus, if energies are restricted to the Compton region, the gamma-ray flux at some distance from the source must be a function only of the number of electrons per unit volume, and cannot depend on the chemical nature, or 2, of the medium. By the same reasoning, the gamma-ray flux in the photoelectric

-

-

-

*

19.

547


energy region is a very sensitive function of 2, and 2 nearly always reflects the dominant elements in the rock matrix rather than the pore fluids (Section 2.4), as can be seen in Fig. 35 and Table IV. For Compton interactions the expression relating the gamma-ray energy after scattering, E, to that before scattering, E’, is (3.40) where 8 is the scatteringangle. This is the gamma-ray analog of Eq. (3.27) for neutrons, but here the relative energy after scattering is not independent of the energy before scattering. Hence, defining a logarithmic energy measure such as lethargy serves no usehl purpose. Since mOcZis a natural constant, E depends only on E’ and 8. By evaluating (AE)= [ l/a,(E’)] Jf’ (E’ E)tsAE’ + E ) dE we obtain an expression for the mean energy loss as a function ofenergy before scattering.s3Here a&’) is the total Compton cross section for scatteringat energy E’ and ts,-(E’ + E) is the partial cross section for scattering from E’ into a unit energy interval at E. Figure 36 shows how the mean relative energy loss decreaseswith decreasing gamma-ray energy.s4 From this we (correctly)surmise that in an extended medium the multiplyscattered gamma-ray flux increases as the energy is reduced, until photoelectric absorption becomes the countervailing mode of interaction. Several measures of gamma-ray interaction are in common use, often leading to confusion. Thus, it may be helpful to define the most common ones and explicitly note their relation to one another. (1) The cross section, usually designated by ts (although z is commonly used for phtotelectric

4

FIG.36. Mean relative energy loss of the photon in a Compton-scatteringcollision. Abscissa is initial energy. [Adapted from J. A. Czubek Znt. J. Appl. Radiat. Zsotop. 34(1), 153 (1983). Reprinted with permission. Copyright 1983, Pergamon Press, Ltd.]

548

JAY TITTMAN

absorption), is a measure of the effective area of the target particle, just as for neutrons. Thus, a plane parallel beam of projectile particles incident on a slab of target material will exit from the opposite side with a fraction e-””fl having had no interaction (transmission). Here n is the number of target particles per unit volume, a, is the total cross section, and t is the slab thickness. (2) The linear absorption coeflcient, usually designated by p, is defined as na,(or I: n p , , when appropriate). This corresponds to the macroscopic cross section used with neutrons. (3) The mass absorption coeficient pm, is defined as p/p, where p is the density of the target material. Thus, pmp = p = na. This quantity is especially useful in the Compton energy region, where density is essentially the only parameter characterizing the target material. Sometimes the symbol p is used to represent the rnass-absorption coefficient,as in Figs. 35 and 37. Ifthe meaning is not made explicit, dimensions or context will usually indicate which coefficient is intended. The subscript “0” or “o”, when used, indicates that p is evaluated at the source energy. 3.3.2. Gamma-Ray Transport and Diffusion. Gamma-ray transport is governed by the same Boltzmann equation as neutron transport, Eq. (3.30). With a few changes in notation it can be written as

R *VF(r, R,E ) + ZXr, E)F(r, Q, E ) =

IEGI, a’, F(r,

E’)&(r,

*

al,E’

E ) dn’ dE’

+

+ S(r, E). (3.41)

Here the energy variable E is used, rather than lethargy, and the constant photon velocity c (velocity of light) is merged with the number density to form flux F = Nc. The meanings of the terms correspond to those given in the discussion following Eq. (3.30).Again, there are many approaches to the solution of Eq. (3.41) involving approximations or numerical methods or both. Usually a particular approach is tailored to a specific type of problem. For example, some methods yield both the spatial and spectral features of F, but are intrinsically limited to infinite homogeneous isotropic media.s5A!ternatively, the one-group diffusion approximation permits the solution of problems in media with boundaries, such as the borehole problem, but is incapable of predicting spectral characteristics of the flux.s6 The Monte Carlo approach can solve both problems, as well as highly practical ones including the effects of sonde materials, detector energy dependence, collimation, mudcake, etc5’ Its utility is often limited, however, by the need to trade off computer running time against statistical uncertaintiesin following photon histories. The diffusion approximation,despite its limitations,yields

19.


549

an easily understood and heuristically useful description. Thus, we outline here the approach of reference 56. Integrating Eq. (3.41) over S2 yields

V j(r, E )

+ &(r, E)f(r, E ) =

r

f(r, E’)Zs(r, E‘ -,E ) dE’

+ S(r, E ) , (3.42)

where the new symbols have the following meanings: f ( r , E ) = Jn F(r, n, E) is the energy spectrum of the photon scaZarJux, i.e., the number of photons per unit energy interval piercing the surface of an imaginary sphere of unit cross section centered at r; j(r, E) = In W(r,Q, E) dQ is the photon current density, representingthe net vector flow rate of photons per unit area and per unit energy interval at r. A somewhat intricate integration of Eq. (3.42) over E yields (3.43) where J(r) = Jp j(r, E ) dE, S(r) 3 15 S(r, E ) dE, and Za= 2,- Z, since the only interactions are scattering and absorption. Two assumptions are made in simplifying Eq. (3.43): the number flux is separable, f(r, E ) = fi(r)fi(E),*and Fick’s law applies, i.e., J(r) = -D(r)V’(r). Validation for these assumptions is given below. Substitution into Eq. (3.43) then gives the stationary-state diffusion equation, DVzfr(r) -

A W ( W ) +w

= 0,

(3.44)

where (Z,(r)) = Jph(E)Z,(r,E ) dE is the macroscopic absorption cross section averaged over the flux spectrum at r. If detection of the gamma-ray flux has a cutoff at some energy Ed > 0, as is effectivelythe case in practice, then the integrationsthroughout thisdevelopment arelfrom Ed to Eo rather than from 0 to Eo.In this case (Z,) becomes the average cross section for the removal of gamma rays from the energy band between Emin and EOby both photoelectric absorption and Compton scattering.? We digress briefly to justify the separability assumption and the use of Fick’s law. Figure 37 shows spectra of the scattered flux about a 0.5-MeV

* Subscripts I and 2 should not be confused with those used in describing epithermal and thermal neutron fluxes in Sections 3.2.3-3.2.5. Also, it is assumed here that .@f2(E) dE is normalized to unity. Since & 0 for the upper part of the spectrum (Figs. 35 and 37), (Z,)is determined essentially by I, values at low energies. On the other hand, values at higher energies contribute significantly to the spectral averaging of D.

-

550

JAY TITTMAN

FIG.37. Calculated spectra of the multiply scattered number-flux about a point 0.5-MeV source in infinite homogeneous media. [Drawn from data tabulated in H. Goldstein and J. E. Wilkins, Jr., USAEC Report NYO-3075(1954).]

point source in infinite homogeneous media. The densities and 2-values of aluminum and water roughly bracket those found in sedimentary formations, so the main features apply to materials of interest in logging. It is seen that the spectra above the photoelectric region are almost identical for the two media and that as distance increases the spectral shape stabilizes. In particular, since practical sonde spacings fall in the range 4 < h r < 10, this observation implies separability. The applicability of Fick’s law is indicated by the fact that most of the flux is at low energy, where Compton scattering is only mildly a n i s o t r ~ p i c . ~ ~ Confining the discussion to infinite homogeneous isotropic media, and setting S(r) = Q6(r), where Q is the total source strength in photons per unit time, and L2 = D / ( C , ) , Eq. (3.44) becomes exactly the same as Eq. (3.33). However, here L retains its identity as a diffusion length rather than becoming a slowing-down length. Similarly, the expression for the flux in an infinite homogeneous medium is determined by Eq. (3.34), with D and L replacing D 1and L, . Since both D and L are inversely proportional to bulk density PB , this becomes

(3.45)

,

where K and K , are proportionality constants. This expression exhibits the main features that are experimentally observed, i.e., fincreasing with


55 1

density at sufficiently small values of h r / K 2 , reaching a maximum at p,r/K2= 1, and falling nearly exponentiallywithp, at large values ofp,r/K,. Despite the fact that this development ignores the presence of borehole, mud, mudcake, sonde materials, etc., Eq. (3.45) predicts the spacing and density dependence of actual sondes surprisingly well. 3.3.3. The Single-Detector Sonde. The earliest density sondes suffered from limited sensitivity to density and excessive borehole influence. Both defects were related to the values of the mass-absorption coefficient in the sonde shielding and in the formation at the energy of the source that was chosen. In order to achieve deep penetration, Co60gamma-ray sources were used. However, the 1.25-MeV gamma-rays penetrated the rear shielding material too easily, thus entering the borehole where they found a low attenuation path to the detector. This produced a “background” component in the detector signal which was not only essentially independent of formation density, but varied with borehole she and mud density. In addition, because of the small source-detector spacings used, the component of the detected flux coming from the formation was not sufficiently sensitive to density changes to provide a high-accuracy measurement.’ The borehole problem is solved by using a gamma-ray source of lower energy, Cs13’ with Eo = 662 KeV.59This permits us to shield more effectively the gamma rays from entering the mud column directly from the sourceand, thus, to reduce the “borehole signal.” In addition, the sensitivity to formation density, for a given source -detector spacing, is increased by the use of the lower-energy source because the Compton cross section increases with decreasing energy (Fig. 35). (The use of still lower-energy sources, e.g., HgZo3, with Eo = 280 keV, provides even higher density sensitivity, but shows excessive photoelectric influen~e.~~) Reduction of borehole-she effects to nearly ignorable levels is achieved by the use of extensive tungsten-alloy or lead shielding “behind” the source, between source and detector, and “behind” the detector. The face of the sonde that rides on the borehole wall is of tungsten alloy also, with holes cut through at the positions of the source and detector. This disposition of shielding material creates a logging geometry corresponding approximately to that of an idealized source and detector located on the plane vertical interface between two semi-infinitemedia. One

-

* Equation (3.45) can be used to illustrate this point. The sensitivityof the flux to changes in formation density is Thus, when h r J K 2> 1, the usual case in practice, the magnitude of the sensitivity increases with source-detector spacing. Note also that K T ’ is an effective mass-absorption coefficient that increases with decreasing gamma-ray energy.

552

JAY TITTMAN

of the media (the sonde pad) is an ideal absorber, or sink, and the other is the formation, as indicated schematically in Fig. 38. Application of Eq. (3.44)to this arrangementis complicated by an incompatibility of boundary conditionsthat results from locating the source on the interface. This mathematical problem is circumvented by replacing the point source by a dipole.56The detector response is then taken to be the photon current density crossing the interface from the formation into the sonde at the location of the detector. Ignoring a minus sign, we get (3.46)

where M is the dipole moment of the fictitious source and r is the source-

FIG. 38. Diffusion-theory solution for the problem of two semi-infinite media with plane interface; left-hand medium is a perfect absorber. Photon current density is identified with the sonde counting rate. Experimental points are normalized at r = 40 cm andpB = 2 g/crn3. [From J. Tittman and J . S . Wahl, Geophysics 30, 284 (1965).]

19.


553

detector spacing (Fig. 38).* J is dominated by the exponential factor at practical working distances. A featureof interest is that the current density in Eq. (3.46), in contrast to the flux in Eq. (3.43, depends on only a single diffusion parameter, i.e., the current density is a function of p, through L only. Figure 38 presents curves calculated according to Eq. (3.46) and normalized to experimental data taken at a source-detector spacing of 40 cm. It is seen that curve-shape agreement is excellent. At smaller spacings the theoretically predicted magnitude of Jfalls above the experimental data, but the dependence on pe remains in quite good agreement at 30 cm and is nearly as good at 20 cm. Unfortunately, no measurements are presented at larger spacings, where there is reason to expect theoretical and experimental results to better retain the agreement seen at 40 cm. The model presented above, although suppressing spectral features, yields a reasonably complete physics description of the density logging measurement. However, in real boreholes mudcake or rugosity can exist between the pad and the wail, thus constitutinga layer of material with transport properties different from those of formation. As may be surmised from the discussions of multicoil induction and two-detector neutron sondes, this obstacle can be overcome by the addition of a detector at shorter spacing. 3.3.4.The Two-Detector Borehole-Compensated Density Sonde. To the best of the author’s knowledge there is no published general theory of the two-detectordensity sonde in borehole geometry, with mudcake. Therefore, we will describe its mode of operation qualitatively and then use experimental data to present its characteristics.60 Figure 39 illustrates schematically the totally absorbing sonde forced against a borehole wall covered with mudcake. The arrows indicate pictorially that the long- and short-spacingdetectors have different depths of investigation. The formation is characterizedby its densityp, and average atomic number Z,, while the mudcake introduces the parameters pmc,Z,,, and thickness t , . Thus, most generally, five independent parameters can influence the responses of the two detectors. By appropriate mass-absorption filtering of the spectrum incident on the primary, or “far”, scintillation detector, the effects of photoelectric absorption in the formation can be eliminated for most practical purposes. However, in some recently designed sondes this filter has been replaced by a beryllium window that transmits the low-energy part of the spectrum for use in simultaneousphotoelectric lithology logging (Section 3.4).61In these sondes, for the density measurement the part of the spectrum influenced by formation photoelectric effect is removed

* A typographical error appears in reference 56, from which Eq. (3.46) is taken. All the j’s from equation (7) on should be replaced by J’s since it is the totalcurrentdensity that is under discussion rather than the current density per unit energy.

554

JAY TITTMAN

Mudcake

Formation

Long-Spacing Detector

ShortSpacing Detector

Source 1

FIG.39. Schematic drawing of a two-detector density sonde pressed against a borehole wall covered with mudcake. The sonde pad and backup arm,shown in black, are articulated relative to the tool structural member in white. [From J. S. Wahl,J. Tittman, C. W. Johnstone, and R. P. Alger, J. Pet. Tech. 16, 141 1 (1964). Copyright 1964 SPE-AIME.]

by pulse-height discrimination at the output of the scintillation detector. Thus, in both types of sonde the dependence on 2, is essentially eliminated. (We ignore the small Z/A effect in this discussion.) Zmcis more of a problem, however. Since many drilling muds contain barite as a weighting material, barium is often present in the mudcake. Because of barium’s relatively high Z value (56), its photoelectric massabsorption coefficient can be significant even above 100 keV. The mudcake layer thus approximatesa mass-absorption filter for the photon current streaming from the formation into the detector window and reduces the detector counting-rate. Fortunately, it has been found experimentally that barite-containing mudcakes can be assigned apparent densities pZc, that combine the effects of both pmcand Zmcinto a single parameter insofar as density logging is concerned.* (For mudcakes free of high-Z materials, Pmc = pmc .) There now remain only three independent variables to contend with. It is found experimentally that for not-too-large values of the product pZctmc,the effect of mudcake on the detectorcounting rates can be characterized by a single parameter y~ = (A- p&)fmc.Then, since responses of the

-

* How well this is achieved depends upon the spectral sensitivities of the long- and shortspacing detectors. This remark is amplified near the end of Section 3.3.4.1.

19.


555

two detectors are different functions of p~ and v/, it is possible to extract p~ from the two measurements. The short-spacingdetector was originally a small Geiger- Mueller counter laid in a longitudinal, protected slot in the tungston-alloy pad face. Newer designs use scintillation detectors at both near and far spacings.61 CROSS-PLOT. The solution to the mud3.3.4.1. THESPINE-AND-RIBS cake problem is conveniently described by a graphical presentation consisting of a cross-plot of counting rates from the long-spacing detector versus those from the short-spacing detector. In order that the representation not depend on source strength or detector sensitivity, all counting rates are normalized by measurement in a standard formation. The development of the correction scheme will be made by reference to the schematic diagrams in Fig. 40. The response of each detector to p~ is determined by observing the counting rates when the sonde is placed in a series of laboratory mock-up formations of accurately known density. These responses can be plotted against one another as in Fig. 40a. The result is very nearly a straight line, known as the spine, corresponding to the essentially exponential response of each detector to p ~ . Refemng next to Fig. 40b, consider the presence of a mudcake with, for example, pgC= 1.5 g/cm3 on a formation with p~ = 2.5 g/cm3. As fmc increases, both counting rates increase since part of the medium immediately in front of the sonde pad contains the less dense mudcake. Each detector exhibitsa rate of increasewith tmcthat is different from its rate with respect to . the representative points fall away from the spine, decreasing p ~ Thus, initially in a "northeasterly" direction. As tmcincreases further, the curve (C)

i I

=1.E

Short-Spacing Detector Counting Rate FIG.40. Schematic study of effects of variation in petKc,and hmc on countingrates of longand short-spacing detectors. Both abscissa and ordinate scales are logarithmic. Numerical values are for illustrative purposes only. [FromJ. S. Wahl, J. Tittman, C. W. Johnstone, and R. P. Alger, J. Pet. Tech. 16, 1141 (1964). Copyright 1964 SPE-AIME.]

556

JAY TITTMAN

traced by the loci of representative points bends upward toward the spine. It terminates at the point pe = pzc = 1.5 g/cm3, because when t,, -,03 both detectors are sensing the same infinite homogeneous medium. Figure 40c illustrates the development of the curves for a single value of pzcand three different values ofp,. It is apparent that asp, andp;, approach one another in value the mudcake curve arcs back to the spine more tightly. In addition, the short cross-marksrepresenting points with tmc= 4 in., in., and3 in.(6.35 mm, 12.7 mm,and 19.1 mm)crowdclosertotheoriginsince less arc length is available to represent the complete range of tmc, 0 to CQ. Figure 40d permits comparison of the curves traced when mudcakes with two different values ofp& ( I .5 g/cm3and 2.0 g/cm3)build up on a formation with p, = 2.5 g/cm3.As the representativepoints leave the spine they initially follow the same curve, then separate in order to terminate back on the spine at their respective pzc values. That the curves for different values of pzc are initially congruent corresponds to the existence, for small departures from the spine, of the characterizingparameter v/. This is made explicit by the fact that a single point on a mudcake curve represents different pairs of values for pzc and kc.Thus far we have considered only cases where p:c < p,, a conditionwhich prevails when little or no barite is present. For barite-loaded mudcakes, p;, > p, usually. In this case the mudcake curve follows a course typified by the trajectory for p;, = 3.0 g/cm3because both detector counting rates decrease with increasing t,,. As for the lighter mudcakes, the curve returns to the spine at the value p, = pz, . When the mudcake curves originating at a particular p, value are truncated where they start to separate, they give the appearance of stubs, or “ribs”, as shown with laboratory data in Fig. 4 1. Hence, this format is known as a spine-and-ribs plot. The experimental points in Fig. 4 1 were taken with a particular sonde in laboratory formations. Synthetic mudcakes made of neoprene sheets loaded with various fillers, some including barite, were used for the measurements. usually It is fortunate that in field practice the values of the product pzCtmc fall in a range that allows successful application of the spine-and-ribs method. In extreme cases, with certain barite-loaded mudcakes and highdensity formations, this approach loses reliability. An example can be seen in Fig. 41 where the northeast ribs emanating from the spine at p, = 2.7 g/cm3 separate very quickly. Even though the two highly curved ribs are for bariteloaded mudcakes, and usually fall on the left-hand side of the spine, they start out in a northeasterly direction. This occurs because of the high density of the formation, i.e., ,p: < p,. We digress briefly to mention two measurement features, ignored in the preceding discussion, which can contribute to departures from the simple model of Fig. 40 when barium or some other high-Zelement is present in the

+


557

gl

a

m

.-

C c

C

3 0

u

/Without Mudcake Barite 2.7 P

C

4 Short -Spacing Detector Counting Rate FIG.4 1 . Spine-and-ribscross-plot developedon experimental data. Both scales are logarithmic. Artificial mudcakes consisted of neoprene sheets weighted with filler materials, including barite. [From J. S. Wahl, J. Tittman, C. W. Johnstone, and R. P. Alger, J. Pet. Tech. 16, 1 141 (1964). Copyright 1964 SPE-AIME.]

mudcake: ( 1) The two detectors, because they are at different spacings from the source, are exposed to slightly different spectra even from the bare formation. Thus, barium photoelectric absorption changes each of the counting rates differently. (2) Even more important, the two detectors have somewhat different spectral sensitivities, especially when one is a GeigerMueller counter, but even when both are scintillation crystals (of different size). Because of these features, the two detectors may impute different p& values to the same mudcake material. Hence, an interestingexception to the simple model used in developing Fig. 40 occurs: The mudcake curve does not return to the spine when tmc+ m, but terminates at some point off the spine determined by the counting rates corresponding to &(far) and p&(near). In the event of pad standoff from the wall, when barite-weighted rnud(rather than mudcake) intervenes, it is possible to generate ribs leaving the spine in an initially northwesterly direction. The far counting rate can , the near response can decrease increase because pzud(far)< p ~ whereas because &,,(near) > p~ owing to the stronger influence of photoelectric absorption on its detected spectrum. Luckily, these conditions occur only

558

JAY TITTMAN

occasionally. They are noted here principally to deepen the reader’s understanding of the physics involved in making the standoff correction. In practice, the spine-and-ribs correction is applied on-line in the surface computer. The outputs recorded on the log are the corrected value ofp, and &,the magnitude of the correction which was applied. Figure 41 shows that the far counting rate alone determines an initial mudcake-perturbed value for p, from the spine. Introduction of the near counting rate then specifiesa point on a rib. Sliding along the rib to the point where it intersects the spine locates the corrected value for p,. Then, & = b(corrected) p,(initial). Reliable corrections can usually be made when Ap,, 5 0.1 g/cm3. The recording of ApBpermits the log analyst to assess how much confidence to place in the corrected value ofp,, provides a continuous record of the state of sonde contact with the wall, provides a qualitative indication of mudcake thickness, and can flag anomalies such as those discussed in the preceding paragraph. An alternative to the use of Fig. 41 is to construct a plot of Ap versus uncorrected values for p,(far) - p,(near) and to fair an average rib through all the data?’ In this format each counting rate is converted into an apparent density before entering the correction chart. Then, the value of Ap determined from the chart is added to the long-spacing apparent density to yield the corrected value for p,. Another variation is to construct a spineand-ribs plot like that of Fig. 4 l except that the abscissa and ordinate are long-spacing and short-spacingapparent densities, respectively.62 3.3.4.2. BOREHOLE-SIZE EFFECT.The tungsten-alloy face ofthe sondeis curved to fit snugly against a 6-in. or 8-in. diameter (1 5 or 20 cm) borehole and internal shielding is provided, as mentioned in Section 3.3.3. These two design elements nearly eliminate hole-size effects. For example, measurements made in a 12-in. (30 cm) hole, using a traditional sonde calibrated in a . ~ aO9-in. (23 cm) hole 6-in. hole, require correction of only -0.02 g / ~ m ~ In the correction is essentially zero. A more recent sonde design has roughly halved the effect.61 3.3.4.3. DEPTHOF INVESTIGATION. The depth of investigation of even the two-detector density sonde is fairly small, the principal consequence being the occasional residual effect of mudcake or standoff. As in the case of neutron measurements, invasion seldom produces much error in the log output because the density of the mud filtrate is usually close to that of the connate liquid. However, the effect of filtrate invasion into gas-bearing formations is significant. One published measurement of pseudo-geometric factors was performed in the same environment described for neutron sondes in Section 3.2.6, i.e., water invasion of air-filled porosity in a 35%porous quartz sand. This shows a 90%-point depth of investigation for this low-density (2.07 g/cm3) formation as 5 in. (1 3 ~ r n ) . 4Other ~ measure¶

-

19. GEOPHYSICALWELL LOGGING

559

ments, using sondes of presumably different design, result in reported values of 7 cm and 11 cm for a 2.7-g/cm3 formation when spacings of 15 cm and 32 cm, respectively, were used.53Since neither the sonde design details nor the formation arrangement are available for the latter measurements, no valid comparisonwith the result reported above can be made. In the author’s experience, conventional sondes with principal spacing in the range 35 40 cm generally have depths of investigation of about 5 to 7 cm in formations of this density.

-

-

3.4. The Gamma-Ray Photoelectric-Absorption Method

The use of photoelectric absorption in measuring formation average atomic number, which then contributes to the identification of lithology, was introduced in Sectiun 2.4. This measurement is made simultaneously with a density log, each utilizing a different part of the spectrum of the gamma-ray current incident on the far detector. The pressure housing is fitted with a beryllium window in front of the detector, so the complete low-energy portion of the spectrum is available for analysis. The pulse-height spectrum produced by the scintillation detector is windowed in two energy bands. One window straddles the high-ehergy Compton, or “hard”, part of the spectrum; the other is set on the low-energy photoelectric, or “sofl”, part. The hard window, H, can be located between 180 and 540 keV, for example, thereby yielding a counting rate which is a function of density only, and the soh.window, S, from 40 to 8 0 keV, where the counting rate responds to both formation density and Z.63(Since all 2’s in this discussion are “effective”, the notation will be simplifiedby using Zin place of ZeE;see Section 2.4.) Taking the ratio S/H produces a quantity in which the pe dependence is effectively ~ a n c e l l e d Figure . ~ ~ ~42~ ~ shows ~~~ typical locations of the two windows on schematic spectra from three formations of different 2 value but the same pe.66 A similar picture is produced when spectra from formationsof direringpe are normalized in the Compton region, since the spectral shapes in the Compton energy band are the same (cf. Fig. 37). In practice, the position of the lower edge of the H window is determined by a trade-off between high counting rates and the requirement that photoelectric absorption not perturb the density measurement. The position and width of the S window can be chosen so as to optimize the accuracy of the Z measurement with respect to counting statistics and instrumental drift. Measurements of S/H ratios in laboratory mock-up formations spanning a wide range of 2 and p~ values are shown in Fig. 43. The choice of the

560

JAY TITTMAN

CounvSec’keV

Region of Photoelectric Effect ( p and 2 Information

/

Region of Compton Scattering ( p Information Only)

(Low Z) (Med Z) (High Z)

Energy

FIG.42. Schematic spectra from formations of the same density but having different Z values. Sindicates the “soft”window and Hthe “hard.” The small peak at 662 keV represents the weak CsI3’ reference source placed on the scintillation crystal. [Adapted from W. Bertozzi, D. V. Ellis, and J. S. Wahl, Geophysics46, 1439 (1981).]

abscissa variable is based on the theoretical surmise* that S/H has the form6354 -=-

H

A +c, T+B

(3.47)

where B is a constant related to the average energy of the S window and C represents a background counting rate in the S window, independent of Z and A.The background arises from a scintillation-detector effect know as the “Compton tail” (see Section 3.5.1.3). The freedom of S/H from dependence on p~ in the usual range of interest is confirmed by the variety of densities covered in the data of Fig. 43. 3.4.1. Gain Stabilization. A noteworthy feature of the instrumentation is the high degree of gain stabilization required. Figures 42 and 37 show that some of the window edges are located on rapidly varying parts of the spectra. Thus, relatively small gain drifts in the scintillator, photomultiplier tube, or amplifier produce significantcounting-rate changes, especially in the S window, and it is necessary to provide a gain-stabilizationsystem that compensates for gain changes anywhere in the detection chain. This is achieved by

* Equation (3.47) is shown in references 63 and 64 to be correct for the infinite-mediumwith-uniformly-distributed-source(IMUDS) problem. The “surmise” enters in applying it to the more difficultpoint-source problem. The derivation of Eq. (3.47) for the IMUDS problem consists of a calculation of the fluxes expected in the hard and soft windows. Equation (3.42) is thestartingpoint, withsimplificationresultingfromuseofthefactsthatV j = OintheIMLJDS case,and S(E) = S(E - Eo).

-

19.

561


A 1.0

-

0.9 0.8

-

0.7 -

S -

”

0.6 -

0.5 -

3 AlMg (2.81)

0.4 -

4 Al (2.57)

0.3 -

5 SiC/Epoxy (1.83)

2

14

6 Si02/Epoxy (1.36)

7 Water (0.358) (Corrected for Z/A)

0.1 0.5

1 .o

1.5

-

1/(?+B)

FIG.43. Experimental values of sofi/hard ratio measured by a photoelectric-effect sonde in laboratory mock-up formations of known ‘5 and density. The two data points near the intercept were taken in barite-loaded muds. The abscissa scale, omitted on the original, has been inserted by the author. It is consistent with the value B = 0.39.Then Eq. (3.47),with A = 0.69 and C = 0.105, describes the straight line drawn through the data. [Adaptedfrom D. Ellis, C. Flaum, C. Roulet, E. Marienbach, and B. Seeman, SOC.Pet. Eng. 58th Ann. Fall Tech. Conj, Oct. 5 -8, 1983,San Fransisco, paper number SPE 12048.Copyright 1983 SPE-AIME.]

irradiating the scintillator with a small CsI3’ source that provides the reference signal for an automatic control loop.61As shown schematicallyin Fig. 42, two voltage-fixed windows are placed so as to straddle the pulse-height peak produced by the reference source. The feedback control signal, generated by any difference between the counting rates in the windows, alters the photomultiplier high-voltage so as to compensate for a gain change anywhere in the detection chain. 3.4.2. Mudcake Effect. The most important environmental perturbation on the photoelectric measurement is the mudcake effect. When Z,, < Zfomti0, the mudcake influence is small, in general, and is usually ignored. However, when 2, > Zfomtio,the effect can range from moderate to overpowering. This occurs when a mud-weighting material such as barite is present. Two correction methods have been used at various times.

562

JAY TITTMAN

One method treats the mudcake as an absorptive sheet covering the borehole wall and uses an independently determined relation between the cor(See Section 2.4 for the definition of U.) rection AT,and the product Vmctmc. V,, is calculated from knowledge of the mud solids present and the assumption of a value for mudcake porosity, usually -50%. The simultaneous caliper log or & is used to estimate tmc. The second method depends upon the special character of the T-versus-E curve in the vicinity of the K absorption edge.67Figure 44 shows the linear absorption coefficient for a very heavily barite-loaded mudcake in this energy region. The high absorption just above the Kedge at 37 keV produces a valley in the spectrum incident on the detector. The “transmission window” just below the Kedge produces a peak, which may contain K-shell fluorescence photons from the barium also. Thus, by taking the ratio of the counting rate Pin a spectral window straddling the peak (e.g., 24- 33 keV) to the rate Vin a window straddling the valley (e.g., 39 - 5 1 keV) an indicator, P/ V, of mudcake effect is established. Laboratory measurements made under a variety of conditions then permit the creation of a cross-plot, S/H versus P/ V,that contains curves of constant 2and constant mudcake parameter.67 This technique has been used successfully for tmc-valuesup to -4 in. 100

E

V

t c

10

C

.-00

g 0

V

0.1

ENERGY (keV) FIG.44. Photoelectric linear absorption coefficient for a heavily barite-loaded mudcake. [From D. C. Moore and J. Tittman,U.S.Patent 3,858,037, 1974.1

19.


563

(9.5 mm) in muds lightly weighted with barite. Unfortunately, mudcakes heavily loaded with barite, often only -$ in. (6 mm) thick, produce such

severe absorption that the low-energy part of the spectrum is essentially obliterated and photoelectric measurements of the formation cannot be made reliably. 3.5. Gamma-Ray Spectrometric Methods

An overview of spectrometric methods was given in Section 2.6. Here we describe the technique in somewhat more detail, leaning on the material in Sections3.2 and 3.3. An adequate foundation for understandingthe physics involved in natural-gamma-ray spectrometry is found in Sections 2.6.2 and 3.3.2. Consequently, we limit ourselves here to the subject of neutron-induced gamma-ray spectrometry. Although this technique currently finds its principal application in cased holes, where resistivity methods are unable to measure oil saturation, its potential for element and mineral analysis in open-hole formation evaluation is just beginning to be realized.68 3.5.1. Relation of Spectra to Formation Properties. Consider a point source of monoenergeticfast neutrons in an infinite homogeneousisotropic medium. We calculate the magnitude and spectrum of the unscattered gamma-ray flux at a detector spaced a distance C from the neutron source. The gamma rays are produced by inelastic neutron scattering and by the eventual capture of neutrons thermalized in the medium. These calculated unscattered spectra closely represent the spectra separately measured by the use of pulsed neutron sources and time-gated detectors introduced in Section 2.6.1. 3.5.1.1. CAPTURE SPECTRA.We examine first the development of the capture spectrum. Figure 45 illustratesthe geometry of the problem in cylindrical coordinates. The rate of production of capture gamma-rays per unit volume at (r, z) is

w h e r e 2 is the thermal neutron flux, Ni is the number of atoms of the ith kind per unit volume, b8,is their thermal-neutron absorption cross section, and vt(Ey)is the number of gamma rays of energy Eythat the ith element produces per neutron capture. This product is just the spatially distributed gamma-ray spectral source. The gamma-rays are attenuated (almost exclusively by Compton scattering) on their way to the detector by the factor exp[-pm(E,)dr2 4n[r2

+ (4 - z)’I

+ (C -

z)2]

564

JAY TITTMAN

FIG.45. Geometry for calculation of unscattered gamma-ray flux at the detector in a neutron-induced-spectrometrysonde. Induced gamma-rays are produced in unit volume element at (r, z ) and detected at D a distance C above the neutron source S located at the origin.

Multiplying the production rate by the attenuation factor and integrating over all space yields

for the unscattered gamma-ray spectral flux at the position of the detector, produced by thermal-neutron capture. The double integral can be replaced by the symbol W(D2,L,, L, E y ,h),where the neutron transport parameters appearing in the argument come from Eq. (3.35) and Eyenters through its effect on pm.This makes explicit the fact thatc(E,) is a function of the neutron transport parameters and h.Furthermore, it shows that the dependence on Eyoccurs not only through the obvious linear factor vi(Ey),but also through pm(Ey)in the exponent of the integrand. The integrand in Eq. (3.48) plays a role similar to that of a generalized geometricfactor. [See Eq. (3.2 l).] Except for two brief discussions,C9the dependence ofe(E,,) on neutron and gamma-ray transport parameters does not appear to have received any attention in the English-languagelogging literature. In a single formation the ratio of the intensities of two gamma-ray lines

19.


565

with energies Ey, and Eyk,characteristic of two different elements i = p, q, respectively, becomes

Since the formation is fixed, the only variable in the argument of W that is different in numerator and denominator is EyeIf the gamma-ray energiesare (See Fig. 35.), and W approximately not too far apart, pm(EYj) pum(EY,) cancels out of Eq. (3.49). This leaves the ratio NJNq directly determinable from a measurement of the ratio of the detected line intensities. Values for a, and v(Ey)for most of the elements can be found in the physics literature, if necessary. In logging practice, however, the proportionality constant between thef-ratio and the N-ratio is, in effect, determined empirically in a laboratory mock-up formation. (Although these measurements are usually performed to provide corrections for borehole and other perturbing environmental effects, they implicitly include determination of the proportionality constant.) However, when this is done it is no longer necessary to assume , pm(EyJbecause the W's are automatically included in the that p m ( E y= experimental determination. What is apparent from the dependence of W on the neutron transport parameters is that the proportionality constant In practice variesfrom oneformation to another even if pm(E,,,= pm(Eyk). this, too, is normally accounted for by empirically calibrating, in known formations, the measured ratios of spectral intensities against varying values of descriptors of interest. 3.5.1.2. INELASTIC SPECTRA. The line of reasoning for calculating the inelastic gamma-ray spectrum is the same as for the thermal, but a few changesmust be made in Eq. (3.48): (1) a, is replaced by o,(E,), the inelastic scatteringcross section as a function of neutron energy. (2) v(Ey)becomes the average number of gamma rays produced per inelastic neutron-scattering, v,(E,, E,,), which depends on neutron energy as well as upon Ey. (3) The energy dependence of the fast-neutron flux fL(E,) is made explicit because both ai,and v, are functions of neutron energy. (4) An integration must be performed over neutron energy. Then the expression for the inelastic gamma-ray spectral flux at the detector becomes I :

where E,is the neutron source energy. Here we write the triple integral as

566

JAY TITTMAN

&(neutron parameters, PB,E,).* The subscript i is required on W now because Oinj and Vini are in the integrand, making W element-specific. Then the analog of Eq. (3.49)is (3.51)

Here again, sondes may be calibrated in a set of laboratory formationswhose properties span the ranges of the parameters in the argument of W.This is the equivalent of determiningthe proportionality constantsbetweenfratios and Nratios. Monte‘Carlocalculationscan be carried out for the same although their use has been rather limited at this writing. Since these calculations pertain only to the unscattered component of the total gamma-ray spectral flux at the detector, they ignore the effects of Compton scattering in the transport medium. [That the pm’sin Eqs. (3.48) and (3.50) are predominantly Compton in geological materials of usual interest may be seen in Fig. 35.1 Some of these scattered gamma-rays, especially muItiply scattered ones, appear in the detected flux spectrum. This is manifested as a buildup ofthe continuous spectrum toward low energies, but the effect is not so pronounced as in density and natural gamma-ray spectrometry logging because only the energy region above 1.5 MeV is used. (See footnote in Section 2.6.2.) Nevertheless, every unscattered spectral line at the detector is superimposed on the degraded continuum from all sources at higher energy. This degradation occurs in the gamma-ray flux spectrum itself, and should not be confused with the instrumental Compton-tail effect discussed in the next section. 3.5.1.3. SCINTILLATION SPECTROMETRY7’. The actual Output Of the spectrometeris only a poor representation of the gamma-ray spectrum at the position of the detector, even in the idealized case we have been examining. We now briefly consider the reasons for this, since problems caused by instrumental degradation of spectra are of importance in sonde design, data processing, and log interpretation.

-

Photoelectric and Pair Peaks. If a gamma ray suffers one or more Compton scatterings in the scinti~Iutioncrystaf and is then photoelectrically absorbed in the crystal, all its energy is converted into a single light flash. (We ignore the small amount which may be lost through recoil electrons that escape the crystal). This is the maximum-intensity light flash possible for gamma-rays of that particular energy. When transformed into a voltage pulse at the output of the photomultiplier,this flash contributesone count to

* We use the term “neutron parameters” hereto representthe result ofthe integration over En since the parameters D,and L, are energy dependent and do not appear explicitly after the integration.


LOGGING

567

thefull-energy peak in the pulse-height spectrum. (It is this peak that determines the commonly used scintillation energy scale, which really refers to deposited energy rather than gamma-ray energy, as will be seen below.) Another contribution to the full-energypeak comes from pair production in the crystal. This interaction converts all the gamma ray’s energy into the mass-plus-kinetic energy of the electron and positron. In general, the electron transfers all its kinetic energy into light as it slows down and is eventually trapped. The positron does likewise until it annihilates with an electron in the crystal. The annihilation produces two 0.5 1-MeV photons which, in turn, may be photoelectricallyabsorbed in the crystal. In the event that both annihilation quanta are absorbed, the total energy of the initialgamma-ray is converted into a single lightflash. (Only one light flash is registered because all these processes produce excitation in the crystal in a time -=sz0.25 ps, the lifetime of the excited state that emits the light.) But this is the same result as occurs for photoelectric absorption, and the light flash so produced has the same intensity. Hence, these pulses contribute, also, to the full-energy peak shown in the schematic spectrum of Fig. 46. When one of the annihilation photons escapes the crystal without interaction, the amount of energy converted to light is 0.5 1 MeV less, and a oneescape peak appears 0.5 1 MeV lower on the energy scale. Similarly, when both annihilation photons escape the crystal without scattering, a two-escape peakis formed 1.02 MeV below the full-energypeak. The three peaks, shown in Fig. 46,are referred to as thepairpeaks, even though the full-energy peak contains a component arising from Compton scattering-plus-photoelectricabsorption. The Compton Tail. In NaI, the nearly universally used logging scintillator, the Compton and pair-production mass-absorption-coefficient curves cross at about 6 MeV, roughly the middle of the energy range of interest. Thus, eyery capture and inelastic gamma ray interacting in the crystal has an appreciable probability of making a Compton scattering. Consider what happens when a scattered gamma-ray escapes the crystal after a single scattering, in contrast to the eventual photoelectric absorption discussed above. The recoiling electron usually transfers all of its kinetic energy into the light flash. (If the scattering occurs sufficiently close to the crystal surface to permit the electron to escape, then only a fraction of its kinetic energy is converted.) Because energy transferred to recoiling electrons varies with the scattering angle, the electrons have a continuous energy distribution ranging from zero up to the maximum permitted by Eq.(3.40)J2Thus, even monoenergetic gamma rays produce a continuous spectrum of light-flash amplitudes extending down to zero. An idealized picture of this Compton tail in the pulse-height spectrum resulting from the single scatteringof mono-energetic gamma rays is shown schematicallyin Fig. 46.The addition ofsecond,

568

JAY TITTMAN OneEscape Peak

TWOEscape Peak

I

I

I

FullEnergy Peak

Tail Cornpton Edge -1 1

Pulse Height (Volts)

I

E~

FIG.46. Schematicrepresentation of NaI scintillationspectrometer response to monoenergetic gamma-rays of energy Ey 2 2 MeV. Dashed lines show idealized pulse-height spectrum produced by pair, Compton,and photoelectric interactions.Solid curve illustrates typical result of convolution of the idealized response with the instrumental resolution function.

third, etc., scatterings, followed by escape of the scattered gamma rays, produces pulses that may fall between the sharp Compton edge and the full-energy peak. Since gamma-rays of all energies can produce Compton tails, the pair peaks of every spectral line are superimposedon the sum of the tails generated by every gamma ray of higher energy. For borehole scintillation spectrometry this is a major obstacle to the accurate measurement of emission-line intensities. System Resolution. The development of the peaks and Compton tail make the single-crystal scintillation spectrometer inherently a one-to-many device, i.e., mono-energetic gamma rays produce many different pulseheights. Nevertheless, because of the regularities observed in Fig. 46, even this would not prohibit accurate determination of line intensities when many lines are present. However, the idealized response to mono-energetic gamma rays, shown dashed in Fig. 46, must be convolved with the system resolution of the ~pectrometer.’~Typically, for the spectrometers and gamma-ray energies under discussion here, the resolution function is moreor-less Gaussian and 2 5% wide at half-height. A discussion of the factors that produce the resolution function is outside the scope of this article, but the solid curve in Fig. 46 shows schematically the result of the convolution, even for mono-energetic gamma rays. The combination of instrumental effects and Compton degradation in the formation produces real pulse-height spectra such as those in Fig. 47. The degree of spectral degradation occuring can best be appreciated by observing that most of the lines have widths < 1 eV when emitted. (However, note


LOGGING

569

that a linear ordinate scale in Fig. 47 would make visually clearer the spectral character that does remain.) 3.5.2. Measurement Technique. In general, sondes measuring neutroninduced spectra employ sealed-tube ion acceleratorsas the neutron source.73 These accelerate deuterium ions across a high-voltage(- 100kV)gap onto a tritium-loaded target. The neutrons produced by the (D, T) reaction are

8

E, (MeV) FIG.47. Multichannel pulse-height spectra recorded by a sonde in a laboratory mock-up formation consisting of oil-saturated quartz sand. The 10-in. (25-cm) borehole was cased. Identifiable peaks are marked by the elements producing them. Unprimed symbols designate full-energy peaks, primed indicate one-escape peaks, and double-primed two-escape. (a) Capture spectra and (b) inelastic spectra. Peaks from oxygen caused by inelastic scattering (n,n') and fast-neutron reaction (n, a) are separately marked. Approximate locations of carbon and oxygen windows used in the broad-window-ratiomethod are shown hatched. [Adapted from R. C. Hertzog, SOC.Pet. Eng. J. 20, No. 5,327 (1980), Copyright 1980SPE-AIME; D. W. Oliver, E.Frost, and W. H. Fertl, SPWLA 22ndAnn. LoggingSymp.Trans., Mexico City, June 23 -27, 1981, Vol. 2, paper TT.]

570

JAY TITTMAN

emitted nearly isotropically at approximately 14 MeV. Three properties of these sources make them exceptionally well suited for neutron-induced gamma-ray spectrometry in the borehole: (1) The 14-MeV neutrons have sufficient energy to excite inelastic gamma rays from carbon and oxygen, and to adequately penetrate into the formation through casing and cement. (2) They can be pulsed, thereby permitting the separation of inelastic and capture spectra by time-gated detection. (3) At acceleratingvoltages achievable downhole they generate at least an order of magnitude greater neutron output than conventional encapsulated sources in current field use. Although, in addition, nearly all these sondes employ NaI scintillation detectors as the basic spectrometer element, two radically different approaches are used in the treatment of the data. We will refer to them as the broad-window-ratio method and the spectral fitting method. However, as will be seen in the ensuing sections, the two embodiments of these methods currently in the field exhibit more differences than the names imply. Both respond to inelastic and, separately, to capture gamma-ray spectra. 3.5.2.1. THE BROAD-WINDOW-RATIO METHOD74. Sondes using this technique operate on a repetitive cycle 50 ,us long, as shown in Fig. 48. The neutron source is pulsed on for 5 to 8 ps, during which time the scintillation detector is gated on also. Since it generally takes about 5 -25 ps for neutrons to thermalize in materials found in the borehole environment, this timing permits the segregated acquisition of inelastic spectra. If only one burst were used, this would be adequate for complete spectral separation. However, the thermal-neutron population decays with a time constant in the range of roughly 100- 1000,us, so each inelasticdetection gate contains a background consisting of thermal and activation spectra created by the capture of neutrons emitted by earlier source bursts. To correct for this a detector gate is opened halfway between bursts in order to record the background spectrum, which is totally free of inelastic gamma rays. This is then subtracted from the raw inelastic-gate spectrum to provide a “net inelastic” spectrum. In addition, the system outputs the background as a capture spectrum. The activation contribution to this gate is said to be small, and is ign~red.’~ The output pulses from the scintillation detector are amplified and transmitted directly up the cable in analog form. At the surface their amplitudes are digitized by a pulse-height analysis system. The resulting digital spectra are then processed, as described below, into logging curves representing element ratios such as C/O. Spectra can also be accumulated and dumped onto tape at selected intervals, such as 1 ft (30 cm), for later data processing. Capture peaks from hydrogen and iron are used to monitor the calibration and linearity of the gamma-ray energy scale. Although taped spectra are recorded in 256 pulse-height channels, the continuously recorded log uses the ratios of counting rates in broad pulse-

19.

57 1


INELASTIC DETECTION GATES 5 lo 8 p SEC

U

n

U

n

U

n

U

n

BACKGROUND DETECTION GATES 5 to 8p SEC NEUTRON SOURCE BURST 5 lo 8 p SEC

1

0

100 200 TIME, MICROSECONDS ( p SEC)

FIG.48. Timing cycle used in the broad-window-ratiomethod. [FromD. W. Oliver, E. Frost, and W. H. Fertl, SPWLA 22ndAnn. Logging Symp. Trans., Mexico City, June 23-21, 1981, Vol. 2, paper TT.]

height windows. Figure 47 shows approximate positions of the oxygen and carbon windows on a net inelastic spectrum. If there were no borehole contributions and if the windows responded only to gamma rays from the elements for which they are named, this method would exhibit the dynamic range of the C/O atom-ratio curves in Fig. 11. However, the carbon window contains gamma rays resulting from a fast-neutron (n,a) reaction with oxygen, as well as inelastic gamma rays from silicon and calcium (Fig. 49) and iron.76In addition, Compton tails produced in the crystal principally by oxygen inelastic gamma rays contribute to the counting rate in the carbon window. There are two distinct consequencesof these facts. First, the broadwindow ratio, also referred to (confusingly) as C/O, has a severely reduced dynamic range. Thus, both systematic and random errors in the measured ratio are magnified as they are propagated into errors in estimates of the C/O atom ratio or oil saturation. This can be seen clearly by comparing the ordinate scales in Figs. 50 and 1 1. Second, the measured ratio is susceptible to variations in the concentrations of elements other than C and 0 as the sonde passes from one formation to another. In the actual borehole environment the measured C/Oratio is also influenced by oil or water in the casing and elements in the cement, including oxygen. In practice, some of these effects, both intrinsic and environmental, may be removed by referenceto chartsbased upon laboratory measurements in different borehole sizes, casing sizes, lithologies, fluids, e t ~The . ~use ~ of these charts requires either local knowledge or independent measurement of the perturbing variable.

300

572

JAY TITTMAN

t

Relative ?-Ray Yield

E,(MeV) + FIG.49. Emission line spectrum induced by fast neutrons in a medium containing typical elements of interest. Intensities are calculated for an oil-saturated 36% porous rock whose matrix is halfquartz sand and halfcalcite.[FromR. C. Hertzog, SOC.Pet. Eng. J. 20, No. 5,327 (1980). Copyright 1980 SPE-AIME.]

In addition to the C- and 0-window measurements, this system sets windows covering the bands 1.54- I .94 MeV and 2.5 - 3.3 MeV, where inelastic gamma rays from silicon and calcium, respectively,fall. Counting-rate ratios from these windows may be recorded as representations of Si/Ca or Ca/Si atom ratios. Similarly, outputs from windows set on the capture spectrum at 3.17-4.65 MeV for Si and 4.86-6.62 MeV for Ca can be used. Both methods provide information on the composition of the rock matrix. In addition, they sometimes are able to supply the independent measurements needed to correct for Si and Ca interference in the C and 0 windows, e.g., through a cross-plot of logged measurements of C/O versus inelastic Si/Ca. The inelastic measurements of Si/Ca or Ca/Si bear the burden of small dynamic range and interference from carbon and oxygen, but are independent of fluid salinity. The capture ratios have roughly twice the dynamic range and are free of carbon and oxygen interference, but suffer from interference by chlorine and iron capture gamma rays. Depth of investigation for the C/O and inelastic Ca/Si measurements are reported to be approximately equal.” For water invasion of a high-porosity oil sand the 90% point is given as about 8.5 in. (22 cm) for a sonde of

19.

0

573


20

10

Porosity,

30

YO

FIG. 50. Response of broad-window ratio to oil saturation and porosity in sand and in limestone. Measurements were taken in a *in. (16.8-mm) well bore with fresh water in the borehole.Note large offset and small dynamic range ofthe measuredratio. [From D. W. Oliver, E. Frost, and W. H. Fertl, SPWLA ZZndAnn. LoggingSymp. Trans.,Mexico City, June 23-27, 198 1 , Vol. 2, paper TT.]

unspecified design. Five-minute station measurements can be made or continuous logging at speeds s 180 ft/h (55 m/h). 3.5.2.2. THESPECTRAL FITTING METHOD.The timing cycle and some other aspects of tools using this method fall naturally into two classes, one for capture gamma-ray spectrometry and the other for inelastic. Therefore, we will treat them separatelyeven though the spectral-fittingfeature is common to both. For reasons that will become clear in the discussion, the two classes of operation are referred to as the capture-2 mode and the inelastic mode. 3.5.2.2.1. The Capture-7 Mode. If only capture spectrometry is to be performed, and not inelastic, there become available several sonde design avenues that lead to significant advantages. These include faster logging speed, reduced borehole effect, reduced inter-element interference, and increased measurement precision. The timing cycle used in the capture-z mode depends upon the nearly exponentialdie-away ofthe thermal neutron population in the vicinity ofthe

514

JAY TITTMAN

sonde. A complete discussion of this phenomenon and its measurement is a separate subject that we will not cover.78However, a brief description of the scafe-factormethod of measurement will be given because it is necessary for understanding the time-base used for spectrometryin the capture-7 mode. It can be shown that the die-away time constant 7* can be determined by taking the ratio of the counting rates in two properly placed time gates of ~ * first gate opens 22 after the neutron source-burstends, width 2 and 2 ~ .The as shown in the lower sequence of Fig. 5 1. In general, the delay of 22 allows the thermal-neutron population inside the borehole to decay substantially before the measurement of the exponential die-away in the formation is begun. The measurement time-baseitself is clocked in units of 7 by the use of a variable-frequency oscillator controlled by a feedback loop. Since the ratio of the counting rates in gates I and II equals 2 when thegates are placedon an exponential with time constant 2, the control signal depends on the difference between the “currently” measured ratio and the value 2. This difference drives the clock frequency up or down, as required, until the latter has the value l/z, at which time the control signal drops to zero. Thus, the equilibrium value of the oscillator period is equal to the T of the formation being traversed. In performing this measurement, all the counts in the pulse-height spectrum produced by the detector are used. It can be shown that the sequence illustrated in Fig. 5 1 is approximately optimized for maximum precision with respect to counting statistics. The subcycle of width 62 occurs a total of eight times, the counts in gatesI and I1being accumulated separately. Finally, gate I11is opened for 122. This gate measures the background, which is essentiallyconstant over the cycle period of 622. The background consists principally of a 7-s activity induced through a fast-neutron reaction with oxygen and a 25-min activity from iodine in the crystal. Appropriate fractions of the contents of gate HI are subtracted from the accumulated counts in gate I and in gate 11, yieldingthe net counting rates whose ratio governs the clock frequency. We turn now to the upper timing program of Fig. 5 1. This presents the parallel cycle used in capture spectrometry itself. Here the “capture gate” is opened after a delay of only 2, a compromise between high counting rates and small borehole influence. The subcycles occur a total of eight times, as for the 7 measurement, followed by a background gate 137long. After a dead interval of length 7 the whole cycle is repeated. The gamma rays detected in the capture and background gates are pulse-height analyzed downhole and accumulated in separate multichannel memories. The contents of the memories are digitally telemetered uphole periodically, e.g., every 6 in. (15 cm).

* This T should not be confused with the symbol for the photoelectric absorption cross section.

575

19. GEOPHYSICAL WELL LOGGING NEUTRON CAPTURE

- T

SPECTROMETRY

BACKGROUND 61

MULTIPLES flF T

SIMULTANEOUS T

MEASUREMENT

____t

-

111

61

OF T FIG.5 1 . Timing cycle for capture-z mode. Lower diagram uses the scale-factor method to determine thermal-neutron die-away time constant T. Upper diagram shows the simultaneous positions of the gates used for capture gamma-ray spectrometry. [From P. Westaway, R. Hertzog, and R. E. Plasek, SOC.Pet. Eng. 55th Ann. Fall Tech.Con& Sept. 2 1 24,1980, Dallas, paper number SPE 9461. Copyright 1983 SPE-AIME.] MULTIPLES

-

There, the background spectrum is appropriately subtracted in the computer, yielding a net capture spectrum similar to that shown in Fig. 47. The complete cycle of 62.r is repeated roughly 20 - 200 times per second, depending on the value of z measured by the system. The influence of borehole or casing fluid on the recorded capture spectrum can be further reduced by surrounding the pressure housing with a sleeve containing a thermal neutron absorber such as boron. The presence of the absorber forces a faster die-away of the thermal-neutron flux in the borehole, thus making the d o n g delay even more effective. The combination of the delay and the absorbing sleeve cuts the sensitivity of the spectrum to borehole fluid roughly in half, the exact amount depending upon hole size and fluid ~alinity.’~ It is the processing of the net capture spectrum that most clearly distinguishes the spectral fitting method from the broad-window-ratio method. The ~pectral~tting method is an optimal estimation procedure consisting of . matching the measured spectrum, in a least-squaressense, with the sum of a set of basis spectra, or standards, representing the sonde’s response toformations consisting of pure elements. The element yield coeflcient, or weight, of each standard in the linear superposition measures that standard‘s contribution to the total spectrum. Given the set of basis spectra, a weighted leastsquares procedure permits the construction of a matrix that operates on the net borehole spectrum (treated as a vector) while logging. This operation produces the yield coefficients for each ~tandard.’~ Under ideal conditions the ratios among these weights correspond to the left-hand side of Eq.(3.49) and are, thus, proportional to the ratios of atom concentrations among the elements represented in the set of standards.

576

JAY TITTMAN

If significant contributions to the logged spectrum are made by elements not in the basis set, errors result. However, in practice this is not a serious problem because the set of standards usually includes the important elements in the environment. If necessary, the taped spectra can be refit, after logging, with a set of basis spectra including other elements whose presence is suspected. When the set of standards contains elements that are absent or very weakly present downhole, the statistical uncertainty in the estimates of all the fitting coefficientsis unnecessarily increased, although the estimates themselves are unbiased. This situation, too, can be improved by refitting after logging, using a set of basis spectra from which the undesired standards have been removed.E0Similarly, standard pulse-height spectra that have been appropriately smeared can be used for post-log fitting when temperature effectson the scintillator require it. A set of capture basis-spectra is shown in Fig. 52. These are developed by using the sonde to measure spectra in laboratory formations containing a high proportion of the element in question. Then the contribution from

‘

L k

M

1

I

0

1

2

3

Fe

4

5

6

7

6

Gamma Ray Energy (MeV) FIG.52. Typical capture-gamma-raybasis spectraused in the spectral fitting method. [From P. Westaway, R. Hertzog, and R. E. Plasek, Soc. Pet. Eng. 55th Ann. Fall Tech. Con$, Sept. 21-24, 1980, Dallas, paper number SPE 9461. Copyright 1983 SPE-AIME.]

19.


577

extraneous elements, activation, etc., are stripped out, leaving the desired pure basis-spectrum. Relatively featurelessdownhole spectra, such as that in Fig. 47, are capable of yielding surprisingly precise estimates of element ratios if the basis spectra are sufficiently orthogonal, i.e., do not have many characteristicfeatures such as peaks or valleys at the same energy. When this condition does not prevail, evidence of cross-correlation appears in the spectral linear estimators representing the affected elements.76 If the W factors in Eq. (3.49) were independent of E,,, the spectral fitting method would be completely free of line interference and the effects of Compton tails in the crystal. The basis spectra would be, in fact, complete signatures of the standard elements. Although two lines from different elements might be close to one another, producing cross-correlation that reduces statistical precision, no systematicerror would result from their proximity. Since the W’s are not exactly equal, the measured intensities of individual lines are not in exactly the same ratio as the emitted intensities, even for a single standard. Taken by itself, however, even this would not cause concern because the measured spectrum would still be the signature of the standard. But each basis spectrum, created as described above, is not exactly the one to be used in the analysis of formations with different trunsport properties. The usual use of element yield coefficients makes the tacit assumption that the consequences of this are ignorable. The effects do not appear to be serious at porosities above about 15% or 20%, presumably the condition under which the basis spectra were measured. However, they do require special attention in low-porosity (high-density) formation^.^^ The whole question of the influence of Won the spectral fitting method is not resolved theoreticallyin the literature at present. However, as with other logging methods in which departures from an idealized theoretical model occur, empirical calibration providesa practical solution. For the problem at hand it is customary, first, to refer to the measured element-yield ratios as indexes, or indicator ratios: Cl/H is the salinity-indicator ratio SIR, H/ (Si Ca) the porosity-indicatorratio PIR, Si/(Si Ca) the lithology-indicator ratio LIR, etc. Then, for example, plotting laboratory measurements of PIR versus 4 in different size boreholes provides an empirical calibration of PIR for field m e a ~ u r e m e n t s .The ~ ~ .influence ~~ of borehole salinity on SIR is calibrated in a similar manner, as can be done for other perturbations and other indicator ratios. If necessary, this procedure can even be used to override the Weffect by calibratingthe indicator ratios with respect to formation constituents, e.g., rock matrix minerals. Although no studies of pseudo-geometric factors appear in the literature, depth of investigation is reported to lie in the range 8 - 12 in. (20 - 3 1 cm).80 Unfortunately, both the definition, i.e., 50%-point or 90%-point, and the environmental conditions are unspecified. It is presumed that the 90Yo-point is intended.

+

+

578

JAY TITTMAN

3.5.2.2.2. The Inelastic Mode. Although this mode ofoperation takes its name from its timing cycle, which is designed primarily for inelastic spectrometry, capture spectra also are recorded between neutron bursts, in a manner similar to that used in the broad-window-ratio method. Figure 53 illustrates the timing program. The 20-ps background gate follows the neutron burst immediately, permitting acquisition of a background spectrum that includes capture gamma-rays produced by neutrons from that burst, as well as from earlier ones. This is necessary because the inelastic detection gate is on long enough for some of the neutrons to be thermalized and captured before the gate is closed, in contrast to the situation prevailing in the broad-window-ratiotool. The “late capture” gate is open for 44 ps to record the capture spectrum, free of inelasticgamma rays. This cycle is repeated lo4 times per second. As in the capture-7 mode, the outputs of the three gates are pulse-height analyzed downhole, accumulated, memorized, and periodically telemetered to the surface computer. There the background spectrum is appropriately weighted and subtracted from the burst spectrum to yield the net inelastic spectrum. A weighted least-squares fitting procedure is then carried out as described in the preceding section, but in this case inelastic basis spectra from C, 0,Si, CayFe, and S are used. Typical basis spectra are shown in Fig. 54. The only element yield coefficients that are normally used for interpretation, however, are those of C and 0. The relatively featureless spectra of the other standardsproduce yield coefficientswhose precision is inferior to those of capture spectrometry for Si, CayFeyand S. Nevertheless, the use of these elements in the inelastic spectral fitting reduces substantially systematic errors in the C/Oestimate that would otherwise be produced by their presence in the environment. Comparison of an element-yield-ratio plot of C/O versus 4, based on NEUTRON BURST INELASTIC MODE

I

I 0

TIME

-

I

/ 0

20

[-\

BURST

CAPTURE

40

BKGND

84

LATE CAPTURE

I

too

Lf INELASTIC h CAPTURE

Y-RAY

co‘UNT RATE SPECTRAL GATES

(FSBC)

100

I 1

BURST

FIG.53. Timing cycle for the inelastic mode of the spectral fitting method. Middle diagram representsschematically the time-evolution of gamma rays from inelasticand thermal capture interactions. [From P. Westaway, R. Hertzog, and R. E. Plasek, Soc. Pet. Eng. 55th Ann. Full Tech. Cant, Sept. 21-24, 1980, Dallas, paper number SPE-9461. Copyright 1983 SPEAIME.]

19.


579

Gamma Ray Energy (MeV)

FIG.54. Typical inelastic-gamma-raybasis spectra used in the spectral fittingmethod. [From P. Westaway, R. Hertzog, and R. E. Plasek, SOC.Pet. Eng. 55th Ann. Fall Tech. ConJ, Sept. 21-24, 1980, Dallas,paper number SPE-9461. Copyright 1983 SPE-AIME.]

laboratory data,82with the actual-atom-ratio plot of Fig. I 1 shows the dynamic range of this measurement to be nearly as large as that of the actual atom ratio. (The residual difference arises primarily from the effect of the borehole on the measurements.) This feature can be misleading, however, because statistical uncertainties in the counting rates are propagated into uncertainties in estimates of oil saturation, So,with similar enlargement. Because of this, the spectral fitting method can impose statistical uncertainties on Soas large as or larger than the broad-window-ratio method. On the other hand, systematic errors produced by variations in the Si, Ca, and 0 inelastic gamma-rays in the broad C window are absent from the spectral fitting results. The comments in the preceding section concerning the Weffect apply to inelastic spectra also. In fact, because of an intrinsically shallower depth of investigation [5 - 10 in. ( 13 - 25 ~ m ) ]the , ~W ~ effect makes borehole-size

580

JAY TITTMAN

influence on inelastic measurements consistently greater than on capture.* In addition, at least partly because of the larger background in the inelastic spectra, longer logging times are usually required to achieve a given level of precision in the estimation of S, .Typically, continuous capture-z-modelogs can be made at a speed of 600 ft/h (183 m/h). Inelastic-modemeasurements are run at an effective speed of 30 ft/h (9.1 m/h) by averagingfive passes, each made at 150 ft/h (45.8 m/h), or by taking several-minute station measurements. Although the timing cycle of Fig. 53 permits the measurement of both inelastic and capture spectra simultaneously, the capture spectra are taken under less optimal conditions than in the capture-z mode. For example, the r-long delay before opening the detection gate in the capture-z mode is reduced here to only 20 ps, thus increasing borehole contributions to the inelastic-mode capture s p e ~ t r a . ~ ~ . ~ ~ 3.5.2.3. THE GERMANIUM SPECTROMETER. Borehole spectrometers using germanium-crystaldetectors have found only limited field application thus far, principally in coal and metals exploration, where boreholes are shallow and bottom-hole temperatureslow. However, the tremendous superiority in resolution over NaI scintillators suggests more widespread use in the f ~ t u r e . ~ ~ ~ ~ ~ The modes of interaction ofgamma rays with Ge are the same as with NaI, i.e., photoelectric,Compton, and pair. However, the ionization produced in Ge by the resulting recoil electrons is not converted to a light flash. Instead, under the action of an applied electric field, the ionizationproducts are swept to collecting electrodes mounted on the crystal surface, very much like the action of a gas-filled ionization chamber. A voltage pulse appears across the electrodes, with an amplitude that is proportional to the number of electronhole pairs created by the initial ionizing event. The number of pairs is, in turn, proportional to the energy deposited in the crystal by the interacting gamma ray. The downhole Ge spectrometer is capable of 0.1% resolution, which is to be compared to 2 5% for NaI in the energy range of interest in capture and inelastic gamma-ray spectrometry. This resolution is a consequence primarily of the large number of electron-hole pairs produced per electron volt deposited by the gamma ray. Two major technological obstacles have limited the field use of Ge spectrometry to date: (1) In order to achieve the low recombination rate and low level of conduction noise needed for the realization of the high intrinsic resolution, crystals must be operated at temperatures5 1 15 K. This requires the use of a freon or propane cryogen, usually frozen to liquid nitrogen temperature. The crystal and cryogen must be kept in a cryostat capable of maintaining this internal low temperature reliably for 10-20 hours while

-


58 1

exposed to bottom-hole temperatures that may range above 150°C. (2) Germanium crystals usable for logging have volumes 5 lo2 cm3, roughly one-tenth that of typical NaI crystals employed in borehole spectrometers. This limits the area exposed to the incident gamma-ray flux and the thickness available for interaction. Consequently, counting rates are low, implying slow logging speeds in order to achieve the desired statistical precision. Figure 55a,b shows a typical capture gamma-ray spectrum recorded by a Ge borehole spectrometer. To date, both capture and inelastic spectra have been recorded on a more-or-less experimental basis, using 14-MeV(D, T) or low-energy encapsulated CfzSzsources. Recordings have been made also of natural-radioactivity spectra, and artificial-activationspectrometryhas permitted the in-situ identification and quantitative analysis of some trace elements.85 It is apparent from the line resolution of Fig. 55a,b that the analysis of Ge spectra is more straighforward than for NaI. Computerized search routines can scan the spectra for pair-peak groups corresponding to known gamma rays from specific elements. Quantitative analysis is readily performed by subtracting the Compton tails, then determining the residual areas under the peaks. Naturally the W effect exists in Ge spectrometry, too, since it is a property intrinsic to the incident spectrum, not to the detector.

3.6. Sonic Methods

Short descriptions of the conventional measurement of interval transittime (ITT) were given in Sections 1.4.4 and 2.5. We now examine in somewhat more detail both the body waves (compressionaland shear) and waves produced by the presence of the cylindrical borehole wall. In addition, some aspects of sound-wave propagation unique to porous media will be discussed. However, it is outside the scope of this article to consider the influence of anisotropy, bed boundaries, and layering on sonic logging measurements. The reader interested in a more detailed treatment of vertical seismic the use of sonic methods for determining the quality of cement between casing and formation wallYa7 and the sonic borehole televiewerasis referred to the cited literature. 3.6.1. Body Waves. Body waves can propagate in unbounded media and are distinguishedfrom head waves (to be discussed in Section 3.6.2. I), which occur as a result of boundaries such as the borehole wall. The existence of body waves in an unbounded continuous isotropic elastic medium can be derived by applying Newton’s second law to a stressed infinitesimal volume

582

JAY TITTMAN

I

30000

20000

-

H

0

1 .o

2.0

Ey(MeV)

1000

I

0

0 6.0

5.0

Ey(MeV) FIG. 55a,b. Thermal-neutron capture gamma-ray spectra recorded by a germanium-spectrometer sonde using a 108-neutrons-per-secondCfZs2source. Strong full-energy lines are identified by the elements producing them. Single primes indicate one-escape peaks and double primes two-escape peaks. Counting time was 10 minutes. The 10-in. borehole contained a fresh-water-filled7-in. casing. Formation was a water sand of salinity lo5ppm NaCl. (Courtesy of J. S. Schweitzer.)

-

19.


583

element in the medium.* We neglect body forces, such as gravity, which usually do not play a significant role in elastic wave motion, and assume infinitesimal strains. Then

+

p(a2uldt2)= ( A p)V(V * u)

+p v u ,

(3.52)

where p is the density and u is the vector displacement of a “particle” in the medium from its equilibrium position. The left-hand side is just the mass times acceleration for a unit volume element; the right-hand represents all the stresses on the element. The stresses are expressed in terms of the displacements and are derived by using the stress-strain relations for an isotropic solid. Here we use the Lami coefficients1andp as the elastic constants characterizing the medium.

The Compressional Wave. Taking the divergence of Eq. (3.52) immediately yields

-

p(a2A/t3t2)= ( A

+ 2p)V2A,

(3.53)

where A = V u is the local fractional increase in volume, or d i l a t a t i ~ n , ~ ~ and use is made of the relation V * V2u = V * VA. This equation predicts the This wave, known varpropagation of a wave with velocity iously as the dilatational, compressional, irrotational, longitudinal, or P (primary) wave, is the one referred to in Eq. (2.20). (The bulk modulus is related to the Lami constants by B = 1 & L . ~ ) The Shear Wave. Applying the curl operation to Eq. (3.52) leads to a different wave equation,

m. +

(3.54) where 8 = V X u, and we have used the vector identity V X VA = 0. It can be shown that for a particle of the medium, @1/2 is the angular displacement from equilibrium in a plane perpendicular to the direction of€k91This wave, known as the torsional, rotational, shear, transverse,or S (secondary) wave, propagates with velocity which is slower than the compressional. It is the shear wave discussed in Section 2.5.

mp,

Since every rotational motion can be decomposed into two perpendicular linear motions in the same plane, it is readily seen that particle motions perpendicularto the direction of propagation correspond to transversewaves which travel with the shear velocity. Particledisplacementsin the directionof

* Although the requirementsof microscopic continuity and ideal elasticity are not rigorously met by earth formations, this brief sketch will make clear the origins of the body waves. The effects of fluid-filled porosity are discussed in Section 3.6.3.

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JAY TITTMAN

propagation correspond to longitudinal waves travelling with the compressional velocity.92This can be seen also by decomposing Eq. (3.52) into rectangular coordinates and assuming that the wave is propagated in the direction of one coordinate only. In nonviscous liquids or gases shear cannot be supported,p = 0, and only permits recognition that the dilatationalwave is predicted. Its velocity, for a liquid 3, is the bulk modulus. The compressional and shear body-wave velocities are non-dispersive, i.e., constant with frequency, since 3, andp are assumed to be independent of frequency in the range used in logging.*Although these waves are often small compared to the amplitudes of the guided waves discussed in the next section, they are usually large enough to permit detection by conventional sonic transducers. Their velocities are the most sought-after in the borehole because of their use in seismicinterpretation,lithology determination,porosity measurement, mechanical properties evaluation, and in deriving the elastic constants B and p. 3.6.2.Borehole Waves. The mathematical derivation of the properties of waves set up in a liquid-filled borehole is quite lengthy even when the porous nature of the formation is i g n ~ r e d .Hence, ~ ~ - ~we ~ will merely sketch the origins and principal characteristicsof the several kinds of waves that can appear in the wavetrain at a sonde receiver. Because of their simplicity and usefulness in predicting the times of first amvals, ray paths will be used in some cases. We do this despite the fact that sonic wavelengths in both the mud and the formation are of the same order as borehole diameters. (See footnote in Section 2.5.) The separate wave components will be discussed more or less in the order oftheir appearance in the idealized wavetrain of Fig. 10. The general solution of Eq. (3.52) in borehole geometry leads to the prediction of several unique wavetrain components, or propagation modes. Usually, in solids, the solution is found by the use of scalar and vector potentials land Wand the relation u = Vc V X W. In liquids and gases only is needed. It can be shown that the potentials satisfy the wave equations V2( = ( l/uc)(a2Z;/dt2)and V 2 y = ( 1/u,)(a2y/dt2). Four boundary conditions are imposed: (1) Radial stress and displacement are continuous across the borehole wall. (2) Tangential stress is zero on the wall. (3) Waves vanish at infinity. (4) Stresses and displacementsare finite on the borehole axis (except at point sources). The resulting radial solutions for displacement and pres-

mp,

+

* However, there is some evidence that real rocks can show a small dispersion arising from viscosity-induced dissipation mechanisms. See, for example, T. Jones and A. Nur, Geophys. Res. Lett. 10(2), 140 (1983).

19.


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sure are cast in terms of modified Bessel functions; vertical solutions are in the form exp[i(kz - ut)]. 3.6.2.1. THEHEADWAVES.In the loggingproblem head waves are body waves that travel vertically in the formation near the wall with compressional or shear velocity and are radiated into the borehole fluid at the critical angle (Fig. 9). Although for logging they are the most important arrivals in the wavetrain, their origin and properties are easily understood and relatively little discussion is required. Compressional. The first wave to arrive at the receiver is the compressional. It not only propagates fastest in the formation, but has the shortest path through the mud (Fig. 9). Refraction at the critical anglePCbends the ray path vertically along the borehole wall, where the wave travels with the formation compressionalspeed. The energy radiated back into the borehole, also at the critical angle, produces the pressure variations sensed by receivers located on the borehole axis. Shear. It is readily shown that for elastic solids the ratio B/p 2 2/3 by expressing it in terms of Poisson’s ratio, which for all materials of interest to us is 20.Then Eq. (2.22) predicts v, > \j2u,, and one expects shear-wave arrivals to be observed only for ITT values t, > fit,.* This, of course, corresponds to the shear wave excited in the formation at the shear criticalangle& = sin-l(uf/va), where vf is the velocity in the borehole fluid. (See Fig. 9.) In the great majority of formations vf < u, and the critically refracted shear head-wave can exist. (Note that this condition for existence sheds no light on the level ofexcitation.) It sometimes occurs that the formation is so soft that vf > .,v Then shear body-waves can propagate, but no shear headwave can be induced by critical-angle refraction. In this case it is possible to measure v, or p by other means described in Sections 3.6.2.5 and 3.6.6. 3.6.2.2. THE LEAKYMODE. The second arrival in the wavetrain is known as the leaky mode for reasons that will become apparent in the following discussion. When present, it is primarily an annoyance, sometimes interfering with or obscuring the shear amval. Its origin and characteristics are simply described by ray-tracingconsiderations.A portion of the approximately spherical wave leaving the transmitter is incident on the formation wall at angles between P, and P, .This incident wave undergoes both internal reflection (not shown in Fig. 9) and conversion into shear waves which are refracted into the formation. The reflected part-we temporarily ignore the presence of the sonde body-then proceeds vertically in the borehole as a conical wave by repeated reflection at the wall. Its phase velocity normally

* Since the theoretical literature usually refers to wave velocities and the experimental refers to ITT observations, it will be necessary to switch freely between these in the sonicsdiscussion.

586

JAY TITTMAN

lies between those of the formation compressional and shear w a v e ~ ? ~ AtJ ~ each reflection, that part of the energy “leaked” into the formation by the conversion into shear reduces the amount available for internal reflection. Hence, the amplitude of the leaky mode decreases with vertical distance from the tran~mitter.9~ It increases with the value of Poisson’s ratio for the f0rmation,9~,~*J~~ presumably because Poisson’s ratio determines the fraction of the energy that is converted to shear. (Consequently, it has been suggested that leaky-mode amplitude measurement may be used as an indicator of Poisson’s ratio for the formation.97)The leaky mode is, thus, expected to be most visible in soft formations(large Poisson’s ratio), especially since the shear wave, which follows it in the wavetrain, is smallest under these conditions. Mode-trapping resonance occurs only for wavelengths satisfyingconditionsfor reinforcement, and not for destructiveinterference. I , , the borehole size and transmitter freThus, for given values of p, and / quency spectrum determine whether the leaky mode exists in any particular case. When the mode is present in the wavetrain it usually appears between the P and S arrivals shown in Fig. 10. Dispersion curvesshowingthe first few modes in a sandstone and in a slow shale have been calculated.*@’ To qualitatively appreciate the influence of the sonde body, coaxial in the borehole, on the low-frequency components of the leaky mode we consider the sonde’s reflection characteristics.For example, the sonde has been modeled as being “hard” (density and sonic velocities larger than those of the formation)or “soft” (density and velocitieslying between those of formation and borehole Reflection coefficients calculated for normal incidence in these two cases are 90% and 30%,respectively, and we might suspect that the character of the leaky mode could be predicted accordingly. If the sonde is considered to be highly absorbing, the leaky mode should attenuate even more rapidly with distance than would be expected solely from reflections at the borehole wall. Whether the sonde body is considered predominantly reflecting or absorbing, that part of the wave confined between the sonde and the borehole wall should exhibit a higher cutoff frequency than that for the wave in a borehole without sonde. The actual effect of the sonde body is not currently available in the logging literature. Although theoretical predictions are made, they are based on “hard” or “soft” models with sonic properties that have not been experimentally verified. This point is especially important because sonde sleeves in common use contain circumferential slots staggered in the vertical direction (Fig. 57b) and the interiors are usually oil-filled. Structuressuch as this are quite different from that assumed in the models, and their behavior may differ accordingly. 3.6.2.3. THEPSEUDO-RAYLEIGH WAVE. When the shear head-wave is present (when u, > of), as is usually the case, a phenomenon somewhat analogousto the leaky mode is permitted. That portion of the wave incident

-

-

19.


587

on the borehole wall at angles >/Ish suffers total internal reflection and propagates up the hole as a conical wave. However, since refraction into the formation at angles greater than Bahis prohibited, there is no energy leakage on each reflection and the wave does not exhibit the attenuation of the leaky mode. This wave is referred to by a variety of names: reflectedconical wave, trapped mode, guided wave, normal mode, and pseudo-Rayleigh wave. The last name is used by analogy with a surface wave that can propagate on the plane interface between two semi-infinite media, one an isotropic elastic solid and the other an inviscid fluid. The borehole pseudo-Rayleigh is actually a hybrid consisting of the reflected conical wave in the fluid coupled to a surface wave travelling vertically on the borehole wall.96When it is excited it follows the shear arrival immediately in the wavetrain (Fig. 10). As will be seen below, the observableproperties depend upon dispersion characteristics created principally by the borehole diameter and the densities and elastic constants of borehole fluid and formation, and upon the frequency characteristics of the instrumentation. Because no energy is refracted into the formation these waves can exhibit relatively large amplitude, as shown in Fig. 10. The existence of the pseudo-Rayleighwave is predicted by solution of Eq. (3.52) in borehole g e ~r n e t r y . ~These ' - ~ ~ solutionslead to dispersion curves, an example of which is shown in Fig. 56 for a particular set of characterizing

0.7

I

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588

JAY TITTMAN

parameters and borehole diameter.’@’ The first two allowed modes are shown. The dispersion curves exhibit several significant features: (1) The pseudo-Rayleigh phase and group velocities for each mode are bounded on the high side by u, and show a low-frequency cutoff. Thus, the lowest-frequency component in each mode travels with a speed equal to u, and it is of no consequencewhether the velocity of the shear arrival is measured or that of the fastest component in the pseudo-Rayleigh wave.98(2) The high-frequency asymptote of both phase and group velocities is vf .99 However, the group-velocity minimum (the Airy phase) is less than uf, so high-frequency components of the pseudo-Rayleigh can appear at the very end of the wavetrain. (3) If the instrumental frequencies are below the cutoff of the first mode, no pseudo-Rayleigh wave can be observed. Even if the instrumental frequency band overlaps the allowed pseudo-Rayleighspectrum,the appearance of the pseudo-Rayleigh wave in the wavetrain can be quite variable, depending as it does upon hole diameter, v,, u, and the angular sensitivity patterns of transmitter and receiver. The predicted effect of the sonde body (assumed coaxial with the borehole, as for the leaky-mode discussion) depends again upon the sonic properties assigned to the If the tool is assumed to be “hard”, the dispersion curves are altered by the trapping of the pseudo-Rayleigh wave in the toolborehole annulus. The cutoff frequency increases somewhat and both the phase- and group-velocity curves are stretched in the direction of higher freq~encies.~’ However, the general character of the dispersion does not change radically. Assigning the tool “soft” properties leaves the frequencies of the cutoff and the Airy phase nearly unchanged from the “no-tool” case. But the Airy-phase minimum takes a value significantly less than uf. From the nature of the tool sleeve shown in Fig. 57b it is again questionable whether these “hard” or “soft” models apply. There is some evidencethat includingattenuation in the theoreticalmodel improves agreement between calculated and observed wave train^.^^ In particular, the assumption that the attenuation coefficient increases with frequency leads to the prediction that the leaky and pseudo-Rayleigh modes suffer greater attenuation than the body waves. This is consistent with the fact that these guided waves are forced to higher frequencies by their resonance conditions. As noted above, the absorption characteristics of real sonde bodies are absent from the published literature. However, there are indications that sonde absorption may be significant,’0’thus affecting the reflections necessary for propagation of the leaky and pseudo-Rayleigh waves. 3.6.2.4. REVERBERANT WAVES.Although the leaky and pseudo-Rayleigh wave have received greater attention in the literature, another propagation mode common to each of the head waves is present in the borehole.

19.


589

Reverberant waves have some properties that are similar to those of the leaky and pseudo-Rayleigh,and indeed, they constitute limiting cases of these two waves, suffering reflection at the compressional and shear critical angles, respectively. The energy radiated back into the borehole, as the critically refracted compressional and shear waves move vertically along the wall, is reflected from the sonde body or both reflected and criticallyrefracted at the opposite side of the hole if the sonde’s presence is ignorable. (See the short arrows in Fig. 9.) The reflected wave impinges on the wall again, suffers critical-anglerefraction, and interfereswith its “parent” wave alreadytravelling along the wall. This interference is constructive for certain wavelengths and destructivefor others, depending upon the time spent in the borehole by the reflected wave and the ratio of vf to the velocity of the body wave under consideration. Since formation head-waves are continuously shedding energy into the borehole, this process is a continuous one as the head-waves move along the wall. At the receiver the reverberant waves appear in the tails following the compressionaland shear arrivals. Since v, and vh are different, the constructive interference condition produces a central-frequency separation in the overlappingFourier spectra of the compressionaland shear wavetrains. This feature is put to use in a logging technique described in Section 3.6.5.2. WAVE.In addition to the pseudo-Rayleigh 3.6.2.5. THE STONELEY wave, a pure surface wave with different velocity and dispersion is predicted by borehole solution ofthe wave equationsin Section 3.6.2. It is referred to as the tube wave, guided wave, or Stoneley wave. The last name is given by analogy with a surface wave discovered by Stoneley on the plane interface between two semi-infinite elastic solids.lo2 In Fig. 56 is shown a typical example of dispersion curves for the Stoneley wave in a formation for which v, > vf.lo0 The difference from the behavior of the pseudo-Rayleighis striking: (1) No cutoff frequency exists. (2) Dispersion is very mild. (3) For all frequencies, us, < vf. (4) Group velocity nearly equals phase velocity over the whole frequency range [because of (2)J.These characteristicsare connected with the fact that the Stoneleywave radiates no energy that can create conical waves in the b o r e h ~ l e . ~ ~ As the frequency approaches zero, the Stoneley becomes a true tube wave corresponding to the classical water hammer. Its velocity approaches the value (3.55) where Bfand pfare the bulk modulus and density of the borehole fluid, andp and p, are the shear modulus and density of the formation.99Thus, measure-

590

JAY TITTMAN

ment of u, at very low frequencypermits the determination ofp or upeven in formations so soft that u, < u, and no shear head-wave can be excited. The values of p,, pB,and u, can usually be ascertained with good accuracy. However, errors in the measured value of u, are magnified as they propagate into errors in the derived value of u,. Thus, if the extraction of u, is the objective, us, must be measured with considerable accuracy. The use of Eq.(3.55) is described to illustrate the principle. In practice, logging frequenciesare too high to permit this approximation;numerical methods are used to solve the more complicated expression that relates vst to u, and other borehole and formation parameters.lo3-lo6 The comments concerning accuracy remain operative, however. In the high-frequency limit, us*approaches an asymptoticvalue somewhat smaller than vf. In this case the ratio of wavelength to hole diameter is < 1, and the velocity becomes identical with that of the Stoneley wave on a plane interface. The Stoneleywave in Fig. 10showstwo interesting features: (1) Because of the small dispersion it appears more or less as a pulse in a relatively confined portion of the wavetrain, (provided, of course, that the transmitter emits a pulse, and not a continuous output). (2) Since there is no cutoff and the Stoneley is easily excited at low frequencies, if low frequenciesare present in the source they will also be in the received wave. This produces the sudden shift to lower frequency visible in the vicinity of the Stoneley “pulse”. The presence of the sonde in the borehole is expected to have less effect on the Stoneleywave than on the p~eudo-Rayleigh.~’ Stoneley dispersion curves computed for a “hard” sonde in the borehole are very similar to those of Fig. 56; only the zero-frequency limit is lowered a few percent. A “soft” sonde is predicted to have a somewhat larger effect. It reduces the zero-frequency limit to vsJuf = 0.82 and the high-frequency asymptote to -0.8, and generates a mild undulation between them. However, dispersion remains small. As was noted for the leaky mode and pseudo-Rayleigh, real sondes may depart substantially from the “hard” and “soft” models, and direct experimental confirmation of the theoretical predictions is absent from the literature. The amplitude of particle displacement in both the formation and the borehole decays approximately exponentially with radial distance from the ~ a l l ? Thus, ~ J ~ all other things being equal, the Stoneley amplitude at a receiver becomes smaller as the borehole diameter increases, because the axial sonde is moving further from the wall. Whereas at usual logging frequencies u, > uf > ust, in the seismic or VSP range it is possible for the velocity sequence to become u, > us, > u, in very soft media, such as ocean s e d i m e n t ~ .In ~~ this ~ Jcase, ~ ~ however, the Stoneley is no longer a true guided wave, but is strongly attenuated by radiation into the formation.

19.


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3.6.3.Waves in Porous Media. The waves treated to this point occur in materials that are homogeneous, microscopically continuous, isotropic, and ideally elastic. Real sedimentary rocks, on the other hand, usually contain fluid-filled pores, cracks, or vugs, so some attention must be given to the consequences of these departures from the ideal. Here we briefly discuss only compressional and shear waves in media with fluid-filled, interconnected porosity. Fortunately, most of the ideas developed above with respect to wave velocities are preserved when fluid-filled porosity is included in the model of the propagating medium. This is true for the leaky mode, reverberant, pseudo-Rayleigh, and Stoneley waves also.’O’ 3.6.3.1. THEGASSMANN MODELIO~. The first attack on the porous-medium problem made the outright assumption (justified in Section 3.6.3.2) that at sufficientlylow frequenciesthe relative motion between the fluid and the solid parts of the composite is small enough to be ignored. Under this assumption, expressions for the compressional- and shear-wave velocities are derived. The rock skeleton, orframe, is viewed as consistingof a homogeneous isotropic elastic solid material characterized by the constants p1 and B1. The evacuated skeleton is characterized by 9, BZ, and p2 ; and the pore fuid by pf and Bf.The problem, then, is to calculate for the composite, or saturatedrock, the average propertiesp, p,and B in terms of the constituents’ parameters. This, in turn, permits direct calculation of the compressional and shear velocities. The assumption that fluid and skeleton sufferno relative motion provides one of these quantities directly, p = (1 - 4)pl +pr. (This assumption is not so superfluous as first appears. It is noted in the next section that an apparent density increase manifests itself when relative motion occurs.) Also, since it is assumed that the fluid and skeleton do not interact chemically, the plausible surmise is made that the shear modulus of the composite is not affected by the presence of the interstitial fluid, i.e., that p = p 2 .Thus, two of the composite parameters, p and p, are immediately in hand. The heart of the problem then becomes a detailed and somewhat intricate calculation of what happens to a sealed cube of the saturated rock when placed under hydrostatic pressure.1o8The expression for B is found to be

+

B = B,

+f(4 B2 9

9

B f , 4),

(3.56)

where (3.57)

Adding 4p/3 to the left-hand side and 4pd3 to the right-hand side of Eq. (3.56), dividing both sides by p, and rearranging yields (3.58)

592

JAY TITTMAN

+

+

where v; = 4(B2 4,u2/3)/p and g =fl(B2 4 ~ ~ / 3The ) . compressional velocity in the composite is thus seen to be the velocity u; of a material having the elastic constants of the skeleton and the density of the composite, corrected by a factor that is influenced by the porosity and the fluid's bulk modulus. Porosity and fluid density both affect v;, but there is no influence from Bf.The validity of the assumption that v, = d(B 4 ~ / 3 ) / prequires that the composite be statistically isotropic and homogeneousover distances short compared to a wavelength. This condition also permits one to write for the shear-wave velocity

+

-

v , = & - d ,11.

7

P2

(3.59)

The same equivalent medium appears, but the value ofB,does not influence u, at all. Porosity influences p z , of course, while both porosity and fluid density affect p, as in Eq. (3.58). 3.6.3.2. THEBIOTTHEORY.An objection to the description sketched above is the assumption, rigorously valid only in the zero-frequency limit, that the relative motion between fluid and skeleton can be ignored at low frequencies. Furthermore, although the low-frequencyproviso is intuitively attractive,the theory is incapable of saying how low this must be. Both these problems, and many more related to wave propagation in porous media, were addressed in a series of classic papers by Biot.*09 We outline here in very condensed fashion the main thrust of Biot's approach and the conclusions which quantify the limits of validity of the Gassmann model described above. Biot sets up the stress- strain relations for a statistically isotropic fluidfilled elastic solid. The two chemically noninteracting phases are individually continuous and interpenetrate one another. There is introduced a fluid displacement vector U that averagesover a volume element large compared to a pore or grain size but small compared to a wavelength. Here U is chosen so that its product with the cross-sectional area for fluid flow yields the correct volume flow. A similar displacement, u, is defined for the skeleton. The stress-strain relations for the composite now involve two elastic constants in addition to the Lame coefficients. One of these measures the work required to make a unit change in fluid dilatation, or the pressure needed to force an additional volume of fluid into the composite under conditions of constant composite volume. The other represents coupling between a volume change in the skeleton material and the concomitant volume change of the fluid. (Suggestions are made how these additional constants can be measured under static conditions.) It is shown that the coupling can be described by the addition of an apparent mass pa to both +pf, the fluid mass fraction, and p z [ = ( I - r$)pl], the skeleton mass fraction. pa

19.


593

takes into account the inertial drag that the fluid exerts on the solid and the solid on the fluid. Dissipation is included by the introduction of a term proportional to (d/dt)(u - U)in the Lagrangian formulation of the equations of motion of the system. The coefficient of this term is &'/ic, where q and ic are the viscosity and permeability, respectively. The resulting wave equationspredict uncoupled shear and compressional waves with the new feature that an additional, slow compressional wave appears. The velocities of the shear and ordinary compressional waves are shown to be negligibly diferentfrom those given by Eqs. (3.58) and (3.59) i f thefrequency is much smaller than a criticalfrequency, f, = &/2Wf.

(3.60)

For practical values of the variables in Eq. (3.60) the condition f -SKf, is usually satisjed at conventional logging frequencies and Eqs. (3.58) and (3.59) yield velocities accurate to within better than afew percent, and often to within afraction of onepercent. Thus, the Biot theory solves both problems posed at the beginning of this section: At sufficiently low frequencies the fluid is locked to the skeleton by its viscosity, and the low-frequency condition is quantified by the requirement that f 4f,. In addition, the theory predicts the values of wave velocities at all frequencies. Whereas in the ordinary compressional wave the fluid and skeleton move in phase at low frequencies, the slow wave corresponds to 180"-out-of-phase motion between the two. In the low-frequency limit the slow wave becomes strongly overdamped and degenerates into a diffusion phenomenon with very high attenuation.IwJlo Although this type of wave has been observed in the laboratory at both high and low frequencies,"' no field application is reported in the literature at present. 3.6.4. Direct Measurement of Compressional Interval-Transit-Time. 3.6.4.1. THE FOOT, 5-FOOT SONDE. The conventional measurement of compressional interval-transit-time tc,is made by detecting the earliest arrival at each of two receivers, as described briefly in Section 1.4.4. Fig. 57a shows the most common transducer configuration used for borehole-compensated logging.112The measurement cycle begins with the firing of the upper transmitter T,, which starts a high-frequency clock in the cartridge. The clock counts time until the output of receiver R,exceeds a pre-set low threshold and signals the first amval of the compressionalwave. The threshold crossing stops the clock and the elapsed time is stored digitally. About 50 ms later T, is fired again and a similar time measurement is made by the compressional arrival at R, . Subtraction of the latter time from the former and division by the 2-ft (6 1-cm) span between R,and R, yields directly the formation ITT in ps/R or pslm, by cancelling the transit times in the mud.

JAY TITTMAN

T

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To effect borehole compensation for caves and sonde tilt the complete cycle is repeated using the lower transmitter TLand receivers & and R, .The “up” and “down” measurements are averaged to provide the value of tcthat is presented on the log. Sinceabout 5 averages are made per secondand the tool is normally run at 4 ft/s (1 5 cm/s), the sonde moves about 1 in. (2.5 cm) during the complete measurement cycle. Slight displacement of the “up” and “down” receivers (Fig. 57a) from one another compensatesfor the static depth-shift that would otherwise result from the inclination of the ray path through the mud. A straightforward calculation shows the total transit time for any ofthe ray paths in Fig. 57a to be

-

(3.61)

where L is the transmitter -receiver spacing and s is the distance from the transducer surfacesto the borehole wall. The earliestpossible arrival through the borehole fluid can appear at a time tf = L/vf.Since the logging measurement requires that the compressionalbe the first amval detected, the spacing must satisfy the condition (3.62)

The “near” spacing of 3 ft (9 1.4 cm) is chosen so that it satisfiesthis condition for most combinations of s and the ratio vJvffound in the field. It fails, however, when hole diameter is too large because of either bit size or caving. In soft shales, in addition, v, is relatively small and the radical in Eq. (3.62) can become large. It would be attractive to eccenter the tool under these conditionsand reduce the effectivevalue of s, i.e., the smallest distance to the wall. However, the receiver signal falls very rapidly as the sonde is moved off the borehole axis because arrivals traveling different paths, thus coming from different azimuths, no longer appear at the same time and suffer destructive interference. Most sondes use magnetostrictiveor piezoelectric transducers. These are supported by high-compliance mountings to isolate them acoustically from the sonde sleeve. They are usually jacketed in thin-wall metal cans so they FIG.57. The conventional borehole-compensated sonic sonde. (a) Arrangement of transducers and critical-refraction paths for head waves. “Up” and “down” measurements are averaged to produce a borehole-compensated value of interval transit-time. (b) Schematic exterior view illustratingthe slotted sleeve. Centralizationis effected by use of two or more sets of bow springs on sliding collars. Only the lowest centralizer is shown. [Adapted from D. H. Thomas, Log Anal. 19,23 (1978).]

5 96

JAY TITTMAN

can communicate readily with the borehole fluid. The protective slottedsleeve is pictured in Fig. 57b. The transducers are immersed in a continuous oil bath that, through the use ofbellowsor pistons, providespressure balance with the mud column. Also, the oil bath permits good acoustic coupling to the mud, while insulating the transducers electrically and protecting them from chemical attack and mechanicaldamage. The slotsin the sleevegive it a very low sonic velocity, thus preventing waves travelling up or down the sleeve from interfering with the head waves arriving from the formation. Signal strength falls with spacing more rapidly than noise, so the signal-tonoise ratio limits the maximum permissable spacing. The largest noise source, by far, is road noise. This consists of random spikes caused by the scraping of the centralizer springs against the borehole wall and the centralizer collars sliding on their mandrel. Even after precautions are taken, such as specially coating the sliding surfaces, road noise remains the major source of system noise. Its primary manifestation is an early stopping of the timing even though receivers are gated off for a period of time (usually 100- 200ps) after transmitter firing. The clock turnoff can be triggered when a noise spike appearsabove the detection threshold before the compressional first-arrival. When this occurs at a “far” receiver it produces an erroneously short reading for tc,and at a “near” receiver an erroneously “long” reading. Since these are sudden departures from a sequence of presumably correct values of tc,they appear on the log as randomly occuring spikes of varying amplitude. The effect of a single noise-triggering on the spike amplitude is reduced by the averagingof the “up” and “down” measurements. Some tool systems have incorporated “despiking” circuits or algorithms that reject single measurements that depart excessively from preceding ones. Since the maximum real change that can occur from one measurement oft, to the next is calculable a priori, valid rejection criteria can be established. Another source of error in tcarises from the use of threshold crossing to stop the clock. When signal strength falls sufficiently the effect is felt first at the far receiver because the signal is smaller there to begin with. The time of threshold crossing then suffers a short, but observable, delay because of the finite rise time of the first amval.’ l 3 In practice, at 20 kHz this timestretching can produce an error as large as 6 ps/ft (- 20 ps/m) in a single boreholecompensated output of tc if the effect occurs on both far receivers. If the signal amplitude falls still further, no part of the first cycle crosses the threshold. Later cycles are usually larger (Fig. lo), so some part of the second cycle may cross the threshold and stop the clock. Thus, the measured time for that transit is too long by more than one cycle, 50 ps for a 20-kHz wave. After averaging, this results in a cycle-skip spike of 2 13 ps/ft (43 p / m ) appearing on the recorded log. When both far receivers cycle-skip the effect is twice as large. If the signal amplitude falls even further, two or more cycles may be

-

-

+

19.


5 97

skipped before the threshold is crossed, initially producing spikes that are multiples of about 13 ps/ft. The despiking methods mentioned above in regard to road noise can remove cycle-skip spikes also. To alleviate both the noise and cycle-skip problems automatic-gain-control systems and noise-controlled threshold settings have been employed. These work significant improvement, but both effects still appear when the signal is attenuated sufficiently. SONDE. Equation (3.62) made clear that 3.6.4.2. THELONG-SPACING transmitter-receiver spacing can limit the range of measurement as hole diameter increases. A problem similar to that of hole size arises when an altered zone surrounds the borehole (Section 2.5). This zone is customarily modeled as a cylindrical annulus between the borehole wall and the unaltered, or virgin, formation. Its compressional ITT is longer than that of the virgin formation, sometimes by as much as 40 ps/ft (1 3 1 ps/m). Consequently, the ray paths shown in Fig. 57a yield the tcvalue of the altered zone, not that of the virgin formation. However, some ray incident at an angle less than pc for the altered zone continues and is critically refracted at the cylindrical interface between the altered zone and unaltered formation. Simple ray tracing shows that this path through the virgin formation can eventually terminate at a receiver as a least-time path if L is large enough.'" In this case, the effect of the altered zone is similar to that of an increase in hole diameter. The same ray-tracing argument shows that for sufficiently large L the tworeceiver technique removes the travel time through both borehole fluid and altered zone. Thus, the remaining time is that spent travelling the critically refracted path in the virgin formation, and division by the receiver span yields the true formation tc. Although the effect of an altered zone was known rather early,114sonde spacingswere usually limited by low signal-to-noiseratios. The development of strongertransmitters,road-noise reduction methods, etc., led to the intro~ ~ to preserve duction of long-spacing sondes in the m i d - l 9 7 0 ~ . ' * ~In, 'order the borehole compensation feature it would be attractive to use the same configuration as that in Fig. 57a, but with the spacings increased to, for example, 10 ft (3.05 m) or more. This increases the sonde length by at least 14 ft (4.27 m), an unattractive step for a variety of practical reasons. Instead, a different technique, depth-derived borehole compensation (DD/BHC), which increases sonde length in this example by only 4 ft ( 1.89 m), is used. A typical long-spacing sonde is shown schematically in Fig. 58. The two vertical positions will be used later in describing the DD/BHC technique. The rays drawn are schematic also, and do not represent actual refraction paths through borehole fluid and altered zone. This sonde produces two long-spacing measurements oft,, one with 8-ft, 1 0 4 (2.44-m, 3.05-m) spacing and the other with .lO-ft, 1 2 4 (3.05-m, 3.66-m) spacing. When the two

+

-

598

JAY TITTMAN

w

/

rn

, B

....

....

j; ................... F . Y t

...................

E ....(

, ’a

M

1)

...........................

K ....

............................

J...

, ’lz Y

FIG.58. Transducer arrangement for a typical long-spacing sonde. Lower sonde position shows schematically the critical-refraction paths for 10-A, 12-A and 8 4 10-A “receiver” measurementsof interval transit-time.Upper position illustrates “transmitter”measurements sampling the same vertical interval. Averaging transmitterand receiver measurements corresponding to the same spacing provides depth-derived borehole-compensation.

DD/BHC values for tcare equal, they are correct either for the virgin formation or for an altered zone thicker than these spacings can “see” through, Usually local knowledge, sometimes including a 3-ft, 5-fl measurement, allows ready discriminationbetween these alternatives.When the two values are not equal, the difference provides a clue as to how close the 10-ft, 12-ft reading is to the true tcof the unaltered formation. Again, knowledge of the 3-ft, 5-ft reading is helpful in making this estimate. The DD/BHC measurement sequence starts with the sonde in the lower position in Fig. 58. The firing of transmitter TL starts the clock, and the compressionalwave that followsthe path TLABCRU stops the clock when it arrives at Ru. T, is fired again and the wave followingthe path TLABRL stops


599

LOGGING

the clock. If the spacing TLRL is large enough, the difference corresponds to the total transit time in the virgin formation between levels B and C,as measured by the 10-ft,1 2 4 subsonde. The individual times are tagged with the depth in the hole and memorized. This subcycle is now repeated, using T, as the transmitter. The arrivals follow the ray paths shown on the righthand side of the sonde. If the 8-ftspacing between TITand RLis large enough, these measurements also correspond to total transit time in the virgin formation between C and B since BC = EF. (The small movement of the tool between sub-cycles is ignored.) These times, too, are stored in memory. As the sonde is pulled up through the hole this complete cycle is repeated 5 times per second, as for the 3 4 5-ft sonde. In the upper diagram of Fig. 58 the tool has moved uphole 9.67 feet (2.95 m). Now the transmitterrays are criticallyrefracted at depths such that the same vertical interval is sampled as before. The rays drawn on the leftand right-hand sides of the sonde correspond to 10-ft, 12-ft and 8 4 , 10-ft spacings, respectively. Now, however, the critically refracted rays which define the sampled depth interval are entering the formation, whereas before they were leaving it. This is the feature that provides borehole compensation. Again, the times to travel the four paths TLGHIR”, TuHIRu, TLJKMRL, and TUKMRL, are stored and depth-flagged. The 10-ft, 12-ft difference is averaged with the difference between the corresponding times memorized 9.67 feet earlier. Then it is divided by the span to yield the borehole compensated value oftc(10, 12). The same procedure is carried out with the 8-ft,lO-ft data to provide tc(8, 10). Of course, as the tool is drawn up the hole, the time-measurement cycles are repeated continuously and all the data are stored in memory until used. A complete DDIBHC output of tcis provided about every in. (2.5 cm). Since the DD/BHC result depends upon the averaging of two measurements presumed to be sampling at the same depth, as measured by the cable spooling device at the surface, cable stretch and yo-yo (Section 1.5)can cause wrong pairs of data to be averaged. Although it was originally anticipated that this would be a serious problem, in practice it appears surprisingly seldom. Another peculiarity of the DD/BHC results from the fact that every transit is used twice. Thus, a noise spike or cycle-skip affecting a single transit-time measurement can appear on the log at depths 10 ft apart. A computer algorithm correcting for this effect has been developed. l6 An experimentallong-spacing sonde consistingofa single-transmitterand two receivers,with 15-ft(4.57-m) “near”-spacing and 5-ft (1.52-m) span, has recently been described.’l7 This tool does not employ DD/BHC. However, the use of wideband receivers, extended spacing, and uphole waveformstacking are reported to permit good separation of the shear arrival, in addition to measurement of tc.

-

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600

JAY TITTMAN

3.6.5. SHEAR-WAVE MEASUREMENT AND WAVEFORM PROCESSING. The borehole measurement oft, is appreciably more difficult than that oft,. When conventional transducers are used the shear-wave first-arrival is often smaller than the largest excursion of the compressionaltrain, as is illustrated in Fig. 10. Therefore, threshold-crossing methods have proved uniformly unreliable except in some very hard formationswhere the shear-waveamplitude is quite large. Further, at 3-ft or 5-ft spacing the shear wave is often mixed with the tail of the compressional and is unrecognizable. The ready accessibility of high-speed computers has made possible a fruitful attack on these problems through waveform data-processing techniques of increasing sophistication. The following three sections briefly describe some of these methods. All of them are essentially correlation schemes comparing waveforms at two or more receivers. The instrumentation details vary somewhat,but certain features are common to nearly all the tool systems employing wavetrain analysis. The systems tend to be broadband in order to permit the use of signals relatively confined in time, and spacings are usually 10 ft (-3 m) or larger. Together, these two features accentutate the time separation of the wavetrain components. The number of receivers ranges from two to around fifteen, with equal spans between them. The spans have values between -3 in. (- 7.5 cm) and 1 ft (30 cm), depending upon the particular tool under consideration. Sondes with small spans and large numbers of receivers are expected to produce more reliable results because of the high degree of spatial sampling they bring to the correlation estimate. Usually, each receiver’s wavetrain output is subjected to high-speed analog-to-digitalconversion and temporary bufferingdownhole. It is then telemetered to the surface during dead periods between receptions. There, real-time signal processing may be executed in the truckborne computer and/or the raw data may be taped for later, more complicated analysis. Logging speeds are usually slower than for conventional ITT measurements (Section 3.6.4), so that a complete data set can be acquired every few inches. 3.6.5.1. FOURFOLDCORRELATION^'^. Although the fourfold-correlation method was quickly superseded, a short review of it is instructive as an introduction to waveform data-processing. For this discussion refer to the wavetrains of Fig. 59. Let the wavetrain outputs of successive receivers be represented by the time functionsgi(t).In a thick, homogeneous bed the g’s should be identical or very similar, at least over a finite portion of the waveforms, except for time shifts, or moveouts, from one receiver output to the next. The moveouts are assumed to be integer multiples of a basic time-shift proportional to the (constant)interreceiver span. The correlation coefficientbetween waveforms from any two receivers designated by sub-

-

19.


601

FIG.59. Waveforms from eight broadband receivers spaced 6 in. (1 5.2 cm) apart. Spacing of transmitterto first receiver is 8 A (2.44 m).The broadband transmitterhas central frequency at 12 kHz.[FromC. F. Morris, T. M. Little, and W. LettonIII, SOC.Pet. Eng. 59thAnn. FUN Tech. Conf,Sept. 16- 19, 1984, Houston, paper number SPE 13285. Copyright 1984 SPE-AIME.]

scripts 1 and 2, in a window of width Tw,is (3.63) where Tois the starting time for the correlation window and 7 is a variable time shift. The use of

&(t) =

(*

gi(t)

I, q2 TO+T w

sT(t)

normalizes the g’s. This, in turn, normalizes C(’)to unity when gl and gz are identical time functions in the window, even if they differ in amplitude. Here C(2)has a maximum value when 7 = 7mo, where is the actual moveout. Thus, a well-positioned window can permit the determination of the moveout for a particular wavetrain component, e.g., the shear wave. Since the interreceiver span is known, knowledge of t m o allows calculation of that component’s ITT. is often fairly broad in the vicinity of the maxiThe function C(2)(~) mum, but increasing the number of waveforms used in the correlation reduces the width. Thus, a generalized correlation coefficient using the

602

JAY TITTMAN

wavetrains from four (any four) equally spaced receivers is defined as

where Q is unity when all four 2’s have the same algebraic sign and is zero otherwise. The insertion of Q prevents spuriouspositive contributionsto the integral at times when two g’s are positive and two are negative. As a refinement, a shaping function may be placed in the integrand to remove window end-effects. A problem in using this correlation method is that auxiliary maxima can appear in correlation coefficients, depending upon the waveform shape (in particular, if it is oscillatory) and the window width and position. Setting Twequal to 1.5 or 2 cycles of the amval under analysis has been used fruitfully in avoiding this difficulty. Although the fourfold-correlation method can be used as a vernier on threshold-crossingmeasurement of k ,its principal value is in finding the ITT of later arrivals, in particular tsh . In this case the correlation window can be (Section 3.6.2.1), and an initial positioned with To corresponding to value of T,,,,,established, from which an initial value of &shis derived. The process is then repeated, moving To to earlier and later times in steps of waveform half-cycles, thus establishing additional candidate values for ksh . The value chosen is one that satisfies certain acceptance criteria. A modification of the DD/BHC technique has been used to borehole-compensate the log outputs. 3.6.5.2. DIRECT PHASEDETERMINATION (DPD)l19. The DPD method for establishing interval transit-times consists essentially of determiningthe phase differences between the same frequency components in windowed waveforms from spatially separated receivers. Also, use is made of the fact that the compressional and shear wavetrains contain reverberant waves whose frequency spectra are somewhat different. This aids in isolating the shear phase-difference from the compressional even when the two waves overlap in the wavetrain. The method uses both window positioning and frequency analysis to separate the waves, and phase differenceto vernier the ITTs. The data-processing procedure will be illustrated by application to wave trains from a pair of receivers in the long-spacing sonde (Section 3.6.4.2).DPD is being extended to the processing ofwaveforms from multireceiver tools, also. Its usefulnessis greatest when the shear wavetrain is large, and diminishes as the amplitude falls, such as in soft formations.

-a&

Compressional Measurement. Although compressional ITT is usually measured satisfactorily on analog waveforms by threshold detection of the first amval, larger sonde-spacingsimply poorer signal-to-noise ratios. Thus, more sophisticatedtechniques become desirableas the use of larger spacings by sondes employingconventional transincreases. The determination of tsh ducers makes these techniques a necessity. Nevertheless, the DPD method

19.


603

will be described principally by applying it to the compressional wave because the procedure is simpler than that for measurement of tahand better exposes the essential ideas. Any pair of waveforms in Fig. 59 can be used for reference. The steps in the data processing will be discussed in order. (1) The outputs from the receivers are digitized and passed through a digital filter approximately matched to the transmitter wavelet. The crossing of a noise-controlled threshold then supplies a first estimate of the compressional arrival-time at each receiver. Accuracy of the threshold-crossing times need be only good enough to make the first-arrival "picks" within the first half-cycle of the compressional wave. (2) Next, windows are positioned on the compressional arrivals from both receivers. The windows start at times T o 1 and Tot, each set shortly before the respective first-arrival picks and extend nearly to the earliest possible time for the shear arrival. For the 8-ft, 10-ft sonde the window width is -4OOps, so roughly a half-dozen compressional-wave cycles are included. The first estimate window moveout ATo , for the far receiver relative to the near is just the difference in their first-arrival picks, T o 2 - Tol. (3) The windowed portion of each waveform is normalized as in the preceding Section and subjected to a discrete Fourier analysis. This provides two spectra, Gl(o) and G2(o),where o is the angular frequency. The crossspectrum X(w) E G?(w)Gz(w),where * represents the complex conjugate, is then computed. An instructive relation between the cross-spectrum and the correlation coefficient can be derived by inserting the Fourier representations of the Q in Eq. (3.63). This leads to G?(o)Gz(w)e-'"'dw =

5

+m

X(o)e-'"'do,

(3.65)

-m

where we see that the correlation coefficient is the Fourier transform of the cross-spectrum and vice versa. Equation (3.65) makes clear the relation (shown below) between trial values of r and the phase difference for each frequency component in the windowed segments of the waveforms. (4) The measured total phase difference at a given frequency is the sum of the window moveout contribution and a frequency-dependent remainder B(w),i.e., 05 = o [ A G + ~ ( c u ) ](The . use of this remainder is why it was not necessary that the first-arrival pick be accurate.) Division by the interreceiver span s then yields

for the estimate of tc provided by the phase difference the frequency o.

604

JAY TITTMAN

Generally, 0 makes a small and only mildly frequency-dependentcontribution to the waveform moveout. The smallness is a result of good first-arrival picks. The modest dependence on w results from the small dispersion, i.e., arrival of most of the frequency components at the far receiver with nearly the same time delay. ( 5 ) The final estimate of tc is made by averaging tc(w)over a band of frequencies - 600 Hz wide near the peak of IX(w)l.

Shear Measurement. DPD processing to extract tsh is the same as that fort, in principle, but is complicated by two problems: (1) Picking the correct window location is made difficult because the shear arrival is so often buried in the compressional tail. (2) Choosing the best band of frequencies is not straightforwardbecause the compressionalreverberant wave can run into or through the shear. These problems are handled by a multistep procedure that contains several new elements, the nature of which will only be sketched here. On the near-receiver waveform the shear window starts 15Ops after the compressional arrival and extends nearly to the Stoneleywave, thus includingmuch of the compressionalreverberation. On the far-receiver waveform the window is located initially at a position fitimes the compressional moveout. The window position is shifted about eight times, with increasing moveout. At each position, IX(o)l and the phase-difference 00 are computed. JX(o)J shows moderately well resolved peaks at 14 kHz and 16 kHz, for example. The different window positions provide also a family of curves of phase difference versus w. Some of these have near-zero phase difference in the shear frequency-band. Several searches are executed to find the best shear frequency-band and to determine the best window moveout for that band from the criterion that we = 0. The average moveout finally selected yields tshfrom what is essentially a relation like Eq. (3.66) with 0 = 0. At present, DPD processing must be performed after logging, but with the appearance of truck-borne array processors DPD logs will probably be made in real time. 3.6.5.3. SEMBLANCE CORRELATION.The correlation-coefficient and phase-difference methods deal only with the functional forms of the wavetrains and take no account of the amplitudes. Another likeness measure, semblance, includes this feature. The way in which semblance does this is made especially visible by considering its relation to the correlation coefficient between two waveforms. Extension of the definition of semblance to any number of waveforms will be made later. Using the terminology of Section 3.6.5.1, the semblance for two windowed segments is defined as

-

-

-

S(’) =

/

+ + ?)I2

[gl(t) gz(t

df/2

[

g g t ) dt

-

+

/

g$(t

+ 7 )d t ] ,

(3.67)

19.

605


+

where the limits on the integrals here and in what followsare Toand To T,. We note that J g: dt is proportional to the energy in the windowed segment of waveform i, a fact that will be used later. Expanding the numerator and using the normalizing factors of Section 3.6.5.1 yields

(3.68)

+

where y1 = J g:(t) dt and yz = I g$(t z) dt. Equation (3.68) shows the linear relation between semblance and correlation coefficient, and explicitly exhibits the ratio of the waveform energies. The bracketed factor can be written alternatively as [a/(l a2)],where a is the ratio of the waveform root-mean-squareamplitudes. The bracket has its maximum value of3when a = 1; it falls as a when a + 0 and as l/a when a + 00. Thus, it gives maximum weight to the correlation coefficient when waveform amplitudes are equal, and decreasing weight as they become more different. A practical consequence of weighting C(2) by the bracketted factor in Eq. (3.68)is the tendency to suppressauxiliary maxima. For example, when one can window is on a compressional wavetrain and the other on a shear, UZ) exhibit a maximum because the frequencies of the two wavetrains are not very different. However, the amplitudes are often quite different (Figs. 10 and 59), so the resulting maximum in Sfz)may be small or absent, whereas the compressional-compressional and shearshear maxima retain their large values. This characteristic of semblance is utilized in the algorithm for the data-processing technique described below. The inclusion of amplitude considerations in the calculation of semblance also tends to make shearshear and compressional-compressionalsemblance peaks sharper than the corresponding ones of the correlation coefficient. Semblancehas been applied to waveforms from the two receivers of longspacing sondes of the kind described in Section 3.6.4.2.lZ0The approach was similar to that for the twofold correlationcoefficient(Section 3.6.5.1) except that semblance was used as the measure of likeness. Differences in detail included the use of special filtering techniques, locating the compressionalwave window by visual inspection, application of special criteria for identifying the shear wave, etc.

+

Slowness-Time Coherence.1z1Currently, semblance is employed in the slowness-timecoherence (STC) data-processingof waveforms from an array of receivers. The STC method is probably the most highly developed of the wavetrain-analysis techniques in use at the present time. Although it could use likeness measures other than semblance, this does not appear to have been done. The latest logging tool used with STC employs PZT (lead zirconate tita-

606

JAY TITTMAN

nate) transmitters operating in the 5- to 18-kHz range, with central frequency at 12 kHz. The receiver section consists of 8 wideband piezoelectric transducers 6 in. (15.2 cm) apart, with minimum spacing of 8 ft (2.44 m). Eight-bit waveform digitization is increased to an effective 1 1 bits downhole by use of two parallel channels operating at different gains. The last feature also provides sufficient dynamic range to prevent waveform clipping. Stepped gain changes are made between transmitter firings, by telemetered commands from the truck-borne computer. Waveform-displacement sampling can be performed as frequently as every 5 ps, and up to 5 ms worth of data are buffered in the tool for transmission to the surface between transmitter firings. Alternatively, less frequent sampling permits data-buffering further into the wavetrain. An array processor in the surfaceunit performs in real time the STC data-processing described below. The expression for the semblance of waveforms from n equally spaced receivers is readily written by inspection of Eq. (3.67) and reference to Eq. (3.64):

syt, To)=

where the window-moveout, z, has been replaced by ts, with t an assumed ITT, or slowness,and s the interreceiverdistance. (The absence of a subscript reflects the trial nature o f t and emphasizes that no particular arrival is yet associated with it.) In the following discussion Torepresents, as before, the window starting time at the nearest receiver. In Fig. 59 there are shown three wave-amval T,-positions on the first-receiver waveform; the slopes of the dashed lines correspond to 7, or t.All eight waveforms are recorded with the tool at a particular depth in the hole. Then, using awindow whose width, T,, is determined according to the arrival being sought, a first value of To is chosen conveniently early. As will be seen below, this choice is not critical. Simultaneously, a value for t is assumed and S(t, To)is computed from the recorded waveforms. This process is repeated until all physically interesting values of the pair (t,To)have been covered. Fig. 60 shows the contour map of S(t, To) that results from applyingthis procedure to the waveforms ofFig. 59. The semblancepeaks select, from all the trial values oft, those that are tc,t& , and tst. The STC procedure measures the ITTs of all coherent arrivals traveling with constant speed across the receiver array. Computation of the data for one contour plot like Fig. 60 is made for each depth in the hole and a computer routine is used to find the semblancepeaks. If desired, and this is not always the case, another data-processing program

19. GEOPHYSICAL. WELL LOGGING

607

240

200

-.

q-3 Stoneley

L

v

160

30, c

6

m

120

R

80 40

5 D

\LO)) - compressional 1000 1500 2000 2500 3000 3500 4000 Time (FS)

Fro.60. Contour plot of semblance on the slowness-timeplane, derived from the waveforms of Fig. 59. Interval transit-times (slowness) for compressional,shear, and Stoneley waves are determined from the slowness coordinate of the respective peaks. [From C. F. Moms, T. M. Little, and W. Letton 111, SOC.Pet. Eng. 59th, Ann. Fall Tech. ConL, Sept. 16-19, 1984, Houston, paper number SPE 13285. Copyright 1984 SPE-AIME.]

can associate each peak with a particular arrival. First, the peak locations are determined by a search algorithm based on two criteria: (1) S must be greater than some predetermined threshold value. (2) The S value of the peak selected must be greater than all other S values in a specified rectangular (t,T,)-neighborhood centered on the peak. With the locations of the peaks established in this way, a small set of peak &values is made available for association with particular arrivals. This association can be executed automatically also, by selecting only those t-values that satisfy certain criteria: (1 ) The choice of tccomes from that peak whose values o f t and To minimize To CI To- tL,(,where cis a constant, typically chosen to be 0.5, and L 1is the spacing of the first receiver. (2) The value selected for tahcomes from the highest semblance peak whose t value satisfies fitczst 5 Min(tf, 2. ltc), where the 2. It, limit is empirical. (3) The chosen value of%tcomes from the peak that correspondsto the largest-amplitudearrival witht > 4.From these computations at each depth, logs of the ITTs and amplitudes of the three waves and of derived quantities such as Poisson’s ratio can be recorded. Since STC processing imputes to each wave a constant slownessacross the receiver array, vertical inhomogeneities such as bed boundaries, caves, or thin beds can reduce log quality. However, simulation studies suggest that tc estimates under these conditions are not greatly different from those derived

+

608

JAY TITTMAN

using threshold-detection.122 Generally, logs derived from any of the fullwaveform data-processing methods are relatively immune to the conditions that produce time stretch, cycle skipping, and road-noise spikes. 3.6.6. Direct Measurement of Shear Interval-Transit-Time. The difficulty in measuring t& stems directly from the conventional use of short cylindrical pressure-sensitivetransducers (piezoelectricor magnetostrictive) located on the borehole axis. Thus, transmitters produce in the borehole the axially symmetric pressure waves (Fig. 9) that are refracted into compressional and shear head-waves. Furthermore, as has been noted earlier, because critical-angle refraction is required, the creation of a shear head-wave is prevented when v, < u,. A way around this limitation is to produce formation shear-wavesdirectly, i.e., without using refraction. One method for doing this proposed using a sonde with multiple pads contacting the borehole wall.lZ3Another used a horizontal “dipole” transmitter consisting of a flexiblediaphragm driven by an electromagnetically actuated piston.lZ4 This created asymmetric distortion of the borehole wall. The resulting vertical movement of the distortion shear-wave was detected by a neutral-buoyancy geophone, which is sensitive to displacement rather than pressure. This sonde required point-by-point logging and, consequently, was not widely used. Most recently, there have been reported field tests of an experimental direct shear-wave sonde that can be run at normal logging speeds.lZ5The sonde is shown schematicallyin Fig. 6 1. Sonic isolation of the transducers is achieved by hanging them on segments of 7-conductor logging cable rather than by supporting them in a slotted sleeve. All the transducers are identical (proprietary)“dipole” devices working in the 1 to 3-kHz range. The dipole receivers are sensitive to motion rather than to pressure. Thus, they respond essentially only to the borehole fluid‘s horizontal motion created by the upward passage of the wall distortion. Spacings can be varied from 6 ft (1.83 m) to 15 ft (4.57 m) and spans from 3 ft (0.92 m) to 5 ft (1.52 m). Also shown in Fig. 6 1 are two waveforms recorded in soft sand formations. The upper and lower waveforms are from formations characterized by tc= 103 ,us/ft (338 ,us/m), tsh = 206 p/ft (677 p / m ) and tc= 115 pus/& (377 ps/m), tah= 250 p / f t (820 ps/m), respectively. Conventional sondes would produce no shear head-wave in these formations since tsh2 6. [Usually, 6 = 2 lOps/ft (690ps/m).] Core analysis of the lower formation showed it to be poorly consolidated or loose sand. (The existence of the shear wave reflects the role of overburden pressure in establishing a non-zero shear modulus.) The earliest arrival ofsignijicant amplitude is theshear wave, and it is likely that tshcould be extracted by means of either threshold-crossing

-

detection or full-waveform data-processing techniques. In hard-rock formations it was found that best results were achieved at a frequency of 3 kHz.

-

19.

609


h

@r ASYMMETRIC RECEIVER 1

k“

7-CONDUCTOR LOGGING CABLE

WEIGHT

FIG.6 1. Schematic configuration of a sonde providing direct shear-wave measurement by use of dipole transducers. Upper waveform recorded in a sand formation havingt, = 206 p s / A (677 ps/m), almost exactly equal to tf. Lower wdveform from an unconsolidated sand with t,,,= 250 ps/R (820 ps/m) roughly 25%larger than t p Both are recorded at 1 lcHz with 15 ft (4.57 m) spacing. [Adapted from J. Zemanek, F. A. Angona, D. M. Williams, and R. L. Caldwell, SPWLA 25th Ann. Logging Symp. Trans., New Orleans, June 10- 13, 1984, Vol. I , paper U.]

-

References 1. For a history of the development of borehole logging see L. AUaud and M. Martin,

“Schlumberger, the History of a Technique,” Wiley, New York, 1977. 2. Ibid., pp. IOlff. 3. H. G . Doll, Pet. Trans. AIME 186, 148 (1949). 4. For a briefexplanation see “Interpretation Handbook for Resistivity Logs-Document 4,” p. 9. Schlumberger, Houston, 1951. 5. Ibid., p. 13. 6. Ibid., pp. 37ff.

610

JAY TITTMAN

7. Ibid., pp. 40K 8. See,for example, “Resistivity Departure Curves--Document 3,” Schlumberger,Houston, 1949. 9. Reference 4, pp. 137ff. 10. Ibid., pp. 142K 1 1 . H. G. Doll, Pet. Trans. AIME 192, 305 (195 1); first published in the J. Pet. Technol. SPE-AIME (1951). 12. For results of laboratory experiments illustrating these features see Reference 11, Figs. 3 and 6. 13. J. Suau, P. Grimaldi, A. Poupon, and P. Souhaite, SOC. Pet. Eng.Ann. FallMeet. 47th, Sun Antonio, SPE Pap. 4018 (1972). 14. N. A. Schuster,J. D. Baden, and E. R. Robbins, Gulfcoast Assoc. Geol. Soc. Trans.Meet., 21st, New Orleans, 177 (1971). 15. For actual current maps, computed by use of a finiteelement solution to Laplace’s equation, see R. Chemali, S. Gianzero, R. Strickland, and S.M. Tijani, Soc. Prof: WellLog Anal. Ann. Logging Symp. Trans., 24th, Calgav, II, Pap. UU (1 983). 16. H. G. Doll, U.S.Patent 2,582,314, 1952. Pet. Eng. Ann. Tech. Conf:, Sdth, Sun Fransisco, SPE Pap. 12049 17. T. D. Barber, SOC. (1983). See also R. Woodhouse, D. N. Greet, and C. R. Mohundro, J. Pet. Technol. 36, 993 (1984); D. W. Martin, M. C. Spencer, H. Patel, SOC.Prof: Well Log Anal. Ann. Logging Symp. Trans., 25th. New Orleans, 1, Pap. M (1984); K. A. Teague and R. Yarlagadda, ibid., Pap. P;E. P. Howell and T. E. Fisher, SOC.Prof: Well Log Anal. Ann. Logging Symp. Trans., 23rd, Corpus Christi. 1, Pap. H (1982); S. G. Thadani and G. A. Merchant, SOC.Pet. Eng. Ann.ITech,.Conj, 57rh, New Orleans, SPE Pap. 10986 (1982). (Most of these papers use sonde response hnctions derived from electromagnetictheory, developed in Section 3.1.2.3, rather than from the simple geometric factors discussed to this point.) 18. Reference 17, Martinet al. (1984). 19. J. H. M o m and K. S. Kunz, Geophysics27,829 (1962);W. C . Duesterhoeft, Jr., Geophysics 26,192 ( 1961); A. J. DeWitte and D. Lowitz, Soc. Prof: Well Log Anal. Ann. Logging Symp. Trans., Znd, Dallas, (1961); W. C. Duesterhoefi, Jr., R. E. Hartline, and H. S . Thomsen, J. Pet. Technol. 13,1137 (196 1). For a treatise on this subject, in English, see A. A. Kaufman, “Theory of Induction Logging”. Siberian Dept. of Science Press, Novosibersk, 1965. 20. J. A. Stratton, “Electromagnetic Theory,” p. 2, Eqs. (1) and (2); p. 6, Eqs. (16) and (1 9). McGraw-Hill, New York, 1941. 2 1 . Reference 19, Duesterhoeft et al. (196 1). 22. R. Woodhouse, P. Threadgold, and P. A. Taylor, Log Anal. 16( l), 3 (1975). 23. J. H. Moran, US.Patent 3,147,429, 1964. 24. S. Gianzero and B. Anderson, LogAnal. 23( l), 20 (1982);G. S. Thadani and H. E. Hall, Jr., SOC.Prof: WellLogAnal. Ann. LoggingSymp. Trans.,22nd. Mexico City, 2, Pap. WW (1981). 24a. For a discussion of some subtle problems related to the definition and interpretation of the generalized geometric factor, see J. H. Moran, Log Anal. 24(6), 4 (1982); M. L. Oristaglio, Log Anal. 24(3), 3 (1983); J. H. Moran, Log Anal. 24(3), 4 (1983). 25. Reference 17, Barber (1983). 26. See, for example, Reference 24, Gianzero and Anderson (1982) and Reference 17, Barber (1983). 27. B. Anderson and W. C. Chew, SOC.Pro/: WellLogAnal. Ann. LoggingSymp. Trans.,25th, New Orleans, Pap. HH (1984);B. Anderson and S . K. Chang, LogAnal. 23(6), 17 (1982). 28. H. G . Doll, U.S.Patent 3,166,709, 1965.

19.


61 1

29. P. Souhaite, A. Misk, and A. Poupon, SOC.ProJ Well Log Anal. Ann. Logging Symp. Trans., 16th. New Orleans, Pap. LL (1975). 30. Theoreticalcurves of GILversusd, may be found in Reference29 and in R. Woodhouse, P. Threadgold,and P. A. Taylor, Log Anal. 16(l), 3 (1975).In both references the curves are parameterized by resistivity although the calculations were performed through the use of Eq.(3.24). Woodhouse et al. appear to define the term “pseudo-geometrical factor” as GIL in Eq. (3.24)when skin effect is included. The terminology defined in the present volume is more commonly used.

3I. For a three-page description of the Monte Car10 method see G. I. Bell and S. Gladstone, “Nuclear Reactor Theory,” p. 53, Krieger Publ., Malabar, Florida, 1968. For a more complete, mathematical treatment see M. H.Kalos, F. R. Nakache, and J. Celnik in “Computing Methods in Reactor Physics” (H. Greenspan, C. N. Kelber, and D. Okrent, eds.), p. 365,Gordon & Breach, New York, 1968.A mathematical introduction to the method of discrete ordinates is found in B. G. Carlson and K.D. Lathrop, ibid.,p. 171, and in Bell and Gladstone (1968)p. 214. 32. For a derivation of Eq. (3.27)and other relations involved in the slowing-down of neutrons see, for example, S. Glasstone and M. C. Edlund, pp. 137ff, Van NostrandReinhold, Princeton, New Jersey, 1952;H.Soodak and E. C. Campbell, “Elementary Pile Theory”, pp. 1 - 10,Wiley, New York, 1950. 33. A. M. Weinberg and E. P. Wigner, “The Physical Theory of Neutron Chain Reactors,” Chapter 9. Univ. of Chicago Press, Chicago, 1958;Reference 31, Bell and Gladstone ( 1968), Chapter 1. 34. Ibid.,Weinberg and Wigner (1958);Bell and Gladstone (1968),Chapters 2-6. 35. Section 3.3.2 outlines a derivation of Eq.(3.31) for gamma rays, starting from the transport equation. Although the problem is not identical with that for neutrons, the salient features are similar. Results for the thermal-neutron group in limestone, with varying porosity, calculated using 2, 7,and 25 groups can be found in J. A. Czubek, Znst. Nucl. Phys. (Cracow), Report 1222/AP,p. 41,(1983). 36. C. W. Tittle, Geophysics26,27(1961).Noteconigendum appearingatendofc. W. Tittle and L. S. Allen, Geophysics31,214 (1966). 37. Reference 32,Glasstone and Edlund (1952),p. 106. 38. Reference 36,Tittle and Allen (1966). 39. Reference 32, Glasstone and Edlund (1952),p. 172ff. For specificapplication to multigroup theory and for tabulation of values for logging use see A. Kreft, Inst. Nucl. Techniq. (Cracow), Report 32/I (1972). 40. J. T. Dewan, J. Pet. Technol. 8(2), 50 (1956). 41. J. Tittman, H.Sherman, W. A. Nagel, and R. P. Alger, J. Pet. Technol. 18, 1351 (1966). 42. S. Locke, H.Sherman, and J. S. Wahl, US.Patent 3,483,376,1969;L. S. Allen, C. W. Tittle, W. R. Mills, and R. L. Caldwall, Geophysics 32,60 (1967). 43. R.P. Alger, S. Locke, W. A. Nagel, and H. Sherman, J. Pet. Technol. 24, 1073 (1972); Ibid.,Allen et al. (1967). 44. D.V. Ellis and C. R. Case, Soc. Prof. Well Log Anal. Ann. Logging Symp. Trans., 24th, Calgary, Pap. S (1983);P. J. McDaniel, J . M. Harris, and D. H. Widman, ibid., Pap. LL; H. D. Scott, C. Flaum, and H. Sherman, SOC.Pet. Eng. Ann. Fa// Tech. Conf., 57th. New Orleans, SPE Pap. 11146 (1982); L. S. Allen, W. R. Mills, K. P. Desai, and R. L. Caldwell, Soc. Prof. Well Log Anal. Ann. Logging Symp. Trans. 13th, Tulsa, Pap. G (1972). 45. Reference 44,H.D. Scott et al. (1982). 46. R. R.Davis, J. E. Hall,Y. L. Boutemy, andC. Flaum, SOC.Pet. Eng. Ann. Fall Technol. ConJ, 56th, San Antonio, SPE Pap. 10296 (1981). 47. Reference 44,H.D. Scott et al. (1982);D. V. Ellis, J. Ullo, and H. Sherman, Soc. Per. Eng.

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Ann. Fall Tech. Con& 56th, San Antonio, SPE Pap. 10294 (198 1); J. Ullo, ibid., SPE Pap. 10295. 58th. San Fransisco. SPE 48. J. Ullo and J. Chiaramonte, SOC.Pet. Eng. Ann. Tech. Conf:, Pap. 12 137 ( 1 983). 49. H. Sherman and S. Locke, Soc. Prof: Well Log Anal. Ann. Logging Symp. Trans., 16th. New Orleans, Pap. Q (1975). 50. C. R. Case, Soc. Prof: Well Log Anal. Ann. Logging Symp. Trans., 23rd, Corpus Christi, Pap. L (1982). 5 1. W. Heitler, “The Quantum Theory of Radiation,” (3rd Ed.), p. 217. Oxford Univ. Press, London, 1954. 52. R. D. Evans, “The Atomic Nucleus,” pp. 672ff.,McGraw-Hill, New York, 1955. 53. J. A. Czubek, Znt. J. Appl. Radiat. Zsotop. 34(1), 153 (1983). 54. For a compendium of detailed graphs of Compton energy-angle relationships and the Klein-Nishina cross-section formula see A. T. Nelms, US.Natl. Bur. Stand. Circ. 542 (1953). 55. W. Bertozzi, D. V. Ellis, and J. S.Wahl, Geophysics46,1439 (1981);L. V. Spencer andU. Fano, J. Res. Natl. Bur. Stand. (US.)46,446 (1 95 1). 56. J. Tittman and J. S. Wahl, Geophysics 30,284 (1965). 57. The published literature on Monte Carlo applications to the density logging problem is rather sparse. However, see C. C. Watson, SOC.Pet. Eng. Ann. Tech. ConJ, 58th, San Fransisco, SPE Pap. 1205 1 (1983)and Reference 55, Bertozzi et al. (1981). For a hybrid approach see D. C. Minette, Soc. Prof: Well Log Anal. Ann. Logging Symp. Trans., 24th, Calgary, 2, Pap. ZZ (1983). Reference 53 contains some Monte Carlo results from the Russian literature, as well as a bibliography of Polish and Russian work on the physics of density and photoelectric-effect logging. 58. Reference 54, Nelms (1953), p. 38. 59. F. F. Johnson and J. Tittman, U.S. Patent 3,263,083, 1966. 60. J. S. Wahl, J. Tittman, C. W. Johnstone, and R. P. Alger, J. Pet. Technol. 16, 141 1 ( 1964). 61. D. Ellis, C. Flaum, C. Roulet, E. Marienbach, and B. Seeman, SOC.Pet. Eng. Ann. FaZl Tech. Conf:, 58th, San Francisco, SPE Pap. 12048 (1983); J. Tittman, U.S. Patent 3,521,063, 1970. 62. J . R. Samworth, J. Pet. Technol. 32, 1316 (1980). 63. Reference 55, Bertozzi et al. (1981). 64. Reference 57, Watson (1983). 65. Reference 6 1, Tittman ( 1970). 66. For additional calculated and measured spectra reported in the Russian and Polish literature, see Reference 53, Czubek (1983)and J . A. Czubek, in “Radioisotope Instruments in Industry and Geophysics,” Proc. Symp., Warsaw, 2,249 ( 1 9 6 9 , M A , Vienna (1966). 67. D. C. Moore and J. Tittman, US.Patent 3,858,037, 1974. 68. C. Flaum and G. Pine, SOC.ProJ Well Log Anal. Ann. Logging Symp. Trans., 22nd, Mexico City, 1, Pap. H (1981). 69. J . A. Grau, S. Antkiw, R. C. Hertzog, R. A. Manente, and J. S.Schweitzer, Znt. Symp. Capture Gamma-Ray Spectroscopy and Related Topics, Sth, Knoxville ( 1 984); R. C. Hertzog, Soc. Pet. Eng. J. 20(5), 327 (1980). 58th, San Fransisco, SPE 70. J. S.K. Tsang and M. L. Evans, SOC.Pet. Eng. Ann. Tech. Conf:, Pap. 12052 (1983). 7 1. For a more complete discussion of scintillation spectrometry, see G. F. Knoll, “Radiation Detection and Measurement,” pp. 328ff.,Wiley, New York, 1979. 72. For graphs of recoil-electron energy distributions produced by gamma rays of different energies, see Reference 54, Nelms (1953), Fig. VII.

19.


613

73. L. A. Shope, R. S.Berg, M. L. ONeal, and B. E. Barnaby, IEEE Trans. Nucl. Sci. NS-28, 1696 (1981); A. H. Frentrop and H. Sherman, Nucleonics 18,72 (1960). 74. W. F. Schultz and H. D. Smith, Jr., J. Pet. Techno/. 26, 1103 (1974); G. A. Lock and W. A. Hoyer, J. Pet. Techno/. 26, 1044 (1974); D. W. Oliver, E. Frost, and W. H. Fertl, SOC. Prof. Well Log Anal. Ann. Logging Symp. Trans,, 22nd. Mexico City, Pap. TT (1981); R. B. Culver, E. C. Hopkinson, and A. H. Youmans, SOC.Pet. Eng. J. 14, 463 (1974); B. L. Lawson, C. F. Cook, and J . D. Owen, SOC.Pet. Eng. J . 11, 129 (1971). 75. Private communication from E. C. Hopkinson. 76. Reference 69, Hertzog (1980). 77. Chapter 2, reference 74, Oliver et al. 78. J. S. Wahl, W. B. Nelligan, A. H. Frentrop, C. W. Johnstone, and R. J. Schwartz, Sot. Pet. Eng. J. 10,365 (1970). For a collection of 25 papers on thermal-neutron die-away logging through 1975, see “Pulsed Neutron Logging,” SPWLA reprint volume, SPWLA, Houston, 1976. Some more recent papers on this subject are:R. R. Randall and E. C. Hopkinson, SOC.Prof: Well Log Anal. Ann. Logging Symp. Trans., 22nd, Mexico City, 2, Pap. JJ (1 98 1); J. A. Czubek, K. Drozdowicz, E. Krynicka-Drozdowin, A. Igielski, and U. Woznicka, ibid., 1,Pap. A; R. Randall, E. Hopkinson, and A. H. Youmans, Sot. Pet. Eng. Ann. Fall Tech. Conf:,52nd, Denver, SPE Pap. 6786 (1977); W. B. Nelligan and S. Antkiw, Sot. Pet. Eng. Ann. Full Tech. Conf:,5Ist, New Orleans,SPE Pap. 6 156 (1 976); S. Antkiw, Sot. Prof: Well Log Anal. Ann. Logging Symp. Trans., I7th, Denver, Pap. CC (1976). 79. Reference 69, Grau et al. (1984). 80. P. Westaway, R. Hertzog, and R. E. Plasek, Soc. Pet. Eng. Ann. Fall. Technol. Conf:, 55th. Dallas, SPE Pap. 9461 (1980). 8 1. J. S. Schweitzer, R. A. Manente, and R. C. Hertzog, J. Pet. Technol. 36, 1527 (1 984). 82. R. C. Hertzog and R. E. Plasek, IEEE Trans. Nucl. Sci. NS-26, 1558 (1979). 83. For examplesof applications see: J. S. Schweitzerand R. A. Manente, Int. Symp. Capture Gamma-Ray Spectrosc. Related Top., Zth, Knoxville, (1984); L. H. Goldman, and H. E. Man, SOC.Prof: Well Log Anal. Logging Symp. Trans. 20th, Tulsa, 2, pap. GG (1 979). F. E. Senftle,A. B. Tanner, P. W. Philbin, G. R. Boynton, and C. Schram, U. S. Geol. Surv. Open-File Report 77-162 (1977). 84. For the physics of Ge detection, see G. F. Knoll, “Radiation Detection and Measurement,” pp. 492 ff., Wiley, New York, 1979. 85. Reference 81, Schweitzer et a/. (1984). 86. J. F. Lewkowicz, R. Reischman, and J. J. Walsh, SOC.Prof: Well Log Anal. Ann. Logging Symp., 24th. Calgary, 2, Pap. M M (1983); “Well Evaluation Conference, South East Asia” (A. Winchester, ed.)p. 74. Schlumberger Tech. Sen.,Paris, 1981; K. D. Wyatt, Geophysics46,880 (198 I); P. Kennett, R. L. Ireson, and P. J. Conn, Geophys. Prospect. 28,676 (1 980); E. I. Gal‘perin, “VerticalSiesmicProfiling,” SOC.Explor. Geophys., Tulsa, 1974. 87. B. Froelich, D. Pittman, and B. Seeman, Soc. Pet. Eng. Ann. Full Tech. Conf:, 56th, Sun Antonio, SPE Pap. 10207 (1981); H. D. Brown, V. E. Grijalva, and L. L. Raymer, Soc. Prof: Well Log Anal. Ann. Logging Symp. Trans., llth, Los Angeles, Pap. F (1970); T. Walker, J. Pet. Technol. 20,8 11 ( 1968); G. H. Pardue, R. L. Moms, L. H. Gollwitzer, and J. H. Moran, Trans. AIME 228, 545 (1963); M. Grosmangin, F. P. Kokesh, and P. Majani, J. Pet. Technol. 13, 165 (1961). 88. R. A. Broding, Sot. Prof: Well Log Anal. Ann. Logging Symp. Trans., 24th, Calgary, 1, Pap. B ( 1983); E. S. Pasternackand W. P. Goodwill, ibid., Pap. X; R. A. Broding, Sac.ProJ Well Log Anal. Ann. Logging Syrnp. Trans., ZZnd, Mexico City, 1, Pap. B (198 1); R. Wiley, SOC.Prof: Well Log Anal. Ann. Logging Symp. Trans., Zlst, Lafayette, La., Pap. HH (1980); J. Zemanek, R. L. Caldwell, E. E. Glen, Jr., S. V. Holcomb, L. J. Norton, and A. J. D. Strauss, J. Pet. Technol. 246, 762 (1969).

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89. C. B. Officer, “Introduction to the Theory of Sound Transmission,” pp. 1 - 13,M a r a w Hill, New York, 1958. 90. Ibid., p. 8. 91. Ibid., p. 4. 92. G. Joos, “Theoretical Physics,” p. 170,Blackie, London, 1934. 93. J. E.White, “Underground Sound,” Chapter 5. Elsevier, Amsterdam, 1983;J. E. White and R. E. Zechman, Geophysics33,302 (1968);J. E. White, Geophysics37,327 (1962). 94. J. W.Minear and C. R. Fletcher, SOC.Prof: Well Log Anal. Ann. Logging Symp., 24th. Calgary, 2, Pap. EE (1983). 95. F. L.Paillet and J. E. White, Geophysics 47, 1215 (1982). 96. F.L.Paillet, SOC. Prof:WeNLogAnal.Ann. Logging Symp. Trans., 22nd, Mexico City, 2, Pap. SS (1981). 97. C. H.Cheng and M. N. Toksiiz, Geophysics 46, 1042 (1981);SOC.Prof: Well Log Anal. Ann. Logging Symp. Trans., 21st, Lafayette, La., Pap. J. (1980). 98. L.Tsang and D. Rader, Geophysics44,1706 (1979). 99. M. A. Biot, J.Appl. Phys. 23, 997 (1952). 100. See Reference 99,Biot (1952),for a collection of dispersion curves covering a wide range of parameters.

101. Private communication from C. F. Moms. 102. J. G.Scholte,K. Ned. Akud. Wet. 51,533(1948);R. Stoneley,Proc.R.SOC.Ser. A 106,414 ( 1924). 103. S.T. Chen and D. E. Willen, SOC.Prof: Well LogAnal. Ann. Logging Symp. Trans.. 25th. New Orleans, 1, Pap. DD (1984). 104. 0.Liu, ibid., 2, Pap. ZZ. 105. C. H.Cheng and M. N. Toksliz, SOC.Prof: Well Log Anal. Ann. Logging Symp. Trans., 24th, Calgary, 1,Pap. V (1983). 106. C. H. Cheng and M. N. Tokdz, Soc. Prof: Well Log Anal. Ann. Logging Symp. Trans., 23rd. Corpus Christi, 1,Pap. P (1982). 107. J. H. Rosenbaum, Geophysics39, 14 (1974). 108. F.Gassmann, VierteljahrsschriftNaturforsch. Ges. Zurich 96(1),1 (1951);Geophysics16, 673 (1951);ibid., 18, 269 (1951). The outline in the text follows the very readable condensation found in Reference 93,White (1983)p. 57ff. 109. M. A.Biot, J.Acoust. SOC.Am. 34,1254(1962);J.Appl. Phys. 33, 1482 (1962);J.Acoust. SOC. Am. 28, 179 (1956);ibid., p. 168.The last mentioned paper is the most relevant for this discussion. See also J. Geertsma and D. C. Smit, Geophysics 26, 169 (1961).A more detailed summary of the Biot theory than is given in the text can be found in D. L. Johnson and T. J. Plona, J.Acousf. SOC.Am. 72(2),558 (1982). 110. R. N. Chandler and D. L. Johnson, J. Appl. Phys. 52,3391 (1981). 1 1 1. T. J. Plona, Appl. Phys. Lett. 36,259 (1980);R.N.Chandler, J.Acoust. SOC. Am. 70,116 (1981). 1 12. D. H.Thomas, Log Anal. 19,23(1978);F.P. Kokesh, R. J. Schwartz, W.B. Wall, and R. L. Moms, J. Pet. Technol. 17,282 (1965). 113. Ibid. 114. W. G. Hicks, Geophysics 24,451 (1959);F. P. Kokesh and R. P. Blizard, ibid., p. 64. 1 15. D. M.Williams, J. Zemanek, F.A. Angona, C. L. Dennis, and R. L. Caldwell, SOC.Prof: Well Log Anal. Ann. Logging Symp. Trans., 25th, New Orleans, 1, Pap. T (1984);“The Long Spacing Sonic,” technical pamphlet. Schlumberger, Houston, 1980;P. E. F.Goetz, L. Dupal, and J. Bowler, Aust. Pet. Explor. Assoc. J. 19, 131 (1979). 1 16. C. C. Purdy, SOC.Prof: Well Log Anal. Ann. Logging Symp. Trans., 23rd, Corpus Christi, 2, Pap. V (1982).

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117. Reference 115, Williams et al. (1984). 1 18. J. Aron, J. Murray, and B. Seeman, Soc.Pet. Eng. Ann. Fall. Tech. Conf:,53rd, Houston, SPE Pap. 7446 (1 978); J. D. Ingram, U.S.Patent 4,2 10,965,1980. 119. J. D. Ingram, C. F. Moms, E. E. Macknight, and T. W. Parks, Ann. Int. Soc. Explor.

Geophys. Meet., 51st, Los Angeles, Pap. 5113 (1981). 120. C. H. Cheng, M. N. Toksbz, and M. E. Willis, Soc. Prof: Well Log Anal. Ann. togging Symp Trans., 2 2 ~ 4 Mexico City, 1, Pap. 0 (1981); R. W. Siegfried and J. P. Castagna, SOC.Prof: Well Log Anal. Logging Symp. Trans., 23rd, Corpus Christi. Pap. I (1982). 121. C. F. Moms, T. M. Little, and W. Letton UI,SOC.Pet. Eng. Ann. Fall Tech. ConJ, 59th, Houston, SPE Pap. 13285 (1984); C. V. Kimball and T. L. M m t t a , Geophysicc49,274 (1984). 122. P. T. Wu, SOC.Pet. Eng. Ann. FaN Tech. ConJ. 59th, Houston, SPE Pap. 13286 (1984). 123. J. E. White, SOC.Prof: Well Log Anal. Ann. Logging Symp. Trans., Bth, Denver, Pap. I (1967). 124. C. Kitsunezaki, Geophysics 45, 1489 (1980). 125. J. Zemanek, F. A. Angona, D. M. Williams,and R. L. Caldwell, Sac.Prof:WellLogAnal. Ann. Logging Symp. Trans., 25th, New Orleans, 1, Pap. U (1984).


Coseismic strain. 414 Critical pressure, see Breakdown pressure Crustal deformation, 410-1 11 Cryogenic magnetometer, 328 CSEM, see Controlled source electromagnetics Current channeling, 318-320, 323, 341.

A Accelerograph, strong motion, 11 Accelerometer, force-balanced. 7, 65-73 Acoustic waves, see Seismic waves Amarada temperature recorder, 192, 194 AMT, see Audiomagnetotellurics Anelastic strain recovery, 404 Apparent resistivity, 335-338, 504 Archie’s Law, 284, 291, 462 Arrays, resistivity, and induced polarization electromagnetic coupling, 304, 311-313 resolution, 304, 311 selection, 301, 304 types, 303

Audiomagnetotellurics. 313, 315, see also Magnetotellurics

B Benioff short-period seismometer, 51-52 Body waves, 11-12, 581-584 Boltzmann’s constant, 286, 290 Borehole breakouts, 401-403 Borehole deformation gage, 383-386 Borehole thermal equilibration, 192-193. 245-246

Bouguer correction, 151 Bourdon tube, see Amarada temperature recorder Breakdown pressure, 393 Broad-window-ratio method, 570-573 Bullard probe, 231-232, 238

364-365

Current focusing, 308-3 10

D Damping of seismometer, see also Pendulum, damped critical, 26, 52 damping ratio, 26 resistive, 29 Deep Sea Drilling Project, 243-246 Deep Tow System, %, 100 Differential strain curve analysis, 404 Dilatometer, 430-432, see also Strainmeter Dipmeter. 402-403 Dipoledipole array, 2%-299, 303-305 Discriminator. 43 Doorstopper gage, 386-389 DPD, see Waveform processing, direct phase determination Drilling mud, see also Spine-and-ribs cross plot effect on well logs, 445, 447448 invasion into borehole wall, 448 DSDP, see Deep Sea Drilling Project Dual laterolog, see Laterolog

E

C

Compton scattering, 549-550 Controlled source electromagnetics data acquisition, 352-357 exploration method. 346-349 FFM and TEM systems, 352-354 frequency bands, 357 source configurations, 354-357 sources of noise in exploration, 361-363 Convection borehole, 202-206 seafloor, 257-259

Earth tides, 143-146 Echo sounders beamwidth, 87-90 conversion of traveltime to depth, 94 design considerations, 87-97 frequency of operation, 87, 95 multibeam, 90-94 pulse length, 87 recording, 88 resolution, 88-89 single beam, 94-96 617

618

INDEX

Eddy current induction, 351 Elastic waves, see also Free oscillations amplitude, 11-12 body, 12 frequency range, 11 surface, 12 Electric and magnetic fields, 306-307, 317-318, see also Maxwell’s equations Electrical resistivity apparent resistivity. 295, 504 arrays, 2%, 301, 303-305 departure curves, 507-511 dependence on pore fluids, 460-463 effect of clays, 284-285 effect of temperature, 283 formation factor, 460 measurement boreholes, 502-530 surface arrays, 291-302 properties of earth materials, 283 Surveys, principals, 292-299 Electrode polarization, 285-287, see also Induced polarization, mechanisms Electrolyte conduction, 283 Electromagnetic wave equations boundary conditions for earth,275-277 derivation, 270-272 impedances. 277-283, 315-316. 320-327, 333-335

solutions, 272-275 Electromagnetic wave propagation in well logging, 491 Eotv(is effect, 130 Epithermal neutron detection, 540-541 Excess mass, 156 Extensometer. 381-382, see also Strainmeter

F FBA, see Accelerometer, force-balanced Finite amplitude acoustics, 90 Flat jack, 381-383 Flattening of spheroid, 132 Free& correction, 151 Free oscillations. 11. 14 Frequency modulation in seismic data transmission, 3 5 4 1 well logging, 198 G g, 128-129 Gal. 128

Galvanometer amplitude and phase, 34 attenuation factor, 33 coupled to pendulum, 32-35 coupling factor, 34 equations of motion, 32-34 use for signal magnification, 32 Gamma-ray detectors, 551-555. 569-581 formation density, 545-559 interactions, 545-548 photoelectic absorption, 473-476.559-563 radioactivity, 450, 491 scattering, 467-472 spectra, 491, 582 spectrometry natural source, 489491 neutron excited, 483-489, 563-581 GDSN, see Global Digital Seismic Network Geodetic satellites cameras, 166-168

orbital characteristics, 164-166 orbit dynamics disturbing function, 173 equations of motion, 170-173 forced perturbations. 175-177 resonance. 177-178 radar altimetry, 169, 181-185 range rate, 168-170 Geodetic techniques, 409 Geoelectric section. 346-348 Geoid, 130. 146-148, 182 Geostationary orbiting earth satellite, 50 Germanium spectrometer, 580-581 Global Digital Seismic Network, 7 Global Positioning System, 93, 119. 168 Global SeismographicNetwork, 71 Gloria, 103 GOES, see Geostationary orbiting earth satellite GPS,see Global Positioning System Gravimeter absolute, 133-135 free-fall. 134-135 LaCoste-Romberg , 137-1 39 moving platform, 138-142 pendulum, 133, 140 relative, 135-138 superconducting, 138 Gravity anomaly B O U W ~ , 152-153

downward continuation, 154-155

619

INDEX interpretation, 153-1 S6 isostatic, 152 monitoring, 156-159 Gravity measurement, see uho Gravimeter at sea, 138-141 from aircraft. 141-142 in boreholes, 155-156 satellite, 163-164 GSN, see Global Seismographic Network Gyroscope, 2, 15

H Heat flow calculation, 216-217, 254-255 corrections, 191 environmental disturbances, 255-259 in shallow boreholcs. 218-219 refraction, 256-257 Holographic interferometry, 403404 Hydraulic fracturing, 393-401 Hydrofrac, see Hydraulic fracturing Hydrophone, 15 Hydrothermal circulation, see Convection Hydrothermal vents, detection, 240

I Induced polarization arrays, 296, 301, 303-305 data acquisition, 300-302 data processing, 302-303 mechanisms, 285 surveys, principals, 292-299 uses, 292 Inertial mass, 4, 14, 15, see uho Pseudostationary point Interfacial impedance, 286-288 Interferometer Fabry-Perot, 417-419.433 Michelson, 417-418, 432 use in measuring g, 134 International Gravity Formula, 132 Interval transit time, 476 IP, see Induced polarization Isostasy, see Isostatic compensation Isostatic compensation, 148-150 J

Johnson-Matheson seismometer, 54-59

K Kepler ellipse, 170 KS-36ooo seismometer, 67

Kuster temperature recorder, 192

L Large-aperture seismic array. 10, 36 LASA. see Large-aperture seismic array Laterolog. 51 1-516 Leaky mode, 585-586, see afso PseudoRayleigh wave; Rverberant wave; Stonely wave Lehner-Griffith short-period seismometer, 52-54 Linear variable differential transducer, 7 4 417,421 LVDT, see Linear variable differential transducer

M Magnetotellurics data acquisition, 327-332 exploration method, 313-314 sensor deployment, 328-330 sources of fields. 314-315 sources of noise in exploration, 339-340 TE and TM modes, 317 Manganese nodules, 100 Marine sediments characterization by acoustic backscatter, 97 sound transmission attenuation, 82 penetration, 94 velocity, 85 Maximum reading thermometer, 192-194 Maxwell’s equations, 266-267, 524 Mean harmonic thermal conductivity, 216 Melt conduction, 291 Membrane polarization, 288-290, see &o Induced polarization, mechanisms Mgal, see Gal Michelson-Gale tiltmeter, 425, 528 Microclimatic effect on geothermal gradient. 191 Microearthquake network basic elements, 36-37 short-period telemetered, 11 U. S. Geological Survey, 36 Moho, 3 MT, see Magnetotellurics MT transfer function, 333-335 Multibeam sonar, see also Sea Beam; Sonar artifacts, 92-93 navigation, 93 performance parameters, 92

620

INDEX

Multiplexing, see Telemetry, modes

N Neutron detection. 538-542 moderation. 463-466 scattering, 531-534 slowing down length, 464-466. 537-538 transport and diffusion, 534-538 Nonpolarizing electrodes, 328 Nuclear magnetic resonance application to borehole fluid properties, 494-497

principal, 494 spin relaxation time, 494-495 0

Oil saturation by C/O ratio, 484-486 Optimal telemetry system, 41 OTS, see Optimal telemetry system Outrigged probe, 232-233 Overcoring, 383-389

Pinger, 110-1 11 Pogo probe technique, 239-240 Poisson’s ratio, 378, 3% Pole-dipole array, 2%, 303-305 Pore pressure, 450 Poroelastic parameter, 396 Positioning, acoustic, 110-120. see also Transponder; Pinger Press-Ewing long-period seismometer, 56-58 Pressure breakdown, 393 critical, see Pressure, breakdown pore, 3% shut-in, 395 Pseudogeometric factor, see Sonde, geometric factor Pseudo-Rayleigh wave, 586-588, see also Leaky mode; Stonely wave; Reverberant wave Pseudosection, 298, 309 Pseudostationary point, 1

R P Packer impression, 397. 399 inflatable. 393, 397 straddle, 397-398 Pendulum, see also Pseudostationary point; Inertial mass absolute measurement of g, 133, 140 dam@ forced oscillation, 27-28 motion, 25-27 galvanometer coupling, 32-35 garden gate, 6, 23 horizontal equations of motion, 22-23 period, 22, 24 period lengthening, 24 inverted, 6, 24 natural period. 5 use in seismic system, 2, 4-6 vertical angular frequency, 16, 17 equation of motion, 15 equilibrium condition, 15 period, 16. 17, 19 period lengthening, 16-21

Radioactivity, measurement, 213 Ranger seismometer, 61 Reverberant wave, 588-589, see also Leaky mode; Pseudo-Rayleigh wave; Stonely wave Richter magnitude, 56

S

San Andreas fault, 405 Satellites, see Geodetic satellites Schlumberger array, 296-297, 303-305 Sea Beam echo sounder, 8 1.91, 97 Sea MARC, 98. 103 Seismic array, 36, see also Large aperture seismic array Seismic background noise detection, 38 filtering, 38 power spectral density, 12-14 sources, 12 Seismic data compression, see Seismic event detection Seismic event detection, 48 Seismic monitoring system, see seismograph, system

INDEX

621

slant-range transformation, 105-106 Seismic network, 7, 35-37, see also Seismotypes, 103-104 graph, network; World-Wide StandardSidewall fracturing, borehole, see Borehole, ized Seismograph Network; Microearthquake network breakouts Seismic recorder, see also Seismograph Skin effect, 523-528 analog tape, 4,47-48 Sonar, see also Echo sounder digital, 4 background noise photosensitive, 4, 45, 47 ambient, 78 smoked paper, 4, 45 bubble sweepdown problem, 81 Seismic Research Observatory, 7, 67 Knudsen curves, 79-80 Seismic signals, telemetered local, 80 conditioning, 38 reduction, 81-82 modes of reception, 41-43 relation to operating frequency, 87 recording, 43-48 sea-surface reverberation, 82-83 timing, 49-50 sources of interference, 88 transmission, 3842, see also Telemetry use of directional transmitters, 82 Seismic waves, see atso Elastic waves voIume reverberations, 82 boreholes, 581-590 beam, angular resolution, 84 porous media bottom imaging, 97-1 10 Biot theory, 592-593 multibeam swath mapping, 90-93 Gassman model, 591-592 phase difference bottom mapping, velocity of compressional waves in 108-1 10 various materials. 479 scanning, 98, 107 Seismogram, 1 side-looking, see Side-looking sonar Seismograph systems, 77 damping, 25-28 Sonde, 441 dynamic range, 12 borehole compensated density, 553-555 early devices, 2-3 combination, 456-458 electromagnetic, 6, 51-65 focused, 520-523 long-period, 14, 56-65 geometric factor, 451453, 516-520, magnification, 5-6 network, 1 528-530, 542-545 multiple sensor, 453-456 output, 61-63 Short-period, 14, 51-56, 61-63 types, 449-458 system, 2-3 Sonic measurements in boreholes Seismometer, see also Seismograph compressional interval-transit-time, Seismoscope, 3 593-599 Semiconduction in earth, 289 depthderived borehole compensation, Shut-in pressure, 395 591-599 Side-looking sonar, see also Sonar, bottom shear interval-transit-time, 608-609 imaging wave forms, 600-608 method of imaging, 99 Sonic velocity, see Sound, velocity nature, 98 Sound naval systems. 104 absorption in sea water, 78-79 on-line processing, 105 reflectivity patch width, 103 frequency dependence, 86 recording sea floor, 85 digitization, 106 velocity facsimile, 93 borehole, 476, see also Interval transit time varying gain (TVG),98 time effect of porosity, 477-478

622

INDEX

Sound, velocity, (cunt.) in ocean, 83 Matthew’s tables, 84 sea floor, 85 shadow zone, 84 sounding, 83-84 Spectral fitting method capture- 7 mode, 573-577 inelastic mode, 578-580 Spine-and-ribs cross plot, 555-558 Spontaneous potential. 449 Squid, see Cryogenic magnetometer SRO. see Seismic Research Observatory Stoke’s integral, 147-148 Stonely wave, 589-590, see also Leaky mode; Pseudo-Rayleigh wave; Reverberant wave Strain, see ufso Crustal deformation gage, 379-380, 383, 386-393,404 principal, 380 rosette, 379-381, 386-387, 390-391 triaxial, 389 Strainmeter Benioff, 429430 comparisons, 434 design, 415, 429433 drift, 413 hydraulic, 431432 installation, 421424 borehole, 423-424 cavity effect, 422 optical anchor, 423 thermal stability, 422, 427-428 underground, 422-423 laser, 432-433 materials, properties, 416 mechanical, 429-431 noise power spectra, 414 sources, 412 quantity measured, 410 seismic Stress in-situ, 377

principal, 377, 380, 382, 386, 388, 394-397

shear, 405 state of, 377-378.404-405 Stress relief method, 379, 383, see ulso Strain, rosette; Overcoring

STS seismometer, 67-71

T Tectonic deformation, 412-413 Telemetry acoustic, 237 links, 35 modes, 35,38 seismic, 2, 5, 36 Televiewer, borehole sonic, 399-401, 403 Temperature measurement, see ulso Thermistor; Thermocouple at high temperature, 197 during drilling, 217-218. 241-245 semiconductor integrated circuit, 197 Temperature recording, 235-238 Tensile strength, hydrofracturing, 396-397 Thermal conductivity driil cuttings, 209 effect of porosity, 213-215 estimates from well logging, 212, 254 in-sifu measurement, 251-253 mean harmonic, 216 measurement divided bar, 207, 25 1 needle probe, 210, 248-251 transient techniques. 210-212, 251 relation to water content in sediments, 254

Thermal neutron detection, 541-542 Thermistor. see uku Temperature measurement marine heat flow, 230-231 precision. 200-206 use in borehole temperature measurement, 195-1% Thermocouple, 197, 229-230 Tilt, see Crustal deformation Tiltmeter, see also Strainmeter bubble, 425-427 design. 415, 424429 drift, 413 indicator equation, 424 installation. see Strainmeter, installation long baseline, 427-429 materials, properties, 416 noise power spectra. 413

sources, 412 pendulum, 425-426

623

INDEX quantity measured, 410 short baseline, 424427 Time code broadcasting, 49-50 Tipper, 323. 325-326, 332 Topographic effect gravity exploration, 151-152 heat flow, 191 magnetotelluricsurvey, 341 Transducer, see also Interferometer capacitive, 417, 420 displacement, 416-421 LVDT, see Linear variable differential transducer optical lever, 417 seismic electromagnetic, 29-32 electromotive, 29 parametric, 29 use for magnification, 29 strain/tilt design criteria, 41I economic criteria, 415 Transponder accuracy of systems for positioning, 112-114, 117 arrays, 117-120

determining relative coordinates, 118-120 interrogation, 114, 116 long baseline system, 111-1 12 nature, 111 navigation configuration, 115-1 16 navigation system, 122 obstacle avoidance, 122 short baseline systems, 111-1 12

up-looking sounder, 122 Triaxial strain cell, 389-393 Tube wave, see Stonely wave

V VCO, see Voltagecontrolled oscillator Vertical seismic profile, 477 Violin bow probe, 233, 253 Voltage-controlled oscillator, 39-41

W Warburg impedance, 287-288 Waveform processing borehole sonic data, 6o0608 direct phase determination, 602-604 fourfold correlation, 600-602 semblance correlation. 604-608 Wenner array, 2%. 303-304 Wirehe logging, 192-200,441-443 cable resistance, 195 high-temperature insulation, 195 Wood-Anderson torsion seismometer, 54-56

World-Wide Standardized Seismograph Network, 7, 21, 58 WWSSN,see World-Wide Standardized Seismograph Network

wwv, 49

WWVB, 50 Z

Zero-length spring, 6, 18-21, 136-137

CONTENTS OF VOLUME 24, PART A 1. Elastic Properties of Rocks and Minerals DONALD

J. WEIDNER

2. Laboratory Measurement of Internal Friction in Rocks and Minerals at Seismic Frequencies

LOUISPESELNICK AND HSI-PINGLIU

3. Measurement of Rock Deformation at High Temperatures

D. L. KOHLSTEDTAND P.N. CHOPRA 4. Diffusion Measurements: Experimental Methods

F. J. RYERSON

5. Rock Fracture and Frictional Sliding HARTMUT

SPETZLER

6. Shock Wave Techniques for Geophysics and Planetary Physics

THOMASJ.

AHRENS

7. The Mutianvil Press

E. K. GRAHAM 8. Thermal Conductivity of Rocks and Minerals

K. HORAIAND T.SHANKLAND 9. Experimental Methods in Rock Magnetism and Paleomagnetism M. FULLER INDEX

624

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