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 ii Table of Contents iii List of Figures v List of Tables vii 1 An Experimental Study of the Stress and Pore Pressure Dependence 1 of Poroelastic Constants for Berea Sandstone and Indiana Limestone. 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 Pressure 81 Test 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 Constants 108 and Flow Parameters in Rocks with Low Hydraulic Conductivities 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
= σ + 1 eij sijkl kl 3 CBij p (1)
ζ = + 1 σ Cp()3 Bij ij (2)
σ where eij is the strain tensor, sijkl is the drained compliance tensor, kl 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 ≡−σ =−σ =−σ in confining stress, Pc 11 22 33 , 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.
= σ eij sijkl kl . (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 = 1 σ 3 CBij 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.
=−1 σ p ()3 Bij 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.
= − 1 σ eij ()sijkl 9 CBij Bkl kl (6)
An undrained compliance tensor can then be defined as
u = − 1 sijkl sijkl 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. σ ∆ =∆ =− kk Setting p Pc 3 in equation 1, the result is
= − 1 σ eij ()sijkk 3 CBij kk (8)
The unjacketed compliance tensor is then equal to
s = − 1 sijkk sijkk 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): tensor 11 22 33 23, 32 31,13 12, 21 notation matrix 123456 notation
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. tensor 11 22 33 23, 32 31, 13 12, 21 notation matrix xyzxyz notation
The strain tensor is written in vector form as:
= γ γ γ T e []ex ey ez x y z (10)
and the stress tensor is written in vector form as: 8 σ = σ σ σ τ τ τ T []x y z x y z (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:
σ ex s11 s12 s13 s14 s15 s16 x B1 σ ey s21 s22 s23 s24 s25 s26 y B2 e s s s s s s σ B z = 31 32 33 34 35 36 z + 1 3 γ τ C p (12) x s41 s42 s43 s44 s45 s46 x 3 B4 γ s s s s s s τ B y 51 52 53 54 55 56 y 5 γ τ z s61 s62 s63 s64 s65 s66 z B6
σ x σ y σ z ζ = Cp+ 1 []B B B B B B (13) 3 1 2 3 4 5 6 τ x τ y τ z
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:
σ ex s11 s12 s13 00 0 x B1 s s s 00 0 σ ey 12 11 13 y B1 σ ez s13 s13 s33 00 0 z 1 B3 = + (14) γ τ C p x 000s44 00 x 3 0 γ 0000s 0 τ 0 y 44 y γ ()− τ z 000002s11 s12 z 0
σ x σ y σ z ζ = Cp+ 1 []B B B 000 (15) 3 1 1 3 τ x τ y τ z
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 core with respect to bedding and applied stresses and the pore fluid boundary condition. Poroelastic Constants Core Axis Applied Stress Pore Fluid Orientation (w.r.t. core Boundary w.r.t.bedding axis) Condition + + perpendicular hydrostatic drained s11 s12 s13 + perpendicular hydrostatic drained 2s12 s33 u + u + u perpendicular hydrostatic undrained s11 s12 s13 u + u perpendicular hydrostatic undrained 2s13 s33 1 ()+ perpendicular hydrostatic undrained 3 2B1 B3 11 perpendicular uniaxial drained s13 perpendicular uniaxial drained s33 u perpendicular uniaxial undrained s13 u perpendicular uniaxial undrained s33 1 perpendicular uniaxial undrained 3 B3 s + s + s perpendicular hydrostatic unjacketed ()s11 s12 s13 s + s perpendicular hydrostatic unjacketed ()2s13 s33 + + parallel hydrostatic drained s11 s12 s13 + parallel hydrostatic drained 2s12 s33 u + u + u parallel hydrostatic undrained s11 s12 s13 u + u parallel hydrostatic undrained 2s13 s33 1 ()+ parallel hydrostatic undrained 3 2B1 B3 parallel uniaxial drained s11 parallel uniaxial drained s12 parallel uniaxial drained s13 u parallel uniaxial undrained s11 u parallel uniaxial undrained s12 u parallel uniaxial undrained s13 1 parallel uniaxial undrained 3 B1 + + 45 degrees hydrostatic drained s11 s12 s13 1 ()+ + + 1 ()+ 45 degrees hydrostatic drained 2 s11 s12 s13 2 2s13 s33 u + u + u 45 degrees hydrostatic undrained s11 s12 s13 1 u + u + u + 1 u + u 45 degrees hydrostatic undrained 2 ()s12 s13 s11 2 ()2s13 s33 1 ()+ 45 degrees hydrostatic undrained 3 2B1 B3 1 ()+ 45 degrees uniaxial drained 2 s12 s13 1 ()+ + − 45 degrees uniaxial drained 4 s11 s33 2s13 s44 1 ()+ + + 45 degrees uniaxial drained 4 s11 s33 2s13 s44 1 u + u 45 degrees uniaxial undrained 2 ()s12 s13 1 u + u + u − 45 degrees uniaxial undrained 4 ()s11 s33 2s13 s44 1 u + u + u + 45 degrees uniaxial undrained 4 ()s11 s33 2s13 s44 1 1 ()+ 45 degrees uniaxial undrained 3 2 B1 B3 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 γ limestone ( A = mP ), 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 grains by 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 carbonate cement by light grey, and the grains by darker grey. These grains are all cemented with few long 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
Profile View Core Axis Core Axis Core Axis at Perpendicular Parallel 45 degrees to Bedding to Bedding to Bedding
Strain gage pairs
Top View Perpendicular Parallel 45 degrees
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 Axial Strain Gage
Tygon Circumferential Sleeves Strain Gage Silicone Jacket
Bottom Endcap with Pressure Transducer
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
Outlet Line
Top Piece
O-ring seal
Retaining Spring
Bottom Piece
Plunger moves up O-ring seal and down to open and close the Inlet to valve Sample
Sample
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 σ apply the uniaxial load, 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 Top View
Inner Ring Load Surface Strain Gages 1
Spoke number 4 2
Spokes with Bending Moment 3
Outer Ring Load Surface Profile View Upper surface Spoke strain gage
4 2
Lower surface Inner Ring Load Surface strain gage
Figure 1.5a. Diagram of the internal load cell. The diagram is approximately to scale. 21 Wiring Schematic V+ Gage location by spoke number and surface
Lower 1 Upper 2
Lower 3 Amplifier Upper 4
+ out - out Upper 1 Lower 2
Upper 3 Lower 4
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 p = 1 ()+ matrix, 3 2B1 B3 as can be seen from equation 15. This slope is found by a Pc 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, ex =− u + u + u ()s11 s12 s13 , and the tangential slope of the confining stress and the strains Pc perpendicular to bedding (the axial strains) give another linear combination of elements of
ez =− u + u the compliance matrix, ()2s13 s33 , as can be seen from equation 14. The Pc 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 18 (Pc=31 MPa, Pp=17 MPa)
17.5
17
16.5
16 Pore Pressure (MPa)
15.5
15 29.5 30 30.5 31 31.5 32 32.5 Confining Stress (MPa) Figure 1.6a. Pore pressure as a function of confining stress under undrained pore fluid and hydrostatic stress boundary conditions. 26
Run 11 Undrained (Pc=31MPa, Pp=17 MPa) Strains Ax1 30 Strains Ci1 Strains Ax2 20 Strains Ci2 Strains Ax3 10 Strains Ci3 l/l x 10^6) ∆ 0 29.5 30 30.5 31 31.5 32 32.5 -10
-20 Microstrain ( -30 Confining Stress (MPa) Figure 1.6b. Strain as a function of confining stress for an undrained pore fluid and hydrostatic stress boundary condition. 27
Run 31 Uniaxial Drained (Pc=31 MPa, Pp=17 MPa) 40
20
0 0 0.2 0.4 0.6 0.8 -20 Ax1 -40 Microstrain l/l x 10 ^6) Ci1 ∆ ( -60 Ax2 Ci2 -80 Ax3 -100 Ci3 Uniaxial Stress above hydrostatic (MPa) Figure 1.6c. Strain as a function of uniaxial stress for a drained pore fluid and triaxial stress 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 of the compliances are 1/GPa and Skempton's B coefficient are Pa/Pa. Confining Stress (MPa) 41 41 31 20 Pore 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 -0.0081 1 ()+ + + 1 ()+ 2 s11 s12 s13 2 2s13 s33 0.0257 0.0314 0.0377 0.0508 1 ()+ 2 s12 s13 -0.0050 -0.0058 -0.0071 -0.0105 1 ()+ + − 4 s11 s33 2s13 s44 -0.0110 -0.0125 -0.0148 -0.0175 1 ()+ + + 4 s11 s33 2s13 s44 0.0757 0.0874 0.0996 0.1305 u + u + u s11 s12 s13 0.0193 0.0200 0.0213 0.0233 u + u 2s13 s33 0.0202 0.0220 0.0245 0.0284 u s13 -0.0068 -0.0077 -0.0094 -0.0123 u s33 0.0372 0.0407 0.0483 0.0625 u s11 0.0319 0.0347 0.0366 0.0411 u s12 -0.0100 -0.0117 -0.0136 -0.0165 1 u + u + u + 1 u + u 2 ()s12 s13 s11 2 ()2s13 s33 0.0196 0.0218 0.0242 0.0282 1 u + u 2 ()s12 s13 -0.0122 -0.0144 -0.0090 -0.0132 1 u + u + u − 4 ()s11 s33 2s13 s44 -0.0072 -0.0089 -0.0102 -0.0149 1 u + u + u + 4 ()s11 s33 2s13 s44 0.0400 0.0454 0.0529 0.0638 1 ()+ 3 2B1 B3 0.3956 0.4562 0.4813 0.5398 1 3 B3 0.1421 0.1580 0.1912 0.2054 1 3 B1 0.1321 0.1531 0.1547 0.1728 1 1 ()+ 3 2 B1 B3 0.1326 0.1709 0.1819 0.2048 s + s + s + s + s ()s11 s12 s13 ()2s13 s33 0.0171 0.0197 0.0215 0.0204 30 Table 1.2 (continued). Measured values of the poroelastic constants for Berea sandstone. The units of the compliances are 1/GPa and Skempton's B coefficients are Pa/Pa. Confining Stress (MPa) 35 26 20 9.7 Pore 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 -0.0121 1 ()+ + + 1 ()+ 2 s11 s12 s13 2 2s13 s33 0.0503 0.0669 0.0741 0.0811 1 ()+ 2 s12 s13 -0.0079 -0.0116 -0.0124 -0.0145 1 ()+ + − 4 s11 s33 2s13 s44 -0.0176 -0.0255 -0.0280 -0.0167 1 ()+ + + 4 s11 s33 2s13 s44 0.1251 0.1628 0.1878 0.1988 u + u + u s11 s12 s13 0.0230 0.0244 0.0244 0.0254 u + u 2s13 s33 0.0282 0.0304 0.0297 0.0294 u s13 -0.0153 -0.0204 -0.0239 -0.0300 u s33 0.0654 0.0788 0.0924 0.1217 u s11 0.0417 0.0785 0.0547 0.0579 u s12 -0.0167 -0.0166 -0.0252 -0.0268 1 u + u + u + 1 u + u 2 ()s12 s13 s11 2 ()2s13 s33 0.0281 0.0285 0.0316 0.0323 1 u + u 2 ()s12 s13 -0.0144 -0.0201 -0.0229 -0.0259 1 u + u + u − 4 ()s11 s33 2s13 s44 -0.0159 -0.0228 -0.0258 -0.0312 1 u + u + u + 4 ()s11 s33 2s13 s44 0.0675 0.0826 0.0914 0.0996 1 ()+ 3 2B1 B3 0.5927 0.6678 0.7163 0.7532 1 3 B3 0.2246 0.2292 0.2547 0.2710 1 3 B1 0.1662 0.1789 0.1991 0.1873 1 1 ()+ 3 2 B1 B3 0.2254 0.2382 0.2636 0.2714 s + s + s + s + s ()s11 s12 s13 ()2s13 s33 0.0219 0.0242 0.0247 0.0283 31 Table 1.3. Measured values of the poroelastic constants for Indiana limestone. The units of the compliances are 1/GPa and Skempton's B coefficient are Pa/Pa. Confining Stress (MPa) 40 40 35 31 Pore 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 u + u + u s11 s12 s13 0.0114 0.0114 0.0117 0.0115 u + u 2s13 s33 0.0108 0.0106 0.0107 0.0108 u s13 -0.0078 -0.0064 -0.0075 -0.0074 u s33 0.0247 0.0239 0.0245 0.0250 u s11 0.0220 0.0229 0.0214 0.0220 u s12 -0.0113 -0.0111 -0.0108 -0.0103 1 u + u + u + 1 u + u 2 ()s12 s13 s11 2 ()2s13 s33 0.0103 0.0103 0.0104 0.0102 1 u + u 2 ()s12 s13 -0.0375 -0.0168 -0.0128 -0.0289 1 u + u + u − 4 ()s11 s33 2s13 s44 -0.0211 -0.0091 -0.0062 -0.0179 1 u + u + u + 4 ()s11 s33 2s13 s44 0.0893 0.0454 0.0345 0.2177 1 ()+ 3 2B1 B3 0.2844 0.2998 0.3390 0.3044 1 3 B3 0.1871 0.2868 0.1739 0.1404 1 3 B1 0.1015 0.1424 0.1429 0.1187 1 1 ()+ 3 2 B1 B3 0.4090 0.1659 0.0917 0.3121 s + s + s + s + s ()s11 s12 s13 ()2s13 s33 0.0087 0.0088 0.0089 0.0088 32 Table 1.3 (continued). Measured values of the poroelastic constants for Indiana limestone. The units of the compliances are 1/GPa and Skempton's B coefficient are Pa/Pa. Confining Stress (MPa) 26 20 20 9.6 Pore 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 u + u + u s11 s12 s13 0.0123 0.0116 0.0120 0.0119 u + u 2s13 s33 0.0110 0.0108 0.0110 0.0104 u s13 -0.0075 -0.0070 -0.0076 -0.0076 u s33 0.0268 0.0258 0.0277 0.0264 u s11 0.0216 0.0225 0.0220 0.0219 u s12 -0.0120 -0.0114 -0.0110 -0.0110 1 ()u + u + u + 1 ()u + u 2 s12 s13 s11 2 2s13 s33 0.0109 0.0108 0.0105 0.0105 1 ()u + u 2 s12 s13 -0.0135 -0.0143 -0.0271 -0.0137 1 ()u + u + u − 4 s11 s33 2s13 s44 -0.0068 -0.0071 -0.0159 -0.0070 1 ()u + u + u + 4 s11 s33 2s13 s44 0.0350 0.0696 0.0933 0.1038 1 ()+ 3 2B1 B3 0.3483 0.3218 0.3386 0.3972 1 3 B3 0.1653 0.2314 0.1790 0.1514 1 3 B1 0.1403 0.1206 0.1444 0.1471 1 1 ()+ 3 2 B1 B3 0.1146 0.1642 0.3179 0.1318 ()s + s + s + ()s + s s11 s12 s13 2s13 s33 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, u 1 u + u b,for Berea sandstone. The poroelastic constants, s12 and 2 ()s12 s13 , 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 1 u + u 1 1 ()+ Indiana limestone data. The poroelastic constants, 2 ()s12 s13 and 3 2 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, 1 u + u 2 ()s12 s13 , 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 1 1 ()+ large error associated with the constant, 3 2 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 1 (b−M(x)) ()= is used to calculate a root mean squared error, = ()()()T 2 , the F x b rms F x F x 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 the compliances are 1/GPa and Skempton's B coefficient are Pa/Pa. Confining Stress (MPa) 41 41 31 20 Pore 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 -0.0087 1 ()+ + + 1 ()+ 2 s11 s12 s13 2 2s13 s33 0.0246 0.0277 0.0313 0.0379 1 ()+ 2 s12 s13 -0.0051 -0.0062 -0.0071 -0.0090 1 ()+ + − 4 s11 s33 2s13 s44 -0.0082 -0.0099 -0.0113 -0.0156 1 ()+ + + 4 s11 s33 2s13 s44 0.0315 0.0364 0.0410 0.0514 u + u + u s11 s12 s13 0.0221 0.0235 0.0250 0.0275 u + u 2s13 s33 0.0226 0.0262 0.0300 0.0369 u s13 -0.0060 -0.0072 -0.0086 -0.0113 u s33 0.0345 0.0405 0.0471 0.0595 u s11 0.0338 0.0377 0.0416 0.0491 1 u + u + u + 1 u + u 2 ()s12 s13 s11 2 ()2s13 s33 0.0224 0.0248 0.0275 0.0322 1 u + u + u − 4 ()s11 s33 2s13 s44 -0.0089 -0.0109 -0.0126 -0.0175 1 u + u + u + 4 ()s11 s33 2s13 s44 0.0371 0.0428 0.0484 0.0605 1 ()+ 3 2B1 B3 0.4039 0.4783 0.5099 0.5752 1 3 B3 0.1402 0.1627 0.1946 0.2152 1 3 B1 0.1318 0.1578 0.1576 0.1800 1 1 ()+ 3 2 B1 B3 0.1360 0.1603 0.1761 0.1976 s + s + s + s + s ()s11 s12 s13 ()2s13 s33 0.0169 0.0193 0.0210 0.0197 37 Table 1.4 (continued). Best-fit values of the poroelastic constants for Berea sandstone. The units of the compliances are 1/GPa and Skempton's B coefficient are Pa/Pa. Confining Stress (MPa) 35 26 20 9.7 Pore 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 -0.0124 1 ()+ + + 1 ()+ 2 s11 s12 s13 2 2s13 s33 0.0389 0.0442 0.0456 0.0488 1 ()+ 2 s12 s13 -0.0088 -0.0123 -0.0131 -0.0147 1 ()+ + − 4 s11 s33 2s13 s44 -0.0162 -0.0237 -0.0267 -0.0202 1 ()+ + + 4 s11 s33 2s13 s44 0.0524 0.0674 0.0718 0.0689 u + u + u s11 s12 s13 0.0275 0.0321 0.0310 0.0331 u + u 2s13 s33 0.0373 0.0403 0.0420 0.0432 u s13 -0.0119 -0.0164 -0.0182 -0.0206 u s33 0.0611 0.0731 0.0784 0.0845 u s11 0.0490 0.0616 0.0628 0.0685 1 ()u + u + u + 1 ()u + u 2 s12 s13 s11 2 2s13 s33 0.0324 0.0362 0.0365 0.0381 1 ()u + u + u − 4 s11 s33 2s13 s44 -0.0185 -0.0265 -0.0299 -0.0240 1 ()u + u + u + 4 s11 s33 2s13 s44 0.0617 0.0775 0.0823 0.0799 1 ()+ 3 2B1 B3 0.6073 0.6623 0.7353 0.7558 1 3 B3 0.2473 0.2613 0.2883 0.3270 1 3 B1 0.1800 0.2005 0.2235 0.2144 1 1 ()+ 3 2 B1 B3 0.2136 0.2309 0.2559 0.2707 ()s + s + s + ()s + s s11 s12 s13 2s13 s33 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 the compliances are 1/GPa and Skempton's B coefficient are Pa/Pa. Confining Stress (MPa) 40 40 35 31 Pore 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 u + u + u s11 s12 s13 0.0111 0.0115 0.0112 0.0115 u + u 2s13 s33 0.0109 0.0109 0.0116 0.0116 u s13 -0.0068 -0.0064 -0.0065 -0.0061 u s33 0.0245 0.0238 0.0246 0.0239 u s11 0.0255 0.0262 0.0268 0.0254 u s12 -0.0075 -0.0082 -0.0091 -0.0078 1 ()u + u + u + 1 ()u + u 2 s12 s13 s11 2 2s13 s33 0.0110 0.0112 0.0114 0.0116 1 ()u + u + u − 4 s11 s33 2s13 s44 -0.0237 -0.0098 -0.0064 -0.0191 1 ()u + u + u + 4 s11 s33 2s13 s44 0.0419 0.0284 0.0256 0.0376 1 ()+ 3 2B1 B3 0.3406 0.3989 0.4039 0.3487 1 3 B3 0.1636 0.1983 0.1595 0.1334 1 3 B1 0.0885 0.1003 0.1222 0.1076 ()s + s + s + ()s + s s11 s12 s13 2s13 s33 0.0086 0.0088 0.0088 0.0086 39 Table 1.5 (continued). Best-fit values of the poroelastic constants for Indiana limestone. The units of the compliances are 1/GPa and Skempton's B coefficient are Pa/Pa. Confining Stress (MPa) 26 20 20 9.6 Pore 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 u + u + u s11 s12 s13 0.0119 0.0119 0.0117 0.0119 u + u 2s13 s33 0.0121 0.0116 0.0121 0.0117 u s13 -0.0066 -0.0065 -0.0068 -0.0070 u s33 0.0252 0.0247 0.0257 0.0257 u s11 0.0268 0.0264 0.0271 0.0277 u s12 -0.0083 -0.0080 -0.0085 -0.0088 1 ()u + u + u + 1 ()u + u 2 s12 s13 s11 2 2s13 s33 0.0120 0.0118 0.0119 0.0118 1 ()u + u + u − 4 s11 s33 2s13 s44 -0.0071 -0.0076 -0.0175 -0.0073 1 ()u + u + u + 4 s11 s33 2s13 s44 0.0266 0.0266 0.0371 0.0271 1 ()+ 3 2B1 B3 0.4055 0.3984 0.4082 0.4322 1 3 B3 0.1548 0.1947 0.1627 0.1489 1 3 B1 0.1253 0.1019 0.1228 0.1416 ()s + s + s + ()s + s s11 s12 s13 2s13 s33 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 C −C Berea sandstone, exp bestfit ×100. Cexp 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 -7.9 1 ()+ + + 1 ()+ 2 s11 s12 s13 2 2s13 s33 4.1 11.6 16.9 25.4 1 ()+ 2 s12 s13 -1.4 -7.6 -0.2 14.5 1 ()+ + − 4 s11 s33 2s13 s44 25.7 20.8 23.7 11.0 1 ()+ + + 4 s11 s33 2s13 s44 58.4 58.4 58.8 60.6 u + u + u s11 s12 s13 -14.5 -17.3 -17.4 -17.8 u + u 2s13 s33 -11.7 -19.3 -22.4 -30.0 u s13 12.1 6.9 8.8 8.5 u s33 7.3 0.6 2.6 4.8 u s11 -6.1 -8.5 -13.8 -19.6 1 u + u + u + 1 u + u 2 ()s12 s13 s11 2 ()2s13 s33 -14.6 -13.8 -13.5 -14.2 1 u + u + u − 4 ()s11 s33 2s13 s44 -24.2 -22.9 -23.4 -17.1 1 u + u + u + 4 ()s11 s33 2s13 s44 7.2 5.8 8.4 5.1 1 ()+ 3 2B1 B3 -2.1 -4.8 -6.0 -6.6 1 3 B3 1.3 -3.0 -1.8 -4.8 1 3 B1 0.2 -3.1 -1.9 -4.2 1 1 ()+ 3 2 B1 B3 -2.6 6.2 3.2 3.5 s + s + s + s + s ()s11 s12 s13 ()2s13 s33 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 C −C constants for Berea sandstone, exp bestfit ×100. Cexp 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 -2.5 1 ()+ + + 1 ()+ 2 s11 s12 s13 2 2s13 s33 22.7 33.9 38.5 39.8 1 ()+ 2 s12 s13 -10.8 -6.2 -5.3 -1.5 1 ()+ + − 4 s11 s33 2s13 s44 8.1 6.9 4.8 -20.9 1 ()+ + + 4 s11 s33 2s13 s44 58.1 58.6 61.8 65.3 u + u + u s11 s12 s13 -19.8 -31.6 -27.2 -30.5 u + u 2s13 s33 -32.3 -32.4 -41.2 -47.1 u s13 22.1 19.8 23.9 31.3 u s33 6.6 7.2 15.2 30.6 u s11 -17.5 21.5 -14.9 -18.4 1 u + u + u + 1 u + u 2 ()s12 s13 s11 2 ()2s13 s33 -15.1 -26.8 -15.6 -18.1 1 u + u + u − 4 ()s11 s33 2s13 s44 -16.5 -16.1 -16.1 23.1 1 u + u + u + 4 ()s11 s33 2s13 s44 8.7 6.1 10.0 19.8 1 ()+ 3 2B1 B3 -2.5 0.8 -2.7 -0.3 1 3 B3 -10.1 -14.0 -13.2 -20.7 1 3 B1 -8.3 -12.1 -12.2 -14.4 1 1 ()+ 3 2 B1 B3 5.2 3.1 2.9 0.2 s + s + s + s + s ()s11 s12 s13 ()2s13 s33 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, C −C exp bestfit ×100. Cexp
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 u + u + u s11 s12 s13 -2.6 1.3 -3.9 -0.2 u + u 2s13 s33 1.1 2.9 8.9 7.2 u s13 -12.6 0.4 -13.9 -17.1 u s33 -0.9 -0.3 0.3 -4.3 u s11 16.0 14.4 25.1 15.2 u s12 -33.3 -26.0 -16.0 -24.3 1 u + u + u + 1 u + u 2 ()s12 s13 s11 2 ()2s13 s33 7.0 9.2 9.5 13.5 1 u + u + u − 4 ()s11 s33 2s13 s44 12.5 7.2 3.8 6.9 1 u + u + u + 4 ()s11 s33 2s13 s44 -53.1 -37.5 -25.8 -82.7 1 ()+ 3 2B1 B3 19.8 33.0 19.2 14.5 1 3 B3 -12.6 -30.9 -8.3 -5.0 1 3 B1 -12.8 -29.6 -14.5 -9.3 s + s + s + s + s ()s11 s12 s13 ()2s13 s33 -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 C −C limestone, exp bestfit ×100. Cexp
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 u + u + u s11 s12 s13 -2.9 2.4 -2.1 0.4 u + u 2s13 s33 10.3 7.1 9.9 12.5 u s13 -12.0 -6.6 -10.1 -8.4 u s33 -6.1 -4.4 -7.2 -2.5 u s11 24.0 17.1 22.9 26.7 u s12 -30.6 -30.1 -22.9 -20.2 1 u + u + u + 1 u + u 2 ()s12 s13 s11 2 ()2s13 s33 10.2 9.7 12.9 12.3 1 u + u + u − 4 ()s11 s33 2s13 s44 4.6 6.4 10.3 4.5 1 u + u + u + 4 ()s11 s33 2s13 s44 -24.0 -61.8 -60.2 -73.9 1 ()+ 3 2B1 B3 16.4 23.8 20.6 8.8 1 3 B3 -6.4 -15.9 -9.1 -1.6 1 3 B1 -10.7 -15.5 -15.0 -3.7 s + s + s + s + s ()s11 s12 s13 ()2s13 s33 -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 d = + + + + measured values of the linear compressibilities, C 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, 1 ()+ 3 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 bulk compressibility, Cu; the unjacketed bulk compressibility, Cs, and Skempton's B coefficient, B. Sample Id Confining Pore Cd Cu Cs B Stress Pressure (1/GPa) (1/GPa) (1/GPa) (Pa/Pa) (MPa) (MPa) 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 bulk compressibility, Cu; the unjacketed bulk compressibility, Cs, and Skempton's B coefficient, B. Sample Id Confining Pore Cd Cu Cs B Stress Pressure (1/GPa) (1/GPa) (1/GPa) (Pa/Pa) (MPa) (MPa) 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 0.926 630.224 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 C −C sandstone, exp bestfit ×100. Cexp Sample Id Confining Pore C Cu Cs B rms Stress Pressure (% error) (% error) (% error) (% error) error (MPa) (MPa) 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 63 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 Pore Cd Cu Cs B Stress Pressure (1/GPa) (1/GPa) (1/GPa) (Pa/Pa) (MPa) (MPa) 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 Pore Cd Cu Cs B Stress Pressure (1/GPa) (1/GPa) (1/GPa) (Pa/Pa) (MPa) (MPa) 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, C −C exp bestfit ×100 Cexp Sample Id Confining Pore Cd Cu Cs B rms Stress Pressure (% error) (% error) (% error) (% error) error (MPa) (MPa) 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 transverse isotropy 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 -0.0162 1 ()+ + + 1 ()+ 2 s11 s12 s13 2 2s13 s33 0.0293 0.0351 0.0406 0.0538 1 ()+ 2 s12 s13 -0.0082 -0.009 -0.0093 -0.0116 1 ()+ + − 4 s11 s33 2s13 s44 -0.0246 -0.0279 -0.0322 -0.0413 1 ()+ + + 4 s11 s33 2s13 s44 0.0621 0.072 0.0822 0.1067 C −C Percent Difference between Measured and Best-fit, exp bestfit ×100 Cexp + + 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 -100.9 1 ()+ + + 1 ()+ 2 s11 s12 s13 2 2s13 s33 -14.2 -12.0 -7.8 -5.8 1 ()+ 2 s12 s13 -63.0 -56.2 -31.3 -10.1 1 ()+ + − 4 s11 s33 2s13 s44 -122.8 -123.3 -117.3 -135.7 1 ()+ + + 4 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 sandstone assuming transverse isotropyand the percent difference between the measured and best-fit values. 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 -0.0186 1 ()+ + + 1 ()+ 2 s11 s12 s13 2 2s13 s33 0.0541 0.0698 0.0788 0.0892 1 ()+ 2 s12 s13 -0.0115 -0.0141 -0.0165 -0.0223 1 ()+ + − 4 s11 s33 2s13 s44 -0.0385 -0.0522 -0.0602 -0.052 1 ()+ + + 4 s11 s33 2s13 s44 0.1042 0.1361 0.1556 0.1635 C −C Percent Difference between Measured and Best-fit, exp bestfit ×100 Cexp + + 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 -53.7 1 ()+ + + 1 ()+ 2 s11 s12 s13 2 2s13 s33 -7.5 -4.3 -6.3 -9.9 1 ()+ 2 s12 s13 -44.8 -21.7 -32.7 -54.0 1 ()+ + − 4 s11 s33 2s13 s44 -118.4 -105.0 -114.6 -211.2 1 ()+ + + 4 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 Berea sandstone samples measured in this experiment. Confining Pore Linear compressibility parallel to bedding Stress Pressure + + 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 Ratios perpendicular to bedding/parallel to bedding 1.5 drained 1.4 undrained
1.3 unjacketed
1.2
Ratio 1.1
1 0 10203040 0.9 Linear Compressibility
0.8 Terzaghi Effective Stress (MPa)
Figure 1.7. Anisotropy ratios of linear compressibilities perpendicular to bedding over + + + linear compressibilities parallel to bedding, ()2s13 s33 ()s12 s13 s11 , for Berea sandstone.
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 Ratios perpendicular to bedding/parallel to bedding 1.5
1.4 drained 1.3 undrained unjacketed 1.2
Ratio 1.1
1.0 0 10203040
Linear Compressibility 0.9
0.8 Terzaghi Effective Stress (MPa)
Figure 1.8. Anisotropy ratios of linear compressibilities perpendicular to bedding over linear 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
0.35
0.3 Drained Undrained 0.25 Unjacketed 0.2
0.15
0.1
0.05 Compressibility (1/GPa) 0 0 10203040 Terzaghi Effective Stress (MPa) Figure 1.9. Plot of the bulk compressibilities as a function of Terzaghi effective stress for Berea sandstone. 58
Skempton's B Coefficient as a Function of Terzaghi Effective Stress
0.9 0.8 0.7
Pc) 0.6 ∆ 0.5
Ppore/ 0.4 ∆ ( 0.3 0.2 0.1 Skempton's B Coefficient 0 0 10203040 Terzaghi Effective Stress (MPa) Figure 1.10. Skempton's B coefficient as a function of Terzaghi effective stress for Berea sandstone.
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 0.05
0.04
0.03 Drained Undrained 0.02 Unjacketed (1/GPa)
Compressibility 0.01
0 0 10203040 Terzaghi effective stress (MPa) Figure 1.11. Bulk Compressibilities as a function of terzaghi effective stress for Indiana limestone. There is little dependence on effective stress. 60
Skempton's B Coefficient as a Function of Terzaghi Effective Stress 0.5
0.4 Pc) ∆ 0.3 Ppore/ ∆
( 0.2
0.1
Skempton's B Coefficient 0 0 10203040 Terzaghi Effective Stress (MPa) Figure 1.12. Skempton's B Coefficient as a Function of Effective Stress for Indiana limestone.
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 0.5
0.4
Experimental Compressibilities 0.3 Hertzian Model Compressibilities 0.2 (1/GPa)
Compressibility 0.1
0.0 0 10203040 Terzaghi Effective Stress (MPa) Figure 1.13. Nonlinearity of the drained compressibilities compared to Hertzian contact model 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).
2 ()Cd − Cs Cd − Cu = (16) []()C f − Cφ φ + ()Cd − Cs
where C f is the fluid compressibility and φ is the porosity. To use this equation, the pore φ compressibility, C , 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 φ so C need not equal Cs even for a monomineralic rock. φ Because measurement of C is technically difficult, no direct measurement has been φ made to date so that the assumption might be tested. Calculated values of C , 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 measured unjacketed bulk compressibility. Sample Confining Pore Pressure φ s Stress (MPa) (MPa) C C 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 0.50 Berea Sandstone Cphi 0.40 Cs 0.30
0.20
0.10 (1/GPa) 0.00 Compressibility 0 10203040 -0.10
-0.20 Terzaghi Effective Stress (MPa) Figure 1.14. Comparison of calculated pore compressibilities to measured unjacketed compressibilities for Berea sandstone. The error bars represent errors of 5% in the measurement of the bulk constants and the 10% in the measurement of porosity.
Comparison of Pore Compressibilities to Measured Unjacketed Compressibilities for Indiana limestone Cphi 0.25 Cs 0.20 0.15 0.10 0.05
(1/GPa) 0.00 -0.05 0 5 10 15 20 25 30 35 Compressibility -0.10 -0.15 Terzaghi Effective Stress (MPa) Figure 1.15. Comparison of pore compressibilities to measured unjacketed compressibilities for Indiana limestone. The error bars represent errors of 5% in the measurement of the bulk constants and the 10% in the measurement of porosity. 67 φ Figures 1.14 and 1.15 compare the calculated pore compressibility, C , values to the φ measured unjacketed values, Cs . Values of C 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 ∂C ()∂p ∆C = ∆p + ∆P = 0 and n =− . (18) ∂ ∂ c ∂C p Pc ()∂ Pc 69
Figure 1.16a. Contour plot of the measured drained bulk compressibilities for Berea sandstone. 70
Figure 1.16b. Contour plot of the measured undrained bulk compressibilities for Berea sandstone. 71
Figure 1.16c. Contour plot of the measured unjacketed bulk compressibilities for Berea sandstone. 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 = α confining stress, p Pc 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
=− d + d − s ev C Pc ()C C 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
Cd − Cs e =−Cd P − p . (20) v c Cd
s so the effective stress coefficient is α = 1 − C . If the material is non-linear, this equation Cd d s α holds only for small values of p and Pc because C and C and thus 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.
=−∫ d +∫ d − s ev C (Pc , p)dPc ()C (Pc , p) C (Pc , p) dp (22)
d s The forms of C (Pc , p) and C (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.
d = − −0.378 C (Pc , p) 0.302()Pc p (23)
d where C (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
− − 1 0.378 − − 1−0.378 ()Pcf p f ()Pci pi e =−0.302 − Cs ()p − p (24) v ()− f i 1 0.378
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
Cs 0.036 α = 1 − = 1 − = 0.84 (25) Cd 0.22 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 in Engineering, Springer-Verlag. Berge, P.A., Wang, H.F. & Bonner, B.P. 1993. Pore pressure build-up coefficient in synthetic and natural 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 Barre Granite. 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 on the 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 of Berea sandstone in triaxial compression. U.S. Dept. of the Army Tech. Report, Engng. Study No. 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 on dynamic elastic properties of Berea sandstone. Geophysics. 50:207-213. Endres, A.L. 1997. Geometrical models for poroelastic behavior. Geophys. J. Int. 128:522-532.
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Hart, D.J. & Wang, H.F. 1999. Pore pressure and confining stress dependence of poroelastic linear 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 Williston basin. Geophysics. 46:288-297. Kachanov, M. 1992. Effective elastic properties of cracked solids: critical review of some basic concepts. Appl. Mech. Rev. 45:304-335. Lo, T., Coyner, K.B., & Toksoz, M.N. 1986. Experimental determination of the elastic anisotropy of 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 and Building, Indianapolis, Indiana. Love, A.E. 1944. A Treastise on the Mathematical Theory of Elasticity. pp.195-198. Dover Publications, 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, San Diego, 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 with fluids. 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 crack problem. 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 fractal dimension of pores in sedimentary rocks and its influence on permeability. Engineering Geology 48: 199-215. 80 Talesnick, M.L., Haimson, B.C., & Lee, M.Y. 1997. Development of radial strains in hollow cylinders of rock subjected to radial compression. International Journal of Rock Mechanics and Mining 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 for Berea 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 permeability and 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: K ∂p ∇2 p = (1) Sσ ∂t 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)
∞ ∆ =∆ +∆ − 2 π + 2 p(x,t) po po ∑{exp[ (K /Sσ )(t / L ) /4(2n 1) ] n=1 (2) × cos[(2n +1)π(x /2L)][4(−1)n+1 /(2n +1)π]}
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 2 ()t / L ()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:
u G ∂ε ∂p G ∇2u − r + kk − α = 0 (3) r r 2 1 − 2υ ∂r ∂r
G ∂ε ∂p G()∇2u + kk − α = 0 (4) z 1 − 2υ ∂z ∂z 84 and the axisymmetric form of the fluid diffusion equation:
∂ε ∂ k 2 kk p ()∇ p − α = Sε (5) µ ∂t ∂t
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).
Constant Stress Constant Pore Pressure Deformed Region of Uncoupled Sample Diffusive Flow Sample Before Pressure Step Negative Negative Axial Strains { Region of { Axial Strains Negative PorePressure
No Flow No Flow Constant Stress Constant Stress
No Flow No Displacement
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
1.2 1 0.8 0.6 0.4 Diffusion Alone 0.2 Fully Coupled 0
Normalized Pressure -0.2 0 0.5 1 1.5 Dimensionless time (t/L^2)(K/S)
Early Time Response versus Normalized Pressure at the Sample Bottom 0.2
0.15 Diffusion Alone Fully Coupled 0.1
0.05
0 0 0.05 0.1 0.15 Normalized Pressure -0.05 Dimensionless time (t/L^2)(K/S)
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.
Negative Pressure Region as a Function of Changing Radius
Radius = 0.00254 m Radius = 0.0254 m Length = 0.0309 m Length = 0.0309 m
Radius = 0.254 m Length = 0.0309 m
Key - Normalized Pore Pressure
0.5 - 1.0 0.0 - 0.5 less than 0.0 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.
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 = 0.0254 m Length = 0.0309 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
Radius = 0.254 m Length = 0.309 m
Key - Normalized Pore Pressure 0.5 - 1.0 0.0 - 0.5 less than 0.0 Figure 2.4. Comparison of the pressure profiles at the same dimensionless time with variation of the 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 Normalized Pore Pressure (Pressure Step at Sample Top Equal to 1) 1.000 to 0.000
0.000 to -0.025
-0.025 to -0.050
Case 2 G=3.9 GPa υ=0.20 -0.050 to -0.075
-0.075 to -0.100
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's ratio (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 1.2
1
0.8
0.6 Axial Strain 0.4 Circumferential Strain Diffusive Flow Solution 0.2
Normalized Strain 0 0 200 400 600 800 1000 -0.2
-0.4 Dimensionless time (t/L^2)(K/S)
Figure 2.6. Comparison of axial and circumferential strains with the pressure calculated from diffusive 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 0.2 Case 1 (G=6.6 GPa, v=0.05) 0.15 Case 2 (G=3.9 GPa, v=0.20) 0.1 Case 3 (G=1.1 GPa, v=0.40) 0.05 0 -0.05 0 0.02 0.04 0.06 0.08 0.1 -0.1 -0.15
Normalized Strain -0.2 -0.25 -0.3 Dimensionless time (t/L^2)/(K/S) Figure 2.7. Comparison of early time axial strains located midway along the model length for the three 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
1
0.8
0.6
0.4 Experimental Pressures Diffusion Alone Pressures 0.2 Fully Coupled Pressures
Normalized Pore Pressure 0 0 200 400 600 800 1000
-0.2 Time (seconds) Figure 2.8a. Comparison of the experimental normalized pressure at the sample bottom for the short sample Ca3 with pore pressures calculated from the fully coupled poroelastic model and from diffusive flow alone. 97
Early time response - short sample Ca3
0.2
0.15
0.1 Experimental Pressures
0.05 Diffusion Alone Pressures
Fully Coupled Pressures 0 Normalized Pore Pressure 0 50 100 150 200
-0.05 Time (seconds)
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 coupled poroelastic model. The early time pore pressure reversal is present and is another example of a Mandel-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 1 0.9 0.8 0.7 0.6 0.5 0.4 Experimental Pressures 0.3 Diffusion Alone Pressures
0.2 Fully Coupled Pressures Normalized Pressures 0.1 0 -0.1 0 10000 20000 30000 40000 Time (seconds) Figure 2.9a. Comparison of the experimental normalized pressure at the sample bottom for the long sample Ca1 with pore pressures calculated from the fully coupled poroelastic model and from diffusive flow alone. 99
Early time response - long sample Ca1 0.2
0.15 Experimental Pressures
0.1 Diffusion Alone Pressures Fully Coupled Pressures
0.05 Normalized Pressures 0 0 500 1000 1500 2000
-0.05 Time (seconds)
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 coupled poroelastic model. The early time pore pressure reversal is negligible in both the experimental data and 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 Time Short Sample Ca3 1.2
1
0.8
0.6 Axial strain 0.4 Diffusion Alone Pressure
0.2 Fully Coupled Strains Normalized Axial Strain 0 0 200 400 600 800 1000 -0.2
-0.4 Time (seconds)
Figure 2.10a. Normalized axial strain for the short sample, Ca3, as a function of time and compared to normalized axial strains from the poroelastic model and normalized pore pressure from the diffusive flow equation. The measuring point is 1.3 cm from the bottom of sample.Ca3. 102
Early time axial strains Short Sample Ca3
0.2
0.15
0.1
0.05
0 0 20406080100 -0.05
-0.1 Axial strain
Normalized Axial Strain -0.15 Diffusion Alone Pressure
-0.2 Fully Coupled Strains
-0.25
-0.3 Time (seconds) Figure 2.10b. The early time portion of the normalized axial strain for the short sample, Ca3, as a function of time and compared to normalized axial strains from the poroelastic model and normalized pore pressure from the diffusive flow equation. 103
Normalized Axial and Circumferential Strains versus time - (Long sample Ca1) 1
0.8
0.6 Axial Strain Circumferential Strain 0.4 Fully Coupled Solution Diffusion alone 0.2 Normalized strain
0 0 10000 20000 30000 40000 -0.2 time (seconds) Figure 2.11a. Normalized axial and circumferential strain for the long sample, Ca1, as a function of time and compared to normalized axial and circumferential strains from the poroelastic model and normalized pore pressure from the diffusive flow equation. The measuring point is 5.0 cm from the bottom of sample Ca1. 104
Early time axial and circumferential strains (Long Sample Ca1)
0.2 Axial Strain Circumferential Strain Fully Coupled Solution 0.15 Diffusion alone
0.1
0.05 Normalized strain
0 0 500 1000 1500 2000
-0.05 time (seconds) Figure 2.11b. The early time portion of the normalized axial and circumferential strains for the long sample, Ca1, as a function of time and compared to normalized axial strains from the poroelastic 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
∂σ ∂ B kk + p = ∇2 Sσ K p (6) 3 ∂t ∂t
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)
4ηB S = Sσ 1 − (7) s 3
1 − 2ν K where η = 1 − and ν is Poisson's ratio. These values, including Poisson's ratio, ()− ν 21 Ks
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 method to 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.] Clarendon Press. Cryer, C.W. 1963, A comparison of the three-dimensional consolidation theories of Biot and Terzaghi. 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 Flow Pump 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. Water Resources Research, 26, pp. 1631-1637. Hart, D. J. and Wang, H. F. 1995, Laboratory measurements of a complete set of poroelastic moduli 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. Peter Sandstone 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 Solid Deformation 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 Response From 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 Resources Research, 22, pp. 1163-1195. Olsen, H. W., Morin, R. H., and Nichols, R. W., 1988, Flow Pump Applications in Triaxial Testing, 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 elastic porous media with compressible constituents, Rev. Geophys., 14, 227-241. Walder, J. and Nur, A. 1986, Permeability Measurement by the Pulse-decay Method: Effects of Poroelastic 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 from Pulse 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
Step 1 Step 2 Step 3 Short Time Long Time Undrained Response Transient Pore Pressure Test Drained Response
Confining Stress
Pressure at Sample Bottom Stress and Pressure
Pressure in External Pore Pressure System
0 10 1000 2000 3000 4000 Time (seconds)
Figure 3.1. Pressure and stress history for an idealized run. The three stresses and pore pressures recorded 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
18 17 16 15 14 13 12 11 Pore pressure (MPa) 10 12 14 16 18 20 22 Confining stress (MPa)
Figure 3.2a. Pore pressure as a function of confining stress. The slope of the line gives a Skempton's B coefficient of 0.84. 113
Volumetric Strain versus Confining Stress
50
0
-50
-100
V/V)x 10^6 -150 ∆ (
Volumetric Strain -200 12 14 16 18 20 22 Confining Stress (MPa)
Figure 3.2b. Volumetric strain as a function of confining stress. The negative of the slope of the line 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.
∞ −αφ 2 ()φ ζ 1 exp()m cos m px(),t =∆P + 2∑ (1) + β ()+ β ()φ − φ ()φ 1 m=1 1 cos m m sin m
where
px(),t is the pore pressure as a function of distance from the sample bottom and time,
∆P is the initial pressure step, α = Kt 2 is a dimensionless time, L Sσ Sσ AL β = is the ratio of sample storage to reservoir storage, Sres x ζ = is a dimensionless length from the sample bottom, and L φ φ is a root of the equation tan()φ = m . m m −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
px(),t 1 = . (2) ∆P 1 + β
px(),t Because the final ratio of pressure change, the normalized pore pressure, , is very well ∆P 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 0.2
Experimental 0.0 Calculated Best-Fit
-0.2 sample bottom
-0.4 strain gage
-0.6
reservoir -0.8 normalized pore pressure
-1.0 0 1000 2000 3000 4000 time (seconds) Figure 3.3 Normalized pore pressures at the sample bottom and reservoir and normalized strains midway along the sample length. Note the early time reverse pore pressure response at the sample bottom.
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
()Cd − Cu Cs = Cd − (4) B 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 a bulk 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)
Cd − Cs Sσ = ρg . (5) B
where the grain compressibility used was Cs =0.015 1/GPa. The resulting estimate for the specific -6 -1 storage is Sσ =1.05x10 m . This result is in good agreement, to within 20%, of the value of 8.6 x Sσ Al 10-7 m-1 calculated using the ratio β = and the experimentally determined reservoir storage. Sres 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,
-7 -1 the value of the three-dimensional unconstrained specific storage, Sσ =8.6x10 m , is 1.5 times
-7 -1 larger than the value calculated for the one-dimensional specific storage, Ss =5.8x10 m , 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.
References Aoki, T., Tan, C.P., Cox, R.H.T., & Bamford, W.E. 1995, Determination of anisotropic poroelastic parameters of a transversly isotropic shale by means of consolidated undrained triaxial tests. Proceedings of the 8th International Congress of Rock Mechanics. 1, pp. 173-176. 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 saturated sedimentary rock. Geophysics, 51, pp. 948-956. Hart, D.J. and Wang, H.F. 1995, Laboratory measurements of a complete set of poroelastic moduli for Berea sandstone and Indiana limestone, J. Geophys. Res. 100, pp. 17,741-17,751. 122 Hart, D.J. and Wang, H.F. 1999, Pore pressure and confining stress dependence of poroelastic linear 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. Hsieh, P.A., Tracy, J.V., Neuzil, C.E., Bredehoft, J.D. and Silliman, S.E. 1981, A transient laboratory method for determining the hydraulic properties of 'tight' rocks --I Theory. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 18, pp. 245-252. Senseny, P.E., Cain, P.J., and Callahan, G.D., 1983, Influence of deformation history on permeability and specific storage of Mesaverde sandstone. Proc. 24th U.S. Symposium on Rock Mechanics, pp. 525-531 Trimmer, D.A. 1982, Laboratory measurements of ultralow permeability of geological materials. Rev. Sci. Instrumen. 53, pp. 1246-1254. Tokunaga, T., Hart, D.J., and Wang, H.F. 1998, Complete set of anisotropic poroelastic moduli for Berea sandstone. In Thimus et al. (eds), Poromechanics, 629-634 Rotterdam: Balkema. Walder, J. and Nur, A. 1986, Permeability Measurement by the Pulse-decay Method: Effects of Poroelastic Phenomena and Non-linear Pore Pressure Diffusion. Int. J. Rock Mech. Min. Sci. and Geomech. Abstr. 23, pp. 225-232. Yilmaz, O., Nolen-Hoeksema, R.C., & Nur, A. 1994, Pore pressure profiles in fractured and compliant rocks, Geophysical Prospecting, 42, pp. 693-714. Zimmerman, R.W., Somerton, W.H., and King, M.S. 1986, Compressibility of porous rocks, J. Geophys. Res., 91, pp. 12,765-12,777. Zoback, M.D. and Byerlee, J.D. 1975, Permeability and Effective Stress. Bulletin of the American Association of Petroleum Geologists. 59; 1, pp. 154-158.