The Rock Physics Handbook: Tools for Seismic Analysis of Porous Media - 2nd edition

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The Rock Physics Handbook, Second Edition Tools for Seismic Analysis of Porous Media

The science of rock physics addresses the relationships between geophysical observations and the underlying physical properties of rocks, such as composition, porosity, and pore fluid content. The Rock Physics Handbook distills a vast quantity of background theory and laboratory results into a series of concise, self-contained chapters, which can be quickly accessed by those seeking practical solutions to problems in geophysical data interpretation. In addition to the wide range of topics presented in the First Edition (including wave propagation, effective media, elasticity, electrical properties, and pore fluid flow and diffusion), this Second Edition also presents major new chapters on granular material and velocity–porosity–clay models for clastic sediments. Other new and expanded topics include anisotropic seismic signatures, nonlinear elasticity, wave propagation in thin layers, borehole waves, models for fractured media, poroelastic models, attenuation models, and cross-property relations between seismic and electrical parameters. This new edition also provides an enhanced set of appendices with key empirical results, data tables, and an atlas of reservoir rock properties expanded to include carbonates, clays, and gas hydrates. Supported by a website hosting MATLAB routines for implementing the various rock physics formulas presented in the book, the Second Edition of The Rock Physics Handbook is a vital resource for advanced students and university faculty, as well as in-house geophysicists and engineers working in the petroleum industry. It will also be of interest to practitioners of environmental geophysics, geomechanics, and energy resources engineering interested in quantitative subsurface characterization and modeling of sediment properties. Gary Mavko received his Ph.D. in Geophysics from Stanford University in 1977

where he is now Professor (Research) of Geophysics. Professor Mavko co-directs the Stanford Rock Physics and Borehole Geophysics Project (SRB), a group of approximately 25 researchers working on problems related to wave propagation in earth materials. Professor Mavko is also a co-author of Quantitative Seismic Interpretation (Cambridge University Press, 2005), and has been an invited instructor for numerous industry courses on rock physics for seismic reservoir characterization. He received the Honorary Membership award from the Society of Exploration Geophysicists (SEG) in 2001, and was the SEG Distinguished Lecturer in 2006. Tapan Mukerji received his Ph.D. in Geophysics from Stanford University in 1995 and

is now an Associate Professor (Research) in Energy Resources Engineering and

a member of the Stanford Rock Physics Project at Stanford University. Professor Mukerji co-directs the Stanford Center for Reservoir Forecasting (SCRF) focusing on problems related to uncertainty and data integration for reservoir modeling. His research interests include wave propagation and statistical rock physics, and he specializes in applied rock physics and geostatistical methods for seismic reservoir characterization, fracture detection, time-lapse monitoring, and shallow subsurface environmental applications. Professor Mukerji is also a co-author of Quantitative Seismic Interpretation, and has taught numerous industry courses. He received the Karcher award from the Society of Exploration Geophysicists in 2000. Jack Dvorkin received his Ph.D. in Continuum Mechanics in 1980 from Moscow

University in the USSR. He has worked in the Petroleum Industry in the USSR and USA, and is currently a Senior Research Scientist with the Stanford Rock Physics Project at Stanford University. Dr Dvorkin has been an invited instructor for numerous industry courses throughout the world, on rock physics and quantitative seismic interpretation. He is a member of American Geophysical Union, Society of Exploration Geophysicists, American Association of Petroleum Geologists, and the Society of Petroleum Engineers.

The Rock Physics Handbook, Second Edition Tools for Seismic Analysis of Porous Media

Gary Mavko Stanford University, USA

Tapan Mukerji Stanford University, USA

Jack Dvorkin Stanford University, USA

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521861366 © G. Mavko, T. Mukerji, and J. Dvorkin 2009 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2009 ISBN-13

978-0-511-65062-8

eBook (NetLibrary)

ISBN-13

978-0-521-86136-6

Hardback

Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Preface

page xi

1

Basic tools

1.1 1.2 1.3 1.4

The Fourier transform The Hilbert transform and analytic signal Statistics and probability Coordinate transformations

1 6 9 18

2

Elasticity and Hooke’s law

21

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

21 23 35 39 40 43 47

2.9 2.10 2.11 2.12

Elastic moduli: isotropic form of Hooke’s law Anisotropic form of Hooke’s law Thomsen’s notation for weak elastic anisotropy Tsvankin’s extended Thomsen parameters for orthorhombic media Third-order nonlinear elasticity Effective stress properties of rocks Stress-induced anisotropy in rocks Strain components and equations of motion in cylindrical and spherical coordinate systems Deformation of inclusions and cavities in elastic solids Deformation of a circular hole: borehole stresses Mohr’s circles Static and dynamic moduli

3

Seismic wave propagation

81

3.1 3.2

Seismic velocities Phase, group, and energy velocities

81 83

v

1

54 56 68 74 76

vi

Contents

3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16

NMO in isotropic and anisotropic media Impedance, reflectivity, and transmissivity Reflectivity and amplitude variations with offset (AVO) in isotropic media Plane-wave reflectivity in anisotropic media Elastic impedance Viscoelasticity and Q Kramers–Kronig relations between velocity dispersion and Q Waves in layered media: full-waveform synthetic seismograms Waves in layered media: stratigraphic filtering and velocity dispersion Waves in layered media: frequency-dependent anisotropy, dispersion, and attenuation Scale-dependent seismic velocities in heterogeneous media Scattering attenuation Waves in cylindrical rods: the resonant bar Waves in boreholes

138 146 150 155 160

4

Effective elastic media: bounds and mixing laws

169

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17

Hashin–Shtrikman–Walpole bounds Voigt and Reuss bounds Wood’s formula Voigt–Reuss–Hill average moduli estimate Composite with uniform shear modulus Rock and pore compressibilities and some pitfalls Kuster and Tokso¨z formulation for effective moduli Self-consistent approximations of effective moduli Differential effective medium model Hudson’s model for cracked media Eshelby–Cheng model for cracked anisotropic media T-matrix inclusion models for effective moduli Elastic constants in finely layered media: Backus average Elastic constants in finely layered media: general layer anisotropy Poroelastic Backus average Seismic response to fractures Bound-filling models

169 174 175 177 178 179 183 185 190 194 203 205 210 215 216 219 224

5

Granular media

229

5.1 5.2

Packing and sorting of spheres Thomas–Stieber model for sand–shale systems

229 237

86 93 96 105 115 121 127 129 134

vii

Contents

5.3 5.4 5.5

Particle size and sorting Random spherical grain packings: contact models and effective moduli Ordered spherical grain packings: effective moduli

245 264

6

Fluid effects on wave propagation

266

6.1 6.2 6.3 6.4

266 272 273

6.20 6.21

Biot’s velocity relations Geertsma–Smit approximations of Biot’s relations Gassmann’s relations: isotropic form Brown and Korringa’s generalized Gassmann equations for mixed mineralogy Fluid substitution in anisotropic rocks Generalized Gassmann’s equations for composite porous media Generalized Gassmann equations for solid pore-filling material Fluid substitution in thinly laminated reservoirs BAM: Marion’s bounding average method Mavko–Jizba squirt relations Extension of Mavko–Jizba squirt relations for all frequencies Biot–squirt model Chapman et al. squirt model Anisotropic squirt Common features of fluid-related velocity dispersion mechanisms Dvorkin–Mavko attenuation model Partial and multiphase saturations Partial saturation: White and Dutta–Ode´ model for velocity dispersion and attenuation Velocity dispersion, attenuation, and dynamic permeability in heterogeneous poroelastic media Waves in a pure viscous fluid Physical properties of gases and fluids

331 338 339

7

Empirical relations

347

7.1

Velocity–porosity models: critical porosity and Nur’s modified Voigt average Velocity–porosity models: Geertsma’s empirical relations for compressibility Velocity–porosity models: Wyllie’s time-average equation Velocity–porosity models: Raymer–Hunt–Gardner relations

6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17 6.18 6.19

7.2 7.3 7.4

242

282 284 287 290 292 295 297 298 302 304 306 310 315 320 326

347 350 350 353

viii

Contents

7.5

7.8 7.9 7.10 7.11 7.12 7.13 7.14

Velocity–porosity–clay models: Han’s empirical relations for shaley sandstones Velocity–porosity–clay models: Tosaya’s empirical relations for shaley sandstones Velocity–porosity–clay models: Castagna’s empirical relations for velocities VP–VS–density models: Brocher’s compilation VP–VS relations Velocity–density relations Eaton and Bowers pore-pressure relations Kan and Swan pore-pressure relations Attenuation and quality factor relations Velocity–porosity–strength relations

358 359 363 380 383 383 384 386

8

Flow and diffusion

389

8.1 8.2 8.3 8.4 8.5 8.6 8.7

Darcy’s law Viscous flow Capillary forces Kozeny–Carman relation for flow Permeability relations with Swi Permeability of fractured formations Diffusion and filtration: special cases

389 394 396 401 407 410 411

9

Electrical properties

414

9.1 9.2 9.3 9.4 9.5

Bounds and effective medium models Velocity dispersion and attenuation Empirical relations Electrical conductivity in porous rocks Cross-property bounds and relations between elastic and electrical parameters

414 418 421 424 429

Appendices

437

Typical rock properties Conversions Physical constants Moduli and density of common minerals Velocities and moduli of ice and methane hydrate

437 452 456 457 457

7.6 7.7

A.1 A.2 A.3 A.4 A.5

355 357

ix

Contents

A.6 A.7 A.8

Physical properties of common gases Velocity, moduli, and density of carbon dioxide Standard temperature and pressure

468 474 474

References Index

479 503

Preface to the Second Edition

In the decade since publication of the Rock Physics Handbook, research and use of rock physics has thrived. We hope that the First Edition has played a useful role in this era by making the scattered and eclectic mass of rock physics knowledge more accessible to experts and nonexperts, alike. While preparing this Second Edition, our objective was still to summarize in a convenient form many of the commonly needed theoretical and empirical relations of rock physics. Our approach was to present results, with a few of the key assumptions and limitations, and almost never any derivations. Our intention was to create a quick reference and not a textbook. Hence, we chose to encapsulate a broad range of topics rather than to give in-depth coverage of a few. Even so, there are many topics that we have not addressed. While we have summarized the assumptions and limitations of each result, we hope that the brevity of our discussions does not give the impression that application of any rock physics result to real rocks is free of pitfalls. We assume that the reader will be generally aware of the various topics, and, if not, we provide a few references to the more complete descriptions in books and journals. The handbook contains 101 sections on basic mathematical tools, elasticity theory, wave propagation, effective media, elasticity and poroelasticity, granular media, and pore-fluid flow and diffusion, plus overviews of dispersion mechanisms, fluid substitution, and VP–VS relations. The book also presents empirical results derived from reservoir rocks, sediments, and granular media, as well as tables of mineral data and an atlas of reservoir rock properties. The emphasis still focuses on elastic and seismic topics, though the discussion of electrical and cross seismic-electrical relations has grown. An associated website (http://srb.stanford.edu/books) offers MATLAB codes for many of the models and results described in the Second Edition. In this Second Edition, Chapter 2 has been expanded to include new discussions on elastic anisotropy including the Kelvin notation and eigenvalues for stiffnesses, effective stress behavior of rocks, and stress-induced elasticity anisotropy. Chapter 3 includes new material on anisotropic normal moveout (NMO) and reflectivity, amplitude variation with offset (AVO) relations, plus a new section on elastic impedance (including anisotropic forms), and updates on wave propagation in stratified media, and borehole waves. Chapter 4 includes updates of inclusion-based effective media models, thinly layered media, and fractured rocks. Chapter 5 contains xi

xii

Preface

extensive new sections on granular media, including packing, particle size, sorting, sand–clay mixture models, and elastic effective medium models for granular materials. Chapter 6 expands the discussion of fluid effects on elastic properties, including fluid substitution in laminated media, and models for fluid-related velocity dispersion in heterogeneous poroelastic media. Chapter 7 contains new sections on empirical velocity–porosity–mineralogy relations, VP –VS relations, pore-pressure relations, static and dynamic moduli, and velocity–strength relations. Chapter 8 has new discussions on capillary effects, irreducible water saturation, permeability, and flow in fractures. Chapter 9 includes new relations between electrical and seismic properties. The Appendices has new tables of physical constants and properties for common gases, ice, and methane hydrate. This Handbook is complementary to a number of other excellent books. For indepth discussions of specific rock physics topics, we recommend Fundamentals of Rock Mechanics, 4th Edition, by Jaeger, Cook, and Zimmerman; Compressibility of Sandstones, by Zimmerman; Physical Properties of Rocks: Fundamentals and Principles of Petrophysics, by Schon; Acoustics of Porous Media, by Bourbie´, Coussy, and Zinszner; Introduction to the Physics of Rocks, by Gue´guen and Palciauskas; A Geoscientist’s Guide to Petrophysics, by Zinszner and Pellerin; Theory of Linear Poroelasticity, by Wang; Underground Sound, by White; Mechanics of Composite Materials, by Christensen; The Theory of Composites, by Milton; Random Heterogeneous Materials, by Torquato; Rock Physics and Phase Relations, edited by Ahrens; and Offset Dependent Reflectivity – Theory and Practice of AVO Analysis, edited by Castagna and Backus. For excellent collections and discussions of classic rock physics papers we recommend Seismic and Acoustic Velocities in Reservoir Rocks, Volumes 1, 2 and 3, edited by Wang and Nur; Elastic Properties and Equations of State, edited by Shankland and Bass; Seismic Wave Attenuation, by Tokso¨z and Johnston; and Classics of Elastic Wave Theory, edited by Pelissier et al. We wish to thank the students, scientific staff, and industrial affiliates of the Stanford Rock Physics and Borehole Geophysics (SRB) project for many valuable comments and insights. While preparing the Second Edition we found discussions with Tiziana Vanorio, Kaushik Bandyopadhyay, Ezequiel Gonzalez, Youngseuk Keehm, Robert Zimmermann, Boris Gurevich, Juan-Mauricio Florez, Anyela Marcote-Rios, Mike Payne, Mike Batzle, Jim Berryman, Pratap Sahay, and Tor Arne Johansen, to be extremely helpful. Li Teng contributed to the chapter on anisotropic AVOZ, and Ran Bachrach contributed to the chapter on dielectric properties. Dawn Burgess helped tremendously with editing, graphics, and content. We also wish to thank the readers of the First Edition who helped us to track down and fix errata. And as always, we are indebted to Amos Nur, whose work, past and present, has helped to make the field of rock physics what it is today. Gary Mavko, Tapan Mukerji, and Jack Dvorkin.

1

Basic tools

1.1

The Fourier transform Synopsis The Fourier transform of f(x) is defined as Z 1 FðsÞ ¼ f ðxÞe
i2pxs dx
1

The inverse Fourier transform is given by Z 1 f ðxÞ ¼ FðsÞeþi2pxs ds
1

Evenness and oddness A function E(x) is even if E(x) ¼ E(–x). A function O(x) is odd if O(x) ¼ –O(–x). The Fourier transform has the following properties for even and odd functions:  Even functions. The Fourier transform of an even function is even. A real even function transforms to a real even function. An imaginary even function transforms to an imaginary even function.  Odd functions. The Fourier transform of an odd function is odd. A real odd function transforms to an imaginary odd function. An imaginary odd function transforms to a real odd function (i.e., the “realness” flips when the Fourier transform of an odd function is taken). real even (RE) ! real even (RE) imaginary even (IE) ! imaginary even (IE) real odd (RO) ! imaginary odd (IO) imaginary odd (IO) ! real odd (RO) Any function can be expressed in terms of its even and odd parts: f ðxÞ ¼ EðxÞ þ OðxÞ where EðxÞ ¼ 12½ f ðxÞ þ f ð
xÞ OðxÞ ¼ 12½ f ðxÞ
f ð
xÞ 1

2

Basic tools

Then, for an arbitrary complex function we can summarize these relations as (Bracewell, 1965) f ðxÞ ¼ reðxÞ þ i ieðxÞ þ roðxÞ þ i ioðxÞ

FðxÞ ¼ REðsÞ þ i IEðsÞ þ ROðsÞ þ i IOðsÞ As a consequence, a real function f(x) has a Fourier transform that is hermitian, F(s) ¼ F*(–s), where * refers to the complex conjugate. For a more general complex function, f(x), we can tabulate some additional properties (Bracewell, 1965): f ðxÞ , FðsÞ f  ðxÞ , F ð
sÞ f  ð
xÞ , F ðsÞ f ð
xÞ , Fð
sÞ 2 Re f ðxÞ , FðsÞ þ F ð
sÞ 2 Im f ðxÞ , FðsÞ
F ð
sÞ f ðxÞ þ f  ð
xÞ , 2 ReFðsÞ f ðxÞ
f  ð
xÞ , 2 ImFðsÞ The convolution of two functions f(x) and g(x) is Z þ1 Z þ1 f ðzÞ gðx
zÞ dz ¼ f ðx
zÞ gðzÞ dz f ðxÞ  gðxÞ ¼
1


1

Convolution theorem If f(x) has the Fourier transform F(s), and g(x) has the Fourier transform G(s), then the Fourier transform of the convolution f(x) * g(x) is the product F(s) G(s). The cross-correlation of two functions f(x) and g(x) is Z þ1 Z þ1   f ðxÞ ? gðxÞ ¼ f ðz
xÞ gðzÞ dz ¼ f  ðzÞ gðz þ xÞ dz
1


1

where f* refers to the complex conjugate of f. When the two functions are the same, f*(x) ★ f(x) is called the autocorrelation of f(x).

Energy spectrum The modulus squared of the Fourier transform jF(s)j2 ¼ F(s) F*(s) is sometimes called the energy spectrum or simply the spectrum. If f(x) has the Fourier transform F(s), then the autocorrelation of f(x) has the Fourier transform jF(s)j2.

3

1.1 The Fourier transform

Phase spectrum The Fourier transform F(s) is most generally a complex function, which can be written as FðsÞ ¼ jFjei’ ¼ Re FðsÞ þ i Im FðsÞ where jFj is the modulus and ’ is the phase, given by ’ ¼ tan
1 ½Im FðsÞ=Re FðsÞ The function ’(s) is sometimes also called the phase spectrum. Obviously, both the modulus and phase must be known to completely specify the Fourier transform F(s) or its transform pair in the other domain, f(x). Consequently, an infinite number of functions f(x) , F(s) are consistent with a given spectrum jF(s)j2. The zero-phase equivalent function (or zero-phase equivalent wavelet) corresponding to a given spectrum is FðsÞ ¼ jFðsÞj Z 1 f ðxÞ ¼ jFðsÞj eþi2pxs ds
1

which implies that F(s) is real and f(x) is hermitian. In the case of zero-phase real wavelets, then, both F(s) and f(x) are real even functions. The minimum-phase equivalent function or wavelet corresponding to a spectrum is the unique one that is both causal and invertible. A simple way to compute the minimumphase equivalent of a spectrum jF(s)j2 is to perform the following steps (Claerbout, 1992): (1) Take the logarithm, B(s) ¼ ln jF(s)j. (2) Take the Fourier transform, B(s) ) b(x). (3) Multiply b(x) by zero for x < 0 and by 2 for x > 0. If done numerically, leave the values of b at zero and the Nyquist frequency unchanged. (4) Transform back, giving B(s) + i’(s), where ’ is the desired phase spectrum. (5) Take the complex exponential to yield the minimum-phase function: Fmp(s) = exp[B(s) þ i’(s)] ¼ jF(s)jei’(s). (6) The causal minimum-phase wavelet is the Fourier transform of Fmp(s) ) fmp(x). Another way of saying this is that the phase spectrum of the minimum-phase equivalent function is the Hilbert transform (see Section 1.2 on the Hilbert transform) of the log of the energy spectrum.

Sampling theorem A function f(x) is said to be band limited if its Fourier transform is nonzero only within a finite range of frequencies, jsj < sc, where sc is sometimes called the cut-off frequency. The function f(x) is fully specified if sampled at equal spacing not exceeding Dx ¼ 1/(2sc). Equivalently, a time series sampled at interval Dt adequately describes the frequency components out to the Nyquist frequency fN ¼ 1/(2Dt).

Basic tools

2

2 Π(s) Boxcar(s)

sinc(x)

Sinc(x)

4

0

−2 −3

−2

−1

0 x

1

2

3

0

−2

−2

−1

0 s

1

2

Figure 1.1.1 Plots of the function sinc(x) and its Fourier transform
(s).

The numerical process to recover the intermediate points between samples is to convolve with the sinc function: 2sc sincð2sc xÞ ¼ 2sc sinðp2sc xÞ=p2sc x where sincðxÞ 

sinðpxÞ px

which has the properties:
sincð0Þ ¼ 1 n ¼ nonzero integer sincðnÞ ¼ 0 The Fourier transform of sinc(x) is the boxcar function
(s): 8 1 > < 0 jsj > 2
ðsÞ ¼

> :

12

/

jsj ¼ 12

1

jsj < 12

Plots of the function sinc(x) and its Fourier transform
(s) are shown in Figure 1.1.1. One can see from the convolution and similarity theorems below that convolving with 2sc sinc(2scx) is equivalent to multiplying by
(s/2sc) in the frequency domain (i.e., zeroing out all frequencies jsj > sc and passing all frequencies jsj < sc.

Numerical details Consider a band-limited function g(t) sampled at N points at equal intervals: g(0), g(Dt), g(2Dt), . . . , g((N – 1)Dt). A typical fast Fourier transform (FFT) routine will yield N equally spaced values of the Fourier transform, G( f ), often arranged as     N N þ1 þ2    ðN
1Þ N 1 2 3  2 2 Gð0Þ Gðf Þ Gð2f Þ    Gð fN Þ Gð
fN þ f Þ    Gð
2f Þ Gð
f Þ

5

1.1 The Fourier transform

time domain sample rate Dt Nyquist frequency fN ¼ 1/(2Dt) frequency domain sample rate Df ¼ 1/(NDt) Note that, because of “wraparound,” the sample at (N/2 þ 1) represents both fN.

Spectral estimation and windowing It is often desirable in rock physics and seismic analysis to estimate the spectrum of a wavelet or seismic trace. The most common, easiest, and, in some ways, the worst way is simply to chop out a piece of the data, take the Fourier transform, and find its magnitude. The problem is related to sample length. If the true data function is f(t), a small sample of the data can be thought of as  f ðtÞ; atb fsample ðtÞ ¼ 0; elsewhere or  1  t
2ða þ bÞ fsample ðtÞ ¼ f ðtÞ
b
a where
(t) is the boxcar function discussed above. Taking the Fourier transform of the data sample gives Fsample ðsÞ ¼ FðsÞ  ½jb
aj sincððb
aÞsÞe
ipðaþbÞs  More generally, we can “window” the sample with some other function o(t): fsample ðtÞ ¼ f ðtÞ oðtÞ yielding Fsample ðsÞ ¼ FðsÞ  WðsÞ Thus, the estimated spectrum can be highly contaminated by the Fourier transform of the window, often with the effect of smoothing and distorting the spectrum due to the convolution with the window spectrum W(s). This can be particularly severe in the analysis of ultrasonic waveforms in the laboratory, where often only the first 1 to 112 cycles are included in the window. The solution to the problem is not easy, and there is an extensive literature (e.g., Jenkins and Watts, 1968; Marple, 1987) on spectral estimation. Our advice is to be aware of the artifacts of windowing and to experiment to determine the sensitivity of the results, such as the spectral ratio or the phase velocity, to the choice of window size and shape.

Fourier transform theorems Tables 1.1.1 and 1.1.2 summarize some useful theorems (Bracewell, 1965). If f(x) has the Fourier transform F(s), and g(x) has the Fourier transform G(s), then the Fourier

6

Basic tools

Table 1.1.1

Fourier transform theorems.

Theorem

x-domain

s-domain

Similarity

f(ax)

,

1 s F jaj a

Addition

f(x) þ g(x)

,

F(s) þ G(s)

Shift

f(x – a)

,

Modulation

f(x) cos ox

,

Convolution

f(x) * g(x)

,

e–i2pasF(s) 1  ! 1  ! F s
þ F sþ 2 2p 2 2p F(s) G(s)

Autocorrelation

f(x) * f (–x)

,

jF(s)j2

Derivative

f 0 (x)

,

i2psF(s)

Table 1.1.2

*

Some additional theorems.

Derivative of convolution Rayleigh Power ( f and g real)

d ½ f ðxÞ  gðxÞ ¼ f 0 ðxÞ  gðxÞ ¼ f ðxÞ  g0 ðxÞ dx Z 1 Z 1 j f ðxÞj2 dx ¼ jFðsÞj2 ds
1
1 Z 1 Z 1 f ðxÞ g ðxÞ dx ¼ FðsÞ G ðsÞ ds
1
1 Z 1 Z 1 f ðxÞ gð
xÞ dx ¼ FðsÞ GðsÞ ds
1


1

transform pairs in the x-domain and the s-domain are as shown in the tables. Table 1.1.3 lists some useful Fourier transform pairs.

1.2

The Hilbert transform and analytic signal Synopsis The Hilbert transform of f(x) is defined as Z 1 1 f ðx0 Þ dx0 FHi ðxÞ ¼ p
1 x0
x which can be expressed as a convolution of f(x) with (–1/px) by FHi ¼


1  f ðxÞ px

The Fourier transform of (–1/px) is (i sgn(s)), that is, þi for positive s and –i for negative s. Hence, applying the Hilbert transform keeps the Fourier amplitudes or spectrum the same but changes the phase. Under the Hilbert transform, sin(kx) is converted to cos(kx), and cos(kx) is converted to –sin(kx). Similarly, the Hilbert transforms of even functions are odd functions and vice versa.

7

1.2 The Hilbert transform and analytic signal

Table 1.1.3

Some Fourier transform pairs. sin px

     i 1 1 d sþ
d s
2 2 2

cos px

     1 1 1 d sþ þd s
2 2 2

d(x)

1

sinc(x)


(s)

sinc2(x)

L(s)

e–px

2

e–ps

2

–1/px

i sgn(s)

x0 x20 þ x2

p exp(–2px0jsj)

e–jxj

jxj–1/2

2 1 þ ð2psÞ2 jsj–1/2

The inverse of the Hilbert transform is itself the Hilbert transform with a change of sign: Z 1 1 FHi ðx0 Þ dx0 f ðxÞ ¼
p
1 x0
x or   1 f ðxÞ ¼

 FHi px The analytic signal associated with a real function, f(t), is the complex function SðtÞ ¼ f ðtÞ
i FHi ðtÞ As discussed below, the Fourier transform of S(t) is zero for negative frequencies.

8

Basic tools

The instantaneous envelope of the analytic signal is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EðtÞ ¼ f 2 ðtÞ þ F2Hi ðtÞ The instantaneous phase of the analytic signal is ’ðtÞ ¼ tan
1 ½
FHi ðtÞ=f ðtÞ ¼ Im½lnðSðtÞÞ The instantaneous frequency of the analytic signal is     d’ d l dS o¼ ¼ Im lnðSÞ ¼ Im dt dt S dt Claerbout (1992) has suggested that o can be numerically more stable if the denominator is rationalized and the functions are locally smoothed, as in the following equation: E3 2D S ðtÞ dSðtÞ dt 5  ¼ Im4  o hS ðtÞ SðtÞi where 〈·〉 indicates some form of running average or smoothing.

Causality The impulse response, I(t), of a real physical system must be causal, that is, IðtÞ ¼ 0;

for t < 0

The Fourier transform T( f ) of the impulse response of a causal system is sometimes called the transfer function: Z 1 IðtÞ e
i2pft dt Tð f Þ ¼
1

T( f ) must have the property that the real and imaginary parts are Hilbert transform pairs, that is, T( f ) will have the form Tð f Þ ¼ Gð f Þ þ iBð f Þ where B( f ) is the Hilbert transform of G( f ): Z

1

Gð f 0 Þ df 0 0
1 f
f Z 1 1 Bð f 0 Þ df 0 Gð f Þ ¼
p
1 f 0
f 1 Bð f Þ ¼ p

9

1.3 Statistics and probability

Similarly, if we reverse the domains, an analytic signal of the form SðtÞ ¼ f ðtÞ
iFHi ðtÞ must have a Fourier transform that is zero for negative frequencies. In fact, one convenient way to implement the Hilbert transform of a real function is by performing the following steps: (1) Take the Fourier transform. (2) Multiply the Fourier transform by zero for f < 0. (3) Multiply the Fourier transform by 2 for f > 0. (4) If done numerically, leave the samples at f ¼ 0 and the Nyquist frequency unchanged. (5) Take the inverse Fourier transform. The imaginary part of the result will be the negative Hilbert transform of the real part.

1.3

Statistics and probability Synopsis The sample mean, m, of a set of n data points, xi, is the arithmetic average of the data values: n 1X xi n i¼1

m¼

The median is the midpoint of the observed values if they are arranged in increasing order. The sample variance, s2, is the average squared difference of the observed values from the mean:
2 ¼

n 1X ðxi
mÞ2 n i¼1

(An unbiased estimate of the population variance is often found by dividing the sum given above by (n – 1) instead of by n.) The standard deviation, s, is the square root of the variance, while the coefficient of variation is s/m. The mean deviation, a, is a¼

n 1X jxi
mj n i¼1

Regression When trying to determine whether two different data variables, x and y, are related, we often estimate the correlation coefficient, r, given by (e.g., Young, 1962) 1 Pn i¼1 ðxi
mx Þðyi
my Þ ; ¼n
x
y

where jj  1

10

Basic tools

where sx and sy are the standard deviations of the two distributions and mx and my are their means. The correlation coefficient gives a measure of how close the points come to falling along a straight line in a scatter plot of x versus y. jrj ¼ 1 if the points lie perfectly along a line, and jrj < 1 if there is scatter about the line. The numerator of this expression is the sample covariance, Cxy, which is defined as Cxy ¼

n 1X ðxi
mx Þðyi
my Þ n i¼1

It is important to remember that the correlation coefficient is a measure of the linear relation between x and y. If they are related in a nonlinear way, the correlation coefficient will be misleadingly small. The simplest recipe for estimating the linear relation between two variables, x and y, is linear regression, in which we assume a relation of the form: y ¼ ax þ b The coefficients that provide the best fit to the measured values of y, in the leastsquares sense, are
y ;
x

a¼

b ¼ my
amx

More explicitly, a¼

n

P

P P xi yi
ð xi Þð yi Þ ; P P n x2i
ð xi Þ2

slope

P P 2  P P ð yi Þ xi
ð xi yi Þð xi Þ ; b¼ P P n x2i
ð xi Þ2

intercept

The scatter or variation of y-values around the regression line can be described by the sum of the squared errors as E2 ¼

n X

ðyi
y^i Þ2

i¼1

where y^i is the value predicted from the regression line. This can be expressed as a variance around the regression line as
^y2 ¼

n 1X ðyi
y^i Þ2 n i¼1

The square of the correlation coefficient r is the coefficient of determination, often denoted by r2, which is a measure of the regression variance relative to the total variance in the variable y, expressed as

11

1.3 Statistics and probability

variance of y around the linear regression total variance of y Pn
^y2 ðyi
y^i Þ2 ¼ 1
¼ 1
Pni¼1 2
y2 i¼1 ðyi
my Þ

r 2 ¼ 2 ¼ 1


The inverse relation is
^y2 ¼
y2 ð1
r 2 Þ Often, when doing a linear regression the choice of dependent and independent variables is arbitrary. The form above treats x as independent and exact and assigns errors to y. It often makes just as much sense to reverse their roles, and we can find a regression of the form x ¼ a0 y þ b0 Generally a 6¼ 1/a0 unless the data are perfectly correlated. In fact, the correlation pffiffiffiffiffiffi coefficient, r, can be written as  ¼ aa0 . The coefficients of the linear regression among three variables of the form z ¼ a þ bx þ cy are given by Cxz Cyy
Cxy Cyz 2 Cxx Cyy
Cxy Cxx Cyz
Cxy Cxz c¼ 2 Cxx Cyy
Cxy b¼

a ¼ mz
mx b
my c The coefficients of the n-dimensional linear regression of the form z ¼ c0 þ c1 x1 þ c2 x2 þ    þ cn xn are given by 2 3 2 3 c0 ð1Þ z 6 c1 7 6 7

ð2Þ 7 
1 6 6 c2 7 6z 7 6 7 ¼ MT M MT 6 .. 7 6 .. 7 4 . 5 4 . 5 zðkÞ cn where the k sets 2 ð1Þ 1 x1 6 6 1 xð2Þ 1 6 M¼6 .. 6 .. 4. . 1

ðkÞ

x1

of independent variables form columns 2:(n þ 1) in the matrix M: 3 ð1 Þ x2    xðn1Þ 7 ð2 Þ x2    xðn2Þ 7 7 .. .. 7 7 . . 5 ðk Þ

x2



xðnkÞ

12

Basic tools

Variogram and covariance function In geostatistics, variables are modeled as random fields, X(u), where u is the spatial position vector. Spatial correlation between two random fields X(u) and Y(u) is described by the cross-covariance function CXY(h), defined by CXY ðhÞ ¼ Ef½XðuÞ
mX ðuÞ½Yðu þ hÞ
mY ðu þ hÞg where E{} denotes the expectation operator, mX and mY are the means of X and Y, and h is called the lag vector. For stationary fields, mX and mY are independent of position. When X and Y are the same function, the equation represents the auto-covariance function CXX(h). A closely related measure of two-point spatial variability is the semivariogram, g(h). For stationary random fields X(u) and Y(u), the cross-variogram 2gXY(h) is defined as 2gXY ðhÞ ¼ Ef½Xðu þ hÞ
XðuÞ½Yðu þ hÞ
YðuÞg When X and Y are the same, the equation represents the variogram of X(h). For a stationary random field, the variogram and covariance function are related by gXX ðhÞ ¼ CXX ð0Þ
CXX ðhÞ where CXX(0) is the stationary variance of X.

Distributions

 A population of n elements possesses nr (pronounced “n choose r”) different subpopulations of size r  n, where   ðnÞr nðn
1Þ    ðn
r þ 1Þ n! n ¼ ¼ ¼ r r! 1  2    ðr
1Þr r!ðn
rÞ! Expressions of this kind are called binomial coefficients. Another way to say this is

 that a subset of r elements can be chosen in nr different ways from the original set. The binomial distribution gives the probability of n successes in N independent trials, if p is the probability of success in any one trial. The binomial distribution is given by   N n fN; p ðnÞ ¼ p ð1
pÞN
n n The mean of the binomial distribution is given by mb ¼ Np and the variance of the binomial distribution is given by
2b ¼ Npð1
pÞ

13

1.3 Statistics and probability

The Poisson distribution is the limit of the binomial distribution as N ! 1 and p ! 0 so that l ¼ Np remains finite. The Poisson distribution is given by fl ðnÞ ¼

ln e
l n!

The Poisson distribution is a discrete probability distribution and expresses the probability of n events occurring during a given interval of time if the events have an average (positive real) rate l, and the events are independent of the time since the previous event. n is a non-negative integer. The mean of the Poisson distribution is given by mP ¼ l and the variance of the Poisson distribution is given by
2P ¼ l The uniform distribution is given by 8 > < 1 ; axb f ðxÞ ¼ b
a > : 0; elsewhere The mean of the uniform distribution is m¼

ða þ bÞ 2

and the standard deviation of the uniform distribution is jb
aj
¼ pffiffiffiffiffi 12 The Gaussian or normal distribution is given by 2 1 2 f ðxÞ ¼ pffiffiffiffiffiffi e
ðx
mÞ =2

2p

where s is the standard deviation and m is the mean. The mean deviation for the Gaussian distribution is rffiffiffi 2 a¼
p When m measurements are made of n quantities, the situation is described by the n-dimensional multivariate Gaussian probability density function (pdf):   1 1 T
1 exp
C ðx
mÞ ðx
mÞ fn ðxÞ ¼ 2 ð2pÞn=2 jCj1=2

14

Basic tools

where xT ¼ (x1, x2, . . . , xn) is the vector of observations, mT ¼ (m1, m2, . . . , mn) is the vector of means of the individual distributions, and C is the covariance matrix: C ¼ ½Cij  where the individual covariances, Cij, are as defined above. Notice that this reduces to the single variable normal distribution when n ¼ 1. When the natural logarithm of a variable, x ¼ ln(y), is normally distributed, it belongs to a lognormal distribution expressed as "  # 1 1 lnðyÞ
a 2 f ðyÞ ¼ pffiffiffiffiffiffi exp
2 b 2p by where a is the mean and b2 is the variance. The relations among the arithmetic and logarithmic parameters are m ¼ eaþb =2 ; 2

2


2 ¼ m2 ðeb
1Þ;

a ¼ lnðmÞ
b2=2

 b2 ¼ ln 1 þ
2=m2

The truncated exponential distribution is given by (1 x 0 X expð
x=XÞ; PðxÞ ¼ 0; x 1. When k ¼ 3, the Weibull distribution is a good approximation to the normal distribution, and when k ¼ 1, the Weibull distribution reduces to the exponential distribution.

Monte Carlo simulations Statistical simulation is a powerful numerical method for tackling many probabilistic problems. One of the steps is to draw samples Xi from a desired probability distribution function F(x). This procedure is often called Monte Carlo simulation, a term made popular by physicists working on the bomb during the Second World War. In general, Monte Carlo simulation can be a very difficult problem, especially when X is multivariate with correlated components, and F(x) is a complicated function. For the simple case of a univariate X and a completely known F(x) (either analytically or numerically), drawing Xi amounts to first drawing uniform random variates Ui between 0 and 1, and then evaluating the inverse of the desired cumulative distribution function (CDF) at these Ui: Xi ¼ F –1(Ui). The inverse of the CDF is called the quantile function. When F
1(X) is not known analytically, the inversion can be easily done by

Basic tools

1

Uniform [0 1] random number

0.8 CDF F(X)

16

0.6 0.4 Simulated X 0.2 0 0

5

10 X

15

20

Figure 1.3.1 Schematic of a univariate Monte Carlo simulation.

table-lookup and interpolation from the numerically evaluated or nonparametric CDF derived from data. A graphical description of univariate Monte Carlo simulation is shown in Figure 1.3.1. Many modern computer packages have random number generators not only for uniform and normal (Gaussian) distributions, but also for a large number of well-known, analytically defined statistical distributions. Often Monte Carlo simulations require simulating correlated random variables (e.g., VP, VS). Correlated random variables may be simulated sequentially, making use of the chain rule of probability, which expresses the joint probability density in terms of the conditional and marginal densities: P(VP, VS) ¼ P(VSjVP) P(VP). A simple procedure for correlated Monte Carlo draws is as follows:  draw a VP sample from the VP distribution;  compute a VS from the drawn VP and the VP–VS regression;  add to the computed VS a random Gaussian error with zero mean and variance equal to the variance of the residuals from the VP–VS regression. This gives a random, correlated (VP, VS) sample. A better approach is to draw VS from the conditional distributions of VS for each given VP value, instead of using a simple VP–VS regression. Given sufficient VP–VS training data, the conditional distributions of VS for different VP can be computed.

Bootstrap “Bootstrap” is a very powerful computational statistical method for assigning measures of accuracy to statistical estimates (e.g., Efron and Tibshirani, 1993). The general idea is to make multiple replicates of the data by drawing from the original data with replacement. Each of the bootstrap data replicates has the same number of samples as the original data set, but since they are drawn with replacement, some of the data may be represented more than once in the replicate data sets, while others might be

17

1.3 Statistics and probability

missing. Drawing with replacement from the data is equivalent to Monte Carlo realizations from the empirical CDF. The statistic of interest is computed on all of the replicate bootstrap data sets. The distribution of the bootstrap replicates of the statistic is a measure of uncertainty of the statistic. Drawing bootstrap replicates from the empirical CDF in this way is sometimes termed nonparametric bootstrap. In parametric bootstrap the data are first modeled by a parametric CDF (e.g., a multivariate Gaussian), and then bootstrap data replicates are drawn from the modeled CDF. Both simple bootstrap techniques described above assume the data are independent and identically distributed. More sophisticated bootstrap techniques exist that can account for data dependence.

Statistical classification The goal in statistical classification problems is to predict the class of an unknown sample based on observed attributes or features of the sample. For example, the observed attributes could be P and S impedances, and the classes could be lithofacies, such as sand and shale. The classes are sometimes also called states, outcomes, or responses, while the observed features are called the predictors. Discussions concerning many modern classification methods may be found in Fukunaga (1990), Duda et al. (2000), Hastie et al. (2001), and Bishop (2006). There are two general types of statistical classification: supervised classification, which uses a training data set of samples for which both the attributes and classes have been observed; and unsupervised learning, for which only the observed attributes are included in the data. Supervised classification uses the training data to devise a classification rule, which is then used to predict the classes for new data, where the attributes are observed but the outcomes are unknown. Unsupervised learning tries to cluster the data into groups that are statistically different from each other based on the observed attributes. A fundamental approach to the supervised classification problem is provided by Bayesian decision theory. Let x denote the univariate or multivariate input attributes, and let cj, j ¼ 1, . . . , N denote the N different states or classes. The Bayes formula expresses the probability of a particular class given an observed x as Pðcj j xÞ ¼

Pðx; cj Þ Pðx j cj Þ Pðcj Þ ¼ PðxÞ PðxÞ

where P(x, cj) denotes the joint probability of x and cj; P(x j cj) denotes the conditional probability of x given cj; and P(cj) is the prior probability of a particular class. Finally, P(x) is the marginal or unconditional pdf of the attribute values across all N states. It can be written as PðXÞ ¼

N X j¼1

PðX j cj Þ Pðcj Þ

18

Basic tools

and serves as a normalization constant. The class-conditional pdf, P(x j cj), is estimated from the training data or from a combination of training data and forward models. The Bayes classification rule says: classify as class ck if Pðck j xÞ > Pðcj j xÞ for all j 6¼ k: This is equivalent to choosing ck when P(x j ck)P(ck) > P(x j cj)P(cj) for all j 6¼ k. The Bayes classification rule is the optimal one that minimizes the misclassification error and maximizes the posterior probability. Bayes classification requires estimating the complete set of class-conditional pdfs P(x j cj). With a large number of attributes, getting a good estimate of the highly multivariate pdf becomes difficult. Classification based on traditional discriminant analysis uses only the means and covariances of the training data, which are easier to estimate than the complete pdfs. When the input features follow a multivariate Gaussian distribution, discriminant classification is equivalent to Bayes classification, but with other data distribution patterns, the discriminant classification is not guaranteed to maximize the posterior probability. Discriminant analysis classifies new samples according to the minimum Mahalanobis distance to each class cluster in the training data. The Mahalanobis distance is defined as follows:  T   M2 ¼ x
mj 
1 x
mj where x is the sample feature vector (measured attribute), mj are the vectors of the attribute means for the different categories or classes, and S is the training data covariance matrix. The Mahalanobis distance can be interpreted as the usual Euclidean distance scaled by the covariance, which decorrelates and normalizes the components of the feature vector. When the covariance matrices for all the classes are taken to be identical, the classification gives rise to linear discriminant surfaces in the feature space. More generally, with different covariance matrices for each category, the discriminant surfaces are quadratic. If the classes have unequal prior probabilities, the term ln[P(class)j] is added to the right-hand side of the equation for the Mahalanobis distance, where P(class)j is the prior probability for the jth class. Linear and quadratic discriminant classifiers are simple, robust classifiers and often produce good results, performing among the top few classifier algorithms.

1.4

Coordinate transformations Synopsis It is often necessary to transform vector and tensor quantities in one coordinate system to another more suited to a particular problem. Consider two right-hand rectangular Cartesian coordinates (x, y, z) and (x0 , y0 , z0 ) with the same origin, but

19

1.4 Coordinate transformations

with their axes rotated arbitrarily with respect to each other. The relative orientation of the two sets of axes is given by the direction cosines bij, where each element is defined as the cosine of the angle between the new i0 -axis and the original j-axis. The variables bij constitute the elements of a 3 3 rotation matrix [b]. Thus, b23 is the cosine of the angle between the 2-axis of the primed coordinate system and the 3-axis of the unprimed coordinate system. The general transformation law for tensors is 0 MABCD... ¼ bAa bBb bCc bDd . . . Mabcd...

where summation over repeated indices is implied. The left-hand subscripts (A, B, C, D, . . .) on the bs match the subscripts of the transformed tensor M0 on the left, and the right-hand subscripts (a, b, c, d, . . .) match the subscripts of M on the right. Thus vectors, which are first-rank tensors, transform as u0i ¼ bij uj or, in matrix notation, as 0 01 0 10 1 u1 b11 b12 b13 u1 B 0C B CB C B u2 C ¼ B b21 b22 b23 CB u2 C @ A @ A@ A 0 u3 b31 b32 b33 u3 whereas second-rank tensors, such as stresses and strains, obey
0ij ¼ bik bjl
kl or ½
0  ¼ ½b½
½bT in matrix notation. Elastic stiffnesses and compliances are, in general, fourth-order tensors and hence transform according to c0ijkl ¼ bip bjq bkr bls cpqrs Often cijkl and sijkl are expressed as the 6 6 matrices Cij and Sij, using the abbreviated 2-index notation, as defined in Section 2.2 on anisotropic elasticity. In this case, the usual tensor transformation law is no longer valid, and the change of coordinates is more efficiently performed with the 6 6 Bond transformation matrices M and N, as explained below (Auld, 1990): ½C0  ¼ ½M½C½MT ½S0  ¼ ½N½S½NT

20

Basic tools

The elements of the 6 6 M and N matrices are given in terms of the direction cosines as follows: 2

b211

6 2 6 b 6 21 6 2 6 b 6 31 M¼6 6b b 6 21 31 6 6b b 4 31 11 b11 b21

3

b212

b213

2b12 b13

2b13 b11

b222

b223

2b22 b23

2b23 b21

b232

b233

2b32 b33

2b33 b31

b22 b32

b23 b33

b22 b33 þ b23 b32

b21 b33 þ b23 b31

b32 b12

b33 b13

b12 b33 þ b13 b32

b11 b33 þ b13 b31

7 7 7 7 7 2b31 b32 7 7 b22 b31 þ b21 b32 7 7 7 b11 b32 þ b12 b31 7 5

b12 b22

b13 b23

b22 b13 þ b12 b23

b11 b23 þ b13 b21

b22 b11 þ b12 b21

2b11 b12 2b21 b22

and 2

b211

6 6 b2 21 6 6 6 b2 31 6 N¼6 6 2b b 6 21 31 6 6 2b b 4 31 11 2b11 b21

3

b212

b213

b12 b13

b13 b11

b222

b223

b22 b23

b23 b21

b232

b233

b32 b33

b33 b31

2b22 b32

2b23 b33

b22 b33 þ b23 b32

b21 b33 þ b23 b31

2b32 b12

2b33 b13

b12 b33 þ b13 b32

b11 b33 þ b13 b31

7 7 7 7 7 b31 b32 7 7 b22 b31 þ b21 b32 7 7 7 b11 b32 þ b12 b31 7 5

2b12 b22

2b13 b23

b22 b13 þ b12 b23

b11 b23 þ b13 b21

b22 b11 þ b12 b21

b11 b12 b21 b22

The advantage of the Bond method for transforming stiffnesses and compliances is that it can be applied directly to the elastic constants given in 2-index notation, as they almost always are in handbooks and tables.

Assumptions and limitations Coordinate transformations presuppose right-handed rectangular coordinate systems.

2

Elasticity and Hooke’s law

2.1

Elastic moduli: isotropic form of Hooke’s law Synopsis In an isotropic, linear elastic material, the stress and strain are related by Hooke’s law as follows (e.g., Timoshenko and Goodier, 1934):
ij ¼ ldij eaa þ 2meij or eij ¼

1 ½ð1 þ nÞ
ij
ndij
aa  E

where eij ¼ elements of the strain tensor
ij ¼ elements of the stress tensor eaa ¼ volumetric strain (sum over repeated index)
aa ¼ mean stress times 3 (sum over repeated index) dij ¼ 0 if i 6¼ j and dij ¼ 1 if i ¼ j In an isotropic, linear elastic medium, only two constants are needed to specify the stress–strain relation completely (for example, [l, m] in the first equation or [E, v], which can be derived from [l, m], in the second equation). Other useful and convenient moduli can be defined, but they are always relatable to just two constants. The three moduli that follow are examples. The Bulk modulus, K, is defined as the ratio of the hydrostatic stress, s0, to the volumetric strain:
0 ¼ 13
aa ¼ Keaa The bulk modulus is the reciprocal of the compressibility, b, which is widely used to describe the volumetric compliance of a liquid, solid, or gas: b¼

21

1 K

22

Elasticity and Hooke’s law

Caution

Occasionally in the literature authors have used the term incompressibility as an alternate name for Lame´’s constant, l, even though l is not the reciprocal of the compressibility. The shear modulus, m, is defined as the ratio of the shear stress to the shear strain:
ij ¼ 2meij ;

i 6¼ j

Young’s modulus, E, is defined as the ratio of the extensional stress to the extensional strain in a uniaxial stress state:
zz ¼ Eezz ;


xx ¼
yy ¼
xy ¼
xz ¼
yz ¼ 0

Poisson’s ratio, which is defined as minus the ratio of the lateral strain to the axial strain in a uniaxial stress state: v¼


exx ; ezz


xx ¼
yy ¼
xy ¼
xz ¼
yz ¼ 0

P-wave modulus, M ¼ VP2 , defined as the ratio of the axial stress to the axial strain in a uniaxial strain state:
zz ¼ Mezz ;

exx ¼ eyy ¼ exy ¼ exz ¼ eyz ¼ 0

Note that the moduli (l, m, K, E, M) all have the same units as stress (force/area), whereas Poisson’s ratio is dimensionless. Energy considerations require that the following relations always hold. If they do not, one should suspect experimental errors or that the material is not isotropic: K ¼lþ

2m 0; 3

m 0

or
1 < n  12;

E 0

In rocks, we seldom, if ever, observe a Poisson’s ratio of less than 0. Although permitted by this equation, a negative measured value is usually treated with suspicion. A Poisson’s ratio of 0.5 can mean an infinitely incompressible rock (not possible) or a liquid. A suspension of particles in fluid, or extremely soft, watersaturated sediments under essentially zero effective stress, such as pelagic ooze, can have a Poisson’s ratio approaching 0.5. Although any one of the isotropic constants (l, m, K, M, E, and n) can be derived in terms of the others, m and K have a special significance as eigenelastic constants (Mehrabadi and Cowin, 1989) or principal elasticities of the material (Kelvin, 1856). The stress and strain eigentensors associated with m and K are orthogonal, as discussed in Section 2.2. Such an orthogonal significance does not hold for the pair l and m.

23

2.2 Anisotropic form of Hooke’s law

Relationships among elastic constants in an isotropic material (after Birch, 1961).

Table 2.1.1

K

E

l

v

M

m

l þ 2m/3

m 3lþ2m lþm

—

l 2ðlþmÞ

l þ 2m

—

—

l 3K
l

3K – 2l

3(K – l)/2

3K
2m 2ð3KþmÞ

K þ 4m/3

— —

K
l 3K
l

—

9K

—

9Km 3Kþm

K – 2m/3

Em 3ð3m
EÞ

—

E
2m m ð3m
EÞ

E/(2m) – 1

m 4m
E 3m
E

—

—

3K 3K
E 9K
E

3K
E 6K

3K 3KþE 9K
E

3KE 9K
E

l 1þn 3n

l ð1þnÞð1
2nÞ n

—

—

l 1
n n

l 1
2n 2n

2ð1þnÞ m 3ð1
2nÞ

2m(1 þ v)

2n m 1
2n

—

m 2
2n 1
2n

—

n 1þn

—

3K 1
n 1þn

3K 1
2n 2þ2n

—

3K(1 – 2v)

3K

E 3ð1
2nÞ

—

En ð1þnÞð1
2nÞ

—

Eð1
nÞ ð1þnÞð1
2nÞ

E 2þ2n

M
43 m

—

M
2m

M
2m 2ðM
mÞ

—

—

Table 2.1.1 summarizes useful relations among the constants of linear isotropic elastic media.

Assumptions and limitations The preceding equations assume isotropic, linear elastic media.

2.2

Anisotropic form of Hooke’s law Synopsis Hooke’s law for a general anisotropic, linear, elastic solid states that the stress sij is linearly proportional to the strain eij, as expressed by
ij ¼ cijkl ekl in which summation (over 1, 2, 3) is implied over the repeated subscripts k and l. The elastic stiffness tensor, with elements cijkl, is a fourth-rank tensor obeying the laws of tensor transformation and has a total of 81 components. However, not all 81 components are independent. The symmetry of stresses and strains implies that cijkl ¼ cjikl ¼ cijlk ¼ cjilk

24

Elasticity and Hooke’s law

reducing the number of independent constants to 36. In addition, the existence of a unique strain energy potential requires that cijkl ¼ cklij further reducing the number of independent constants to 21. This is the maximum number of independent elastic constants that any homogeneous linear elastic medium can have. Additional restrictions imposed by symmetry considerations reduce the number much further. Isotropic, linear elastic materials, which have maximum symmetry, are completely characterized by two independent constants, whereas materials with triclinic symmetry (the minimum symmetry) require all 21 constants. Alternatively, the strains may be expressed as a linear combination of the stresses by the following expression: eij ¼ sijkl
kl In this case sijkl are elements of the elastic compliance tensor which has the same symmetry as the corresponding stiffness tensor. The compliance and stiffness are tensor inverses, denoted by cijkl sklmn ¼ Iijmn ¼ 12ðdim djn þ din djm Þ The stiffness and compliance tensors must always be positive definite. One way to express this requirement is that all of the eigenvalues of the elasticity tensor (described below) must be positive.

Voigt notation It is a standard practice in elasticity to use an abbreviated Voigt notation for the stresses, strains, and stiffness and compliance tensors, for doing so simplifies some of the key equations (Auld, 1990). In this abbreviated notation, the stresses and strains are written as six-element column vectors rather than as nine-element square matrices: 2 3 2 3
1 ¼
11 e1 ¼ e11 6
2 ¼
22 7 6 e2 ¼ e22 7 6 7 6 7 6
3 ¼
33 7 6 e3 ¼ e33 7 7 6 7 T¼6 6
4 ¼
23 7 E ¼ 6 e4 ¼ 2e23 7 6 7 6 7 4
5 ¼
13 5 4 e5 ¼ 2e13 5
6 ¼
12 e6 ¼ 2e12 Note the factor of 2 in the definitions of strains, but not in the definition of stresses. With the Voigt notation, four subscripts of the stiffness and compliance tensors are reduced to two. Each pair of indices ij(kl) is replaced by one index I(J) using the following convention:

25

2.2 Anisotropic form of Hooke’s law

ijðklÞ IðJÞ 11 1 22 2 33 3 23; 32 4 13; 31 5 12; 21 6 The relation, therefore, is cIJ ¼ cijkl and sIJ ¼ sijkl N, where 8 > < 1 for I and J ¼ 1; 2; 3 N ¼ 2 for I or J ¼ 4; 5; 6 > : 4 for I and J ¼ 4; 5; 6 Note how the definition of sIJ differs from that of cIJ. This results from the factors of 2 introduced in the definition of strains in the abbreviated notation. Hence the Voigt matrix representation of the elastic stiffness is 0

c11 B c12 B B c13 B B c14 B @ c15 c16

c12 c22 c23 c24 c25 c26

c13 c23 c33 c34 c35 c36

c14 c24 c34 c44 c45 c46

c15 c25 c35 c45 c55 c56

1 c16 c26 C C c36 C C c46 C C c56 A c66

and similarly, the Voigt matrix representation of the elastic compliance is 0

s11 B s12 B B s13 B B s14 B @ s15 s16

s12 s22 s23 s24 s25 s26

s13 s23 s33 s34 s35 s36

s14 s24 s34 s44 s45 s46

s15 s25 s35 s45 s55 s56

1 s16 s26 C C s36 C C s46 C C s56 A s66

The Voigt stiffness and compliance matrices are symmetric. The upper triangle contains 21 constants, enough to contain the maximum number of independent constants that would be required for the least symmetric linear elastic material. Using the Voigt notation, we can write Hooke’s law as 0 1 0 10 1
1 c11 c12 c13 c14 c15 c16 e1 B
2 C B c12 c22 c23 c24 c25 c26 CB e2 C B C B CB C B
3 C B c13 c23 c33 c34 c35 c36 CB e3 C B C¼B CB C B
4 C B c14 c24 c34 c44 c45 c46 CB e4 C B C B CB C @
5 A @ c15 c25 c35 c45 c55 c56 A@ e5 A
6 c16 c26 c36 c46 c56 c66 e6

26

Elasticity and Hooke’s law

It is very important to note that the stress (strain) vector and stiffness (compliance) matrix in Voigt notation are not tensors. Caution

Some forms of the abbreviated notation adopt different definitions of strains, moving the factors of 2 and 4 from the compliances to the stiffnesses. However, the form given above is the more common convention. In the two-index notation, cIJ and sIJ can conveniently be represented as 6 6 matrices. However, these matrices no longer follow the laws of tensor transformation. Care must be taken when transforming from one coordinate system to another. One way is to go back to the four-index notation and then use the ordinary laws of coordinate transformation. A more efficient method is to use the Bond transformation matrices, which are explained in Section 1.4 on coordinate transformations.

Voigt stiffness matrix structure for common anisotropy classes The nonzero components of the more symmetric anisotropy classes commonly used in modeling rock properties are given below in Voigt notation.

Isotropic: two independent constants The structure of the Voigt elastic stiffness matrix for an isotropic linear elastic material has the following form: 2 3 c11 c12 c12 0 0 0 6 c12 c11 c12 0 0 0 7 6 7 6 c12 c12 c11 0 0 0 7 6 7; c12 ¼ c11
2c44 6 0 0 0 c44 0 0 7 6 7 4 0 0 0 0 c44 0 5 0 0 0 0 0 c44 The relations between the elements c and Lame´’s parameters l and m of isotropic linear elasticity are c11 ¼ l þ 2m;

c12 ¼ l;

c44 ¼ m

The corresponding nonzero isotropic compliance tensor elements can be written in terms of the stiffnesses: s11 ¼

c11 þ c12 ; ðc11
c12 Þðc11 þ 2c12 Þ

s44 ¼

1 c44

Energy considerations require that for an isotropic linear elastic material the following conditions must hold: K ¼ c11
43 c44 > 0;

m ¼ c44 > 0

27

2.2 Anisotropic form of Hooke’s law

Cubic: three independent constants When each Cartesian coordinate plane is aligned with a symmetry plane of a material with cubic symmetry, the Voigt elastic stiffness matrix has the following form: 2

c11 6 c12 6 6 c12 6 6 0 6 4 0 0

c12 c11 c12 0 0 0

c12 c12 c11 0 0 0

0 0 0 c44 0 0

0 0 0 0 c44 0

3 0 0 7 7 0 7 7 0 7 7 0 5 c44

The corresponding nonzero cubic compliance tensor elements can be written in terms of the stiffnesses: s11 ¼

c11 þ c12 ; ðc11
c12 Þðc11 þ 2c12 Þ

s12 ¼


c12 ; ðc11
c12 Þðc11 þ 2c12 Þ

s44 ¼

1 c44

Energy considerations require that for a linear elastic material with cubic symmetry the following conditions must hold: c44 0;

c11 > jc12 j;

c11 þ 2c12 > 0

Hexagonal or transversely isotropic: five independent constants When the axis of symmetry of a transversely isotropic material lies along the x3-axis, the Voigt stiffness matrix has the form: 2

c11 6 c12 6 6 c13 6 6 0 6 4 0 0

c12 c11 c13 0 0 0

c13 c13 c33 0 0 0

0 0 0 c44 0 0

0 0 0 0 c44 0

3 0 0 7 7 0 7 7; 0 7 7 0 5 c66

c66 ¼ 12ðc11
c12 Þ

The corresponding nonzero hexagonal compliance tensor elements can be written in terms of the stiffnesses: c33 1 ; s11
s12 ¼ 2 c11
c12 c33 ðc11 þ c12 Þ
2c13 c13 c11 þ c12 ¼
; s33 ¼ ; c33 ðc11 þ c12 Þ
2c213 c33 ðc11 þ c12 Þ
2c213

s11 þ s12 ¼ s13

s44 ¼

1 c44

Energy considerations require that for a linear elastic material with transversely isotropic symmetry the following conditions must hold: c44 0;

c11 > jc12 j;

ðc11 þ c12 Þc33 > 2c213

28

Elasticity and Hooke’s law

Orthorhombic: nine independent constants When each Cartesian coordinate plane is aligned with a symmetry plane of a material with orthorhombic symmetry, the Voigt elastic stiffness matrix has the following form: 2 3 c11 c12 c13 0 0 0 6 c12 c22 c23 0 0 0 7 6 7 6 c13 c23 c33 0 7 0 0 6 7 6 0 0 0 c44 0 0 7 6 7 4 0 0 0 0 c55 0 5 0 0 0 0 0 c66 Monoclinic: 13 independent constants When the symmetry plane of a monoclinic medium is orthogonal to the x3-axis, the Voigt elastic stiffness matrix has the following form: 2 3 c11 c12 c13 0 c15 0 6 c12 c22 c23 0 c25 0 7 6 7 6 c13 c23 c33 0 c35 0 7 6 7 6 0 7 0 c 0 0 c 44 46 6 7 4 c15 c25 c35 0 c55 0 5 0 0 0 c46 0 c66

Phase velocities for several elastic symmetry classes For isotropic symmetry, the phase velocity of wave propagation is given by rffiffiffiffiffiffi rffiffiffiffiffiffi c11 c44 VS ¼ VP ¼   where r is the density. In anisotropic media there are, in general, three modes of propagation (quasilongitudinal, quasi-shear, and pure shear) with mutually orthogonal polarizations. For a medium with transversely isotropic (hexagonal) symmetry, the wave slowness surface is always rotationally symmetric about the axis of symmetry. The phase velocities of the three modes in any plane containing the symmetry axis are given as quasi-longitudinal mode (transversely isotropic): pffiffiffiffiffi VP ¼ ðc11 sin2 y þ c33 cos2 y þ c44 þ MÞ1=2 ð2Þ
1=2 quasi-shear mode (transversely isotropic): pffiffiffiffiffi VSV ¼ ðc11 sin2 y þ c33 cos2 y þ c44
MÞ1=2 ð2Þ
1=2

29

2.2 Anisotropic form of Hooke’s law

pure shear mode (transversely isotropic):  1=2 c66 sin2 y þ c44 cos2 y VSH ¼  where M ¼ ½ðc11
c44 Þ sin2 y
ðc33
c44 Þ cos2 y2 þ ðc13 þ c44 Þ2 sin2 2y and y is the angle between the wave vector and the x3-axis of symmetry (y ¼ 0 for propagation along the x3-axis). The five components of the stiffness tensor for a transversely isotropic material are obtained from five velocity measurements: VP(0 ), VP(90 ), VP(45 ), VSH(90 ), and VSH(0 ) ¼ VSV(0 ): c11 ¼ VP2 ð90 Þ 2 c12 ¼ c11
2VSH ð90 Þ

c33 ¼ VP2 ð0 Þ 2 c44 ¼ VSH ð0 Þ

and c13 ¼
c44 þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 2 42 VPð45  Þ
2VPð45 Þ ðc11 þ c33 þ 2c44 Þ þ ðc11 þ c44 Þðc33 þ c44 Þ

For the more general orthorhombic symmetry, the phase velocities of the three modes propagating in the three symmetry planes (XZ, YZ, and XY) are given as follows: quasi-longitudinal mode (orthorhombic, XZ plane):  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=2 2 2 VP ¼ c55 þ c11 sin y þ c33 cos y þ ðc55 þ c11 sin2 y þ c33 cos2 yÞ2
4A ð2Þ
1=2

quasi-shear mode (orthorhombic, XZ plane):  VSV ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=2 c55 þ c11 sin y þ c33 cos y
ðc55 þ c11 sin2 y þ c33 cos2 yÞ2
4A ð2Þ
1=2 2

2

pure shear mode (orthorhombic, XZ plane):  1=2 c66 sin2 y þ c44 cos2 y VSH ¼  where A ¼ ðc11 sin2 y þ c55 cos2 yÞðc55 sin2 y þ c33 cos2 yÞ
ðc13 þ c55 Þ2 sin2 y cos2 y quasi-longitudinal mode (orthorhombic, YZ plane):  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=2 2 2 VP ¼ c44 þ c22 sin y þ c33 cos y þ ðc44 þ c22 sin2 y þ c33 cos2 yÞ2
4B ð2Þ
1=2

30

Elasticity and Hooke’s law

quasi-shear mode (orthorhombic, YZ plane):  VSV ¼

c44 þ c22 sin2 y þ c33 cos2 y


qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=2 ðc44 þ c22 sin2 y þ c33 cos2 yÞ2
4B ð2Þ
1=2

pure shear mode (orthorhombic, YZ plane): VSH

 1=2 c66 sin2 y þ c55 cos2 y ¼ 

where B ¼ ðc22 sin2 y þ c44 cos2 yÞðc44 sin2 y þ c33 cos2 yÞ
ðc23 þ c44 Þ2 sin2 y cos2 y quasi-longitudinal mode (orthorhombic, XY plane): VP ¼

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=2 c66 þ c22 sin2 ’ þ c11 cos2 ’ þ ðc66 þ c22 sin2 ’ þ c11 cos2 ’Þ2
4C ð2Þ
1=2

quasi-shear mode (orthorhombic, XY plane):  VSH ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=2 c66 þ c22 sin ’ þ c11 cos ’
ðc66 þ c22 sin2 ’ þ c11 cos2 ’Þ2
4C ð2Þ
1=2 2

2

pure shear mode (orthorhombic, XY plane): VSV

 1=2 c55 cos2 ’ þ c44 sin2 ’ ¼ 

where C ¼ ðc66 sin2 ’ þ c11 cos2 ’Þðc22 sin2 ’ þ c66 cos2 ’Þ
ðc12 þ c66 Þ2 sin2 ’ cos2 ’ and y and ’ are the angles of the wave vector relative to the x3 and x1 axes, respectively.

Kelvin notation In spite of its almost exclusive use in the geophysical literature, the abbreviated Voigt notation has several mathematical disadvantages. For example, certain norms of the fourth-rank stiffness tensor are not equal to the corresponding norms of the Voigt stiffness matrix (Thomson, 1878; Helbig, 1994; Dellinger et al., 1998), and the eigenvalues of the Voigt stiffness matrix are not the eigenvalues of the stress tensor. The lesser-known Kelvin notation is very similar to the Voigt notation, except that each element of the 6 6 matrix is weighted according to how many elements of the actual stiffness tensor it represents. Kelvin matrix elements having indices 1, 2, or 3 each map to only a single index pair in the fourth-rank notation, 11, 22, and 33,

31

2.2 Anisotropic form of Hooke’s law

respectively, so any matrix stiffness element with index 1, 2, or 3 is given a weight of 1. Kelvin elements with indices 4, 5, or 6 each represent two index pairs (23, 32), (13, 31), and pffiffiffi(12, 21), respectively, so each element containing 4, 5, or 6 receives a weight of 2. The weighting must be applied for each pffiffiffi Kelvin index. For example, the Kelvin notation would map c^11 ¼ c1111 , c^14 ¼ 2c1123 , c^66 ¼ 2c1212 . One way to convert a Voigt stiffness matrix into a Kelvin stiffness matrix is to pre- and post-multiply by the following weighting matrix (Dellinger et al., 1998): 2 3 1 0 0 0 0 0 60 1 0 0 0 0 7 7 6 60 0 1 0 0 0 7 6 pffiffiffi 7 60 0 0 7 2 0 0 6 7 pffiffiffi 40 0 0 0 2 p0ffiffiffi 5 0 0 0 0 0 2 yielding 0

c^11 B c^ B 12 B B c^13 B B c^ B 14 B @ c^15

c^12 c^22

c^13 c^23

c^14 c^24

c^15 c^25

c^23 c^24 c^25

c^33 c^34 c^35

c^34 c^44 c^45

c^35 c^45 c^55

1 c^16 c^26 C C C c^36 C C c^46 C C C c^56 A

c^16

c^26

c^36

c^46

c^56

c^66

0

c11

B c B 12 B B c13 ¼B B pffiffi2ffic B 14 B pffiffiffi @ 2c15 pffiffiffi 2c16

c12

c13

c22

c23

c23 pffiffiffi 2c24 pffiffiffi 2c25 pffiffiffi 2c26

c33 pffiffiffi 2c34 pffiffiffi 2c35 pffiffiffi 2c36

pffiffiffi 2c14 pffiffiffi 2c24 pffiffiffi 2c34 2c44

pffiffiffi 2c15 pffiffiffi 2c25 pffiffiffi 2c35 2c45

2c45

2c55

2c46

2c56

pffiffiffi 1 2c16 pffiffiffi C 2c26 C pffiffiffi C 2c36 C C 2c46 C C C 2c56 A 2c66

where the c^ij are the Kelvin elastic elements and cij are the Voigt elements. Similarly, the stress and strain elements take on the following weights in the Kelvin notation: 2


^1 ¼
11

6
^ 6 2 6 6
^3 T^ ¼ 6 6
^ 6 4 6 4
^5

3

7 7 7 ¼
33 7 pffiffiffi 7 ; ¼ 2
23 7 7 pffiffiffi 7 ¼ 2
13 5 pffiffiffi
^6 ¼ 2
12 ¼
22

2

^e1 6 ^e 6 2 6 6 ^e3 E^ ¼ 6 6 ^e 6 4 6 4 ^e5 ^e6

¼ e11

3

7 7 7 ¼ e33 7 pffiffiffi 7 ¼ 2e23 7 pffiffiffi 7 7 ¼ 2e13 5 pffiffiffi ¼ 2e12 ¼ e22

32

Elasticity and Hooke’s law

Hooke’s law again takes on 0 1 0 c^11 c^12 c^13
^1 B
^2 C B c^12 c^22 c^23 B C B B
^3 C B c^13 c^23 c^33 B C¼B B
^4 C B c^14 c^24 c^34 B C B @
^5 A @ c^15 c^25 c^35 c^16 c^26 c^36
^6

the familiar form in the Kelvin notation: 10 1 ^e1 c^14 c^15 c^16 B ^e2 C c^24 c^25 c^26 C CB C B C c^34 c^35 c^36 C CB ^e3 C C C c^44 c^45 c^46 CB B ^e4 C A @ ^e5 A c^45 c^55 c^56 ^e6 c^46 c^56 c^66

The Kelvin notation for the stiffness matrix has a number of advantages. The Kelvin elastic matrix is a tensor. It preserves the norm of the 3 3 3 3 notation: X X c^2ij ¼ c2ijkl i;j¼1;...;6

i;j;k;l¼1;...;3

The eigenvalues and eigenvectors (or eigentensors) of the Kelvin 6 6 C-matrix are geometrically meaningful (Kelvin, 1856; Mehrabadi and Cowin, 1989; Dellinger et al., 1998). Each Kelvin eigentensor corresponds to a state of the medium where the stress and strain ellipsoids are aligned and have the same aspect ratios. Furthermore, a stiffness matrix is physically realizable if and only if all of the Kelvin eigenvalues are non-negative, which is useful for inferring elastic constants from laboratory data.

Elastic eigentensors and eigenvalues The eigenvectors of the three-dimensional fourth-rank anisotropic elasticity tensor are called eigentensors when projected into three-dimensional space. The maximum number of eigentensors for any elastic symmetry is six (Kelvin, 1856; Mehrabadi and Cowin, 1989). In the case of isotropic linear elasticity, there are two unique eigentensors. The total stress tensor
~ and the total strain tensor ~e for the isotropic case can be decomposed in terms of the deviatoric second-rank tensors (~
dev and ~edev ) and ~ scaled unit tensors [1 trð~
ÞI~ and 1 trð~eÞI]: 3

3


ÞI~
~ ¼
~dev þ 13 trð~ ~e ¼ ~edev þ 1 trð~eÞI~ 3

For an isotropic material, Hooke’s law can be written as
~ ¼ l trð~eÞI~ þ 2m~e where l is Lame´’s constant and m is the shear modulus. However, if expressed in terms of the stress and strain eigentensors, Hooke’s law becomes two uncoupled equations: ~ tr ð
~ÞI~ ¼ 3K tr ð~eÞI;


~dev ¼ 2m~edev

33

2.2 Anisotropic form of Hooke’s law

Table 2.2.1

Eigenelastic constants for several different symmetries.

Symmetry

Eigenvalue

Multiplicity of eigenvalue

Isotropic

c11 þ 2c12 ¼ 3K 2c44 ¼ 2m

1 5

Cubic

c11 þ 2c12 c11
c12 2c44 pffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi  c33 þ 2c13 b þ b2 þ 1 ; b ¼ 4c132 ðc11 þ c12
c33 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi  c33 þ 2c13 b
b2 þ 1 c11
c12 2c44

1 2 3

Transverse Isotropy

1 1 2 2

where K is the bulk modulus. Similarly, the strain energy U for an isotropic material can be written as

 2U ¼ 2m tr ~e2dev þ K ½tr ð~eÞ2 where m and K scale two energy modes that do not interact. Hence we can see that, although any two of the isotropic constants (l, m, K, Young’s modulus, and Poisson’s ratio) can be derived in terms of the others, m and K have a special significance as eigenelastic constants (Mehrabadi and Cowin, 1989) or principal elasticities of the material (Kelvin, 1856). The stress and strain eigentensors associated with m and K are orthogonal, as shown above. Such an orthogonal significance does not hold for the pair l and m. Eigenelastic constants for a few other symmetries are shown in Table 2.2.1, expressed in terms of the Voigt constants (Mehrabadi and Cowin, 1989).

Poisson’s ratio defined for anisotropic elastic materials Familiar isotropic definitions for elastic constants are sometimes extended to anisotropic materials. For example, consider a transversely isotropic (TI) material with uniaxial stress, applied along the symmetry (x3-) axis, such that
33 6¼ 0


11 ¼
12 ¼
13 ¼
22 ¼
23 ¼ 0

One can define a transversely isotropic Young’s modulus associated with this experiment as E33 ¼


33 2c231 ¼ c33
e33 c11 þ c12

34

Elasticity and Hooke’s law

where the cij are the elastic stiffnesses in Voigt notation. A pair of TI Poisson ratios can similarly be defined in terms of the same experiment: n31 ¼ n32 ¼


e11 c31 ¼ e33 c11 þ c12

If the uniaxial stress field is rotated normal to the symmetry axis, such that
11 6¼ 0


12 ¼
13 ¼
22 ¼
23 ¼
33 ¼ 0

then one can define another TI Young’s modulus and pair of Poisson ratios as E11 ¼


11 c2 ð
c11 þ c12 Þ þ c12 ð
c33 c12 þ c231 Þ ¼ c11 þ 31 e11 c33 c11
c231 e33 c31 ðc11
c12 Þ n13 ¼ n23 ¼
¼ e11 c33 c11
c231 n12 ¼ n21 ¼


e22 c33 c12
c231 ¼ e11 c33 c11
c231

Caution

Just as with the isotropic case, definitions of Young’s modulus and Poisson ratio in terms of stresses and strains are only true for the uniaxial stress state. Definitions in terms of elastic stiffnesses are independent of stress state. In spite of their similarity to their isotropic analogs, these TI Poisson ratios have several distinct differences. For example, the bounds on the isotropic Poisson ratio require that
1  nisotropic 

1 2

In contrast, the bounds on the TI Poisson ratios defined here are (Christensen, 2005) jn12 j  1;

jn 31 j 

 1=2 E33 ; E11

jn13 j 

 1=2 E11 E33

Another distinct difference is the relation of Poisson’s ratio to seismic velocities. In an isotropic material, Poisson’s ratio is directly related to the VP/VS ratio as follows: ðVP =VS Þ2 ¼

2ð1
nÞ ð1
2nÞ

or, equivalently, ðVP =VS Þ2
2 i n¼ h 2 ðVP =VS Þ2
1

35

2.3 Thomsen’s notation for weak elastic anisotropy

However, in the TI material, the ratio of velocities propagating along the symmetry (x3-) axis is  2 VP c33 ¼ VS c44 and is not simply related to the n31 Poisson ratio. Caution

The definition of Young’s modulus and Poisson’s ratio for anisotropic materials, while possible, can be misinterpreted, especially when compared with isotropic formulas. For orthorhombic and higher symmetries, the “engineering” parameters – Poisson ratios, nij, shear moduli, Gij, and Young’s moduli, Eij – can be conveniently defined in terms of elastic compliances (Jaeger et al., 2007): E11 ¼ 1=s11

E22 ¼ 1=s22

E33 ¼ 1=s33

n21 ¼
s12 E22

n31 ¼
s13 E33

n32 ¼
s23 E33

G12 ¼ 1=s66

G13 ¼ 1=s55

G23 ¼ 1=s44

Existence of a strain energy leads to the following constraints: G12 ; G13 ; G23 ; E11 ; E22 ; E33 > 0 E11 E11 ðn21 Þ2 < 1; ðn31 Þ2 < 1; E22 E33

E22 ðn32 Þ2 < 1 E33

Assumptions and limitations The preceding equations assume anisotropic, linear elastic media.

2.3

Thomsen’s notation for weak elastic anisotropy Synopsis A transversely isotropic elastic material is completely specified by five independent constants. In terms of the abreviated Voigt notation (see Section 2.2 on elastic anisotropy) the elastic constants can be represented as 2 3 c11 c12 c13 0 0 0 6 c12 c11 c13 0 0 0 7 6 7 6 c13 c13 c33 0 0 0 7 6 7; where c66 ¼ 1ðc11
c12 Þ 2 6 0 0 7 0 0 c44 0 6 7 4 0 0 0 0 c44 0 5 0 0 0 0 0 c66

36

Elasticity and Hooke’s law

and where the axis of symmetry lies along the x3-axis. Thomsen (1986) suggested the following convenient notation for a TI material that is only weakly anisotropic. His notation uses the P- and S-wave velocities (denoted by a and b, respectively) propagating along the symmetry axis, plus three additional constants: pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi c11
c33 a ¼ c33 =; b ¼ c44 =; e ¼ 2c33 g¼

c66
c44 ; 2c44

d¼

ðc13 þ c44 Þ2
ðc33
c44 Þ2 2c33 ðc33
c44 Þ

In terms of these constants, the three phase velocities can be approximated conveniently as VP ðyÞ að1 þ d sin2 y cos2 y þ e sin4 yÞ   a2 2 2 VSV ðyÞ b 1 þ 2 ðe
dÞ sin y cos y b VSH ðyÞ bð1 þ g sin2 yÞ where y is the angle of the wave vector relative to the x3-axis; VSH is the wavefront velocity of the pure shear wave, which has no component of polarization in the symmetry axis direction; VSV is the pseudo-shear wave polarized normal to the pure shear wave; and VP is the pseudo-longitudinal wave. Berryman (2008) extends the validity of Thomsen’s expressions for P- and quasiSV-wave velocities to wider ranges of angles and stronger anisotropy with the following expressions:   2 sin2 ym sin2 y cos2 y 2 VP ðyÞ a 1 þ e sin y
ðe
dÞ 1
cos 2ym cos 2y  2   a 2 sin2 ym sin2 y cos2 y ðe
dÞ VSV ðyÞ b 1 þ 1
cos 2ym cos 2y b2 where tan2 ym ¼

c33
c44 c11
c44

Berryman’s formulas give more accurate velocities at larger angles. They are designed to give the angular location of the peak (or trough) of the quasi-SV-velocity closer to the correct location; hence, the quasi-SV-velocities are more accurate than those from Thomsen’s equations. Thomsen’s P-wave velocities will sometimes be more accurate than Berryman’s at small y and sometimes worse. For weak anisotropy, the constant e can be seen to describe the fractional difference between the P-wave velocities parallel and orthogonal to the symmetry axis (in the weak anisotropy approximation):

37

2.3 Thomsen’s notation for weak elastic anisotropy

e

VP ð90 Þ
VP ð0 Þ VP ð0 Þ

Therefore, e best describes what is usually called the “P-wave anisotropy.” Similarly, for weak anisotropy the constant g can be seen to describe the fractional difference between the SH-wave velocities parallel and orthogonal to the symmetry axis, which is equivalent to the difference between the velocities of S-waves polarized parallel and normal to the symmetry axis, both propagating normal to the symmetry axis: g

VSH ð90 Þ
VSV ð90 Þ VSH ð90 Þ
VSH ð0 Þ ¼ VSV ð90 Þ VSH ð0 Þ

The small-offset normal moveout velocity is affected by VTI (transversely isotropic with vertical symmetry axis) anisotropy. In terms of the Thomsen parameters, NMO (normal moveout) velocities, VNMO;P , VNMO;SV , and VNMO;SH for P-, SV-, and SHmodes are (Tsvankin, 2001): pffiffiffiffiffiffiffiffiffiffiffiffiffiffi VNMO;P ¼ a 1 þ 2d  2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a VNMO;SV ¼ b 1 þ 2
;
¼ ðe
dÞ b pffiffiffiffiffiffiffiffiffiffiffiffiffi VNMO;SH ¼ b 1 þ 2g An additional anellipticity parameter, , was introduced by Alkhalifah and Tsvankin (1995): ¼

e
d 1 þ 2d

 is important in quantifying the effects of anisotropy on nonhyperbolic moveout and P-wave time-processing steps (Tsvankin, 2001), including NMO, DMO (dip moveout), and time migration. The Thomsen parameters can be inverted for the elastic constants as follows: c33 ¼ a2 ;

c44 ¼ b2

c11 ¼ c33 ð1 þ 2eÞ; c66 ¼ c44 ð1 þ 2gÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c13 ¼  2c33 ðc33
c44 Þd þ ðc33
c44 Þ2
c44 Note the nonuniqueness in c13 that results from uncertainty in the sign of (c13 þ c44). Tsvankin (2001, p. 19) argues that for most cases, it can be assumed that (c13 þ c44) > 0, and therefore, the þsign in the equation for c13 is usually appropriate. Tsvankin (2001) summarizes some bounds on the values of the Thomsen parameters:

  the lower bound for d: d
1
b2 =a2 =2;

  an approximate upper bound for d: d  2= a2 =b2
1 ; and  in VTI materials resulting from thin layering, e > d and g > 0.

38

Elasticity and Hooke’s law

Ranges of Thomsen parameters expected for thin laminations of isotropic materials (Berryman et al., 1999).

Table 2.3.1

e d g e–d

m ¼ const

l ¼ const m 6¼ const

lþm ¼ const m 6¼ const

lþ2m ¼ const m 6¼ const

n ¼ const l, m 6¼ const

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

Transversely isotropic media consisting of thin isotropic layers always have Thomsen (1986) parameters, such that e – d 0 and g 0 (Tsvankin, 2001). Berryman et al. (1999) find the additional conditions summarized in Table 2.3.1, based on Backus (1962) average analysis and Monte Carlo simulations of thinly laminated media. Although all of the cases in the table have d  0, Berryman et al. find that d can be positive if the layers have significantly varying and positively correlated shear modulus, m, and Poisson ratio, n. Bakulin et al. (2000) studied the Thomsen parameters for anisotropic HTI (transversely isotropic with horizontal symmetry axis) media resulting from aligned vertical fractures with crack normals along the horizontal x1-axis in an isotropic background. For example, when the Hudson (1980) penny-shaped crack model (Section 4.10) is used to estimate weak anisotropy resulting from crack density, e, the dry-rock Thomsen parameters in the vertical plane containing the x1-axis can be approximated as ðVÞ

edry ¼ ðVÞ ddry ðVÞ

gdry

ðVÞ

dry

c11
c33 8 ¼
e  0; 2c33 3

  ðc13 þ c55 Þ2
ðc33
c55 Þ2 8 gð1
2gÞ ¼
e 1þ ¼  0; 2c33 ðc33
c55 Þ 3 ð3
2gÞð1
gÞ c66
c44 8e ¼ ¼
 0; 2c44 3ð3
2gÞ   ðVÞ ðVÞ edry
ddry 8 gð1
2gÞ ¼ ¼ e 0 ðVÞ 3 ð3
2gÞð1
gÞ 1 þ 2ddry

where g ¼ VS2 =VP2 is a property of the unfractured background rock. In the case of fluid-saturated penny-shaped cracks, such that the crack aspect, a, is much less than the ratio of the fluid bulk modulus to the mineral bulk modulus, Kfluid/Kmineral, then the weak-anisotropy Thomsen parameters can be approximated as ðVÞ

esaturated ¼ ðVÞ

dsaturated ¼

c11
c33 ¼0 2c33 ðc13 þ c55 Þ2
ðc33
c55 Þ2 32ge ¼
0 2c33 ðc33
c55 Þ 3ð3
2gÞ

39

2.4 Tsvankin’s extended Thomsen parameters

ðVÞ

c66
c44 8e ¼
0 2c44 3ð3
2gÞ 32ge ðVÞ ¼
dsaturated ¼ 0 3ð3
2gÞ

gsaturated ¼ ðVÞ

saturated

In intrinsically anisotropic shales, Sayers (2004) finds that d can be positive or negative, depending on the contact stiffness between microscopic clay domains and on the distribution of clay domain orientations. He shows, via modeling, that distributions of clays domains, each having d  0, can yield a composite with d > 0 overall if the domain orientations vary significantly from parallel. Rasolofosaon (1998) shows that under the assumptions of third-order elasticity an isotropic rock obtains ellipsoidal symmetry with respect to the propagation of qP waves. Hence e ¼ d in the symmetry planes (see Section 2.4) on Tsvankin’s extended Thomsen parameters.

Uses Thomsen’s notation for weak elastic anisotropy is useful for conveniently characterizing the elastic constants of a transversely isotropic linear elastic medium.

Assumptions and limitations The preceding equations are based on the following assumptions:  material is linear, elastic, and transversely isotropic;  anisotropy is weak, so that e, g, d 1.

2.4

Tsvankin’s extended Thomsen parameters for orthorhombic media The well-known Thomsen (1986) parameters for weak anisotropy are well suited for transversely isotropic media (see Section 2.3). They allow the five independent elastic constants c11, c33, c12, c13, and c44 to be expressed in terms of the more intuitive P-wave velocity, a, and S-wave velocity, b, along the symmetry axis, plus additional constants e, g, and d. For orthorhombic media, requiring nine independent elastic constants, the conventional Thomsen parameters are insufficient. Analogs of the Thomsen parameters suitable for orthorhombic media can be defined (Tsvankin, 1997), recognizing that wave propagation in the x1–x3 and x2–x3 symmetry planes (pseudo-P and pseudo-S polarized in each plane and SH polarized normal to each plane) is analogous to propagation in two different VTI media. We once again define vertically propagating (along the x3-axis) P- and S-wave velocities, a and b, respectively: pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi a ¼ c33 =; b ¼ c55 =

40

Elasticity and Hooke’s law

Unlike in a VTI medium, S-waves propagating along x3-axis in an orthorhombic pthe ffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi medium can have two different velocities, bx2 ¼ c44 = and bx1 ¼ c55 =, for waves polarized in the x2 and x1 directions, respectively. pffiffiffiffiffiffiffiffiffiffiffi Either polarization can be chosen as a reference, though here we take b ¼ c55 = following the definitions of Tsvankin (1997). Some results shown in later sections will use redefined polarizations in the definition of b. For the seven constants, we can write eð2Þ ¼

c11
c33 2c33

ðc13 þ c55 Þ2
ðc33
c55 Þ2 2c33 ðc33
c55 Þ c66
c44 ¼ 2c44

eð1Þ ¼

ðc23 þ c44 Þ2
ðc33
c44 Þ2 2c33 ðc33
c44 Þ c66
c55 ¼ 2c55

dð2Þ ¼

dð1Þ ¼

gð2Þ

gð1Þ

dð3Þ ¼

c22
c33 2c33

ðc12 þ c66 Þ2
ðc11
c66 Þ2 2c11 ðc11
c66 Þ

Here, the superscripts (1), (2), and (3) refer to the TI-analog parameters in the symmetry planes normal to x1, x2, and x3, respectively. These definitions assume that one of the symmetry planes of the orthorhombic medium is horizontal and that the vertical symmetry axis is along the x3 direction. These Thomsen–Tsvankin parameters play a useful role in modeling wave propagation and reflectivity in anisotropic media.

Uses Tsvankin’s notation for weak elastic anisotropy is useful for conveniently characterizing the elastic constants of an orthorhombic elastic medium.

Assumptions and limitations The preceding equations are based on the following assumptions:  material is linear, elastic, and has orthorhombic or higher symmetry;  the constants are definitions. They sometimes appear in expressions for anisotropy of arbitrary strength, but at other times the applications assume that the anisotropy is weak, so that e, g, d 1.

2.5

Third-order nonlinear elasticity Synopsis Seismic velocities in crustal rocks are almost always sensitive to stress. Since so much of geophysics is based on linear elasticity, it is common to extend the familiar linear elastic terminology and refer to the “stress-dependent linear elastic moduli” – which can

41

2.5 Third-order nonlinear elasticity

have meaning for the local slope of the strain-curves at a given static state of stress. If the relation between stress and strain has no hysteresis and no dependence on rate, then it is more accurate to say that the rocks are nonlinearly elastic (e.g., Truesdell, 1965; Helbig, 1998; Rasolofosaon, 1998). Nonlinear elasticity (i.e., stress-dependent velocities) in rocks is due to the presence of compliant mechanical defects, such as cracks and grain contacts (e.g., Walsh, 1965; Jaeger and Cook, 1969; Bourbie´ et al., 1987). In a material with third-order nonlinear elasticity, the strain energy function E (for arbitrary anisotropy) can be expressed as (Helbig, 1998) E ¼ 12 cijkl eij ekl þ 16 cijklmn eij ekl emn where cijkl and cijklmn designate the components of the second- and third-order elastic tensors, respectively, and repeated indices in a term imply summation from 1 to 3. The components cijkl are the usual elastic constants in Hooke’s law, discussed earlier. Hence, linear elasticity is often referred to as second-order elasticity, because the strain energy in a linear elastic material is second order in strain. The linear elastic tensor (cijkl) is fourth rank, having a minimum of two independent constants for a material with the highest symmetry (isotropic) and a maximum of 21 independent constants for a material with the lowest symmetry (triclinic). The additional tensor of third-order elastic coefficients (cijklmn) is rank six, having a minimum of three independent constants (isotropic) and a maximum of 56 independent constants (triclinic) (Rasolofosaon, 1998). Third-order elasticity is sometimes used to describe the stress-sensitivity of seismic velocities and apparent elastic constants in rocks. The apparent fourth-rank stiffness tensor, c~eff , which determines the speeds of infinitesimal-amplitude waves in a rock under applied static stress can be written as c~eff ijkl ¼ cijkl þ cijklmn emn where emn are the principal strains associated with the applied static stress. Approximate expressions, in Voigt notation, for the effective elastic constants of a stressed VTI (transversely isotropic with a vertical axis of symmetry) solid can be written as (Rasolofosaon, 1998; Sarkar et al., 2003; Prioul et al., 2004) 0 ceff 11 c11 þ c111 e11 þ c112 ðe22 þ e33 Þ 0 ceff 22 c11 þ c111 e22 þ c112 ðe11 þ e33 Þ 0 ceff 33 c33 þ c111 e33 þ c112 ðe11 þ e22 Þ 0 ceff 12 c12 þ c112 ðe11 þ e22 Þ þ c123 e33 0 ceff 13 c13 þ c112 ðe11 þ e33 Þ þ c123 e22 0 ceff 23 c13 þ c112 ðe22 þ e33 Þ þ c123 e11 0 ceff 66 c66 þ c144 e33 þ c155 ðe11 þ e22 Þ 0 ceff 55 c44 þ c144 e22 þ c155 ðe11 þ e33 Þ 0 ceff 44 c44 þ c144 e11 þ c155 ðe22 þ e33 Þ

42

Elasticity and Hooke’s law

Table 2.5.1 Experimentally determined third-order elastic constants c111, c112, and c123 and derived constants c144, c155, and c456, determined by Prioul and Lebrat (2004), using laboratory data from Wang (2002). Six different sandstone and six different shale samples are shown. c111 (GPa)

c112 (GPa)

c123 (GPa)


10 245
9 482
6 288
8 580
8 460
12 440


966
1745
1744
527
1162
3469


966
1745
1744
527
1162
3094


6 903
4 329
7 034
4 160 1 294
1 203


976
2122
2147
2013
510
637


976
1019 296
940
119
354

c144 (GPa)

c155 (GPa)

c456 (GPa)

0 0 0 0 0
188


2320
1934
1136
2013
1825
2243


1160
967
568
1006
912
1027

0
552
1222
536
196
141


1482
552
1222
536
196
141


741 0 0 0 0 0

Sandstones

Shales

where the constants c011 , c033 , c013 , c044 , c066 are the VTI elastic constants at the unstressed reference state, with c012 ¼ c011
2c066 . e11, e22, and e33 are the principal strains, computed from the applied stress using the conventional Hooke’s law, eij ¼ sijkl
kl . For these expressions, it is assumed that the direction of the applied principal stress is aligned with the VTI symmetry (x3-) axis. Furthermore, for these expressions it is assumed that the stress-sensitive third-order tensor is isotropic, defined by the three independent constants, c111, c112, and c123 with c144 ¼ ðc112
c123 Þ=2, c155 ¼ ðc111
c112 Þ=4, and c456 ¼ ðc111
3c112 þ 2c123 Þ=8. It is generally observed (Prioul et al., 2004) that c111 < c112 < c123, c155 < c144, c155 < 0, and c456 < 0. A sample of experimentally determined values from Prioul and Lebrat (2004) using laboratory data from Wang (2002) are shown in the Table 2.5.1. The third-order elasticity as used in most of geophysics and rock physics (Bakulin and Bakulin, 1999; Prioul et al., 2004) is called the Murnaghan (1951) formulation of finite deformations, and the third-order constants are also called the Murnaghan constants. Various representations of the third-order constants that can be found in the literature (Rasolofosaon, 1998) include the crystallographic set (c111, c112, c123) presented here, the Murnaghan (1951) constants (l, m, n), and Landau’s set (A, B, C) (Landau and Lifschitz, 1959). The relations among these are (Rasolofosaon, 1998)

43

2.6 Effective stress properties of rocks

c111 ¼ 2A þ 6B þ 2C ¼ 2l þ 4m c112 ¼ 2B þ 2C ¼ 2l c123 ¼ 2C ¼ 2l
2m þ n Chelam (1961) and Krishnamurty (1963) looked at fourth-order elastic coefficients, based on an extension of Murnaghan’s theory. It turns out from group theory that there will only be n independent nth-order coefficients for isotropic solids, and n2 – 2n þ 3 independent nth-order constants for cubic systems. Triclinic solids have 126 fourth-order elastic constants.

Uses Third-order elasticity provides a way to parameterize the stress dependence of seismic velocities. It also allows for a compact description of stress-induced anisotropy, which is discussed later.

Assumptions and limitations 





2.6

The above equations assume that the material is hyperelastic, i.e., there is no hysteresis or rate dependence in the relation between stress and strain, and there exists a unique strain energy function. This formalism assumes that strains are infinitesimal. When strains become finite, an additional source of nonlinearity, called geometrical or kinetic nonlinearity, appears, related to the difference between Lagrangian and Eulerian descriptions of motion (Zarembo and Krasil’nikov, 1971; Johnson and Rasolofosaon, 1996). Third-order elasticity is often not general enough to describe the shapes of real stress–strain curves over large ranges of stress and strain. Third-order elasticity is most useful when describing stress–strain within a small range around a reference state of stress and strain.

Effective stress properties of rocks Synopsis Because rocks are deformable, many rock properties are sensitive to applied stresses and pore pressure. Stress-sensitive properties include porosity, permeability, electrical resistivity, sample volume, pore-space volume, and elastic moduli. Empirical evidence (Hicks and Berry, 1956; Wyllie et al., 1958; Todd and Simmons, 1972; Christensen and Wang, 1985; Prasad and Manghnani, 1997; Siggins and Dewhurst, 2003, Hoffman et al., 2005) and theory (Brandt, 1955; Nur and Byerlee, 1971; Zimmerman, 1991a; Gangi and Carlson, 1996; Berryman, 1992a, 1993; Gurevich, 2004) suggest that the pressure dependence of each of these rock properties, X, can be

44

Elasticity and Hooke’s law

represented as a function, fX, of a linear combination of the hydrostatic confining stress, PC, and the pore pressure, PP: X ¼ fX ðPC
nPP Þ;

n1

The combination Peff ¼ PC
nPP is called the effective pressure, or more generally, C the tensor
eff ij ¼
ij
nPP dij is the effective stress. The parameter n is called the “effective-stress coefficient,” which can itself be a function of stress. The negative sign on the pore pressure indicates that the pore pressure approximately counteracts the effect of the confining pressure. An expression such as X ¼ fX ðPC
nPP Þ is sometimes called the effective-stress law for the property X. It is important to point out that each rock property might have a different function fi and a different value of ni (Zimmerman, 1991a; Berryman, 1992a, 1993; Gurevich, 2004). Extensive discussions on the effective-stress behavior of elastic moduli, permeability, resistivity, and thermoelastic properties can be found in Berryman (1992a, 1993). Zimmerman (1991a) gives a comprehensive discussion of effective-stress behavior for strain and elastic constants. Zimmerman (1991a) points out the distinction between the effective-stress behavior for finite pressure changes versus the effective-stress behavior for infinitesimal increments of pressure. For example, increments of the bulk-volume strain, eb, and the pore-volume strain, ep, can be written as   1 @VT eb ðPC ; PP Þ ¼
Cbc ðPC
mb PP ÞðdPC
nb dPP Þ; Cbc ¼
VT @PC PP   1 @VP ep ðPC ; PP Þ ¼
Cpc ðPC
mp PP ÞðdPC
np dPP Þ; Cpc ¼
VP @PC PP where the compressibilities Cbc and Cpc are functions of PC
mPP . The coefficients mb and mp govern the way that the compressibilities vary with PC and PP. In contrast, the coefficients nb and np describe the relative increments of additional strain resulting from pressure increments dPC and dPP. For example, in a laboratory experiment, ultrasonic velocities will depend on the values of Cpc, the local slope of the stress–strain curve at the static values of PC and PP. On the other hand, the sample length change monitored within the pressure vessel is the total strain, obtained by integrating the strain over the entire stress path. The existence of an effective-stress law, i.e., that a rock property depends only on the state of stress, requires that the rock be elastic – possibly nonlinearly elastic. The deformation of an elastic material depends only on the state of stress, and is independent of the stress history and the rate of loading. Furthermore, the existence of an effective-stress law requires that there is no hysteresis in stress–strain cycles. Since no rock is perfectly elastic, all effective-stress laws for rocks are approximations. In fact, deviation from elasticity makes estimating the effective-stress coefficient from laboratory data sometimes ambiguous. Another condition required for the existence of an effective-pressure law is that the pore pressure is well defined and uniform throughout the pore space. Todd and Simmons (1972) show that the effect of pore

45

2.6 Effective stress properties of rocks

pressure on velocities varies with the rate of pore-pressure change and whether the pore pressure has enough time to equilibrate in thin cracks and poorly connected pores. Slow changes in pore pressure yield more stable results, describable with an effective-stress law, and with a larger value of n for velocity. Much discussion focuses on the value of the effective-stress constants, n (and m). Biot and Willis (1957) predicted theoretically that the pressure-induced volume increment, dVT, of a sample of linear poroelastic material depends on pressure increments ðdPC
ndPP Þ. For this special case, n ¼ a ¼ 1
K=KS , where a is known as the Biot coefficient or Biot–Willis coefficient. K is the dry-rock (drained) bulk modulus and KS is the mineral bulk modulus (or some appropriate average of the moduli if there is mixed mineralogy), defined below. Explicitly,     dVT 1 K dPP ¼
dPC
1
VT K KS where dVT, dPC, and dPP signify increments relative to a reference state.

Pitfall

A common error is to assume that the Biot–Willis effective-stress coefficient a for volume change also applies to other deformation-related rock properties. For example, although rock elastic moduli vary with crack and pore deformation, there is no theoretical justification for extrapolating a to elastic moduli and seismic velocities. Other factors determining the apparent effective-stress coefficient observed in the laboratory include the rate of change of pore pressure, the connectivity of the pore space, the presence or absence of hysteresis, heterogeneity of the rock mineralogy, and variation of pore-fluid compressibility with pore pressure. Table 2.6.1, compiled from Zimmerman (1984), Berryman (1992a, 1993), and H. F. Wang (2000), summarizes the theoretically expected effective incremental stress laws for a variety of rock properties. These depend on four defined rock moduli:   1 1 @VT ; K ¼ modulus of the drained porous frame; ¼
K VT @Pd PP   1 1 @VT ¼
; KS ¼ unjacketed modulus; if monomineralic; KS VT @PP Pd KS ¼ Kmineral ; otherwise KS is a poorly understood average of the mixed mineral moduli   1 1 @V ¼
; if monomineralic; KS ¼ K ¼ Kmineral K V @PP Pd     1 1 @V 1 1 1 ¼
¼
KP V @Pd PP  K KS

46

Elasticity and Hooke’s law

Theoretically predicted effective-stress laws for incremental changes in confining and pore pressures (from Berryman, 1992a).

Table 2.6.1

Property

General mineralogy

Sample volumea

dVT VT dV V

Pore volumeb Porosityc Solid volumed Permeabilitye Velocity/elastic modulif

¼
K1 ðdPC
adPP Þ

¼
K1P ðdPC
bdPP Þ   d a
  ¼
K ðdPC
dPP Þ 1 ¼
ð1
ÞK ðdPC

dPP Þ S h   i a
 dk 2 k ¼
h K þ 3K ðdPC
kdPP Þ dVS VS

dVP VP

¼ f ðdPC
yPP Þ

Notes: VT is the total volume. a ¼ 1
K=KS , Biot coefficient; usually in the range   a  1; if monomineralic, a ¼ 1
K=Kmineral . b V ¼ VT , pore volume. b ¼ 1
KP =K , usually,     b  1, but it is possible that b > 1. c  ¼ b
 a
 a; if monomineralic,  ¼ 1.

 2ð1
aÞ d e
¼ a

a 1
a ða
Þ; if monomineralic,
¼ .
 a  b. k ¼ 1
3hða
Þþ2  1: a

S Þ=@PC h 2 þ m 4, where m is Archie’s cementation exponent. f y ¼ 1
@ð1=K @ð1=KÞ=@PC ; if monomineralic,

y ¼ 1.

where Pd ¼ PC
PP is the differential pressure, VT is the sample bulk (i.e., total) volume, and Vf is the pore volume. The negative sign for each of these rock properties follows from defining pressures as being positive in compression and volumes positive in expansion. There is still a need to reconcile theoretical predictions of effective stress with certain laboratory data. For example, simple, yet rigorous, theoretical considerations (Zimmerman, 1991a; Berryman, 1992a; Gurevich, 2004) predict that nvelocity ¼ 1 for monomineralic, elastic rocks. Experimentally observed values for nvelocity are sometimes close to 1, and sometimes less than one. Speculations for the variations have included mineral heterogeneity, poorly connected pore space, pressure-related changes in pore-fluid properties, incomplete correction for fluid-related velocity dispersion in ultrasonic measurements, poorly equilibrated or characterized pore pressure, and inelastic deformation.

Uses Characterization of the stress sensitivity of rock properties makes it possible to invert for rock-property changes from changes in seismic or electrical measurements. It also provides a means of understanding how rock properties might change in response to tectonic stresses or pressure changes resulting from reservoir or aquifer production.

47

2.7 Stress-induced anisotropy in rocks

Assumptions and limitations  

2.7

The existence of effective-pressure laws assumes that the rocks are hyperelastic, i.e., there is no hysteresis or rate dependence in the relation between stress and strain. Rocks are extremely variable, so effective-pressure behavior can likewise be variable.

Stress-induced anisotropy in rocks Synopsis The closing of cracks under compressive stress (or, equivalently, the stiffening of compliant grain contacts) tends to increase the effective elastic moduli of rocks (see also Section 2.5 on third-order elasticity). When the crack population is anisotropic, either in the original unstressed condition or as a result of the stress field, then this condition can impact the overall elastic anisotropy of the rock. Laboratory demonstrations of stress-induced anisotropy have been reported by numerous authors (Nur and Simmons, 1969a; Lockner et al., 1977; Zamora and Poirier, 1990; Sayers et al., 1990; Yin, 1992; Cruts et al., 1995). The simplest case to understand is a rock with a random (isotropic) distribution of cracks embedded in an isotropic mineral matrix. In the initial unstressed state, the rock is elastically isotropic. If a hydrostatic compressive stress is applied, cracks in all directions respond similarly, and the rock remains isotropic but becomes stiffer. However, if a uniaxial compressive stress is applied, cracks with normals parallel or nearly parallel to the applied-stress axis will tend to close preferentially, and the rock will take on an axial or transversely isotropic symmetry. An initially isotropic rock with arbitrary stress applied will have at least orthorhombic symmetry (Nur, 1971; Rasolofosaon, 1998), provided that the stress-induced changes in moduli are small relative to the absolute moduli. Figure 2.7.1 illustrates the effects of stress-induced crack alignment on seismicvelocity anisotropy discovered in the laboratory by Nur and Simmons (1969a). The crack porosity of the dry granite sample is essentially isotropic at low stress. As uniaxial stress is applied, crack anisotropy is induced. The velocities (compressional and two polarizations of shear) clearly vary with direction relative to the stressinduced crack alignment. Table 2.7.1 summarizes the elastic symmetries that result when various applied-stress fields interact with various initial crack symmetries (Paterson and Weiss, 1961; Nur, 1971). A rule of thumb is that a wave is most sensitive to cracks when its direction of propagation or direction of polarization is perpendicular (or nearly so) to the crack faces. The most common approach to modeling the stress-induced anisotropy is to assume angular distributions of idealized penny-shaped cracks (Nur, 1971; Sayers, 1988a, b; Gibson and Tokso¨z, 1990). The stress dependence is introduced by assuming or inferring distributions or spectra of crack aspect ratios with various orientations.

48

Elasticity and Hooke’s law

5.0 Stress (bars) 300 4.6 200 VP (km/s)

4.8

4.4 100 4.2 4.0 0 3.8 3.6

0

20 40 60 80 Angle from stress axis (deg) 3.2

3.2

300

2.9

200

2.8

100

2.7

Vs (SV) (km/s)

Vs (SH) (km/s)

400 3.0

2.6

Stress (bars) 3.1 400

Stress (bars)

3.1

200

2.9

2.8 100 2.7

0 0

300

3.0

20 40 60 80 Angle from stress axis (deg)

2.6

0 0

20 40 60 80 Angle from stress axis (deg)

Figure 2.7.1 The effects of stress-induced crack alignment on seismic-velocity anisotropy measured in the laboratory (Nur and Simmons, 1969a).

The assumption is that a crack will close when the component of applied compressive stress normal to the crack faces causes a normal displacement of the crack faces equal to the original half-width of the crack. This allows us to estimate the crack closing stress as follows:
close ¼

3pð1
2nÞ p aK0 ¼ am 4ð1
n2 Þ 2ð1
nÞ 0

where a is the aspect ratio of the crack, and n, m0, and K0 are the Poisson ratio, shear modulus, and bulk modulus of the mineral, respectively (see Section 2.9 on the deformation of inclusions and cavities in elastic solids). Hence, the thinnest cracks will close first, followed by thicker ones. This allows one to estimate, for a given aspect-ratio distribution, how many cracks remain open in each direction for any applied stress field. These inferred crack distributions and their orientations can be put into one of the popular crack models (e.g., Hudson, 1981) to estimate the resulting effective elastic moduli of the rock. Although these penny-shaped crack models have been relatively successful and provide a useful physical interpretation, they are limited to low crack concentrations and may not effectively represent a broad range of crack geometries (see Section 4.10 on Hudson’s model for cracked media).

49

2.7 Stress-induced anisotropy in rocks

Dependence of symmetry of induced velocity anisotropy on initial crack distribution and applied stress and its orientation.

Table 2.7.1

Symmetry of initial crack distribution

Applied stress

Random

Hydrostatic Uniaxial Triaxiala

Axial

Hydrostatic Uniaxial Uniaxial Uniaxial Triaxiala Triaxiala

Orthorhombic

Hydrostatic Uniaxial Uniaxial Uniaxial Triaxiala Triaxiala Triaxiala

Orientation of applied stress

Parallel to axis of symmetry Normal to axis of symmetry Inclined Parallel to axis of symmetry Inclined Parallel to axis of symmetry Inclined in plane of symmetry Inclined Parallel to axis of symmetry Inclined in plane of symmetry Inclined

Symmetry of induced velocity anisotropy

Number of elastic constants

Isotropic Axial Orthorhombic

2 5 9

Axial Axial

5 5

Orthorhombic

9

Monoclinic Orthorhombic

13 9

Monoclinic

13

Orthorhombic Orthorhombic

9 9

Monoclinic

13

Triclinic Orthorhombic

21 9

Monoclinic

13

Triclinic

21

Note: a Three generally unequal principal stresses.

As an alternative, Mavko et al. (1995) presented a simple recipe for estimating stress-induced velocity anisotropy directly from measured values of isotropic VP and VS versus hydrostatic pressure. This method differs from the inclusion models, because it is relatively independent of any assumed crack geometry and is not limited to small crack densities. To invert for a particular crack distribution, one needs to assume crack shapes and aspect-ratio spectra. However, if rather than inverting for a crack distribution, we instead directly transform hydrostatic velocity–pressure data to stress-induced velocity anisotropy, we can avoid the need for parameterization in terms of ellipsoidal cracks and the resulting limitations to low crack densities. In this sense, the method of Mavko et al. (1995) provides not only a simpler but also a more general solution to this problem, for ellipsoidal cracks are just one particular case of the general formulation.

50

Elasticity and Hooke’s law

The procedure is to estimate the generalized pore-space compliance from the measurements of isotropic VP and VS. The physical assumption that the compliant part of the pore space is crack-like means that the pressure dependence of the generalized compliances is governed primarily by normal tractions resolved across cracks and defects. These defects can include grain boundaries and contact regions between clay platelets (Sayers, 1995). This assumption allows the measured pressure dependence to be mapped from the hydrostatic stress state to any applied nonhydrostatic stress. The method applies to rocks that are approximately isotropic under hydrostatic stress and in which the anisotropy is caused by crack closure under stress. Sayers (1988b) also found evidence for some stress-induced opening of cracks, which is ignored in this method. The potentially important problem of stress–strain hysteresis is also ignored. The anisotropic elastic compliance tensor Sijkl(s) at any given stress state s may be expressed as Sijkl ð
Þ ¼ Sijkl ð
Þ
S0ijkl Z p=2 Z 2p 0

0 ^ T
mÞ ^
4W2323 ^ T
mÞ ^ mi mj mk ml sin y dy d ¼ W3333 ðm ðm ¼0

y¼0

Z

þ

p=2 y¼0

Z

0 ^ T
mÞ ^ dik mj ml þ dil mj mk þ djk mi ml W2323 ðm ¼0

þdjl mi mk sin y dy d 2p

where 1 Siso ðpÞ 2p jjkk 1 0 iso W2323 ðpÞ ¼ ½Siso abab ðpÞ
Saabb ðpÞ 8p 0 W3333 ðpÞ ¼

The tensor S0ijkl denotes the reference compliance at some very large confining hydrostatic pressure when all of the compliant parts of the pore space are closed. The iso 0 expression Siso ijkl ðpÞ ¼ Sijkl ðpÞ
Sijkl describes the difference between the compliance under a hydrostatic effective pressure p and the reference compliance at high pressure. These are determined from measured P- and S-wave velocities versus the 0 0 hydrostatic pressure. The tensor elements W3333 and W2323 are the measured normal and shear crack compliances and include all interactions with neighboring cracks and pores. These could be approximated, for example, by the compliances of idealized ellipsoidal cracks, interacting or not, but this would immediately reduce the general^  ðsin y cos ; sin y sin ; cos yÞT denotes the unit normal to the ity. The expression m crack face, where y and f are the polar and azimuthal angles in a spherical coordinate system, respectively.

51

2.7 Stress-induced anisotropy in rocks

An important physical assumption in the preceding equations is that, for thin 0 0 0 cracks, the crack compliance tensor Wijkl is sparse, and thus only W3333 , W1313 , and 0 W2323 are nonzero. This is a general property of planar crack formulations and reflects an approximate decoupling of normal and shear deformation of the crack and decoupling of the in-plane and out-of-plane deformations. This allows us to write 0 0 Wjjkk W3333 . Furthermore, it is assumed that the two unknown shear compliances 0 0 are approximately equal: W1313 W2323 . A second important physical assumption is that for a thin crack under any stress field, it is primarily the normal component of ^ T
m, ^ resolved on the faces of a crack, that causes it to close and to have stress,
¼ m a stress-dependent compliance. Any open crack will have both normal and shear deformation under normal and shear loading, but it is only the normal stress that determines crack closure. For the case of uniaxial stress s0 applied along the 3-axis to an initially isotropic rock, the normal stress in any direction is sn ¼ s0 cos2y. The rock takes on a transversely isotropic symmetry, with five independent elastic constants. The five independent components of DSijkl become Z Suni 3333

p=2

¼ 2p 0

Z

p=2

þ 2p

0 p=2

Z Suni 1111 ¼ 2p

0

Z

0 p=2

Z Z

0

Z

0

Suni 1133 ¼ 2p Suni 2323

p=2

p=2

¼ 2p 0

Z

þ 2p þ 2p 0

0 cos2 yÞ
4W2323 ð
0 cos2 yÞ sin4 y sin y dy

0 2W2323 ð
0 cos2 yÞ sin2 y sin y dy

0 1 8 ½W3333 ð
0

0 cos2 yÞ
4W2323 ð
0 cos2 yÞ sin4 y sin y dy

0 1 2 ½W3333 ð
0

0 cos2 yÞ
4W2323 ð
0 cos2 yÞ sin2 y cos2 y sin y dy

0 1 2 ½W3333 ð
0

0 cos2 yÞ
4W2323 ð
0 cos2 yÞ sin2 y cos2 y sin y dy

p=2

0

Z

0 4W2323 ð
0 cos2 yÞ cos2 y sin y dy

0 3 8 ½W3333 ð
0

p=2

þ 2p Suni 1122 ¼ 2p

0 0 ½W3333 ð
0 cos2 yÞ
4W2323 ð
0 cos2 yÞ cos4 y sin y dy

p=2

0 1 2 W2323 ð
0

cos2 yÞ sin2 y sin y dy

0 W2323 ð
0 cos2 yÞ cos2 y sin y dy

0 Note that in the above equations, the terms in parentheses with W2323 ðÞ and 0 0 0 W3333 ðÞ are arguments to the W2323 and W3333 pressure functions, and not multiplicative factors. Sayers and Kachanov (1991, 1995) have presented an equivalent formalism for stress-induced anisotropy. The elastic compliance Sijkl is once again written in the form

52

Elasticity and Hooke’s law

Sijkl ¼ Sijkl ð
Þ
S0ijkl where S0ijkl is the compliance in the absence of compliant cracks and grain boundaries and DSijkl is the excess compliance due to the cracks. DSijkl can be written as Sijkl ¼ 14 ðdik ajl þ dil ajk þ djk ail þ djl aik Þ þ bijkl where aij is a second-rank tensor and bijkl is a fourth-rank tensor defined by 1 X ðrÞ ðrÞ ðrÞ ðrÞ B n n A V r T i j  1 X ðrÞ ðrÞ ðrÞ ðrÞ ðrÞ ðrÞ ¼ BN
BT ni nj nk nl AðrÞ V r

aij ¼ bijkl

In these expressions, the summation is over all grain contacts and microcracks within ðrÞ ðrÞ the rock volume V. BN and BT are the normal and shear compliances of the rth discontinuity, which relate the displacement discontinuity across the crack to the ðrÞ applied traction across the crack faces; ni , is the ith component of the normal to the discontinuity, and A(r) is the area of the discontinuity. A completely different strategy for quantifying stress-induced anisotropy is to use the formalism of third-order elasticity, described in Section 2.5 (e.g., Helbig, 1994; Johnson and Rasolofosaon, 1996; Prioul et al., 2004). The third-order elasticity approach is phenomenological, avoiding the physical mechanisms of stress sensitivity, but providing a compact notation. For example, Prioul et al. (2004) found that for a stressed VTI (transversely isotropic with vertical symmetry axis) material, the effective elastic constants can be approximated in Voigt notation as 0 ceff 11 c11 þ c111 e11 þ c112 ðe22 þ e33 Þ 0 ceff 22 c11 þ c111 e22 þ c112 ðe11 þ e33 Þ 0 ceff 33 c33 þ c111 e33 þ c112 ðe11 þ e22 Þ 0 ceff 12 c12 þ c112 ðe11 þ e22 Þ þ c123 e33 0 ceff 13 c13 þ c112 ðe11 þ e33 Þ þ c123 e22 0 ceff 23 c13 þ c112 ðe22 þ e33 Þ þ c123 e11 0 ceff 66 c66 þ c144 e33 þ c155 ðe11 þ e22 Þ 0 ceff 55 c44 þ c144 e22 þ c155 ðe11 þ e33 Þ 0 ceff 44 c44 þ c144 e11 þ c155 ðe22 þ e33 Þ

where the constants c011 , c033 , c013 , c044 , c066 are the VTI elastic constants at the unstressed reference state, with c012 ¼ c011
2c066 . The quantities e11, e22, and e33 are the principal strains, computed from the applied stress using the conventional Hooke’s law, eij ¼ sijkl
kl . For these expressions, it is assumed that the direction of

53

2.7 Stress-induced anisotropy in rocks

the applied principal stress is aligned with the VTI symmetry (x3-) axis. Furthermore, for these expressions it is assumed that the stress-sensitive third-order tensor is isotropic, defined by the three independent constants, c111, c112, and c123 with c144 ¼ ðc112
c123 Þ=2 and c155 ¼ ðc111
c112 Þ=4. In practice, the elastic constants (five for the unstressed VTI rock and three for the third-order elasticity) can be estimated from laboratory measurements, as illustrated by Sarkar et al. (2003) and Prioul et al. (2004). For an intrinsically VTI rock, hydrostatic loading experiments provide sufficient information to invert for all three of the third-order constants, provided that the anisotropy is not too weak. However, for an intrinsically isotropic rock, nonhydrostatic loading is required in order to provide enough independent information to determine the constants. Once the constants are determined, the stress-induced elastic constants corresponding to any (aligned) stress field can be determined. The full expressions for stress-induced anisotropy of an originally VTI rock are given by Sarkar et al. (2003) and Prioul et al. (2004). Sarkar et al. (2003) give expressions for stress-induced changes in Thomsen parameters of an originally VTI rock, provided that both the original and stressinduced anisotropies are weak: c155 c155 ð
22

33 Þ; eð2Þ ¼ e0 þ 0 0 ð
11

33 Þ 0 0 c33 c55 c33 c55 c155 c155 0 ð2Þ 0 d þ 0 0 ð
22

33 Þ; d ¼ d þ 0 0 ð
11

33 Þ c33 c55 c33 c55 c456 c456 g0 þ 0 0 ð
22

33 Þ; gð2Þ ¼ g0 þ 0 0 ð
11

33 Þ 2c55 c55 2c55 c55 c155 ð
22

11 Þ c033 c055 1 1 c456 ¼ ðc111
3c112 þ 2c123 Þ ðc111
c112 Þ; 4 8

eð1Þ ¼ e0 þ dð1Þ ¼ gð1Þ ¼ dð3Þ ¼ c155 ¼

The results are shown in terms of Tsvankin’s extended Thomsen parameters (Section 2.4). Orthorhombic symmetry occurs when
11 6¼
22 . (Strictly speaking, three different principal stresses applied to a VTI rock do not result in perfect orthorhombic symmetry. However, orthorhombic parameters are adequate in the case of weak anisotropy.)

Uses Understanding or, at least, empirically describing the stress dependence of velocities is useful for quantifying the change of velocities in seismic time-lapse data due to changes in reservoir pressure, as well as in certain types of naturally occurring overpressure. Since the state of stress in situ is seldom hydrostatic, quantifying the impact of stress on anisotropy can often improve on the usual isotropic analysis.

54

Elasticity and Hooke’s law

Assumptions and limitations   



2.8

Most models for predicting or describing stress-induced anisotropy that are based on cracks and crack-like flaws assume an isotropic, linear, elastic solid mineral material. Methods based on ellipsoidal cracks or spherical contacts are limited to idealized geometries and to low crack densities. The method of Mavko et al. (1995) has been shown to sometimes under predict stress-induced anisotropy when the maximum stress difference is comparable to the mean stress magnitude. More generally, there is evidence (Johnson and Rasolofosaon, 1996) that classical elastic formulations (nonlinear elasticity) can fail to describe the behavior of rocks at low stresses. The methods presented here assume that the strains are infinitesimal. When strains become finite, an additional source of nonlinearity, called geometrical or kinetic nonlinearity, appears, related to the difference between Lagrangian and Eulerian descriptions of motion (Zarembo and Krasil’nikov, 1971; Johnson and Rasolofosaon, 1996).

Strain components and equations of motion in cylindrical and spherical coordinate systems Synopsis The equations of motion and the expressions for small-strain components in cylindrical and spherical coordinate systems differ from those in a rectangular coordinate system. Figure 2.8.1 shows the variables used in the equations that follow. In the cylindrical coordinate system (r, f, z), the coordinates are related to those in the rectangular coordinate system (x, y, z) as pffiffiffiffiffiffiffiffiffiffiffiffiffiffi y r ¼ x2 þ y2 ; tanðÞ ¼ x x ¼ r cosðÞ; y ¼ r sinðÞ; z ¼ z The small-strain components can be expressed through the displacements ur, uf, and uz (which are in the directions r, f, and z, respectively) as @ur 1 @u ur @uz ; e ¼ þ ; ezz ¼ @r r @z r @   1 1 @ur @u u þ
¼ @r r 2 r @     1 @u 1 @uz 1 @uz @ur ¼ þ ; ezr ¼ þ @z 2 @z r @ 2 @r

err ¼ er ez

The equations of motion are @
rr 1 @
r @
zr
rr

 @ 2 ur þ þ þ ¼ 2 @r @z r @t r @

55

2.8 Strain components and equations of motion

z

z r

Spherical coordinates

Cylindrical coordinates

θ

y

y

f

x

f

x

r

Figure 2.8.1 The variables used for converting between Cartesian, spherical, and cylindrical coordinates.

@
r 1 @
 @
z 2
r @ 2 u þ þ þ ¼ 2 @r @z r @t r @ @
rz 1 @
z @
zz
rz @ 2 uz þ þ þ ¼ 2 @r @z r @t r @ where r denotes density and t time. In the spherical coordinate system (r, f, y) the coordinates are related to those in the rectangular coordinate system (x, y, z) as r¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ y2 þ z2 ;

x ¼ r sinðyÞ cosðÞ;

tanðÞ ¼

y ; x

z cosðyÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 x þ y2 þ z2

y ¼ r sinðyÞ sinðÞ;

z ¼ r cosðyÞ

The small-strain components can be expressed through the displacements ur, uf, and uy (which are in the directions r, f, and y, respectively) as @ur 1 @u ur uy ; e ¼ þ þ ; @r r r tanðyÞ r sinðyÞ @   1 1 @ur @u u þ
er ¼ @r r 2 r sinðyÞ @   1 1 @u u 1 @uy
þ ey ¼ 2 r @ r tanðyÞ r sinðyÞ @   1 @uy uy 1 @ur ery ¼
þ r 2 @r r @y err ¼

eyy ¼

1 @uy ur þ r r @y

The equations of motion are @
rr 1 @
r 1 @
ry 2
rr þ
ry cotðyÞ



yy @ 2 ur þ þ þ ¼ 2 @r r @t r sinðyÞ @ r @y @
r 1 @
 1 @
y 3
r þ
y cotðyÞ @ 2 u þ þ þ ¼ 2 @r @t r sinðyÞ @ r @y r @
ry 1 @
 1 @
y 3
ry þ ð
yy

 ÞcotðyÞ @ 2 uy þ þ þ ¼ 2 @r @t r sinðyÞ @ r @y r

56

Elasticity and Hooke’s law

Uses The foregoing equations are used to solve elasticity problems where cylindrical or spherical geometries are most natural.

Assumptions and limitations The equations presented assume that the strains are small.

2.9

Deformation of inclusions and cavities in elastic solids Synopsis Many problems in effective-medium theory and poroelasticity can be solved or estimated in terms of the elastic behavior of cavities and inclusions. Some static and quasistatic results for cavities are presented here. It should be remembered that often these are also valid for certain limiting cases of dynamic problems. Excellent treatments of cavity deformation and pore compressibility are given by Jaeger and Cook (1969) and by Zimmerman (1991a).

General pore deformation Effective dry compressibility Consider a homogeneous linear elastic solid that has an arbitrarily shaped pore space – either a single cavity or a collection of pores. The effective dry compressibility (i.e., the reciprocal of the dry bulk modulus) of the porous solid can be written as  1 1  @p  ¼ þ Kdry K0 p @
dry where Kdry is the effective bulk modulus of the dry porous solid, K0 is the bulk modulus of the solid mineral material,  is the porosity, p is the pore volume, @p =@
jdry is the derivative of the pore volume with respect to the externally applied hydrostatic stress. We also assume that no inelastic effects such as friction or viscosity are present. This is strictly true, regardless of the pore geometry and the pore concentration. The preceding equation can be rewritten slightly as 1 1  ¼ þ Kdry K0 K where K ¼ p =ð@p =@
Þjdry is defined as the dry pore-space stiffness. These equations state simply that the porous rock compressibility is equal to the intrinsic mineral compressibility plus an additional compressibility caused by the pore space.

57

2.9 Deformation of inclusions and cavities

Caution: “Dry rock” is not the same as gas-saturated rock

The dry-frame modulus refers to the incremental bulk deformation resulting from an increment of applied confining pressure with pore pressure held constant. This corresponds to a “drained” experiment in which pore fluids can flow freely in or out of the sample to ensure constant pore pressure. Alternatively, the dry-frame modulus can correspond to an undrained experiment in which the pore fluid has zero bulk modulus, and in which pore compressions therefore do not induce changes in pore pressure. This is approximately the case for an air-filled sample at standard temperature and pressure. However, at reservoir conditions, the gas takes on a non-negligible bulk modulus and should be treated as a saturating fluid. An equivalent expression for the dry rock compressibility or bulk modulus is Kdry ¼ K0 ð1
bÞ where b is sometimes called the Biot coefficient, which describes the ratio of porevolume change Dup to total bulk-volume change DV under dry or drained conditions:  p  Kdry b¼ ¼  V dry K

Stress-induced pore pressure: Skempton’s coefficient If this arbitrary pore space is filled with a pore fluid with bulk modulus, Kfl, the saturated solid is stiffer under compression than the dry solid, because an increment of pore-fluid pressure is induced that resists the volumetric strain. The ratio of the induced pore pressure, dP, to the applied compressive stress, ds, is sometimes called Skempton’s coefficient and can be written as B

dP 1 ¼ d
1 þ K ð1=Kfl
1=K0 Þ 1 ¼


1 1 þ ð1=Kfl
1=K0 Þ 1=Kdry
1=K0

where Kf is the dry pore-space stiffness defined earlier in this section. For this definition to be true, the pore pressure must be uniform throughout the pore space, as will be the case if: (1) there is only one pore, (2) all pores are well connected and the frequency and viscosity are low enough for any pressure differences to equilibrate, or (3) all pores have the same dry pore stiffness.

58

Elasticity and Hooke’s law

Given these conditions, there is no additional limitation on pore geometry or concentration. All of the necessary information concerning pore stiffness and geometry is contained in the parameter Kf.

Saturated stress-induced pore-volume change The corresponding change in fluid-saturated pore volume, up, caused by the remote stress is  1 dp  1 dP 1=Kfl ¼ ¼ p d
sat Kfl d
1 þ K ð1=Kfl
1=K0 Þ

Low-frequency saturated compressibility The low-frequency saturated bulk modulus, Ksat, can be derived from Gassmann’s equation (see Section 6.3 on Gassmann). One equivalent form is 1 1  1  ¼ þ þ K K 0 fl Ksat K0 K þ K
K K0 K þ Kfl 0

fl

where, again, all of the necessary information concerning pore stiffness and geometry is contained in the dry pore stiffness Kf, and we must ensure that the stress-induced pore pressure is uniform throughout the pore space.

Three-dimensional ellipsoidal cavities Many effective media models are based on ellipsoidal inclusions or cavities. These are mathematically convenient shapes and allow quantitative estimates of, for example, Kf, which was defined earlier in this section. Eshelby (1957) discovered that the strain, eij, inside an ellipsoidal inclusion is homogeneous when a homogeneous strain, e0ij , (or stress) is applied at infinity. Because the inclusion strain is homogeneous, operations such as determining the inclusion stress or integrating to obtain the displacement field are straightforward. It is very important to remember that the following results assume a single isolated cavity in an infinite medium. Therefore, substituting them directly into the preceding formulas for dry and saturated moduli gives estimates that are strictly valid only for low concentrations of pores (see also Section 4.8 on self-consistent theories).

Spherical cavity For a single spherical cavity with volume p ¼ 43 pR3 and a hydrostatic stress, ds, applied at infinity, the radial strain of the cavity is

59

2.9 Deformation of inclusions and cavities

dR 1 ð1
vÞ ¼ d
R K0 2ð1
2vÞ where v and K0 are the Poisson ratio and bulk modulus of the solid material, respectively. The change of pore volume is dp ¼

1 3ð1
vÞ p d
K0 2ð1
2vÞ

Then, the volumetric strain of the sphere is eii ¼

dp 1 3ð1
vÞ ¼ d
p K0 2ð1
2vÞ

and the single-pore stiffness is 1 1 dp 1 3ð1
vÞ ¼ ¼ K p d
K0 2ð1
2vÞ Remember that this estimate of Kf assumes a single isolated spherical cavity in an infinite medium. Under a remotely applied homogeneous shear stress, t0, corresponding to remote shear strain e0 ¼ t0/2m0, the effective shear strain in the spherical cavity is e¼

15ð1
vÞ t0 2m0 ð7
5vÞ

where m0 is the shear modulus of the solid. Note that this results in approximately twice the strain that would occur without the cavity.

Penny-shaped crack: oblate spheroid Consider a dry penny-shaped ellipsoidal cavity with semiaxes a b ¼ c. When a remote uniform tensional stress, ds, is applied normal to the plane of the crack, each crack face undergoes an outward displacement, U, normal to the plane of the crack, given by the radially symmetric distribution UðrÞ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4ð1
v2 Þc 1
ðr=cÞ2 3pK0 ð1
2vÞ

d


where r is the radial distance from the axis of the crack. For any arbitrary homogeneous remote stress, ds is the component of stress normal to the plane of the crack. Thus, ds can also be thought of as a remote hydrostatic stress field. This displacement function is also an ellipsoid with semiminor axis, U(r ¼ 0), and semimajor axis, c. Therefore the volume change, du, which is simply the

60

Elasticity and Hooke’s law

integral of U(r) over the faces of the crack, is just the volume of the displacement ellipsoid: 4 16c3 ð1
v2 Þ dp ¼ pUð0Þ c2 ¼ d
9K0 ð1
2vÞ 3 Then the volumetric strain of the cavity is eii ¼

dp 4ðc=aÞ ð1
v2 Þ ¼ d
p 3pK0 ð1
2vÞ

and the pore stiffness is 1 1 dp 4ðc=aÞ ð1
v2 Þ ¼ ¼ K p d
3pK0 ð1
2vÞ An interesting case is an applied compressive stress causing a displacement, U(0), equal to the original half-width of the crack, thus closing the crack. Setting U(0) ¼ a allows one to compute the closing stress:
close ¼

3pð1
2vÞ p aK0 ¼ am 4ð1
v2 Þ 2ð1
vÞ 0

where a ¼ (a/c) is the aspect ratio. Now, under a remotely applied homogeneous shear stress, t0, corresponding to remote shear strain e0 ¼ t0/2m0, the effective shear strain in the cavity is e ¼ t0

2ðc=aÞ ð1
vÞ pm0 ð2
vÞ

Needle-shaped pore: prolate spheroid Consider a dry needle-shaped ellipsoidal cavity with semiaxes a  b ¼ c and with pore volume  ¼ 43pac2 . When a remote hydrostatic stress, ds, is applied, the pore volume change is dp ¼

ð5
4vÞ p d
3K0 ð1
2vÞ

The volumetric strain of the cavity is then eii ¼

dp ð5
4vÞ d
¼ p 3K0 ð1
2vÞ

and the pore stiffness is 1 1 dp ð5
4vÞ ¼ ¼ K p d
3K0 ð1
2vÞ

61

2.9 Deformation of inclusions and cavities

Note that in the limit of very large a/c, these results are exactly the same as for a two-dimensional circular cylinder. Estimate the increment of pore pressure induced in a water-saturated rock when a 1 bar increment of hydrostatic confining pressure is applied. Assume that the rock consists of stiff, spherical pores in a quartz matrix. Compare this with a rock with thin, penny-shaped cracks (aspect ratio a ¼ 0.001) in a quartz matrix. The moduli of the individual constituents are Kquartz ¼ 36 GPa, Kwater ¼ 2.2 GPa, and vquartz ¼ 0.07. The pore-space stiffnesses are given by K-sphere ¼ Kquartz K-crack ¼

2ð1
2vquartz Þ ¼ 22:2 GPa 3ð1
vquartz Þ

3paKquartz ð1
2vquartz Þ ¼ 0:0733 GPa ð1
v2quartz Þ 4

The pore-pressure increment is computed from Skempton’s coefficient: Ppore 1 ¼B¼
1
1 Þ Pconfining 1 þ K ðKwater
Kquartz Bsphere ¼ Bcrack ¼

1 ¼ 0:095 1 þ 22:2½ð1=2:2Þ
ð1=36Þ 1 ¼ 0:970 1 þ 0:0733½ð1=2:2Þ
ð1=36Þ

Therefore, the pore pressure induced in the spherical pores is 0.095 bar and the pore pressure induced in the cracks is 0.98 bar.

Two-dimensional tubes A special two-dimensional case of long, tubular pores was treated by Mavko (1980) to describe melt or fluids arranged along the edges of grains. The cross-sectional shape is described by the equations   1 x ¼ R cos y þ cos 2y 2þg   1 sin 2y y ¼ R
sin y þ 2þg where g is a parameter describing the roundness (Figure 2.9.1).

62

Elasticity and Hooke’s law

g =0

g =1

g =∞

2R

2R

2R

Figure 2.9.1 The cross-sectional shapes of various two-dimensional tubes.

Consider, in particular, the case on the left, g ¼ 0. The pore volume is 12 paR2 , where a  R is the length of the tube. When a remote hydrostatic stress, ds, is applied, the pore volume change is dp ¼

ð13
4v
8v2 Þ p d
3K0 ð1
2vÞ

The volumetric strain of the cavity is then eii ¼

dp ð13
4v
8v2 Þ ¼ d
3K0 ð1
2vÞ p

and the pore stiffness is 1 1 dp ð13
4v
8v2 Þ ¼ ¼ K p d
3K0 ð1
2vÞ In the extreme, g ! 1, the shape becomes a circular cylinder, and the expression for pore stiffness, Kf, is exactly the same as that derived for the needle-shaped pores above. Note that the triangular cavity (g ¼ 0) has about half the pore stiffness of the circular one; that is, the triangular tube can give approximately the same effective modulus as the circular tube with about half the porosity. Caution

These expressions for Kf, dup, and eii include an estimate of tube shortening as well as a reduction in pore cross-sectional area under hydrostatic stress. Hence, the deformation is neither plane stress nor plane strain.

Plane strain The plane-strain compressibility in terms of the reduction in cross-sectional area A is given by ( 6ð1
vÞ g!0 1 1 dA m0 ; ¼ ¼ 2ð1
vÞ K0 A d
; g!1 m 0

63

2.9 Deformation of inclusions and cavities

The latter case (g ! 1) corresponds to a tube with a circular cross-section and agrees (as it should) with the expression given below for the limiting case of a tube with an elliptical cross-section with aspect ratio unity. A general method of determining K0 for nearly arbitrarily shaped two-dimensional cavities under plane-strain deformation was developed by Zimmerman (1986, 1991a) and involves conformal mapping of the tube shape into circular pores. For example, pores with cross-sectional shapes that are n-sided hypotrochoids given by 1 cosðn
1Þy ðn
1Þ 1 y ¼
sinðyÞ þ sinðn
1Þy ðn
1Þ

x ¼ cosðyÞ þ

(see the examples labeled (1) in Table 2.9.1) have plane-strain compressibilities 1 1 dA 1 1 þ ðn
1Þ
1 ¼ ¼ K0 A d
K0cir 1
ðn
1Þ
1 where 1=K0cir is the plane-strain compressibility of a circular tube given by 1 2ð1
nÞ ¼ 0cir m0 K Table 2.9.1 summarizes a few plane-strain pore compressibilities.

Two-dimensional thin cracks A convenient description of very thin two-dimensional cracks is in terms of elastic line dislocations. Consider a crack lying along –c < x < c in the y ¼ 0 plane and very long in the z direction. The total relative displacement of the crack faces u(x), defined as the displacement of the negative face (y ¼ 0–) relative to the positive face (y ¼ 0þ), is related to the dislocation density function by BðxÞ ¼


@u @x

where B(x) dx represents the total length of the Burger vectors of the dislocations lying between x and x þ dx. The stress change in the plane of the crack that results from introduction of a dislocation line with unit Burger vector at the origin is
¼

m0 2pDx

where D ¼ 1 for screw dislocations and D ¼ (1 – v) for edge dislocations (v is Poisson’s ratio and m0 is the shear modulus). Edge dislocations can be used to describe mode I and mode II cracks; screw dislocations can be used to describe mode III cracks.

1

(1)

1.581

3

(1)

1.188

2

(1)

p/8a

Plane-strain compressibility normalized by the compressibility of a circular tube.

1/2a

(2)

2/3a

(3)

Notes: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i3=2 (1) 1 1 x ¼ cosðyÞ þ ðn
1Þ cosðn
1Þy, where n ¼ number of sides; y ¼
sinðyÞ þ ðn
1Þ sinðn
1Þy. (2) y ¼ 2b 1
ðx=cÞ2 ðellipseÞ. (3) y ¼ 2b 1
ðx=cÞ2 (nonelliptical, “tapered” crack).

m0 2ð1
nÞK0

Table 2.9.1

65

2.9 Deformation of inclusions and cavities

The stress here is the component of traction in the crack plane parallel to the displacement: normal stress for mode I deformation, in-plane shear for mode II deformation, and out-of-plane shear for mode III deformation. Then the stress resulting from the distribution B(x) is given by the convolution Z c m Bðx0 Þ dx0
ðxÞ ¼ 0 2pD
c x
x0 The special case of interest for nonfrictional cavities is the deformation for stressfree crack faces under a remote uniform tensional stress, ds, acting normal to the plane of the crack. The outward displacement distribution of each crack face is given by (using edge dislocations) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c
ð1
vÞ 1
ðx=cÞ2 2c
ð1
v2 Þ 1
ðx=cÞ2 UðxÞ ¼ ¼ m0 3K0 ð1
2vÞ The volume change is then given by dp 2pc2 a ð1
v2 Þ ¼ pUð0Þca ¼ d
3K0 ð1
2vÞ It is important to note that these results for displacement and volume change apply to any two-dimensional crack of arbitrary cross-section as long as it is very thin and approximately planar. They are not limited to cracks of elliptical cross-section. For the special case in which the very thin crack is elliptical in cross-section with half-width b in the thin direction, the volume is u ¼ pabc, and the pore stiffness under plane-strain deformation is given by 1 1 dA ðc=bÞð1
vÞ 2ðc=bÞ ð1
v2 Þ ¼ ¼ ¼ K0 A d
m0 3K0 ð1
2vÞ Another special case is a crack of nonelliptical form (Mavko and Nur, 1978) with initial shape given by "

 2 #3=2 x U0 ðxÞ ¼ 2b 1
c0 where c0 is the crack half-length and 2b is the maximum crack width. This crack is plotted in Figure 2.9.2. Note that unlike elliptical cracks that have rounded or blunted ends, this crack has tapered tips where faces make a smooth, tangent contact. If we apply a pressure, P, the crack shortens as well as thins, and the pressure-dependent length is given by  1=2 2ð1
vÞ c ¼ c0 1
P 3m0 ðb=c0 Þ

66

Elasticity and Hooke’s law

b 0 −b

b 0 −b −c

0

−c

c

0

c

Figure 2.9.2 A nonelliptical crack shortens as well as narrows under compression. An elliptical crack only narrows.

Then the deformed shape is  3  x2 3=2 c Uðx; PÞ ¼ 2b 1
; c0 c

jxj  c

An important consequence of the smoothly tapered crack tips and the gradual crack shortening is that there is no stress singularity at the crack tips. In this case, crack closure occurs (i.e., U ! 0) as the crack length goes to zero (c ! 0). The closing stress is
close ¼

3 3 a0 E0 a0 m0 ¼ 2ð1
vÞ 4ð1
v2 Þ

where a0 ¼ b/c0 is the original crack aspect ratio, and m0 and E0 are the shear and Young’s moduli of the solid material, respectively. This expression is consistent with the usual rule of thumb that the crack-closing stress is numerically ~a0E0. The exact factor depends on the details of the original crack shape. In comparison, the pressure required to close a two-dimensional elliptical crack of aspect ratio a0 is
close ¼

1 a0 E0 2ð1
v2 Þ

Ellipsoidal cracks of finite thickness The pore compressibility under plane-strain deformation of a two-dimensional elliptical cavity of arbitrary aspect ratio a is given by (Zimmerman, 1991a)     1 1 dA 1
v 1 2ð1
v2 Þ 1 ¼ aþ ¼ ¼ aþ K0 A d
m0 a 3K0 ð1
2vÞ a where m0, K0, and v are the shear modulus, bulk modulus, and Poisson’s ratio of the mineral material, respectively. Circular pores (tubes) correspond to aspect ratio a ¼ 1 and the pore compressibility is given as 1 1 dA 2ð1
vÞ 4ð1
v2 Þ ¼ ¼ ¼ 0 K A d
m0 3K0 ð1
2vÞ

67

2.9 Deformation of inclusions and cavities

Rp

Ri

Rc

Rp

Rc

Figure 2.9.3 Deformation of a single- and double-layer spherical shell.

Deformation of spherical shells The radial displacement ur of a spherical elastic shell (Figure 2.9.3) with inner pore radius Rp and outer radius Rc under applied confining pressure Pc and pore pressure Pp is given by (Ciz et al., 2008) ur ¼

R3p r ðPp
Pc Þ ðPp
Pc Þ þ 2 4r mð1
Þ 3Kð1
Þ

where r is the radial distance from the center of the sphere, K is the bulk modulus of the solid elastic material, m is the shear modulus of the solid elastic material, and f is the porosity. The associated volume change of the inner spherical pore is dV ¼

3V 1




     1 1 1 þ dPp
þ dPc : 3K 4m 3K 4m

The volume change of the total sphere (shell plus pore) is dV ¼

      3V 1 1 1   þ dPp
þ dPc : 1
 3K 4m 3K 4m

Similarly, the radial displacement of a double-layer sphere (Figure 2.9.3) is given by (Ciz et al., 2008) ur ¼

An r Bn þ 2 r 3

where Rp is the pore radius, Rc is the outer sphere radius, and Ri is the radius at the boundary between the inner and outer elastic shells. The elastic moduli of the inner solid shell are K1 and m1, and the moduli of the outer shell are K2 and m2:

1 n A1 ¼ ð4m1 m2 þ 3K2 m2 ÞR3i þ ð3K2 m1
3K2 m2 ÞR3c R3p Pp  o
ð3K2 m1 þ 4m1 m2 ÞR3c R3i Pc

68

Elasticity and Hooke’s law

B1 ¼

A2 ¼

B2 ¼

i  3 1 nh 3 3 3 3 K K þ K m þ ðK m þ K m ÞR R 2 2 1 2 2 2 i Rp Ri Pp c 4 1 2  o


34 K1 K2 þ K1 m2 R3c R3i R3p Pc i 1 nh ð3K1 m1
3K1 m2 ÞR3p
ð3K1 m2 þ 4m1 m2 ÞR3i R3c Pc  o þ ð3K1 m2
4m1 m2 ÞR3i R3p Pp

 i 1 nh ðK1 m1
K2 m1 ÞR31
34K1 K2 þ K2 m1 R3p R3c R3i Pc  o

 þ 34 K1 K2 þ K1 m1 R3c R3i R3p Pp

 ¼ K2 m1 ð3K1 þ 4m2 ÞR3c R3i
K1 m2 ð3K2 þ 4m1 ÞR3i R3p þ 4m1 m2 ðK1
K2 ÞR6i þ 3K1 K2 ðm2
m1 ÞR3c R3p

Uses The equations presented in this section are useful for computing deformation of cavities in elastic solids and estimating effective moduli of porous solids.

Assumptions and limitations The equations presented in this section are based on the following assumptions.  Solid material must be homogeneous, isotropic, linear, and elastic.  Results for specific geometries, such as spheres and ellipsoids, are derived for single isolated cavities. Therefore, estimates of effective moduli based on these are limited to relatively low pore concentrations where pore elastic interaction is small.  Pore-pressure computations assume that the induced pore pressure is uniform throughout the pore space, which will be the case if (1) there is only one pore, (2) all pores are well connected and the frequency and viscosity are low enough for any pressure differences to equilibrate, or (3) all pores have the same dry pore stiffness.

2.10

Deformation of a circular hole: borehole stresses Synopsis Presented here are some solutions related to a circular hole in a stressed, linear, elastic, and poroelastic isotropic medium.

69

2.10 Deformation of a circular hole: borehole stresses

Hollow cylinder with internal and external pressures The cylinder’s internal radius is R1 and the external radius is R2. Hydrostatic stress p1 is applied at the interior surface at R1 and hydrostatic stress p2 is applied at the exterior surface at R2. The resulting (plane-strain) outward displacement U and radial and tangential stresses are U¼

ðp2 R22
p1 R21 Þ ðp2
p1 ÞR21 R22 1 rþ 2 2 2ðl þ mÞðR2
R1 Þ 2mðR22
R21 Þ r


rr ¼

ðp2 R22
p1 R21 Þ ðp2
p1 ÞR21 R22 1
ðR22
R21 Þ r 2 ðR22
R21 Þ


yy ¼

ðp2 R22
p1 R21 Þ ðp2
p1 ÞR21 R22 1 þ ðR22
R21 Þ ðR22
R21 Þ r2

where l and m are the Lame´ coefficient and shear modulus, respectively. If R1 ¼ 0, we have the case of a solid cylinder under external pressure, with displacement and stress denoted by the following: U¼

p2 r 2ðl þ mÞ


rr ¼
yy ¼ p2 If, instead, R2 ! 1, then p2 r ðp2
p1 ÞR21 þ 2mr 2ðl þ mÞ   R21 p1 R2
rr ¼ p2 1
2 þ 2 1 r r   2 R p1 R2
yy ¼ p2 1 þ 21
2 1 r r U¼

These results for plane strain can be converted to plane stress by replacing v by v/(1 þ v), where v is the Poisson ratio.

Circular hole with principal stresses at infinity The circular hole with radius R lies along the z-axis. A principal stress, sxx, is applied at infinity. The stress solution is then    
xx R2
xx 4R2 3R4
rr ¼ 1
2 þ 1
2 þ 4 cos 2y 2 r 2 r r     2 4
xx R
xx 3R 1þ 2
1 þ 4 cos 2y
yy ¼ 2 r 2 r

70

Elasticity and Hooke’s law

 
xx 2R2 3R4 1 þ 2
4 sin 2y
ry ¼
2 r r     8mUr r 2R R2 ¼ ð
1 þ 2 cos 2yÞ þ 1 þ  þ 1
2 cos 2y R
xx r R r    2 8mUy 2r 2R R sin 2y ¼
þ 1

2 R
xx r R r where y is measured from the x-axis, and  ¼ 3
4v; 3
v ¼ ; 1þv

for plane strain for plane stress

At the cavity surface, r ¼ R,
rr ¼
ry ¼ 0
yy ¼
xx ð1
2 cos 2yÞ Thus, we see the well-known result that the borehole creates a stress concentration of syy ¼ 3sxx at y ¼ 90 .

Stress concentration around an elliptical hole If, instead, the borehole is elliptical in cross-section with a shape denoted by (Lawn and Wilshaw, 1975) x2 y2 þ ¼1 b2 c2 where b is the semiminor axis and c is the semimajor axis, and the principal stress sxx is applied at infinity, then the largest stress concentration occurs at the tip of the long axis (y ¼ c; x ¼ 0). This is the same location at y ¼ 90 as for the circular hole. The stress concentration is
yy ¼
xx ½1 þ 2ðc=Þ1=2  where r is the radius of curvature at the tip given by ¼

b2 c

When b c, the stress concentration is approximately rffiffiffi
yy 2c c ¼2
xx b 

71

2.10 Deformation of a circular hole: borehole stresses

z⬘

i

sv

z

y y⬘ a

sh

θ

sH x⬘

x

Figure 2.10.1 The coordinate system for the remote stress field around an inclined, cylindrical borehole.

Stress around an inclined cylindrical hole We now consider the case of a cylindrical borehole of radius R inclined at an angle i to the vertical axis, in a linear, isotropic elastic medium with Poisson ratio n in a nonhydrostatic remote stress field (Jaeger and Cook, 1969; Bradley, 1979; Fjaer et al., 2008). The borehole coordinate system is denoted by (x, y, z) with the z-axis along the axis of the borehole. The remote principal stresses are denoted by
n ; the vertical stress;
H ; the maximum horizontal stress; and
h ; the minimum horizontal stress The coordinate system for the remote stress field (shown in Figure 2.10.1) is denoted by (x0 , y0 , z0 ) with x0 along the direction of the maximum horizontal stress, and z0 along the vertical. The angle a represents the azimuth of the borehole x-axis with respect to the x0 -axis, while y is the azimuthal angle around the borehole measured from the x-axis. Assuming plane-strain conditions with no displacements along the z-axis, the stresses as a function of radial distance r and azimuthal angle y are given by (Fjaer et al., 2008): ! !  
0xx þ
0yy
0xx

0yy R2 R4 R2
rr ¼ 1
2 þ 1 þ 3 4
4 2 cos 2y 2 r 2 r r   R4 R2 R2 þ
0xy 1 þ 3 4
4 2 sin 2y þ pw 2 r r r ! !  
0xx þ
0yy
0xx

0yy R2 R4 1þ 2
1 þ 3 4 cos 2y
yy ¼ 2 r 2 r   R4 R2

0xy 1 þ 3 4 sin 2y
pw 2 r r

72

Elasticity and Hooke’s law


zz ¼
ry ¼
yz ¼
rz ¼

    R2 2 0 0 0 R
v 2
xx

yy 2 cos 2y þ 4
xy 2 sin 2y r r !    0 0 4 2
yy

xx R R R4 R2 0 1
3 4 þ 2 2 sin 2y þ
xy 1
3 4 þ 2 2 cos 2y 2 r r r r    2 R

0xz sin y þ
0yz cos y 1 þ 2 r    2 R 0 0
xz cos y þ
yz sin y 1
2 r
0zz

In the above equations pw represents the well-bore pressure and
0ij is the remote stress tensor expressed in the borehole coordinate system through the usual coordinate transformation (see Section 1.4 on coordinate transformations) involving the direction cosines of the angles between the (x0 , y0 , z0 ) axes and the (x, y, z) axes as follows:
0xx ¼ b2xx0
H þ b2xy0
h þ b2xz0
n
0yy ¼ b2yx0
H þ b2yy0
h þ b2yz0
n
0zz ¼ b2zx0
H þ b2zy0
h þ b2zz0
n
0xy ¼ bxx0 byx0
H þ bxy0 byy0
h þ bxz0 byz0
n
0yz ¼ byx0 bzx0
H þ byy0 bzy0
h þ byz0 bzz0
n
0zx ¼ bzx0 bxx0
H þ bzy0 bxy0
h þ bzz0 bxz0
n and bxx0 ¼ cos a cos i;

bxy0 ¼ sin a cos i;

bxz0 ¼
sin i

byx0 ¼
sin a;

byy0 ¼ cos a;

byz0 ¼ 0

bzx0 ¼ cos a sin i;

bzy0 ¼ sin a sin i;

bzz0 ¼ cos i

Stress around a vertical cylindrical hole in a poroelastic medium The circular hole with radius Ri lies along the z-axis. Pore pressure, pf, at the permeable borehole wall equals the pressure in the well bore, pw. At a remote boundary R0  Ri the stresses and pore pressure are
zz ðR0 Þ ¼
n
rr ðR0 Þ ¼
yy ðR0 Þ ¼
h pf ðR0 Þ ¼ pf0

73

2.10 Deformation of a circular hole: borehole stresses

The stresses as a function of radial distance r from the hole are given by (Risnes et al., 1982; Bratli et al., 1983; Fjaer et al., 2008) "  2 # R2i R0 1
2n
ðpf0
pw Þ 1

rr ¼
h þ ð
h
pw Þ 2 2 r 2ð1
nÞ R0
Ri ( ) " #   R2 R0 2 lnðR0 =rÞ

a 2 i 2 1
þ r lnðR0 =Ri Þ R0
Ri "  # R2i R0 2 1
2n 1þ
yy ¼
h þ ð
h
pw Þ 2
ðpf0
pw Þ 2 r 2ð1
nÞ R0
Ri ( ) " #  2 R2i R0 lnðR0 =rÞ
1 1þ

a 2 þ lnðR0 =Ri Þ r R0
R2i R2 1
2n
zz ¼
n þ 2nð
h
pw Þ 2 i 2
ðpf0
pw Þ 2ð1
nÞ R0
Ri   2 2R 2 lnðR0 =rÞ
n

a n 2 i 2þ lnðR0 =Ri Þ R0
Ri where n is the dry (drained) Poisson ratio of the poroelastic medium, a ¼ 1
Kdry =K0 is the Biot coefficient, Kdry is effective bulk modulus of dry porous solid, and K0 is the bulk modulus of solid mineral material. In the limit R0 =Ri ! 1 the expressions simplify to (Fjaer et al., 2008) "  #  2 Ri 1
2n Ri 2 lnðR0 =rÞ þ ðpf0
pw Þ
a
rr ¼
h
ð
h
pw Þ r r 2ð1
nÞ lnðR0 =Ri Þ "  #  2 Ri 1
2n Ri 2 lnðR0 =rÞ
yy ¼
h þ ð
h
pw Þ
ðpf0
pw Þ þ a r r 2ð1
nÞ lnðR0 =Ri Þ
zz ¼
n
ðpf0
pw Þ

1
2n 2 lnðR0 =rÞ
n a 2ð1
nÞ lnðR0 =Ri Þ

Uses The equations presented in this section can be used for the following:  estimating the stresses around a borehole resulting from tectonic stresses;  estimating the stresses and deformation of a borehole caused by changes in borehole fluid pressure.

Assumptions and limitations The equations presented in this section are based on the following assumptions:  the material is linear, isotropic, and elastic or poroelastic.

74

Elasticity and Hooke’s law

Extensions More complicated remote stress fields can be constructed by superimposing the solutions for the corresponding principal stresses.

2.11

Mohr’s circles Synopsis Mohr’s circles provide a graphical representation of how the tractions on a plane depend on the angular orientation of the plane within a given stress field. Consider a stress state with principal stresses s1 s2 s3 and coordinate axes defined along the corresponding principal directions x1, x2, x3. The traction vector, T, acting on a plane with outward unit normal vector, n ¼ (n1, n2, n3), is given by Cauchy’s formula as T ¼ sn where s is the stress tensor. The components of n are the direction cosines of n relative to the coordinate axes and are denoted by n1 ¼ cos ;

n2 ¼ cos g;

n3 ¼ cos y

and n21 þ n22 þ n23 ¼ 1 where f, g, and y are the angles between n and the axes x1, x2, x3 (Figure 2.11.1).

x3

q

n T g

f

x2

x1

Figure 2.11.1 Angle and vector conventions for Mohr’s circles.

75

2.11 Mohr’s circles

P 45⬚ 60⬚

30⬚ 15⬚

45⬚

30⬚

75⬚ 60⬚ 75⬚

A

B

Figure 2.11.2 Three-dimensional Mohr’s circle.

The normal component of traction, s, and the magnitude of the shear component, t, acting on the plane are given by
¼ n21
1 þ n22
2 þ n23
3 t2 ¼ n21
21 þ n22
22 þ n23
23

2

Three-dimensional Mohr’s circle The numerical values of s and t can be read graphically from the three-dimensional Mohr’s circle shown in Figure 2.11.2. All permissible values of s and t must lie in the shaded area. To determine s and t from the orientation of the plane (f, g, and y), perform the following procedure. (1) Plot s1 s2 s3 on the horizontal axis and construct the three circles, as shown. The outer circle is centered at (s1 þ s3)/2 and has radius (s1 – s3)/2. The left-hand inner circle is centered at (s2 þ s3)/2 and has radius (s2 – s3)/2. The right-hand inner circle is centered at (s1 þ s2)/2 and has radius (s1 – s2)/2. (2) Mark angles 2y and 2f on the small circles centered at A and B. For example, f ¼ 60 plots at 2f ¼ 120 from the horizontal, and y ¼ 75 plots at 2y ¼ 150 from the horizontal, as shown. Be certain to include the factor of 2, and note the different directions defined for the positive angles. (3) Draw a circle centered at point A that intersects the right-hand small circle at the mark for f. (4) Draw another circle centered at point B that intersects the left-hand small circle at the point for y. (5) The intersection of the two constructed circles at point P gives the values of s and t. Reverse the procedure to determine the orientation of the plane having particular values of s and t.

76

Elasticity and Hooke’s law

P 45⬚ 60⬚ 75⬚

30⬚ f = 158

Figure 2.11.3 Two-dimensional Mohr’s circle.

Two-dimensional Mohr’s circle When the plane of interest contains one of the principal axes, the tractions on the plane depend only on the two remaining principal stresses, and using Mohr’s circle is therefore simplified. For example, when y ¼ 90  (i.e., the x3-axis lies in the plane of interest), all stress states lie on the circle centered at B in Figure 2.11.2. The stresses then depend only on s1 and s2 and on the angle f, and we need only draw the single circle, as shown in Figure 2.11.3.

Uses Mohr’s circle is used for graphical determination of normal and shear tractions acting on a plane of arbitrary orientation relative to the principal stresses.

2.12

Static and dynamic moduli In a uniaxial stress experiment (Figure 2.12.1), Young’s modulus E is defined as the ratio of the axial stress s to the axial strain ea, while Poisson’s ratio n is defined as the (negative) ratio of the radial strain er to the axial strain: E¼


; ea

n¼


er ea

It follows from these definitions that Poisson’s ratio is zero if the sample does not expand radially during axial loading and Poisson’s ratio is 0.5 if the radial strain is half the axial strain, which is the case for fluids and incompressible solids. Poisson’s ratio must lie within the range
1 < n  0:5. The speeds of elastic waves in the solid are linked to the elastic moduli and the bulk density r by the wave equation. The corresponding expressions for Poisson’s ratio and Young’s modulus are:

77

2.12 Static and dynamic moduli

s

ea

er

Figure 2.12.1 Uniaxial loading experiment. Dashed lines show undeformed sample, and solid lines show deformed sample.

n¼

1 ðVP =VS Þ2
2 ; 2 ðVP =VS Þ2
1

E ¼ 2VS2 ð1 þ nÞ

where VP and VS are the P- and S-wave velocities, respectively. The elastic moduli calculated from the elastic-wave velocities and density are the dynamic moduli. In contrast, the elastic moduli calculated from deformational experiments, such as the one shown in Figure 2.12.1, are the static moduli. In most cases the static moduli are different from the dynamic moduli for the same sample of rock. There are several reasons for this. One is that stress–strain relations for rocks are often nonlinear. As a result, the ratio of stress to strain over a large-strain measurement is different from the ratio of stress to strain over a very small-strain measurement. Another reason is that rocks are often inelastic, affected, for example, by frictional sliding across microcracks and grain boundaries. More internal deformation can occur over a large-strain experiment than over very small-strain cycles. The strain magnitude relevant to geomechanical processes, such as hydrofracturing, is of the order of 10
2, while the strain magnitude due to elastic-wave propagation is of the order of 10
7 or less. This large strain difference affects the difference between the static and dynamic moduli. Relations between the dynamic and static moduli are not simple and universal because: (a) the elastic-wave velocity in a sample and the resulting dynamic elastic moduli depend on the conditions of the measurement, specifically on the effective pressure and pore fluid; and (b) the static moduli depend on details of the loading experiment. Even for the same type of experiment – axial loading – the static Young’s modulus may be strongly

Elasticity and Hooke’s law

40 Wang hard 30 E static (GPa)

78

20 Eissa and Kazi

10

Wang soft Mese and Dvorkin

0

0

10

20

30

40

50

E dynamic (GPa)

Figure 2.12.2 Comparison of selected relations between dynamic and static Young’s moduli.

affected by the overall pressure applied to the sample, as well as by the axial deformation magnitude. Some results have been reviewed by Scho¨n (1996) and Wang and Nur (2000). Presented below and in Figure 2.12.2 are some of their equations, where the moduli are in GPa and the impedance is in km/s g/cc. In all the examples, Estat is the static Young’s modulus and Edyn is the dynamic Young’s modulus. Data on microcline-granite, by Belikov et al. (1970): Estat ¼ 1:137Edyn
9:685 Igneous and metamorphic rocks from the Canadian Shield, by King (1983): Estat ¼ 1:263Edyn
29:5 Granites and Jurassic sediments in the UK, by McCann and Entwisle (1992): Estat ¼ 0:69Edyn þ 6:4 A wide range of rock types, by Eissa and Kazi (1988): Estat ¼ 0:74Edyn
0:82 Shallow soil samples, by Gorjainov and Ljachowickij (1979) for clay: Estat ¼ 0:033Edyn þ 0:0065 and for sandy, wet soil: Estat ¼ 0:061Edyn þ 0:00285

79

2.12 Static and dynamic moduli

Empirical relations between static and dynamic bulk moduli in dry tight sandstones from the Travis Peak formation.

Table 2.12.1

Kstat ¼ a þ bKdyn (GPa) Pressure (MPa)

a (GPa)

b

5 20 40 125

0.98 3.16 1.85
1.85

0.490 0.567 0.822 1.13

Wang and Nur (2000) for soft rocks (defined as rocks with the static Young’s modulus < 15 GPa): Estat ¼ 0:41Edyn
1:06 Wang and Nur (2000) for hard rocks (defined as rocks with the static Young’s modulus > 15 GPa): Estat ¼ 1:153Edyn
15:2 Mese and Dvorkin (2000) related the static Young’s modulus and static Poisson’s ratio (nstat) to the dynamic shear modulus calculated from the shear-wave velocity in shales and shaley sands: Estat ¼ 0:59mdyn
0:34;

nstat ¼
0:0208mdyn þ 0:37

where mdyn is the dynamic shear modulus mdyn ¼ VS2 . The same data were used to obtain relations between the static moduli and the dynamic S-wave impedance, IS dyn : Estat ¼ 1:99IS

dyn


3:84;

nstat ¼
0:07IS

dyn

þ 0:5

as well as between the static moduli and the dynamic Young’s modulus: Estat ¼ 0:29Edyn
1:1;

nstat ¼
0:00743Edyn þ 0:34

Jizba (1991) compared the static bulk modulus, Kstat and the dynamic bulk modulus, Kdyn, and found the following empirical relations as a function of confining pressure in dry tight sandstones from the Travis Peak formation (Table 2.12.1).

Assumptions and limitations There are only a handful of well-documented experimental data where large-strain deformational experiments have been conducted with simultaneous measurement of the dynamic P- and S-wave velocity. One of the main uncertainties in applying

80

Elasticity and Hooke’s law

laboratory static moduli data in situ comes from the fact that the in-situ loading conditions are often unknown. Moreover, in most cases, the in-situ conditions are so complex that they are virtually impossible to reproduce in the laboratory. In most cases, static data exhibit a strongly nonlinear stress dependence, so that it is never clear which data point to use as the static modulus at in-situ conditions. Because of the strongly nonlinear dependence of static moduli on the strain magnitude, the isotropic linear elasticity equations that relate various elastic moduli to each other might not be applicable to static moduli. Finally, this section refers only to isotropic descriptions of static and dynamic moduli.

3

Seismic wave propagation

3.1

Seismic velocities Synopsis The velocities of various types of seismic waves in homogeneous, isotropic, elastic media are given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffi K þ 43  l þ 2 VP ¼ ¼   rffiffiffi  VS ¼  sffiffiffi E VE ¼  where VP is the P-wave velocity, VS is the S-wave velocity, and VE is the extensional wave velocity in a narrow bar. In addition, r is the density, K is the bulk modulus, m is the shear modulus, l is Lame´’s coefficient, E is Young’s modulus, and v is Poisson’s ratio. In terms of Poisson’s ratio one can also write VP2 2ð1
vÞ ¼ ð1
2vÞ VS2 VE2 ð1 þ vÞð1
2vÞ ¼ 2 ð1
vÞ VP VE2 ¼ 2ð1 þ vÞ VS2 V 2
2VS2 V 2
2V 2  ¼ E 2 S v ¼ P 2 2 2VS 2 VP
VS

81

82

Seismic wave propagation

The various wave velocities are related by VP2 4
VE2 =VS2 ¼ VS2 3
VE2 =VS2 VE2 3V 2 =V 2
4 ¼ 2P 2S 2 VS VP =VS
1 The elastic moduli can be extracted from measurements of density and any two wave velocities. For example,  ¼ VS2

 K ¼  VP2
43 VS2 E ¼ VE2 V 2
2VS2  v ¼ P 2 2 VP
VS2 The Rayleigh wave phase velocity VR at the surface of an isotropic homogeneous elastic half-space is given by the solution to the equation (White, 1983) 

2  1=2  1=2 VR2 VR2 VR2 2
2
4 1
2 1
2 ¼0 VS VP VS

(Note that the equivalent equation given in Bourbie´ et al. (1987) is in error.) The wave speed is plotted in Figure 3.1.1 and is given by "  2 1=2 #1=3 "  2 1=2 #1=3 VR2 q q p3 q q p3 8 þ þ ¼
þ þ

þ 2 4 27 4 27 2 2 3 VS  2  q p3 þ for >0 4 27 " # 
p1=2 VR2 
cos
1 ð
27q2 =4p3 Þ1=2 8 ¼
2 cos þ 2 3 3 3 VS for

 2  q p3 þ