LABORATORY MEASUREMENTS OF POROELASTIC CONSTANTS

AND FLOW PARAMETERS AND SOME ASSOCIATED PHENOMENA

by

DAVID J. HART

A dissertation submitted in partial fulfillment of the

requirements for the degree of

DOCTOR OF PHILOSOPHY

(Geophysics)

at the

UNIVERSITY OF WISCONSIN — MADISON

2000

i

Abstract

In this investigation, three laboratory experiments were conducted to better characterize the

coupling between fluid pressure, stress, and strain in porous rock. In the first experiment, complete

sets of poroelastic constants were measured for Berea sandstone and Indiana limestone at eight

different pore pressure and confining stress pairs. The Berea sandstone was most compliant at low

effective stresses (drained bulk compressibility, 0.22 1/GPa; undrained bulk compressibility, 0.080

1/GPa; Skempton's B Coefficient, 0.73; and unjacketed bulk compressibility, 0.030 1/GPa at an

effective stress of 3 MPa) and approached less compliant asymptotic values at higher effective

stresses (drained bulk compressibility, 0.080 1/GPa; undrained bulk compressibility, 0.061 1/GPa;

Skempton's B Coefficient, 0.38; and unjacketed bulk compressibility, 0.030 1/GPa at 33 MPa

effective stress) whereas the poroelastic constants of Indiana limestone showed little dependence on

the pore pressure and confining stress state of the sample (drained bulk compressibility, 0.048

1/GPa; undrained bulk compressibility, 0.034 1/GPa; Skempton's B Coefficient, 0.40; and

unjacketed bulk compressibility, 0.013 1/GPa). The Berea sandstone was transversely isotropic at

effective stresses less than 20 MPa and approached isotropy at higher effective stresses whereas the

Indiana limestone remained isotropic at all effective stresses applied.

In the second experiment, the poroelastic coupling during a transient pore pressure test was

investigated. For a positive pressure step, a small pore pressure decrease developed within the

sample at early times. This induced pore pressure of opposite sign is an example of a Mandel-

Cryer effect. The poroelastic response is nearly identical to the diffusive flow response after the

early time interval has passed.

In the third experiment, four independent poroelastic constants (drained bulk

compressibility, 0.10 1/GPa; undrained bulk compressibility, 0.021 1/GPa; Skempton's B

Coefficient, 0.84; and Biot's 1/H, 0.093 1/GPa) and the hydraulic flow parameters (hydraulic

conductivity, 1.3 x 10-12 m/s and specific storage, 8.6 x 10-7 1/m) were determined from a single

hydrostatic loading test on Barre granite.

ii

Acknowledgements

I would like to thank my advisor, Dr. Herbert F. Wang, for giving me the opportunity to

return to school and complete my Ph.D. His gentle guidance and steady support are much

appreciated. I could ask for no better mentor in this pursuit of science. I also wish to thank my

committee members: Dr. Charlie Bentley, Dr. Jean Bahr, Dr. Nik Christensen, and Dr. Chuck

DeMets, for reviewing this manuscript and for all they have taught me. Other mentors who should

be mentioned are Dr. Patricia Berge, Dr. Tomochika Tokunaga, and William Unger.

This research was supported by the Office of Basic Energy Sciences, Department of Energy

under Award DE-FG02-98ER14852, by the National Science Foundation under Award

EAR9614558, and by Lewis G. Weeks Teaching and Research Assistantships.

I would also like to acknowledge the wonderful support of my wife, Kristin, who urged me

to return to school and who might possibly enjoy hearing about poroelasticity as much as I enjoy

talking about it.

iii

Table of Contents

Abstract i

Acknowledgements i i

Table of Contents iii

List of Figures v

List of Tables vii

1 An Experimental Study of the Stress and Pore Pressure Dependenceof Poroelastic Constants for Berea Sandstone and IndianaLimestone.

1

1.1 Introduction 2

1.2 Theory 3

1.3 Experiment 10

1.4 Experimental Results 24

1.5 Discussion 60

1.6 Conclusion 76

1.7 References 78

2 Poroelastic Effects During a Laboratory Transient Pore PressureTest

81

2.1 Introduction 82

2.2 Model 82

2.3 Model Results 85

2.4 Experimental Results 94

2.5 Discussion 104

2.6 Conclusion 105

2.7 References 106

iv

3 A Single Test Method for Determination of Poroelastic Constantsand Flow Parameters in Rocks with Low Hydraulic Conductivities

108

3.1 Introduction 109

3.2 Experiment 109

3.3 Discussion 118

3.4 Conclusion 121

3.5 References 121

v

List of Figures

Chapter 1

1.1a Thin section of Berea sandstone 13

1.1b Thin section of Indiana limestone 13

1.2 Sample and gage orientation 15

1.3 Sample assembly 16

1.4 Diagram of the checkvalve 18

1.5a Diagram of the internal load cell 20

1.5b Schematic of the internal load cell circuitry 21

1.6a Pore pressure as a function of confining stress 25

1.6b Strain as a function of confining stress 26

1.6c Strain as a function of uniaxial stress 27

1.7 Anisotropy ratios of linear compressibility for Berea sandstone 55

1.8 Anisotropy ratios of linear compressibility for Indiana limestone 56

1.9 Bulk compressibilities for Berea sandstone 57

1.10 Skempton's B Coefficient for Berea sandstone 58

1.11 Bulk compressibilities for Indiana limestone 59

1.12 Skempton's B Coefficient for Indiana limestone 60

1.13 Drained compressibilities compared to hertzian contact model 62

1.14 Comparison of pore and grain compressibilities for Berea sandstone 66

1.15 Comparison of pore and grain compressibilities for Indiana limestone 66

1.16 Contours of the bulk compressibilities and Skempton's B Coefficient 69-72

vi

Chapter 2

2.1 Cartoon of the fully-coupled poroelastic response 86

2.2 Normalized Pore Pressure versus dimensionless time at the sample bottom 87

2.3 Early time pore pressure response as a function of radius 88

2.4 Comparison of the pressure profiles at the same dimensionless time 89

2.5 Pore pressure sensitivity to the shear modulus and Poisson's ratio 91

2.6 Comparison of axial and circumferential strains with diffusive pressure 92

2.7 Axial strains for three cases of shear modulus and Poisson's ratio 93

2.8a Experimental normalized pressure at the bottom of short sample Ca3 96

2.8b Early time response of the pore pressure at the bottom of sample Ca3 97

2.9a Experimental normalized pressure at the bottom of long sample Ca1 98

2.9b Early time response of the pore pressure at the bottom of sample Ca1 99

2.10a Normalized axial strain for the short sample Ca3 101

2.10b The early time response of the normalized axial strain for sample Ca3 102

2.11a Normalized axial and circumferential strain for the long sample Ca1 103

2.11b The early time response of the normalized strains for sample Ca1 104

Chapter 3

3.1 Pressure and stress history for an idealized run 111

3.2a Pore pressure as a function of confining stress 112

3.2b Volumetric strain as a function of confining stress 113

3.3 Normalized pore pressures and strains as a function of time 116

vii

List of Tables

Chapter 1

1.1 Poroelastic constants measured in this experiment 10

1.2 Measured values of the poroelastic constants for Berea sandstone 29

1.3 Measured values of the poroelastic constants for Indiana limestone 31

1.4 Best-fit values of the poroelastic constants for Berea sandstone 36

1.5 Best-fit values of the poroelastic constants for Indiana limestone 38

1.6 Difference between measured and best-fit values for Berea sandstone 40

1.7 Differences between measured and best fit values for Indiana limestone 42

1.8 Experimental values of the bulk compressibilities for Berea sandstone 45

1.9 Best-fit values of the drained bulk compressibilities for Berea sandstone 46

1.10 Difference between measured and best fit values for Berea sandstone 47

1.11 Measured values of the bulk compressibilities for Indiana limestone 48

1.12 Best-Fit values of the bulk compressibilities for Indiana limestone 48

1.13 Difference between measured and best-fit values for Indiana limestone 49

1.14 Best-Fit results for drained triaxial measurements 50

1.15 Linear compressibilities parallel to the sample bedding 53

1.16 Comparison of the pore compressibility to the unjacketed compressibility 65

Chapter 2

2.1 Parameters used in the base model 85

Chapter 3

3.1 Measured bulk poroelastic constants and flow parameters for Barre granite 118

3.2 Comparisons of grain compressibilities 119

1

Chapter 1

An Experimental Study of the Stress and Pore Pressure Dependence of

Poroelastic Constants for Berea Sandstone and Indiana Limestone.

God made the bulk; surfaces were invented by the devil.—Wolfgang Pauli

ABSTRACT: Complete sets of poroelastic constants were measured for Berea sandstone

and Indiana limestone at eight different pore pressure and confining stress pairs. Confining

pressures were between 9.6 and 40 MPa. Pore pressures were between 7.0 and 26 MPa.

Three cores from both rocks, oriented parallel, perpendicular, and at an angle of 45 degrees

to bedding, were tested under drained, undrained, and unjacketed pore fluid boundary

conditions and under hydrostatic and uniaxial stresses in a triaxial vessel. Thirty six stress-

strain and stress-pore pressure curves were found at each of the eight pore pressure and

confining stress pairs. The following results were found: 1) The Berea sandstone was

most compliant at low effective stresses and approached less compliant asymptotic values at

higher effective stresses while the poroelastic constants of Indiana limestone showed little

dependence on the pore pressure and confining stress state of the sample. 2) The Berea

sandstone behaved in a transversely isotropic manner at low effective stresses and

approached isotropy at higher effective stresses while the Indiana limestone remained

isotropic at all the pore pressure and confining stress pairs.

2

1.1 Introduction

The poroelastic behavior of rocks is often more complicated than the common

assumptions of linearity and isotropy. Rocks exhibit nonlinear elastic behavior due to

closing of cracks or fractures and an increase in contact area between grain to grain contacts

(Brace, 1965; Nur and Simmons, 1969, Warpinski and Teufel, 1992). In addition they are

often anisotropic due to bedding surfaces, foliations, and microcracks (Friedman and Bur,

1974; Jones and Wang, 1981; Lo et al.; 1986, Sayers and Kachonov, 1995). Both

nonlinearity and anisotropy may be due to the orientation of microcracks or grain to grain

contacts.

In this study we measured complete sets of poroelastic constants for a transversely

isotropic rock which has many grain to grain contacts and microcracks, Berea sandstone,

and for comparison, a well cemented uncracked rock, Indiana limestone. The poroelastic

constants were measured at eight different pore pressure and confining stress pairs under

drained, undrained, unjacketed pore fluid boundary conditions and under hydrostatic and

uniaxial stress conditions. This was done so that the rock would be in the same strain state

for each pore pressure and confining stress so that the relationships between the poroelastic

constants would hold and so that the pore pressure and stress dependence of the poroelastic

elastic constants and of the anisotropy could be reliably determined for the two rocks

(Warpinski and Teufel, 1992). The pore fluid and stress boundary conditions were all

incremental about the target pore pressure and confining stress.

Because 36 constants were measured but only eight constants are needed to

characterize a transversely isotropic poroelastic material, the constants were overdetermined.

A nonlinear inversion scheme was used to find a best fit set of data and to test the goodness

of fit of the data to the model, providing a check of the assumption of transverse isotropy

and a test of systematic biases due to the measurement technique or to the model not being

representative of the rock (Hart and Wang, 1995; Tokunaga et al., 1998).

3

This research was conducted to provide better estimates of the stress and pore

pressure dependence of poroelastic constants for larger-scale studies, such as how the

velocities within a petroleum reservoir may increase as fluid is withdrawn or how stress

changes in a fault zone may alter the velocities and fluid pressures near the fault . Another

goal is to better understand the underlying physics of the nonlinear and anisotropic

poroelastic behavior of rocks and improve the models, crack or contact, which can predict

that behavior (Roeloffs, 1982; Endres, 1996).

1.2 Theory

Constitutive Equations

The constitutive equations for an anisotropic poroelastic material may be written

(Cheng, 1997) as

eij = sijklσkl + 13 CBij p (1)

ζ = C p + 13 Bijσ ij( ) (2)

where eij is the strain tensor, sijkl is the drained compliance tensor, σkl is the stress tensor,

C is the three dimensional storage coefficient, Bij is the generalized Skempton's pore

pressure build-up coefficient tensor, p is the pore pressure, and ζ is the variation of fluid

content. The sign convention follows that of general elasticity whereby tensile stress,

extensive strain, and pore pressure and fluid content increases are taken as positive. The

strains, stresses, pore pressure, and fluid contents in these equations are considered to be

incremental relative to a reference state about some pore pressure and confining stress pair.

If the poroelastic material behaves in a nonlinear fashion then each pore pressure and

confining stress pair will have an associated set of poroelastic constants: the compliance

4

tensor, sijkl ; the three dimensional storage coefficient, C ; and the generalized Skempton's

pore pressure buildup tensor, Bij . These constants will vary as a function of the pore

pressure and the confining stress.

Pore Fluid Boundary Conditions

In the measurements of the poroelastic constants, three pore fluid boundary

conditions are applied: (1) drained ( ∆p = 0 ) defined by no change in the pore pressure,

(2) undrained ( ∆ζ = 0 ) defined by no change in the fluid content of the rock, and (3)

unjacketed ( ∆p = ∆Pc ) defined by the change in pore pressure being equal to the change

in confining stress, Pc ≡ −σ11 = −σ22 = −σ33 , i.e., the stress field is isotropic. Note

that the confining stress, Pc , is defined to be positive in compression. This boundary

condition is termed unjacketed because it was originally applied by placing the sample in a

pressure vessel without a jacket in direct contact with the confining fluid. The definition has

been expanded in this work to mean the change in pore pressure is equal to the change in

confining stress. Relationships between the compliances measured under the different pore

fluid boundary conditions are developed next.

Drained

When the drained pore fluid boundary condition is applied, ∆p = 0 , the first

constitutive equation, equation 1, becomes the usual equation of elasticity with no

contribution from the pore fluid.

eij = sijklσkl . (3)

Equation 2 becomes a measure of the amount of fluid that moves into and out of the rock as

the stress is varied.

5

ζ = 13 CBijσ ij . (4)

Undrained

Under the undrained pore fluid boundary condition, where no fluid is allowed to

flow into or out of the sample, ∆ζ = 0 , equation 2 becomes a measure of how the pore

pressure varies with an external stress.

p = − 13 Bijσ ij( ) (5)

If pore pressure and the stress are measured during an undrained test, then the ratio between

the two quantities gives the Skempton's B tensor. If the right hand side of equation 5 is

substituted into equation 1, then an expression for the relationship between the strain and

stress tensors under an undrained pore fluid boundary condition is the result. This equation

describes how the strains will vary as the external stress is varied under an undrained pore

fluid boundary condition.

eij = sijkl − 19 CBij Bkl( )σkl (6)

An undrained compliance tensor can then be defined as

sijklu = sijkl − 1

9 CBij Bkl . (7)

This equation shows that an undrained material is less compliant than a drained material as

would be expected if the pores of the rock are filled with a fluid of finite compressibility.

6

Unjacketed

In a manner similar to the undrained case, an unjacketed compliance tensor can also

be defined as a function of the constants given in the basic constitutive equations 1 and 2.

Setting ∆p = ∆Pc = − σkk3 in equation 1, the result is

eij = sijkk − 13 CBij( )σkk (8)

The unjacketed compliance tensor is then equal to

sijkks = sijkk − 1

3 CBij (9)

Using these definitions of the pore fluid boundary condition, the relationships between the

measured poroelastic constants can be found so that a set of measured constants can be

checked for self consistency. For example, the measured undrained compliance tensor

should be within some experimental error of a calculated value using equation 7 along with

the drained compliance tensor, the storage coefficient, and Skempton's B tensor. These

relationships are also used to create the model used for an inversion which calculates a best-

fit set of constants from the measured constants.

Matrix Notation

In order to better visualize the relationships between the variables (stress, strain, pore

pressure, and fluid content) matrix notation will be used. Because sijkl = s jikl and

sijkl = sijlk due to the symmetry of the stress and strain tensors, it is possible to write the

compliance tensors in matrix form (Nye, 1998). This form is also referred to as

"engineering notations" (Cheng, 1997). Also, because the Skempton's B tensor is

symmetric if the rock and fluid are elastic, it is possible to substitute that tensor for a vector.

7

The compliance tensors and Skempton's B tensor can be rewritten as follows. In matrix

notation the first two indices are replaced by a single one and the last two by another. The

conventions used in this paper are as follows and follow that of Nye (1998):

tensornotation

11 22 33 23, 32 31,13 12, 21

matrixnotation

1 2 3 4 5 6

Also for the compliance tensor, factors of 2 and 4 are introduced as follows:

sijkl = smn where m and n are 1, 2, or 3

2sijkl = smn when either m or n are 4, 5, or 6.

4sijkl = smn when both m and n are 4, 5, or 6.

The stress and strain tensors can be rewritten as vectors by replacing the two indices with a

single letter corresponding to a coordinate axes and using separate symbols for shear and

normal stresses and strains.

tensornotation

11 22 33 23, 32 31, 13 12, 21

matrixnotation

x y z x y z

The strain tensor is written in vector form as:

e = ex ey ez γ x γ y γ z[ ] T(10)

and the stress tensor is written in vector form as:

8

σ = σ x σ y σ z τ x τ y τ z[ ] T(11)

where e and σ are normal components of strain and stress, respectively, γ and τ are the

shear components of strain and stress, respectively, and the superscript T stands for matrix

transpose (Cheng, 1997)

The constitutive equations 1 and 2 can then be written in matrix form as:

ex

ey

ez

γ x

γ y

γ z

=

s11 s12 s13 s14 s15 s16

s21 s22 s23 s24 s25 s26

s31 s32 s33 s34 s35 s36

s41 s42 s43 s44 s45 s46

s51 s52 s53 s54 s55 s56

s61 s62 s63 s64 s65 s66

σ x

σ y

σ z

τ x

τ y

τ z

+ 13

C

B1

B2

B3

B4

B5

B6

p (12)

ζ = C p + 13 B1 B2 B3 B4 B5 B6[ ]

σ x

σ y

σ z

τ x

τ y

τ z

(13)

If the deformation is completely reversible, a potential energy function can be written and it

can be shown that smn = snm . This reduces the number of elastic constants in the

compliance matrix from 36 to 21. The number of independent poroelastic constants in

these equations is then 21 from the compliance matrix, 6 from the Skempton's B vector, and

the three dimensional storage coefficient, for a total of 28 constants.

9

Transverse Isotropy

The above equations are the most general case for an anisotropic porous material

and to completely characterize the rock, the 28 poroelastic constants in equations 12 and 13

would need to be measured. In this work, transverse isotropy with the axis of symmetry

aligned perpendicular to the bedding plane of the rocks is assumed. Here the axis of

symmetry axis is aligned with the z-axis. This reduces the number of independent

poroelastic constants to eight: five in the compliance matrix, two in the Skempton's B vector

and the three dimensional storage coefficient. Equations 12 and 13 are reduced to:

ex

ey

ez

γ x

γ y

γ z

=

s11 s12 s13 0 0 0

s12 s11 s13 0 0 0

s13 s13 s33 0 0 0

0 0 0 s44 0 0

0 0 0 0 s44 0

0 0 0 0 0 2 s11 − s12( )

σ x

σ y

σ z

τ x

τ y

τ z

+ 13

C

B1

B1

B3

0

0

0

p (14)

ζ = C p + 13 B1 B1 B3 0 0 0[ ]

σ x

σ y

σ z

τ x

τ y

τ z

(15)

These two equations were used to describe the poroelastic behavior of the two rock types

tested in this experiment.

10

1.3 Experiment

Experiment Overview

Strain gages were applied to three samples each of Indiana limestone and Berea

sandstone. The sample core axis orientations were parallel, perpendicular, and at an angle of

45 degrees to bedding. The samples were saturated with deionized water, jacketed, and

placed in a triaxial vessel. Drained, undrained, and unjacketed hydrostatic measurements

were made at eight different pore pressure and confining stress pairs. Each pair

corresponded to a different effective stress. Drained and undrained uniaxial measurements

were also made at the same eight pore pressure and confining stress pairs. A total of 36

strain-stress and pore pressure-stress curves were recorded at each pore pressure and

confining stress pair. The slopes of these curves correspond with either the poroelastic

constants given in equation 12 or some function of those constants. The 36 measurements

were inverted to a best fit set of measurements using the relationships between the

poroelastic constants under the different pore fluid boundary conditions given in equations

7 and 9. These constants were thus determined at the eight different pore pressure and

confining stress pairs so that the stress and pore pressure dependence of the anisotropy and

nonlinearity could be determined. Table 1.1 shows the 36 poroelastic constants, the

associated pore fluid boundary conditions, the orientation of the core axis, and the applied

stresses with respect to the bedding plane.

Table 1.1. Poroelastic constants measured in this experiment, orientation of the sample corewith respect to bedding and applied stresses and the pore fluid boundary condition.

Poroelastic Constants Core AxisOrientation

w.r.t.bedding

Applied Stress(w.r.t. core

axis)

Pore FluidBoundaryCondition

s11 + s12 + s13perpendicular hydrostatic drained

2s12 + s33perpendicular hydrostatic drained

s11u + s12

u + s13u perpendicular hydrostatic undrained

2s13u + s33

u perpendicular hydrostatic undrained

13 2B1 + B3( ) perpendicular hydrostatic undrained

11

s13perpendicular uniaxial drained

s33perpendicular uniaxial drained

s13u perpendicular uniaxial undrained

s33u perpendicular uniaxial undrained

13 B3

perpendicular uniaxial undrained

s11s + s12

s + s13s( ) perpendicular hydrostatic unjacketed

2s13s + s33

s( ) perpendicular hydrostatic unjacketed

s11 + s12 + s13parallel hydrostatic drained

2s12 + s33parallel hydrostatic drained

s11u + s12

u + s13u parallel hydrostatic undrained

2s13u + s33

u parallel hydrostatic undrained

13 2B1 + B3( ) parallel hydrostatic undrained

s11parallel uniaxial drained

s12parallel uniaxial drained

s13parallel uniaxial drained

s11u parallel uniaxial undrained

s12u parallel uniaxial undrained

s13u parallel uniaxial undrained

13 B1

parallel uniaxial undrained

s11 + s12 + s1345 degrees hydrostatic drained

12 s11 + s12 + s13( ) + 1

2 2s13 + s33( ) 45 degrees hydrostatic drained

s11u + s12

u + s13u 45 degrees hydrostatic undrained

12 s12

u + s13u + s11

u( ) + 12 2s13

u + s33u( ) 45 degrees hydrostatic undrained

13 2B1 + B3( ) 45 degrees hydrostatic undrained

12 s12 + s13( ) 45 degrees uniaxial drained

14 s11 + s33 + 2s13 − s44( ) 45 degrees uniaxial drained

14 s11 + s33 + 2s13 + s44( ) 45 degrees uniaxial drained

12 s12

u + s13u( ) 45 degrees uniaxial undrained

14 s11

u + s33u + 2s13

u − s44( ) 45 degrees uniaxial undrained

14 s11

u + s33u + 2s13

u + s44( ) 45 degrees uniaxial undrained

13

12 B1 + B3( ) 45 degrees uniaxial undrained

12

Sample Description

Berea sandstone and Indiana limestone were chosen as the rock samples because

they had both been previously used to measure poroelastic constants other than the drained

values (Green and Wang, 1986; Berge et al., 1993; Fredrich et al., 1995; Hart and Wang,

1995 and 1999). These two rocks have very different pore and frame structures, which

contribute to the difference in the poroelastic behavior between the two rocks. The Berea

sandstone has many long thin cracks and angular pores. The Indiana limestone has few

cracks and the pores are rounded. A measure of the "roundness" of the pores is the power

law relationship between the area and perimeter of the pores of Berea sandstone and Indiana

limestone ( A = mPγ ), determined by Schlueter et al. (1997). The exponent γ would be 2

for perfectly smooth pores and decreases as the pore boundary becomes more jagged. The

exponents for Berea sandstone and Indiana limestone were found to be 1.43 and 1.67

respectively (Schlueter, 1997) showing that the pores of Indiana limestone are smoother.

Thin sections of the two rocks are shown below, Figures 1.1a and b.

Berea sandstone is a medium-grained, Mississippian age graywacke. It is

composed of quartz (~80%), feldspar (~5%), clay (predominately kaolinite) (~8%), and

calcite (~6%). Its grains are well sorted (~155µm) and subangular with quartz overgrowths

(Bruhn, 1972; Friedman and Bur, 1974; Winkler, 1983). The Berea sandstone tested has a

porosity of 21% measured by dry and immersed weights.

Indiana limestone, also known as Salem limestone, is of Mississippian age. It is

composed of calcium carbonate (~98.2%), magnesium carbonate (~0.1%), and ferric oxide

(~0.2%). The rest of the constituents are insoluble. Some grains are actual oolites, while

others are merely coated with calcium carbonate. The grains are cemented with calcium

carbonate (Logan et al., 1922). The Indiana limestone tested had a porosity of 15% as

measured by dry and immersed weights.

13

Figure 1.1a. Berea sandstone thin section. The porosity is shown by black and the grainsby grey. Note that most of the grains are separated by cracks.

Figure 1.1b. Indiana limestone thin section. The porosity is shown by black, the carbonatecement by light grey, and the grains by darker grey. These grains are all cemented with fewlong thin cracks evident.

14

Sample Preparation

Three samples of each rock were cored from the same Berea sandstone and Indiana

limestone blocks. The samples were cored, cut, and ground into right cylinders with

dimensions of approximately 10 cm length and 5.08 cm in diameter with the ends ground to

within 0.003 cm of parallel. To obtain all eight poroelastic constants for an anisotropic

poroelastic material, it was necessary to core the samples for the two rocks so that the core

axes were aligned perpendicular, parallel, and at an angle of 45 degrees (Amadei, 1983).

The core with its axis aligned at an angle of 45 degrees to bedding allows measurement of

the s44 constant.

Three pairs of metal foil strain gages (Micro-Measurements EA-06-125TF-120)

were glued to the bedding perpendicular samples. Each gage pair was aligned so that one of

the gages of the pair was aligned parallel to the bedding plane, perpendicular to the core

axis, and the other was aligned perpendicular to the bedding plane and parallel to the core

axis. The three pairs were placed at 120 degree intervals around the cylinder midway on the

length of the sample. Four pairs of strain gages were applied to the bedding parallel

samples. Two of the gage pairs were located so that one of the gages of the pair was

aligned with the bedding and core axis and the other gage was aligned perpendicular to the

bedding and core axis. The other two gage pairs were aligned so that one of the gages was

aligned parallel to bedding and the core axis and the other gage was aligned parallel to

bedding and perpendicular to the core axis. Four pairs of strain gages were also applied to

the samples oriented at 45 degrees to bedding. Two of the gage pairs were aligned so that

one of the gages was and parallel to the core axis while the other gage was also aligned at

45 degree to bedding but perpendicular to the core axis. The other two gage pairs were

aligned so that one of the gages was aligned parallel to the bedding but perpendicular to the

core axis and the other gage was aligned at 45 degrees to bedding and parallel to the core

axis. Figure 1.2 shows the orientation of the strain gages.

15

Core Axis Perpendicular

to Bedding

Profile View

Top ViewPerpendicular Parallel 45 degrees

Strain gage pairs

Core Axis Parallel

to Bedding

Core Axis at 45 degrees to Bedding

Figure 1.2. Sample and gage orientation.

RTV silicone gel was applied to the cylindrical core's side and two Tygon sleeves

were fitted and clamped over the ends of the core. This arrangement prevented leakage

around the strain gage leads. The samples were evacuated and saturated with deionized

water. Figure 1.3 shows the sample assembly.

16

Top Endcap with

Check Valve

Tygon Sleeves

Silicone Jacket

Bottom Endcap with

Pressure Transducer

Axial Strain Gage

Circumferential Strain Gage

Figure 1.3. Sample assembly.

Testing Apparatus

Pore pressure measurements were made with an endcap containing a pressure

transducer (Kulite HKM-375) inserted into a Tygon sleeve at the sample bottom. The

transducer was recessed slightly to prevent damage from contact with the sample. This

transducer measured the pore pressure without significantly increasing the pore space of the

sample (Green and Wang, 1986).

The pore fluid boundary condition was controlled by using a checkvalve located in

the upper endplug (Hart and Wang, 1999). This endplug was designed so that it did not

significantly increase the pore space of the sample and so removed the need for corrections

(Wissa, 1969). The check valve was developed so that the three different pore fluid

17

boundary conditions defined above could be applied at the same pore pressure and

confining stress. The checkvalve, when closed by imposing an outlet pressure greater than

the sample pore pressure, prevented fluid from flowing from the sample and so applied an

undrained pore fluid boundary condition. Alternatively, when the valve was opened by

decreasing the outlet pressure below the sample pore pressure, fluid could be injected or

withdrawn from the sample so that the pore pressure was held constant to give the drained

pore fluid boundary condition. Finally, the outlet pressure could be controlled so that the

change in confining stress was equal to the change in pore pressure to give the unjacketed

pore fluid boundary condition. Figure 1.4 is a schematic of the checkvalve.

18

O-ring seal

Top Piece

Bottom Piece

Inlet to Sample

Outlet Line

Retaining Spring

O-ring seal

Sample

Plunger moves up and down to open and close the valve

Figure 1.4. Diagram of the checkvalve used to control the pore fluid boundary condition.

The samples were placed in a 4 inch inner-diameter triaxial pressure vessel.

Hydraulic oil was used to apply the hydrostatic stress, Pc , and a hydraulic ram was used to

apply the uniaxial load, σ z . Because friction between the triaxial piston and the vessel

caused significant hysteresis, an internal load cell was designed and built. This load cell is

based on the low profile "wagon wheel" load cell design used in industry (Hannah and

19

Reed, 1992). It consists of an inner ring, an outer ring, and four "spokes" that radiate from

the inner disk to the outer load surface. Strain gages were applied to the top and bottom of

the spokes to measure the strain. The thickness and length of the spokes were designed by

calculating the bending moments to measure small axial stresses, around 1 MPa, above the

hydrostatic stress. The strain gages were placed on both the top and bottom of the spokes

to provide both temperature and hydrostatic pressure compensation. The load cell was first

calibrated under no hydrostatic stress using a BLH Load Cell. It was then placed in the

pressure vessel in line with an 6 inch length aluminum plug with strain gages applied to the

side of the plug and the Young's modulus was measured at a confining stresses of 10 MPa

and 33 MPa. The measured values of the Young's modulus were 74.0 GPa at a confining

stress of 10 MPa and 72.6 GPa at the confining stress of 33 MPa. This difference is less

than 2 % and will be seen to be negligible when compared to the errors present in the

uniaxial data. Figure 1.5 is a diagram of the load cell and a schematic of the gage circuitry.

20

Spokes with Bending Moment

Inner Ring Load Surface

Strain GagesInner Ring Load Surface

Top View

Outer Ring Load Surface

Lower surfacestrain gage

Upper surface strain gageSpoke

Profile View

2

3

4

1

24

Spoke number

Figure 1.5a. Diagram of the internal load cell. The diagram is approximately to scale.

21

V+

Upper 2

Upper 4

Lower 2

Lower 4

Upper 1

Upper 3

Lower 3

Lower 1

- out+ out

Amplifier

Gage location by spoke number and surface

Wiring Schematic

Figure 1.5b. Schematic of the Wheatstone bridge used in the internal load cell.

The hydrostatic confining stress and external pore line pressure were measured using BLH

pressure transducers. The data from the strain gages, the pore pressure transducer, the axial

load cell, the confining pressure transducer, and the external pore line transducer were

digitized and recorded by a National Instruments board and Labview software.

Testing Procedure

Seasoning

After the samples were saturated and the endplug clamped in place, the sample was

placed in the triaxial vessel. The confining pressure was increased to ~1MPa and the check

valve was closed so that the undrained pore fluid boundary condition was applied. The

samples were then seasoned by increasing the confining stress from 0 MPa to 40 MPa and

then reducing the confining stress to 20 MPa or less. The seasoning was completed by

cycling the confining stress between 40 MPa and 20 MPa or less two to three more times.

22

The pore pressure varied between 20 MPa at the largest confining stress and 10 MPa or

less at the smaller confining stress during the seasoning. The seasoning was conducted so

that the measurements would have a greater repeatability. Because the rock is slightly

altered during seasoning, most likely due to the crushing of asperities, it provides a

consistent matrix structure so that the "same" rock poroelastic constants and pore pressure

and stress dependence are present at the start and the end of the measurements (Warpinski

and Teufel, 1992).

Undrained

Following the seasoning cycle, the undrained poroelastic constants were measured.

The confining stress was increased to a target value and the pore pressure was adjusted so

that it too matched a target value. Adjustment of the pore pressure was done by careful

manipulation of the checkvalve. Decreasing the line pressure below the pore pressure

allowed pore fluid to flow from the sample. When the line pressure was then increased, the

check valve closed and prevented fluid from entering the sample so the reduced fluid content

resulted in a lower pore pressure. If too much pore fluid was removed, it was necessary to

crack the valve open again, and then slowly increase the pore fluid content by slowly

increasing the line pressure, a task that required a "feel" for the check valve. This design

might be improved with the implementation of an electromechanical valve, making it easier

and more reliable to drain and inject fluid into the sample (Tokunaga, 1999). The

hydrostatic measurements at the target pore pressure and confining stress pair were then

conducted. The confining stress was cycled about the target value +/- 1 MPa two to three

times at a rate of approximately 1 MPa/minute. The confining stress, pore pressure, and six

or seven axial and circumferential strains were all digitally recorded. This constituted an

undrained hydrostatic run. Next an undrained uniaxial run was conducted. These uniaxial

tests were conducted at low differential stresses so that the sample would be in nearly the

23

same strain state as when under the hydrostatic confining stresses. The ram of the triaxial

vessel was brought into contact with the sample assembly so that a small axial load, less

than 0.1 MPa, was applied above the hydrostatic stress. The axial load was then cycled

from 0 to 1.5 MPa above the hydrostatic stress three times at a rate of approximately 1

MPa/minute. Again the confining stress, pore pressure, axial stress, and the six or seven

strains were recorded. This constituted a uniaxial undrained run. Each of the runs

consisted of several hundred to over a thousand data points. The confining stress was then

changed so that another target value was reached and the pore pressure then adjusted so that

data at another pore pressure and confining stress pair could be recorded.

Drained and Unjacketed

After undrained data were obtained for the eight pore pressure and confining stress

pairs, the drained and unjacketed measurements were conducted. To apply drained and

unjacketed pore fluid boundary conditions it was necessary to open the check valve and

keep it open. This was accomplished either by quickly dropping the line pressure to the

check valve so that the plunger "cocked" and stayed open or by draining the vessel,

removing the plunger from the endplug, and then refilling and pressurizing the vessel. The

open valve allowed the pore pressure to be controlled externally with a screw-type pressure

generator so that the pore pressure could be held constant to give the drained pore fluid

boundary condition or varied with the confining stress to give the unjacketed pore fluid

boundary condition. The confining stress and pore pressure were adjusted to a target pore

pressure and confining stress, cycled +/- 1 MPa two or three times first while the pore

pressure was held constant to give the drained pore fluid boundary condition and then

cycled +/- 1 MPa two to three times again while the pore pressure and confining stress were

changed simultaneously equal amounts to give the unjacketed pore fluid boundary. The

strains and confining stresses were all recorded. The uniaxial drained test was then

24

conducted in a manner similar to the undrained measurements. The ram of the triaxial

vessel was brought into contact with the sample assembly with a small axial load, less than

0.1 MPa, above the hydrostatic stress. The axial load was cycled from 0 to 1.5 MPa above

the hydrostatic stress three times while the pore pressure and hydrostatic stress were held

constant. As in the undrained case, the stresses were applied at a rate of approximately 1

MPa per minute. The strains, pore pressure, hydrostatic stress, and axial stresses were all

recorded. After completing the three runs, the hydrostatic drained, the hydrostatic

unjacketed, and the uniaxial drained, the pore pressure and confining stress were again

adjusted to the next stress and pore pressure target pair until all the runs had been collected

at the eight pore pressure and confining stress data pairs.

1.4 Experimental Results

Figures 1.6a and b are examples of the data for an undrained hydrostatic run at a

pore pressure of 17 MPa and a confining stress of 31 MPa for the Berea sandstone

bedding perpendicular sample, Be3. Figure 1.6c is an example of data for an drained

triaxial run for the same pore pressure and confining stress pair of pore pressure equal to

17 MPa and confining pressure equal to 31 MPa. These plots are comprised of several

hundred to a thousand data points. The best-fit slope of the pore pressure curve versus the

confining stress, Figure 1.6a, gives a linear combination of elements of the Skempton’s B

matrix, pPc

= 13 2B1 + B3( ) as can be seen from equation 15. This slope is found by a

linear least-squares fit to the data. Figure 1.6b shows the strain-stress curves. The best-fit

slope of the curve of confining stress and the strains parallel to bedding, the circumferential

strains, give a linear combination of elements of the compliance matrix,

exPc

= − s11u + s12

u + s13u( ), and the tangential slope of the confining stress and the strains

perpendicular to bedding (the axial strains) give another linear combination of elements of

the compliance matrix, ez

Pc= − 2s13

u + s33u( ), as can be seen from equation 14. The

25

negative signs are introduced because the sign of confining stress is opposite to that of the

external stress tensor. Figure 1.6c shows the strain versus stress curves for a triaxial run.

The axial strain versus stress slopes from the different gages were averaged to give a value

for s33 and similarly the circumferential strain versus stress slopes were averaged to give a

value for s13.

Run 11 Undrained (Pc=31 MPa, Pp=17 MPa)

1 5

15.5

1 6

16.5

1 7

17.5

1 8

29.5 3 0 30.5 3 1 31.5 3 2 32.5

Confining Stress (MPa)

Por

e P

ress

ure

(MP

a)

Figure 1.6a. Pore pressure as a function of confining stress under undrained pore fluid andhydrostatic stress boundary conditions.

26

Run 11 Undrained (Pc=31MPa, Pp=17 MPa)

-30

-20

-10

0

1 0

2 0

3 0

29.5 3 0 30.5 3 1 31.5 3 2 32.5

Confining Stress (MPa)

Mic

ros

tra

in

(∆l/

l x

1

0^

6)

Strains Ax1

Strains Ci1

Strains Ax2Strains Ci2

Strains Ax3

Strains Ci3

Figure 1.6b. Strain as a function of confining stress for an undrained pore fluid andhydrostatic stress boundary condition.

27

Run 31 Uniaxial Drained(Pc=31 MPa, Pp=17 MPa)

-100

-80

-60

-40

-20

0

20

40

0 0.2 0.4 0.6 0.8

Uniaxial Stress above hydrostatic (MPa)

Mic

ros

tra

in

(∆l/

l x

1

0

^6

)

Ax1

Ci1

Ax2

Ci2

Ax3

Ci3

Figure 1.6c. Strain as a function of uniaxial stress for a drained pore fluid and triaxialstress boundary condition.

Measured Sets of Poroelastic Constants

The set of measured values for Berea sandstone consisted of thirty-six tangential

slopes of the stress-strain and stress-pore pressure curves for each of the eight pore

pressure and confining stress pairs which combined for a total of 288 slopes. Only 32

tangential slopes of the stress-strain and stress-pore pressure curves were found for Indiana

limestone for each of eight pore pressure and confining stress pairs for a total of 256

slopes. The drained measurements were not conducted on the Indiana limestone sample

oriented with its core at 45 degrees to bedding because the sample sleeve failed and the

sample was contaminated with oil during pressurization for the drained tests. Because

many of the slopes are measurements of the same set of constants, those slopes were

averaged. For example, the bedding parallel strains as a function of the confining stress are

the same within sample variation and experimental error whether the core of the sample is

28

parallel, perpendicular or at an angle of 45 degrees to bedding. After averaging the slopes

that correspond to the same group of compliances or Skempton’s B measurements, the

number of slopes for a single pore pressure and confining stress pair is reduced from 36 to

25 for the Berea sandstone data and from 32 to 21 for the Indiana limestone data. These

sets of measured poroelastic constants are shown in Tables 1.2 and 1.3.

29

Table 1.2. Measured values of the poroelastic constants for Berea sandstone. The units ofthe compliances are 1/GPa and Skempton's B coefficient are Pa/Pa.Confining Stress (MPa) 41 41 31 20Pore Pressure (MPa) 7.6 22 17 10

s11 + s12 + s13 0.0247 0.0284 0.0319 0.0399

2s12 + s33 0.0257 0.0330 0.0396 0.0548

s13 -0.0046 -0.0057 -0.0065 -0.0079

s33 0.0353 0.0439 0.0509 0.0699

s11 0.0487 0.0524 0.0552 0.0670

s12 -0.0048 -0.0061 -0.0068 -0.008112 s11 + s12 + s13( ) + 1

2 2s13 + s33( ) 0.0257 0.0314 0.0377 0.050812 s12 + s13( ) -0.0050 -0.0058 -0.0071 -0.0105

14 s11 + s33 + 2s13 − s44( ) -0.0110 -0.0125 -0.0148 -0.017514 s11 + s33 + 2s13 + s44( ) 0.0757 0.0874 0.0996 0.1305

s11u + s12

u + s13u

0.0193 0.0200 0.0213 0.0233

2s13u + s33

u0.0202 0.0220 0.0245 0.0284

s13u

-0.0068 -0.0077 -0.0094 -0.0123

s33u

0.0372 0.0407 0.0483 0.0625

s11u

0.0319 0.0347 0.0366 0.0411

s12u

-0.0100 -0.0117 -0.0136 -0.016512 s12

u + s13u + s11

u( ) + 12 2s13

u + s33u( ) 0.0196 0.0218 0.0242 0.0282

12 s12

u + s13u( ) -0.0122 -0.0144 -0.0090 -0.0132

14 s11

u + s33u + 2s13

u − s44( ) -0.0072 -0.0089 -0.0102 -0.014914 s11

u + s33u + 2s13

u + s44( ) 0.0400 0.0454 0.0529 0.063813 2B1 + B3( ) 0.3956 0.4562 0.4813 0.5398

13 B3 0.1421 0.1580 0.1912 0.205413 B1 0.1321 0.1531 0.1547 0.1728

13

12 B1 + B3( ) 0.1326 0.1709 0.1819 0.2048

s11s + s12

s + s13s( ) + 2s13

s + s33s( ) 0.0171 0.0197 0.0215 0.0204

30

Table 1.2 (continued). Measured values of the poroelastic constants for Berea sandstone. Theunits of the compliances are 1/GPa and Skempton's B coefficients are Pa/Pa.Confining Stress (MPa) 35 26 20 9.7Pore Pressure (MPa) 26 20 16 6.9

s11 + s12 + s13 0.0400 0.0507 0.0560 0.0627

2s12 + s33 0.0566 0.0738 0.0833 0.0925

s13 -0.0084 -0.0109 -0.0120 -0.0128

s33 0.0758 0.0997 0.1182 0.1646

s11 0.0668 0.0806 0.0908 0.1062

s12 -0.0083 -0.0105 -0.0112 -0.012112 s11 + s12 + s13( ) + 1

2 2s13 + s33( ) 0.0503 0.0669 0.0741 0.081112 s12 + s13( ) -0.0079 -0.0116 -0.0124 -0.0145

14 s11 + s33 + 2s13 − s44( ) -0.0176 -0.0255 -0.0280 -0.016714 s11 + s33 + 2s13 + s44( ) 0.1251 0.1628 0.1878 0.1988

s11u + s12

u + s13u

0.0230 0.0244 0.0244 0.0254

2s13u + s33

u0.0282 0.0304 0.0297 0.0294

s13u

-0.0153 -0.0204 -0.0239 -0.0300

s33u

0.0654 0.0788 0.0924 0.1217

s11u

0.0417 0.0785 0.0547 0.0579

s12u

-0.0167 -0.0166 -0.0252 -0.026812 s12

u + s13u + s11

u( ) + 12 2s13

u + s33u( ) 0.0281 0.0285 0.0316 0.0323

12 s12

u + s13u( ) -0.0144 -0.0201 -0.0229 -0.0259

14 s11

u + s33u + 2s13

u − s44( ) -0.0159 -0.0228 -0.0258 -0.031214 s11

u + s33u + 2s13

u + s44( ) 0.0675 0.0826 0.0914 0.099613 2B1 + B3( ) 0.5927 0.6678 0.7163 0.7532

13 B3 0.2246 0.2292 0.2547 0.271013 B1 0.1662 0.1789 0.1991 0.1873

13

12 B1 + B3( ) 0.2254 0.2382 0.2636 0.2714

s11s + s12

s + s13s( ) + 2s13

s + s33s( ) 0.0219 0.0242 0.0247 0.0283

31

Table 1.3. Measured values of the poroelastic constants for Indiana limestone. The units of thecompliances are 1/GPa and Skempton's B coefficient are Pa/Pa.Confining Stress (MPa) 40 40 35 31Pore Pressure (MPa) 7.8 20 26 17

s11 + s12 + s13 0.0126 0.0137 0.0149 0.0145

2s12 + s33 0.0124 0.0133 0.0142 0.0139

s13 -0.0061 -0.0060 -0.0058 -0.0051

s33 0.0245 0.0247 0.0264 0.0257

s11 0.0260 0.0266 0.0244 0.0254

s12 -0.0063 -0.0070 -0.0086 -0.0068

s11u + s12

u + s13u

0.0114 0.0114 0.0117 0.0115

2s13u + s33

u0.0108 0.0106 0.0107 0.0108

s13u

-0.0078 -0.0064 -0.0075 -0.0074

s33u

0.0247 0.0239 0.0245 0.0250

s11u

0.0220 0.0229 0.0214 0.0220

s12u

-0.0113 -0.0111 -0.0108 -0.010312 s12

u + s13u + s11

u( ) + 12 2s13

u + s33u( ) 0.0103 0.0103 0.0104 0.0102

12 s12

u + s13u( ) -0.0375 -0.0168 -0.0128 -0.0289

14 s11

u + s33u + 2s13

u − s44( ) -0.0211 -0.0091 -0.0062 -0.017914 s11

u + s33u + 2s13

u + s44( ) 0.0893 0.0454 0.0345 0.217713 2B1 + B3( ) 0.2844 0.2998 0.3390 0.3044

13 B3 0.1871 0.2868 0.1739 0.140413 B1 0.1015 0.1424 0.1429 0.1187

13

12 B1 + B3( ) 0.4090 0.1659 0.0917 0.3121

s11s + s12

s + s13s( ) + 2s13

s + s33s( ) 0.0087 0.0088 0.0089 0.0088

32

Table 1.3 (continued). Measured values of the poroelastic constants for Indiana limestone. Theunits of the compliances are 1/GPa and Skempton's B coefficient are Pa/Pa.Confining Stress (MPa) 26 20 20 9.6Pore Pressure (MPa) 20 10 15 7

s11 + s12 + s13 0.0156 0.0149 0.0157 0.0166

2s12 + s33 0.0149 0.0142 0.0155 0.0159

s13 -0.0057 -0.0057 -0.0061 -0.0063

s33 0.0263 0.0261 0.0260 0.0266

s11 0.0256 0.0260 0.0263 0.0290

s12 -0.0072 -0.0067 -0.0075 -0.0077

s11u + s12

u + s13u

0.0123 0.0116 0.0120 0.0119

2s13u + s33

u0.0110 0.0108 0.0110 0.0104

s13u

-0.0075 -0.0070 -0.0076 -0.0076

s33u

0.0268 0.0258 0.0277 0.0264

s11u

0.0216 0.0225 0.0220 0.0219

s12u

-0.0120 -0.0114 -0.0110 -0.011012 s12

u + s13u + s11

u( ) + 12 2s13

u + s33u( ) 0.0109 0.0108 0.0105 0.0105

12 s12

u + s13u( ) -0.0135 -0.0143 -0.0271 -0.0137

14 s11

u + s33u + 2s13

u − s44( ) -0.0068 -0.0071 -0.0159 -0.007014 s11

u + s33u + 2s13

u + s44( ) 0.0350 0.0696 0.0933 0.103813 2B1 + B3( ) 0.3483 0.3218 0.3386 0.3972

13 B3 0.1653 0.2314 0.1790 0.1514

13 B1 0.1403 0.1206 0.1444 0.1471

13

12 B1 + B3( ) 0.1146 0.1642 0.3179 0.1318

s11s + s12

s + s13s( ) + 2s13

s + s33s( ) 0.0086 0.0087 0.0084 0.0085

33

Best Fit Sets of Poroelastic Constants

In general, the two sets of constants given in the tables above are not self consistent.

For example, using values for Berea sandstone from Table 1.2 at a pore pressure of 7.6

MPa and a confining stress of 41 MPa, the value of the bedding parallel linear

compressibility, s12 + s13 + s11, calculated from the individual compliances, s12 , s13, and

s11, is equal to 0.0393 1/GPa. This value is not equal to the measured value of 0.0247

1/GPa. In order to reconcile these differences, a nonlinear least squares inversion was made

for an independent, complete set of poroelastic constants. In addition to resulting in a self

consistent set of poroelastic constants with a minimum least squares error, a "best-fit set",

the inversion may indicate model and experimental method bias (Hart and Wang, 1995).

The least squares inversion assumes that the experimental error is normally distributed

about zero error. If the calculated best-fit values for a measurement are either all less than

or all greater than the measured values then a strong bias is present and errors have been

introduced, not by random chance, but by the measurement technique or by application of a

model that does not represent the behavior of the system. This bias can be explored by

selecting subsets of the data that are themselves overdetermined for certain of the stress or

pore fluid boundary conditions. In this work, the complete data set, the hydrostatic

measurements under the three pore fluid boundary conditions and the drained

measurements under the hydrostatic and triaxial stress boundary conditions are analyzed to

determine sources of error.

Inversion Description

Complete Set

Twenty-three of the twenty five measured constants were chosen as the data vector,

b ,for Berea sandstone. The poroelastic constants, s12u and 1

2 s12u + s13

u( ) , were not used for

the inversion of the Berea sandstone data because the errors associated with those two

34

constants were nearly twice those of the other constants for all eight pore pressure-

confining stress pairs when these two constants were included in the inversion. This meant

that these values were outliers and so either the measurement technique or the model was

incorrect. Because it is difficult to conduct a truly uniaxial test, it was concluded that the

measurement technique is the probable reason for the large error. There is nearly always a

bending moment present when a uniaxial stress is applied, due to nonalignment of the

loading pistons and frame or even to heterogeneity in the sample itself. This bending

moment can be seen in the example of uniaxial test plot, Figure 1.6c, where one of the three

axial strains actually shows an increase for a compressive stress. This source of error is

generally reduced by averaging strains measured 180 degrees apart on the sample. In the

case of the two constants eliminated from the complete set inversion, only one strain gage

was available or working when the measurement was made, so it is likely the error was due

to not having an average value for the strains.

Nineteen of the 21 measured constants were chosen as the data vector, b , for the

Indiana limestone data. The poroelastic constants, 12 s12

u + s13u( )and 1

312 B1 + B3( ), were

not used. These constants were not used for the same reasons as above. They had larger

associated error calculated from an inversion which included them. The constant,

12 s12

u + s13u( ) , was likely unreliable for the same reason as in the case of Berea sandstone.

It was the result of a single strain measurement and not an average value. The reason for the

large error associated with the constant, 13

12 B1 + B3( ), is unknown but may be in part to

bending of the sample during a "uniaxial" test.

The parameter vector, x , is the set of eight independent constants given in equations 14 and

15. Each element of the model vector, M x( ), corresponds to an element of the data vector. These

values are calculated as a functions of the eight parameters. The relationships used in those

calculations are given by equations 7, 9, 14, and 15. The normalized residual vector

F x( ) = b−M x( )( )b

is used to calculate a root mean squared error, rms = F x( )F x( )T( )12 , the

35

function that is minimized through the nonlinear least squares inversion scheme (Tokunaga et al.,

1998; Menke, 1989). The results of the inversions are given in Tables 1.4 and 1.5. The root mean

squared errors and the percentage differences between the measured and best-fit values are given

for each pore pressure and confining stress pair in Tables 1.6 and 1.7. Tables 1.6 and 1.7 show

that there is nearly always a consistent difference between the measured slope and the best-fit values

for all the pore pressure and confining stress pairs. For example, the percentage error between the

measured and best-fit values is always positive for the drained linear compressibility parallel to

bedding, s11 + s12 + s13, starting at a lower value of 1.7% at the highest effective stress (Pc=41

MPa and Ppore=7.6 MPa) and increasing to 34% as the effective stress decreases to the lowest

effective stress (Pc=9.7 MPa and Ppore=6.9 MPa). This bias is present for all the measured

poroelastic constants. This means that bias has been introduced into the data not from random

measurement error, but either by the measurement technique or by not applying the correct model.

Two additional inversions, one using only the hydrostatic data and the other, using only the drained

uniaxial stress data, were completed to determine the source of the bias.

36

Table 1.4. Best-fit values of the poroelastic constants for Berea sandstone. The units of thecompliances are 1/GPa and Skempton's B coefficient are Pa/Pa.Confining Stress (MPa) 41 41 31 20Pore Pressure (MPa) 7.6 22 17 10

s11 + s12 + s13 0.0243 0.0264 0.0284 0.0327

2s12 + s33 0.0249 0.0291 0.0342 0.0432

s13 -0.0052 -0.0062 -0.0073 -0.0093

s33 0.0353 0.0415 0.0487 0.0618

s11 0.0345 0.0387 0.0427 0.0507

s12 -0.0050 -0.0061 -0.0070 -0.008712 s11 + s12 + s13( ) + 1

2 2s13 + s33( ) 0.0246 0.0277 0.0313 0.037912 s12 + s13( ) -0.0051 -0.0062 -0.0071 -0.0090

14 s11 + s33 + 2s13 − s44( ) -0.0082 -0.0099 -0.0113 -0.015614 s11 + s33 + 2s13 + s44( ) 0.0315 0.0364 0.0410 0.0514

s11u + s12

u + s13u

0.0221 0.0235 0.0250 0.0275

2s13u + s33

u0.0226 0.0262 0.0300 0.0369

s13u

-0.0060 -0.0072 -0.0086 -0.0113

s33u

0.0345 0.0405 0.0471 0.0595

s11u

0.0338 0.0377 0.0416 0.049112 s12

u + s13u + s11

u( ) + 12 2s13

u + s33u( ) 0.0224 0.0248 0.0275 0.0322

14 s11

u + s33u + 2s13

u − s44( ) -0.0089 -0.0109 -0.0126 -0.017514 s11

u + s33u + 2s13

u + s44( ) 0.0371 0.0428 0.0484 0.060513 2B1 + B3( ) 0.4039 0.4783 0.5099 0.5752

13 B3 0.1402 0.1627 0.1946 0.215213 B1 0.1318 0.1578 0.1576 0.1800

13

12 B1 + B3( ) 0.1360 0.1603 0.1761 0.1976

s11s + s12

s + s13s( ) + 2s13

s + s33s( ) 0.0169 0.0193 0.0210 0.0197

37

Table 1.4 (continued). Best-fit values of the poroelastic constants for Berea sandstone. The unitsof the compliances are 1/GPa and Skempton's B coefficient are Pa/Pa.Confining Stress (MPa) 35 26 20 9.7Pore Pressure (MPa) 26 20 16 6.9

s11 + s12 + s13 0.0330 0.0390 0.0390 0.0415

2s12 + s33 0.0448 0.0493 0.0523 0.0560

s13 -0.0097 -0.0137 -0.0151 -0.0170

s33 0.0641 0.0767 0.0824 0.0900

s11 0.0506 0.0637 0.0653 0.0709

s12 -0.0080 -0.0110 -0.0112 -0.012412 s11 + s12 + s13( ) + 1

2 2s13 + s33( ) 0.0389 0.0442 0.0456 0.048812 s12 + s13( ) -0.0088 -0.0123 -0.0131 -0.0147

14 s11 + s33 + 2s13 − s44( ) -0.0162 -0.0237 -0.0267 -0.020214 s11 + s33 + 2s13 + s44( ) 0.0524 0.0674 0.0718 0.0689

s11u + s12

u + s13u

0.0275 0.0321 0.0310 0.0331

2s13u + s33

u0.0373 0.0403 0.0420 0.0432

s13u

-0.0119 -0.0164 -0.0182 -0.0206

s33u

0.0611 0.0731 0.0784 0.0845

s11u

0.0490 0.0616 0.0628 0.068512 s12

u + s13u + s11

u( ) + 12 2s13

u + s33u( ) 0.0324 0.0362 0.0365 0.0381

14 s11

u + s33u + 2s13

u − s44( ) -0.0185 -0.0265 -0.0299 -0.024014 s11

u + s33u + 2s13

u + s44( ) 0.0617 0.0775 0.0823 0.079913 2B1 + B3( ) 0.6073 0.6623 0.7353 0.7558

13 B3 0.2473 0.2613 0.2883 0.3270

13 B1 0.1800 0.2005 0.2235 0.2144

13

12 B1 + B3( ) 0.2136 0.2309 0.2559 0.2707

s11s + s12

s + s13s( ) + 2s13

s + s33s( ) 0.0208 0.0224 0.0229 0.0250

38

Table 1.5. Best-fit values of the poroelastic constants for Indiana limestone. The units of thecompliances are 1/GPa and Skempton's B coefficient are Pa/Pa.Confining Stress (MPa) 40 40 35 31Pore Pressure (MPa) 7.8 20 26 17

s11 + s12 + s13 0.0119 0.0124 0.0122 0.0124

2s12 + s33 0.0123 0.0128 0.0130 0.0128

s13 -0.0065 -0.0059 -0.0061 -0.0057

s33 0.0252 0.0247 0.0252 0.0243

s11 0.0257 0.0264 0.0271 0.0257

s12 -0.0073 -0.0080 -0.0088 -0.0076

s11u + s12

u + s13u

0.0111 0.0115 0.0112 0.0115

2s13u + s33

u0.0109 0.0109 0.0116 0.0116

s13u

-0.0068 -0.0064 -0.0065 -0.0061

s33u

0.0245 0.0238 0.0246 0.0239

s11u

0.0255 0.0262 0.0268 0.0254

s12u

-0.0075 -0.0082 -0.0091 -0.007812 s12

u + s13u + s11

u( ) + 12 2s13

u + s33u( ) 0.0110 0.0112 0.0114 0.0116

14 s11

u + s33u + 2s13

u − s44( ) -0.0237 -0.0098 -0.0064 -0.019114 s11

u + s33u + 2s13

u + s44( ) 0.0419 0.0284 0.0256 0.037613 2B1 + B3( ) 0.3406 0.3989 0.4039 0.3487

13 B3 0.1636 0.1983 0.1595 0.1334

13 B1 0.0885 0.1003 0.1222 0.1076

s11s + s12

s + s13s( ) + 2s13

s + s33s( ) 0.0086 0.0088 0.0088 0.0086

39

Table 1.5 (continued). Best-fit values of the poroelastic constants for Indiana limestone. The unitsof the compliances are 1/GPa and Skempton's B coefficient are Pa/Pa.Confining Stress (MPa) 26 20 20 9.6Pore Pressure (MPa) 20 10 15 7

s11 + s12 + s13 0.0130 0.0129 0.0129 0.0132

2s12 + s33 0.0136 0.0136 0.0136 0.0131

s13 -0.0061 -0.0060 -0.0064 -0.0065

s33 0.0258 0.0256 0.0264 0.0262

s11 0.0271 0.0266 0.0274 0.0281

s12 -0.0080 -0.0077 -0.0082 -0.0084

s11u + s12

u + s13u

0.0119 0.0119 0.0117 0.0119

2s13u + s33

u0.0121 0.0116 0.0121 0.0117

s13u

-0.0066 -0.0065 -0.0068 -0.0070

s33u

0.0252 0.0247 0.0257 0.0257

s11u

0.0268 0.0264 0.0271 0.0277

s12u

-0.0083 -0.0080 -0.0085 -0.008812 s12

u + s13u + s11

u( ) + 12 2s13

u + s33u( ) 0.0120 0.0118 0.0119 0.0118

14 s11

u + s33u + 2s13

u − s44( ) -0.0071 -0.0076 -0.0175 -0.007314 s11

u + s33u + 2s13

u + s44( ) 0.0266 0.0266 0.0371 0.027113 2B1 + B3( ) 0.4055 0.3984 0.4082 0.4322

13 B3 0.1548 0.1947 0.1627 0.1489

13 B1 0.1253 0.1019 0.1228 0.1416

s11s + s12

s + s13s( ) + 2s13

s + s33s( ) 0.0085 0.0086 0.0083 0.0083

40

Table 1.6. Percent difference between measured and best-fit values of the poroelastic constants for

Berea sandstone, Cexp −Cbestfit

Cexp×100.

Confining Stress (MPa) 41 41 31 20

Pore Pressure (MPa) 7.6 22 17 10

s11 + s12 + s13 1.7 7.1 10.9 18.0

2s12 + s33 3.3 11.9 13.5 21.2

s13 -12.5 -8.4 -12.3 -17.9

s33 -0.1 5.5 4.4 11.6

s11 29.2 26.1 22.7 24.3

s12 -3.2 0.4 -3.0 -7.912 s11 + s12 + s13( ) + 1

2 2s13 + s33( ) 4.1 11.6 16.9 25.412 s12 + s13( ) -1.4 -7.6 -0.2 14.5

14 s11 + s33 + 2s13 − s44( ) 25.7 20.8 23.7 11.014 s11 + s33 + 2s13 + s44( ) 58.4 58.4 58.8 60.6

s11u + s12

u + s13u

-14.5 -17.3 -17.4 -17.8

2s13u + s33

u-11.7 -19.3 -22.4 -30.0

s13u

12.1 6.9 8.8 8.5

s33u

7.3 0.6 2.6 4.8

s11u

-6.1 -8.5 -13.8 -19.612 s12

u + s13u + s11

u( ) + 12 2s13

u + s33u( ) -14.6 -13.8 -13.5 -14.2

14 s11

u + s33u + 2s13

u − s44( ) -24.2 -22.9 -23.4 -17.114 s11

u + s33u + 2s13

u + s44( ) 7.2 5.8 8.4 5.113 2B1 + B3( ) -2.1 -4.8 -6.0 -6.6

13 B3 1.3 -3.0 -1.8 -4.813 B1 0.2 -3.1 -1.9 -4.2

13

12 B1 + B3( ) -2.6 6.2 3.2 3.5

s11s + s12

s + s13s( ) + 2s13

s + s33s( ) 1.0 2.0 2.4 3.6

Root Mean Squared Error 3.5 3.5 3.7 4.1

41

Table 1.6 (continued). Percent difference between measured and best-fit values of the poroelastic

constants for Berea sandstone, Cexp −Cbestfit

Cexp×100.

Confining Stress (MPa) 35 26 20 9.7

Pore Pressure (MPa) 26 20 16 6.9

s11 + s12 + s13 17.5 23.1 30.3 33.8

2s12 + s33 20.8 33.2 37.2 39.4

s13 -16.0 -25.9 -26.2 -32.8

s33 15.5 23.1 30.3 45.3

s11 24.3 21.0 28.0 33.3

s12 4.2 -4.6 -0.1 -2.512 s11 + s12 + s13( ) + 1

2 2s13 + s33( ) 22.7 33.9 38.5 39.812 s12 + s13( ) -10.8 -6.2 -5.3 -1.5

14 s11 + s33 + 2s13 − s44( ) 8.1 6.9 4.8 -20.914 s11 + s33 + 2s13 + s44( ) 58.1 58.6 61.8 65.3

s11u + s12

u + s13u

-19.8 -31.6 -27.2 -30.5

2s13u + s33

u-32.3 -32.4 -41.2 -47.1

s13u

22.1 19.8 23.9 31.3

s33u

6.6 7.2 15.2 30.6

s11u

-17.5 21.5 -14.9 -18.412 s12

u + s13u + s11

u( ) + 12 2s13

u + s33u( ) -15.1 -26.8 -15.6 -18.1

14 s11

u + s33u + 2s13

u − s44( ) -16.5 -16.1 -16.1 23.114 s11

u + s33u + 2s13

u + s44( ) 8.7 6.1 10.0 19.813 2B1 + B3( ) -2.5 0.8 -2.7 -0.3

13 B3 -10.1 -14.0 -13.2 -20.713 B1 -8.3 -12.1 -12.2 -14.4

13

12 B1 + B3( ) 5.2 3.1 2.9 0.2

s11s + s12

s + s13s( ) + 2s13

s + s33s( ) 5.1 7.3 7.1 11.7

Root Mean Squared Error 4.1 4.8 5.2 6.2

42

Table 1.7. Percent differences between measured and best fit values for Indiana limestone,Cexp −Cbestfit

Cexp×100.

Confining Stress (MPa) 40 40 35 31

Pore Pressure (MPa) 7.8 20 26 17

s11 + s12 + s13 -5.6 -9.4 -18.1 -14.7

2s12 + s33 -1.0 -3.5 -8.5 -8.0

s13 6.7 -2.1 5.1 12.0

s33 2.8 0.2 -4.5 -5.4

s11 -1.3 -0.8 11.1 1.3

s12 15.6 14.8 2.6 12.4

s11u + s12

u + s13u

-2.6 1.3 -3.9 -0.2

2s13u + s33

u1.1 2.9 8.9 7.2

s13u

-12.6 0.4 -13.9 -17.1

s33u

-0.9 -0.3 0.3 -4.3

s11u

16.0 14.4 25.1 15.2

s12u

-33.3 -26.0 -16.0 -24.312 s12

u + s13u + s11

u( ) + 12 2s13

u + s33u( ) 7.0 9.2 9.5 13.5

14 s11

u + s33u + 2s13

u − s44( ) 12.5 7.2 3.8 6.914 s11

u + s33u + 2s13

u + s44( ) -53.1 -37.5 -25.8 -82.713 2B1 + B3( ) 19.8 33.0 19.2 14.5

13 B3 -12.6 -30.9 -8.3 -5.013 B1 -12.8 -29.6 -14.5 -9.3

s11s + s12

s + s13s( ) + 2s13

s + s33s( ) -0.9 -0.4 -1.0 -1.9

Root Mean Squared Error 3.9 4.0 3.0 5.0

43

Table 1.7 (continued). Percent differences between measured and best fit values for Indiana

limestone, Cexp −Cbestfit

Cexp×100.

Confining Stress (MPa) 26 20 20 9.6

Pore Pressure (MPa) 20 10 15 7

s11 + s12 + s13 -16.7 -13.5 -17.8 -20.4

2s12 + s33 -8.7 -4.1 -12.4 -17.8

s13 6.8 5.0 5.7 3.6

s33 -2.0 -1.7 1.4 -1.3

s11 5.9 2.3 4.1 -3.2

s12 11.3 14.8 9.4 8.9

s11u + s12

u + s13u

-2.9 2.4 -2.1 0.4

2s13u + s33

u10.3 7.1 9.9 12.5

s13u

-12.0 -6.6 -10.1 -8.4

s33u

-6.1 -4.4 -7.2 -2.5

s11u

24.0 17.1 22.9 26.7

s12u

-30.6 -30.1 -22.9 -20.212 s12

u + s13u + s11

u( ) + 12 2s13

u + s33u( ) 10.2 9.7 12.9 12.3

14 s11

u + s33u + 2s13

u − s44( ) 4.6 6.4 10.3 4.514 s11

u + s33u + 2s13

u + s44( ) -24.0 -61.8 -60.2 -73.913 2B1 + B3( ) 16.4 23.8 20.6 8.8

13 B3 -6.4 -15.9 -9.1 -1.613 B1 -10.7 -15.5 -15.0 -3.7

s11s + s12

s + s13s( ) + 2s13

s + s33s( ) -1.3 -0.8 -1.1 -2.1

Root Mean Squared Error 3.1 4.3 4.2 4.7

44

Hydrostatic Inversion

Experimental values of the volumetric compressibilities under the three pore fluid

boundary conditions, the drained bulk compressibility, Cd ; the undrained bulk

compressibility, Cu ; and the unjacketed bulk compressibility, Cs , were calculated from the

measured values of the linear compressibilities, Cd = 2 s11 + s12 + s13( ) + 2s13 + s33( ).

Using equations 14 and 15, it can be shown (Brown and Korringa, 1975, Tokunaga, 1998)

that only three compressibilities are needed to completely characterize the hydrostatic

response of the poroelastic medium. If the hydrostatic Skempton's B coefficient,

13 2B1 + B3( ) , is included in the set, then the data are overdetermined and the same

nonlinear least squares technique used to invert the complete set can also be used to find a

best-fit set of hydrostatic data. The experimental values, the calculated best-fit set and

percentage error are given in Tables 1.8, 1.9, and 1.10 for Berea sandstone and Tables 1.11,

1.12, and 1.13 for Indiana limestone. Values from Hart and Wang (1999) are included in

Tables 1.8, 1.9, and 1.10. These samples were cored from the same block and the same

measurement technique was used, except no uniaxial testing was done.

45

Table 1.8. Experimental values of the drained bulk compressibility, Cd; the undrained bulkcompressibility, Cu; the unjacketed bulk compressibility, Cs, and Skempton's B coefficient,B.

Sample Id ConfiningStress(MPa)

PorePressure(MPa)

Cd(1/GPa)

Cu(1/GPa)

Cs(1/GPa)

B(Pa/Pa)

This study-Be3 and Be6

41 8 0.075 0.059 0.029 0.396

41 22 0.090 0.062 0.029 0.456

31 17 0.103 0.067 0.032 0.481

20 10 0.135 0.075 0.031 0.540

35 26 0.137 0.074 0.031 0.593

26 20 0.175 0.079 0.029 0.668

20 16 0.195 0.079 0.030 0.716

10 7 0.218 0.080 0.029 0.753

Be1 - (Hart and Wang, 1999)

34.8 32.1 0.239 0.070 0.035 0.749

35 22 0.106 0.071 0.031 0.488

34 25 0.131 0.074 0.030 0.546

22 13 0.138 0.076 0.032 0.533

18.3 17.1 0.309 0.065 0.040 0.834

17 12 0.174 0.081 0.032 0.610

12 7 0.172 0.084 0.034 0.611

7.9 7.3 0.263 0.059 0.043 0.893

6 3 0.225 0.085 0.037 0.736

1.1 0.7 0.281 0.071 0.036 0.866

Be4 (Hart and Wang, 1999)

41 6 0.070 0.056 0.028 0.390

41 10 0.072 0.057 0.028 0.405

41 16 0.078 0.059 0.028 0.424

31 10 0.083 0.061 0.029 0.441

21 10 0.124 0.075 0.029 0.531

21 17 0.230 0.087 0.031 0.684

10 8 0.299 0.090 0.034 0.748

46

Table 1.9. Best-fit values of the drained bulk compressibility, Cd; the undrained bulkcompressibility, Cu; the unjacketed bulk compressibility, Cs, and Skempton's B coefficient,B.

Sample Id ConfiningStress(MPa)

PorePressure(MPa)

Cd(1/GPa)

Cu(1/GPa)

Cs(1/GPa)

B(Pa/Pa)

This study-Be3 and Be6

41 8 0.080 0.061 0.030 0.385

41 22 0.090 0.062 0.029 0.457

31 17 0.100 0.066 0.032 0.492

20 10 0.129 0.074 0.031 0.559

35 26 0.137 0.074 0.031 0.591

26 20 0.177 0.079 0.029 0.658

20 16 0.196 0.079 0.030 0.709

10 7 0.221 0.080 0.029 0.730

Be1 - (Hart and Wang, 1999)

34.8 32.1 0.233 0.073 0.035 0.806

35 22 0.107 0.070 0.031 0.485

34 25 0.130 0.075 0.030 0.550

22 13 0.135 0.079 0.032 0.545

18.3 17.1 0.305 0.066 0.039 0.899

17 12 0.171 0.083 0.032 0.627

12 7 0.170 0.086 0.034 0.621

7.9 7.3 0.262 0.059 0.043 0.926

6 3 0.224 0.085 0.037 0.743

1.1 0.7 0.281 0.071 0.037 0.860

Be4 (Hart and Wang, 1999)

41 6 0.071 0.055 0.028 0.387

41 10 0.074 0.055 0.028 0.401

41 16 0.080 0.058 0.028 0.421

31 10 0.084 0.060 0.029 0.438

21 10 0.124 0.074 0.029 0.528

21 17 0.226 0.089 0.031 0.705

10 8 0.295 0.092 0.034 0.779

47

Table 1.10. Percent difference between measured and best fit values for Berea

sandstone,Cexp −Cbestfit

Cexp×100.

Sample Id ConfiningStress(MPa)

PorePressure(MPa)

C(% error)

Cu(% error)

Cs(% error)

B(% error)

rmserror

This study-Be3 and Be6

41 8 -9.3 -3.7 -2.3 4.5 2.80

41 22 0.8 0.4 0.1 -0.4 0.25

31 17 4.0 1.7 1.2 -2.5 1.29

20 10 3.7 1.2 1.1 -3.4 1.32

35 26 0.9 0.1 0.5 -0.9 0.35

26 20 1.7 0.6 0.7 -2.7 0.84

20 16 1.2 0.3 0.5 -2.6 0.73

10 7 1.4 0.4 1.1 -3.7 1.04

Be1 - (Hart and Wang, 1999)

34.8 32.1 2.2 -4.0 1.4 -7.6 2.24

35 22 -0.8 1.0 -0.2 0.6 0.36

34 25 0.8 -1.1 0.2 -0.8 0.39

22 13 2.5 -3.6 0.7 -2.3 1.24

18.3 17.1 1.1 -2.4 1.2 -7.8 2.08

17 12 2.0 -2.9 0.6 -2.8 1.14

12 7 1.3 -1.8 0.4 -1.7 0.71

7.9 7.3 0.4 -1.1 0.7 -3.7 0.99

6 3 0.4 -0.6 0.2 -0.9 0.28

1.1 0.7 -0.1 0.2 -0.1 0.6 0.17

Be4 (Hart and Wang, 1999)

41 6 1.3 -2.0 0.5 -4.1 1.19

41 10 1.5 -2.1 0.5 -3.1 1.02

41 16 -0.6 0.7 -0.2 0.5 0.26

31 10 -1.2 1.5 -0.3 0.6 0.51

21 10 -1.5 1.8 -0.4 0.8 0.63

21 17 -2.4 2.8 -0.6 0.8 0.96

10 8 -2.1 2.4 -0.5 0.7 0.83

48

Table 1.11. Measured values of the bulk compressibilities for Indiana limestone.Sample Id Confining

Stress(MPa)

PorePressure(MPa)

Cd(1/GPa)

Cu(1/GPa)

Cs(1/GPa)

B(Pa/Pa)

Indiana limestone - Measured Bulk Compressibilities

40 7.8 0.038 0.034 0.013 0.284

40 20 0.041 0.033 0.013 0.300

31 17 0.043 0.034 0.013 0.304

20 10 0.044 0.034 0.013 0.322

26 20 0.046 0.036 0.013 0.348

35 26 0.044 0.034 0.013 0.339

20 15 0.047 0.035 0.013 0.339

9.6 7 0.049 0.034 0.013 0.397

Table 1.12. Best-Fit values of the bulk compressibilities for Indiana limestone.Sample Id Confining

Stress(MPa)

PorePressure(MPa)

Cd(1/GPa)

Cu(1/GPa)

Cs(1/GPa)

B(Pa/Pa)

Indiana limestone - Best-Fit Bulk Compressibilities

40 7.8 0.044 0.036 0.013 0.269

40 20 0.042 0.034 0.013 0.296

31 17 0.043 0.034 0.013 0.305

20 10 0.044 0.034 0.013 0.322

26 20 0.048 0.036 0.013 0.343

35 26 0.045 0.034 0.013 0.337

20 15 0.046 0.035 0.013 0.340

9.6 7 0.048 0.034 0.013 0.401

49

Table 1.13. Percent difference between measured and best-fit values for Indiana limestone,Cexp −Cbestfit

Cexp×100

Sample Id ConfiningStress(MPa)

PorePressure(MPa)

Cd(% error)

Cu(% error)

Cs(% error)

B(% error)

rmserror

Indiana limestone

40 7.8 -14.3 -5.7 -2.1 5.6 4.7

40 20 -4.1 -1.5 -0.3 1.4 1.2

31 17 0.2 0.2 0.3 -0.1 0.1

20 10 -0.2 -0.1 0 0.1 0.1

26 20 -3.8 -1.2 -0.7 1.6 1.1

35 26 -1.6 -0.4 -0.5 0.6 0.5

20 15 1.5 0.4 0 -0.5 0.4

9.6 7 2.1 0.6 0.1 -1 0.6

Drained Transverse Isotropy Inversion

An inversion of the drained measurements was conducted to determine whether the

bias seen in the complete set is introduced by bias in the triaxial measurements or from

application of the pore fluid boundary conditions. The first ten measured values of the

drained compliances and linear combinations of those compliances shown in Table 1.2 form

the data set, b , for Berea sandstone. Because all of the relationships between the constants

are linear for the transversely isotropic measurements under drained conditions, the

inversion is linear and no iterations are necessary. The best-fit results and the differences

between the measured and best-fit set and the rms errors are shown in Table 1.14.

50

Table 1.14. Best-Fit results for drained measurements for Berea sandstone assuming transverseisotropy and the percent difference between the measured and best-fit values.

Confining Stress (MPa) 40 40 35 31

Pore Pressure (MPa) 7.8 20 26 17

Best-Fit Tranverse Elastic Constants

s11 + s12 + s13 0.0309 0.0344 0.0379 0.0471

2s12 + s33 0.0277 0.0358 0.0433 0.0606

s13 -0.0052 -0.0056 -0.0056 -0.0069

s33 0.0382 0.047 0.0545 0.0745

s11 0.0474 0.0523 0.0565 0.0702

s12 -0.0112 -0.0123 -0.0131 -0.016212 s11 + s12 + s13( ) + 1

2 2s13 + s33( ) 0.0293 0.0351 0.0406 0.053812 s12 + s13( ) -0.0082 -0.009 -0.0093 -0.0116

14 s11 + s33 + 2s13 − s44( ) -0.0246 -0.0279 -0.0322 -0.041314 s11 + s33 + 2s13 + s44( ) 0.0621 0.072 0.0822 0.1067

Percent Difference between Measured and Best-fit, Cexp −Cbestfit

Cexp×100

s11 + s12 + s13 -24.9 -21.1 -18.9 -18.0

2s12 + s33 -7.6 -8.4 -9.5 -10.6

s13 -12.5 2.1 13.8 12.6

s33 -8.3 -7.0 -7.0 -6.6

s11 2.7 0.2 -2.3 -4.8

s12 -131.1 -100.9 -92.8 -100.912 s11 + s12 + s13( ) + 1

2 2s13 + s33( ) -14.2 -12.0 -7.8 -5.812 s12 + s13( ) -63.0 -56.2 -31.3 -10.1

14 s11 + s33 + 2s13 − s44( ) -122.8 -123.3 -117.3 -135.714 s11 + s33 + 2s13 + s44( ) 18.0 17.7 17.4 18.3

rms error 19.7 17.2 15.6 17.2

51

Table 1.14 (continued). Best-Fit results for drained measurements for Berea sandstoneassuming transverse isotropyand the percent difference between the measured and best-fitvalues.

Confining Stress (MPa) 26 20 20 9.6

Pore Pressure (MPa) 20 10 15 7

Best-Fit Tranverse Elastic Constants

s11 + s12 + s13 0.0462 0.0575 0.0638 0.069

2s12 + s33 0.062 0.0821 0.0939 0.1094

s13 -0.0084 -0.0106 -0.0137 -0.0259

s33 0.0789 0.1033 0.1214 0.1613

s11 0.0692 0.0857 0.0968 0.1135

s12 -0.0146 -0.0175 -0.0193 -0.018612 s11 + s12 + s13( ) + 1

2 2s13 + s33( ) 0.0541 0.0698 0.0788 0.089212 s12 + s13( ) -0.0115 -0.0141 -0.0165 -0.0223

14 s11 + s33 + 2s13 − s44( ) -0.0385 -0.0522 -0.0602 -0.05214 s11 + s33 + 2s13 + s44( ) 0.1042 0.1361 0.1556 0.1635

Percent Difference between Measured and Best-fit, Cexp −Cbestfit

Cexp×100

s11 + s12 + s13 -15.5 -13.3 -13.9 -10.0

2s12 + s33 -9.6 -11.3 -12.7 -18.3

s13 -0.4 2.6 -14.5 -102.3

s33 -4.1 -3.6 -2.7 2.0

s11 -3.5 -6.3 -6.7 -6.8

s12 -74.9 -66.5 -72.5 -53.712 s11 + s12 + s13( ) + 1

2 2s13 + s33( ) -7.5 -4.3 -6.3 -9.912 s12 + s13( ) -44.8 -21.7 -32.7 -54.0

14 s11 + s33 + 2s13 − s44( ) -118.4 -105.0 -114.6 -211.214 s11 + s33 + 2s13 + s44( ) 16.7 16.4 17.1 17.7

rms error 15.1 12.9 14.3 24.9

52

The error associated with the triaxial drained measurements is much larger than the

error associated with the hydrostatic measurements under the three pore fluid boundary

conditions. This can be seen in the root mean squared (rms) error which is greater than 10

percent for all of the transverse isotropy measurements and only rarely greater than 5

percent for the hydrostatic measurements. The maximum percent difference in the

transverse set is greater than 100 percent while the maximum percent difference for the

hydrostatic set is only 9 percent. From these two inversions of the subsets of the complete

sets, we can now see that the bias is introduced not from the application of the pore fluid

boundary conditions but from the triaxial measurements .

Although it may be possible that the bias is due to the model not representing the

material, i.e., that the assumption of transverse isotropy is wrong, it is unlikely that is the

cause. If the material were isotropic, then the data should show that and no bias or error

would be introduced. If the material showed a degree of anisotropy greater than transverse

isotropy, then measurements of the linear compressibilities parallel to sample bedding for

the three samples should show significant differences. Because the strain gages were

applied without regard to any azimuthal orientation, differences between the linear

compressibilities measured parallel to bedding should appear if the material has a strong

anisotropy in the plane of the bedding. It is unlikely the gages were aligned all with the

same azimuthal orientation. No significant difference appears between these measurements.

Table 1.15 shows the values of the measured linear compressibilities measured parallel to

bedding as a function of decreasing effective stress. The linear compressibilities parallel to

the sample bedding of the three samples are similar for all stresses and pore pressure

implying that there is no anisotropy in the bedding plane. Furthermore, the error for the s12

compliance, given from the drained transverse isotropic inversion, decreases as the

anisotropy increases. It is expected that if the degree of anisotropy were the cause of the

53

bias, that as the sample became more isotropic, the bias would decrease, not increase, as is

the case.

Although neither of the previous arguments is a proof that the model is adequate,

they do present a strong case for the introduction of bias to be from the measurement

method, not the model assumption. It is likely that a bending moment is always present

because the boundary condition of uniaxial stress is not met. Another possibility for the

increased error may be due to friction between grain surfaces during the uniaxial loading.

This friction effect would produce hysteresis during the loading and unloading cycles.

However, Figure 1.6c shows little hysteresis and is representative of the hysteresis seen in

the other uniaxial curves. The conclusion is that true uniaxial stresses are difficult to apply,

especially in a pressure vessel, and that is the cause of the bias.

Table 1.15. Linear compressibilities parallel to the sample bedding for the three Bereasandstone samples measured in this experiment.Confining

StressPore

PressureLinear compressibility parallel to bedding

s11 + s12 + s13

(MPa) (MPa) Sample Be6 Sample Be3 Sample Be11

41 7.6 0.024 0.025 0.025

41 22 0.028 0.029 0.029

31 17 0.031 0.032 0.033

20 10 0.039 0.042 0.038

35 26 0.039 0.043 0.038

26 20 0.05 0.052 0.049

20 16 0.055 0.058 0.055

9.7 6.9 0.066 0.058 0.065

Because the error and bias is much less for the hydrostatic measurements, their

reliability is greater than the other measurements in Tables 1.2 and 1.3 and so they will be

used to discuss the anisotropy and non-linear behavior of the rocks.

54

Anisotropy

Berea sandstone is anisotropic at low effective stresses and becomes isotropic at

higher effective stresses. At a low effective stress, pore pressure = 6.9 MPa and confining

pressure = 9.7 MPa, the measured drained linear compressibility perpendicular to bedding,

2s13 + s33 , is nearly one and one-half times larger than the measured drained linear

compressibility parallel to bedding, s12 + s13 + s11. At the highest effective stress value

where pore pressure = 7.6 MPa and confining pressure = 41 MPa, the ratio approaches

unity. This same phenomenon is also seen in Table 1.1 in the ratio of s33 divided by s11.

Figure 1.7 is a graph which shows the ratios of linear compressibilities perpendicular to

bedding over the linear compressibilities parallel to bedding for drained, undrained, and

unjacketed pore fluid boundaries as a function of Terzaghi effective stress (confining stress

minus pore pressure), Pc − p , for Berea sandstone. The plot also includes data from Hart

and Wang (1999). The ratio of drained compressibilities is always greater than the

undrained ratio, as would be expected when filling the voids of an anisotropic material, the

Berea sandstone matrix, with an isotropic material, water. The ratio of unjacketed

compressibilities, often called the grain compressibilities, has a ratio near unity for all

measurements. Because the drained and undrained linear compressibility ratios approach

unity as the effective stress is increased to a value greater than 30 MPa, Berea sandstone

then behaves as an isotropic material.

55

Linear Compressibility Ratiosperpendicular to bedding/parallel to bedding

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0 1 0 2 0 3 0 4 0

Terzaghi Effective Stress (MPa)

Lin

ea

r C

om

pre

ss

ibil

ity

R

ati

o

drained

undrained

unjacketed

Figure 1.7. Anisotropy ratios of linear compressibilities perpendicular to bedding over

linear compressibilities parallel to bedding, 2s13 + s33( ) s12 + s13 + s11( ) , for Bereasandstone.

The absence of anisotropic behavior in Indiana limestone is in sharp contrast to

Berea sandstone. Indiana limestone behaves in an isotropic manner throughout the entire

range of effective stresses where measurements were taken. The anisotropy ratios do not

show any stress or pore pressure sensitivity and the ratios of the compressibilities under the

drained, undrained, and unjacketed pore fluid boundary conditions are all near unity within

experimental error. This isotropy is also evident in Table 1.3. The values of the ratio of s33

divided by s11 are nearly always equal to unity as are the values of the linear

compressibilities perpendicular to bedding divided by the linear compressibilities parallel to

bedding. Figure 1.8 shows the ratios of the linear compressibilities perpendicular to

bedding divided by the linear compressibilities parallel to bedding. Values of the ratios of

56

the linear compressibilities under the unjacketed pore fluid boundary conditions are not

included at the two greatest effective stresses because not enough data were available to give

a consistent average.

Linear Compressibility Ratiosperpendicular to bedding/parallel to bedding

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

0 1 0 2 0 3 0 4 0

Terzaghi Effective Stress (MPa)

Lin

ea

r C

om

pre

ss

ibil

ity

R

ati

o

drained

undrained

unjacketed

Figure 1.8. Anisotropy ratios of linear compressibilities perpendicular to bedding overlinear compressibilities parallel to bedding for Indiana limestone.

Nonlinearity

In addition to anisotropy, nonlinear elastic behavior is present in Berea sandstone at

low effective stresses. Again, the hydrostatic measurements will be used to illustrate this

point, but a look at any of the compliances in Table 1.2 also shows this. As the effective

stress decreases, moving from left to right in Table 1.2, the absolute values of all of the

compliances and Skempton's B coefficient increase. Berea sandstone becomes more

compliant as the effective stress decreases. Figure 1.9 shows this trend of decreasing

compressibility as the effective stress is increased. Figure 1.10 shows the decrease of the

57

bulk Skempton's B coefficient as a function of effective stress. Figures 1.9 and 1.10

include data from Hart and Wang (1999).

Bulk Compressibilities as a Function of Terzaghi Effective Stress

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 1 0 2 0 3 0 4 0

Terzaghi Effective Stress (MPa)

Co

mp

res

sib

ilit

y

(1/G

Pa

)

Drained

Undrained

Unjacketed

Figure 1.9. Plot of the bulk compressibilities as a function of Terzaghi effective stress forBerea sandstone.

58

Skempton's B Coefficient as a Function of Terzaghi Effective Stress

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 10 20 30 40

Terzaghi Effective Stress (MPa)

Sk

em

pto

n's

B

C

oe

ffic

ien

t (∆

Ppor

e/∆P

c)

Figure 1.10. Skempton's B coefficient as a function of Terzaghi effective stress for Bereasandstone.

The drained bulk compressibilities are most sensitive to the Terzaghi effective stress,

decreasing by a factor of around one-third from the lowest effective stress to the highest

effective stress. The undrained and unjacketed bulk compressibilities also decrease but not

as dramatically, especially in the case of the unjacketed bulk compressibilities. In the

undrained case, the matrix has been stiffened and made more linear by the pore fluid, water.

As the effective stress increases, the compressibilities and Skempton's B coefficient appear

to approach asymptotic values and Berea sandstone behaves in a more linear elastic fashion.

Indiana limestone differs from Berea sandstone with respect to nonlinear behavior.

There is little dependence on the effective stress as inspection of Table 1.3 shows. The

linear behavior of Indiana limestone is shown graphically in Figure 1.11, where the values

of the bulk compressibilities do not vary a great deal as the effective stress increases. The

59

values of Skempton's B coefficient, shown in Figure 1.12, seem to be more sensitive,

decreasing from ~0.4 to ~0.3, but do not show nearly the dependence on effective stress

shown by the values of Skempton's B coefficient for Berea sandstone which decrease from

~0.9 to ~0.4 over a similar effective stress range as shown in Figure 1.10. Indiana

limestone is very nearly a linear elastic material, a result previously shown by Talesnick et

al. (1997).

Bulk Compressibilities as a Function of Terzaghi Effective Stress

0

0.01

0.02

0.03

0.04

0.05

0 1 0 2 0 3 0 4 0

Terzaghi effective stress (MPa)

Co

mp

res

sib

ilit

y

(1/G

Pa

) Drained

Undrained

Unjacketed

Figure 1.11. Bulk Compressibilities as a function of terzaghi effective stress for Indianalimestone. There is little dependence on effective stress.

60

Skempton's B Coefficient as a Function of Terzaghi Effective Stress

0

0.1

0.2

0.3

0.4

0.5

0 10 20 30 40

Terzaghi Effective Stress (MPa)

Sk

em

pto

n's

B

C

oe

ffic

ien

t (∆

Ppor

e/∆P

c)

Figure 1.12. Skempton's B Coefficient as a Function of Effective Stress for Indianalimestone.

1.5 Discussion

The above results show that Berea sandstone behaves anisotropically and in a

nonlinear elastic fashion at low effective stresses (less than 20 MPa) and becomes isotropic

and linear at higher effective stresses (greater than 30 MPa). They also show that Indiana

limestone behaves an isotropic and linear fashion over the effective stress range of this

experiment (5 to 30 MPa). These results can be explained not by looking at the anisotropy

of the grains that make up the two rocks, but by looking at grain contact and microcrack

models, which have been presented by other researchers to explain the phenomena of stress

dependent non-linearity and anisotropy (Mindlin, 1948; Walsh, 1965; Nur, 1971;

Kachonov, 1992).

61

Nonlinearity

Microcrack Model

The nonlinearity can be explained by assuming the porosity is composed of cracks

and pores with very different aspect ratios (a=height/length). When the effective stress is

low, both the long-thin cracks and the round pores are open and the rock is more compliant.

As the stress is increased, the more compliant low aspect ratio cracks, those cracks with a

length much larger than their height, close and the rock becomes stiffer because those

cracks can no longer contribute to strains (Mavko and Nur, 1978). When all of the lower

aspect ratio cracks are closed, the compliances no longer decrease with an increase in

effective stress and only cracks with aspect ratios near unity, the more circular pores, are

left. The rock now behaves in a nearly linear poroelastic fashion.

Grain Contact Model

The explanation of nonlinear poroelastic behavior using grain contact models is

similar to that of the microcrack model. At low effective stresses, fewer grains are in contact

and the area of contact of those grains is small. As the effective stress is increased, more

grains come into contact with each other and the area of contact between grains already in

contact increases. Both of these mechanisms make the rock stiffer and so the compliances

decrease as the effective stress increases. As the effective stress increases beyond a certain

value, all of the grains are in contact and the surface areas have increased to such a point that

an increase in stress produces little additional strain. The result is the same as in the case

of microcracks; the rock is now in a linear poroelastic regime. Figure 1.13 compares

compressibilities calculated assuming Hertzian contact (Love, 1944) with the experimental

drained compressibilities for Berea sandstone. Although the shape of the curves are similar

it should be noted that the Young's Modulus used to calculate the curve is over an order of

magnitude greater than the value given for quartz. This graph is only meant to illustrate the

62

point that nonlinear behavior can be produced by contact models, not to provide an exact

model of the observed nonlinear behavior.

Hertzian Contact Model Compressibilities Compared to Experimental Drained

Compressibilities

0.0

0.1

0.2

0.3

0.4

0.5

0 1 0 2 0 3 0 4 0

Terzaghi Effective Stress (MPa)

Co

mp

res

sib

ilit

y

(1/G

Pa

) Experimental Compressibilities

Hertzian Model Compressibilities

Figure 1.13. Nonlinearity of the drained compressibilities compared to Hertzian contactmodel compressibilities.

Anisotropy

Because the unjacketed or grain linear compressibilities show no significant

anisotropy, there is little or no contribution to anisotropy from mineral alignment. In

addition to the unjacketed measurements, a study of the C-axis alignment of the quartz

grains in Berea sandstone was conducted. A universal stage was used to corroborate the

results of an earlier study (Friedman and Bur, 1974) which showed no alignment of the C-

axis of the quartz grains. Also, ultrasonic velocity measurements conducted on dry samples

at Terzaghi effective stresses greater than 40 MPa showed Berea sandstone to be essentially

63

isotropic at those stresses, verifying the result reported in Thomsen (1986). The anisotropy

observed in Berea sandstone is not due to mineral alignment.

Microcrack Model

The anisotropy can also be explained by assuming the porosity is composed of

cracks and pores with very different aspect ratios. In the case of anisotropy, a larger

population of cracks aligned along an axis will increase the compliance in a direction

perpendicular to that axis. If a larger population of cracks is aligned parallel to the bedding

plane, then the compliance perpendicular to the bedding will be less than in the bedding

plane and the material will exhibit transverse isotropy. In addition to setting the degree of

anisotropy, microcracks might also produce other behaviors. For example, the peak in the

anisotropy ratios of the drained and undrained linear compressibilities at 5 MPa in Figure

1.7 might be explained by a more random orientation of the very longest microcracks which

close at effective stresses below 5 MPa. After those longest cracks close, the population of

cracks that are left are more strongly aligned with the bedding plane. As the effective stress

is further increased, those aligned cracks close and only the less compliant, more circular

pores are left, and the rock shows little anisotropy.

Grain Contact Model

Again instead of assuming the anisotropy is produced by alignment of microcracks,

the grain contact model produces anisotropy by assuming alignment of the grain contacts.

If few grains are initially in contact in some plane, the compliance measured perpendicular

to that plane will be larger than compliances measured in other planes and if the surface area

of the grains in contact in a plane is small, the compliance measured perpendicular to that

plane will be larger than compliances measured in other planes. As the effective stress is

increased, more strain occurs on the more compliant surfaces, so that more grain contacts

64

are formed and the areas of the existing contacts increase. The more compliant surface

becomes stiffer more quickly and approaches the value of the compliance of the stiffer

planes. In this way, the compliances of the planes may approach isotropy as the effective

stress is increased.

Estimation of Pore Compressibility

Gassmann's equation (Brown and Korringa, 1975) is commonly used in the oil and

gas industry to estimate how different pore fluids change the bulk modulus of the rock

(Berge, 1998).

Cd − Cu =Cd − Cs( )2

C f − Cφ( )φ + Cd − Cs( )[ ] (16)

where C f is the fluid compressibility and φ is the porosity. To use this equation, the pore

compressibility, Cφ , is often assumed to equal the grain or unjacketed compressibility, Cs .

This section tests that assumption.

If the grains are homogenous, then the pore compressibility will equal the grain

compressibility. When the unjacketed hydrostatic stress is applied, the pore pressure and

confining stress are both increased equally and so each grain will experience the exact same

strain with the result that the volumetric change in porosity will exactly equal the volumetric

change in the grains. The porosity will not change either in shape or magnitude. To

understand how the pore compressibility might be greater than the grain compressibility, it

is possible to imagine some more compliant grains located in pores and not tightly bound in

the rock matrix. When the unjacketed hydrostatic stress is applied, those grains will strain

at a greater rate than the stiffer grains with the result that the porosity increases because the

more compliant grains now contribute less to the volume of the solid portion of the rock.

65

Two possible mechanisms for more compliant grains are either more compliant minerals

like clays or microcracks isolated within some of the grains. The microcracks might

significantly soften a grain and not be connected to the primary porosity which transmit the

pore pressure. In this way those more compliant grains would experience more strain and

so Cφ need not equal Cs even for a monomineralic rock.

Because measurement of Cφ is technically difficult, no direct measurement has been

made to date so that the assumption might be tested. Calculated values of Cφ , using

equation 16, for Berea sandstone and Indiana limestone are in Table 1.16.

Table 1.16. Comparison of the calculated pore bulk compressibility to the measuredunjacketed bulk compressibility.Sample Confining

Stress (MPa)Pore Pressure

(MPa) Cφ Cs

Berea sandstone porosity φ=0.21

41 8 0.074 0.026

41 22 0.092 0.030

31 17 0.070 0.032

20 10 0.013 0.031

35 26 0.095 0.033

26 20 0.105 0.036

20 16 0.136 0.037

10 7 0.161 0.042

Indiana Limestone porosity φ=0.15

40 7.8 -0.042 0.013

40 20 -0.057 0.013

35 26 -0.025 0.013

31 17 -0.090 0.013

26 20 -0.043 0.013

20 10 -0.068 0.013

20 15 -0.081 0.013

9.6 7 0.010 0.013

66

Comparison of Pore Compressibilities to Measured Unjacketed Compressibilities for

Berea Sandstone

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

0 1 0 2 0 3 0 4 0

Terzaghi Effective Stress (MPa)

Co

mp

res

sib

ilit

y

(1/G

Pa

)

Cphi

Cs

Figure 1.14. Comparison of calculated pore compressibilities to measured unjacketedcompressibilities for Berea sandstone. The error bars represent errors of 5% in themeasurement of the bulk constants and the 10% in the measurement of porosity.

Comparison of Pore Compressibilities to Measured Unjacketed Compressibilities for

Indiana limestone

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0 5 1 0 1 5 2 0 2 5 3 0 3 5

Terzaghi Effective Stress (MPa)

Co

mp

res

sib

ilit

y

(1/G

Pa

)

Cphi

Cs

Figure 1.15. Comparison of pore compressibilities to measured unjacketedcompressibilities for Indiana limestone. The error bars represent errors of 5% in themeasurement of the bulk constants and the 10% in the measurement of porosity.

67

Figures 1.14 and 1.15 compare the calculated pore compressibility,Cφ , values to the

measured unjacketed values, Cs . Values of Cφ and Cs from Hart and Wang (1999) are

included in Figure 1.14. The error bars represent error of +/- 5% in the bulk

compressibilities and Skempton's B coefficient, and an error of +/- 10% in the porosity to

account for any variation caused by the testing. 5% error for the unjacketed compressibility

is less than dimensions of the symbol representing the unjacketed compressibility and so is

not shown. The error of 5% used for the compressibilities was chosen because only a few

of the differences between the measured and the best-fit sets of compressibilities had an

error greater than 5%. In general the error was less than 2% . The graph of

compressibilities as a function of effective stress shows that the pore compressibility for

Berea sandstone is probably not equal to the unjacketed or grain compressibility at low

effective stresses and possibly not at higher effective stresses as well and that the pore

compressibility is consistently larger than the grain compressibility. However, for Indiana

limestone, all of the error bars do intersect the values for the unjacketed compressibilities, so

in the case of Indiana limestone, it is likely that the pore compressibility is equal to the grain

compressibility.

Effective Stress

Effective stress is defined in this work to mean a set of pore pressure and confining

stress pairs that hold some property or process invariant (Carroll, 1979). Very often

effective stress is written as Peff = Pc − np , where n is an emperical effective stress

coefficient and Pc and p are a pair of the set of confining stress and pore pressure pairs

that hold the property or process invariant (Christensen and Wang, 1984). Although the

emperical effective stress coefficient, n , may be a constant, it may also depend on the pore

pressure and the confining stress (Warpinski and Teufel, 1992). If n is a function of the

confining stress and pore pressure, the usefulness of finding an effective stress coefficient

68

may be diminished as it is no longer possible to reduce the number of state variables, here

Pc and p , from two to one, Peff for all values of Pc and p (Gangi and Carlson, 1996).

Effective Stress Law for Poroelastic Constants

The effective stresses for the four hydrostatic poroelastic constants, Cd , Cu , Cs ,

and B , are discussed next. Figures 1.16 a,b,c, and d are plots of the four best-fit

hydrostatic poroelastic constants found in this experiment and from Hart and Wang (1999)

for Berea sandstone contoured in pore pressure and confining stress space. Contour lines

represent constant values of the poroelastic constants and so constitute sets of effective

stress pairs of pore pressure and confining stress. The emperical effective stress coefficient,

n , can be found empirically by setting n equal to the inverse of the slope of the contour

lines because for a contour (Bernabe, 1986; Warpinski and Teufel, 1992),

∆C = ∂C

∂p∆p + ∂C

∂Pc

∆Pc = 0 and n = −∂C∂p( )∂C∂Pc

( ) . (18)

69

Figure 1.16a. Contour plot of the measured drained bulk compressibilities for Bereasandstone.

70

Figure 1.16b. Contour plot of the measured undrained bulk compressibilities for Bereasandstone.

71

Figure 1.16c. Contour plot of the measured unjacketed bulk compressibilities for Bereasandstone.

72

Figure 1.16d. Contour plot of the measured Skempton's B coefficient for Berea sandstone.

From the contours of the drained bulk compressibility, Figure 1.16a and the

Skempton's B coefficient, Figure 1.16d, it appears that the emperical effective stress

coefficient is close to unity over most of the pore pressure and confining stress range.

This result of the emperical effective stress coefficient equal to unity agrees with the

theoretical analysis by Zimmerman (1986) for rock containing microcracks. It also explains

why the values of these compressibilities and Skempton's B coefficient all show little scatter

when plotted as function of the Terzaghi effective stress, Figures 1.9-1.12. The effective

stress coefficient for the Terzaghi effective stress is equal to 1.

These contour plots also show more strongly the errors associated with the

measurements. It is likely that the peaks and valleys, seen in the undrained and unjacketed

compressibility plots, Figures 1.16 b and c, are due to measurement error and not from

73

behavior in the rock. Both of the mechanisms given above for nonlinearity, microcracks and

grain contacts, would give only single valued responses to increases in either the pore

pressure or confining stress.

Two general trends can be seen from these four contour plots. The first is that an

increase in pore pressure causes the rock to be more compliant and an increase in confining

stress causes the rock to be less compliant; as effective stress increases, the rock becomes

less compliant. The second is that the rock behaves in a nearly linear fashion at higher

effective stresses, the portions of the plots that are to the bottom and to the right.

The range of stresses here correspond to shallow burial. Under lithostatic stress

and pore pressure conditions, and assuming a rock density of 2.5 g/cm3, a depth of one

kilometer corresponds to a pore pressure of 10 MPa and a confining stress of 25 MPa. It

is an easy matter to find the corresponding values of the poroelastic constants that would

correspond to that pore pressure and stress pair, approximately 0.5 for Skempton's B

coefficient. Because the effective stress coefficient is approximately one for the Skempton's

B plot, the effective stress is Terzaghi's effective stress and is equal to 15 MPa. Using

Figure 1.10, it is also possible to find that same value of 0.5 for Skempton's B coefficient at

a Terzaghi effective stress of 15 MPa.

Effective Stress Law for Strain

The effective stress laws for the bulk compressibilities and Skempton's B coefficient

are not commonly used or measured. It is much more common to find an effective stress

law for strain, assuming linear poroelastic behavior. The effective stress law coefficient for

an effective stress for volumetric strain is called the Biot-Willis parameter, α (Biot and

Willis, 1957). In this law, the strain is held invariant for pairs of pore pressure and

confining stress, p = αPc and Pc . Nur and Byerlee (1971) found the exact form of the

law for a linear elastic material. An extension of that law is given here for a nonlinear

74

material for which the poroelastic constants have been measured as functions of pore

pressure and confining stress. It is possible to rewrite the constitutive equation, Equation

12, under hydrostatic stress conditions using the bulk compressibilities. The result is

ev = −Cd Pc + Cd − Cs( )p. (19)

where ev is the volumetric strain. If the material is linear, then the Biot-Willis parameter

can be found by rewriting the equation in the form of an effective stress so that equation 19

becomes

ev = −Cd Pc − Cd − Cs

Cd p

. (20)

so the effective stress coefficient is α =1 − Cs

Cd . If the material is non-linear, this equation

holds only for small values of p and Pc because Cd and Cs and thus α now depend on

the confining stress and the pore pressure. The total volumetric strain will be an integral

consisting of contributions from confining stress and pore pressure and can be written as

below. The total volumetric strain considered here will be less than 1% and so can be

treated as infinitesimal strain. If the material is elastic, then the final volumetric strain is path

independent and so it is possible to write the volumetric strain as the sum of two separate

integrals, one integrated with respect to confining stress and the other integrated with respect

to pore pressure.

ev = − ∫Cd (Pc , p)dPc + ∫ Cd (Pc , p) − Cs (Pc , p)( )dp (22)

The forms of Cd (Pc , p) and Cs (Pc , p) can be estimated from the contour plots and

either fit to some model or to some polynomial surface. A simple power law fit is used

75

below to illustrate how the effective stress pairs of confining stress and pore pressure can be

found. The regression correlation coefficient, r2, is equal to 0.88 for this model.

Cd (Pc , p) = 0.302 Pc − p( )−0.378(23)

where Cd (Pc , p) has units of 1/GPa and Pc and p have units of MPa. The value of the

unjacketed compressibility is relatively constant with confining stress and pore pressure and

so will be modeled as such, Cs =0.03 1/GPa. Substitution of the power law fit into

equation 22 and subsequent integration yields

ev = −0.302Pcf − p f( )1−0.378

− Pci − pi( )1−0.378

1 − 0.378( )

− Cs p f − pi( ) (24)

where ev has units of millistrain and the subscripts f and i of the pore pressure and the

confining stress stand for final and initial pressures and stresses.

If values for the initial and final confining stress are given as 10 and 30 MPa

respectively and the initial pore pressure is 5 MPa, then the final pore pressure that results

in zero strain can be found and is approximately 28.5 MPa. The pairs of pore pressure and

confining stress, (5, 10) and (28.5, 30), are two pairs of the infinite set for that particular

effective stress or curve of zero strain.

These two effective stress pairs can be used to test whether or not using the linear

Biot-Willis parameter would give the same result or if using equation 22 is necessary. The

linear Biot-Willis parameter is equal to

α =1 − Cs

Cd =1 − 0.0360.22

= 0.84 (25)

76

where Cd = 0.22 1/GPa and Cs =0.036 1/GPa are estimated from the contour plots,

Figures 1.16a and c, at the initial effective stress pair, Pc = 10 MPa and p=5 MPa. If

∆Pc = 20 MPa then equation 20 yields a ∆p = 23.8 MPa and a final effective stress pair,

Pc = 30 MPa and p=28.8 MPa.. The results for ∆p are nearly the same whether using

the nonlinear solution or the linear solution suggesting that using the linearized Biot-Willis

effective stress law will not result in large errors when calculating effective stresses for

volumetric strain for nonlinear materials, even at the low effective stresses considered here.

1.6 Conclusion

Sets of transversely isotropic poroelastic constants were measured under drained,

undrained, and unjacketed pore fluid boundary conditions for Berea sandstone and Indiana

limestone. This was done while the sample was at the same reference pore pressure and

confining stress so that the internal structure of the sample was the same for all of the

measurements. The eight sets of constants corresponding to eight different pore pressure

and confining stress pairs were measured so that the variation of the poroelastic constants

with pore pressure and confining stress could be determined. The measurements showed

that Berea sandstone is transversely isotropic at low effective stresses (less than 20 MPa)

and becomes isotropic at higher effective stresses (greater than 30 MPa) whereas Indiana

limestone is isotropic for all of the effective stresses. The anisotropic behavior of Berea

sandstone is not due to grain anisotropy but is caused by oriented porosity or grain to grain

contacts not present in the Indiana limestone.

In addition to anisotropy, Berea sandstone exhibits nonlinear poroelastic behavior.

As the effective stress increases, the compliances and Skempton's B coefficient decrease and

approach asymptotic values at effective stresses greater than 30 MPa. Berea sandstone

becomes less compliant as the effective stress increases with most of the change occurring

at the lower effective stresses. The values of the compliances vary relatively little for Indiana

77

limestone as a function of effective stress so Indiana limestone is a linear poroelastic

material for all of the effective stresses in this study. The nonlinear behavior of the Berea

sandstone can be explained by the same cracks or grain to grain contacts that caused the

anisotropic behavior.

Estimates of the pore compressibility were made and suggest that the assumption

often made when using Gassmann's equation that the pore compressibility is equal to the

grain compressibility may not hold for Berea sandstone. That assumption does appear to

hold for Indiana limestone. The effective stress coefficient for the hydrostatic poroelastic

constants was found to be equal to one, reconfirming the result found by Zimmerman

(1986). The effective stress law for strain was also investigated. There was little difference

between effective stresses calculated assuming nonlinear behavior of the poroelastic

constants compared to effective stresses calculated using initial values of the constants and

holding them invariant.

78

References

Amadei, B. 1983. Rock anisotropy and the theory of stress measurement. Lecture Notes inEngineering, Springer-Verlag.

Berge, P.A., Wang, H.F. & Bonner, B.P. 1993. Pore pressure build-up coefficient in synthetic andnatural sandstones. Int. J. Rock Mech. Min. Sci. and Geomech. Abstr. 30:1135-1141.

Berge, P.A. 1998. Pore compressibility in rocks. In Thimus et al. (eds.), Poromechanics: 351-356. Rotterdam: Balkema.

Bernabe, Y. 1986. The effective pressure law for permeability in Chelmsford Granite and BarreGranite. Int. J. Rock Mech. Min. Sci. and Geomech. Abstr. 23:267-275.

Biot, M.A. & Willis, D.G. 1957. The elastic coefficients of the theory of consolidation. J. Appl.Mech. 24:594-601.

Brace, W.F. 1965. Some new measurements of linear compressibility in rocks. J. Geophys. Res.70:391-398.

Brown, R.J. & Korringa, J. 1975. On the dependence of the elastic properties of a porous rock onthe compressibility of the pore fluid. Geophysics. 40:608-616.

Bruhn, R.W. 1972. A study of the effects of pore pressure on the strength and deformability ofBerea sandstone in triaxial compression. U.S. Dept. of the Army Tech. Report, Engng. StudyNo. 552.

Carroll, M.M. 1979. An effective stress law for anisotropic elastic deformation. J. Geophys. Res.84:7510-7512.

Cheng, A.H. 1997. Material coefficients of anisotropic poroelasticity. Int. J. Rock Mech. Min. Sci.34:199-205.

Christensen, N.I. & Wang, H.F. 1985. The influence of pore pressure and confining pressure ondynamic elastic properties of Berea sandstone. Geophysics. 50:207-213.

Endres, A.L. 1997. Geometrical models for poroelastic behavior. Geophys. J. Int. 128:522-532.

Fredrich, J.T., Martin, J.W. & Clayton, R.B. 1994. Induced pore pressure response duringundrained deformation of tuff and sandstone. Mechanics of Materials 20:95-104.

Friedman, M. & Bur, T.R. 1974. Investigations of the relations among residual strain, fabric,fracture and ultrasonic attenuation and velocity in rocks. Int. J. Rock Mech. Min. Sci. &Geomech. Abstr. 11:221-234.

Green, D.H. & Wang, H.F. 1986. Fluid pressure response to undrained compression in saturatedsedimentary rock. Geophysics. 51:948-956.

Hannah, R.L. & Reed, S.E. 1992. Strain Gage User's Handbook; 300-307. London. Elsevier.

Hart, D.J. & Wang, H.F. 1995. Laboratory measurements of a complete set of poroelastic modulifor Berea sandstone and Indiana limestone. J. Geophys. Res. 100:17,741-17,751.

79

Hart, D.J. & Wang, H.F. 1999. Pore pressure and confining stress dependence of poroelasticlinear compressibilities and Skempton's B coefficient for Berea sandstone. In Amadei et al.(eds.), Rock Mechanics for Industry. pp. 365-371. Rotterdam, Balkema.

Jones, L.E. & Wang, H.F. 1981. Ultrasonic velocities in Cretaceous shales from the Willistonbasin. Geophysics. 46:288-297.

Kachanov, M. 1992. Effective elastic properties of cracked solids: critical review of some basicconcepts. Appl. Mech. Rev. 45:304-335.

Lo, T., Coyner, K.B., & Toksoz, M.N. 1986. Experimental determination of the elastic anisotropyof Berea sandstone, Chicopee shale, and Chelmsford granite. Geophysics. 51:164-171.

Logan, W.N., Visher, S.S., Cumings, E.R., Tucker, W.M., Malott, C.A. & Reeves, J.R. 1922.Handbook of Indiana Geology. pp. 772-773. W.B. Buford, Contractor for State Printing andBuilding, Indianapolis, Indiana.

Love, A.E. 1944. A Treastise on the Mathematical Theory of Elasticity. pp.195-198. DoverPublications, New York.

Mavko, G.M. & Nur, A. 1978. The effect of nonelliptical cracks on the compressibility of rocks.J. Geophys. Res. 83:4459-4468.

Menke, W. 1989 Geophysical Data Analysis: Discrete Inverse Theory. p. 289. Academic, SanDiego, California.

Mindlin, R.D. 1949. Compliance of elastic bodies in contact. J. Appl. Mech. 16:259-268.

Nur, A. & Simmons, G. 1969. Stress-induced velocity anisotropy in rock: an experimental study.J. Geophys. Res. 74:6667-6674.

Nur, A. 1971. Effects of stress on velocity anisotropy in rocks with cracks. J. Geophys. Res.76:2022-2034.

Nur, A. & Byerlee, J.D. 1971. An exact effective stress law for elastic deformation of rocks withfluids. J. Geophys. Res. 76:6414-6419.

Nye, J. F. Physical properties of crystals, their representation by tensors and matrices. -- Oxford,Clarendon Press, 1998.

Roeloffs, E.A. 1982. Elasticity of saturated porous rocks: laboratory measurements and a crackproblem. Ph.D. Thesis, University of Wisconsin-Madison.

Sayers, C.M. & Kachanov, M. 1995. Microcrack-induced elastic wave anisotropy of brittle rocks.J. Geophys. Res. 100:4149-4156.

Schlueter, E.M., Zimmerman, R.W., Witherspoon, P.A., and Cook, N.G.W. 1997. The fractaldimension of pores in sedimentary rocks and its influence on permeability. EngineeringGeology 48: 199-215.

80

Talesnick, M.L., Haimson, B.C., & Lee, M.Y. 1997. Development of radial strains in hollowcylinders of rock subjected to radial compression. International Journal of Rock Mechanics andMining Sciences & Geomechanics Abstracts. 34:1229-1236. 1997.

Thomsen, L. 1986. Weak elastic anisotropy. Geophysics. 51:1954-1966.

Tokunaga, T., Hart, D.J. & Wang, H.F. 1998. Complete set of anisotropic poroelastic moduli forBerea sandstone. In Thimus et al. (eds.), Poromechanics: 629-634. Rotterdam: Balkema.

Tokunaga, T. 1999. Personal Communication.

Walsh, J.B. 1965. The effect of cracks on the compressibility of rocks. J. Geophys. Res. 70:381-389.

Warpinski, N.R. & Teufel, L.W. 1992. Determination of the effective stress law for permeabilityand deformation in low-permeability rocks. SPE Formation Evaluation. 7; 2, Pages 123-131.1992.

Winkler, K. 1983. Frequency-dependent ultrasonic properties of high-porosity sandstones. J.Geophys. Res. 88:9493-9499.

Wissa, A.E. 1969. Pore pressure measurements in saturated stiff soils. J. Soil Mech. & Foun. Div.Am. Soc. Civ. Eng. 95:1063-1073.

Zimmerman, R.W., Somerton, W.H., & King, M.S. 1986. Compressibility of porous rocks. J.Geophys. Res. 91:12,765-12,777.

81

Chapter 2

Poroelastic Effects During a Laboratory Transient Pore Pressure Test.

From near to far, from here to there, funny things are everywhereãDr. Seuss

ABSTRACT: Transient pore pressure tests are used to determine fluid transmission and storage

behavior of porous material. In this study we investigated the coupling between stress, strain, and

pore pressure during a transient pore pressure test in which a pressure step was applied to one end

of a cylindrical core and a no-flow boundary was applied to the other end while the external stress

was held constant. The following results, applicable to most transient pore pressure tests, were

obtained from modeling and verified by experiment. 1) For a positive pressure step, a small pore

pressure decrease develops within the sample at early times. This induced pore pressure of

opposite sign is an example of a Mandel-Cryer effect. 2) For a positive pressure step, the axial

strain decreases at early times. Subsequently, the axial strain increases as the sample expands in

response to diffusion. The circumferential strain does not show the decrease at early time. 3) The

fully-coupled poroelastic response is nearly identical to an uncoupled diffusive flow response after

the early time interval has passed. 4) Strain gages can be used to better constrain the hydraulic

parameters by measuring pore pressure responses along the sample length. 5) Because the sample

is under constant stress, the unconstrained specific storage, Sσ , is the parameter measured in most

transient pore pressure tests in the laboratory.

82

2.1 Introduction

Transient pore pressure tests have been developed to measure hydraulic flow parameters in

low permeability rocks. The pulse-decay method (Brace et al., 1968) and the constant rate injection

test (Olsen et al., 1988) are two examples of this type of test. In these tests, a pore pressure or flow

rate is introduced at one end of a sample and the differential pore pressure between the sample ends

is measured with respect to time. It is possible to invert the pressure-time data either numerically or

graphically to find the permeability and the specific storage of the sample (Hsieh, 1981; Senseny et

al., 1983; Hart and Wang, 1993; and Esaki et al., 1996). Poroelastic effects during transient pore

pressure tests have been discussed by Walder and Nur (1986), Morin and Olsen (1987), Adachi

and Detournay (1997) and briefly by Neuzil (1986). The goals of the present study are to provide

a qualitative understanding of poroelastic phenomena during a transient pore pressure test and

numerically model the phenomena. The results of the poroelastic model will be compared to a one-

dimensional diffusive flow solution and an experiment designed to give confirmation of the model.

2.2 Model

The one-dimensional diffusive flow and the fully-coupled poroelastic models are developed

in this section for a cylindrical sample with no-flow boundaries applied to the sample bottom and

sides. A pore pressure step is applied to the sample top. The external stress (confining pressure) is

held constant and the sample bottom is fixed in the vertical direction. These boundary conditions

were chosen because they are typical in a laboratory setting.

The usual analysis of a transient pore pressure test is based on the one-dimensional

diffusion equation, the result of Darcy's Law and fluid continuity:

∇ 2 p = K

Sσ

∂p

∂t(1)

83

Here K, the hydraulic conductivity, and Sσ , the three dimensional unconstrained specific storage,

are the hydraulic parameters and p is the pore fluid pressure. Gravity head is neglected here

because of the short sample length and large pressure steps.

When the above hydraulic boundary conditions are applied to the governing equation, the

resulting solution for pore pressure as a function of time is (Carslaw and Jaeger, 1947)

∆p(x, t) = ∆po + ∆po {exp[−(K /n=1

∞

∑ Sσ )(t / L2 )π / 4(2n +1)2 ]

×cos[(2n +1)π(x / 2L)][4(−1)n+1 / (2n +1)π]}

(2)

Here L is the sample length, x is measured from the sample bottom, and K, Sσ , µ, and p are the

same as in equation 1. It should be noted that K / Sσ is a hydraulic diffusivity and

t / L2( ) K / Sσ( )is a dimensionless time.

A poroelastic response occurs in the sample because the pore pressure step creates strains

in the sample. If those strains cause changes in pore pressure elsewhere in the sample the response

is said to be fully coupled. When stress equilibrium, the poroelastic constitutive equations, Darcy's

Law, and fluid continuity are combined, the result is four coupled nonhomogeneous differential

equations. For an axisymmetric problem, the number of equations reduces to three (Hsieh, 1994).

The three equations are two axisymmetric displacement equations:

G ∇ 2ur − ur

r2

+ G

1 − 2υ∂ε kk

∂r− α ∂p

∂r= 0 (3)

G ∇ 2uz( ) + G

1 − 2υ∂ε kk

∂z− α ∂p

∂z= 0 (4)

84

and the axisymmetric form of the fluid diffusion equation:

k

µ∇ 2 p( ) − α ∂ε kk

∂t= Sε

∂p

∂t(5)

Here G, υ, and α are the shear modulus, Poissons ratio, and the Biot-Willis parameter respectively,

ur and uz are the radial and axial displacements, εkk is the volumetric strain, and Sε is the specific

storage at constant strain.

These equations were solved numerically using a finite element code, Biot2 (Hsieh, 1994)

developed for axisymmetric poroelastic problems. A 21-by-24 nodal array was used to model a

sample with a radius of 0.0254 m and a length of 0.0309 m. The values of the poroelastic constants

in equations 3-5 were calculated for an argillaceous sandstone, Calgary sandstone, from an

experimental determination of the undrained bulk modulus and Skempton's B coefficient, and

estimates of the grain compressibility and Poisson's ratio. The porosity and hydraulic diffusivity of

Calgary sandstone were found to be 0.05 and 1.4 x10-6 m2/s, respectively. Those values were used

in the base model to understand the poroelastic responses. Table 2.1 shows the measured,

estimated, and calculated parameters for the base model.

85

Table 2.1. Parameters used in the base model.

Parameter Units Values

Measured Poroelastic

Undrained Bulk Modulus-Ku (GPa) 17.9

Skempton's B Coefficient-B (Pa/Pa) 0.86

Porosity-φ 0.05

Estimated Poroelastic

Grain Modulus-Ks (GPa) 30

Poisson's Ratio-υ 0.20

Calculated Poroelastic

Bulk Modulus-K (GPa) 5.2

Shear Modulus-G (GPa) 3.9

Measured Flow Parameter

Hydraulic Diffusivity-D (m2/s) 1.4x10-6

Calculated Flow Parameters

Specific Storage-Sσ (m-1) 1.8x10-6

Hydraulic Conductivity-K (m/s) 2.6x10-12

2.3 Model Results

Qualitative Explanation

Figure 2.1 provides a qualitative understanding of the fully-coupled poroelastic response.

The pore pressure step reduces the effective stress and "pulls" the sample apart. This creates a

region of pore pressure change opposite in sign to the pressure step as shown in Figure 2.1. Also,

for a positive pressure step, a region of negative axial strain occurs that is associated with the region

of pore pressure decrease. Subsequently, the axial strain increases as the sample expands as pore

pressure diffusion overtakes the initial pore pressure decrease. The effect of a pore pressure

86

generating a strain which in turn induces a pore pressure of sign opposite to the initial pore

pressure is termed a Mandel-Cryer effect (Mandel, 1953; Cryer, 1963).

NegativeAxial Strains

No Flow No Displacement

{ {

Sample Before Pressure Step

Deformed Sample

Region of Uncoupled Diffusive Flow

No Flow Constant Stress

No Flow Constant Stress

Constant StressConstant Pore Pressure

Region of Negative

PorePressure

NegativeAxial Strains

Figure 2.1. Cartoon of the fully coupled poroelastic response.

After the early-time fully-coupled poroelastic response, diffusive flow dominates the

poroelastic response. At this time and location in the sample, the fully-coupled poroelastic response

for pore pressure calculated by Biot2 and the one-dimensional diffusive flow solution converge.

The problem is now uncoupled and the strains and pore pressure are linearly related through the

constitutive equations (Biot, 1941; Rice and Cleary, 1976). This allows measured strains to record

the pressure diffusion. This region of diffusive flow is shown in Figure 2.1. Figure 2.2 compares

87

the pore pressure at the sample bottom calculated for the fully-coupled poroelastic response and the

one-dimensional diffusive flow solution. After the initial early-time fully-coupled response, the two

solutions converge and it is no longer necessary to solve the fully-coupled problem.

Dimensionless Time versus Normalized Pressure at the Sample Bottom

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5

Dimensionless time (t/L^2)(K/S)

No

rma

lize

d

Pre

ss

ure

Diffusion Alone

Fully Coupled

Early Time Response versus Normalized Pressure at the Sample Bottom

-0.05

0

0.05

0.1

0.15

0.2

0 0.05 0.1 0.15

Dimensionless time (t/L^2)(K/S)

No

rma

lize

d

Pre

ss

ure

Diffusion Alone

Fully Coupled

Figure 2.2. Normalized Pore Pressure versus dimensionless time at the sample bottom.

88

Diameter-to-Length Responses

Figure 2.3 shows the dependence of the early-time pore pressure response for several

diameter-to-length ratios. The smallest diameter-to-length case shows little response and the larger

diameter-to-length case shows the greatest response, because the bending moment is greatest in the

latter case. It should be noted that the one-dimensional diffusive flow solution and the fully-

coupled solution still converge after the early-time response, even for the widest sample.

Radius = 0.254 m

Length = 0.0309 m

0.5 - 1.00.0 - 0.5less than 0.0

Key - Normalized Pore Pressure

Radius = 0.00254 m

Length = 0.0309 m

Radius = 0.0254 m

Length = 0.0309 m

Negative Pressure Region as

a Function of Changing Radius

Figure 2.3. Early time pore pressure response as a function of radius.

89

Dimensionless Time Comparisons

Figure 2.4 compares two models with the same length to radius ratio. In the first model, the

hydraulic conductivity, length, and time of observation are an order of magnitude less than for the

second model. The storage is held constant. When the two results are compared at the same

dimensionless time, the pressure profiles are identical. This means that solutions for all lengths and

all radii do not need to be calculated, only solutions for every length to radius ratio, to have all the

possible pressure profiles, assuming that the elastic properties of the material do not vary.

0.5 - 1.00.0 - 0.5less than 0.0

Key - Normalized Pore Pressure

Radius = 0.0254 m Length = 0.0309 m

Dimensionless Variable = (K/S) (t/L^2)= (2.57e-12m/s)/(1.83e-6 1/m)(2.85 s)/(0.0309 m)^2=4.2e -3

Radius = 0.254 m Length = 0.309 m

Dimensionless Variable = (K/S) (t/L^2)= (2.57e-11 m/s) / (1.83e-6 1/m) (28.5 s)/ (0.309 m)^2 = 4.2 e -3

Figure 2.4. Comparison of the pressure profiles at the same dimensionless time with variation ofthe sample size and hydraulic conductivity holding the storage and elastic parameters constant.

90

Sensitivity to Shear Modulus and Poisson's Ratio

Figure 2.5 shows the sensitivity of the early time pore pressure response to variation of

shear modulus and Poisson's ratio with a constant bulk modulus for three cases. Case 2 is the base

case given by the parameters in Table 2.1. The shear modulus and Poisson's ratio were varied while

the value of the bulk modulus was held constant at 5.2 GPa. No other input parameters were varied

between the models, i.e., the hydraulic diffusivity, the hydraulic conductivity, the porosity, the grain

bulk modulus, and the storage were all held constant for the three tests shown below. Although

there is little variation in the extent of the negative pore pressure response between the three cases,

the maximum negative pore pressure response is sensitive to the shear modulus and Poisson's ratio.

A larger shear modulus and a smaller Poisson's ratio result in the largest negative pore pressure

response, Case 1, while a smaller shear moduli and larger Poisson's ratios result in a smaller

negative pore pressure response as shown by Case 3. This result is understood by looking at the

result of smaller shear moduli and larger Poisson's ratios. As the shear modulus decreases and the

Poisson's ratio increases, the material will act more and more like a fluid and so will not resist any

shear. When the pore pressure step is applied, it "pulls" the sample apart and strain is transmitted

ahead of the diffusive pressure. If the material has no resistance to shear, the strain will not be

transmitted as is shown in the modeling. This result is intriguing because it suggests that the early

time negative pore pressure response might be used to find shear moduli and Poisson's ratios

without needing to apply a uniaxial load. The test can be done in a hydrostatic vessel.

91

Case 1 G=6.6 GPa υ=0.05 Legend

1.000 to 0.000

0.000 to -0.025

-0.025 to -0.050

-0.050 to -0.075

-0.075 to -0.100

Normalized Pore Pressure (Pressure Step at Sample Top Equal to 1)

Case 2 G=3.9 GPa υ=0.20

Case 3 G=1.1 GPa υ=0.40

Figure 2.5. Early time pore pressure response sensitivity to the shear modulus (G) and Poisson'sratio (υ) with a constant bulk modulus. All other elastic and flow parameters were held constant.

92

Strain Response

Figure 2.6 shows the axial and circumferential strains calculated by the fully-coupled model

and the pore pressure calculated from the diffusive flow equation for a location midway along the

model. The axial strain shows an early time negative response quite strongly while the

circumferential strain does not. The circumferential strain and the pressure calculated by diffusive

flow are proportional throughout the entire test. The axial strain is slightly less than the

circumferential strain until the end of the test because the sample edge remains curved until the

sample reaches steady state. The dilation from the increasing pore pressure seen by both the

circumferential and axial strains is decreased in the axial strain by the bending of the sample.

Normalized Axial and Circumferential Strains Compared to Diffusive Flow Pore Pressures

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 200 400 600 800 1000

Dimensionless time (t/L^2)(K/S)

No

rma

lize

d

Str

ain

Axial Strain

Circumferential Strain

Diffusive Flow Solution

Figure 2.6. Comparison of axial and circumferential strains with the pressure calculated fromdiffusive flow.

93

Strain Sensitivity to Shear Modulus and Poisson's Ratio

Figure 2.7 shows the early time axial strain response midway along the sample length for

the same three cases shown in Figure 2.5 of varying shear modulus and Poisson's ratio while

holding the bulk modulus constant. However, unlike the pore pressure profiles in Figure 2.5, there

is negligible variation between the three cases. The negative axial strain response is not sensitive to

variations of the shear modulus and Poisson's ratio. The negative axial strain is due to bending

from the outward bulge of the sample as shown in Figure 2.1. The mode of the bulge is volumetric,

not shear, and so variation of the shear modulus while holding the bulk modulus constant has little

effect. It is the bulk moduli that will control the amount of negative axial strain in the early time

response. The pore pressure profiles in Figure 2.5 are sensitive to the shear modulus because the

bulge at the top of the sample creates shear farther ahead of the bulge.

Early time axial strain sensitivity to shear modulus and Poisson's ratio

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 0.02 0.04 0.06 0.08 0.1

Dimensionless time (t/L^2)/(K/S)

No

rma

lize

d

Str

ain

Case 1 (G=6.6 GPa, v=0.05)Case 2 (G=3.9 GPa, v=0.20)

Case 3 (G=1.1 GPa, v=0.40)

Figure 2.7. Comparison of early time axial strains located midway along the model length for thethree cases of shear modulus and Poisson's ratio.

94

2.4 Experimental Results

Description of Experiment

To test the results of the fully coupled model an experiment was conducted using a tight

argillaceous sandstone, Calgary sandstone. Two samples were prepared, Ca1 and Ca3, both with

diameters of 5.04 cm and lengths of 8.1 and 3.1 cm, respectively. Axial and circumferential strain

gages of length 4 mm and width 3 mm were applied 5.0 cm from the bottom of sample Ca1. Axial

strain gages were applied 1.3 cm from the bottom of sample Ca3. The samples were then prepared

in a manner similar to those in Chapter 1. RTV silicone was applied to the sample sides and Tygon

sleeves were fitted over the ends. The samples were placed under a vacuum and then saturated with

tap water. The low volume pore pressure endplug used in Chapter 1 was inserted into the Tygon

sleeve at the sample bottoms. A plug, connected to the pore pressure system when the sample was

loaded into the pressure vessel, was placed in the Tygon sleeve at the sample tops. The samples

were placed in the pressure vessel and allowed to reach equilibrium before testing began. A test

consisted of rapidly increasing or decreasing the pore pressure at the sample top. The experimental

setup was designed to approximate the boundary conditions used in the fully-coupled model. The

endplug located at the sample top was to apply the step change in pore pressure at the sample top

and be a constant stress boundary. The bottom endplug was to apply no-flow and no

displacements to the sample bottoms and the Silicone and Tygon sleeves were to apply no-flow and

and constant stress boundaries to the sample sides.

Pore Pressure Results

Figures 2.8a and 2.8b show the normalized pore pressure at the bottom of the short sample

Ca3 for a pore pressure step of -6.9MPa from an initial pore pressure of 28 MPa and a confining

stress of 41 MPa. The data were normalized by subtracting the initial pore pressure and then

dividing by the pressure step. The data have been normalized so that comparisons can be made

95

with strain data and results from the long sample Ca1. These data show two of the results predicted

by the model. The first result, shown in Figure 2.8a, is that the experimental pore pressure matches

the pore pressure calculated from diffusive flow for most ot the test. The hydraulic diffusivity used

to match the data was 1.4x10-6 m2/s.

The second result, seen in Figure 2.8b, is that a small pore pressure, opposite in sign to that

of the pressure step, developed at the sample bottom at early times as predicted by the fully coupled

poroelastic model. The experimental early time pore pressure response is less than that predicted

by the model. The reason for this may be that the constant stress boundary conditions at the

sample top and bottom are not being met in this experimental setup. When the pore pressure step

is applied and the sample top bulges upward at the sample top, the sample top center will encounter

the steel endplug before the top edge. Instead of a constant stress boundary, a point back stress will

be applied to damp the upward bulge that contributes to the reverse pore pressure response. At the

sample bottom, a similar damping effect will occur. The center of the sample will warp upward and

so the bottom edges of the sample will encounter the steel bottom plug and again instead of a

constant stress boundary at the sample bottom, a back stress is applied by the outer rim of the

endplug that will damp the upward bulge, reducing the reverse pore pressure response.

96

Normalized Pore Pressure versus Time at the Sample Bottom - Short Sample Ca3

-0.2

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000

Time (seconds)

No

rma

lize

d

Po

re

Pre

ss

ure

Experimental Pressures

Diffusion Alone Pressures

Fully Coupled Pressures

Figure 2.8a. Comparison of the experimental normalized pressure at the sample bottom for theshort sample Ca3 with pore pressures calculated from the fully coupled poroelastic model and fromdiffusive flow alone.

97

Early time response - short sample Ca3

-0.05

0

0.05

0.1

0.15

0.2

0 50 100 150 200

Time (seconds)

No

rma

lize

d

Po

re

Pre

ss

ure

Experimental Pressures

Diffusion Alone Pressures

Fully Coupled Pressures

Figure 2.8b. The early time response of the pore pressure at the sample bottom in the short sample,Ca3, compared to the pore pressures calculated from diffusive flow and the fully coupledporoelastic model. The early time pore pressure reversal is present and is another example of aMandel-Cryer type effect.

Figures 2.9a and 2.9b show the normalized pore pressure response at the sample bottom for

the long sample, Ca1 for a pore pressure step of 18 MPa to 0 MPa at a confining stress of 27 MPa.

The pore pressures measured at the sample bottom were normalized in the same way as for the Ca3

data. Figure 2.9a shows that, as was the case for the data from Ca3, the experimental pore pressure

matches the pore pressure calculated from the diffusive flow alone. The hydraulic diffusivity for

sample Ca1 was found to be 2.5x10-7 m2/s.

Unlike the pore pressure data from Ca3 shown in Figure 2.8b, pore pressure data from Ca1,

shown in Figure 2.9b, show there is a negligible reverse pore pressure response at early time for the

98

long sample. This result is predicted by the fully coupled poroelastic model shown in Figure 2.9b

and confirms the results shown in Figure 2.3 that the reverse pore pressure pulse is damped by the

time the pressure front reaches the sample bottom for samples with large length to diameter ratios.

Normalized pore pressure versus time at the sample bottom - long sample Ca1

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10000 20000 30000 40000

Time (seconds)

No

rma

lize

d

Pre

ss

ure

s

Experimental Pressures

Diffusion Alone Pressures

Fully Coupled Pressures

Figure 2.9a. Comparison of the experimental normalized pressure at the sample bottom for thelong sample Ca1 with pore pressures calculated from the fully coupled poroelastic model and fromdiffusive flow alone.

99

Early time response - long sample Ca1

-0.05

0

0.05

0.1

0.15

0.2

0 500 1000 1500 2000

Time (seconds)

No

rma

lize

d

Pre

ss

ure

s

Experimental Pressures

Diffusion Alone Pressures

Fully Coupled Pressures

Figure 2.9b. The early time response of the pore pressure at the sample bottom in the long sample,Ca1, compared to the pore pressures calculated from diffusive flow and the fully coupledporoelastic model. The early time pore pressure reversal is negligible in both the experimental dataand the fully coupled poroelastic model.

Strain gage results

Figures 2.10a and 2.10b show the axial strains in the short sample, Ca3, measured by strain

gages located 1.3 cm from the sample bottom. The strain data were collected in the same test as the

pore pressure data shown Figures 2.8a and 2.8b as a function of time. The strains were normalized

by subtracting the initial strain and then dividing by the final strain at steady state. Figures 2.11a

and 2.11b show the normalized axial and circumferential strains in the long sample, Ca1, measured

by strain gages located 5.0 cm from the sample bottom. The data were collected in the same test as

the pore pressure data shown in Figures 2.9a and 2.9b.

100

The model result that the strains are linearly proportional to the pore pressure calculated

from diffusive flow alone after the early time response shown in Figure 2.6 is shown here in

Figures 2.10a and 2.11a experimentally. The diffusivity used to match the experimental pore

pressure at the sample bottom in Figure 2.8a, is the same as the diffusivity used to match the

normalized strains 1.3 cm from the bottom of sample Ca3 in Figure 2.10a, D=1.4x10-6 m2/s.

Likewise the diffusivity used to match the experimental pore pressure at the sample bottom in

Figure 2.9a is the same as the diffusivity used match the normalized strains 5.0 cm from the sample

bottom of sample Ca2 in Figure 2.11a, D=2.5x10-7 m2/s.

The fully coupled model also predicts that for a positive pore pressure step, the axial strains

will initially decrease during early time, and then increase in proportion to the pore pressure

predicted by diffusion alone while the circumferential strains do not decrease. This model result is

seen in the experimental data from both samples, Figures 2.10b and 2.11b. The experimental

normalized axial strains initially decrease and then increase proportionally with the pore pressures

calculated by diffusion alone. The reverse axial strain, although present in the Ca3 data, Figure

2.10b, does not match the fully coupled poroelastic axial strains as well as the data from Ca1,

Figure 2.11b. The greater difference between the poroelastic model axial strains and the

experimental axial strains for Ca3 is likely due to the same discrepancy between the model and

experimental boundary conditions discussed above for the case of pore pressure. The strains in

Ca1 were measured farther from the sample ends and so the differences between the model and

experimental boundary conditions might play less of a role. Figure 2.11b shows the

circumferential strain does not exhibit the early time response reverse fluctuation of the axial strain,

another result predicted by the poroelastic model.

Strain gages applied along the sample length allow tracking of the pore pressure pulse along

the entire length of the sample, not just at the sample ends, as the pore pressure pulse diffuses

through the sample. This provides additional data for estimation of the hydraulic flow parameters.

101

Normalized Axial Strains Versus TimeShort Sample Ca3

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 200 400 600 800 1000

Time (seconds)

No

rmal

ized

A

xial

S

trai

n

Axial strain

Diffusion Alone Pressure

Fully Coupled Strains

Figure 2.10a. Normalized axial strain for the short sample, Ca3, as a function of time andcompared to normalized axial strains from the poroelastic model and normalized pore pressurefrom the diffusive flow equation. The measuring point is 1.3 cm from the bottom of sample.Ca3.

102

Early time axial strainsShort Sample Ca3

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 20 40 60 80 100

Time (seconds)

No

rmal

ized

A

xial

S

trai

n

Axial strain

Diffusion Alone Pressure

Fully Coupled Strains

Figure 2.10b. The early time portion of the normalized axial strain for the short sample, Ca3, as afunction of time and compared to normalized axial strains from the poroelastic model andnormalized pore pressure from the diffusive flow equation.

103

Normalized Axial and Circumferential Strains versus time - (Long sample Ca1)

-0.2

0

0.2

0.4

0.6

0.8

1

0 10000 20000 30000 40000

time (seconds)

No

rma

lize

d

str

ain

Axial Strain

Circumferential Strain

Fully Coupled Solution

Diffusion alone

Figure 2.11a. Normalized axial and circumferential strain for the long sample, Ca1, as a function oftime and compared to normalized axial and circumferential strains from the poroelastic model andnormalized pore pressure from the diffusive flow equation. The measuring point is 5.0 cm from thebottom of sample Ca1.

104

Early time axial and circumferential strains(Long Sample Ca1)

-0.05

0

0.05

0.1

0.15

0.2

0 500 1000 1500 2000

time (seconds)

No

rma

lize

d

str

ain

Axial Strain

Circumferential Strain

Fully Coupled Solution

Diffusion alone

Figure 2.11b. The early time portion of the normalized axial and circumferential strains for thelong sample, Ca1, as a function of time and compared to normalized axial strains from theporoelastic model and normalized pore pressure from the diffusive flow equation.

2.5 Discussion

Because the fully-coupled poroelastic solution quickly converges to the one-dimensional

diffusive flow solution, it is not necessary to correct the traditional analysis of a transient pore

pressure test as long as the early-time response is not used in the analysis. In addition, because a

reservoir of a known volume is usually placed at the sample bottom instead of the no-flow

boundary, the early time pore pressure response will be muted and is probably not present above

the level of the data noise. There is no evidence in the literature of any observations of the early

time response and the response was not observed by Hart and Wang (1992) when a reservoir was

105

placed at the sample bottom, so no correction is necessary for any time when a reservoir is present

at the sample bottom.

In a hydrostatic pressure vessel, the storage coefficient found in the experiment is Sσ, the

specific storage under constant stress or the unconstrained specific storage, not the usual one-

dimensional specific storage defined in hydrogeology, Ss (Fetter, 1994; Green and Wang, 1990),

because the fluid diffusion equation (5) can be rewritten as

SσB

3

∂σkk

∂t+ ∂p

∂t

= K∇ 2 p (6)

When the one-dimensional diffusive flow solution and the fully-coupled solution converge after the

early time response, the coupling term, (B/3)( σkk/ t) is negligible and the diffusion equation with

Sσ as the storage term remains. The three-dimensional unconstrained specific storage,Sσ, is related

to the one-dimensional specific storage, Ss, by (Wang, 1993)

Ss = Sσ 1 − 4ηB

3

(7)

where η = 1 − 2ν2 1 − ν( )

1 − K

Ks

and ν is Poisson's ratio. These values, including Poisson's ratio,

can be calculated from the parameters given in Table 2.1. The values for the two specific storages

are Sσ equal to 1.8x10-6 m-1 and Ss equal to 1.2x10-6 m-1. Sσ is 1.5 times greater than Ss.

2.6 Conclusion

A qualitative explanation of the fully-coupled poroelastic response has been presented

which gives a framework to understand the poroelastic response of most common transient pore

pressure tests. Because the fully-coupled poroelastic solution varies little from the one-dimensional

106

diffusive flow solution except at early times, negligible error is introduced when using the one-

dimensional analysis to estimate hydraulic conductivities and storage.

Difficulties in applying the experimental stress boundary conditions at the top and bottom

of the sample probably account for the differences between the experimentally observed and

predicted early time pore pressure and axial strain reversals. If the model stress conditions could be

rigorously tested and applied in an experimental setup then it would be possible to determine shear

moduli and Poisson's ratio using only pore pressure and a hydrostatic vessel. There would be no

complications caused by bending moments due to misalignment of the uniaxial loading apparatus.

It is important to understand the role of the stress boundary conditions when analyzing the

data for a specific storage. If the sample is placed in a constant stress condition then the specific

storage at constant stress, Sσ, is measured, not the one-dimensional specific storage, Ss. After the

early time response, strain is linearly related to the pore pressure, so it is possible to apply strain

gages along the sample length and use those gages as surrogate pressure transducers. Pore

pressure can then be measured along the sample length instead of only at the sample ends, resulting

in better constraints on the estimates of the hydraulic flow parameters.

References

Adachi, J. I. and Detournay, E. 1997, A poroelastic solution of the oscillating pore pressure methodto measure permeabilities of tight rocks. Int. J. Rock Mech and Min. Sci. 34:3-4, Paper No.062.

Biot, M. A. 1941, General theory of three-dimensional consolidation. J. Appl. Phys. 12, pp. 155-164.

Brace W. F., Walsh, J. B. and Frangos, W. T. 1968, Permeability of granite under high pressure.J. Geophys. Res. 73, pp. 2225-2236.

Carslaw, H. S. and Jaeger, J. C., 1947, Conduction of heat in solids. Oxford [Eng.] ClarendonPress.

Cryer, C.W. 1963, A comparison of the three-dimensional consolidation theories of Biot andTerzaghi. Quart. J. Mech. Appl. Math. 16. pp. 401-412.

107

Esaki, T., Zhang, M., Takeshita, A., and Mitani, Y. 1996, Rigorous Theoretical Analysis of a FlowPump Permeability Test. Geotechnical Testing Journal. GTJODI., 19, pp. 241-246.

Fetter, C. W. 1994, Applied Hydrogeology. Macmillan College Publishing, New York.

Green, D. H. and Wang, H. F. 1990, Specific Storage as a Poroelastic Coefficient. WaterResources Research, 26, pp. 1631-1637.

Hart, D. J. and Wang, H. F. 1995, Laboratory measurements of a complete set of poroelasticmoduli for Berea sandstone and Indiana limestone, J. Geophys. Res., 100, pp. 17,741-17,751.

Hart, D. J. and Wang, H. F. 1992, Core Permeabilities of Diagenetically-Banded St. PeterSandstone from the Michigan Basin. Eos, Transactions, American Geophysical Union., 73, p.514.

Hsieh, P. A. 1994, A Finite Element Model to Simulate Axisymmetric/Plane-Strain SolidDeformation and Fluid Flow in a Linearly Elastic Porous Medium. U. S. Geological Survey,Menlo Park, California.

Mandel, J. 1953, Consolidation des sols ('etude math'ematique). Geotechnique. 3, pp 287-299.

Morin, R. H. and Olsen, H. W. 1987, Theoretical Analysis of the Transient Pressure ResponseFrom a Constant Flow Rate Hydraulic Conductivity Test. Water Resources Research, 23, pp.1461-1470.

Neuzil, C. E. 1986, Groundwater Flow in Low-Permeability Environments. Water ResourcesResearch, 22, pp. 1163-1195.

Olsen, H. W., Morin, R. H., and Nichols, R. W., 1988, Flow Pump Applications in TriaxialTesting, in Advanced Triaxial Testing of Soil and Rock, ASTM STP 977, R.T. Donaghe, R. C.Chaney, and M. L. Silver, Eds., pp. 68-81.

Rice, J. R. and Cleary, M. R. 1976, Some basic stress-diffusion solutions for fluid-saturated elasticporous media with compressible constituents, Rev. Geophys., 14, 227-241.

Walder, J. and Nur, A. 1986, Permeability Measurement by the Pulse-decay Method: Effects ofPoroelastic Phenomena and Non-linear Pore Pressure Diffusion. Int. J. Rock Mech. Min. Sci.and Geomech. Abstr. 23, pp. 225-232.

Wang, H. F. 1993, Quasi-static poroelastic parameters in rock and their geophysical applications.Pure and Applied Geophysics.141, pp. 269-286.

Wang, H. F. and Hart, D. J. 1993, Experimental Error for Permeability and Specific Storage fromPulse Decay Measurements. Int. J. Rock Mech. Min. Sci. and Geomech. Abstr., 30, pp. 1173-1176.

108

Chapter 3

A Single Test Method for Determination of Poroelastic Constants and Flow

Parameters in Rocks with Low Hydraulic Conductivities

All things come round to him who will but wait.— Longfellow

Abstract: Measurement of poroelastic constants and hydraulic flow parameters of tight rocks is

important for modeling of many geologic processes, from post-earthquake deformation models to

elastic responses of aquifer and aquitard systems due to fluid withdrawal. Few measurements of

both poroelastic constants and hydraulic flow parameters conducted on the same rock are available

in the literature. This chapter presents a method for determining three independent poroelastic

constants and hydraulic flow parameters from a single test. The method makes use of the very

property that makes measurement of "tight" rocks difficult, their long time constant. When an

external stress is imposed on the sample, it will initially react in an undrained manner because it

does not have time to drain even if an outlet is provided. It will then "relax" to a drained state,

assuming the pressure at the outlet is held constant from the start of the test. If the sample is

instrumented with strain gages, the short term or undrained elastic response can be measured; then

as the sample drains, the hydraulic diffusivity and, depending on the apparatus setup, the hydraulic

conductivity and storage can be determined; and finally, when the sample has reached a steady state

condition, the drained elastic constants can be determined. The strain gages, in addition to giving

strains from the short term (undrained) and long term (drained) responses to the external stress, can

also be used to give information on the transient response. They respond to the change in effective

stress as the pore pressure diffuses through the sample and so can add information to that provided

by the pressure transducers in a traditional pulse decay permeability test.

109

3.1 Introduction

The transient pulse decay test provides estimates of flow parameters such as hydraulic

conductivity and specific storage (Brace, 1968; Zoback and Byerlee, 1975; Hsieh, 1981). Quasi-

static measurements of strain, confining stresses, and pore pressure result in estimates of

poroelastic constants (Zimmerman et al., 1986; Hart and Wang, 1995; Aoki et al., 1995; Tokunaga

et al., 1998; Hart and Wang, 1999). Because both measurements require pressure vessels and

variation of pore pressures and confining stresses, it is advantageous to measure the flow

parameters and the poroelastic constants simultaneously. This reduces variation due to loading

history and multiple samples and should result in more consistent results. Also, the poroelastic

constants of rocks with very low permeabilities cannot be measured using the technique described

in Chapter 1 because it is not possible to control the pore pressure throughout all of the sample in a

timely manner. The time needed for the pore pressure to equilibrate in a laboratory scale sample is

too long for a low permeability sample. This test method is an attempt to overcome this problem.

Measurements of strain while conducting transient pulse decay tests have been made

(Trimmer, 1982; Senseny et al., 1983; Walder and Nur, 1986) but no estimates of both the flow

parameters and the quasistatic poroelastic constants were found. In this experiment a method for

determining estimates of the flow parameters, hydraulic conductivity and specific storage, and the

poroelastic constants, the drained and undrained bulk compressibilities and Skempton's B

coefficient, is presented for Barre granite at a low effective stress of around 8 MPa.

3.2 Experiment

Experimental Procedure

A Barre granite sample was cored and ground into a right cylinder with a length of 8.40 cm

and a diameter of 5.08 cm with the ends ground within 0.003 cm of parallel. The porosity of the

core was measured to be 0.0085 using the dry and immersed weights (Green and Wang, 1986).

Three pairs of metal foil strain gages were applied to the sample with 120 degrees between the

110

gages at 4.2 cm from the sample bottom. Each gage pair had one gage aligned parallel and one

perpendicular to the core axis for measurement of axial and circumferential strains. RTV silicone

was applied to the cylindrical core's side and two Tygon sleeves were fitted and clamped over the

ends of the core. Figure 1.3 shows the sample assembly.

The sample was placed in a vacuum for 24 hours and then submerged in deionized water

while still under the vacuum. The vacuum was released and the sample was allowed to saturate for

another 24 hours. The zero-volume pressure transducer endplug, described in Chapter 1 was

inserted into the Tygon sleeve and clamped in place at the sample bottom. An endplug with slots in

the surface in contact with the sample to distribute the applied pore pressure was inserted and

clamped in place at the sample top. The sample was placed in the pressure vessel, the strain gage

leads and bottom endplug pressure transducer leads were connected and finally the external pore

pressure line was connected to the endplug at the sample top. Following pressurization of the

pressure vessel, the sample was allowed to drain for 24 hours and so reach equilibrium.

Figure 3.1 shows the pore pressure, confining stress, and external pore pressure system

pressure versus time for a typical test. The test consists of three steps. In the first step, the

confining stress is quickly increased, giving an undrained response, and the pore pressure and

strains are measured during the increase, giving Skempton's B coefficient and the undrained bulk

compressibility. In the second step, fluid slowly drains from the sample into the external pore fluid

pressure system and the pressures and strains are recorded as a function of time. The pressure and

strain versus time curves can then be inverted to find the hydraulic conductivity and storage of the

sample. In the last step, the external pore pressure is adjusted and allowed to equilibrate until the

pore pressure throughout the sample is equal to the pore pressure in the sample before the

confining stress step was imposed. The sample has now reached a drained state with respect to the

initial stress state and the change in volumetric strain over the change in confining stress gives the

drained bulk compressibility. Several adjustments of the external pore fluid system pressure were

usually adequate to nearly match the initial pore pressure at the sample bottom.

111

Confining Stress

Pressure at Sample Bottom

Pressure in External Pore Pressure System

Stre

ss a

nd P

ress

ure

Time (seconds)0 10 40001000 2000 3000

Step 1Short Time

Undrained Response

Step 2

Transient Pore Pressure Test

Step 3Long Time

Drained Response

Figure 3.1. Pressure and stress history for an idealized run. The three stresses and pore pressuresrecorded during a test are labeled on the graph. The strains are not shown here.

The first step was the undrained portion of the test. In that step, the confining stress was

increased at a rate of ~1MPa/second while the pore pressure at the sample outlet was not varied.

Because the hydraulic diffusivity of the sample was so low, negligible amounts of fluid drained

from the sample top during the nine or ten seconds that the confining stress was increased so the

sample was in an undrained state. The pore pressure at the sample bottom and the strains midway

down the sample were recorded during this confining pressure step. Figures 3.2a and 3.2b show

the pore pressure and strain response to the confining pressure step.

The slopes of the lines correspond to a Skempton's B coefficient of 0.84 and the negative of

the bulk undrained compressibility equal to 0.021 1/GPa. The possibility that the undrained

response might be altered by an early time Mandel-Cryer type effect caused by drainage at the

112

sample outlet was checked using Biot2, the modeling software used in Chapter 2. The effect was

not apparent at the location of the strain gages nor at the sample bottom in the first 10 seconds of

measurements. There is an early time reverse pore pressure response seen in the pore pressure data

in Figure 3.3 with a response about 5% greater than the expected pore pressure rise due only to the

confining pressure increase but this effect is negligible in the first 10 seconds of the test and does

not reach a maximum until over 100 seconds into the test and so it is neglected.

Pore Pressure versus Confining Stress

1011

121314

1516

1718

12 14 16 18 20 22

Confining stress (MPa)

Po

re

pre

ss

ure

(M

Pa

)

Figure 3.2a. Pore pressure as a function of confining stress. The slope of the line gives aSkempton's B coefficient of 0.84.

113

Volumetric Strain versus Confining Stress

-200

-150

-100

-50

0

50

12 14 16 18 20 22

Confining Stress (MPa)

Vo

lum

etr

ic

Str

ain

(∆

V/V

)x

10

^6

Figure 3.2b. Volumetric strain as a function of confining stress. The negative of the slope of theline gives the bulk undrained compressibility.

The first step gave the undrained response of the rock to an applied confining stress and

resulted in a larger fluid pressure in the sample than in the external pore fluid system. In the

second step, shown in Figure 3.1, the confining stress was held constant while the pore fluid was

allowed to drain from the sample to reach equilibrium with the external pore fluid system. The

volume of the external pore pressure system was not varied for the initial pore pressure decay so

that it acted as a constant volume reservoir. The storage of the external pore pressure system was

found experimentally so that the amount of fluid draining from the sample could be found.

This set of boundary conditions, an applied pore pressure step at the sample end, constant

external stress, a homogenous pore pressure throughout the sample, and a constant volume

reservoir at the sample end, was now the same as that used for a pulse decay transient pore pressure

test. The fluid pressure of the external pore pressure system, the pore pressure at the sample

bottom, and the strains midway up from the sample bottom were all recorded with time as fluid

114

flowed from the sample into the external pore pressure system. Equation 1 below is taken from

Hsieh et al. (1981). It gives the pore pressure as a function of time and location along the sample

for the initial and boundary conditions stated above.

p x, t( ) = ∆P1

1 + β+ 2

exp −αφm2( )cos φmζ( )

1 + β( )cos φm( ) − φm sin φm( )m=1

∞

∑

(1)

where

p x, t( ) is the pore pressure as a function of distance from the sample bottom and time,

∆P is the initial pressure step,

α = Kt

L2Sσ is a dimensionless time,

β = Sσ AL

Sres

is the ratio of sample storage to reservoir storage,

ζ = x

L is a dimensionless length from the sample bottom, and

φm is a root of the equation tan φm( ) = φm

−β.

The physical parameters used to compute the above dimensionless variables are the hydraulic

conductivity, K , the three dimensional unconstrained specific storage, Sσ , the compressive storage

of the external pore fluid pressure system, Sres , the sample area, A, the sample length, L, the

distance from the sample bottom, x , and the time elapsed after the pressure step, t .

The pore pressure and external reservoir pressure versus time data were reduced to

normalized data by first subtracting the initial pore pressures before the pressure step, then dividing

those values by the total pore pressure step. The sense of the pressure step in equation 1 is usually

positive; the pressure in the reservoir increases over the sample pressure. However, in this test the

115

sample pressure was increased and so the sense of the pressure step is reversed, as shown in Figure

3.2. The strain-time data were normalized in a similar manner.

In the limit as time goes to infinity, equation 1 reduces to

p x, t( )

∆P= 1

1 + β. (2)

Because the final ratio of pressure change, the normalized pore pressure, p x, t( )

∆P, is very well

constrained as shown in Figure 3.3, the ratio of the sample storage to the reservoir storage, β , is

also well constrained and can be found. The ratio of sample storage to reservoir storage, β , for this

test was found to be 0.90. The storage of the reservoir, 1.63 x 10-10 m2, was found experimentally

by measuring the pressure change as the volume of fluid in the reservoir was varied. The specific

storage of the sample was then calculated to be 8.6 x 10-7 m-1 using the definition of β given

above.

The three pressure and strain versus time curves were inverted together using the sample

storage calculated above to give a best-fit hydraulic conductivity of 1.3 x 10-12 m/s. The

experimental data and the best-fit curves are shown in Figure 3.3. It should be noted that any one

of the three curves could have been used to give estimates of the flow parameters. If only the pore

pressures at the sample bottom were used, the best-fit hydraulic conductivity was 1.4 x 10-12 m/s

and if only the strains midway along the sample were used, the best-fit value was 1.2 x 10-12 m/s.

The best-fit value using only the external pore pressure system pressure-time curve gives a value of

1.0 x 10-12 m/s.

116

Transient Responses after Confining Stress Step

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0 1000 2000 3000 4000

time (seconds)

no

rma

lize

d

po

re

pre

ss

ure Experimental

Calculated Best-Fit

reservoir

strain gage

sample bottom

Figure 3.3 Normalized pore pressures at the sample bottom and reservoir and normalized strainsmidway along the sample length. Note the early time reverse pore pressure response at the samplebottom.

After the sample pore pressure and the reservoir fluid pressure had equilibrated, the

reservoir pressure was adjusted downward and the sample again was allowed to drain to

equilibrium. This adjustment was repeated several times until the sample and reservoir pressures

nearly reached the value of pore pressure before the first confining pressure step. The sample had

now nearly drained and the change in volumetric strain (729.4 x 10-6 ∆V/V) divided by the

confining pressure step (7.23 MPa) gave an initial estimate of the drained bulk compressibility

equal to 0.101 1/GPa.

Because the time needed to reach equilibrium was so great, over an hour, a check was done

to determine the amount of drift that might have occurred in the strain gages. In a separate test, the

117

confining pressure was increased to 14 MPa and the pore pressure was set to 10 MPa and the

sample was allowed to reach equilibrium. After the pore pressure had equilibrated, after about 5000

seconds, the confining stress and pore pressure were held nearly constant and strains were recorded

for an additional 15 hours. The maximum variation of the volumetric strain was found to be 5

microstrain after 10 hours. This variation is probably due to temperature fluctuations. This

variation of the volumetric strain (∆V/V) of 5 x 10-6 would affect the estimate of the drained bulk

compressibility by less than 1% and so is neglected.

Another source of error might be due to not reaching the pore pressure that was present

before the confining pressure step, i.e. the sample was not yet in a drained state. This can be

corrected by calculating the strain that would occur from the change in pore pressure to reach the

drained state. The change in volumetric strain divided by the change in pore pressure after the

initial confining stress step gives another bulk poroelastic constant, Cbp (Zimmerman, 1986; Aoki,

1996) equal to 1/H, a constant introduced by Biot (1941). The value of Cbp is

(567.8 x 10-6)/(6.09 MPa)=0.093 1/GPa. The difference between the final nearly steady state pore

pressure and the initial pore pressure before the confining pressure step is 0.3 MPa. When this

difference is multiplied by Cbp to extrapolate the strain when the initial and final pore pressures

were equal and so predict the final drained strain, the result is an additional strain (∆V/V) equal to

28.0 x 10-6. Adding this additional strain changes the value of the drained bulk compressibility to

0.105 1/GPa. Table 3.1 shows the measured values for this granite at a confining stress of 21 MPa

and a pore pressure of 13 MPa.

118

Table 3.1 Measured bulk poroelastic constants and flow parameters for Barre granite

Parameter

Drained Compressibility (1/GPa) 0.105

Biot's 1/H (1/GPa) 0.093

Undrained Compressibility (1/GPa) 0.021

Skempton's B Coefficient (MPa/MPa) 0.84

Hydraulic Conductivity (m/s) 1.3 x 10-12

Specific Storage (m-1) 8.6 x 10-7

3.3 Discussion

Grain Compressibility

It is possible to calculate a value for the grain modulus, Cs , using the values in Table 3.1.

Cs may be calculated using

Cs = Cd −1 H (3)

with the result that Cs =0.012 1/GPa. Cs may also be calculated using

Cs = Cd −Cd − Cu( )

B(4)

with the result that Cs =0.005 1/GPa. Assuming an error of +/- 2% in the measurements gives an

error range in the calculated values of +/- 0.004 1/GPa for the grain compressibility calculated by

equation 3 and an error range of +/- 0.002 1/GPa for the grain compressibility calculated by

equation 4. Table 3.2 summarizes these results and compares these values with grain

compressibilities of quartz and a plagioclase and a value of the bulk compressibility of Barre granite

at pressures greater than 200 MPa calculated from velocity measurements (Sano et al., 1992).

Because at this pressure nearly all of the cracks are closed, as shown by the assymptotic behavior of

119

the pressure velocity curves, this compressibility should approach the bulk averaged grain

compressibility.

Table 3.2 Comparisons of the two calculated grain compressibilities to quartz, a plagioclase, and abulk compressibility calculated from velocity measurements.

Grain Compressibility Values (1/GPa)

Equation 3 0.012 +/- 0.004

Equation 4 0.005 +/- 0.002

Quartz (Simmons and Wang, 1971) 0.028

Plagioclase (Simmons and Wang, 1971) 0.013

Bulk Average from Velocity Measurements 0.015

The mineral composition of Barre granite is 25% quartz, 20 % potash feldspar, 35%

plagioclase, 9% biotite, 9% muscovite, and 2% accessories (Chayes, 1952). The bulk grain

compressibility for Barre granite would be expected to lie somewhere between the grain

compressibilities of quartz and plagioclase, the most and least compliant minerals, respectively, in

Barre granite. The grain compressibility calculated from equation 3 is a better estimate than the one

calculated by equation 4 and is similar to the one found from the velocity measurements at a

confining stress of 200 MPa (Sano et al., 1992). The reason for the discrepancy between the

estimates may be that the rock is in a nonlinear elastic regime for these measurements and the

average effective stress for the undrained measurements is less than the average effective stress for

the drained measurement. Because equation 4 includes undrained measurements while equation 3

does not, it may be that the error originates in the undrained measurements.

120

Specific Storage

The three dimensional unconstrained specific storage of the sample was estimated from the

transient portion of the test. It may also be calculated independently using the poroelastic constants

in Table 3.2 and equation 5 (Hart and Wang, 1995)

Sσ = ρgCd − Cs

B

. (5)

where the grain compressibility used was Cs =0.015 1/GPa. The resulting estimate for the specific

storage is Sσ =1.05x10-6 m-1. This result is in good agreement, to within 20%, of the value of 8.6 x

10-7 m-1 calculated using the ratio β = Sσ Al

Sres

and the experimentally determined reservoir storage.

Choosing different values of the grain modulus will alter the result somewhat, but within reasonable

bounds, using the values for the grain modulus of quartz equal to 0.028 1/GPa, and feldspar equal

to 0.013 1/GPa, the result is still within 20%. This gives confirmation that the three dimensional

unconstrained specific storage may be calculated from the associated poroelastic constants. It

should be noted that if a reasonable value for Poisson's ratio for the granite is assumed, ν = 0.25,

the value of the three-dimensional unconstrained specific storage, Sσ =8.6x10-7 m-1, is 1.5 times

larger than the value calculated for the one-dimensional specific storage, Ss =5.8x10-7 m-1, using

equation 7 in Chapter 2.

Hydraulic conductivity and effective stress

In addition to providing additional information for estimation of the hydraulic conductivity

and the specific storage, strain gages located along the sample length may also provide information

on the nonlinearity of the hydraulic conductivity along the sample length (Walder and Nur, 1986;

Yilmaz et al., 1994). In this experiment, the hydraulic conductivity decreased as the measurement

point moved from the sample bottom to the sample top as described in the discussion of Figure 3.3.

121

This decrease corresponds to an increase of the average effective stress over time as a function of

location on the sample. The greater effective stress would have reduced pore throat diameters and

so reduced the hydraulic conductivity (Zoback and Byerlee, 1975). The sample bottom would have

experienced the least effective stress because it was farthest from the drainage at the sample top

while the sample top would have experienced the greatest effective stress. Additional strain gages

would resolve and show this effect more strongly.

3.4 Conclusion

A method for determining the hydraulic conductivity and specific storage in addition to

several (three independent) bulk poroelastic constants during one test on a single sample was

successfully demonstrated. The three dimensional specific storage found from the transient pore

pressure test was found to be within 20% difference of the three dimensional unconstrained specific

storage calculated from the measured poroelastic constants. Strains measured during the transient

pore pressure test provided another measurement point for estimation of the flow parameters and

possibly showed a dependence of the hydraulic conductivity on the effective stress that would not

be apparent otherwise.

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Biot, M. A. 1941, General theory of three-dimensional consolidation. J. Appl. Phys. 12, pp. 155-164.

Brace W. F., Walsh, J. B. and Frangos, W. T. 1968, Permeability of granite under high pressure.J. Geophys. Res. 73, pp. 2225-2236.

Green, D.H. and Wang, H.F., 1986, Fluid pressure response to undrained compression in saturatedsedimentary rock. Geophysics, 51, pp. 948-956.

Hart, D.J. and Wang, H.F. 1995, Laboratory measurements of a complete set of poroelastic modulifor Berea sandstone and Indiana limestone, J. Geophys. Res. 100, pp. 17,741-17,751.

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Hart, D.J. and Wang, H.F. 1999, Pore pressure and confining stress dependence of poroelasticlinear compressibilities and Skempton's B coefficient for Berea sandstone. In Amadei et al.(eds), Proceedings of the 37th U.S. Rock Mechanics Symposium, 365-371. Rotterdam:Balkema.

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Walder, J. and Nur, A. 1986, Permeability Measurement by the Pulse-decay Method: Effects ofPoroelastic Phenomena and Non-linear Pore Pressure Diffusion. Int. J. Rock Mech. Min. Sci.and Geomech. Abstr. 23, pp. 225-232.

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