Thermal Properties and Temperature-Related Behavior of Rock Fluid Systems

Developments in Petroleum Science, 37

thermal properties and temperature-related behavior of rock/fluid systems

DEVELOPMENTS IN PETROLEUM SCIENCE Advisory Editor: G.V. Chilingarian Volumes 1. 3, 4 . 7 and 13 are out of print. W.H. FERTL - Abnormal Formation Pressures T.F. YEN and G.V. CHILINGARIAN (Editors) - Oil Shale D.W. PEACEMAN - Fundamentals of Numerical Reservoir Simulation 8. L.P. DAKE - Fundamentals of Reservoir Engineering 9. K. MAGARA - Compaction and Fluid Migration 10. M.T. SILVIA and E.A. ROBINSON Deconvolution o f Geophysical Time Series in the Exploration for Oil and Natural Gas 11. G.V. CHILINGARIAN and P. VORAHU'I'R - Drilling and Drilling Fluids 12. T.D. VAN GOLF-RACHT - Fundamentals of Fractured Reservoir Engineering 1.1. G. MOZES (Editor) - Paraffin Products 15A. 0. SERKA - Fundamentals of Well-log Interpretation. 1. T h e acquisition of logging dat.a 15R. 0. SERRA - Fundamentals of Well-log Interpretation, 2. T h e interpretation of logging data 16. R.E. CHAPMAN - Petroleum Geology 17.4. E.C. DONALDSON, G.V. CHILINGARIAN and T.F. YEN (Editors) Enhanced Oil Recovery, I. Fundamentals and analyses 17B. E.C. DONALIBON, G.V. CHILINGARIAN and T.F. YEN (Editors) - Enhanced Oil Recovery, 11. Processes and operations 18A. A.P. SZILAS - Production and Transport of Oil and Gas, A. Flow mechanics and production (second completely revised editiun ) 1HR.A.P. SZILAS - Production and Transport of Oil and Gas, €3. Gathering and Transport 2. 5. 6.

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( s x m d and completely revised edition )

19A. (;.V. CHILINGARIAN, J.O. ROBERTSON J r . and S. KUMAR Surface Operations in Petroleum Production, I 19R. G.V. CHILINGARIAN, J.O. ROBERTSON J r . and S.KUMAR Surface Operations in Petroleum Production, I1 20. A..J. DIKKERS - Geology in Petroleum Production 21. F. RAMIREZ - Application of Optimal Control Theory to Enhanced Oil Recovery 22. E.C. DONALDSON, G.V. CHILINGARIAN andT.F. YEN - Microbial Enhanced Oil Recovery 2 3 . J. HAGOORT - Fundamentals of Gas Reservoir Engineering 24. W. LITTMANN - Polymer Flooding 25. N.K. BAIBAKOV and A.R. GARUSHEV - Thermal Methods of Petroleum Production 26. D. MADER - Hydraulic Proppant Fracturing and Gravel Packing 27. G. DA PRAT - Well Test Analysis for Naturally Fractured Reservoirs 28. E.R. NELSON (Editor) - Well Cementing 29. R.W. ZIMMERMAN - Compressibility of Sandstones 80. G.V. CHILINGARIAN, S.J. MAZZULLO and H.H. RIEKE - Carbonate Reservoir Characterization: A Geologic-Engineering Analysis, Part I 8 1 . E.C. DONALDSON (Editor) - Microbial Enhancement of Oil Recovery - Recent Advances 33. E. F J R R , R.M. HOLT, P. HORSRUD, A.M. RAAEN and R. RISNES - Petroleum Related Rock Mechanics 34. M.J. ECONOMIDES - A Practical Companion to Reservoir Stimulation 36. L. DAKE - T h e Practice of Reservoir Engineering ~

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Developments in Petroleum Science, 37

thermal properties and temperaturelrelated behavior of rock/fluid systems W.H. SOMERTON University of California, 5130 Etcheverry Hall, Berkeley, CA 94720, U.S.A.

ELSEVIER, Amsterdam - London - New York - Tokyo

1992

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PREFACE Information on the thermal properties and temperature-related behavior of rocks in their subsurface environments is widespread throughout the literature, covering a span of about 35 years. This is an effort to bring together much of this information into a single source book. A great deal of the research on rock properties and behavior has been done at room temperature. With the increased application of underground processes in high-temperature, and in some cases low-temperature environments, knowledge of temperature-related behavior has become increasingly important. Much of the work reported here has been done by the author, his students and his colleagues at the University of California, Berkeley.

However, references to important

research of dozens of other investigators have been included in this book. Since literature searches are seldom infallible, the author apologizes to those researchers whose works have not been reported here. In the first chapter of this book, some of the important processes are described in which thermal data are needed. Several of the terms used repeatedly in the text are also defined. Later chapters deal with such matters as thermal properties of rocks, including heat capacities thermal conductivities and thermal diffusivities, under conditions simulating subsurface environments. Since thermal properties of rock/fluid systems are difficult to measure, a good deal of effort has gone into developing models that would predict thermal properties from moreeasily measured properties of the system. These models are discussed in Chapter VI. Heating or cooling of rocks can change their structures, alter minerals contained in the aggregate and, consequently, change their physical properties. The effects of thermal reactions in rocks, differential thermal expansion, and thermal alterations are discussed in separate chapters. The effects of temperature on rock properties, as distinct from the irreversible effects of heating, are also reviewed in a separate chapter. Many processes involve flowing fluids in rock pore spaces. Heat transfer characteristics of the rock/fluid system are of particular importance in this case. A final chapter in the book deals with wellbore applications of thermal and hightemperature behavior of rocks and methods of deducing thermal properties from geophysical logs run in boreholes. The appendices include thermal units conversion factors and thermal properties of some typical reservoir rocks and fluids. The author wishes to acknowledge the assistance of many persons who provided material used in the preparation of this book. First, I thank the many graduate students in Petroleum and Mechanical Engineering at University of California, Berkeley who, over the years, have done

VI

much of the research reported herein. This work has been reported in M.S. research reports and theses and PhD. dissertations on file at the University. In particular, I want to express my appreciation to three former Doctoral students at Berkeley who have contributed important sections of this book.

Dr. Michael S. King, currently Oil Industry Professor of Petroleum

Engineering, Royal School of Mines, London, contributed most of the material on lowtemperature behavior of rocks.

Dr. Robert W. Zimmerman, Senior Scientist, Lawrence

Berkeley Laboratory, University of California, provided material on his thermal conductivity model and Dr. Zeng-Guang Yuan, Research Scholar, Department of Mechanical Engineering, University of California, was largely responsible for the material on the thick-walled tube model of thermal behavior with flowing fluids. Much of the research reported herein was funded by the American Petroleum Institute, the U S . Department of Energy, through the Lawrence Berkeley Laboratory, and several oil companies including, in particular, Chevron Oilfield Research Company. The assistance of Dr. Turk Timur, Dr. Frank Campbell and Dr. Don Jones all of Chevron, in providing funding for some the research and arranging for preparation of the initial manuscript and most of the figures in this book, is gratefully acknowledged. Last, but certainly not least, the author thanks his wife, Irma, for her painstaking and critical review of the manuscript and her constant encouragement in the preparation of this book.

W.H. Somerton

VII

TABLE OF CONTENTS Chapter 1 Thermal Processes and Terms 1 Applications requiring thermal data 1 Definition of terms 4 Chapter II Heat Capacities of Rocks 8 Experimental measurements 8 Calculated heat capacities 10 Heat capacities of fluid saturated rocks 14 Generalized calculations of heat capacities 18 Heat capacity of shales 21 Chapter 111 Thermal Reactions in Rocks 22 Experimental methods 23 Results of measurements 27 Chapter IV Thermal Expansion of Rocks 29 Thermal expansion of dry sandstones 30 Thermal expansion of fluid saturated rocks under stress 32 Conclusions on thermal expansion 38 Chapter V

Thermal Conductivities of RocUFluid Systems 39 Methods of measuring thermal conductivities 41 Effects of rocklfluid properties on thermal conductivity 60 Effects of temperature on thermal Conductivity 75 Effects of stress on thermal conductivity 77 Summary 80

Chapter VI Thermal Conductivitiy Models 82 Mixing law models 82 Empirical models 84 Theoretical models 88 Summary 110 Chapter VII Thermal Diffusivities of Rocks 112 Experimental methods of measurement 112 Measured diffusivities of rocks 114 Calculated thermal diffusivities of rocks 117

Vlll Chapter Vlll Heat Transfer with Flowing Fluids 119 Natural convection in porous media 119 Fluid phase changes in porous media (VCC effect) 122 Convective heat transfer with flowing fluids 135 Chapter IX Thermal Alterations of Rocks 148 High temperature alterations 149 Role of fluxing agents in high temperature alterations 156 Reduction in fracture pressures by intensive borehole heating 161 Effects of steaming on rock properties 166 Chapter X

Effects of Temperature on Rock Properties 181 Bulk and pore compressibilities 181 Elastic wave velocities 187 Permeability 190 Formation resistivity factor 194 Summary and conclusions 197

Chapter XI Low Temperature Behavior of RocUFluid Systems 199 P- and S-wave velocities and elastic moduli 199 Electrical properties 202 Thermal conductivity 204 Other low-temperature effects 204 Chapter XI1

Wellbore Applications 206 Thermal data from well logs 206 Thermal gradients in wells 212 Heat losses in wells due to VCC effect 227 Summary and conclusions 229

Appendix A

Thermal Units Conversion Factors 230

Appendix B Thermal Properties Data for Various Rocks 232 Appendix C Thermal Properties of Subsurface Reservoir Fluids 239 References 241 Author Index 247 Subject Index 250

1

Chapter 1 THERMAL TERMS AND PROCESSES

Knowledge of thermal properties and high-temperature behavior of rock/fluid systems has become increasingly important with the wide-spread interest in thermal processes in underground fluid-bearing reservoirs. For meaningful analyses of these processes, in addition to thermal properties, other rock properties and their behavior in high-temperature environments must be known. Low-temperature behavior may also be of importance as in the case of permafrost, or permanently frozen ground. Some of the processes requiring knowledge of thermal properties and other hightemperature behavior of rocklfluid systems include thermal methods of enhanced oil recovery, management of geothermal reservoirs, underground storage of heat, and underground disposal of nuclear waste. Another use of these data is in the interpretation of well logs from deep, hightemperature wells.

Low-temperature behavior of rocks, particularly when ice forms in the

pore spaces, is of major importance in the construction of roads and buildings and in drilling of wells through permafrost zones. Interpretation of geophysical data in permafrost areas may be complicated by the effects of low temperatures on the pertinent physical properties. Thermal properties of greatest importance include thermal conductivity, heat capacity, and thermal diffusivity. If fluids are in motion by natural or forced convection, some form of heat transfer coefficient between rock and fluid may be required.

In high-temperature

processes involving large temperature gradients, heat transfer by radiation might occur in which case an emissivity coefficient would be needed. These properties are generally not invariant but are dependent on temperature and to some extent on pressure. Other temperature-dependent properties which may be important include: flow characteristics such as absolute and relative permeabilities; fluid-storage capacity, expressed in terms of porosity, involving such characteristics as bulk and pore compressibilities and thermal expansions; properties of importance in well-log interpretation, for example, formation electrical resistivity factor and elastic-wave velocities; mechanical properties such as strength and deformation moduli. The above parameters are all temperature and stress dependent and some may also be affected by the types, amounts, and phases of saturating fluids.

2 1 APPLICATIONS REQUIRINGTHERMAL DATA

1.1 Thermal Oil Recoverv

There are two major methods of thermal enhanced recovery of petroleum. The first of these involves the injection of steam into depleted or partially depleted oil reservoirs for the purpose of heating the residual oil, thus reducing its viscosity and increasing its flow into producing wells.

Steam may be injected intermittently into producing wells, permitting the

steam to condense and give up its latent heat to the oil (and to the reservoior water and rock solids) and then, after the soaking period, allowing the well to produce the condensed steam and heated oil. These injection/soaking/production cycles may be repeated several times before the recovery area of the well has been depleted of recoverable oil. Steam may also be injected continuously through injection wells, creating a steam-drive and movement of oil into producing wells.

In addition to heating the residual oil, several other oil-displacement

processes occur including volatilization of lighter components of the oil and "steam-cleaning'' of the rock surfaces. The other major method of thermal enhanced oil recovery is in-situ combustion or "fire flooding".

In this process, air is injected into an injection well, residual oil around the

wellbore is ignited creating a fire-front which drives oil into producing wells.

In addition to

reducing the viscosity of the oil. the complex oil-recovery mechanisms which occur in this method include displacement of oil by cold water, hot water, condensed light fractions of the oil, and by gas, including nitrogen from the injected air and products of Combustion.

An oil

distillationkracking zone exists in front of the combustion zone, residuum from the distillation providing fuel for the combustion.

Behind the combustion zone is dry, heated rock which is

cooled by the injected air. Water may be added to the injected air to improve displacement of heat from the rock into the combustion zone. Analysis of thermal enhanced oil recovery methods requires knowledge of thermal properties of the rocklfluid systems at elevated temperatures and pressures and under variable fluid-saturation conditions.

In addition, knowledge of changes in porosity, permeability and

other properties under changing environmental conditions may also be required. Applications of thermal oil recovery methods to cemented sandstones and limestones is less common than applications to unconsolidated or poorly consolidated sands, but the probability of future applications makes it important to evaluate thermal and high-temperature behavior of all types of producing formations. 1.2 Geothermal Reservoirs

There are three general types of geothermal reservoirs.

Probably the most desirable

for energy-recovery purposes are those underground reservoirs from which high-pressure steam can be produced directly. Important in this type of reservoir is the recharge capacity in

3 which circulating groundwater contacts the underground heat source, converting the water to steam and thus maintaining the producing capacity of the steam production wells. A second type of geothermal reservoir contains high-temperature water which, when produced from wells, is flashed into steam and is then used to generate electric power. The third and probably least important type of geothermal reservoir is hot, dry subsurface rock.

This type of reservoir

rock must generally be fractured before water is injected into it. The water picks up heat as it transits through the fracture system and is then produced as hot water or steam through producing wells that intersect the fracture system. A wide variety of rock types may be found in geothermal reservoirs.

There may be

poorly-consolidated to well-consolidated and tightly cemented sandstones. The latter are often fractured, some showing secondary deposition of dissolved minerals. Hot, dry rock reservoirs generally consist of igneous or metamorphic rocks. These rocks have little storage capacity and limited fluid-flow capabilities unless they contain extensive natural fracture systems.

It is

generally necessary to fracture these reservoir rocks by artificial means to obtain adequate flow capacity. Thermal and fluid-flow properties of geothermal reservoir rocks are needed for analysis of these systems. In addition, those properties required for the interpretation of well logs run in high-temperature wells need also be known.

Knowledge of the mechanical

properties or rocks in high-temperature environments is useful for application of hydraulic fracturing operations. of W

1.3

Underground storage of heat involves the injection of heated fluids into porous rocks during periods of excess heat availability. These heated fluids can then be withdrawn later when heat is required. This is a common way to store solar energy in dealing with the diurnal cycle. Porous, permeable rocks of various types may be used for this purpose. Thermal properties, notably heat capacity and thermal diffusivity, are of particular importance in this application because of the transient or cyclic nature of the process. The underground storage of nuclear waste generally involves the use of long-life containers which are stored in underground chambers in a variety of rock types including granite, basalt, salt, tuff, and sands or sandstones.

Heat released from the nuclear waste is

dissipated through the surrounding rock. The relatively low thermal diffusivity of most rocks, particularly dry rocks, could lead to excessive temperature buildup in the containers and the surrounding rocks, resulting in possible damage to the containers and leakage of radioactve waste material. If such material were to reach moving ground water, disasterous contamination could result. The importance of knowledge of thermal properties of the surrounding rock in this application is obvious. temperature buildup.

Rocks need to be sufficiently conductive to prevent excessive

Mechanical properties of rocks in high-temperature environments are

4

important as are thermal expansion characteristics. Because of the multi-mineral composition of rocks, heating causes micro-fracturing due to differential thermal expansion of mineral grains.

If the rock is cooled at some later time,

porosity and permeability may be substantially increased and mechanical strength may be decreased. Thus, the effects of heating rocks in altering their properties may be even greater than the effects of elevated temperatures on such properties. 1.4 It is estimated that 20 percent of the earth's land mass is covered by permafrost. Thus

in the building of roads, pipelines and structures in such environments, account must be taken of the effects of heating associated with such construction.

Thawing of ice, partcularly in

unconsolidated soil, can have serious consequences on the stability of such soils.

Physical

properties strongly affected by the presence of ice in the rock pore spaces include strength, elastic moduli, elastic-wave velocities, electrical resistivities and thermal properties. Changes in these properties by low temperatures alone are rather minor. Dry rocks may show only small changes in these properties through a wide range of low temperatures.

Properties

other than the above generally show only small changes at low temperatures regardless of whether the rocks are dry or water-saturated. 2 DEFINITION OF TERMS 2.1 Thermal ProDerties (

i

)

. .

m. Thermal conductivity is the capacity of a material to conduct or

transmit heat. This is the coefficient ( h ) in Fourier's Law of heat conduction:

q = -hgrad T

where q

=

(1-1)

heat flux, W (watts)/mZ; h

gradient, K/m.

=

thermal conductivity, W/m-K; grad T

The standard unit of thermal conductivity is W/m-K.

cal/sec-cm-"C and Btulhr-ft-OF.

=

temperature

Other units include

Conversion factors for all thermal units are given in

Appendix A. Thermal conductivities of crystalline rocks decrease with increased temperature.

In

general, the higher the conductivity of the rock, the greater is the decrease with increased temperature. Rocks containing amorphous or poorly crystallized materials, on the other hand, generally have low thermal conductivities and the conductivity may actually increase with increased temperature. The stress to which a rock is subjected has some effect or its thermal conductivity.

In

the case of poorly-consolidated rocks, increasing stress will substantially increase thermal

5 conductivity. Thermal conductivities of well-consolidated and cemented rocks are only slightly affected by increased stress. Increasing pore-fluid pressure has the opposite effect, decreasing the thermal conductivity as pore-fluid pressure is increased with the external stress held constant. The opposite effect is also true. Decline in reservoir pore-fluid pressure causes the effective stress to increase and, consequently, thermal conductivity increases. The amount of liquid saturation in porous rocks and the thermal conductivity of the wetting-phase liquid may have large effects on thermal conductivity, particularly for highly porous rocks. Thermal conductivity of unconsolidated sands saturated with water may be three or more times the conductivity of dry sands.

In multi-fluid saturated rocks, thermal

conductivity of the wetting-phase fluid has the greatest effect on thermal conductivity of the rock.

caDacltv ' .

(ii) Heat

As the name implies, heat capacity of a material is its capacity to store

heat. The SI unit for heat capacity is Joulelkg-K but other commonly used units are cal/g-"C or Btu/lb-"F both of which have the same numerical value. Specific heat is derived from the amount of heat required to raise the temperature of a unit mass of pure water one degree at standard conditions (15OC, atmospheric pressure). This value is 1.OO cal/g-"C, 1.OO Btu/lbO F or 4.184 kJ/kg-K.

Heat capacities of dry rocks are about one-fourth the value for water. Heat capacities of rocks increase with temperature but only to the extent of about 30 percent over a wide temperature range. Since heat capacity is based on mass, it is relatively insensitive to stress. In some cases, volumetric heat capacity may be used.

This latter value is obtained by

multiplying mass heat capacity by mass density. The SI unit for volumetric heat capacity is kJlm3-K but another commonly used unit is cal/crn3-"C.

Volumetric heat capacity may be

affected by stress depending on the compressibility of the rock. (iii) T

h

e

.

r

. .

p.

In transient heat-transfer problems, the coefficient ( a ) . thermal

diffusivity, must be used as given in the diffusivity equation:

where V

=

differential operator dependent on coordinates; q = heat flux, W/m2; a

diffusivity. m2/s; T

=

temperature, K; t = time, s.

=

thermal

Thermal diffusivity is related to ofher

thermal properties as:

a

= WCpp

(1-3)

where a

=

thermal diffusivity, m%;

J/kg-K; p

=

mass density, kg/m3. Thermal diffusivity has units of cm%, ftZ/s, or SI units of

h

=

thermal conductivity, W/m-K; Cp

=

heat capacity,

6 m2/s. Conversion factors for thermal diffusivity units are given in Appendix A. Thermal diffusivities of rocks are generally strong functions of temperature, values decreasing with increased temperature.

Examination of Eq. 1-3 will show that thermal

diffusivity varies in a manner similar to that of thermal conductivity but amplified by the temperature behavior of heat capacity. The effects of temperature on mass density are small.

..

2.2 -!(i

' '

. The ability of a porous medium to transmit fluids through it is known as

permeability. Permeability is the coefficient (k) in Darcy's Law:

v = grad @(kip)

where v

=

apparent velocity

area of flow, m2; grad 0

=

=

(1-4)

Q/A, m/s; Q

=

volumetric flow rate, m3/s; a

potential gradient, Pa/m; k

=

=

permeability, m2; p

cross-sectional =

flowing fluid

viscosity, Pa s. For horizontal linear flow (grad 0) may be replaced by dpldx where (p) is pressure. Absolute permeability refers to fluid-flow capacity of the medium to a single fluid with which the porous medium is fully saturated. Absolute permeability varies with both temperature and stress, generally decreasing with increase in both parameters. The common unit of Permeability is the darcy but since this is derived from a mixed set of units, the preferred SI unit is m2.

If the pm2 unit is used, then 0.987 darcys = 1 .O pm2.

(ii) p o r o s i u . The storage capacity of porous rocks, referred to as porosity, is that fraction of the bulk volume of the rock available for the storage of fluids. "Total" porosity is the fraction of the total pore volume in the bulk volume of the rock. Some of this pore space might be isolated and not available for the storage of removable fluids. The available storage volume divided by bulk volume is referred to as "effective" porosity. Fracture porosity, as the name implies, is the available fluid storage space in the fracture network divided by the bulk volume. Numerically, this later porosity value may be small compared to pore-space porosity but it may have a disproportionately large effect on other properties and behavior of rock/fluid systems. Porosity decreases with increased effective stress and, to a lesser degree, decreases with increased temperature.

The effects of stress and temperature are interrelated in that the

amount of reduction in porosity caused by stress is increased with increased temperature. The effect of heating is quite another matter in that differential thermal expansion of mineral grains making up the rocks may cause intergranular fracturing, effectively increasing the porosity of the heated rock.

7

2.3 m t s of Temperature and

S k s on Other

Rock P r o D e r U

Other properties of rock may also be affected by temperature and stress. Elastic-wave velocities, electrical resistivities, pore and bulk compressibilities and thermal expansions are all. to some extent, temperature and stress dependent. These effects will be discussed in some detail in Chapters X and XI of this book.

8

Chapter II HEAT CAPACITIES OF ROCKS

Heat capacity is by definition the first derivative of heat content of a substance with respect to temperature.

Pure water with a heat capacity of 1.00 cal/g-"C at 15°C (4.184

kJ/kg-K in SI units) has about four times the heat capacity of dry rocks. Specific heat is heat capacity of a substance relative to that of water. Heat contents may be measured by the use of appropriate calorimeters and, from these measurements, heat capacities may be derived. As will be shown in the following, however, heat contents may be more easily calculated from published data based on Kopp's Law of additive properties.

1 EXPERIMENTAL MEASUREMENTS Heat contents of rock samples may be measured by the use of a Bunsen-type calorimeter. Measurement with this type of apparatus is based on the method of mixtures, which is the method generally used for heat content determination (Kelley, 1960). Diphenyl-ether is used for heat absorption. The heat content of the unknown sample is measured relative to the known heat content of platinum. The apparatus is referred to as a relative-error type because of the comparative measurements, and has the advantage of cancelling systematic errors. Experimental results are reproducible within f0.5 percent. In the above measuring method, test specimens are mounted in silver capsules and the known heat content of the empty capsule is subtracted from the total observed heat content to obtain the correct value for the test specimen. It may be necessary to heat the specimen to a temperature above the maximum experimental temperature to eliminate volatile constituents. Assuming the weight loss on preheating is for the most part water, a further correction to the observed heat content may be made to account for the water originally present in the specimen. This procedure was followed in determination of calculated heat contents to be described later. This apparatus and the method of measurement employed do not permit heat content determination of liquid-saturated samples.

However, another method for heat capacity

measurements of liquid-saturated samples will be discussed later in this chapter. A temperature base of 298.16 Kelvin (25%) is taken for heat content measurements. Measurements to be reported here were made at five temperatures (127, 227, 327, 427, and 527%).

Experimental heat contents relative to the base temperature are reported in Table II-

2 for the several rock samples described in Table 11-1.

9 TABLE 11-1 Description of test samples for heat capacity studies. Sample

Description

-

Porosity Quartz

1-Sandstone 2-Sandstone 3-Silty-sdst. 4-Silty-sdst. 5-Siltstone 6-Siltstone 7-Shale 8 - Liin e s to ne 9-Sand 10-Sand

Principal Minerals Clays Others

well consol., med-coarse poor consol., poor sort. poor consol., poor sort. med. hard, poor sort. med. hard, broken hard

0.20

80

tr. kaol.

pyrite

0.27

40

illite

feldsp.

0.21

20

kaol.

feldsp.

0.23

20

kaol.

feldsp.

0.30

20

kaol.

feldsp.

0.20

25

illite

_-

hard, laminated granular, unif. text. Unconsol., fine gr. Unconsol., coarse gr.

0.07

40

illite, kaol

0.19

_ -

__

__

0.38

100

__

__

0.34

100

__

__

Ca. carb.

TABLE 11-2 Comparison of calculated and experimental heat contents. Values in cal/g; temperature base ,298.16 K. Temp.

0 127 227 327 427 527

1-Sandstone Exper. Calc.

20.0 42.8 67.3 93.8 120.7

19.6 42.5 67.7 93.6 126.0

2-Sandstone Exper. Calc. 21.3 45.3 71.1 98.6 127.5

21.5 45.8 71.4 98.1 126.0

3-Silty Sand Exper. Calc. 21.5 45.7 71.7 99.2 128.6

21.9 45.1 72.8 101.0 130.4

4-Silty Sand Exper. Calc. 21.4 45.3 71.3 98.2 127.7

21.8 45.1 71.0 98.0 127.6

Temp. ("C)

5-Siltstone Calc. Exper.

6-Siltstone Calc. Exper.

7-Shale Calc. Exper.

8-Limestone Exper. Calc.

127 227 327 427 527

21.7 45.8 71.8 99.8 129.0

21.3 45.3 71.7 98.5 127.4

20.9 44.2 69.6 96.3 124.8

22.1 45.8 72.1 98.3 126.2

21.3 46.1 71.2 99.4 130.2

21.8 46.0 72.0 97.8 126.5

21.1 44.0 69.3 96.7 124.4

21.5 46.0 71.0 97.3 125.0

10

Heat capacity is represented by the slope of the heat content versus temperature curve at a given temperature.

Values of heat capacity at the several experimental temperatures were

thus determined and are presented in Table 11-3. Values at the temperature extremes (25 and 527°C) are shown in parentheses because of the lack of control in evaluating the slopes.

TABLE 11-3 Experimental heat capacities of several rocks at temperatures indicated. Values within parentheses are less reliable than values without. Sample

1-Sandstone 2-Sandstone 3-Silty sdst. 4-Silty sdst. 5-Siltstone 6-Siltstone 7-Shale 8-Limestone

25°C

127"

(.211) (.197) (.200) (.198) (.200) (.202) (.190) (.200)

0.21 1 0.227 0.228 0.224 0.228 0.228 0.218 0.228

Heat capacity 227" 0.241 0.294 0.254 0.246 0.249 0.251 0.244 0.247

-

cal/g-K 327"

427"

0.255 0.264 0.278 0.269 0.268 0.266 0.263 0.257

0.264 0.274 0.290 0.282 0.288 0.276 0.276 0.271

527" (.272) (.282) (.300) (.288) (.306) (.282) (.281) (.280)

Experimentally determined heat capacity variations with temperature are shown graphically in Figs. 11-1 and 11-2. The maximum differences in the heat capacities of any of the rock samples tested at any given temperature is less than 10 percent. The heavy smooth curves shown in Fig. 11-2 are calculated from heat content values from the literature (Kelley, 1960) for pure quartz and pure calcite.

It will be noted that the limestone curve closely

follows the calcite curve. This is also true for the quartz-rich sandstone sample which closely follows the quartz curve except at the two highest temperatures. This latter difference may be due to experimental errors since fairly significant differences between calculated and experimental values of heat content were observed. For the remaining samples tested, the heat capacity appears to be a function of the silica-alumina ratio, In general, the lower is this ratio the higher is the heat capacity. This is not true for the shale sample; the high iron content in this case appears to have a depressing effect on the heat capacity. 2 CALCULATED HEAT CAPACITIES

According to Kopp's Law the heat content of a compound is equal to the sum of the heat contents of its constituent elements. This law may be extended to minerals when the composition is expressed in terms of oxides (Kelley, 1956).

Further extension of this law for complex

mineral assemblages such as sandstones, shales, and limestones was tested in the work presented herein. Heat content values for the constituent oxides of the rock samples were obtained from a compilation by Kelley (1960). The temperature base for heat content was again taken as 298.16 Kelvin (25%). These heat content values are reported in terms of

11

.32Or

I

9

1

1

I

I

I

- .30

.300Y I

-

.280-

.20 Y m I

m

- 1.10 5

0

Y 7

$ .260I

I

- 1.00 .g

.240.-5 0 a

0 Sandstone

e .220u

A Silty Sand

CI

m

f .200-

d

----------

0 Silty Sand

---.-

x Siltstone

-..--.-

0

a

Q

- 0.90

CI

m Q

- 0.80

v Siltstone

.160 0

100

200

300

400

s

I

500

Temperature "C

Fig. 11-1. Experimental heat capacities for several rocks as function of temperature. Only end data points are shown for purposes of clarity.

0.30 y I

1.30 41.20

0.28 -

rn. -J 0.26 I

5*

0.24 -

a

a

5+

0.22-

f

0.20 -

m

-

A Calcite 0 Limestone-------0 Sandstone

0

AQuartz

,

I

100

,

I

200

,

, 300

,

0.80

-_------

-I

400

,

J c

x Shale

0.18 -

I 0.16

0.90

I

a Q I

10.70

500

Temperature "C

Fig. 11-2. Experimental heat capacities as function of temperature for sandstone, limestone and shale compared to literature values for calcite and quartz (latter values from Kelley, 1960).

12

calories per mole as functions of temperature.

For ease of calculation, these data have been

converted to calories per gram for each oxide commonly reported in a chemical analyses of rocks (see Table 11-4). For combined CO2. heat content values were obtained by subtracting the heat content values of CaO from those of CaCO3. In cases where C02 is not reported in the analysis, an amount of C02 equivalent to the CaO content is subtracted from the "ignition loss". and the heat content is added to the summation. The remainder of the ignition loss is considered to be water.

TABLE 11-4 Heat contents of common oxides at various temperatures. Source: Kelley (1960). Heat contents: AH = HT - H w , callg @ To C.

Oxide

50°C Si02 A1203 Fe203 GO K20

5.00 5.00 3.99 5.00 7.50 Na20 7.50 COz 5.94 (H2O)c 12.10 ( H 2 0 ) f 25.00

75"

100

150"

200"

300"

10.00 10.50 7.98 9.53 14.00 14.00 12.03 24.40 50.00

14.80 16.00 12.02 14.06 21.00 21.02 17.97 37.00 75.00

24.70 27.50 20.52 24.06 36.00 36.02 31.72 63.11 -

35.70 38.70 30.00 33.44 50.50 50.51 45.47 90.54

60.00 63.80 50.26 54.69 80.63 80.47 76.56 150.2

400"

500"

600"

86.60 90.49 72.02 75.00 110.6 110.5 109.4 216.8

114.4 118.0 93.8 96.9 142.5 142.6 145.3 290.6

148.0 147.0 1 1 7.7 1 1 8.8 175.0 176.2 182.8 372.3

700" 175.5 176.0 141.5 142.2 21 0.0 210.2 221.9 461.5

Heat contents of combined water are obtained by subtracting the heat contents of MgO from those of Mg(OH)2. The loss of combined water is considered to be linear between drying temperature (10 5 T ) and ignition

loss temperature (800%).

Kelley (1960)has derived

correlation equations for both heat content and heat capacity as functions of temperature. These equations are of the form:

AH = AT

Cp

where T

=

=

+ B T +~ CT-1 - D

A + 2BT - CT-2

(11-1)

(11-2)

temperature, K; AH = heat content, cal/g; Cp = heat capacity, cal/g-K; A, B, C, D

are constants for each oxide as presented in Table 11-5. Applying the above equations to the chemical analyses of the test samples, heat contents were calculated at each of the experimental temperatures.

Although this calculation is

13

TABLE 11-5 Constants for heat content and heat capacity correlations where:

AH = AT + BT2 + CT-1 - D and Cp = A + 2BT - CT-2; T

=

temperature, K. Source: Kelley (1960).

Oxide

A

B

SiO2 A1203 FeKb QO

0.1867 0.2696 0.1559 0.2081 0.2533 0.3025 0.1589

6.82E-05 1.38E-05 6.17E-05 9.63E-06 4.36E-05 4.73E-05 3.91 E-04

K20 (C02)c

(H20)c

250

C 4.49E+03 8.22E+03 2.36E+03 2.78E+03 0 1.05E+03 -8.22E+03

Experimental Heat Content 400 600

0

D

- kJikg

200 I

800

I

I

76.8 109.2 59.9 72.2 79.4 129.7 54.5

1OO(

I

I

/

1000

P'

/

/ 800

0

5

*

IP

7

I

*

/.*

.*/

C

600

0)

E

0

0 c

m 0)

I

/*

/

/

400

3m -a J 0

/* 200

/#

n -

/ 0

50

100 150 200 Experimental Heat Content - Calig

250

Fig. 11-3. Comparison of calculated and measured heat contents. Diagonal line represents perfect agreement. programmed for computer calculation, Table 11-6 shows details of

the calculation.

Loss of

combined water was equally distributed between drying temperature (105°C) and ignition

14

temperature (800°C). The lowermost calculation shows the effect of free water saturation

on

heat capacity. Figure 11-3 shows a plot of calculated versus measured heat content values. The broken line is the diagonal representing perfect agreement between calculated and measured values. The maximum deviation is less that 2 percent. TABLE 11-6 Heat capacity calculations from a typical sandstone oxide analysis. Oxide

Frac. 50°C

Si02 A120 Fe203 GO K20

75"

CaVg @ To C. Heat contents: AH = HT 100" 150" 200" 300" 400" 500"

.699 .1 41 ,019 .006 ,032 ,026 ,006 ,067 .996 ,062 .992 ,057 ,983 ,048 ,977 ,038 .968 ,029 .958 ,019 ,945

3.50 0.71 0.08 0.03 0.24 0.19 0.04 0.81 5.60

1 .ooo Cp (cal/g-"C)

5.62 ,224

(H20)f

4.02 8.03 12.05 9.64 1 9 . 2 7 28.83 8.30 16.60 24.83 .333 ,332 .332

Nz0 COZ (H20)c

c c

x c

x c z:

,161 1.161 1.000 Cp(cal/g-"C)

C

6.99 10.35 1.48 2.25 0.15 0.23 0.06 0.09 0.45 0.67 0.36 0.54 0.08 0.12 1.63 2.46 1 1 . 2 0 16.71

600"

1 7 . 2 7 24.97 41.95 60.55 7 9 . 9 9 1 0 3 . 4 8 8.99 12.75 16.63 20.71 3.88 5.45 0.40 0.58 0.97 1.39 1.81 2.26 0.15 0.21 0.35 0.62 0.76 0.48 1.15 1.62 2.58 3.53 4.56 2.63 0.92 1.29 2.06 2.83 3.65 4.51 0.20 0.29 0.49 0.70 0.93 1.17

3.90 27.87 5.1 7 39.58 7.15 64.54 8.26 90.50 8.31 116.50 7.1 1 145.63 11.24 16.78 2 8 . 0 9 4 0 . 2 6 6 6 . 0 6 9 3 . 4 9 121.61 153.45 . 2 2 3 .223 ,235 ,246 .227 .266 .278 ,300

3 HEAT CAPACITY OF FLUID SATURATED ROCKS

Based on the excellent agreement between calculated and measured heat contents of mineral aggregates, it may be assumed that the heat contents of fluid-saturated porous rocks can also be calculated by summations of the heat contents of the rock and its fluid constituents, on a mass fraction basis. There is no evidence to indicate that the intimate contacts of the fluids and mineral grains will alter the heat contents of the individual constituents of the saturated rock.

15

Heat capacity of a rock saturated with one or more fluids can best be calculated on a volumetric basis. Thus:

pbcp = (1

where pbCp rock;

@ =

=

-

Q)ps cps + cb(s1p1 c p l + s2p2cp2 + - - - Snpncpn)

volumetric heat capacity of fluid-saturated rock: pb

fractional porosity of rock; ps

solids; Si, - - Sn

=

=

=

(11-3)

bulk density of saturated

density of rock solids; Cps

=

heat capacity of rock

fractional saturation of fluids; p i ,- - pn = densities of fluids; Cpi , - - Cpn

= heat capacities of fluids. For a typical petroleum reservoir formation, the three fluids would

be water, oil, and gas. Because of its relatively low density and low heat capacity, gas can be ignored in the calculation and Eq. 11-3 becomes:

where subscripts (w) and

(0)

refer to water and oil, respectively.

In evaluating the contribution of saturating fluids to the heat capacity of the bulk rock, it is necessary to obtain heat capacity data for the fluids under reservoir conditions of temperature and pressure. The two fluids considered first were water and methane.

Heat

capacities and specific weights for water were obtained from Keenan and Keyes (1936). Heat capacities for methane at atmospheric pressure and experimental temperatures were obtained by interpolation of API-U.S. Bureau of Mines data. Values of the heat capacities of methane at elevated pressures were calculated by methods outlined by Edminister (1948).

Table 11-7

shows the values of heat capacity and density of water and methane at the six experimental temperatures and at pressures of 0.10. 3.5, 10.3 and 20.7 MPa (14.7, 500. 1500, 3000 psia). Calculated values of volumetric heat capacities for dry, methane-saturated, and watersaturated rock samples at one temperature (327°C) and several pressures are shown in Table 11-8.

The effects of liquid and vapor phase water saturation on the heat capacity of sandstone sample 2 are shown in Fig. 11-4. It will be observed that vapor or gas saturation has negligible effect on heat capacities of rocks as was assumed earlier. This is due primarily to the relatively low density of gas or vapor which may occupy the pore spaces of rocks. On the other hand, the presence of liquid water may increase the heat capacity by 35 percent or more, depending upon the prevailing fluid pressures and temperatures. Heat capacity data for crude oils at elevated pressures and temperatures are generally unavailable. Gambill (1 957) has provided some data on the variation of heat capacities of oil with temperature but not with pressure.

From

reported values at atmospheric pressure and temperature, it may be deduced that crude oil saturation will increase the heat capacities of rocks by about half the amount as the increase by

16 water saturation.

TABLE

11-7

Heat capacities and densities of water and methane as functions of temperature and pressure. CP = heat capacity, cal/g-K; p = density, g/cm3.

l!Y!IIm Temp. ("C)

0.1 OMPa(l4.7psia) CD P

3SMPa(500psia) CP

P

25 127 227 327 427 527

1.007 0.475 0.472 0.481 0.494 0.512

0.992 1.016 1.123 0.605 0.546 0.543

0.997 0.941 0.829 0.135 0.111 ,0094

0.994 0.555E-3 0.441E-3 0.376E-3 0.314E-3 0.274E-3

10.3MPa(1500psia) 20.7MPa(3000psia) CD P CD P 0.990 1.012 1.097 1.338 0.695 0.598

1.000 0.944 0.837 ,0522 ,0369 ,0304

0.981 1.005 1.082 1.450 1.200 0.735

1.005 0.949 0.846 0.676 ,0910 .0637

METHANE Temp. ("C)

0.1 OMPa(l4.7psia) CP P

3SMPa(500psia) CP P

25 127 227 327 427 527

0.520 0.604 0.693 0.782 0.864 0.940

0.58 0.62 0.69 0.78 0.87 0.95

0.654E-3 0.487E-3 0.391E-3 0.322E-3 0.279E-3 0.244E-3

.0235 ,0168 ,0132 .0109 ,0094 ,0082

10.3MPa(1500psia) 20.7MPa(3000psia) CP P CP P 0.73 0.67 0.72 0.79 0.88 0.95

,0838 .0548 ,0420 ,0345 ,0292 ,0255

0.95 0.74 0.75 0.81 0.89 0.96

0.161 0.102 .0768 .0625 .0527 ,0458

TABLE 11-8 Calculated volumetric heat capacities of methane- and water-saturated rocks at 327% and pressure indicated, (callcm3-K). Sample

1-Sandstone 2-Sandstone 3-Silty sdst. 4-Silty sdst. 5-Siltstone 6-Siltstone 7-Shale 8-Limestone

Dry

0.545 0.527 0.571 0.537 0.513 0.538 0.634 0.567

O.1OMPa (14.7 psia)

0.545 0.527 0.571 0.537 0.513 0.538 0.634 0.567

0.545 0.527 0.571 0.537 0.513 0.538 0.634 0.567

3.5 MPa (500 psia)

0.546 0.529 0.572 0.537 0.514 0.541 0.634

0.546 0.529 0.572

0.538 0.514 0.541 0.634

10.3 MPa (1500 psia)

0.549 0.533 0.577 0.543 0.521 0.545 0.636

0.559 0.545 0.585 0.553 0.532 0.554 0.641

20.7 MPa (3000 psia)

0.554 0.541 0.582 0.549 0.527 0.549 0.639

0.735 0.763 0.787 0.774 0.780 0.764 0.716

3

17

.380

&+ OQ

.360 -

- 1.50

.340-

I

- 1.40

I

- 1.30

y

-..320-

%

0)

.=r

.300 -

6

5

20.7 MPa (3000 psia)

m

I"

.260

3.45 MPa (500 psia)

-

1.10

.240 .220

0

Sample 2 Sandstone Water Saturated I

1

100

I

I

200

I

I

I

300

I

400

I

-I

I

1.oo

I

500

Temperature "C Fig. 11-4. Calculated heat capacity values at several temperatures and pressures for Sandstone-2 saturated with water in liquid and vapor phases.

3.1

. . p SaturKing (1960) has reported on a method of measuring heat capacities of liquid-saturated

rocks. The method involves coating the liquid-saturated test specimen with paraffin to prevent the loss of fluids and thus the test temperature is strictly limited. The test method involves preparation of the rock sample in the form of a 3.18 cm (1 1/4 in) diameter cylinder, 2.54 cm (1 in) long with two symetrically placed axial holes drilled 1.43 cm (9116 in) apart. A glass-

bead type thermister was mounted in one of the holes and a 1000 ohm carbon resistor was mounted in the other hole; both were held in place with paraffin wax. The water-saturated test specimen was coated with paraffin wax to prevent evaporation, as mentioned earlier, but an oven-dried comparison rock specimen was not coated. The test specimen was suspended in a Dewar flask which was mounted in a constant temperature calorimeter bath. A known current was passed through the resistor for a known period of time and the electrical resistance response of the thermister was recorded with time. Heat capacity was calculated from the plot of electrical resistance versus time. The oven-dried test specimen was run both with the Dewar flask evacuated and at atmospheric pressure conditions. The heat capacities measured under the two conditions were essentially the same, but it was noted that the thermister response drift rates were greater for the atmospheric pressure run. The water-saturated test specimen was run with the Dewar flask at atmospheric pressure to prevent the paraffin coating from cracking with the reduction in external pressure.

18

The results of heat capacity measurements are shown in Fig. 11-5 and are compared with calculated values. Agreement between calculated and experimental values for the oven-dried specimen is within Q percent. The experimental heat capacity values for the water-saturated test specimen are 9 percent higher than the calculated value for an adjacent sample. However, the water content of the adjacent sample was reported as 16.06 percent but for the experimental test specimen the measured water content was 18.2 percent.

Adjusting the

calculated values to the higher water saturation, the discrepancy decreases to only 5 percent. The limits of accuracy of the heat capacity values measured with the reported apparatus are considered to be t5 percent of the absolute magnitude. By the nature of the current measuring technique, the relative values are believed to be much more reliable.

0.40

4

Measured 10.2% H 2 0

1.60

0.35 Y I

-m 1

s l

6,

0.30 11.20

c .0

3

m

0.

sm c

0.25

0

I

0.20

0.15 0

20

40

60

80

100

120

Temperature "C

Fig. 11-5. Comparison of experimentally determined and calculated heat capacities as function of temperature for a dry and water saturated sandstone.

4 GENERALIZED CALCULATION OF HEAT CAPACITIES

Analysis of Fig. 11-1 shows that heat capacity as a function of temperature for all sandstones can be represented by the following equation:

Cps

=

0.108 T'J.155

(11-5)

19

where Cps

=

heat capacity, cal/g-"C; T = temperature, "C.

Error is within f 3 percent for all

but the lowest temperature, at which the calculated value is about 7 percent low. Based on the above analysis, Gomaa (1973) developed a generalized calculation of heat capacities for fluid saturated sandstones. In using Eq. 11-4, all terms must be evaluated at the temperature for which the analysis is being made. For example, according to Holman (1958) the density of liquid water as a function of temperature may be expessed as:

where pw20 = density of water at 20°C; T = temperature, "C;

PW = coefficient

of thermal

expansion for water which may be expressed as:

p w = 2 . 1 1 5 ~ 1 0 -+~ 1.32~10-6T+ 1.09x10-8T2;

(11-7)

In a similar manner, the density of oil at any temperature, according to Standing (1977), may be expressed as:

PO =

p020/[1 + (T - 20)p0]

where p020 = oil density at 20°C; T = temperature,

(11-8)

"C; PO = coefficient of thermal expansion

for oil which may be expressed as:

Po = 4 . 4 2 ~ 1 0 - 4+ 0 . 1 0 3 ~ 1 0 - 4x "API

where "API = 141.5/Go

-

(11-9)

131.5; Go = specific gravity of oil at 20°C.

Density of rock solids as a function of temperature may be expressed as:

PS = Ps201 [ l

+ (T - 2O)Bsl

(11-10)

A value of ps = 0 . 5 0 ~ 1 0 ' ~ Thas - ~ been found in earlier work for sandstones (Somerton and Selim. 1961). Assuming that mineral grain density is 2.65 g/cm3, Eq. 11-10 becomes:

Ps

=

2.65111

+

(T

-

20)0.5Oxl 0 - 4 ]

(11-11)

20

Combining this with Eq. 11-5 gives:

psCps = 0.286 To.155/[1 + (T - 20)0.50~10-4]

(11-12)

Gomaa (1973) has expressed Eq. 11-12 in the form of a polynomial as:

psCps = 0.49 + 0.91x10-3T

+ 0.80 T2

(II - 13)

Variation of heat capacity of oil with temperature is given by Gambill (1957) as:

Cpo

=

(0.389

+

(11-14)

0.81~10-3T)/G00.5

The heat capacity of water is more complicated. Combining data given by Holman (1958) for heat capacity of water with Eqs. 11-6 and 11-7 for density of water gives:

for 70"


---

100

200

300

400

500

600

700

800

900

1000

Temperature - "C

Fig. IV-3. Volumetric thermal expansion of three sandstones compared with quartz 2) Expansion of quartz perpendicular to the C-axis has a predominating effect on the

expansion characteristics of sandstones. Since the expansions of the sandstones in the directions parallel and perpendicular to the bedding were approximately the same, random orientation of the quartz crystals would be expected. 3) Permanent elongation of the test specimens after cooling resulted from deformation of the

test specimen due primarily to differential thermal expansion of the quartz grains.

At

temperatures above the a-b quartz inversion temperature, where the coefficient of expansion for quartz becomes negative, the role of other mineral constituents becomes important in controlling expansion and deformation behavior of the sandstones.

2 EXPANSION OF FLUID-SATURATED ROCKS UNDER STRESS

Measurements of thermal expansion of rocks under simulated subsurface reservoir conditions present a number of difficult problems. Most of these problems were resolved in the work to be reported here with the exception that the temperature range of the tests was limited

33 to 25" to 175°C. In addition to linear thermal expansion, pore-volume contraction was also measured (Somerton et al., 1981). 2.1 Bulk and pore thermal expansion tests were run on 5.1 cm (2 in) diameter by 5.1 cm (2 in) long test specimens of Bandera, Berea, and Boise sandstones. Change in lengths were

measured by strain gauges mounted directly onto the test specimens. Pore volume changes were measured by use of a precision high-pressure, hand-operateddisplacement pump. The jacketed test specimens were mounted in a pressure vessel using Dow-Corning 200 silicone oil as the external pressuring fluid. External heaters were used to raise the temperature of the vessel and the test specimen at a controlled rate of 1°C per minute. Fig. IV-4 is a schematic diagram of the experimental apparatus. The strain gauges used to measure longitudinal strain had an upper limit in operating temperature of 290°C for continuous use. A thin coating of high temperature epoxy cement was used to mount the strain gauges onto the test specimen. A thin coating of silicone rubber was used over the gauges and the lead lines to prevent fluid leakage and provide electrical insulation. The specimens were jacketed with shrink-fit teflon tubing which sealed the specimens to end

Pressure Gauge Volumetric TC

r Temperature Control

-

Pressure Vessel Test Specimen

Electric Heater

-.

T

Insulation

I .

Constant Rate Pump

kQ Vacuum Pump

Fig, IV-4. Schematic diagram of experimental apparatus for measuring thermal expansion of rocks subjected to elevated temperatures, confining pressures and pore-fluid pressures.

34

caps containing pore-fluid tubing connections.

The specimens were vacuum-saturated with

water before being mounted in the pressure vessel. Several pore volumes of water were flowed through the test specimens at a back-pressure of 2.5 MPa (360 psi) to assure that all air had been removed from the pore-pressure system. Changes in pore volume were measured by use of the hand-operated, precision-volume displacement pump.

The pump is capable of detecting a change in volume of as little as

0 . 0 0 0 5 ~ ~The . system was calibrated for dead-fluid volume at several pore pressures and as a function of temperature in ranges used in the experiments (6.9, 13.8, and 20.7 MPa. and 30" to 175°C). Corrections were also made for thermal expansion of pore water by use of I.F.C. data (International Formulation Committee, 1968).

2.2 J inear Thermal F x D a m

Results of the measurements of linear thermal expansions of the three liquid saturated sandstones under a confining stress of 20.7 MPa (3000 psi) and pore pressure of 6.9 MPa (1000 psi), are compared with the dry unstressed test results for the same sandstones in Fig. IV-5. Although the agreement is fair, the slopes of the curves for Boise and Bandera are steeper, compared to earlier data, but the slope for Berea is less steep. This would indicate that the coefficients of thermal expansion are greater for Boise and Bandera but less for Berea, compared with earlier data. Table IV-2 shows comparison of thermal expansion coefficients in the temperature range of 100 - 200°C.

0.8 N

z 4 0.6 -1

Dry-Zero Stress

I I C

:.

I Berea SS

X

0.4

C

m

Q

W

& 0.2 0) C .-1 0

0

/ 200

Bandera SS

Dry-Zero Stress

Dry-Zero Stress

Present Data,

400 0

Boise SS

Present

Present

200

400

0

200

400

Temperature "C

Fig. IV-5. Comparison of linear thermal expansions of dry sandstones and water-saturated sandstones under stress.

35 The thermal expansion values shown in Table IV-2 compare with a value of 16 x 10-6 ("C-1) for quartz in the same temperature range.

No thermal expansion data for liquid-

saturated rocks, and very little data for rocks under stress, have been found in the literature to compare with the present results. Sweet (1978) estimated the effect of stress on thermal bulk expansion based on existing data showing the effects of temperature on rock compressibility. His results indicated that the thermal bulk expansion of rocks should decrease with increased stress. For example, for a low porosity sandstone he estimated that the bulk expansion should decrease by 25 percent upon increasing the stress from atmospheric to 100 MPa (14,500 psi). Of the three sandstones tested in the present work, only Berea showed a decrease of this magnitude.

Wong and Brace (1979) ran tests on Cheshire quartzite at substantially higher

stresses and found a decrease in thermal expansion of about 25 percent between stresses at 100 and 500 MPa (14,500-72,500 psi).

TABLE IV-2 Thermal expansion coefficients for three sandstones in the range 10O-20O0C, ("C-1) after Somerton et al. (1981).

2.3

Condition

Berea SS

Bandera SS

Boise SS

Dry Saturated

15. x 10-6 13. x 10-6

15. x 10-6 20. x 10-6

16. x 10-6 17. x 10-6

m Earlier work by Von Gonten and Choudhary (1969) showed that pore-volume

compressibilities of sandstones increased with increased temperature.

From these results, one

would expect that pore volume would decrease with temperature at constant stress.

Results of

tests by Ashqar (1979) and Janah (1980) showed this to be the case. Figure IV-6 shows porevolume thermal contraction of the three sandstones tested. One should note in particular that the amout of contraction is small until temperatures of about 90-120°C are reached. Above this temperature range the thermal contraction increases to nearly constant rates.

Both the

temperature at which the increased contraction begins and the slopes of the contraction curves are related to the porosity of the sandstones. For the lowest porosity sandstone (Bandera) the increased slope begins at a lower temperature and the slope is greater. The relationship between the amount of pore contraction and the porosity of the sandstone noted above would be expected since it is the thermal expansion of mineral grains into the pore space that causes the decrease in pore volume. Thus the greater the fraction of mineral solids in the bulk volume, the greater would be the reduction in pore space.

Although

differences in mineral composition probably have some effect on the magnitude of the contraction, earlier results of linear thermal expansion tests on the same sandstones showed

36

I

1.6 -

P,

= 20.7 MPa

bORE = 6.9 MPa N

1.2

-

0.8

-

0.4

-

X

a

?

5

Fig. IV-6. Pore-volume contraction of three sandstones as function of temperature at constant stress conditions. that the quartz content of the sandstones had a dominant effect on the magnitude of the expansion (Somerton and Selim, 1961).

However, the importance of the effects of differences in mineral

composition on pore volume contraction needs to be considered. The low coefficient of porevolume contraction for Boise sandstone may be due in part to the relatively low quartz content of this sandstone. The effects of stress on the pore-volume contraction curves are more difficult to explain. The stress-strain curves for sandstone generally show large strains at low stress levels. This is attributed to the "tightening up" of the grain structure as stress is first applied. However, since the tests reported here were run at elevated and constant stress conditions, this should not be a factor. The magnitude of the stresses applied to the sandstones during heating has a rather small effect on pore volume contraction. Figure IV-7 shows the effect of changing confining stress while keeping pore pressure constant for Bandera sandstone.

Increasing confining stress

decreased both the amount and the rate of pore-volume contraction but the decrease is small, close to being within the limits of accuracy of the measuring system. However, the change in the amount of contraction with increased pore pressure at constant confining stress is substantial as shown in Fig. IV-8. The amount of contraction in pore volume decreased with increased pore-fluid pressure. In terms of effective stress (difference between confining stress and pore-fluid pressure), there appears to be anomalous behavior depending on whether the confining stress

37

1.6

1.2

: X

n

0.8

> 4

0.4

0 40

60

80

100

120

140

160

Temperature ("C)

Fig. IV-7. Effect of varying confining stress at constant pore-fluid pressure on pore-volume contraction versus temperature for Bandera sandstone.

Bandera PCoNF= 27.6 MPa

1.6

1.2

z X

n

?

c

0.8

I

0.4

0 40

60

80

100

120

140

160

Temperature ("C)

Fig. IV-8. Effect of varying pore-fluid pressure at constant confining stress on pore-volume contraction versus temperature for Bandera sandstone.

38 or the pore-fluid pressure is changed. In the former case. increase in effective stress leads to a decrease in contraction, whereas in the latter case, increase in effective stress results in increased contraction.

However, since the thermal contraction is considered to be due to

mineral grains expanding into the pore spaces, it would be expected that pore pressure would have the dominant effect. Increase in pore pressure should decrease mineral-grain expansion into the pore space. This is in agreement with Zoback and Byerlee's (1975) observations that pore-fluid pressure changes have a much larger effect on pore-dominated properties (permeability in this case) than do changes in confining stress.

3

coNcLusloNs From the above reported work (Somerton e l al., 1981) a number of conclusions can be

reached: 1) Linear thermal expansions of liquid-saturated sandstones under stress are not much different from values for dry, unstressed samples. The presence of quartz appears to dominate the thermal expansion behavior in both cases. 2) Pore volume of liquid-saturated sandstones under stress decreases with increased

temperature.

Pore-volume thermal contraction is a strong function of porosity of the rock,

decreasing with increased porosity.

Quartz content of the rock probably also has some effect on

pore-volume contraction. 3) The magnitude of pore-volume thermal contraction is only slightly affected by changes in

confining stress.

However, change in pore-fluid pressure has a more pronounced effect,

increase in pore-fluid pressure resulting in decreased thermal contraction. 4)

Changes in porosity of rocks due to reservoir heating are probably not very large. For

example, calculations based on the above data show that for a 20 percent porosity sandstone at a depth of 1000 meters with pore fluid at hydrostatic pressure heated from 100" to 200°C would undergo a reduction in porosity of about one percent. The amount of reduction in porosity would of course increase at lower pore pressures and with greater increase in temperature.

39

Chapter V THERMAL CONDUCTIVITY OF ROCWFLUID SYSTEMS Thermal conductivity is defined as the capacity of a substance to conduct or transmit heat. This is the coefficient (1)in Fourier's Law of heat conduction: q

=-h

grad T

(V-1)

where q

=

thermal

conductivity is W/m-K. Other units include cakec-cm-"C and Btu/hr-ft-"F.

heat flux, watts/m2; grad T

=

temperature gradient, Klm.

The standard unit of

Conversion factors are given in Appendix A. Thermal conductivities of dry rocks have been shown to be functions of density, porosity, grain size and shape, degree of cementation, and mineral composition (Scorer, 1964). The first two properties are easily measured and precise values may be assigned for correlation purposes. Grain size and shape and cementation are difficult to quantify.

However, other

related properties can be used to characterize these properties in developing correlations. Permeability and formation resistivity factors are readily measurable as unique values and are quite dependent on grain size and shape and degree of cementation. Precise mineral composition values are often not available but in cases where they are available, a weighted average value of solids conductivity may be determined for use in correlations. The high thermal conductivity of quartz present in the rock (see Table V-1) has a dominant effect so that if the amount of quartz is known within reasonable limits, a good estimate of solids conductivity can be made, as will be discussed later. Thermal conductivity of a liquid-saturated rock is dependent on the thermal conductivities of the dry rock and the saturating liquid as well as physical properties of the rock. Liquid-saturated rocks have higher conductivities than dry rocks, the amount of increase being a complex function of the amount of pore space, its nature and distribution, and the conductivity of the saturating fluid.

For water-saturated unconsolidated sands, thermal

conductivity can be as much as four times that for the dry sand.

For tight, low porosity

sandstones the increase with water saturation may be as low as 50 percent. Thermal conductivities of most materials having crystalline structures decrease with increased temperature (Powell, et al., 1966). should vary with the reciprocal of temperature.

Theory indicates that thermal conductivity In mixed crystals and highly disordered

40

TABLE V-1 Thermal conductivities of rock-lorming minerals. Source. Horai (1 971 ). Thermal Conductivity

Mineral Wlm-K Quartz Orthoclase Plagioclase Calcite Muscovite Chlorite Hornblende Epidote Sphene Biotite

Btu/ft-hr-"F 4.45 1.34 1.24 2.08 1.28 2.84 1.78 1.15 1.35 1.35

7.70 2.32 2.15 3.60 2.21 4.91 3.08 2.61 2.34 2.34

crystals, conductivity varies more slowly than T-1 and, in fact, may show a slight increase with temperature.

Thermal conductivities of glasses and vitreous materials increase with

temperature. Tikhomirov (1 968) has developed a correlation equation, based on experimental data, for prediction of the effect of temperature on thermal conductivities of rocks. A plot of this equation shows a moderate negative gradient of thermal conductivity with temperature for high conductivity rocks, whereas small positive gradients are shown for low conductivity rocks. A modified form of this equation will be shown to give good agreement with experimental measurements for both dry and liquid-saturated rocks (Anand, et al., 1973). Investigations have shown that thermal conductivity increases with increase in effective stress on the rock. This would be expected since increase in stress on the rock improves the thermal contact between mineral grains, increases the overall density of the rock and, consequently, increases the thermal conductivity of the rock.

In measuring thermal

conductivity in the laboratory, part of the observed increase in conductivity with stress may actually be due to reduction in thermal contact resistance between the heat source and sink, and the temperature measuring devices, and the test specimen.

When good thermal contact is

achieved, the change in thermal conductivity with added stress is generally small. The effect of increasing pore fluid pressure is to reduce the effective stress on the rock. More realistically, a reduction in pore fluid pressure results in increased effective stress on the rock and thus an increase in thermal conductivity.

Pore pressure may also be associated

with phase behavior of saturating fluids. Reduction in pore pressure may result in vaporization of some of the liquid components and this, in turn, may cause a reduction in thermal conductivity. This is a fluid saturation effect and should not be attributed to the effect of pore pressure per se.

41

1 METHODS OF MEASURING THERMAL CONDUCTIVITY

Two groups of methods are used to measure thermal conductivity of rocks. In steadystate methods thermal conductivity is measured directly while in transient methods values of thermal diffusivity are generally measured and from these measurements thermal conductivities are calculated.

Steady-state methods require long periods of time to achieve

equilibrium conditions but results are generally quite accurate. Steady-state tests may readily be run under simulated subsurface environmental conditions of pressure (stress), temperature and fluid saturation. Transient methods of measurement are usually much faster than steadystate methods but results are often less accurate and it may be difficult to run such tests under simulated reservoir conditions.

Densities and heat capacities must be known to convert

measured diffusivity values to thermal conductivity values.

A similar problem exists in

converting steady-state thermal conductivity values to thermal diffusivities. Steady-state methods require carefully prepared test specimens of specific geometries and a means of measuring the amount of heat flowing through the test specimen. Either primary or secondary heat flow measuring techniques may be used. Secondary techniques make use of reference materials of known thermal conductivities as heat meters. Transient methods employ a variety of heat flow measuring techniques.

Methods are

generally based on known solutions of nonsteady-state heat flow equations for specific boundary and initial conditions with specified heat inputs.

Experimental techniques may range from

conditions of constant rate of temperature rise at a specific boundary to constant heat input rates.

So many different apparatuses for measuring thermal conductivity have been reported in the literature that it would be impractical to review all of them here. Instead, a typical steadystate apparatus will be described. This apparatus has been developed and used by the author and his colleagues in obtaining much of the thermal conductivity data presented herein.

Two

transient-type methods used in this work will also be described. CQmparator 9pparatus

1.1 --State

The steady-state comparator method involves applying a temperature gradient across a test specimen and one or more standards of known thermal conductivity until steady-slate temperatures are obtained.

Temperature differentials across the test specimen and the

standards are measured and, if the cross-sectional areas of the specimen and standard are the same, thermal conductivity of the test specimen may be calculated for a linear system from the following equation:

h

=

hst(ATst/Lst)(L/AT)

where h = thermal conductivity of test specimen; L

(V-2)

=

length of test specimen; AT

=

temperature

42

differential across test specimen; kt, Lst, and ATst refer to the standards. In the apparatus used in the work reported here, two disc-shaped standards, 5.08 cm (2.0 in) in diameter and 1.59 cm (0.625 in) thick, of known thermal conduclivity are mounted in holders above and below the test specimen. The test specimen, having the same diameter as the standards, 5.08 cm (2.0 in), is 3.18 cm (1.25 in) thick and is mounted in a matching holder between the two standards. Faces of the test specimen must be smooth and parallel and have tolerances of +.0025 cm/-.0000 cm in thickness and +.OOOO cm/-.0025 cm in diameter. The holders, made of Vespel (a polyamide resin manufactured by DuPont). are 10.16 cm (4.0 in) in OD. This material was selected for the holders because it is machinable, has low thermal conductivity (0.37 wattslm-K) and is stable up to temperatures of 320°C.

The

holders are used to contain the standards and the test specimens and to allow for fluid saturation of the latter. The holders are sealed by use of stainless steel plates 8.90 cm (3.50 in) in diameter and 0.238 cm (0.094 in) thick and nylon screws. Viton O-rings are used between the plates and the specimen holder lo obtain a fluid seal. Although there are positioning rings and grooves on the top and bottom of the holders, the holder thicknesses are slightly less than the standards and test specimen thicknesses so that when axial stress is applied to the stack, the standards and specimen are subjected to the load rather than the holders. The small air gaps between the holders also serve as heat barriers, forcing more of the heat to pass through the standards and the test specimen. The heat source and sink are solid copper cylinders 10.16 cm (4.0 in) in diameter and approximately 25 cm (10 in) long, provided with 400 watt resistance-wire heaters and resistance thermometers which are embedded in them. These cylinders have positioning rings and grooves to match with the top plate of the upper standard holder and the bottom plate of the lower standard holder but the thermal contact is provided by the sealing plates rather than the Vespel holders. Details of the stack are shown in Fig. V-1. The necessary condition of no lateral heat transfer is achieved by use of guard heaters surrounding the stack of standards and the test specimen. A schematic diagram of the entire apparatus is shown in Fig. V-2. The stack, consisting of source and sink heaters and standard and test specimen holders, is mounted in a loading frame provided with a hand-operated hydraulic pump to apply axial stress to the stack.

Manual

operation of the loading system has been found to be entirely satisfactory. Although only direct axial stress is applied to the test specimen, the confinement provided by the holders and the small U D ratio simulate to some degree triaxial stress conditions. Pore pressure and fluid saturation control are provided by means of an outlet through the middle edge of the test specimen holder.

A groove is provided around the inner

circumference of the test specimen holder, intersecting the pore pressure opening, to provide uniform fluid pressure in the test specimen. The pressure outlet is connected to an accumulator

43

I

I

Bottom Heater

Fig. V-1 . Test specimen and standards stack for comparator thermal conductivity apparatus.

Heater Control

n

Axial Loading System

Terminal

I I

Nitrogen

Heater

I Control J Fig. V-2. Schematic diagram of comparator thermal conductivity apparatus.

44

which contains the liquid phase saturating the test specimen. the accumulator,

Pore pressure, applied to

is controlled by a nitrogen supply and precision pressure regulator.

For liquidhapor phase saturation studies, the accumulator may be bypassed and the pore fluid line connected directly to a control valve/bleed line/condenser system. A controlled amount of vapor may be withdrawn from the test specimen, condensed and measured to provide a known partial liquid saturation. The pore pressure then returns to the vapor pressure curresponding to the prevailing temperature in the test specimen. Temperatures of the heat source and sink are controlled by use of Thermotrol temperature controllers. These are capable of controlling temperature within 0.03% at the control points. A total temperature differential of 15°C is used across the stack. Temperatures in the stack are measured by use of K-type chrome-alumel thermocouples embedded in the centers of the stainless steel holder sealing plates. Holes are drilled in the edge of the plates to the center of their diameters. Stainless steel sheathed thermocouples are inserted into the holes and cemented in place with technical-G copper oxide cement. This cement provides excellent thermal contact but is an electric insulator.

The stainless steel thermocouple sheathing is

silver-soldered to the plate at the entry point. The thermcouple plates are carefully calibrated before use. To minimize radial heat loss, the entire stack is surrounded by a split-cylinder furnace which consists of a layer of low conductivity ceramic fiber insulation, a uniformly wound resistance-wire heater on a ceramic cylinder, another layer of ceramic fiber insulation, and a sheet stainless steel outer case. Temperature of the guard heater is controlled to match the temperature at the midpoint of the test specimen. This latter temperature is the average of the temperature at the top and bottom of the test specimen. The temperature controller is operated by feedback to a Digital PDP-11 microcomputer. The primary standard used in the apparatus is Pyroceram brand glass ceramic code 9609 manufactured by Norton Co.

This material has a very stable and well-known thermal

conductivity, within the range of liquid-saturated rocks, which is shown as a function of temperature in Fig. V-3.

Since this material is very expensive, is brittle and subject to

fracture if the axial stress is not applied carefully and uniformly, and has thermal conductivities somewhat higher than some rocks, secondary standards are generally used. The most useful of these standards is fired Lava. Lava is the trade name for an aluminum silicate manufactured by American Lava Corporation. This material is machinable in the unfired state but when fired to about llOO°C, becomes rock-hard and has thermal conductivity in the range of most rocks. For oil or gas saturated unconsolidated sands, shales, and other low conductivity materials, other lower conductivity standards such as pyrex glass may have to be used. Since thermal conductivity of the Vespel holders is finite, some accounting for heat flow in the holders must be made. Also, some radial heat flow does occur since the guard heater is maintained at the midpoint temperature of the stack. Thus at the heat source end, a negative

45

thermal gradient between the stack and the surrounds exists and some heat flows out of the stack. At the sink end a positive thermal gradient exists and some heat flows into the stack from the surrounds. Efforts have been made to develop a variable-wound guard heater which would provide a match to the thermal gradients in the stack but this proved to be a very difficult control problem. Consequently, it was necessary to develop a "holder correction" to account for these possible errors.

4.0

Y

. k B

3.0

r *

.-5 .-* 0

3

0 C

0

0

=E

2.0

5

r

k

1.0

Temperature "C

Fig. V-3. Thermal conductivities of Pyroceram and fired Lava standards as functions of temperature. Fired Lava used as working standard for comparator thermal conductivity apparatus. Assuming a no-heat-flux outer boundary condition and further assuming linear heat flow through parallel and series resistors, heat flow calculations were made for the system (Gomaa, 1973).

Results of these calculations in terms of apparent conductivity (calculated

from experimental data using Eq. V-2) divided by conductivity of the standard used in the test versus the ratio of the true conductivty of the test specimen and the apparent conductivity, are shown as the dashed line of Fig. V-4. Although this gives the general form of the relationship, uncertainties in the assumptions made and the data used in these calculations made it necessary to run standardization tests.

These tests were run with materials of known thermal

conductivities. Results are the data points in Fig. V-4 through which a solid line has been

46

drawn.

This line may be represented by the following equation within the limits,

0.50 2 ha/hst

2

1.50:

htlha

where hi

=

[log(Aa/hst

-

+ 2.11]/1.84

0.37)

true thermal conductivity of test specimen; ha

=

from Eq. V-2; hst

=

(V-3)

=

apparent conductivity calculated

thermal conductivity of the standards. This relation is used as the "holder

correction" in all calculations of thermal conductivity.

A chart of the general form of Fig. V-4

is shown by Tye (1969)but it should be pointed out that each apparatus must be calibrated to obtain the appropriate holder correction.

1.4

3.37

r

1.2 1.o

0.8 qm

1 q

0.6 0.4 0.2

01 0

I

I

I

I

I

I

1

I

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

I

I

I

1.8

2.0

2.2

1 2.4

414 Fig. V-4. Experimental and calculated holder correction chart for comparator thermal conductivity apparatus.

Recent improvements in the thermal conductivity apparatus have been made to enhance the precision and ease of operation of the apparatus and minimize human errors inherent in the experimental process. Data acquisition is provided by a data logging system (Kaye Instruments Digilink-4). The system is interfaced with a microcomputer (Digital PDP-11)for continuous monitoring, storing and processing of the information transmitted by the data logger. Software has been developed for constant monitoring of the experiment, control of the apparatus and calculation of the final results. Table V-2 shows typical output from a test. The above improvements in the apparatus have been reported in detail by Mehos

(1986). In addition to implementing these improvements, Mehos also recalculated the holder correction curve.

He developed a cylindrical finite difference model in which he assumed

47 constant temperature at the outer cylindrical boundary (guard heater outer boundary). Results of his calculations again confirm the general form of the experimental curve.

The most

significant contributions of these calculations were the isotherms through the stack as shown in Fig. V-5.

It will be noted that the isotherms are quite flat (planar) within the diameter of the

thermocouple plates. The relatively high thermal conductivity of the thermocouple plates aids in assuring uniform radial temperature distributions through the standards and the test specimen. TABLE V-2 Computer output for thermal conductivity run with thermal comparator apparatus. Test specimen: Berea sandstone - dry. Snapshot requested for run #1 . . . TC 1 2 12.033 12.02 127.2 131.4 131.4 127.1 12.050 12:03 127.1 131.3 1 2 . 0 6 7 12:04 127.2 131.4 12.083 12:05 127.2 131.4 12.100 12:06

3 131.7 131.7 131.7 131.7 131.7

4 138.9 138.9 138.8 138.9 138.9

5 139.6 139.6 139.5 139.6 139.6

6 142.4 142.4 142.3 142.4 142.4

GH 135.0 135.0 135.0 135.0 135.0

12.117 12.133 12.150 12.167 12.183

12:07 12:08 12:09 12:lO 12:ll

127.1 127.2 127.1 127.1 127.1

131.3 131.4 131.4 131.3 131.4

131.7 131.7 131.7 131.7 131.7

138.8 138.9 138.9 138.9 138.9

139.5 139.6 139.5 139.6 139.6

142.3 142.3 142.3 142.3 142.4

135.0 135.0 135.0 135.0 135.0

12.200 12.217 12.233 12.250 12.267

12:12 12:13 12:14 12:15 12:16

127.1 127.1 127.2 127.2 127.2

131.4 131.4 131.4 131.4 131.4

131.7 131.7 131.7 131.8 131.7

138.9 138.8 138.9 138.9 138.9

139.6 139.6 139.6 139.6 139.6

142.4 142.4 142.4 142.4 142.4

135.0 135.0 135.1 135.1 135.1

12.283 12:17 1 2 . 3 0 0 12:18 12.317 12:19 12.333 12:20 12.350 12:21

127.1 127.1 127.1 127.1 127.1

131.4 131.4 131.4 131.4 131.4

131.7 131.7 131.7 131.7 131.7

138.8 138.8 138.9 138.8 138.8

139.5 139.5 139.6 139.5 139.5

142.3 142.3 142.4 142.3 142.3

135.0 135.0 135.1 135.0 135.0

12.367 12.383

127.1 127.1

131.4 131.3

131.7 131.7

138.9 138.8

139.5

142.4

135.1

12.22 12:23

Log started 10:07:21 Wed. Nov. 20 1985 Last snapshot taken at 12:22 ka/kstd = 0.995 kt/ka = 1.047 Average temperature of test specimen = 135.3"C Apparent thermal conductivity of specimen = 2.193 Wlm-K True thermal conductivity of specimen = 2.297 Wlm-K

48

L \

\

'

1

9 ?

r-

Q)

___.____---

+ 100.0

II

E = I

9

t-

Guard Heater Insulation

T,bottom = 90.0% Heat Sink

1

Fig. V-5. Calculated isotherms in comparator apparatus stack for test specimen and standards having same thermal conductivities. Anand (1 971) has assessed the possible errors in measurements with the comparator apparatus and has concluded that measured values of thermal conductivities should be well within k 5 % of the true values providing calibration has been carefully done.

1.2 Transient Methods Many transient methods for measuring thermal conductivity have been developed, the needle probe method probably being the best known. As with all transient methods, the needle probe suffers from contact resistance problems.

Inserting the probe into unconsolidated or

poorly consolidated sands not only disturbs the arrangement of the mineral grains but also tends to minimize grain to probe contact. A worse situation may arise in application to consolidated rocks, into which it is necessary to drill a hole to insert the probe, unless a means is used to establish good contact between the probe and the rock. Sass, et al., (1971) state that there is a

+ l o percent uncertainty in measurements with the needle probe and that may increase to k20 percent in some cases.

49

Two transient methods for measuring thermal conductivity have been developed by the author and his co-workers.

In these an effort has been made to overcome some of the

disadvantages of the needle probe method and yet preserve the principal advantage of rapid determinations. The main improvement in these methods is that the measuring devices are applied to the outer surface of the test specimen allowing application of stress to minimize contact resistance and to avoid disturbance of the rock structure or mineral grain arrangement. The principal goal of the work was to develop a method which could be used in testing rocks under conditions of elevated temperature and pressure and with variable fluid saturation. An additional goal was to apply the method to multi-properties tests run under simulated subsurface reservoir conditions. (i)-Sour-

The first transient method to be discussed, developed by Somerton

and Mosahebi (1967), employs a ring heat source mounted in a ceramic holder which is applied to the surface of the test specimen and is spring-loaded to assure good thermal contact. The 1.O cm diameter ring of lnconel alloy metal, 0.0175 cm in thickness, provides a sharp contact edge. Some variation in spring tension is necessary depending on the sample hardness and nature of its surface.

A spring-loaded thermocouple is mounted exactly in the center of the ring,

contacting the test specimen surface. Power is applied to the ring heater at a constant rate and the temperaturehime response of the thermocouple is recorded. ( 4 !leYdQPment of a4.utmL

For a point heat source located in an infinite,

homogeneous medium, the temperature rise at a radius (r) from the point is given by Carslaw and Jaeger (1959) as:

T

=

(q/4xrh) e r f ~ [ r / ( 4 a t ) ~ / ~ ] .

(V-4)

Mossahebi (1966) has shown that the identical solution is obtained for a ring heat source in an infinite, homogeneous medium for the temperature rise at the center of and in the same plane as the ring. For a semi-infinite system in which there is no heat flow across the boundary ( Z

=

0)

Crank (1956) gives:

T where T

=

=

(ql2xrh) erfc[rl(4at) l I 2 ] ,

temperature rise at center of ring at time 1, "C; q

(V-5)

=

heat liberated by ring heater, W;

r = radius of ring, m; h = thermal conductivity of test specimen, W/m-K; a = thermal diffusivity of test specimen, m%;

t = time, s; erfc = complementary error function.

For calculation of thermal conductivity from experimental data, Eq. V-5 is expressed in the following form:

50

h

=

(q/2xrAT) erfc(r/(4at) * I 2 ]

(V-6)

In Eq. V-6, (4) is the rate of heat output from the heater, corrected for heat losses (to be discussed later), and (AT) is the temperature rise above ambient at the center of the ring. One difficulty that arises in application of Eq. V-6 is that the argument of (erfc) contains ( a ) which is defined as ( a = WpCp). Thus an explicit solution for (1)is not possible. However, the nature of the function is such that if a reasonable value of ( a ) is used in the expression, the resultant value of ( h ) will be close to the correct value. To test this premise, typical values were taken as a

=

0.89~10-6m2/s, and t

=

60 and 120 s. The term (q/2xrAT)

was calculated for each time and the value of ( a ) was then varied 10-fold and corresponding values of

(A) were calculated. As may be seen in Fig. V-6, a 10-fold change in ( a ) results in

less than a two-fold change in (h).

I

I

I

I

I

I

I

I

1

1

1.20 Y

LL

2.00

0 I . c c

L I

€

3 I

. .-> w .-

1.00

3

z

1.50

I

e,

0.75

CI

0

3

= 8C

f

2 w .-

e,

0

1.00

3

U

0.50

dl /I

; 0.50

1

-

0

Ul

a1

0

E

01

=I

c

0.25

21

I-

21

I-

$1 I

0.40

I

I

0.60

I

I

I

Q,

r

I

I

1.0

2.0

I

1

4.0

0

x1o-6

Thermal Diffusivity - m*/Sec Fig. V-6. Effect of estimating diffusivity in calculating thermal conductivity from Eq. V-6 by ring source method.

A more serious problem in applying Eq. V-6 in the calculation of thermal conductivity is

the difficulty of evaluating heat losses. For example, in Fig. V-7 the experimental temperature rise record for Bandera sandstone is shown as the solid line.

Thermal conductivity and

51

diffusivity for this material have been determined previously (Somerton, 1958).

If heat

output from the ring is taken as equivalent to the power input to the ring, then:

Q

where Q

=

=

V I /4.18

(V-7)

heater output, watts; V

heater voltage, volts; I

=

=

heater current, amps.

Substituting this expression into Eq. V-5 and then inserting known values of V, I, r, h, and a, the following relation between temperature and time is obtained:

AT = 40.6 erf~[2.55/1'/~]

I

I

I

I

Bandera Sandstone Diameter 7.0 cm Length 10.0 cm

360 Experimental

270

Calculated Considering Heat Loss

2 Q,

f i=

180

/

90 / /

0

10

Calculated from Equation V-8

20

30

40

50

60

Temperature, O C Fig. V-7. Experimental and calculated temperatureltime relation for heating by ring source device.

Values calculated from Eq. V-8 are plotted in Fig. V-7 as the dashed line, compared with the experimental values shown as the solid line. The difference between the two curves is due to

52

heat losses from the system. Heat losses from the system are primarily conduction losses to the ceramic ring holder and to the copper lead wire. Convection and radiation losses from the heated surfaces are considered to be small. Mossahebi (1966) has evaluated heat losses in an approximate manner by using a corrected value of (4)obtained from the broken curve in Fig. V-7. Although good agreement between theory and experiment can be obtained, the uncertainty of heat loss calculations led to the development of ring probe calibration techniques which are described in the following section.

04 Probe calibratinn. Approximate heat loss determinations discussed above gave good agreement between theoretical and experimental temperature rise but heat loss depends to some degree upon the thermal conductivity (more properly, thermal diffusivity) of the test specimen.

Therefore it was decided that direct calibration of the probe would be a more

reasonable procedure for interpretation of ring heat source data. Tests were made on five materials of known thermal conductivity. Conductivities of these materials were measured by use of the steady-state apparatus discussed previously

I

:

'

I

I

4 -

' .6

I

I

::

2.5 3.2 4.0

5-

31

I

I

I

l

l

I

1.0 Cm Ring 0.50 Watt Input

120

I

0.8 1.0

I

1.5

I

I

2.0

3.0

I

I

1

4.0 5.06.0

I

8.0 10.

Thermal Conductivity,W/m- K Fig. V-8. Calibration chart for ring source thermal conductivity apparatus for various test specimen sizes.

53 (Somerton, 1958). The temperature rises with the ring source probe at 0.50 W input to the heater for several times (30, 60, 90, and 120 s) were plotted against known thermal conductivities of the five materials on log-log coordinates as shown in Fig. V-8. Good straightline relations result. In using the calibration chart of Fig. V-8, consideration must be given to which time line should be selected for a given test specimen. Actually if the test specimen is large enough (at least five-times the ring diameter in all dimensions), the same value of conductivity should be obtained for all four times. For smaller test specimens and larger times, one of the conditions for which the derivation of Eq. V-5 was made may be violated. That is, temperature at an outer boundary of the test specimen may begin to rise and therefore the condition of a semi-infinite medium would no longer be valid. To test the effect of specimen size on maximum time of validity of assumed boundary conditions, a sample of material of known conductivity, seven ring diameters in minimum dimension, was progressively reduced in size and for each size, ring probe measurements were made. The Table V-3 shows the maximum time for which the boundary conditions were valid for each specimen size. TABLE V-3 Transient time limit for various size test sDecimens.

Ma' 1.9 2.85 3.1 5.1

ring diameters ring diameters ring diameters ring diameters

-.

30 sec. 90 sec. 120 sec. 180 sec.

Additional tests were run to determine the times at which the boundary temperatures began to rise above ambient temperature. These measurements were made by means of a thermocouple placed in a droplet of mercury in a small indentation on the outer boundary of the specimens. Results are shown in Table V-4. TABLE V-4 Times for temperature to rise at boundary for various test SDecimen sizes.

1.9 2.85 3.1 5.1

ring diameters ring diameters ring diameters rina diameters

P 40 sec 80 sec 110 sec 140 sec

54

From the above tests the restrictions on selection of a time line have been noted on the calibration chart. (c) F x D e r i m W l m e t w A drawing of the ring heat source probe is shown in Fig. V-9 and the complete apparatus used in conjunction with the ring probe is shown schematically in Fig. V-10.

The important components, in addition to the ring probe, include a spring-loaded

mThermocouFe/ ,//,////,

Thermocouple Spring Loader

Loading Frame

Fig. V-9. Cross-sectional diagram of ring source probe.

Voltage Stabilizer

Step Down Transformer

Variable Transformer

I

Voltmeter

Ammeter

1

@ Wattmeter 01 It

Cold Junction

I

L'l'

Insulation

I1 I i 1

Sample

Fig. V-1 0. Schematic diagram of ring source probe apparatus.

55

test specimen holder, means of controlling and measuring electric power input to the heater, and a recorder for the thermocouple output. The ring heater, made of lnconel alloy (7 percent Fe, 13 percent Cr. and 80 percent Ni). is 0.986 cm OD, 0.0175 cm in thickness and 0.112 cm wide. Electrical leads to the heater are 18-gauge copper wire spot-welded to the inside of the ring. The thermocouple is made of 30-gauge copper and constantan wire. Commercial mineral-insulated, stainless steel-sheathed thermocouple material is used. A small, well-rounded thermocouple bead was formed at the end of the sheathing and was insulated from the sheathing by means of a small amount of technical "G" copper oxide cement. Both the heater ring and the thermocouple are mounted in a ceramic holder 2 cm in diameter and 10 cm long. The ceramic holder is made of Lava, an aluminum silicate which is machinable in an unfired state but has a stone-like hardness after firing. The ring heater fits into a groove in the end of the Lava holder so that it protrudes 0.056 cm to assure intimate contact with the test specimen. The thermocouple is located in the center of the ring through an axial hole in the center of the holder and is independently spring-loaded so that it contacts the test specimen surface in the same plane as the ring heater. Power is supplied to the ring heater through a 115-volt voltage stabilizer, a 5-volt constant output transformer, and a variable output transformer. A VAW meter is used in the circuit to measure and permit control of power input to the heater. Good thermal contact between the test specimen and the ring heater is obtained by use of a spring-loaded holder. Rock specimens of almost any size or shape may be tested providing parallel flats approximately 4 cm in diameter are cut on opposite sides of the test specimen. The sample is placed in the spring-loaded holder against the flats and a standard spring load of approximately 10 Kg is applied. This magnitude of load was found to give reproducible results. The thermocouple is connected to a recording potentiometer with a standard reference junction.

In running a test, the recorder is started and the thermocouple reading is checked

against room temperature. When the reading is stabilized, constant power is applied to the ring heater and the temperature rise is recorded as shown in Fig. V-7. The run is discontinued after about 180 seconds. The calibration chart shown in Fig. V-8 is then used to evaluate thermal conductivity. (d) m

r deve-

Although the ring heat source probe method was designed

primarily for measurements on dry rock samples at atmospheric pressure conditions and at temperatures only a few degrees above room temperature, these are not limiting conditions. Because of the short period of time required to run the test and the small temperature rise during the test, measurements can be made on liquid-saturated test specimens with minimum disturbance of the fluid distribution.

Tests could also be run on test specimens at elevated

steady-state temperatures by recording the differential temperature resulting from superimposed transient heating from the ring.

For these tests and for tests at elevated

pressures, elaboration of the present equipment would be required.

56

(ii) Disc Heat Sour-

Gomaa (1972) developed a transient method of measuring

thermal conductivity based on a transient heat flow solution by Selim. et al., (1963) for a disc heat source.

This solution gives temperature rise in a semi-infinite medium, which was

initially at a uniform temperature and then was heated by a disc source with heat applied only to a finite part of the free surface, the rest of the surface being insulated. The temperature rise in the medium at the center of the disc heater can be obtained from this solution by setting R and Z

=

0, the result being expressed in dimensionless form as follows:

T'

=

h [T(O,O,t)

where T'

=

-

Ti]/ a qeff

=

erfc(a2/4at)

dimensionless temperature; T(O,O,t)

T I = initial temperature of the medium,

conductivity, W/m-K; a

=

"C;a

= =

+ (4at/na2)[1 - exp(a2/4at)] temperature at R

=

0

(V-9)

0, Z = 0, and time t,

radius of the disc heater, m; h

=

=

"C;

thermal

thermal diffusivity, m?/s; qeff = effective heat flux to the medium

from the disc heater, W/m2. From a time record of temperature rise in the medium at the center of the disc heater and known values of heat flux and radius of the disc heater, both thermal conductivity and thermal diffusivity can be determined.

However, a trial and error solution is required which

makes this method of limited value. Although the disc heat source method described above gave satisfactory results, the trial and error solution was inconvenient and the experimental apparatus was complex and did not lend itself very well to application in a multi-properties apparatus.

Hirsh (1973) and Nguyen

(1974) developed a simplified version of the disc heat source method using a small printedcircuit type disc heater. After extensive investigation by Hirsh, a Thermofoil HK-3946 heater was selected for the tests.

This heater has a radius of 0.635 cm (0.25 in) and weighs only

0.045 g. It may be cemented on either a flat or curved test specimen surface. A thermocouple

is cemented between the heater and the test specimen in the exact center of the heater. A second thermocouple is cemented onto the test specimen surface far enough away so as not to be affected by the heater. This thermocouple is connected to the first thermocouple to give a differential temperature reading. If the heater and thermocouples are applied to the outer surface of a cylindrical test specimen, the assembly may be jacketed with shrink-fit tubing. The assembly is shown in Fig. V - I 1 and a schematic diagram of the entire apparatus is shown in Fig. V-12. In running a test, the test specimen assembly is mounted in the pressure vessel and the pressure and temperature are stabilized at the desired values. The voltage regulator is adjusted to give a constant wattage of 1, 2. or 3 watts (heater is limited to 5 watts).

The power source

57 Heater Leads

7-

Thermocouple Leads

Fig. V-1 1 . Test specimen assembly for paste-on disc heater thermal conductivity apparatus. Cell Heater

Fig. V-12. Schematic diagram of paste-on disc heat source thermal conductivity apparatus.

and differential temperature recorder are started at the same time. The test takes about 50 seconds to run before boundary conditions are violated. Temperature response curves for three different power inputs are shown for Berea sandstone in Fig. V-13. Thermal conductivity and thermal diffusivity are determined from the test results using a curve-fitting technique.

The transient heat flow solution by Selirn, et al. (1969) may be

expressed as:

T'(O,O,.r)

where T' = ( h AT/a q) )i =

=

=

erfc(l/.r)1/2 + ( . r / j ~ ) ~[l' ~- e x p ( - l / r ) ]

dimensionless temperature rise; 7

thermal conductivity, W/m-K; AT = temperature rise,

flux, W/m2; t

=

=

(V-10)

(4a Va2)

"C;a

dimensionless time;

=

= heater radius, m; q = heat

time, s; a = thermal diffusivity, m2/s.

A plot of Eq. V-10 in terms of dimensionless temperature rise and dimensionless time is shown in Fig. V-14.

The experimental time-temperature plot is superimposed onto the

100 80

60 3.05 Watts

40 0 0

I

20

.-%

1.05 Watts

K

g

10

c)

2

8

Q

6

+

4

0

k

Berea Sandstone Diameter - 5.08 cm

2

I

1

2

I

i

4

6

l

l

8 10

I

I

1

20

40

60

1

100

Time - Seconds Fig. V-13. Time-temperature response for paste-on heat source test on Berea sandstone with power input as the parameter.

59

i 2.0 2 a

c

l5i

1.0 0.8

,, 0.6

I-

u) Q)

F 0.4

.-0v) E

Q)

0.2 0.1 L 0.1

I

0.2

I

I

I

1

0.4 0.60.8 1.0

I

I

I

1

1

2.0 3.0 4.0 6.0 8.0 10. Dimensionless Time T

-

Fig. V-14. Solution of Eq. V-10 in terms of dimensionless temperature versus dimensionless time.

4.0

*

100 80 0 60 0

I-

2.0 ?!

z

L

E 0)

1.0 0.8 9) I0.6 $

40

20

Q

0.4

C

0.2 0 0.1

.-

u)

10

0.2

0.4 0.60.81.0

0.1 2.0 3.04.0 6.08.010.

Dimensionless Time -

T

Fig. V-15. Curve matching of experimental time-temperature response with theoretical solution of Fig. V-14.

.-E

n

60

dimensionless solution and is shifted until a match is obtained, as shown in Fig. V-15. Picking any point on the matched curves gives the solutions for thermal conductivity and thermal diffusivity from:

h

=

T' a 4/AT

(V-11)

a= T a2141

(V-12)

Comparison of results of thermal conductivity measurements on dry Berea sandstone at elevated temperatures and pressures for transient and steady-state tests is shown in Fig. V-16. It is apparent that transient values are high, probably because heat losses from the heater were

not taken into account. Further work is needed to account for heat losses. This will need to be done for each experimental setup.

I

Y

4.0

3

I

I

I

c _

0

0

3

u C

8 -

2

I

I

f

Transient Tests

. * __-. - .*.-*

"

100°C

I

1-

L

v

-O-----

/---

.->

1

I

-

& ._.

2.0 -

.-. -&. - .- .-

-.

Steady State Tests

1.0

-

025"C -.4 100"c -

Berea Sandstone (Dry)

tl

S

O

1

I

I

1

I

1

1

.

Fig. V-16. Comparison of transient and steady-state determined thermal conductivities as function of confining stress for Berea sandstone at two temperatures.

2 EFFECTS OF ROCK / FLUID PROPERTIES ON THERMAL CONDUCTIVITY

Thermal conductivities of porous rocks are functions of density, porosity, grain size and shape, cementation, mineral composition and nature of the saturating fluids. With this many variables, it is difficult to quantify the effects of each property on thermal conductivity. Several of these properties are interrelated.

For example, porosity and bulk density are

certainly related to thermal conductivity but in addition to this relationship, density is also related to mineral composition and this in turn is related to thermal conductivity.

The

61 pertinent variable here would be grain density rather than bulk density, as long as porosity is included in the correlation. Grain size (and size distribution), grain shape, and degree of cementation are difficult to quantify but these properties are related to other measurable properties such as permeability and electrical resistivity factor. Tables B-1 through B-9 in Appendix B give thermal conductivity values for several suites of rocks, along with other physical properties. These thermal conductivity data were obtained with the steady-state apparatus described earlier.

Discussion of these results and

their relationship to other physical properties will be divided into consolidated rocks and unconsolidated sands. 2.1 Anand (1971) has investigated the effects of a number of physical properties on thermal conductivities of several dry sandstone samples. He took additional data from Zierfuss and Van der Vliet (1956) to develop correlations.

For example, Fig. V-17 shows a plot of thermal

conductivities versus bulk densities of these rocks. Applying regression analysis to these data, the following equation represents the best fit to the data:

)i =

where

)i

pb419.56~10-3

(V-13)

= thermal conductivity, W/m-K; pb = bulk density, glcms. Note that the data points

for "very fine-grained" test specimens gave generally lower thermal conductivity values. Although Anand did not relate thermal conductivity to porosity directly, he found excellent agreement between bulk density and fractional porosity for the rock samples tested so that the equality,

(V-14)

may be substituted into Eq. V-12 to obtain,

)i

p

5.16(1 - @)4.

(V-15)

To improve the correlation, it is necessary to introduce other variables. The electrical formation resistivity factor may be included in the correlation to account for differences in structural characteristics of rocks having the same porosities. Figure V-18 shows a plot of porosity versus the product of thermal conductivity and formation resistivity factor (F). The best fit equation for these data, with a squared multiple regression coefficient of 0.896, is:

62

4.50

Zierfuss (1956) Data

.

4.00

2.50

Fine Grained Very Fine Grained

2.25

Anand (1971) Data Bandera Berea

A 0

3.50

2.00

Y I

x

7

-

y

SSNo.2 SSNo.3

0

E

F 1.75

3.00

.-z > .-c

I

.-z ;. L

1.50

0

2.50

3 ‘0

C

0

0

-m

E +

!+

0

1.25

aJ c

0 -

E

2.00

z

1.00

f

1.50 0.75

1.oo

/

/

‘ .. /

0.50

1

/

0.50 I

1 . 6 ~ 1 0 3 1.8

I

I

I

1

2.0

2.2

2.4

2.6

Bulk Density

- kglm3

Fig. V-17. Correlation of thermal conductivity with bulk density for a variety of sandstones.

log h

where h

=

=

[475 exp(-’2.3q)]/F

thermal conductivity, W/m-K; F

(V-16)

=

formation resistivity factor.

If permeability is also included in the correlation, multiple regression analysis yields the following relationship between the several pertinent variables and thermal conductivity:

h

where h

=

=

0.60~10-3pb - 5.52 41 + 0.92 ko.10 + 0.22 F

thermal conductivity, W/m-K; k

=

permeability, md.

-

0.054

(V-17)

63

100 --

0

-

-

-

Y

k

s

-

z

c

m

-

..-2

-

0

U 2.

..-> c

LT C

\\

Zierfuss (1956) Data

o

Anand (1971) Data SS No. 3

-

0 Berea 0 SS No. 1

-

A Bandera 0 SS No. 2 V Boise

-

-

a

-

c 0

U

m

-

2.

-- 10 ..-In -

\

@

u)

-

-

r

I-

1.0

m

E0

U

-

X

.-2. 5 c

.4-

0

:

-

\

-

- 1.0 C 0 -0

-

\

I

I

I

0.10

0.20

0.30

E

-

\

\

0

C

-

0

Q)

Q)

T

.-0

0

;

L

- -

C

-

I

z 0

U

\

-

r c ..->

5

c

-

X

L

G

10 -

5 U

F

\ \

0

?

-

\

u) u) Q)

0

0

-

\

z

I

r -

\ a \

0.40

Fig. V-18. Correlation of the product of thermal conductivity and formation resistivity factor with porosity for a variety of sandstones.

In British units, the above coefficients become:

h = 0.346~10-3pb - 3.20 @

+

0.530 ko.10

+

0.013 F - 0.031

(V-17a)

where h = thermal conductivity. Btulhr-ft-"F; other terms as above. The agreement between experimental and calculated values of thermal conductivity, using Eq. V-17, is shown in Fig. V-19.

The squared multiple regression coefficient for the

equation is 0.902 with a standard deviation of 0.240 for the thermal conductivity range of 0.70 to 3.8 W/m-K.

64

4.0

I

I

I

I

0 Zierfuss (1956) Data Anand (1971) Data

Y I

c

/

E

2I

3.0

.-* c

.-> .3-

0 3

U

6

-

2.0

2f

c l-

/$'

P

f

1.0

u)

m

?

- 0.4

/

g

I

0

1

I

1.o

2.0 3.0 Predicted Thermal Conductivity - Wlm-K

3

u)

2

I 0 4.0

Fig. V-19. Agreement between measured thermal conductivities of a variety of sandstones and values calculated from Eq. V-17. Solid diagonal line represents perfect agreement.

One limitation in the above relations is that thermal conductivity of the rock solids was not included in the correlation.

This is a quantity that is diffucult to evaluate.

Quartz has the

highest thermal conductivity of common rock-forming minerals and, consequently, the quartz content generally controls the rock solids conductivity. In modeling work to be discussed later, thermal conductivity of the rock solids was found to be second in importance only to porosity in determining thermal conductivity of the rock. A method of estimating rock solids conductivities from a mineral analysis and application of these values in correlations for unconsolidated sands will be presented in the next section of this chapter. Another physical property that might be expected to correlate well with thermal conductivity is sonic velocity.

Goss, et. al., (1975) included compressional wave velocity with

porosity in their correlation of Imperial Valley core data, obtaining agreement within 10 percent of measured values. Most of their rock samples had low porosities (4-16 percent) and the correlation failed for higher porosity rocks. Sahnine (1979) ran similar tests on a suite of dry and brine-saturated Mid-continent sandstones and siltstones.

Results showed poor

65

correlation when porosity and compressional wave velocities only were included in the regression analysis.

When formation resistivity factor was included

in the analysis, the

correlation was greatly improved for both dry and brine saturated rock samples. The multiple regression coefficient for dry samples was 0.871 and for brine saturated rock samples was 0.967.

In both cases the order of significance of parameters included in the correlation was:

porosity, formation resistivity factor, and compressional wave velocity, the latter being much less significant than the two former parameters. The average error between measured and calculated values of thermal conductivity based on Sahnine's correlation was 11 percent. Using the same data but applying the Goss. et. al. (1975) correlating equation, the average error more than doubled.

It seems reasonable to

conclude that although for a given suite of rocks, it is possible to develop correlations that permit predictions of thermal conductivities within about 10 percent of measured values, general correlations have not yet been achieved. One or more correlating parameters have been overlooked; one of these parameters must certainly be thermal conductivity of the rock solids. 2.2

Measurement of thermal conductivity of unconsolidated sands with the steady-state apparatus required modifications in the top sealing/loading plate for the sample holder. The 0ring seal on the flat holder surface limits freedom of vertical movement as the unconsolidated material compresses under applied stress. A new sealing disc was used which has the O-ring mounted on its outer circumference allowing freedom of vertical movement of the disc and at the same time providing a fluid seal. This modification is shown in Fig. V-20. Keese (1973) and Chu (1973) ran extensive tests on unconsolidated quartz sands, extracted oil sands, and oil sands containing original fluids.

Measured thermal conductivities

and other physical properties of these sands are given in Tables 8-3

-

8-5 in Appendix 8.

Thermal conductivity values for quartz sands are plotted in Fig. V-21 against porosity for both

dry and brine-saturated test specimens. The data are segregated according to grain-size ranges to give some measure of the importance of this variable.

The solid lines represent the

relationship predicted by Krupiczka's (1967) model and the broken lines represent the relationships predicted by Kuni and Smith's (1960) model.

Both of these models will be

discussed in a later section. The increase in thermal conductivity with decrease in porosity is clearly shown in Fig. V-21 for dry quartz sands. The effect of grain size on conductivity is much less clear, although

there is a general tendency for coarser sands of equal porosities to have higher thermal conductivities.

The Krupiczka model fits the experimental data quite well with the Kuni and

Smith model not showing as good a match. The data for brine-saturated test specimens plotted in Fig. V-21 show larger scatter that the data for dry sands. The group of six data points with low conductivities and high porosity values was for crushed Del Monte sand, whereas the other

66 Too Heater

t

Variable Clearance

Fig. V-20. Thermal conductivity test specimen holder modified for unconsolidatedsand. data were for natural Ottawa sands. Since the quartz content of the two sands is nearly the same the lower conductivity of the crushed sand may be attributed to its greater angularity and, probably, reduced grain-to-grain contacts.

I t is not clear, however, why the thermal

conductivity values for the dry Del Monte sands were not also low. The Krupiczka model

CUWE

parallels the trend of the Ottawa sand data, although the Kuni and Smith model curve seems to fit the combined data better. Data for the extracted oil sands are plotted against porosity in Fig. V-22 for dry (airsaturated), petroleum solvent-saturated, and brine-saturated test specimens.

The Krupiczka

relations for the three saturations are plotted on the figure. The effects of porosity and grain size on thermal conductivity are about the same as for quartz sands. The Krupiczka model was compared with the experimental data by using values of solids conductivity ( h s ) for quartz sand of 8.65 Wlm-K and a value of 4.76 W/m-K for oil sands. However, Horai (1971) gives a value of 7.70 W/m-K for quartz as shown in Table V - I . Mineral analysis of the oil sand given in Table V-5 was combined with the mineral conductivity values shown in Table V-1 to obtain a value of solids conductivity for the oil sands of 4.45 W/m-K.

Using these values for

(AS) and modifying the empirical constants in Krupiczka's

model, the dashed lines shown in Figs. V-21 and V-22 result. Ozbek (1 981) has considered the various methods of estimating solids thermal

67

x 0.607 mm 0 0.368 mm

5.00

A

A

0.368-0.445 mm Mix 0.152 mm 0.152-0.246 mm Mix

0 0.0737 mm 0 0.0737-0.0991 m m Mix

4.50

4.00

-

-Kunii & Smith (1960) Krupiczka (1967) A, = 8.65 A, = 7.70-

v

Y

'

-

3.50

-

\ \

--- Krupiczka Mod.

X

2.00

P I.

5 1.60

I 2

m .-c>; .-> c

1.20

0.80 0.40

c +

0.20

Fig. V-21. Experimentally determined thermal conductivities of quartz sand dry and brine saturated compared with model predictions. conductivity for a mineral assemblage. Figure V-23 shows the effect of the averaging method on the estimation.

Quartz, having the highest conductivity of common rock-forming minerals,

dominates thermal behavoir of the assemblage. The weighted arithmetic average, which gives the highest value for the mineral assemblage, seems fully justified and is recommended when a value of solids conductivity is needed.

In cases where a complete mineral analysis is not

available but the quartz content can be estimated, the solids thermal conductivity can be approximated by the following equation:

hs

where hs

=

=

7.70 Q + 2.85(1

-

0)

solids thermal conductivity, W/m-K; Q

(V-18)

=

fractional quartz content.

68

- 1.5 2.4

-Krupiczka (1967) A,

o,;

= 4.76

Brine Saturated A, = 0.650

1.2

c

0.5 I

0.4

o 0.58 - 0.84 mm 00.30 - 0.42 mm I h0.13 - 0.21 mm

I

nsC

O

'

0.L

a

n

I

-I

Air Saturated A, = 0.029

I

I

1

I

0.32

0.36

0.40

0.44

lo

Porosity

Fig. V-22. Measured thermal conductivities of extracted oil sands dry, solvent-saturated and brine-saturated compared with predictions from Krupiczka's (1967) model.

TABLE V-5 Estimation of oil sand solids conductivity from mineral analysis in W/m-K. Mineral Quartz Orthoclase Plagioclase KaoliniteSericite Chlorite Hornblende Sphene Epidote Others Total

h

nx h

0.34 0.01 0.21

7.70 2.32 2.15

2.618 0.023 0.452

0.25 0.07 0.04 0.02 0.02

2.77 4.91 3.08 2.34 2.61 2.34

0.693 0.344 0.123 0.047 0.052

Fraction (n)

p94 1. o o

Q994 4.446

69

8.0

P 1 7.0 E

s j,

x

6.0

v)

.z

5 5.0

v) X 0

0

a 4.0 r 0

.-r > c

.= 3.0 0

a c

U

0"-

Curve Curve Curve

2.0

m

E

2

@) Arithmetic Average @ Geometric Average @ Harmonic Average

1.0

k

0

20 40 60 80 100 Volumetric Quartz Content (Percent)

0

Fig. V-23. Estimated thermal conductivities of rock solids as function of quartz content using several averaging methods.

Figure V-24 shows thermal conductivities of oil sands as they were received, containing original fluids, plotted against fractional interstitial brine saturation. The data have been segregated into the following porosity ranges: 0.27-0.29, 0.30-0.32, 0.33-0.35, and 0.360.38. The solid lines shown on the figure are calculated from the following equation for average

porosities of 0.28, 0.31. 0.34, and 0.37:

Asw

where =

&W

=

=

1.27 - 2.25$

+

0.39AsS~''~

(V-19)

thermal conductivity at brine saturation Sw, W/m-K; @=fractional porosity; AS

thermal conductivity of rock solids, W/m-K; SW = fractional water saturation. Equation V-19 was derived from multiple regression analysis of data shown in Table B-4

70

1.50 !+

2.5

I

Kern River Oil Sands

Y

L

I

f

F3

E 2.0

z

1.00

s

CI

‘5 1.5

.-c 0

.27-.29 0 .30-.32 A .33-.35 0 .36-.38 Solid Lines Calculated From: AS, = 1.27-2.25 (I) + 0.39 A, 0

3 ‘0

g

1.0

0

-

0.5

0

3

U

0.50 g 0

-Q E

s,’

al r

7 r. c ..-c>

Porosity Range

P)

I-

0

I

0

I

I

I

I

.10 .20 .30 .40 50

I

I

I

I

I

o

t

c

.60 .70 .80 .90 1.O

Brine Saturation Fig. V-24. Measured thermal conductivities of unextracted Kern River oil sands compared with predicted values based on Eq. V-19.

in Appendix B. For these analyses, combinations of variables were used including porosity, permeability, formation resistivity factor, median grain size, grain-size distribution and conductivity of rock solids and saturating fluids. Keese (1 973) showed that for unconsolidated sands, porosity and saturation of the wetting-phase fluid had overwhelming effects on thermal conductivity.

Grain-size distribution and formation resistivity factor had small effects on

thermal conductivity and other variables had only moderate effects on this quantity. Results showed a definite relationship between porosity and grain size and between porosity and permeability.

It was concluded that porosity was an adequate correlating parameter for the

matrix structure of unconsolidated sands.

Thus it appears that the pertinent variables for a

single fluid-saturated unconsolidated sand are porosity and conductivities of the rock solids and saturating fluid. The effects of multi-fluid saturation on thermal conductivity will be discussed in the next section.

2.3 Multi-Fluid Satu r

’

m

When more that one fluid is present in a porous rock, conductivity of the wetting-phase fluid will have the greatest effect on thermal conductivity of a multi-fluid saturated rock. Since water is generally the wetting fluid in subsurface reservoir rocks and furthermore, since water has the highest thermal conductivity of any fluid which might occupy the pore spaces in reservoir rocks, water saturation will have a disproportionately large effect on conductivity of the fluid-saturated rock. Thus, application of the mixing laws to obtain an effective composite value of fluid conductivity is probably not valid (see further in Chapter VI).

71

For the thermal conductivity values of unextracted oil sands reported in Table B- 5 and plotted against interstitial brine saturation in Fig. V-24, Chu (1 973) added the additional variable of multi-fluid saturation in the regression analysis. Results of his analysis showed that oil saturation. when included with water saturation in the correlation, had little effect on thermal conductivity. In fact in the range of oil saturations for the unextracted oil sands (0 to

0.67),it appeared to make no appreciable difference whether the non-wetting phase was oil or air. It should be noted that the total liquid saturation for the unextracted oil sands was less than unity in all cases, the remainder being air or gas saturation.

The correlation was improved

when water saturation was taken to a fractional power as given in Eq. V-19. Another approach to predicting thermal conductivity of multi-fluid saturated rocks is to use the Krupiczka (1967) model, referred to earlier, to estimate thermal conductivity at the two end points (SW = 1.0 and SW = 0.0) and then apply a modified form of Eq. V-19 for intermediate values. The Krupiczka equation is given as:

where k = effective conductivity of single-fluid saturated rock; hf = thermal conductivily of saturating fluid; constant

=

XS

-0.057;

=

thermal conductivity of rock solids; A

=

0.362 - 0.650 log I$;B

=

$I = fractional porosity.

Solving for effective conductivity for full saturation with the wetting-fluid and for full saturation with the non-wetting fluid, effective conductivity for the multi-fluid saturated case may be calculated from: XSw = h(SW=O) + [h(SW=l.O) - X ( S W = O ) l

sw”2

(V-21)

Results of calculations for a quartz sand/brine/air system using Eqs. V-19. V-20, and V-21 are compared with experimental data in Fig. V-25. A slight reduction in the exponent in

Eq. V-21 from 0.50 to 0.47 gives a better fit with experimental data.

Increasing thermal

conductivity for quartz from 7.70 to 8.00 Wlm-K also improves the match. With these modifications, the maximum difference between calculated and experimental results is 7 percent. For multi-fluid saturated sandstones,

conductivity

increasingly important. For unconsolidated sands,

(SW) relationship.

of the rock solids becomes

(Ls)controls the slope of the (XSW) versus

For consolidated sandstones, the relative position of the curve on the

thermal conductivity axis depends heavily on the value of (b) as seen in Fig. V-26. Thus, in a correlation equation for consolidated sandstones, (XS) must appear in the first term as follows:

72

Quartz Sand

-

Porosity = 0.335 Median Grain Size 0.61mm Lines Calculated Eq. (V-19) A, = 7.70 Wim-K

-

F

Fig. V-25. Measured thermal conductivities of partially brine-saturated quartz sand compared with predicted values based on Eqs. V-19 and V-21

Note also that the exponent reduces further to (113) for consolidated sandstones. Although Fig. V26 shows that agreement between calculated and experimental values of thermal conductivity is

not as good as might be desired, it is within the limit of accuracy of the experimental apparatus (? 5 percent). Uncertainty in the values of

(Ls)could easily account for differences between

calculated and experimental values. Anand (1971) developed correlations to predict thermal conductivities of liquidsaturated sandstones using the following dimensionless groupings:

A = hsat/hdry;

B = (hliqlhair - 1);

c

=

[Q hliq/(l

-

Q) hdry]; D

=

hsat/hdry;

m

=

Archie's

cementation factor, where subscripts are self-explanatory. A value of (hdry) needed in the correlations may be estimated by use of Eq. V-18. A non-linear multiple regression computer program was applied to literature data (Zierfuss and Van der Vliet, 1956) resulting in the following best fit equation:

73 A

=

1.00

+ 0.30 B0.33 + 4.57

(V-23)

C0.48m D-4.30

or ksat = kdry I l . 0 0 + 0.30 Bo.33

Y

+ 4.57

(V-23a)

C0.48m D-4.30]

3.00 bI

5.00

I

o A Experimental Data Temperature - 91°C Pore Pressure - 0

0

0.20

0.40

0.60

0.80

1.00

Brine Saturation Fig. V-26. Measured thermal conductivities of three outcrop sandstones compared with calculated values based on Eq. V-22.

The agreement between literature values of thermal conductivity ratio (hsadhdry) and values calculated from Eq. V-22 is shown in Fig. V-27.

The solid line represents perfect

agreement between calculated and literature values and the broken lines show the limits of one standard deviation.

For the 52 literature data points used in the correlation, the standard

deviation was 0.179 for the range of (ksat/kdry) ratio values of 1.20 to 2.30.

The agreement

between literature values and calculated values of thermal conductivity was within 10 percent for 56 percent of the values and within 15 percent for 85 percent of the values. Experimental and calculated values of thermal conductivity from Anands work are plotted as open triangles in Fig. V-27. Eleven of the 14 data points are within one standard deviation, one point was out of the range of the correlation and two points showed differences greater than one standard deviation.

The low experimental point may be due to incomplete

saturation of the test specimen with a viscous silicone oil. Since the correlation equation is expressed in terms of dimensionless ratios, m (the exponent in Archie's equation relating porosity and formation resistivity factor) was used in the correlation rather than resistivity factor itself. The effect of (m) in Eq. V-23 is similar to the effect of (F) in Eq. V-17 in that increasing both values increases thermal conductivity.

74

2.E 0 Zierfuss (1956) Data

/

A Anand (1971) Data

2.4

/

/ 2.2

/

/

2.0 u)

i

-

^o 1.8 x

v)

x

1

1.6

One Standard Deviation

1.4

/

/

1.2

1.o

1.0

1.2

I

I

1

I

I

I

1.4

1.6

1.8

2.0

2.2

2.4

2.6

(XS/XD)Calc

Fig. V-27. Comparison of measured and calculated values of ratio of brinesaturated to dty thermal conductivity values for a variety of sandstones. Solid diagonal line represents perfect agreement.

Conductivity of the saturating liquid has the dominant effect on the thermal conductivity of liquid-saturated rocks.

It is difficult to assess the relative importance of the other

parameters in Eq. V-22 because of their complex interrelationships. In the case of saturation with two liquids or liquids and a gas, conductivity of the wetting phase fluid has the dominant effect on the thermal conductivity of the rockfluid system. Thus for water-wet sandstones, the value of liquid conductivity to be used in the correlation should be biased towards the value of thermal conductivity of water. A saturation-weighted arithmetic average thermal conductivity of the two fluid phases has been used for this purpose where:

hliq

=

S w hwf + S n w hnwf

(V-24)

However, since the earlier equations for liquid-saturated rocks were not intended for multifluid saturated rocks, the use of Eq. V-24 for this purpose is not generally recommended.

75

Further information on multi-fluid saturated rocks will be given in Chapter VI on modeling of thermal conductivity. 3 EFFECTS OF TEMPERATURE ON THERMAL CONDUCTIVITY Thermal conductivities of most materials which have crystalline structures decrease with increased temperature (Powell, et al., 1966). Theory indicates that thermal conductivity for these types of materials should vary with the reciprocal of temperature. In the case of mixed crystals or highly disordered crystals, thermal conductivity varies more slowly than T-1 and, in fact, may show a slight increase in thermal conductivity with temperature.

Thermal

conductivities of glasses and vitreous materials increase with temperature. Tikhomirov (1968) has developed a correlation equation based on experimental data for the prediction of the effects of temperature on thermal conductivities of rocks.

A plot of this

equation shows that moderate negative gradients of thermal conductivity with temperature will be predicted for high conductivity rocks, whereas small positive gradients will be predicted for low conductivity rocks. This agrees with theory and with experimental results to be presented later in this chapter. The effects of temperature on thermal conductivities of several sandstones were tested against Tikhomirov's correlation but the results were not very satisfactory. A new family of curves was developed, guided by Tikhomirov's correlation, but modified by the conductivitytemperature trends observed in our experimental work. The equation of this family of curves is as follows:

where l i ~= thermal conductivity at temperature T, Wlm-K: h20° 20°C, W/m-K: T

=

temperature, K = "C

+

=

thermal conductivity at

273".

A plot of Eq. V-25 based on even values of thermal conductivities at 20°C is shown as Fig. V-28. In British units Eq. V-25 becomes:

where

XT

=

thermal conductivity at temperature T, Btulhr-ft-"F: h680 = thermal conductivity

at 68"F, Btu/hr-ft-"F: T

=

temperature,

O R

= OF

+

460'.

76

100

~.

200

300

400

6.0

6.0

5.0

5.0

4.0

4.0

3.0

3.0

2.0

2.0

1 .o

1 .o

0

0

20

100

300

200 Temperature

400

- "C

Fig. V-28. Temperature correction chart for thermal conductivilies based on Eq. V-25 in SI units.

Calculations based on Eq. V-25a for even values of thermal conductivity at 68"F, are shown as Fig. V-29. Results of thermal conductivity measurements at several temperatures are plotted on the same figure. Although there is some scatter in the experimental data, the general agreement is quite good. Tikhomirov's correlation was developed for dry rocks but the present correlation seems to be equally valid for liquid saturated sandstones. Unusual thermal properties of some liquid saturants could cause some deviation of temperature behavior from that predicted by Eq. V-25. particularly for high porosity rocks.

In addition, phase changes of the fluid saturants

may result in discontinuities but this is a fluid-saturation effect rather than a temperature effect per se. For lower thermal conductivity rocks, the change in conductivity with temperature becomes nearly linear and a simplified equation may be used.

For conductivities below 2.5 W/

m-K (1.5 Btu/hr-ft-"F) the following equations are satisfactory:

2.30~10-3(T

h~ = 1200

-

hT

-1.75~1

= h68"

(T

-

-

293) x (h20° -1.38) 529)

X

(h68" - 0.80)

(V-26) (V-26a)

77

3.5

-

--

3.0

P

2c

2.5

Predicted Eq. V-2Sa Experimental STD APP Experimental HT APP

3

5

s 2.0 .-c > .-c 0

3

1.5 0

-

0

2k

1.0

c

+

0.5

Fig. V-29. Temperature correction chart for thermal conductivities based on Eq. V-25a in British units, compared with experimental values. 4 EFFECTS OF STRESS ON THERMAL CONDUCTIVITY Increasing effective stress on a rock would be expected to increase its thermal conductivity since increasing the stress improves the thermal contact between mineral grains and increases the density of the rock. In measuring thermal conductivity in the laboratory, part of the apparent increase in thermal conductivity with stress may actually be due to reduction in thermal contact resistance between heat source and sink and the temperature measuring devices and the test specimen itself. When good thermal contact is established, the change in thermal conductivity with added stress is generally small. These effects may be observed in Fig. V-30 for tests on three sandstones. Edmondson (1961) has reported increases in thermal conductivities of Berea, Bandera, and Boise sandstones to be 7.8, 9.5, and 12.3 percenV1000 psi (6.9 MPa), respectively, in the stress range of 900-3600 psi (6.2-24.8 MPa). These values are high compared to values reported by Woodside and Messmer (1961) which were 11.5 percenW1000 psi (6.9 MPa) in the stress range of 0-1000 psi and 2.5 percenV1000 psi in the range of 2000-4000 psi (13.8-27.6 MPa) for Berea sandstone. From Fig. V-30 an increase in thermal conductivity of only about 1.25 percent/l000 psi for Berea and about 2.0 percenW1000 psi for Boise is

78

Axial Stress, psi

0

i i 1.5 E

1.o

1000

1500 1.4

1 --

0

2.0

4.0 6.0 Axial Stress - MPa

8.0

10.

Fig. V-30. Effects of axial stress on thermal conductivities of three sandstones.

observed above an axial stress of about 500 psi. A relationship between bulk compressibility of porous rocks and change in thermal

conductivity with stress should be expected.

Bulk compressibilities of Berea, Bandera, and

Boise sandstones are reported by Lobree (1968) to be 0.066/GPa, 0.094/Gpa, and 0.131/GPa ( 0 . 4 5 4 ~ O-e/psi, 1 0.646x10-6/psi, and 0 . 9 0 ~ O-G/psi), 1 respectively.

Plotting these data

versus change in thermal conductivity with stress, a linear relationship is obtained as shown in Fig. V-31. The known effects of stress on other physical properties of rocks may be used to obtain an estimate of the magnitude of the effect of stress on thermal conductivity. To accomplish this, Eq. V-17 was differentiated with respect to stress. The derivatives of density, porosity, permeability, and formation resistivity factor with respect to stress have been evaluated by Dobrynin (1962) in terms of bulk compressibilities of rocks.

Since numerical magnitudes of

the bulk compressibilities of rocks are generally not known, numerical coefficients for high, medium, and low compressibility rocks may be substituted into the following general equation:

79

dud0

where d u d 0

=

2 . 5 ~ 1 0 -(A ~ pb @

+ B @ - C k0.'O + D F)

(V-27)

change in conductivity with stress, W/m-K-MPa; pb

=

fractional porosity; k

=

=

bulk density, kg/m3; @

permeability, md; F = formation resistivity factor; values of A. B, C

and D are given in Table V-6 for a range of rock compressibilities.

lo7

Bulk Compressibility, psi-' x 2 I

2.0

1

4

6

8

I

I

I

10 I 16

Boise SS

m

4

l4

- 12

n

5c

.-

v)

1.5 -

-

0

;

(1

10

8 0 T c

n

- 8

I

..< 1.0
1) spheroids, this factor is given by,

m:

M

prolak: M

=

(28 - sin28)/(2 tan0 sin%), where 0 = arcos(a),

=

(l/sinze)

-

(VI-23)

(cos%/2sin38) In[(l+ sinO)/(l- sine)], where 8 = arcos(l/a). (VI-24)

The factor

(p)

is a measure of the amount by which the fluid-saturated pores decrease

the thermal conductivity below the value ( i s ) . This can be seen by noting that for small values of porosity, Eq. VI-21 reduces to (Xlhs = 1-

p 9). The factor (p) is plotted in Fig. VI-22 for a

range of conductivity ratios and aspect ratios. For a fixed aspect ratio, conductivity ratio decreases.

For a fixed conductivity ratio,

(p) is

(P) increases

as the

a minimum for spherical

pores, for which it has a value of 3(1- b)/(2+ b). In the limiting cases of needle-like or cracklike pores,

(p) has the values of (1- b)(5+ b)/3(1+ b), and

Since the factor

(p)

(1- b ) ( l + 2b)/3b. respectively.

is very insensitive to the aspect ratio when (a > l ) , it is therefore more

convenient to model the pores as oblate spheroids, particularly when attempting to invert thermal conductivity values in order to infer aspect ratios. In this regard, it is worth pointing out that as the aspect ratio varies over the range ( O - w ) ,

the thermal conductivities predicted

by using Eqs. VI-21, VI-22, and VI-23, cover the entire range allowed by the theoretical upper and lower bounds derived by Hashin and Shtrikman (1962) (see Fig. VI-23). A partial test of the validity of the Zimmerman model can be made by using it to predict,

for example, the conductivity of liquid-saturated rocks based on laboratory measurements made under air-saturated conditions. Thermal conductivity measurements are much easier to make on air-saturated rocks than on liquid-saturated rocks. Useful data for such a prediction can be found in the work of Woodside and Messmer (1961), who presented results of thermal conductivity measurements on six consolidated sandstones using air, water, and n-heptane as

107

Aspect Ratio, a Fig. VI-22. Average p factors for randomly-oriented spheroidal pores with different values of conductivity ratio (b). Zimmerman (1989).

1.0

c" 0.8 ..

> ..-> 4-

5

0.6

-u

c

0

U

0.4

a, N .-

-

;0.2 z

z 0

0

0.2

0.4 0.6 Porosity ,

+

0.8

1.0

Fig. VI-23. Normalized effective thermal conductivity as a function of porosity for one conductivity ratio over the full range of aspect values within the Hashin-Shtrikman (HS) bounds.

108

pore saturants. The porosities of these sandstones ranged from 0.03 to 0.59, and they each were composed of at least 85% quartz. To simplify the calculations, the value of ( h s ) is taken to be the value for quartz given by Woodside and Messmer as 8.4 W/m-K at a temperature of 300 K. This is stated by them to be an average value of the thermal conductivities in the three

crystallographic directions. given by Horai (1971).

It should be noted that this value is somewhat higher than that Thermal conductivities of the three saturants (air, water, and n-

heptane) at the same temperature are taken as 0.026, 0.63, and 0.13 W/m-K, respectively, corresponding to ( h f / h s )values of 0.0031, 0.075, and 0.016. For each of the sandstones, the air-saturated value of (hlhs) given by Woodside and Messmer was used along with values of (b) and (4) to solve for

(p) from

Eq. VI-21. Equations

VI-22 and VI-23 were then used to solve for (a) which is interpreted as an “average” aspect

ratio. Most of this inversion can be done in closed form, leaving only Eq. VI-23 to be solved numerically. Once the average aspect ratio (a) is solved for, the model equations were used to predict the effective thermal conductivity for other fluid saturants.

Figure VI-24 shows the

predicted thermal conductivity values to be relatively close to the measured values but in each case the values are somewhat lower, an average of 7.5% for the water-saturated samples and 16% lower for the heptane-saturated samples.

10

Data: Woodside 8 Messmer (1961) Saturant.

I

E U

0

I 2 3 4

5

0

2

Berkeley SS St. P e t e r s SS rensleep SS B e r e o SS TeapotSS

4 lpredicted

6

0.03 0 II 0.16 0.22 0.29

a

W/m-K

Fig. VI-24. Predicted values of thermal conductivity for several Iiqu id-saturated sedimentary rocks compared with measured values. Zimmerman (1989).

109

The Zimmerman model has also been used to predict thermal conductivities of lowporosity crystalline rocks, using results of measurements on five air-saturated and watersaturated granitic rocks reported by Scharli and Ryback (1984).

The rocks each had

substantial amounts of quartz, along with K-feldspar, plagioclase and biotite. Based on mineral volume fractions, Scharli and Ryback estimated a value of (hs) as lying between 3.03 and 3.44 W/m-K. This seemingly low value was used in the calculations along with pore-saturant values of 0.026 W/m-K for air and 0.63 W/m-K for water. Using the procedure described above and assuming the "thin-crack limit, the ratio of air-saturated conductivity to water-saturated conductivity can be predicted from Eqs. VI-21, VI-22, and VI-23 to be:

h a s l h w s = [(1-0.661@)(1+1 .37@)]/[(1+41.3@)(1-0.538@)]

(VI-25)

Figure VI-25 shows the prediction to be quite accurate, with an average error of less than 3%. Since the differences between air-saturated and water-saturated conductivities are about 30%, these predictions are non-trivial.

R y b a c h (19841

3.5

t

J"

t

3.0

t

2.5 2.5

3.0

Sample

Q,

I 2 3 5

0.0081 0.0079 0.0083 0.0078

6

0.0063

3.5

Fig. VI-25. Predicted thermal conductivity of water-saturated granitic rocks, based on "infinitely-thin crack" approximation, compared with measured values. Zimmerman (1989). Thermal conductivity measurements made on Casco granite by Walsh and Decker (1966) were also used to predict thermal conductivity using the Zimmerman model. These authors presented both compressibility and thermal conductivity data over a range of confining stress

110

from 0 to 100 MPa. Crack porosity can be estimated from the stress-strain curve as a function of confining stress and then used as input to the thermal conductivity model to predict the variation of effective thermal conductivity as a function of confining stresss. In order to focus on the effect of the closable cracks, the mineral phase plus any non-closable pores are treated as the “rock solids” and only the closable cracks are considered as the inclusion phase. The effective solids conductivity (hs) was estimated to be 3.23 Wlm-K, based on measured values of conductivity at high stress when all of the cracks are assumed to be closed. With the crack porosity ($c) known as a function of stress from compressibility measurements and using the value of (b = 0.008). corresponding to air saturation, Eqs. VI-21, VI-22, and VI-23 yield the following expression:

x

=

(VI-26)

3.23 [l-0.661$ ~ ( 0 ) ] / [ 1 + 4 1 . 3 $ c ( a ) I ,

where $c(o) is crack porosity at applied stress

=

a.

The crack porosity can be fit with an exponential function of stress as:

(V 1-27)

$c(o) = 0.0038 exp-O’*

Figure VI-26 shows the predicted values of thermal conductivity calculated by use of Eqs. VI-26 and VI-27 (solid curve) and experimental data of Walsh and Decker. There is seen to be close agreement between predicted and measured values over the entire range of stresses. The Zimmerman model is seen to be useful for prediction of thermal conductivities of a wide range of liquid-saturated rocks, providing that the appropriate aspect ratio of the particular rock can be estimated. An average effective aspect ratio may be estimated if, for example, effective conductivity of the air-saturated rock is known.

With this value, thermal

conductivities of the rock saturated with other fluids may be estimated.

Other means of

estimating the aspect ratio may include analyses from other physical measurements on the rock such as compressibility. The model seems to predict thermal conductivities of lower porosity rocks better than for higher porosity rocks.

4 SUMMARY

Although Mixing Law models have been widely used to estimate thermal conductivities of composite systems, their application to rocWfluid systems is not satisfactory for they fail to take into account the complex geometries of such systems. Empirical models are often used as an alternative and they are generally satisfactory for a given suite of rocks.

Extrapolation of

111

Y

3.2

I

E \

-

3

3.1

)r

+ ..-> c

0 3

3.0

-

73

Predicted Eqs. VI -26 8 2 7

2.9

E

J=

L D a t a : W a l s h &Decker (1966)

0

c

-

/

2.8

-

t-

0

Casco Granite

I 20

I 40

I 60

I

80

I( 0

Pressure (MPa) Fig. VI-26. Predicted thermal conductivity of Casco granite as a function of applied stress compared with measured values. Crack porosity was inferred from compressibility measurements. Zimmerman (1989).

empirical models to suites of rocks other than those used in developing the correlation equations may not be satisfactory. Reported herein are the results of efforts to develop theoretical models which might have general applicability to estimating thermal conductivities of all types of rock/fluid systems. The main difficulties in developing such models are the complexities of the geometries of the rock structure and of the distribution of fluids within the pore spaces.

Three cubic-packing

models have been developed, each an improvement over the previous model. The third model by Ghaffari shows good promise for use in predicting thermal conductivity of any rocWfluid system.

A set of charts is provided which makes it relatively easy to estimate thermal

conductivity knowing only a few commonly available properties of the system. Further testing is needed to confirm the general applicability of this model. A fourth model by Zimmerman (1989), an extension of the Maxwell model, shows promise particularly for low porosity rocks.

112

Chapter VII THERMAL DlFFUSlVlTY OF ROCKS Thermal diffusivities of rocklfluid systems need to be known for calculations involving transient heating or cooling operations. The basic equation governing heat transfer under transient conditions is given as:

V(hVT)

=

Cp pb LIT/&

where h = thermal conductivity; T

(VII-1)

=

temperature; Cp = heat capacity; pb = bulk density; t

=

time. If the medium through which heat is flowing is homogeneous and isotropic, Eq. VII-1 may be expressed as:

V2T

=

(Cp pblh) iIT/iIt

where a = lJCp pb

=

thermal diffusivity.

=

l / a aT/at

( v II- 2)

Thermal conductivity is independent of temperature only in the case of fairly low thermal conductivity materials.

Neither heat capacity nor bulk density is independent of

temperature. The consequence is that thermal diffusivity is a strong function of temperature. Thermal diffusivity may be measured by transient heat flow tests or it may be calculated from known values of thermal conductivity, heat capacity and bulk density. In the following, some methods of measuring thermal diffusivity developed by the author, his co-workers and others will be discussed and results of measurements will be compared with calculated values. 1 EXPERIMENTAL METHODS OF MEASUREMENT 1.1 In early work by Somerton and Boozer (1961), a method of measuring thermal diffusivity as a function of temperature was developed and tests were run on dry rocks in the temperature range of 150-10OO0C. The test method involved heating a cylindrical rock test specimen at a constant heating rate at the outside circumference and recording the temperature difference between the outer radius and the center of the test specimen.

113

In interpreting data from the above test method for materials in which diffusivity is a function of temperature, thermal conductivity cannot be removed from within the differential operator in Eq. VII-1. However, in the present case, since edge temperature is a linear function of time (constant heating rate) and differential temperature between edge and center of the test specimen is small, diffusivity can be expressed as a function of time:

(Vll-3)

aTiat = a(t) W T . With the following transform:

(V II-4)

Eq. Vll-3 reduces to:

aTia6 = V ~ T

(Vll-5)

Boozer (1959) has obtained a solution for Eq. Vll-5, with appropriate initial and boundary conditions, which will not be repeated here. The solution contains two transient terms, one to account for initial startup of the test and one to account for temperature dependence of diffusivity, both of which are minor in magnitude when considering the low heating rate applied (1.O"C/min).

Consequently, as reported in Chapter 111, thermal diffusivity

may be calculated from the following simplified relation:

a = a2h14AT

where

Q =

thermal diffusivity, W/m-K; a

(Vll-6)

=

radius of the test specimen, m ; h

=

rate of

temperature rise at outer edge, Kls; AT = temperature differential between outer edge and center of test specimen, K. Details of the experimental apparatus and procedure are given elsewhere (Somerton and Boozer, 1961). It is important to note that, due to thermal reactions which may occur during the heating process, as discussed in Chapter 111, preliminary test runs were made to locate resulting AT anomolies.

In actual tests, the temperature was held constant at temperatures

where thermal reactions occur until the reaction was completed and then the constant heating rate was resumed. Although this procedure caused small gaps in the data, the rate of change of diffusivity with temperature was small enough so that interpolation could easily be made. Results of these tests will be reported in a later section.

114

- M 1.2 Other The Mossahebi (1966), Gomaa (1973), and Hirsh (1973)INguyen (1974) transient methods of measuring thermal diffusivities, from which thermal conductivities are derived, have been discussed in Chapter V. Since the principal objective of these methods was to measure thermal conductivity, in the discussion in Chapter V emphasis was placed on that aspect. In all cases, however, thermal diffusivity was also determined.

For example, Nguyen reported

thermal diffusivity values which were obtained by curve-matching techniques. With the pasteon heater method, she developed test techniques that could be run at both elevated temperatures and pressures. Some of these test results will be reported in a later section. It is difficult to find data in the literature on thermal diffusivities of liquid-saturated rocks. This is probably due to difficulties in making such measurements. Heat-pulse methods could lead to problems such as disturbing the liquid-saturation distribution and possibly introducing convective modes of heat transfer.

Good electrical insulation is necessary in

methods involving resistance heating and this can complicate the measurements. Hanley, et al. (1978) have reported on a "flash" method of measuring thermal diffusivity of liquid-saturated rocks. In their method, a laser is used to provide a high energy pulse to the front of a thin discshaped test specimen. Temperature rise at the rear surface of the specimen is recorded. From these data, thermal diffusivity may be calculated. Radiation effects may be important at higher temperatures and various coatings have been applied to minimize this effect. Results of these measurements will be reported in the following section. 2 MEASURED THERMAL DlFFUSlVlTlES OF ROCK Thermal diffusivities for a group of dry sedimentary rocks reported by Somerton and Boozer (1960), are shown in Fig. VII-1. Note the large decrease in diffusivity with increased temperature, particularly at the lower temperatures. The somewhat different behavior of shale is probably due to the slow release of structural-held water contained in the clay minerals present in the shale.

Flatness of the diffusivity/temperature curve for tuff is due to the high

porosity of this rock and the resulting low thermal conductivity values which are only slightly affected by temperature. The individual data points shown for the three sandstones in Fig. VII-1 are values derived from measurements at room temperature using the transient method of Gomaa (1972). These values are consistent with diffusivity values calculated from steadystate thermal conductivity data. Figure Vll-2 shows results of transient heating tests on some very low porosity rocks reported by Mongelli, et al (1982).

Their transient test method is similar to the method

developed by Gomaa described earlier.

These dense rocks have generally higher thermal

diffusivity values but they follow the same trend with temperature as for the sedimentary rocks shown in Fig. VII-1.

The higher diffusivity values are consistent with the fact that thermal

conductivity increases with increased density much more than density itself increases in the

115

.

Gomaa (1972) Boozer (1961) @

0

0

.@

@

-

2.15 x

Berea Sandstone 0--------

@ @ @

--------

--@) ---

1.6

Q kg/m3 -

Bandera Sandstone

I

I

2.16 x

lo3 lo3 lo3

Boise Sandstone

1.91 x

Limestone

2.25

103

Shale

2.20

103

Tuff

1.85

103

I

1

I

1

-

(D

$ 1.4 -*@I

0.0

I

0

I

1

100

200

I

I

300 400 Temperature "C

I

I

500

600

700

Fig. VII-1. Experimentally determined thermal diffusivities of various rocks (dry) as function of temperature by method of Boozer (1960) compared with Gomaa (1973) data at room temperature.

relation (a= h/pbCp). Table VII-1 shows thermal diffusivity data from Hanley, et al., (1978) for a suite of well-known rocks measured by the "flash" method. Measurements were made at room temperature on three sets of test specimens: air-dried, preheated to 1000" K, and watersaturated. Note the large differences in "air-dried'' and "preheated" test specimen values. The air-dried specimens undoubtedly contained hygroscopic moisture which is variable in amount depending on the humidity of the environment in which the tests were run.

This partially

explains why these air-dried specimen values do not agree with the oven-dried (120°C) room temperatures values reported by Mongelli, et al (1982).

The large reductions in thermal

diffusivity for the "preheated" specimens, compared to the values for the air-dried specimens, are consistent with the findings of Somerton and Boozer (1960). Due to irreversible thermal reactions and, in particular, loss of bound and structural water, the preheated specimens are not representative of the original unaltered specimens.

116

I

I

1

I

I

0 Ouartizite A Dolomite

2.60

103

2.71

103

0 Limestone

2.71

103

Mica-Quartzite

0.0

0

40

I

80 120 160 Temperature "C

2.73 x lo3

200

240

Fig. Vll-2. Experimentally measured thermal diffusivities of various rocks (dry) as function of temperature. Source: Mongelli, et al. (1982).

TABLE VII-1 Thermal diffusivities of various rocks. Source: Hanlev, et al. (1978). Rock Type

Barre granite Dresser basalt St. Cbud granodiorite Westerly granite Berea sandstone Holston marble Salem limestone Sioux quartzite

Density (kg/m3) 2.63~103 2.97~103 2.72~103 2.63~103 2.15~103 2.68~103 2.32~103 2.64~103

Thermal Diffusivitv (m2/s x l O 6 ) Air-dried Preheated Saturated 1.37 1.21 1.25 1.40 1.80 1.21 1.14 2.80

0.875 0.91 0 0.870 1.070 1.21 0 0.840 0.745 1.620

1.58 1.33 1.51 1.55 2.23 1.47 1.24

__

Thermal diffusivity values for the water-saturated test specimens reported by Hanley, et al., (1978) may be the most representative of the three sets of data. The amount of increase in thermal diffusivity with water saturation should be a function of the porosity of the rocks. Berea sandstone, which has the highest porosity of all the rocks tested by Hanley, does show the

117

greatest increase in thermal diffusivity for wafer-saturated specimens when compared to the values for the air-dried specimens. Figure Vll-3 shows the effects of stress and of temperature on thermal diffusivity of Berea sandstone. These tests were run on oven-dried test specimens using the Hirsh-Nguyen heat pulse method. Tests were run on jacketed test specimens in a pressure vessel with heating capabilities.

Difficulties of sealing the test specimen from the pressuring fluid limited the

amount of data obtained. The results shown in the figure are consistent with other test data. The effects of confining stress in increasing thermal diffusivities agrees with test data on the effects of stress on thermal conductivity which was shown earlier (see Chapter V).

Stress 0

I

1.4

0.4

1

0

- psi

1000

2000

I

I

3000 I

4

50

100 Temperature "C

150

200

Fig. Vll-3. Effects of temperature and stress on thermal diffusivity of Berea sandstone (dry). 3 CALCULATED THERMAL DlFFUSlVlTlES OF ROCKS

The difficulties inherent in measuring thermal diffusivity, particularly in liquidsaturated rocks, indicate that calculation of thermal diffusivity may be the better approach. The principal term in that calculation, thermal conductivity, is easier to measure accurately and several good models and correlations are available to estimate this quantity. Heat capacity can

118

be calculated either from a mineral analysis or a chemical analysis expressed in terms of oxides.

The third term in the calculation, bulk density, does not change significantly with

temperature or pressure and the effect of change in fluid saturation on this quantity can be easily calculated. Excellent agreement between transient-measured and calculated steady-state thermal diffusivities can be obtained if all measurements and calculations are carefully done. Table V112 shows comparison of results for a group of materials used as standards reported by Gomaa and

Somerton (1972).

There is less than three percent average difference for this group of

carefully tested materials.

TABLE Vll-2 Comparison of thermal diffusivities measured by transient method and calculated from steady state data. Source: Gomaa, 1972. Material

Alundum Pyroceram Pyrex glass Lava (fired) Bandera sandstone Berea sandstone Boise sandstone

Thermal Diffusivity ( m z k ~ 1 0 6 ) Measured Calculated 2.79

2.00 0.68 1.47 1.20 1.40 1 .oo

2.68 1.96 0.67 1.50 1.22 1.41 0.96

In estimating thermal diffusivity when no thermal data are available, it is suggested that thermal conductivity be estimated by using Eq. V-19 for unconsolidated sands and Eq. V-22 for sandstones. More sophisticated model results may also be used to estimate thermal conductivity but the degree of improvement in accuracy may not justify the extra effort. These values should then be corrected for temperature using Eq. V-26. The volumetric heat capacity (pbCp) may be estimated using Eq. 11-15 or 11-16, whichever is appropriate.

119

Chapter Vlll HEAT TRANSFER WITH FLOWING FLUIDS In previous chapters, the tacit assumption has been made that all fluids present in the rock pore spaces are stationary. This may be true in low permeability reservoir cap-rocks or in quiescent, completely undisturbed subsurface reservoirs.

However, whenever production/

injection processes are in operation, fluid motion does occur and this affects the heat transfer character of the rocWfluid system. Several types of fluid motion are possible in porous media. The simplest of these is natural convection in which fluid movement is caused by temperature-gradienvfluid-density differences and the resulting effect of gravity. So-called "convection cells" may be set up by temperature gradients in which the hotter fluid rises and the cooler fluid moves downwards. Conditions necessary for this type of fluid motion to occur will be discussed in a later section. A second and more complex mechanism causing fluid motion is the condition where fluid phase changes occur. Under appropriate temperature and fluid pressure conditions, liquids may vaporize at the higher temperature zone and be driven in vapor phase to the cooler zone by the action of a vapor pressure gradient. At the location where the temperature drops to the appropriate level for the prevailing fluid pressure, the vapor will condense, giving up its latent heat to the surroundings.

The condensed liquid is then caused to move towards the higher

temperature zone, which is low in liquid saturation, under a capillary pressure gradient. This vaporization-condensation-capillary (VCC) effect is equivalent to the so-called "heat-pipe effect" and can have a profound influence on the apparent heat transfer characteristics of the rocWfluid system. The third mechanism causing fluid motion is that due to natural or imposed pressure gradients, the former being moving ground water in a hydrodynamic environment and the latter being fluid motion in productionlinjection operations.

Each of these mechanisms will be

discussed in the following sections, largely in relation to work done by the author and his coworkers but with some reference to work done by other investigators. 1 NATURAL CONVECTION IN POROUS MEDIA

Under certain circumstances, natural convection of fluids may occur in porous media due to fluid density gradients. Heated fluids tend to rise under buoyancy forces and cooler fluids tend to move downwards by the action of gravity. neighborhood of underground heat sources.

So-called "convection cells" may exist in the This is an important mode of heat transfer in

120

geothermal reservoirs.

Combarnous and Bories (1975) have reported on this phenomenon.

They suggest that the Rayleigh number be used as a criteria for the onset of natural convecfion, expressed in the following form:

where k

=

absolute permeability;

p

=

coefficient of thermal expansion of fluid; L = length of

system; AT = temperature difference across length: h = thermal conductivity of fluid-saturated porous media: p = mass density of fluid; Cp

=

heat capacity of fluid; v

=

kinematic viscosity of

fluid. A critical value of 40 for the Rayleigh number is suggested by Combarnous and Bories. In work by Tung (1977) on the effective conductivity of porous media with flowing fluids, he investigated the possible occurrence of natural convection in the steady-state comparator thermal conductivity apparatus discussed in Chapter V.

Providing isothermal

surfaces are flat and horizontal in such a system, as was found to be the case by Mehos (1986) natural convection cannot occur. In addition, the Rayleigh number for such a system would be much lower than the critical value cited above. To check these factors, Tung ran a series of experiments in the above apparatus in which he varied the differential temperature across the test specimen. If natural convection were occurring, the apparent thermal conductivity should increase with the increase in temperature differential. Results of these tests are summarized in Table VIII-1 and are plotted in Fig. VIII-1. Table VIII-1 confirms that the Rayleigh number is three orders of magnitude smaller than the critical value suggested by Combarnous and Bories (1975).

Figure VIII-1 shows that apparent thermal conductivity is constant with changes in

temperature differential when the heat flow direction is vertically upwards. When heat flow is vertically downwards, apparent thermal conductivity is lower but increases with increased temperature differential.

These latter results are surprising in that no convective heat

transfer would be expected, in particular, for the case of downwards heat flow.

If natural

convection were occurring in the upwards heat flow case, it would be expected that the apparent thermal conductivity would be larger than for the downwards heat flow case, which it was, but the apparent thermal conductivity should also have increased with increased temperature differential. which it did not.

As was pointed out above, natural convection cannot occur if the isothermal surfaces in a linear. vertical flow system are horizontal planes.

Mehos (1986) calculated the isotherms for

the comparator apparatus and, as shown in Fig. V-5, these were horizontal planes through the test specimen.

Thus from this and other evidence, natural convection in these thermal

121

.4 . -

5. .-c* .-c5

2

Heat Flow Upwards

3. --

0

s 0

----a----

Tln -

0

E

2.

&TL-R ----- 9---- r.1

4

I

,

Heat Flow Downwards

-

8

c I-

E

5n

1.

-

Comparator Apparatus Water Saturated Ottawa Sand.

D

Temperature Differential (“C) Fig. VIII-1. Experimentally observed effects of temperature differential and direction on apparent thermal conductivities. Source: Tung (1 977). TABLE VIII-1 Experimentally measured apparent thermal conductivities of Ottawa sand as functions of imposed heat flow magnitude and direction and corresponding Rayleigh numbers. Source: Tung (1977). Heat flow direction UP

down

Temperature differential across sample (“C) 4.06 7.50 8.61 12.22 16.39 19.44 4.33 7.78 8.33 12.22 16.50 19.17

Apparent thermal conduct. (W/m-K)

Rayleigh number

3.1 7 3.1 6 3.1 0 3.07 3.1 5 3.1 5 2.71 2.91 2.90 2.98 3.1 3 3.04

0.01 2 0.023 0.026 0.038 0.050 0.060 0.013 0.024 0.026 0.038 0.051 0.059

conductivity measurements can be ruled out. It is probable that the results for downward heat flow shown in Fig. V-1 are in error due to the holder correction factor which was required because of the finite thermal conductivity of the holder material.

(See Chapter V for further

explanation of this effect.) The calibration was done for a fixed temperature differential across

122

the stack and may not be valid for other differentials. In summary, although natural convection may be an important heat transfer mechanism in geothermal reservoirs, study of this effect in a comparator thermal conductivity apparatus is not a viable approach to the problem. The reader is referred to the geothermal literature for further analysis of this phenomenon.

2 FLUID PHASE CHANGES IN POROUS MEDIA (VCCEFFECT)

Anand (1971) first observed anomalous heat-transfer behavior in running tests on the effects of partial liquid saturation on thermal conductivity of porous rocks. His tests were made at temperatures above the boiling point of the saturating liquid. In the tests, liquid saturation was reduced in small increments by momentarily reducing pore-fluid pressure and allowing liquid to be withdrawn from the test specimen in vapor phase. The vapor was condensed and measured so that change in average liquid saturation could be calculated. Upon closing in the pore-fluid system, the pore pressure built back up to the vapor pressure at the prevailing temperature. Because of the lower thermal conductivity of the vapor phase, one would expect the thermal conductivity of the liquid/vapor/rock system to decrease as liquid was withdrawn. Instead the apparent thermal conductivity increased as liquid was withdrawn, to a maximum value somewhat above the irreducible liquid saturation for the particular porous rock. This was recognized as a vaporization-condensation-capillary (VCC) effect, similar in principle to the “heat-pipe’‘ effect (Somerton, 1973).

If the pore pressure is maintained at or near the

vapor pressure of the pore fluid and a temperature gradient is imposed across the porous rock, this VCC effect may occur, substantially increasing the amount of heat transferred. In the heatpipe effect, vaporization of some of the liquid occurs at the higher temperature zone of the system, the vapor travels to the lower temperature zone in the porous system under a vapor pressure gradient, condensing and giving up its heat of vaporization. The condensed liquid is then drawn back to the higher temperature (and lower liquid saturation) zone by capillary suction. The amount of heat transferred by this process may be several times that transferred by conduction alone. The magnitude of the VCC effect in porous rocks is strongly influenced by the heat of vaporization of the saturating fluid, the absolute permeability of the porous rock, and the fluid-saturationlrelative-permeability relations for the system. Gomaa (1973) developed a model of the VCC effect based on heat-pipe theory. Since it is apparent that mass flow rates of liquid and vapor must be equal at equilibrium conditions, the

123

heat-pipe effect can be expressed in terms of average mass flux:

XVCC = mhvU(1

-

(Vlll-2)

$Sv) AT

where hvcc = "thermal conductivity" with heat-pipe; m vapor; hv

=

=

average mass flux of liquid and

heat of vaporization of liquid; L = length of mixing zone; AT = temperature

differential across length L; $

=

fractional porosity of porous medium; Sv

=

pore space

saturation of vapor phase. The mass flux in porous media may be expressed as:

m = - k [ ( d P l d z ) l +(dP/dZ)v]l[(vIlkrl) + (vv/krv)]

where k

=

absolute permeability; (dPldz)

=

pressure gradient; v

(VI 11-3)

=

kinematic viscosity; kr

=

relative permeability; subscripts (I) and (v) for liquid and vapor, respectively. The effective pressure gradient for liquid consists of two components: 1) capillary pressure gradient and, 2) head gradient.

Pressure gradient for the vapor is that due to the

vapor pressure difference at the higher and lower temperature ends of the mixing zone. The difficulty of evaluating these pressure gradient terms limited the usefulness of the theoretical analysis. However, it did show that the VCC effect should be directly proportional to the absolute permeability and to the heat of vaporization of the fluid. Experimental results of measurements on outcrop sandstones and unconsolidated sand run at temperatures above the boiling points of the saturation fluids are shown in Figs. Vlll-2 to Vlll-5. The VCC effect is moderate for the lower permeability sandstones, Bandera and Berea (Fig. Vlll-2), but is of larger magnitude for the high permeability Boise sandstone and Ottawa sand (Fig. Vlll-3).

In Fig. Vlll-4 the magnitude of the VCC effect is shown in comparison with

heat transfer by conduction alone from tests run at a temperature just below the boiling point. The total apparent thermal conductivity may be given by:

ha

=h

+ hvcc

where

ha

~ V C C=

equivalent conductivity due to VCC.

(Vlll-4)

= apparent (measured) thermal conductivity; h = thermal conductivity without VCC;

The ratios of maximum VCC effects for the four materials tested are given in Table VIII2.

If these ratios are compared with absolute permeabilities of the samples, the relative

magnitude of the VCC effect is approximately proportional to the square root of permeability, rather than the linear relation predicted by Eq. Vlll-3.

124

5.0

Y

E

4.0

5

--s

' .-E 3.0 0

a

-0 C

0

0

-

2.0

0

E

c I-

00 A

1.0

v

OI

0

I

I

I

I

I

I

I

I

I

.20

.30

.40

.50

.60

.70

.80

.90

1.O

I

.10

GOMAA Reported Data Run 1 Run2

Brine Saturation Fig. Vlll-2. Apparent thermal conductivities for Berea and Bandera sandstones as function of brinehapor saturation at average temperature of 132°C.

"

0

.I0

.20

.30

.40

.50

60

.70

A0

.90

1.0

24.

Brine Saturation Fig. Vlll-3. Apparent thermal conductivities for Boise sandstone and Ottawa sand as function of brinehapor saturation at average temperature of 132°C

125

TABLE Vlll-2 Maximum VCC effects for the four materials tested normalized relative to thermal conductivities without VCC effects. Values compared to absolute permeability of materials tested.

Bandera Berea Boise Ottawa

1.23 1.26 3.1 5 4.45

38 190 2500 6500

Figure Vlll-4 shows that the magnitude of the VCC effect is a function of liquid saturation

but, more basically, the magnitude must be a function of relative permeability. At equilibrium conditions the average mass flow rate of liquid towards the higher temperature zone must be equal to the average mass flow rate of vapor towards the lower temperature zone. Thus the liquid saturation at which the maximum VCC effect occurs would be at relative permeability values for liquid and vapor which would give the maximum mass flow rate under existing vapor

7.0 6.0

%

E

5

--s

5.0

9

5 4.0

I

a

'0

E

0

3.0

0

E

c

2.0

I-

1.o

Broken Lines: Without VCC Effects

I

0

.10

I

I

I

1

I

1

I

I

I

.20

.30

.40

.SO

.60

.70

.a0

.90

1.O

Brine Saturation Fig. Vlll-4. Comparison of VCC effect for several sandstones as function of brinelvapor saturation at 132°C. Broken lines show thermal conductivities for brinelair saturation for same samples at 98°C without VCC effects.

126

pressure and capillary pressure gradients.

These latter two quantities are also controlling

parameters in determining the VCC effect (see Eq. Vlll-3). Figure Vlll-5 shows the much-reduced VCC effect for the several samples saturated with heptane. This is to be expected due to the much lower heat of vaporization of heptane (321 J/kg (138 Btullb)) compared to water (2256 J/kg (970 Btu/lb)).

For Ottawa sand the ~ V C C / ~

ratio decreases from about 3.5 for brine saturation to about 1.5 for heptane saturation. This is only one-third the magnitude of change that would be expected from the ratio of heats of vaporization as predicted by Eq. Vlll-2.

Y

E

5

3.0

s;

> 0

Bandera

2.0

U

C

0

0

;1.0 0 GOMAA Reported

+

o

0

I

1

.10

.20

1

1

.30 .40

Data

1

I

1

1

I

I

.50

.60

.70

.80

.90

1.O

Heptane Saturation

Fig. Vlll-5. Apparent thermal conductivities for several sandstones as function of heptanehapor saturation at average temperature of 132°C.

In view of the difficulties of matching the experimental results to predictions using Eqs. Vlll-2 and Vlll-3, Gomaa applied a linear multiple regression (ARIEL) analysis to develop an

empirical equation for VCC effect and to order the variables according to their significance. The variables in order of decreasing significance were: k

absolute permeability,

hv

heat of vaporization,

SI

liquid phase saturation,

[(kr/p)l x (kr/p)v]l’2

geometric average of liquid and vapor mobility ratios,

dPc/dSI

capillary pressure gradient.

Since the temperature gradient across the test specimen was not varied in the experiments, no evaluation of the significance of this quantity was possible.

127

Figure Vlll-6 shows the VCC effect on apparent thermal conductivity for Ottawa sand with several saturating liquids and compares the effects above and below the boiling point for brine saturation.

7.0

uE

0.0

s

5.0

'.-.L

>

L)

g

4.0

1 0 - 3.0

e 4)

r

c 2.0

Hrptrnr, 132°C

1.0 Stod. Solvent, 132°C

0

1

1

I

1

I

I

1

1

I

1

.10

.20

.30

.40

50

.60

.70

.80

.90

1.O

Liquid Saturation Fig. Vlll-6. Comparison of the effects of different fluid saturants on VCC effects for Ottawa sand pack. 2.1 -of

VCC Fffect to S

t

.

e

.

e

A preliminary investigation of the possible importance of the VCC effect in subsurface thermal operations was made by Chou (1979). He developed a one-dimensional, two-phase numerical model of the VCC effect that could be converted to a steamflood model by adding a term for steam injection in the first grid block. The VCC model could also be applied to a study of heat losses from a thermal injection or geothermal production well by the use of radial coordinates. The model, patterned after that of Shutler (1969), consists of three parts.

The first

part of the model solves for steam pressure by an implicit pressure-explicit saturation scheme (IMPRES), the second part solves for temperature implicitly, and the third part solves for interphase mass transfer. Boundary conditions for the VCC model were: constant temperatures at the two boundaries, no flow boundary at the hot end, and constant back pressure at the lower temperature boundary.

The steamflood version of the model imposed constant pressure

boundary conditions with constant temperature boundary conditions based on the saturation temperatures at each boundary pressure.

128

Because of limited computer time available, runs were not made to conclusion. Typical VCC model results for the ninth and tenth time steps are shown in Table Vlll-3.

The

temperature and steam-saturation distributions are plotted in Fig. Vlll-7. In the steam plateau or mixing zone, where temperature is nearly constant, heat transfer is almost entirely due to VCC effect. This agrees closely with experimental results to be reported later.

TABLE Vlll-3 Ninth and tenth time step output for Chou's (1979) one-dimensional, two-phase model of heat DiDe effect.

X/L

T ("C)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

103.32 99.56 99.32 99.32 99.32 99.32 99.32 99.26 98.20 88.51

Pv

PI

SS'

(atm.)

(atm.)

(frac.)

1.0073 1.0073 1.0072 1.0070 1.0067 1.0063 1.0057 1.0049 1.0038 1.0022

0.8736 0.9269 0.9565 0.971 0 0.9804 0.9872 0.9920 0.9956 0.9983 0.9997

0.933 0.746 0.632 0.548 0.990 0.439 0.393 0.349 0.300 0.030

1.1 OE-7 7.87E-6 1.01 E - 5 1.11E-5 1.1 3 E - 5 1.1 OE-5 1.04E-5 9.66E-6 5.85E-6 0.00

-4.62E-4 -7.50E-3 -1.1 8 E - 2 -1.38E-2 -1.45E-2 1.41 E - 2 1.31 E - 2 1.22E-2 1.1 9 E - 2 2.38E-2

0.00 .1.86E-4 1.1 9 E - 5 9.43E-6 6.62E-6 3.91 E - 6 7.63E-7 6.73E-6 1.77E-4 2.1 8 E - 4

1.0074 1.0074 1.0073 1.0071 1.0068 1.0063 1.0058 1.0050 1.0039 1.0022

0.8738 0.9294 0.9577 0.971 6 0.9806 0.9871 0.991 9 0.9956 0.9983 0.9998

0.933 0.736 0.617 0.545 0.489 0.440 0.394 0.350 0.300 0.032

2.1 1 E - 7 8.1 OE-6 1.01 E - 5 1.09E-5 1.1 1 E - 5 1.08E-5 1.04E-5 9.84E-6 6.09E-6 0.00

-7.73E-4 -7.87E-3 -1.1 7 E - 2 -1.35E-2 -1.41 E - 2 -1.38E-2 -1.32E-2 -1.25E-2 -1.23E-2 2.37E-2

0.00 - 1.91 E-4

uv

UI

(m/cm'k)

MC"

(gm)

Time step 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .o

104.02 99.59 99.32 99.32 99.32 99.32 99.32 99.26 96.21 87.89

* Steam saturation

1.23E-5 9.74 E- 6 6.94E-6 4.25E-6 1.09E-6 6.83E-6 1.76E-4 2.37E-4

** Mass liquid condensed

Chou made a parameter study using the VCC boundary conditions. Results of the study agreed with experimental findings. Lowering absolute permeability values decreased the magnitude of the VCC effect. Increasing the imposed temperature gradient reduced the length of the mixing zone but increased the relative magnitude of the VCC effect. Setting capillary pressure to zero resulted in no negative mass flux of water, leading to only a gradual movement of the evaporation front without condensation and thus no VCC effect.

129

100 90 80 70

a9

s o .-$

CI

2

50 40

30

!i[ i5

20 10

0

Grid Block No. Fig. Vlll-7. Temperature and steam saturation distributions as function of grid block number from Chou (1979) heat pipe model. The model was converted to the steamflood version by setting injection pressure at the upstream boundary and a back pressure at the downstream boundary. Temperature boundary conditions were set at saturation values corresponding to ths imposed pressures. Since the counterflow of water is essential to the VCC process, a low enough imposed pressure gradient had to be applied so that the capillary pressure would not be overwhelmed. Under no reasonable range of controlling parameters was evaporation ever observed. The heat supply was not sufficient to vaporize water under the imposed pressure gradient. Had the pressure gradient been greater, the possibility of evaporation would have increased but this would have precluded the possibility of counterflow of water. It was concluded that the only way the VCC effect could exist in steam injection would be to use superheated steam. This is not considered to be a viable option in steam injection. Chou concluded from his work that in steam injection operations, the VCC effect would only be important in heat loss to aquifers from the injection well. For this to occur, the casing temperature would have to be higher than the saturation temperature of the water at the fluid pressure in the aquifer. For the casing temperature to be sufficiently high, pressure of the fluids in the borehole adjacent to the aquifer would have to be higher than the aquifer fluid

130

pressure so that if the aquifer and wellbore were at saturation conditions, the wellbore would be at the higher temperature.

This condition would usually be met, particularly for shallow

aquifers. Shallow aquifers could draw excessively large amounts of heat from the wellbore by the VCC process. Thus special care should be taken to properly insulate the shallower part of the injection well or the geothermal production well.

Further analysis of this problem will be

made in Chapter XI. 2.2 w o v e d VCC

With the potential importance of the VCC effect in subsurface thermal processes, Su (1981) developed an improved model of the phenomenon and constructed an experimental apparatus to test the model. The conceptional model of the VCC effect for a one-dimensional, two-phase system is shown in Fig. Vlll-8.

Sufficient heat is applied at the high-temperature end to cause

vaporization of the liquid in the super-heated zone.

The steep temperature gradient in this zone

-x

I I

I

x = o

Super-

I

heated Vapor Zone

I I

I I

Mixing Zone

I

1 I I

I

I

x = a

x = b

Liquid Zone

f

c)

E

a

E

l-

TOP

Length

Bottom

Fig. Vlll-8. Conceptual configuration of a two-phase system in a porous medium as used in Su's (1981) heat pipe model.

131

is due to the low conductivity of the vapor-saturated porous medium. At the lower temperature end of the system, the temperature is low enough so that the fluid is all in the liquid state. The temperature gradient is less steep because of the higher thermal conductivity of the liquid-

-

saturated porous medium. In between the two zones, a mixing zone exists in which liquid is being vaporized at (x

a), the vapor travels lo (x = b) under a vapor pressure gradient where

it is condensed. releasing its heat of vaporization to the surroundings. The condensed liquid is

then drawn back to the vaporizing front (x

-

a) by capillary pressure gradient.

Under the above conditions. for a closed boundary condition the overall mass flow rate of vapor must equal the overall mass flow rate of liquid in the opposite direction. Mass fluxes for the two phases may be expressed by Darcy's law as: mv = kkrv/vv [(dPv/dx) - p v g l

mi

=

(Vlll-5)

kkrilvv [(dPl/dx) - pig]

(VI I1-6)

where k = absolute permeability; kr = relative permeabilities; P = pressure: p = fluid density;

v

=

fluid kinematic viscosity: subscripts (v) and (I), vapor and liquid, respectively. Considering that capillary pressure is given by:

Pc = Pv - PI

(VI I1-7)

and combining with Eqs. Vlll-5 and Vlll-6, the following expression for mass flux results:

Assuming capillary pressure to be a function of liquid saturation only, Eq. Vlll-8 can be reduced to the following form for heat flux (Su. 1981):

where q = heat flux; hv = heat of vaporization; L = length of mixing zone; m = average mass flux; S

=

liquid saturation; dPddS

=

capillary pressure gradient; a, b = limits of mixing zone;

other terms as given earlier. There are several problems in applying Eq. Vlll-9 to a real system. Most notable is the

lack of relative permeability and capillary pressure data for a system of a liquid and its vapor. Udell and his coworkers (1989) are currently working on this problem.

For purposes of

132

testing the model, Su used standard liquid-gas relative permeability and capillary pressure data for unconsolidated sand. Another problem is evaluating the liquid saturations at the limits of the mixing zone for a liquid-vapor system. values, to wit, Sb

=

Su used conventional limiting liquid/gas saturation

(1 - critical gas saturation) and Sa = irreducible liquid saturation.

and R e s m

2.3

The experimental apparatus designed to test the VCC effect model is shown schematically in Fig. Vlll-9.

It consists of a 0.25 m long Lexane tube packed with Ottawa sand. Several

thermocouples are mounted along the axis of the tube. Heat source and sink are provided at either end of the tube with heat meters mounted between source and sink and the sand pack. An external guard heater is provided around the tube to minimize radial heat flow. Liquid saturation is controlled by use of a back-pressure regulator. Pairs of electrodes are mounted at

Brine Water Reservoir

l--l

Heating Coil

Vacuum Pump

To Controller

ElectrodeInsulatiow Thermocouples

Guard Heater Assembly Back Pressure Regulator

,

LIlCopper Block

Fig. Vlll-9. Schematic diagram of linear heat pipe experimental apparatus.

133

several locations across the tube to monitor electrical resistivities. In running a test, the sand-packed tube was fully saturated with a brine solution. Temperature controllers were set for the desired top and bottom temperatures and the porefluid pressure regulator was set for a pressure slightly in excess of the saturation pressure for the highest temperature. When temperature equilibrium was obtained, the back pressure was reduced to drive off the amount of liquid needed to give the desired average liquid saturation for the test. The test was continued until temperature equilibrium was again obtained. Figure Vlll-10 shows the results of a typical test. The temperature gradient is constant through the liquid zone, the temperature is constant through the mixing zone, and the temperature gradient is constant and larger in the vapor zone. The resistance measurements were somewhat erratic but they show clearly a large increase in resistance at the liquid-mixing zone boundary. It is difficult to determine, quantitatively, liquid saturations in a vaporizing system because of the change in salinity of the brine as liquid is vaporized and removed from the system.

15

130

0

120

co 12 110

100

CI

E

r 0

E

9

?!

b

90

Y v-

0

u) u)

: ti Q

B

.-d

-

6

0

a

3

80

Porosity = 38% Permeability = 6 Darcys Grain Size = 65-100Mesh Average Liquid Saturation = 75% Heat Flow = 1.28 Watts

70

60

0 0

0 TOP

E

(

50 Distance (cm)

Bottom

Fig. VIII-10. Experimental measurements of resistance and temperature distributions in heal pipe apparatus for Ottawa sand pack.

134 Comparison of the calculated temperature profile with experimental results is shown in Fig. VIII-11. The agreement is reasonably good. Calculated liquid saturations are also shown on The sharp change in liquid saturation at the liquid-mixing front agrees with

the figure.

experimental resistance measurements shown in the previous figure.

The conclusion to be

reached is that the model adequately describes the VCC effect even though different but similar data were used for relative permeabilities and capillary pressures. The model should be useful for predicting heat transfer behavior in processes where the VCC effect might occur.

An

example will be given in Chapter XI which deals with borehole applications of heat transfer.

130

I

-- A

120

Experimental Temperatures Numerical Temperatures Numerical Saturations 0---------

/

110

oe2

h

5a E

i

I .80

100

90

.60

01

I-"

41.00

80 .40

70

j .20

,/-

60

50 1

TOP

0

Distance (cm)

Bottom

Fig. VIII-11. Temperature and liquid saturation distributions calculated from Su's (1981) heat pipe model compared with experimental data.

3 CONVECTIVE HEAT TRANSFER WITH FLOWING FLUIDS In many practical applications of thermal data in subsurface reservoirs, fluids are not stagnant but are in motion. We must therefore deal with convective heat transfer as well as with conduction of heat.

A number of questions arise as to the proper formulation of the

135 problem. Is there a detectable temperature difference between the solid grains making up the porous medium as the fluids flow through it? Is there a surface resistance or contact resistance between solid and fluid and, if so, how may this be accounted for? In the flowing fluid case is there a mixing action which alters the effects of the above factors on the heat transfer character of the rock-fluid system? Since answers to these questions are generally unknown. the usual practice is to lump all of the effects into a term referred to as the "effectve conductivity".

3.1

with

Steadv-State Fluid Flow

In the case where fluid is flowing through a heated linear system, the temperature gradient is no longer linear. When a mean temperature can be assumed at any point in the medium through which heat and fluid are flowing, steady-state heat transfer may be expressed by: mCp (dT/dx) + he (dZT/dxz) = 0

where mi

=

(VIII-10)

mass fluid flux; CP = heat capacity of flowing fluid; )ie

=

effective thermal

conductivity. Integrating Eq. Vlll-10 for a linear system (Kunii and Smith, 1961) yields the following expression for temperature distribution:

(Tx - TL)/(To - TL)

=

1 - (1 - exp-ax)/(l

-

exp-aL)

(VIII-11)

where a = mCp/he.

Considering that heat flux in this system is given by:

q

=

he (dT/dx)

(VIII-12)

Tung (1977) has solved for effective conductivity as:

he = -mCpL/ln{[q~/(To- T ~ ) m C p ] / [ l+ qL/(To - T ~ ) m C p l l

(Vlll-13)

where To - TL = temperature difference across length L; qL = heat flux leaving the system at x = L. Tung (1977) performed a series of experiments with flowing fluids in unconsolidated sand using the comparator thermal conductivity apparatus described earlier (Chapter 5). Inlet fluid temperature was adjusted to the temperature of the system at the inlet end. Water was used as the fluid flowing in the Ottawa sand-pack. Results of the tests are shown in Table VIII-

136

4.

In calculating effective conductivities using Eq. Vlll-13, the heat flux term (qL) was

determined from the heat meter by noting the temperature differential across the standard, for which thermal conductivity was known.

TABLE Vlll-4

Effective thermal conductivities of sand pack at various fluid flow rates for heat flow directions vertically upwards and downwards. Source: Tung (1977).

Heat flow

Mass flux

he'

direction

(gts-cmz)

(W/m-K)

down

0 1.54E-3 1.70E-3 2.24E-3 2.22E-3 2.51 E - 3 2.96E-3

3.02 5.07 5.30 6.15 6.1 4 6.57 7.1 8

UP

__

0

6.29 6.79 8.32

1.89E-3 2.1 9 E - 3 3.02E-3

Re"

Nu/Pr2"'

0 5.90E-5 6.36E-5 7.30E-5 7.17E-5 7.66E-5 8.52E-5

0 4.15E-2 4.59E-2 6.04E-2 5.99E-2 6.77E-2 7.99E-2

0

0

5.10E-2 5.91 E - 2 8.15E-2

4.87E-5 5.39E-5 6.1 7 E - 5

Effective thermal conductivity Reynolds number *** Nu = Nusselt number; Pr = Prandtl number

**

Effective thermal conductivities are plotted against mass fluid flux in Fig. Vlll-12. Note that the effective conductivities for the upwards flow case (heat and fluid both flowing vertically upwards) were higher that for the downwards fluid flow case.

Tung attributed this

difference to be due to natural convection contributing to total heat flow in the upwards flow case, adding to the effective thermal conductivity. Dimensionless groupings of variables are often employed to characterize convective heat flow. Fluid flow is generally characterized by Reynolds number which for porous media may be expressed as:

Re

where m

=

=

mD/bf

mass flux of flowing fluid; p

porous medium, D

=

(VIII-14)

=

viscosity of flowing fluid; $I

=

fractional porosity of

characteristic length: (1) average grain diameter, or (2) equivalent pore

137

"

0

.04

.08

.12 .16 .20 Mass Flux (g/sec-cmz)

.24

.28

.30

Fig. Vlll-12. Effective thermal conductivity of Ottawa sand pack as function of mass flux of fluid flowing upwards and downwards.

diameter, Dp = (32 k/g)ll2, where k

=

absolute permeability.

The Nusselt number is used to characterize convective heat transfer as given by the following expression: NU = hD/hf

where

h

=

flowing fluid.

(VI 11-15 )

heat transfer coefficient between solid and fluid: If = thermal conductivity of

138 The Prandtl number characterizes the viscouslthermal properties of the flowing fluid in the form: (VIII-16)

Pr = Cf pVXf

Kuni and Smith (1961), seeking a common correlation between these parameters, plotted the quantities NulPr2 versus Re for a large number of flow tests with several different fluids flowing in packed beds of glass beads and unconsolidated sands. Figure Vlll-13 shows the approximate ranges of the experimental values they obtained.

The correlation can be

approximated by the expression:

(VIII- 17)

I

1.02 mm

10-~ Glass Beads-

8

IW

UP Down

Data and N

?I

3 10-5

z

I Limits of Data Cluster Kunii and Smith (1961) Data

Fig. Vlll-13. Kunii and Smith (1961) data showing effects of Reynolds number on Nusselt number for heat transfer in a fluid-flowing system.

139

Tung's (1977) data plotted on Fig. Vlll-13 do not agree well with Kuni and Smith's data. Part of the difficulty is in defining a "characteristic length" although this in itself cannot explain all of the discrepancy.

3.2

Heat T

m with -

T

In addition to the steady-state flow experiments reported above, a number of studies of

transient heat-transfer have been reported. Green (1964) reported on a series of experiments in packed beds in which a step change was made in temperature of the flowing fluid at the entrance to the bed. As the fluid moved through the bed, the temperature front dispersed away from the mean heat-front position due to longitudinal thermal dispersion. The effective thermal conductivities were calculated by comparing the measured temperature profiles with an approximate solution of the energy equations for the solid phase and the fluid phase. Heat exchange between the two phases was treated as a heat source term in both equations. The effective conductivities were found to be a function of the product of the interstitial velocity and the average diameter of the particles. When the value of this product is less than about 0.01, the effective conductivity is essentially equal to the static thermal conductivity. The method of spatial averaging is another approach used to deal with heat transfer in porous media when a convective contribution is involved. In the spatial averaging method the temperature is decomposed at a given point into the local average temperature and the deviation of the temperature from the average at the point. By using this method and assuming that the spatial gradients of the temperature deviation are much greater than the gradients of the local average temperature, Levec and Carbonell (1985) derived equations for the local average temperature of the fluid and the solid phases of a packed bed, from point-wise energy equations of the two phases.

Equations for the average temperatures were then used to analyze the

response of the bed to a pulse temperature disturbance introduced into the fluid phase. As the fluid moved through the bed, the temperature pulse in the fluid phase induced a temperature pulse in the solid phase. For long time periods, the spatial distance between the two pulses was found to be a constant and the two pulses spread about their center points at equal rates. The above authors pointed out that the effective thermal conductivities under steady-state conditions could be significantly different from those in transient processes. This was attributed to heat transfer between the separate temperature pulses of the two phases. To verify the theoretical predictions, Levec and Carbonell conducted experiments in packed beds using specially designed probes each of which measured temperature at three points simultaneously: inside a solid particle, at the solid/liquid interface, and in the fluid region. The temperature response of the bed to a step temperature change at the inlet showed good agreement with the theoretical predictions.

140

.

.

3.3 Heat Transfer with Fluid F l o w in a

Thick-Walled Tube

Yuan (1991) studied heat transfer in a thick-walled tube containing a flowing fluid, as a simplified model of heat transfer in a porous medium. (See Fig. Vlll-14).

Heat transfer in the

tube was formulated with the following two equations:

(hf/r) d(rdTf/dr)/dr + hf(d2Tf/dx2) - vf(r)Cfpf(dTf/dx)

=

C f p f (dTf/dt)

(Vlll-17)

(Vlll-18)

where rw

=

inner radius of tube; ro

=

outer radius of tube: x

=

distance along the tube; vf(r)

=

velocity of fluid flowing at radius r; other terms as defined earlier.

// /

Fig. Vlll-14. Thick-walled tube model with fully developed laminar fluid flow inside and insulated condition outside.

The outer wall of the tube is assumed to be insulated so that,

dTs(ro)/dr

=

0.

(Vlll-19)

At the inner wall, continuity of temperature and heat flux requires that Tf(rw)

hf[dTf(rw)]/dr

=

hs[dTs(rw)l/ar.

=

Ts(rw) and

(VII1-20)

The known values of thermal conductivities of the fluid and the solid are used in the above equations. The velocity field vf(r) in Eq. Vlll-17 is assumed to be known.

141

Treating the tube as a 'uniform' one-dimensional rod, conservation of energy leads to the following equation in which the Darcy velocity is used to describe the convective contribution:

where T is the average temperature of the two phases over an elemental volume; vd is Darcy velocity; hd is the effective thermal conductivity referred to here as the thermal dispersion coefficient. Because of the averaging nature of Darcy velocity, the second term in Eq. Vlll-21 fails to account for all of the convective contribution to heat exchange in the model.

Part of the

convective contribution must be included in the first term of the equation which results in velocity dependence of the effective thermal conductivity.

Thus the term hd reflects the

combined effects of molecular diffusion and hydrodynamic dispersion.

Yuan (1 991) suggests

that hd be referred to as a thermal dispersion coefficient to distinguish it from thermal conductivity that results wholly from molecular diffusion. Instead of using the volume averaging method implied in Eq. Vlll-21, Yuan integrated Eqs. Vlll-17 and Vlll-18 with respect to the radius (r) and added the two resulting equations by invoking conditions given by Eqs. Vlll-19 and Vlll-20. spatial independent variable, (x).

The final equation includes only one

By comparing the final equation with Eq. Vlll-21, the

effective thermal conductivity may be expressed as: ro

rw

h d = {hfj(aTf/ax)d(r/ro)2

+ hsid3Ts/3x)d(rlro)2 (Vlll-22)

Equation Vlll-22 indicates that the effective thermal conductivity, or the thermal dispersion coefficient (hd), depends not only on the inherent properties of the system such as thermal properties of the two phases and 'porosity' of the tube, (9, = (rwlro)2), but also on the temperature field and the velocity field in the tube/fluid system. The temperature field and the velocity field may vary widely from case to case and even in the same case the value of (hd) may vary from one point to another.

Calculation of hd was carried out for the case of constant

temperatures imposed at each end of the tube and the velocity field was assumed to be parabolic in the radial direction.

The temperature fields in both the solid and the fluid regions were

obtained by solving Eqs. Vlll-17 and Vlll-18 numerically.

Based on the temperature fields

obtained, values of (hd) were calculated using Eq. Vlll-22. These values varied in the axial direction showing a concave profile with two peaks near each end but relatively constant in the

142

L + v)

% 1.05 c

i 0.50

.-0 u)

c

1.00

Fig. VIII-15. Calculated values of thermal dispersion coefficient as a function of Peclet number based on Eq. Vlll-22 showing end effects due to imposed boundary conditions.

middle portion of the tube. (See Fig. Vlll-15). The two peaks are end effects due to imposing constant and uniform temperatures at the two ends. Although the nearly constant values of (hd) in the middle portion of the tube approximate the true value, it appears not feasible to obtain a universally valid correlation between ( h d ) and velocity (Vd) without further analysis. Since the region of interest is far removed from end effects, the asymptotic value of (hd), as the length/diameter ratio of the tube approaches infinity, was studied for two commonly encountered cases: steady-state temperature field and a step temperature front. Parabolic velocity distribution was assumed for both cases.

For the case of a steady-state

temperature field, Eq. Vlll-22 yields the following expression for the dimensionless thermal coefficient (hnd):

hnd

=

1+(Ped2/24$)(hf/he){3/8+8[1-$+0.75@ ( h f / h s ) /1(-$)]/[$+B( 1

where hnd = M h e ; h e @ = In@

=

static thermal conductivity for parallel heat flow

+ 4$ - $2 - 3. The parameter

-$)I}

= $

( V I 11-23)

hf + (1- $ ) h s ;

B and the Peclet number (Ped) are defined as follows:

143 B = CspslCfpf

and

Ped

=

2rwvdCfpf/hf

For the case of a progressing temperature front, the dimensionless dispersion coefficient becomes:

(Vlll-24) It is important to note that both Eqs. Vlll-23 and Vlll-24 show that (hnd) increases with

the Peclet number squared. Thus, regardless of the flow direction with respect to the heat flux direction, counter flow or co-current flow, thermal dispersion always increases due to fluid flow.

Figure VIII-16 shows a plot of dimensionless thermal dispersion coefficient versus Peclet number for an air-glass system based on experimental data by Yagi, et al. (1960) and Gunn arid DeSouza (1974). The solid curve is from calculations based on Eq. Vlll-24 for transient fluid flow. Although there is a large spread in the experimental data, the prediction fits the data reasonably well. Figure Vlll-17 is a similar plot of experimental results of Green,

A i r-GI ass

#) = 0.4 p=l.8x1O3

-

X,/xf = 23 X,/xf= 8

i y:og:e+:l.

a dp= 2.6mm (1960) rn dp= 6.0mm Gunn EI De Souzo (1974 o do= 0 . 4 6 m m

I.o

10

to2

to3

Pecle t Number Fig. VIII-16. Dimensionless dispersion coefficient as function of Peclet number for transient flow condition in air-glass system. Source: Yuan, et al., 1991.

144

8.0 + L c

..a_, .-u Y-

u-

Q)

t

-

Water - G I ass

0

c

0 v)

._ 0 v) v) a)

O

t

n

/

1

8o

oofio

Green,et al. ( 1 9 6 0 ) A dp= 0.46mm 8 d p = I. I mm o d p = 3.0mm

I

I .o

I

I

I

I

I

10 Pecle t Number

I

I

lo2

Fig. VIII-17. Dimensionless dispersion coefficient as function of Peclet number for transient flow condition in water-glass system. Source: Yuan, et al., 1991.

+ = 0.4 fi = 6.7 x

rc.

lo3

Kunii €3 Smith (1961) h d p = 0 . 5 7 mm 0 d p = l.02mm Eq. Vlll - 2 5

v) v)

a, .-

c

.0 _

0.5 -

-

v)

c

a,

.-E n

0.

1

I

I

Fig. Vlll-18. Dimensionless dispersion coefficient as function of Peclet number for steady-slate flow condition in water-glass system. Source: Yuan, et al., 1991.

145

et al. (1964) for a water-glass system.

The prediction based on Eq. Vlll-24 shows

considerable deviation from the experimental results. Figure Vlll-18 shows experimental data of Kuni and Smith (1961) for steady-state temperature distribution in a water-glass system at low Peclet numbers. These low Peclet numbers are in the range that would be expected for subsurface fluid-flow rates. The solid line represents predicted values for a steady-slate system using Eq. Vlll-23.

Although the

experimental data shows a small increase in the dimensionless dispersion coefficient beginning at a Peclet number of about 0.12, the model shows essentially no change within the range of the plot. The difficulty with the above analysis is the over-simplification of using a single uniform-diameter tube to model porous media.

Porous media always display large

heterogeneities which need to be taken into account in modeling. A first step in modeling these heterogeneities is to consider a bundle of tubes of different diameters. Equations Vlll-23 and Vlll-24 were extended to investigate the effects of heterogeneities on thermal dispersion coefficients. These two equations may be rewritten in the following form:

And = 1 + Pedz G(@,Xs/)lf,B)

(Vlll-25)

where the particular form of the function G(@,hs/kf,B) can be determined based on the temperature field. Applying Eq. Vlll-25 to a bundle of tubes of the same solid wall material with various radii but with the same ratio of ro/rw, yields the the following dimensionless thermal dispersion coefficient for the bundle of tubes as a whole:

(VI 11-26)

where kndb and Pedb are the thermal dispersion coefficient and the Peclet number for the bundle as a whole, respectively. The Peclet number for the bundle is based on the average diameter of the tubes in the bundle. The term

5 is a function of the distribution function of

the radii of the

tubes in the bundle, defined as follows:

where f(rw) is the distribution function of the tubes in the bundle and rwa is the average radius

of the tubes in the bundle.

146

TABLE Vlll-4

Values of ,€ for certain distribution functions. Source: Yuan (1991).

--

F

Mran

Function

A c

0

Gamma

PI01

a/

-a P

a

4

1 140.0

0

4

P * i

c 0

rSO

-

L

0

W

I

0

L"(P,o2)

0'

L

Normal

+

I

I

10.0

U

5

0.1

1.2 4.6 4.lr10'

2 3

-1 E

r -m

3,000 .

.4 m

2,500 .

C

m

2

Q

k

0

2,000

'

I

1

I

I

I

0

2,000

4,000

6,000

a,ooo

I J

10,ooo

Effective Stress, psi

Fig. X-8. Effects of temperature and stress on compressional-wave velocities of Cerro Prieto sandstone dry and liquid saturated. (Palen, 1978)

The effects of stress on dilatational and shear wave velocities can be represented by an equation of the form:

189

where A = a constant for different rock types;

CT =

effective stress; x

=

exponent for different

rock types; subscripts (p) and (s) refer to dilatational and shear waves, respectively.

Effective Stress, MPa 0

-

10

20

30

50

40

60

70

2,000

u)

E .b 1,750 v

-8 Q

> L

2m

1,500

T("C)

Dry

175

0

Saturated

v)

1,250 0

2,000

4,000

6,000

8,000

10,000

Effective Stress, psi Fig. X-9. Effects of temperature and stress on shear-wave velocities of Cerro Prieto sandstone dry and liquid-saturated. (Palen, 1978) Palen (1978) has found that the value of (A) for shear velocities varies widely for different rock types, ranging from 500 for softer rocks to 3,000 for harder rocks. Constants for compressional wave velocities are approximately 50 percent higher. The exponent (x) ranges from a value of 6 for unconsolidated sands to well over 40 for hard rocks. Although analytical expressions for the effects of temperature and fluid saturation on sonic velocities have not been formulated, it is probable that the constant (A) and exponent (x) in Eq. X-3 can be adjusted to account for these effects. The Cerra Prieto sandstone, which is much more friable than Berea sandstone, has an average exponent value of about 9 in dry condition. When saturated with liquid, this value increases to 20, indicating a much lesser effect of stress when liquid-saturated. This same effect, although to a lesser degree, will be noted for Berea sandstone. Based on well-known expressions relating P- and S-wave velocities to dynamic elastic moduli, Palen (1978) has calculated Young's moduli, bulk moduli and Poisson's ratios for the Cerra Prieto sandstone.

All three moduli show decreases with increased temperature.

Somerton, et al. ( 1 974) have shown that bulk compressibilities calculated from sonic velocities (dynamic) may be converted to static values by the expression:

190

Calculated bulk compressibility values from Palen's data at 20°C using Eq. X-4, are shown as x's in Fig. X-2. Ignoring the effect of temperature on sonic velocities could lead to error in interpreting transit-time logs run in high-temperature reservoirs.

The decrease in sonic velocities with

increased temperature could result in predicting higher porosities in using the time-average relationship. This is contradictory to some evidence that porosity tends to decrease with higher temperatures. 3 PERMEABILITY

A good deal of controversy exists as to the effect of temperature on permeability of rocks. Weinbrandt, et al. (1975) have published data for several sandstones indicating substantial decreases in absolute permeability with increased temperature.

Since the decreases in

permeability were observed to be reversible, this would be interpreted as an effect of temperature rather than of heating.

Further work by Aruna (1976), Danesh, et al. (1978).

and Gobran, et al. (1980) tended to confirm the effect of temperature in decreasing permeability when water was the flowing fluid.

However, other flowing fluids including

nitrogen, oil and organic solvents did not show any significant effect of temperature on absolute permeability.

Following this lead, Potter, et al. (1981) ran flow experiments on St. Peter

sandstone which indicated that the decrease in permeability with increased temperature observed by other investigators was probably due to the formation of colloidal ferric oxides or hydroxides in the stainless steel flow lines, causing plugging of the rock pore spaces. When they used a titanium flow system instead of stainless steel, permeability of St. Peter sandstone was found to be independent of temperature.

In tests with the stainless steel flow lines, Potter, et al. (1981) observed a partial restoration of permeability after stopping flow for a period of time.

This they considered

consistent with the reversibility of permeability upon reducing temperature as reported by earlier investigators. Several studies have been made by the author and his co-workers on the effect of temperature on absolute permeability.

Wong (1 979) ran permeability tests on several

sandstones using an apparatus developed earlier by Mathur (1976). Tests were run on copperjacketed test specimens mounted in a pressure vessel with provisions for external heating. Stainless steel flow lines were used as were stainless steel porous filters to remove particulates from the flowing stream. The flowing fluid was a 6000 ppm KCI solution with flow velocities ranging from about 1 to 10 feet per day. Results of the above permeability tests run on Berea sandstone as a function of temperature and at effective stresses of 3.45 and 6.9 MPa (500 and 1000 psi) are shown in

191

Although there was an apparent decrease in permeability of about 25 percent at the

Fig. X-10.

lower stress, most of this was recovered on the cooling cycle. Increasing the effective stress to 1000 psi resulted in decrease in permeability of the order of 50 percent.

Much less of this

permeability was restored on the cooling cycle. Flow tests on other Berea test specimens and on Boise, Bandera, and Cerra Prieto sandstones showed erratic results similar to the above except that they all showed a greater degree of reversibility.

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40

60

80

100

120

140

160

180

Temperature, "C

Fig. X-10. Relative change in measured permeability of Berea sandstone as function of temperature at two stress levels. (Wong, 1979) In an effort to explain the loss in permeability with increased temperature, Wong analyzed the silicon content of the effluent liquid from the Berea flow tests and found it to increase from about 10 l g / m l at room temperature to about 10 times that amount at 165°C. It was not determined, however, whether the silicon content was in dissolved form from silica dissolution or was in the form of ultra-fine particulates. This work was done before the work of Potter, et al. (1981) reported the colloidal iron plugging effect and iron content of the effluent was not determined. On the basis of the evidence available, Wong concluded that the irreversible portion of the permeability loss was due to particulate plugging and the reversible portion was a temperature effect possibly due to thermal expansion of mineral grains at pore throats.

192

Wong's work on the effects of temperature on permeability of sandstones was followed up by Okoh (1981). An important difference in his experimental apparatus was the removal of the stainless steel filters from the high temperature environment. The effective stress on the test specimens was increased to 13.8 MPa (2000 psi). Flow rates of the 6000 ppm KCI solution were limited to the equivalent of 5 feet per day and heating rate to less than 1°C per minute. This was done to reduce mechanical drag and thermal shock in an effort to minimize transport and plugging by particulates. Berea and Bandera sandstones were selected for the tests in order to have a ten-fold difference in initial permeabilities. With the modified procedures and the special precautions taken in running the flow tests, results of the tests showed no more than 20-25 percent decrease in permeabilities for temperatures up to 165°C. Data taken during the heating cycle were compared with data at equivalent stabilized temperatures and the resulting calculated permeabilities showed essentially no differences. In a few cases there was a slight decline in permeability with time at constant temperature and flow rate. The direction of fluid flow was reversed to test for plugging and in all cases permeability was partially restored for a short period of time. In an effort to explain the reversible portion of the permeability reduction, Okoh applied Marshall's equation and estimated the effects of pore-throat contraction by thermal expansion. The Marshall equation was used in the following form:

k

n = @2/8n2[X(2j - l ) r j Z ] j=1

where k = permeability, Darcys; I$ = fractional porosity; n

=

number of pore throats; rj

=

radius of jth pore throats, microns; j = running index. Assuming the porosity change was negligible within the temperature range of the tests, Eq. X-5 may be expressed as:

The relative value of (k) will then depend only on the distribution of pore-throat radii. The radii for the test specimen at elevated temperature can be expressed as:

rjT = rj -

(p rj AT)

where rjT = pore-throat radius of heated rock; rj

(X-7)

=

pore-throat radius of unheated rock;

P=

coefficient of thermal expansion of rock solids; AT= temperature increase. Using pore-size distribution data for Berea sandstone and thermal expansion data from Ashqar (1979), Okoh calculated the effect of temperature on permeability as shown by the solid

193 line in Fig. X-11.

Experimental values of permeability versus temperature for three Berea

sandstone samples are shown on the same figure. The broken line was calculated based on the observation that the Berea samples tested showed an average decrease in porosity of 12 percent due to compaction by heating and stressing during the flow tests. Porosity was assumed to decrease linearly with increase in temperature. These results show that both the reversible and irreversible decrease in permeability with increased temperature can be explained theoretically on the above basis.

1'2

I A Berea 1 o Berea 2 0 Berea 3

-- 0.2 20

40

Calculation Using Marshall's Eq. Calculation Using Marshall's Eq., Porosity Corrected

60

80

100

120

140

160

180

Temperature, "C Fig. X-11. Relative change in measured permeability of Berea sandstone as function of temperature compared with calculated values based on Marshall's equation. (Okoh, 1981) The problem of silica dissolution at elevated temperatures and its role in reduction of permeability was studied by Lofy (1983). In his experimental work he used unconsolidated Ottawa sand and glass beads, flowing deionized and deaerated water with an applied effective stress on the sand pack of 11.7 MPA (1700 psi) and temperatures from 25" to 150°C. The effluent was monitored for silicon and iron content using Atomic Absorption Spectrophotometry

(AAS). The flow apparatus was similar to that used by Wong and Okoh except that the stainless steel filters were replaced with alundum filters.

Lofy's test results showed that permeability of both the Ottawa sand and the glass beads tended to increase initially at the lower temperatures (50' to 75°C) but that in all cases permeability decreased at the higher temperatures. The decrease in permeability for Ottawa sand was of the order of 30 to 40 percent but the decrease for the glass beads was as much as 90 percent. Effluent silicon content for the Ottawa sand reached a maximum value of about 150

194

ppm at 150°C but for the glass beads it was as high as 4300 ppm at this temperature. The iron content of the effluent was hardly measurable with the AAS, being less than 0.5 ppm. The glass beads were examined with a Scanning Electron Microscope (SEM) before and after the flow tests. Beads before the flow tests had smooth surfaces but after the tests they showed growths of filaments on the surfaces.

If left in the high-temperature, high-stress

environment over long periods of time, the glass beads became a compact mass with essentially no porosity or permeability. The conclusion reach from Lofy's work was that permeability reduction was due to silicdwater reactions. Dissolution of silica at the lower temperatures caused a small increase in permeability but by increasing the temperature under constant stress, the packs became more compact and the permeability decreased.

Since there was no detectable iron in the

effluent, plugging by colloidal iron oxides or hydroxides, as proposed by Potter, et al. (1981). was unlikely. The exaggerated behavior of the glass beads was attributed to the substantially greater solubility of the amorphous silica in the beads than in the crystalline form in the quartz grains of the Ottawa sand. 4 FORMATION RESISTIVITY FACTOR

Formation resistivity factor and permeability can be related by the term "tortuosity" which is common to both. The Kozeny-Carman equation for permeability may be expressed as:

where k = permeability, cm2; q~ = fractional porosity; 'I = tortuosity; SO= specific surface area, cm2tcm3. Formation resistivity factor may be related to tortuosity as:

Thus, where changes in porosity and surface area with temperature are small, permeability would be expected to change with the reciprocal of tortuosity while formation resistivity factor should change directly with change in tortuosity. Tortuosity would be expected to increase with increase in temperature due to thermal expansion of mineral grains into pore spaces and possibly due to plugging of pore spaces by particulates and/or redeposition of dissolved silica. On these bases, permeability would be expected to decrease and formation resistivity factor to increase with increased temperature.

It is also apparent that if porosity decreased with

increased temperature, this would accentuate the effects of temperature on these two properties.

195

In conjunction with tests on the effects of temperature on permeability, Wong (1979) ran tests on the effect of temperature on formation resistivity factor. In these tests Wong used a radial system with a central needle electrode and with the outer copper sheath of the test specimen serving as the outer electrode. The porous stainless steel filters at either end of the test specimen were replaced with insulated discs of Lava. Results of the tests on Berea sandstone run at an effective stress of 6.7 MPa (1000 psi) showed a four-fold increase in formation resistivity factor from room temperature to 165°C. This was a much larger change that was expected or has been reported by others, [Brannan (1973) and Saynal, et al. (1972)]. Reasons for this excessively large increase could not be explained and, consequently, this work was suspended. Rugama (1981) made a thorough study of the literature on the effects of temperature on formation resistivity factor.

The data and the analysis presented were highly contradictory.

The work of Ucok (1979) seemed to be the most reasonable. His experimental results showed a slight decrease in formation resistivity factor with increased temperature to about 160°C, the decrease being greater for rock samples having high clay content. resistivity factor tends to increase.

Above 160°C formation

Ucok attributed this to physical changes in pore

constrictions due to thermal expansion of mineral grains with increased temperature. Rugama (1981) ran tests on three outcrop sandstones (Bandera, Berea and Boise) using a two-electrode system and he also tested the radial system used by Wong. Electrodes used for the two-electrode system were gold-plated to minimize surface resistance. Tests were run at three frequencies - 0.5, 1.0 and 3.88 kHz. Results of Rugama’s measurements on the three sandstones are shown in Fig. X-12. The formation resistivity factor is seen to be essentially independent of temperature to about

26[

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75

100

1 .OkHz

A

I 125

I

150

I 175

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37

200

Temperature, “C

Fig. X-12. Effect of temperature on values of formation resistivity factors for three sandstones measured at frequency of 1.O kHz. (Rugama, 1981)

196 2.0

I

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0.8 I

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I

I

25

50

75

100

125

150

175

200

Temperature, "C

Fig. X-13. Effect of temperature on relative values of formation resistivity factor for Berea sandstone as measured by several investigators. 150°C. Above this temperature, the formation resistivity factor begins to increase, agreeing with Ucoks findings.

Figure X-13 compares the results of Sanyal et al. (1972), Brannan and

Von Gonten (1973) and Ucok (1979) for Berea sandstone, with Rugama's results. There is wide discrepancy in the results but good agreement between Ucok's and Rugama's results. Figure X-14 compares the results of Rugama's measurements at different frequencies. There is seen to be little difference in relative formation resistivity factors at 0.5 and 1.0 kHz and only a slightly greater effect at the highest frequency (3.88 kHz). The highest frequency tests did show higher values of absolute formation resistivity factors. The effect of temperature showed the same trends at all three frequencies. Rugama ran a few tests using the radial electrode system. He found that as long as the needle electrode made good contact with the rock, results of measurements agreed with linear test results within a few percent.

It is believed that Wong's results with the radial electrode

system were in error, perhaps due to inadequate contact of the needle electrode with the rock. There is insufficient information to explain the large differences between Rugama's and Ucok's results and Saynal's and Brannon and Von Gonten's results. Further work will be needed to reconcile these differences.

197

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-

1.8

.

1.6

.

I

1

I

1

7

1

1

1.4

1.2 1.o

1

0.8 1

1

I

I

I

I

I

Fig. X-14. Effect of temperature on relative values of formation resistivity factor for Berea sandstone measured at several frequencies. (Rugama, 1981) 5 SUMMARY AND CONCLUSIONS There is good evidence to show that bulk, pore and rock-solids compressibilities increase with increased temperatures.

Although the effect is modest in amount, fairly

significant errors can be made in estimating reservoir pore-volume changes with pore-fluid pressure decline and in predicting the extent of subsidence, if temperature effects are ignored. If laboratory compressibility measurements are made, tests should be run at reservoir temperatures. Both P- and S-wave velocities decrease with increased temperature.

Dynamic

deformation moduli calculated from velocity data all tend to decrease with increased temperature. These results are consistent with the effects of temperature on compressibilities mentioned above. The temperature effects on interpretation of transit-time logs could lead to errors in estimating porosities from such logs.

Lower velocities would indicate higher

porosities when, in fact, elevated temperatures tend to decrease porosities. The effects of temperature on permeabilities and formation resistivity factors are not clear at this time. tortuosity.

These two properties should be related through the common term of

Earlier work showed that permeability decreased and formation resistivity factor

increased with increased temperature. More recent work shows little or no change in these two properties at temperatures to somewhat in excess of about 120°C. Above this temperature the

198

effects of temperature on the physical behavior of the rock appears to cause decrease in permeablities and increase in formation resistivity factors.

Further work is needed to

reconcile the major differences reported in the literature on these two properties.

199

Chapter XI LOW TEMPERATURE BEHAVIOR OF RCCWFLUID SYSTEMS

There are some cases where low-temperature behavior of rocks is very important. The most notable case is that of permafrost, or permanently frozen ground. According to Kurfurst and King (1972), about 20 percent of the earth's land surface is covered with permafrost. Thus, study of the physical properties and behavior of rocks containing ice is important in many applications.

It is obvious that roads and structures built on permafrost or wells drilled

through permafrost zones can be subjected to serious problems if heating occurs. In addition to 'these potential problems, interpretation of geophysical prospecting data can be complicated if the physical properties being measured are affected by the presence of ice in the pore spaces. It should be pointed out that the effects of low temperatures on rock behavior are similar

to the effects of subjecting rocks to high temperatures. The effects of temperature per se may not be large but the effects of cooling, in particular if ice is formed in the pore spaces, can have a very significant effect on some physical properties. may not be large.

Low temperature effects in dry rocks

In addition, Kurfurst and King (1972) point out that freezing of water-

saturated rocks has little effect on properties such as density, magnetism, or radioactivity. However, properties such as elastic-wave velocities, electrical resistivities, and thermal conductivities may show radical changes for water-saturated rocks subjected to freezing temperatures. The effects of lowering temperatures on these latter properties will be discussed in detail in the following.* 1. P- AND S-WAVE VELOCITIES AND DYNAMIC AND STATIC ELASTIC MODULI Kurfurst and King (1972) have measured P- and S-wave velocities of dry and watersaturated Boise and Berea sandstones at temperatures as low as -20°C. They also measured static elastic constants for the same sandstones under similar conditions and compared these results with dynamic elastic moduli calculated from velocity data. P- and S-wave velocities for both sandstones under dry conditions showed essentially no difference when the temperature was lowered from ambient temperature to -20°C. For water-saturated Boise sandstone both P- and S-wave velocities were increased by more than 50 percent by lowering the temperature to 'It should be pointed out that most of the work reported here has been done by our former colleague, Dr. Michael S. King, now at Imperial College of Science, Technology & Medicine in London. and his co-workers.

200 about the same value. For Berea sandstone the increase in the two velocities was somewhat dependent on the stress level but averaged about 45 percent. These increases are believed to be due to the cementing action of the ice formed in the pore spaces. Figure XI-1 shows values of Young's modulus as a function of temperature for Boise sandstone calculated from velocity data (ED) and as measured by static tests (Es) for both dry (dashed line) and water-saturated (solid lines) test specimens. Since there is some effect of stress level on these results, all tests shown in the figure were run at constant stress of 13.8 MPa. It is apparent that the Young's modulus for the dry samples is only slightly affected by lowering the temperature. The dramatic increase in the dynamic modulus with decrease in temperature might be expected from the elastic-velocity results. However, an actual decrease in static Young's modulus for the water-saturated specimen below freezing temperatures is difficult to explain. Similar results were obtained for Berea sandstone as shown in Fig. XI-2.

40

Axial Stress 13.8 MPa Saturated D rY

0

---

LL

a 30 v)

3 -

ED

3

20 2 cn

ED

0,

C 3

>c"

10

c---a---+

0 Temperature, "C Fig. XI-1. Static and dynamic Young's moduli for Boise sandstone dry and water saturated. Source: Kurfurst and King (1972). Later work by King (1977) confirmed the increase in P- and S-wave velocities for tests run on one sandstone and two shaley sandstones.

Figures XI-3b and XI-4 show P- and S-wave

velocities for these samples and two shale samples as function of temperature at a constant axial Stress of 3.5 MPa. It should be noted that the samples were transported and stored in their original frozen state (-10°C). Description of the five samples tested are given in Table XI-1. It is interesting to note that the shale samples showed no effect on velocities with decrease in temperature below the freezing point. King attributes this to the fine-grained structure of the shales and the salinity of the pore water. attributes it to the same causes.

Timur (1968) has observed this same effect and

20 1

30 0

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0 c

2 3

20

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0

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v)

IC 3

0

>

0

I

I

- 20

I

I

- 10

I

I

Temperature,

I

I

10

0 OC

Fig. XI-2. Static and dynamic Young’s moduli for Berea sandstone dry and water saturated. Source: Kurfurst and King (1972).


”

4000

ss4

> SH I I

.-0 v) v)

F ss5

-

0.

I 20

Temperature, OC Fig. XI-3. P-wave velocity as a function of temperature at axial stress of 3.5 MPa for several sedimentary rocks. Source: King (1977).

3000

-

-

u)

\

E

ss I ss5

c

r.

'3 2000

-

>"

-

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-SSI

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>

ss4

f 1000 -

h

SH2

ss5

L

-

0 Q)

r

cn

-

I

- 10

-15

-5 Temperature, "C

0

5

Fig. XI-4. S-wave velocity as a function of temperature at axial stress of 3.5 MPa for several sedimentary rocks. Source: King (1977). Table XI-1. Description and physical properties of test specimens. Source: King (1977).

No.

Specimen description

SH2 SS1 SS4 SS5

Black shale Clean sandst. Shaleysandst. Shaleysandst.

Bulk Density(gkm31 Frozen Dry 2.33 2.45 2.37 2.33

2.21 2.33 2.24 2.17

Porosity (frac.) 0.1 3 0.1 4 0.1 5 0.1 7

Moisture content (frac. dry wt.) 0.058 0.053 0.06 1 0.074

2. ELECTRICAL PROPERTIES

Early investigators have reported that the electrical resistivities of water-bearing rocks show pronounced increase as the temperature is decreased below 0°C. Collett (1974) concludes that this increase is a function of the water content of the rock, the chemical composition of the water, and the mineral composition and grain size of the rock. King (1977) has measured the electrical resistivities of the several rock samples described in Table XI-1. Results of these tests show electrical resistivities measured at 1 kHz frequency and 0.35 MPa hydrostatic stress, as function of temperature in Fig. XI-5.

These

results show large increases in resistivities as the temperature is lowered below the freezing

203

point for the four sandstones but, as in the case of P- and S-wave velocities, only a relatively small increase for the shale sample. It should be noted that the arrows show the direction of temperature change and that in most cases, the temperature was increased from sub-zero to above freezing. Some hysteresis will be noted between increasing and decreasing temperature runs. King found that there was a general decrease in resistivity with increased frequency at which the measurements were made. This is in agreement with Rugama's (1984) findings for measurements on Berea sandstone in which the frequency was varied. Timur (1968) postulates that ice forms first in the larger pore spaces as the temperature is reduced below freezing, and then forms in progressively smaller pores as the temperature is further reduced.

He further considers that the large interfacial forces

associated with the large surface areas in fine grain-size materials tends to inhibit the formation of ice in the very fine pore spaces. These factors help explain the relatively small effect of temperature on the resistivity of shales and also the delayed effect of lowering temperature on the two shaley sandstones below the freezing point as may be noted in Fig. XI-5.

ss I

\ss5

-15

-10

-5 0 5 Tern pera ture, OC

10

Fig. XI-5. Electrical resistivity as a function of temperature at frequency of 1 kHz and axial stress of 0.35 MPa for several sedimentary rocks. Source: King (1977).

204

3. THERMAL CONDUCTIVITY

King (1979). using a divided-bar apparatus, has measured thermal conductivities of two sandstones and a limestone at permafrost temperatures.

The Boise sandstone had a

fractional porosity of 0.25, Berea porosity was 0.18, and the Salem limestone 0.12. Results of the tests run in the temperature range of -13' to +5"C and an axial stress of 0.7 MPa are shown in Fig. XI-6. A small decrease in thermal conductivity from the lowest test temperalure to close to 0°C for the three samples will be noted. At 0°C a sharp drop in thermal conductivity occurs. As ice in the pore spaces melts, a further small decrease in thermal conductivity occurs as the temperature rises above the freezing point. These results are consistent with the above reported behavior of P- and S-wave velocities and electrical resistivities in the temperature region of 0°C.

The increase in thermal conductivity, when the pore fluid freezes, would be

expected for two reasons. First, the ice tends to make the rocks more completely "cemented" as was considered the case for the other two properties. Second, perhaps of greater importance in the case of thermal conductivities, the thermal conductivity of ice is about four times that of water at 0°C. Clark (1966) gives a value for ice at 0°C of 2.22 Wlm-K and for water at the same temperature of 0.56 Wlm-K. The results of King's measurements on water-saturated Boise and Berea sandstones agree quite closely with those values reported in Chapter V when the latter values are adjusted to 0°C by use of Eq. V-25. There is some concern regarding King's results in that tests were run at a low axial stress, perhaps too low to minimize contact resistance problems in the divided bar apparatus.

King studied this matter and concluded that the measured values of thermal

conductivity were only 1 to 2 percent lower due to this cause than the actual values.

4. OTHER LOW-TEMPERATUREEFFECTS

A number of other studies have been made on the physical properties and behavior of rocks and unconsolidated sediments at permafrost temperatures. Pandit and King (1979) have studied the effects of pore-water salinity on elastic-wave velocities and electrical resistivities of sedimentary rocks at permafrost temperatures.

They found that increased salinity of pore

water diminished the sharp increase in velocity

observed below 0°C.

Similar effect was

observed for electical resistivity. They attribute these effects to the decrease in ice content in the pore spaces at a given temperature for the increased salinity pore fluid. King (1984) studied the influence of clay-sized particles on seismic velocity of Canadian Arctic permafrost. He found that at temperatures below -2"C,

P-wave velocities were strongly dependent on the

fraction of clay-sized particles, decreasing with increased clay content, but essentially independent of porosity. At temperatures above 0°C P-wave velocity of the unconsolidated permafrost was a function of water-filled porosity and essentially unaffected by the original porosity, clay content or temperature.

205 Ternpe rature Drop

5.5

4% H

5.0 Berea Sandstone Y

;4.5 -

\

t

2.5 2.0I

I

I

I

Salem Limestone I

-15

I

I

I

0 -5 Mean Temperature, O C

-10

I

5

Fig. XI-6. Thermal conductivities of water-saturated Berea and Boise sandstones and Salem limestone as function of temperature at 0.35 MPa. Source: King (1979).

King, e l al. (1982) ran tests on Beaufort Sea sub-seabottom permafrost to determine seismic, electrical and thermal properties of these materials. Results agreed in general with those reported above and will not be presented here. Zimmerman and King (1986) and King, el al. (1988) have reported on development of a model to predict seismic velocities and electrical properties from porosity and the extent of freezing of the interstitial water.

One difficulty in

testing the model was the lack of a direct method to determine the extent of freezing. The extent of freezing has been inferred from measured values of the ratio of the resistivity of permafrost

in the frozen state to that in its unfrozen state. Since the topics included in this final section cover a broad area of study beyond the scope of this book, the reader is referred to the original papers as given in the References for further information.

206

Chapter XI1 WELLBORE APPLICATIONS

The difficulties of obtaining reliable data to characterize the thermal properties and thermal behavior of subsurface rock/fluid systems have led to efforts to obtain such data from borehole logging measurements. Two different approaches have been taken. The first of these is to use well log data directly and attempt to correlate these data with measured values of thermal conductivity or, use interpreted well log data and apply these data to previously established The second approach is to derive thermal properties data from temperature

correlations.

gradient surveys in boreholes. These two approaches will be reviewed in the following. In addition to deriving thermal data from borehole measurements, there are other wellbore applications. One of these to be discussed here is that of heat losses in thermal injection wells or in geothermal producing wells. The specific problem to be discussed is the role of the VCC effect (see Chapter VIII) in heat losses from thermal wells. 1 THERMAL DATA FROM WELL LOGS

Some of the early work in predicting thermal conductivity from well log data was reported by Goss. et al. (1975).

They measured thermal conductivity and several other

physical properties for a suite of Imperial Valley rock samples.

They then developed an

empirical correlation from the data and applied this relationship to well-log data for the borehole section from which the core samples had been taken. The correlation equation was expressed as:

h

where h

=

=

1.33 - 0.026 Q

+

0.38 Vp

thermal conductivity, Wlm-K;

(XII-1)

(I =

fractional porosity; Vp

=

compressional (P)

wave velocity, kmls. For this suite of samples, representing a specific geological environment, Eq. XII-1 gave a multiple regression coefficient of 0.965.

Applying this correlation to the well-log data,

including neutron porosity and transit-time logs, the general trend of the predicted and experimental values agreed well but the mean predicted value through the interval tested was 20-25 percent lower than the mean experimental value.

207 It was reported earlier (Chaper V) that Sahnine (1979) found a rather poor correlation

for thermal conductivity of a suite of Mid-Continent sandstones and siltstones when only measured P-wave velocities and porosities were used in the regression analysis.

The

correlation was greatly improved when formation resistivity factor was included in the analysis. This demonstrates the inherent danger in attempting to apply a correlation for one suite of samples from a given geological environment to other suites of samples from other geological environments.

In an effort to obtain a correlation of more general applicability, Dea (1976) applied the following correlation equation (developed in Chapter V) to estimate thermal conductivities from wellbore-log data:

1 = 1.27 - 2.25 4 + 0.39 h ~ S w ' ' ~ where h

=

(XI1-2)

thermal conductivity, W/m-K; I$= fractional porosity; hs

=

thermal conductivity of

rock solids, W/m-K; SW = saturation of wetting-phase fluid. This correlation provides for variable wetting-phase fluid saturation and also takes into account differences in mineral composition of the rock.

If mineral composition is properly

accounted for, it seems reasonable to assume that P-wave velocities would not be necessary in a correlation that contains the end product of this quantity, i.e., porosity. In applying the above correlation to well-log data, Dea (1976) used the well-log

combinations shown in Table XII-1 to evaluate the three correlating quantities in Eq. Xll-2. Appropriate borehole, bed thickness and invasion corrections need to be applied to the log data to obtain the desired correlating quantities. A computer program was developed to obtain values of

(6)and (SW) from digitized well-log data. The evaluation of ( h s) is difficult.

If the quartz

content can be estimated, an approximate value of (hs) for sandstones or siltstones can be obtained from the following relation given earlier (Chapter V):

hs = 7.70 Q + 2.85(1 - Q )

(Xll-3)

where Q = fractional quartz content. Since shale usually has low thermal conduntivity, a value of 3.0 W/m-K for (AS) for shale is suggested. In application, the (hs) value estimated from Eq. Xll-3 is assigned to the maximum value on the SP or gamma log (clean sand) and the shale value is assigned to the minimum value (shale base line) and a linear interpolation is made to evaluate (AS) values for intermediate log values.

208 Table XII-1 Log data needed for estimation of thermal conductivity by method of Dea (1976). Property

Logging Method 1. SP, induction, transit time log. 2. SP, induction andlor laterolog,

Water saturation

neutron andlor density log. 3. 16” short normal, neutron and/or density log. 4. Sp, microlaterolog, induction and/or density log. 1. density log 2. transit time log

Porosity

3. neutron log 1. SP log 2. gammalog

Rock solids conductivity

bepth

Depth

Feet)

(Feet)

8350

8350

8400

8400

8450

8450

8500

8500

c

10

SP (mV)

20

Resistivity (Ohm-m*/m) Deep induction

-----

1 6 Short Normal

-

8550

8550

150

100

Sonic (FSeciFt)

0

50

Predicted Thermal Conductivity, W/m

-K

Fig. XII-1. Thermal conductivity estimated from log data for a Mid-Continent well. (Dea,1976) To test the computer program, well-log data for a Mid-Continent well, shown in Fig. Xll-1, were used. The SP, induction, and transit-time logs were hand-digitized every foot. Calculated thermal conductivities for each foot and for 10-foot averages are shown in Fig. Xll-2. Although measured thermal conductivity values were not available for this well, calculated values are

209

8350

8400

c

A

Q)

a3

k 8450 5 P p"

8500

8550 Predicted Thermal Conductivity W/m - K Fig. Xll-2. Comparison of point values and ten-foot average values for estimating thermal conductivity from well log data. (Dea. 1976) within the range expected for this type of formation. The above analysis was tested with oil sand data from three San Joaquin Valley wells. Well log data including SP, induction, gamma, density, and neutron porosity, are shown in Figs. Xll-3 through Xll-5.

Thermal conductivities of core samples from the wells were measured

using the steady-state comparator apparatus described earlier (Chapter V).

Results of the

measurements and calculations are shown in Table Xll-2. In interpreting the results in Table Xll-2, the reader should note the large differences in porosity and in particular water saturations as determined by the three methods for the same core depth. Figure Xll-6 shows a

plot of measured versus calculated thermal conductivities. Calculations based on measured core data are shown as the open data points and those based on log data calculations are shown as the

210

Depth (fi)

7 122

Depth

Depth

600

-

+ sp (mv)

700

700

800

800

900

900

0

80

40

Resistivity (Ohm-m*/m) Deep Induction

----

1 6 Short Normal

-

2.0

U

Gamma Ray (API Units)

Porosity Density Neutron

(Oh)

-----

3.0

--

K

Thermal Conductivity, W/m Log Predicted Core Calculated 0 Measured X

Fig. Xll-3. Thermal conductivity estimated from log data for a Kern River (A) well. (Dea,1976)

Depth (ft)

Depth (11)

Depth (11)

800

900

1000

1100

5-

,Ps

SP (mV)

120 Resistivity (Ohm-mVm) Deep Induction -----16 Short Normal

-

n Gamma Ray (API Units)

Porosity ('10) Density Neutron -----.

-

~

0 1.o 2.0 3.0 Thermal Conductivity, W/m K

Log Predicted Values

--

Core Calculated Values o X Measured

Fig. Xll-4. Thermal conductivity estimated from log data for a Kern River (SJ) well. (Dea, 1976)

21 1 Depth (11)

Depth (11)

Resistivity (Ohm-m*/m) Deep Induction

sp (mv)

-----

Gamma Ray (API Units)

16" Short Normal-

Depth (A)

-

Porosity (%) Density Neutron

Thermal Conductivity, Wlm Log Predicted Core Calculated 0 Values

-----

Fig. Xll-5. Thermal conductivity estimated from log data for a Mc Kittrick well. (Dea, 1976)

3.0

/ Y I

/'

/'

E

I .-* 2 2.0 c

0

3 0 C

s-

m

9p

Fi

z

I-

1.0

/

Q

-m c

3

-0

s

/ a

0

/

/ 1.o

Kern River A

Core Data o

Kern River SJ

o

McKittrick

a

2.0

3.0

Measured Thermal Conductivity W/m - K Fig. Xll-6. Comparison of measured and calculated values for the several wells.

0

-K

212

TABLE

Xll-2

Comparison of thermal conductivities of several oil sands as measured in the laboratory, estimated from core analysis data, and calculated from well-log data. (Dea, 1976)

Sample

Depth (fi)

Kern R.

@

Laboratorv sw

?L

$

Core Analvisis sw h

$

Well Loas S W

h

715-716 719-720 894-895 898-899

,300 ,334 ,363 ,341

,720 .404 ,484 ,562

2.16 1.73 1.61 1.85

,318 ,229 .339 ,324

,396 ,447 .370 .417

1.66 1.92 1.57 1.68

,302 ,317 ,345 .329

.434 ,461 ,362 ,356

1.80 1.85 1.47 1.51

887-888 891 -892 1013-1014 1 0 16 - 101 7

,339 .378 .344 ,301

.490 .477 .471 .494

1.68 1.59 1.85 1.97

.346 ,336 .276 ,278

.467 ,633 .413 ,468

1.70 1.90 1.78 1.85

.314 ,311 ,289 ,289

,279 .284 .223 ,205

1.51 1.52 1.45 1.44

885-886 886-887 887-888 "Reward" 8 8 8 - 8 8 9 889-890 1212-1213 1213-1214 1214-1215 1215-1216

.331 ,318 ,318 ,269 ,258 ,328 .343 .334 ,303

.586 .437 .433 ,389 ,378 ,095 ,165 .156 .195

2.02 2.15 1.90 2.20 2.1 8 1.42 1.37 1.31 1.28

.262 .324 .300 .268 ,256 ,313 ,342 .334 ,303

.685 .600 ,612 .462 ,722 .216 ,162 .156 ,195

2.42 2.1 6 2.23 2.09 2.47 1.54 1.35 1.35 1.52

,270 ,260 ,270 .300 ,310 ,290 ,255 .225 ,205

.643 ,670 ,639 ,571 .555 .816 ,883 .959 1.00

2.35 2.42 2.35 2.18 2.11 2.54 2.75 2.94 3.04

"A"

Kern R. "SJ"

McKittrick

Thermal conductivity in W/m-K.

solid points. With the exception of the four points based on log data for the deeper McKittrick section, all calculated data points fall within f15 percent of the measured values. 2 THERMAL GRADIENTS IN WELLS

Temperature surveys are run in wells for a number of important applications.

Of

particular interest here is the possiblity of evaluating in-situ thermal conductivities of formations behind casing from thermal gradient measurements in the borehole. Assuming a constant heat flux from the earth's interior to the surface, changes in thermal gradients should reflect changes in thermal conductivity of the formations through which heat is flowing.

If

changes in thermal conductivity can be detected, the possibility exists that differentiation between shales, water sands, and hydrocarbon-bearing formations can be made in cased wells. In the following analysis, interpretation of thermal gradients in wells will be considered but emphasis will be placed on transient gradient measurements and their interpretation.

213

in W&i

2.1

When a well is at thermal equilibrium with its surroundings and there are no disturbing influences such as local heat sources or sinks, a thermal gradient log is analogous to a thermal resistivity log (reciprocal of thermal conductivity) of the surrounding formations.

To convert

the thermal gradient log to thermal conductivity, an estimate of the terrestrial heat flux in the area of lhe wellbore must be available. As an alternate, and perhaps preferable, one or more known values of thermal conductivity may be used in the analysis. Hoang (1980) has shown, as will be discussed later, that in nearly all cases wellbore fluid temperature gradients are within 4 percent of the geothermal gradients within 72 hours of shut-in. To establish reliable steady-state temperature gradients, it would be useful to have a continuous multi-point temperature log.

Since such a log is generally not available,

consideration must be given to the time constant of the temperature sensing device and the logging speed, and appropriate corrections made as discussed by Conaway (1977). With suitably corrected steady-state gradients, the following relationship is used to determine conductivities of the formations adjacent to the borehole: hillj

=

(aT/JZ)j/(aT/aZ)i

(Xll-4)

This assumes that thermal conductivity of the jth formation is known. This might be thermal conductivity of a known shale bed since thermal conductivities of shales do not vary much and are nearly independent of temperature (Willhite, 1967). Hoang (1980) analyzed thermal gradient data for a Mid-Continent well.

No core

samples were available for the well but other well logs were available which made possible the estimation of thermal conductivities using the correlation Eqs. V-29 and V-31. A regional terrestrial thermal flux value was assumed in order to convert temperature gradient values to thermal conductivities.

Figure Xll-7 shows comparison of well-log data estimates of thermal

conductivities with estimates based on thermal gradients. The comparison between the two estimates is quite good on a relative basis but on an absolute basis the mean value from thermal gradient calculations is about 17 percent lower than the mean for well-log derived values. Some adjustment of the terrestrial thermal flux value would bring the estimates into excellent agreement. 2.2

Hoang (1980) developed a model for the study of transient gradients in wellbores drilled through subsurface formations having variable thermal conductivities.

Purpose of the work

was to determine whether thermal conductivities of the formations could be estimated from measured transient gradients. Figure Xll-8 shows the model used in the study: a cased well with tubing through which injected or produced fluids may flow. The formations through which

214

5

0

F

i

dX

’

Thermal Grad. X

!

100 -

150 -

r? Q

2

200-

fj. P

2

250 -

300 -

350 -

1

2 3 1 2 Thermal Conductivity, W/m-K

3

Fig. Xll-7. Estimated thermal conductivities using welllog data and thermal gradient data. (Hoang, 1980)

the well was drilled are assumed to be horizontal layers of variable thicknesses and variable thermal conductivities. The well completion, including casing, cement, tubing, and insulating materials, may be changed to yield any appropriate value of overall heat transfer coefficient. In the injection case, the injected fluid is assumed to enter the top of the well at a fixed temperature and to flow down the tubing to the injection interval at a constant

mass rate. No

account is taken of movement of the temperature front in the injection interval and heat transfer from it to upper formations. The concern in the present analysis is the heat loss from the well into surrounding formations far enough removed from the injection interval to be unaffected by it. When injection is stopped and the well shut-in, fluid in the well becomes quiescent and it loses heat to the surrounding formations by conduction and, possibly, by natural convection.

215

\

Fig. Xll-8. Well-bore model used by Hoang (1980) for study of thermal gradients for differing thermal conductivities of sub-surface strata. In time (-72 hours) the wellbore fluid temperature approaches the geothermal temperature. In the case of production of hot fluids, these fluids are assumed to enter the bottom of the tubing at a constant rate at the temperature of the producing zone. Heat is lost to the surrounding formations as the fluid rises up the tubing. At the beginning of an injection or production cycle, the initial conditions are the

existing geothermal temperatures.

Geothermal gradients are assumed to be inversely

proportional to thermal conductivities of the formations.

The initial condition for the shut-in

period are the temperature gradients, both vertical and radial, that were calculated to exist just prior to shut-in. (i) m i o n Case In developing a mathematical model for temperature distribution during injection, the system was separated into two parts: 1) the tubing through which the heated fluid is flowing vertically downwards, losing

heat radially to the surroundings, 2) the surrounding formations through which heat is transferred both radially and

vertically. Since the injection rate is generally high, for which fluid flow conditions would be turbulent, temperature across the tubing is assumed to be constant. The Peclet number is very high ( > l o o ) so that axial conduction of heat in the tubing is negligible compared to axial

216

convection. The equation for axi-symmetric heat flow in a circular cylinder is obtained by writing a heat balance in the tubing between (Z) and (Z + dZ):

(Xll-5)

where m

=

mass flow rate of injected fluid; CW = heat capacity of flowing fluid; pw

fluid; rt = radius of tubing; q

m

where v

=

=

density of

heat flow rate into formationhnit thickness.

nrt2pwv

=

=

(Xll-6)

fluid velocity.

The injection fluid density (pw) is a function of temperature given by:

pw(T) = 1/(A

+

BT

+ CT2 +

DT3)

(Xll-7)

where T = temperature, K; A, B, C, and D are constants for a given fluid. The fluid velocity (v) is inversely proportional to fluid density:

Within the formation, from the wellbore outwards, heat flows by conduction only. The applicable two-dimensional heat flow equation is given as:

aWat

-

a ( z ) ( l / r ) d(r&/dr)/ar - a(a(z)ae/aZ)aZ = 0

(Xll-9)

where 0 = temperature in the formation; a(Z) = thermal diffusivity of formation = h(z)/pfCf;

x(z) = thermal conductivty of formation;

pf = density of formation; Cr = specific heat of

formation. Since the heat transfer rate must be the same on either side of the wellborelformation interface, the following boundary conditions couple Eqs. Xll-5 and Xll-9:

at r

and,

= rw:

q

=

2xrw

X(z)ae/&

X(Z) ae/& = UT(e - T)

(XII-lo)

(XII-11)

217

The overall heat transfer coefficient (UT) accounts for the net resistance to heat flow from the tubing through the insulation, annulus, casing, and cement to the outer well radius. Using Willhite's (1967) analysis, (UT) may be expressed as:

U T = (l/ri)[ln(rins/ri/)iins)

where ri

=

+ l / r i n s ( h c + h r ) + In(rh/rcol)icern)l

(Xll-12)

outer radius of tubing; rins = outer radius of insulation; rh = radius of wellbore; rco

= outer radius of casing; hc = convective heat transfer coefficient; hr

-

radiation heat transfer

coefficient; kins = thermal conductivity of insulation; kern= thermal conductivity of cement. lriitially the temperature in the tubing is assumed to be the same as the formation temperature:

(XII-13)

(Xll-14)

where Ta = ambient temperature: qs = terrestrial heat flux.

It is further assumed that:

(Xll-15)

Temperature of the entering injection fluid is assumed to be constant:

auAl,

T(t,O) = Tinj

(XII-16)

Boundary conditions in the formation at top (0) and bottom (L) of the well are:

(Xll-17)

auzL.

e(t,r,L) = Ta +

X

qi! AZilXi

Analytical solution of the above equations for temperature in the flowing injection fluid, with the indicated initial and boundary conditions, is probably not possible. Resort, therefore, was made to a numerical solution technique.

Hoang (1980) applied a finite difference method

218

using (m) equally spaced grid nodes to solve for fluid temperatures in the tubing. An implicit backward difference scheme was used for computational efficiency. The wellbore model was coupled with the formation model, the latter being represented by the two-dimensional grid system shown in Fig. XI-9. To obtain temperature distribution in the formation, an alternate direction implicit (ADI) scheme was used. Details of the computations are presented by Hoang

(1980).

Fig. Xll-9. Grid sysem used by Hoang (1980) for numerical simulation of variable conductivity subsurface strata.

Using the above model, a series of calculations was made varying some of the important parameters. Physical properties of the system and dimensions of the well used for the analysis correspond with the range of values given by Boberg (1971) and Ramey (1962). Figure Xll-10 shows the temperature profile after the start of injection for a case where the formations have different thermal conductivities as shown on the plot. The temperature rather quickly stabilizes at this high injection rate and only a small effect of the variable thermal conductivity of the formations is shown at the shortest injection period (1 hr). Figure XII-11 compares the temperature profiles for uniform and variable thermal conductivities of

the surrounding formations. At a shorter injection time than the above case (0.15 hr) a small

219 0

\ 200

-$

400

E 0)

0

---

\

\

x2 = 3.75

---

8

r"rn

m = 40,000 kglhr

\ x1 = 1.25 Wlm-K

\

\\

Geothermal

\

600

A3 = 1.88

_-x4 = 5

\

1 hr

\

8oa

1ooa

\

\

I

I

I

I

I

I

20

40

60

80

100

120

,24 /72

I

140

160

Temperature, OC

Fig. XII-10. Calculated wellbore fluid temperature distributions for variable thermal conductivity strata for various times after start of hot fluid injection. (Hoang, 1980)

Uniform Thermal Conductivity o\ = 1.8 Wlm-K)

--- Variable Thermal Conductivity 0

Temperature, OC

Fig. XII-11. Comparison of wellbore fluid temperature distribution during injection for uniform and variable formation thermal conductivities.

220 0

200

f

-.

400

E"

sn 0"

600

800

1000

Temperature, OC Fig. Xll-12. Effect of injection rate on wellbore fluid temperature distribution for various injection times.

- 200

0 0

$ a

150

[

I-"

- 15OoC