Environmental & Engineering Geoscience
 © 2012 Association of Environmental & Engineering Geologists
Abstract
Skempton's B coefficient is an important characteristic of a porous medium that describes the way in which pore pressure responds to a change in the mean stress under undrained conditions. This number is well known for many sedimentary rocks under water and/or oilsaturated conditions. However, due to the difficulty in conducting laboratory tests on supercritical CO_{2}saturated rock because of the specific requirements in terms of pressure and temperature to ensure the supercritical status of CO_{2}, this number is not available for a supercritical CO_{2}saturated rock. A finite element method (FEM) approach was developed in an effort to solve this problem. We first calibrated our numerical rock models according to the poroelastic properties of Berea sandstone and Indiana limestone, and conducted a series of numerical tests to measure their Skempton's B coefficients under watersaturated conditions. The test results, which were found to be very close to the laboratory measurements, gave us confidence to extend this testing approach to a CO_{2}saturated rock by changing the pore fluid from water to supercritical CO_{2}. The numerical test results showed that the Skempton's B coefficient for supercritical CO_{2}saturated rock is considerably different (lower) than that of watersaturated conditions. This implies that the poromechanical conditions of supercritical CO_{2}saturated rock are significantly different from those of water or oilsaturated rock, and the change of pore fluids from water/oil to supercritical CO_{2} may introduce a significant change in the poromechanical properties of the rock.
 Skempton's B Coefficient
 Pore Pressure
 Poroelasticity
 NonLinear Poroelasticity
 FEM
 Abaqus
 Supercritical CO_{2}
Introduction
Enhanced oilrecovery operations using CO_{2} as an injection solvent have been practiced for more than 40 years, and also have shown potential as a means of CO_{2} sequestration in order to decrease anthropogenic greenhouse gas emissions into the atmosphere (Klara and Byrer, 2003). Total oil production can be increased from 10 percent to 30 percent using CO_{2} flooding (Gaspar et al., 2009). On the other hand, injecting CO_{2} into deep saline aquifers is one of the main options for the geological storage of CO_{2} (Bradshaw et al., 2007). In all these cases, the pore fluids have been or will be changed in deep formations from oil or saline water to CO_{2}, and rock matrix saturated with different pore fluids may behave very differently.
Since the pioneering work of Biot (1941, 1956), poroelasticity theory has played an important role in many branches of engineering, including geosciences, material science, chemical engineering, as well as biomechanics, etc. It has been further developed by many people, such as Zienkiewicz and Shiomi (1984), Lewis and Schrefler (1998), de Boer (2000), Coussy (2004), etc. For example, Schrefler extended Biot's theory to a multiphase medium and made the first application of the theory to land deformation studies in Italy. Zienkiewicz was notable for having recognized the general potential for using the finite element method to resolve problems in areas outside the area of solid mechanics. Coussy provided a unified approach to the fundamental concepts of continuum poromechanics, etc.
An important characteristic of a porous medium is the way in which the pore pressure responds to a change in the mean stress under undrained conditions (Zimmerman, 1986; Fjaer et al., 2008). If a fluidsaturated porous rock undergoes undrained compression, the confining pressure causes the pores to contract, thereby pressurizing the trapped pore fluid. The magnitude of this induced porepressure increment is described by the following equation (Skempton, 1954):
where B is Skempton's B coefficient, P_{f} is pore pressure, P_{c} is confining pressure, and dm_{f} is the change in mass of pore fluid (zero implies undrained conditions). It should be noted that water is usually used as the pore fluid during laboratory testing.
The Skempton's B coefficient allows the coupling between mechanical deformation and pore pressure to be quantified (Jaeger et al., 2007). The relationships among B and compressibilities or bulk moduli of bulk rock, solid grain, and pore fluid, as well as porosity, have been investigated by many researchers (Brown and Korringa, 1975; Rice and Cleary, 1976; Berryman and Milton, 1991; and Berge et al., 1993). Typically, two sets of analytic solutions are used to describe these relationships, i.e.,
and
where n is porosity; K is drained bulk modulus of rock, defined as ; K_{u} is undrained bulk modulus of rock, defined as ; K_{s} is the solid grain bulk modulus, defined as ; 1/K_{n} is the unjacketed pore compressibility, which is defined as ; and K_{f} is pore fluid bulk modulus. If all grains in the rock are composed of the same material, K_{s} = K_{n}, and a minor simplification can be made in Eq. 2 (Berge et al., 1993).
Hart and Wang (1995) conducted laboratory tests on Berea sandstone and Indiana limestone and acquired a complete set of poroelastic moduli for these two types of rocks, including the Skempton's B coefficients. In this paper, two finite element models were calibrated according to the properties of Indiana limestone and Berea sandstone, and their Skempton's B coefficients were measured by conducting numerical hydrostatic compression tests on “Indiana limestone” and “Berea Sandstone” (where the quotes imply numerically derived poromechanical properties of these rock types). The simulated B coefficients correspond quite well to the laboratory tests. This favorable result allowed us to extend this approach to CO_{2}saturated rock by changing the pore fluid from water to supercritical CO_{2}.
The input parameters in the finite element models are different from those used in the analytic solutions in Eq. 2 and Eq. 3; the numerical testing approach has its own significance.
POROELASTICITY IN ABAQUS
The finite element package Abaqus (Hibbitt, Karlsson & Sorensen, Inc., 1998) was used to obtain the results used in this paper. Abaqus was developed as a general FEM package and uses engineering convention with regard to tension (positive), which is opposite to the rock mechanics convention (where compression is positive). Furthermore, it inherently contains its own parameters that describe porous materials, so a short introduction of poroelasticity used in Abaqus is provided here.
In Abaqus, the porous elastic material model is nonlinear and isotropic, in which the pressure stress varies as an exponential function of volumetric strain. The elastic part of the volumetric behavior of porous material is modeled accurately based on the assumption that the elastic part of the change in volume of the material is proportional to the logarithm of the pressure stress. For a poroelastic model, the following parameters are generally needed (Table 1):
The elastic behavior of a sample is mainly determined by three parameters: logarithmic bulk modulus κ, Poisson's ratio ν, and elastic tensile strength . Besides Poisson's ratio, the other two parameters are not commonly used in geoscience disciplines (Wang, 2000; Jaeger et al., 2007). In addition, these two parameters may not be easy to acquire or measure in a laboratory test designed for rock mechanics. The derivation of their relations to some common mechanical parameters such as Young's modulus and the Bulk modulus used in rock mechanics is presented herein.
If the volume change of a sample is defined by J = (1 + e)/(1 + e_{0}), where e_{0} is the initial void ratio, then the elastic part of the volume ratio between the current and reference configurations can be defined by J^{el} as:
where e^{el} is the elastic change of the void ratio. Here, the relationship between the void ratio e and porosity n is e = n/(1 − n). In addition, we also have:
where is the logarithmic measure of the elastic volume change.
In porous materials, during recoverable elastic straining, the change in the void ratio and the change in the logarithm of the equivalent pressure stress are linearly related, i.e.,
where p is the equivalent pressure stress, defined by p = −1/3(σ_{11} + σ_{22} + σ_{33}), and is the elastic tensile strength. Equation 6 shows the relationship between void ratio and elastic tensile strength.
By integrating the linear relation from Eq. 6 and also using Eq. 4, the elasticity relationship is:
where p_{0} is the initial value of the equivalent pressure stress. In Abaqus, the instantaneous shear modulus is defined as:
Since shear modulus relates with Young's modulus by G = E/2(1 + ν) under an isotropic assumption, then there is:
So a porous elastic material's Young's modulus can be derived from its logarithmic bulk modulus κ, Poisson's ratio ν, and elastic tensile strength using Eq. 9. In other words, an elastic material and a porous elastic material can behave externally the same if their Young's modulus and Poisson's ratio are the same.
Because bulk modulus K relates with Young's modulus E by K = E/3(1 − 2ν) under the assumption of isotropy, there is also:
which yields a straightforward relationship between logarithmic bulk modulus κ and bulk modulus K.
In porous materials such as some sedimentary rock, including Indiana limestone and Berea sandstone, E, ν, and e_{0} can all be measured directly. The key problem would be how to determine κ and .
CALIBRATION OF “BEREA SANDSTONE” AND “INDIANA LIMESTONE”
Recall the laboratory test results in Hart and Wang (1995), in which five Berea sandstone samples and two Indiana limestone samples were tested. The poroelastic moduli for Berea sandstone and Indiana limestone are summarized in Table 2.
Note that these tests were conducted with water as the pore fluid. External confining stresses and pore pressure were also applied, with the midpoint mean external stress ranging from 10 to 30 MPa for both types of rock and pore pressures ranging from 6.2 to 26.7 MPa for Berea sandstone and 2.6 to 19.5 MPa for Indiana limestone. The most common condition can be considered as an external confining pressure of approximately 20 MPa and a pore pressure of approximately 10MPa.
To determine the numerical input parameters, besides the three porous elastic parameters, κ, ν, , we also need to know two bulk moduli: i.e., the bulk modulus of solid grains K_{g}, and the bulk modulus of pore fluid K_{f}. Permeability is also usually required, but its value is not important here because it has no influence on the mechanical properties.
For Berea sandstone, which is composed of quartz (80 percent), feldspar (5 percent), clay (predominantly kaolinite, 8 percent), and calcite (6 percent) (Winkler, 1983), the bulk modulus of the solid is assumed to be 28 GPa (Table 2). Note quartz's bulk modulus is 38 GPa, but some clay cements may reduce the effective bulk modulus of the solid grains due to their lower bulk moduli. For Indiana limestone, which is mainly composed of calcite (98 percent), the bulk modulus of the solid part was determined to be 74 GPa. In both samples, the bulk modulus of fluid is given as 2.2 GPa, which corresponds to water at 20°C.
Poisson's ratio in the poroelasticity of Abaqus refers to the drained Poisson's ratio; thus, the problem is left to calibrate κ and , as all the other input parameters can be acquired directly from the laboratory testing results or are preset by nature.
Here, a backcalibration strategy is used, where we begin with a pair of κ and values and then conduct a numerical uniaxial compression test, triaxial drained and undrained compression tests (Figure 1), and a hydrostatic compression test (Figure 2). From these tests, Poisson's ratio, Young's modulus (both drained and undrained), and Skempton's B coefficients can all be measured. By comparing these measured values to the laboratory results, we can verify if the selected pair of κ and adequately describes the expected rock types. If not, further adjustments are made, and so on, until the measured data are sufficiently close to the laboratory results.
After a series numerical tests, the input parameters that characterize “Indiana limestone” were organized in Table 3.
The uniaxial compression test shows that under dry conditions, the sample behaves in a pure linear elastic fashion, as shown in Figure 3. Because in Abaqus compression is negative, to follow the rock mechanics convention, both strain and stress were switched to positive in the stressstrain plot. Young's modulus and Poisson's ratio can all be read from this figure as:
and
It should be noted that the laboratory tests conducted by Hart and Wang (1995) were generally constrained to high confining pressures; triaxial tests are therefore actually more suitable for the purpose of calibration.
During the drained triaxial test, the external confining pressure was kept at 20 MPa, and the pore pressure was kept at 10 MPa. After stabilization, the upper face of the sample was switched from stress control to strain control, and then differential stress conditions were initialized. This is exactly the same as an actual laboratory testing procedure. Figure 4 shows the test results.
Drained Young's modulus can be read from the strainstress plot in Figure 4 as:
This is close to what has been acquired from the unconfined test but not identical. In fact, the minor increase due to the confined conditions agrees with actual observations.
From the axial and lateral strain behavior during the test (lower part in Figure 4), one can see that initially the sample is compressed homogeneously, and then the axial and lateral strain differs by following the differential stress condition. Drained Poisson's ratio can be read as:
This confirms that Poisson's ratio in Abaqus' poroelasticity refers to a drained Poisson's ratio.
At last, the drained bulk modulus can be calculated by the following formula:
For an undrained triaxial test, the procedure is the same except that pore pressure cannot be precisely set at 10 MPa, but somewhere close to it. This is because after the start, the communication of pore fluid with outside control is cut off during the stabilization stage. Figure 5 shows the axial and lateral strain behavior during the test, from which the undrained Poisson's ratio can be read as:
From the strainstress plot in this figure, the undrained Young's modulus can be read as:
and the undrained bulk modulus can be calculated as:
In either Figure 4 or 5, there is a turning point in the strainstress plot to indicate a minor change of the trend between stress and strain. This is caused by the change of compression condition from hydrostatic compression to differential compression. The Young's modulus was calculated after this turning point, as the conditions prior to this turning point are considered as a preparation stage for this type of test.
From this undrained triaxial test, Skempton's B coefficient can also be derived. Figure 6 shows the change of pore pressure and confining stresses (both lateral and axial stress), from which B is calculated as:
Another robust method to acquire Skempton's B coefficient is accomplished by conducting a hydrostatic compression test (Figure 2). In this test, a cubeshaped sample is prepared, and one block in the center of the sample is monitored during the test to minimize boundary noise. The external confining pressure was set at 20 MPa, and pore pressure was close to 10 MPa after sample installation. Following stabilization, the external pressure was increased stepwise and the responses of pore pressure and confining stress were recorded, as shown in Figure 7.
Table 4 gives the results of the stepwise pore pressure and confining stress increments due to an external confining pressure increase, and Figure 8 graphically shows the results of Table 4. The rock mechanics convention for compression (positive) was also used in Figure 8. The Skempton's B coefficient of 0.55 for “Indiana limestone” can be obtained from this figure, which is identical to the value obtained from the triaxial undrained test.
These two approaches provided similar B values; in reality, however, the undrained triaxial compression tests may tend to lower the estimate of B due to the difficulty in equilibrating minor changes of lateral stress. For example, if the lateral stress changes were ignored in this case, the B value from the triaxial test would only be about 0.40.
Following a similar methodology, the poroelastic input parameters of “Berea sandstone” were also calibrated as shown in Table 5.
Similar numerical tests have also been conducted on “Berea sandstone,” and a more detailed testing report can be found in Appendix A.
As a summary, Table 6 lists the measured Young's modulus and Poisson's ratio (both drained and undrained) and Skempton's B coefficient from these numerical tests for both types of rock. These numerical tests were all conducted under an external confining pressure of 20 MPa with pore pressures at 10 MPa in the case of the drained test or approximately 10 MPa in the case of the undrained test. Results from uniaxial compression tests are not included.
By comparing Table 6 with Table 2, one can see that the simulated behaviors of the numerical models correspond quite well with the experimentally measured values. It can be concluded that the match between the numerical testing results and actual laboratory testing results is quite good. Therefore, the entire set of poroelastic input parameters are assumed to have been calibrated successfully for these two types of rock.
The FEM model gives the correct B values, even though different input parameters are used in Eq. 2 and Eq. 3 because poroelasticity is handled differently in Abaqus.
SKEMPTON'S B COEFFICIENTS FOR SUPERCRITICAL CO_{2}SATURATED ROCKS
These two numerical rock models, which behave both externally and internally to resemble the real rock samples very well, provide confidence that we can extend this testing approach to include other types of pore fluids such as supercritical CO_{2}, because it may not be easy to conduct an actual laboratory test to include such a fluid.
Two parameters in Abaqus are required to represent a pore fluid: one is specific weight, and the other is bulk modulus. For supercritical CO_{2}, these parameters will vary with temperature and pressure (Law and Bachu, 1996; Span and Wagner, 1996). In this paper, relatively small bulk modulus (0.058 GPa) and density (660 kg/m^{3}) are used, which probably correspond to a minimum status for supercritical conditions (7.4 MPa and 31.1°C).
Figure 9 shows the hydrostatic numerical compression test result for supercritical CO_{2}saturated “Berea sandstone.” The acquired Skempton's B coefficient is about 0.026. Figure 10 shows the hydrostatic numerical compression test result for supercritical CO_{2}saturated “Indiana limestone.” The acquired Skempton's B coefficient is about 0.018. Please refer to Appendix B for more detailed testing results.
During CO_{2} sequestration, when pore fluid is changed from water to supercritical CO_{2}, the Skempton's B coefficient decreases significantly, from the range of 0.55 to 0.85 to the range of 0.018 to 0.026. Because there are also other waterCO_{2} mixtures present, the variation of Skempton's B coefficient in a CO_{2}influenced region can be very large and complicated. Consequently, the poromechanical behavior of rock with pores predominately filled with CO_{2} can be significantly different from rock behavior when pores are filled by water. Pore pressure under supercritical CO_{2} saturation conditions would be insensitive to the pore space change.
DISCUSSION
Equations 2 and 3 yield analytic solutions for Skempton's B coefficient. However, the undrained bulk modulus K_{u} is used in Eq. 3, which involves the pore fluid effect. In reality, this value may not be available when supercritical CO_{2} is the pore fluid. Consequently, only Eq. 2 can be used. In the case of “Berea sandstone,” this number can be calculated to be:
and for Indiana limestone, this number is calculated as:
These are close to the numerical simulation results but not identical. In addition, both analytical and numerical results show that Skempton's B coefficient for “Berea sandstone” is larger than that of “Indiana limestone.”
Different approaches were used to obtain the numerical and analytical solutions. A nonlinear poroelasticity relation was used in the numerical simulation with Abaqus. The numerical result is in close agreement with the analytical result of Eq. 2, which is based on linear poroelasticity. Confining pressure is an important input parameter in the numerical simulation but does not play a role in analytical solution. Low confining pressures will result in Poisson ratios close to 0.5, which tend to be unrelated to the intrinsic rock property.
Skempton's B coefficient is an important coefficient for the deformation of porous elastic media. The crust behaves not simply as a dry medium but as a multiphasefluidssaturated porous medium (Roeloffs, 1996), and this can explain a variety of hydromechanical phenomena. For example, for sudden faulting, the instantaneously induced porepressure field is given by the undrained condition, and the porepressure change is given by ΔP_{f} = −BΔP_{c}. One can see that the watersaturated rock and CO_{2}saturated rock will have very different porepressure change curves because of the different B values, and, consequently, the fractures may propagate very differently (Berryman, 2012). This has an obvious implication to the safety of CO_{2} sequestration.
During CO_{2} sequestration, saline water/oil and CO_{2} may coexist at very different mixture conditions, and the saturation level of CO_{2} may have a very complex scenario, with higher levels of saturation close to the injection well and along the preferential flow paths and lower levels of saturation in the far field and in the lowerpermeability regions. Since the finite element method provides greater flexibility in modeling complex geometries, the Skempton's B values may yield highly complex contours in response to the CO_{2} saturation levels. More detailed research is needed in this area and is beyond the scope of this paper.
CONCLUSIONS
The finite element package Abaqus was used to simulate two types of rock: Indiana limestone and Berea sandstone, which possess relatively isotropic and homogeneous stratigraphy. The input parameters of porous elasticity were calibrated based on laboratory test results conducted by Hart and Wang (1995). The numerical tests, unconfined uniaxial compression test, triaxial drained and undrained compression tests, and hydrostatic compression tests all showed that the numerical rock models properly resemble the actual rock samples both externally and internally. This positive result allowed us to confidently extend this testing approach and rock modeling to estimate the Skempton's B coefficient for these two types of rock with supercritical CO_{2} as the pore fluid. The results show that the Skempton's B coefficients decrease significantly in the case of CO_{2} saturation. Even confining pressure does not play a role in the analytic solutions based on Eq. 2 and Eq. 3; it is an important input for the numerical testing in this finite element model. Nonlinear poroelasticity of the numerical approach and linear poroelasticity of the analytical approach have shown that the results are close. The numerical results have specific significance that may affect our understanding of reservoirscale CO_{2} injection and storage. To our knowledge, this is the first investigation aimed at addressing the effect of supercritical CO_{2} on Skempton's B coefficient.
Acknowledgments
This technical effort was performed in support of the National Energy Technology Laboratory's ongoing research in geologic CO_{2} sequestration under the RES contract DEFE0004000. Dustin Crandall is greatly appreciated for providing a very helpful review.
DISCLAIMER
This project was funded by the Department of Energy, National Energy Technology Laboratory, an agency of the United States Government, through a support contract with URS Energy &Construction, Inc. Neither the United States Government nor any agency thereof, nor any of their employees, nor URS Energy &Construction, Inc., nor any of their employees, makes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.
APPENDIX A. NUMERICAL TEST RESULTS OF “BEREA SANDSTONE”

Uniaxial compression test (Figure A1)

Triaxial drained compression test (Figure A2)

Triaxial undrained compression test (Figure A3)

Hydrostatic compression test (Figures A4 and A5 and Table A1)
Skempton's B coefficient for “Berea sandstone” saturated with water can be read from Figure A5 as 0.85.
APPENDIX B. HYDROSTATIC COMPRESSION TEST RESULTS ON “BEREA SANDSTONE” AND “INDIANA LIMESTONE”
After hydrostatic compression tests, Skempton's B coefficients for “Berea sandstone” and “Indiana limestone” can all be obtained.
B = 0.026 for “Berea sandstone” with supercritical CO_{2} saturation (Table B1 and Figure B1).
B = 0.018 for “Indiana limestone” with supercritical CO_{2} saturation (Table B2 and Figure B2).