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Laboratory measurements of the effective-stress law for carbonate rocks under deformation
Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol.30, No.7, pp. 1169-1172, 1993 Printed in Great Britain
I)148-9062,'93 $6.00 + 0.00 Pergamon Press Ltd
Laboratory Measurements of the Effective-Stress Law for Carbonate Rocks under Deformation N.R. WARPINSKI* L.W. TEUFEL*
INTRODUCTION The behavior of rocks under the combined effects of confining stress and pore pressure is an important issue for any in situ petroleum process. In order to simplify the difficulties in dealing with two independent parameters, it is customary to introduce an effective-stress law which relates a net, or effective, stress to some combination of confining stress and pore pressure. This law is usually given as Creff= or- ap,
(1)
where cr is the stress, p is the pore pressure, and a is a parameter or function that may depend upon both stress and pressure. Assuming that the effective-stress law is known, then material behavior can be measured at one pore pressure (often zero) and subsequently predicted for any other pore pressure. The major difficulty is obtaining an accurate model for the effective-stress law, since rocks are often heterogeneous, nonlinear and anisotropic. Robin [1] has demonstrated that the effective-stress law is dependent upon the particular property or process that is active (e.g., failure, deformation, permeability), so that different laws will be required for each. In this study, deformation of rocks is of concern, a subject that is often designated poroelasticity. Important developments in poroelasticity include works by Biot [2], Biot and Willis [3] and Rice and Cleary [4], to name a few. An effective-stress-law model for deformation has been developed by Nur and Byerlee [5], although this model requires an isotropic, homogeneous, single-component material with fully interconnected pore space to be valid. For this model, ~ is given as a:l-K/Ks
(2)
,
where K is the drained frame bulk modulus of the rock and K s is the unjacketed bulk modulus. Carroll [6] has developed a model for anisotropic materials, but the *Sandia National Laboratories, PO Albuquerque, New Mexico, USA 87185
Box
5800,
formulation is highly complex requiring several material constants. Berryman [7] has developed equation (2) for multi-component rocks. An approach often taken for deformation behavior is to assume the Nut and Byerlee model is correct and to determine both K and Ks in the laboratory, thus arriving at a calculated value of a. The primary uncertainty with this approach is an unknown accuracy and/or range of validity of this model for real rocks. This study presents some initial laboratory data that quantifies the error found in using the Nur and Byerlee model. Since it is also likely that poroelastic behavior will be a strong function of the rock internal structure, different types of rocks (e.g., sandstones compared to limestones) are likely to exhibit distinct effective-stress laws and any variability in the poroelastic rock behavior will be difficult to understand if disparate rock types are compared. To minimize this complication, the present study deals only with carbonate rocks (chalks and limestones only). EXPERIMENTAL PROCEDURE The experimental procedure 18.9] consists of measuring the volumetric-strain response of a rock subjected to a compression composed of varying hydrostatic confining-stress and pore-pressure levels. A 2.5 cm diameter core plug is instrumented with axial and circumferential strain gages, inserted in a neoprene jacket, and emplaced in an overburden coreholder that has been modified to allow feed-throughs for the straingage wires. Samples are loaded and unloaded several times ("seasoning") with hydrostatic confining stress alone until rock behavior is stabilized (i.e., repeatable results are achieved on successive cycles), and the effects of relaxation microcracks are diminished [8]. The samples are then saturated with nitrogen and measurements are made at constant pore pressure through both loading and unloading cycles in the confining stress. Pore pressures are then changed and data are again taken through loading and unloading cycles in the confining stress [8].
1169
1170
ROCK MECHANICS IN THE 1990s
The analysis procedure [8,91 uses a statistical technique whereby an empirical response surface of volumetric strain is mathematically generated [10]. To do this, the volumetric strains are (1) power-law transformed and (2) fit in a least-squares sense by a quadratic surface in both tr and p [8]. The resultant mathematical form of the response surface is given by 0~
:
x 2+xzcr+xsp+x4~
+ % o p + x6p 2 ,
(3)
where 0 is the volumetric strain and the xi are constant parameters to be fit. The power law employed (the value of 2, a constant) is determined through a maximumlikelihood approach [10]. Given this empirical function, of the form O(tr,p), Bernabe's formula for the tangent a [ 11] can be employed as
i!i ......
~O
re . ' r 0.1 .. ~o.o o>
0
",s .~
1o <~.w 10
20
30
40
50
60
STRESS (MPo) Fig. I. Exampleresponsecurve for 24% porosity Ekofisk chalk
in unloading. (4)
Additionally, standard statistical techniques can be used to determine confidence limits, regression coefficients, and other indicators of the quality of the fit and the variability of the data. The attractive feature of this technique is that there is no a priori need to assume a model for the rock behavior, as is the case when a mechanistic model is fit to the data. This avoids any possibility of biasing the fit. Rock types tested included samples from the Albuskjell and Ekofisk reservoirs in the North Sea, the Austin chalk in Texas, and the Saratoga limestone in Louisiana. Table I gives a list of relevant properties, including both nominal porosity and permeability values and values at 34.5 MPa confining stress and 13.8 MPa pore pressure which will be used later. RESULTS The initial result of the analysis is a response surface that shows the model fit relative to the data. An example of a response surface, for a 24% Ekofisk chalk sample, is shown in Figure 1. This rock shows some nonlinear behavior, yet the response surface provides an excellent representation of the data. Lines on the response surface are at constant pore pressure (along which the data were taken) and constant volumetric strain. Plots such as these are obtained for both loading and unloading cycles (there can be a difference in the two cycles) for all samples.
1,101 I .o51
,oo I o.gs
~,~
÷o
~ o.9ol
< o.ss] 0-801 o.Ts 1 o.7o/ o
i0
20
30
40
50
60
STRESS (IdPo)
Fig.2. Example a behavior for 24% porosity Ekofisk chalk in unloading. The response surface in Figure 1 is differentiated to yield the oc functionality shown in Figure 2. In this example, a has a value near 0.87 at low stress and pore pressure, increasing to near 1.0 at high stress levels. There is only a slight dependence on pore pressure (for this sample) and it is small enough to be statistically insignificant. The theoretical data, shown for zero pore pressure, were obtained by measuring unjacketed volumetric strain as a function of confining stress (Ks), determining K from the measured data of Figure 1, and applying equation (2). Ks was found to be highly linear, and the values are given in Table 1. The theoretical
Table 1 Rock sample information Sample Ekofisk Ekofisk Albuskjell Austin SaraU~a
Porosity (fraction) 0.36 0.24 0.15 0.04 0.11
Porosity at stress (fraction) 0.35 0.24 0.15 0.04 0.11
Permeability
Permeabitit~ at stress
1.OxlO'3
1.OxlO~3
43.0
1.lxlO"4 5.5x10"6
1, lxlO"4 5.4x10"6
55.0 63.0
Ks
(QP,,),
1.5xlO "8
-
64.8
5.0xlO"5
5.0xlO"5
63.4
ROCK MECHANICS IN THE 1990s
1171
Table 2 Response surface parameters Ekofisk (0.36) 4) .08136 0.04676 -0.04682 -4.399E-5 1.371E-4 -5.345E-5 2.6 27.6 27.6 +p
Parameters x1 x2 x3 x4 x5 x6 2 Maximum p (MPa) Maximum tr (MPa)
Ekofisk (0.24) -0.02162 0.009113 0.007861 4.619E-6 -4.005E-5 3.315E-5 1.35 27.6 55.2
value of a is accurate at low stress levels, but at high stress levels it is in error by 25%. The results would be similar for any other pore pressure. The same procedure was carried out for each of the rock samples in Table 1. The response-surface parameters (xi, 2,) are listed in Table 2, and they can be used to reproduce the functional form of a, for these test conditions. However, care must be taken not to extrapolate outside the data domain given in the table (the maximum p and or). Since the measurement of a is a time-consuming, difficult process, it would be helpful if results from a few carbonate samples could be used to estimate a for other carbonates. If the matrix rock material is the same for all samples (calcite, in this case), then it is reasonable to assume that the pore and grain structures are the primary influence on the effective-stress law. Such structures would likely represent themselves in both the porosity and the permeability of the material, and cross plots of a as a function of these parameters could show behavior trends. Figure 3 shows the measured value of a at a selected condition (34.5 MPa stress, 13.8 MPa pore pressure) as a function of porosity. The unloading data are presented here because they are considered to be the most accurate measure of a (as discussed later). The value given by equation (2) (theory) is also shown. The theoretical value is about 15-30% lower than the measured value. 1
SARATOGA
0.8
24%
0.6
38%
ALPHA 0.4
Albuskjell Chalk 0.00545 0007987 4).006882 -8.656E-6 8.181E-6 5.703E-6 1.0 27.6 55.2
Saratoga Limestone -0.01205 0.0047144 -0.004 184 1.776E-6 -1.616E-5 1.643E-5 1.22 27.6 55.2
Austin Chalk 4).00301 0.001862 4).0005416 8.686E-7 -1.802E-6 -2.004E-6 1.04 27.6 55.2
Figure 4 shows the measured and theoretical values of a as a function of permeability. Although this plot is a log scale, the measured data show a more consistent monotonic relationship between a and permeability than between ~ and porosity (Figure 3). The results in Figures 3 and 4 are only valid for one confining-stress and porepressure condition and there is some variability of a with cr and p. However, the results shown here are representative of the general behavior throughout the data domain studied in these experiments.
0.8
0.6 ALPHA 0.4
ALBUSKJELL
0.2
0 0.01
AUSTIN
ALPHA AT 34.5 MPa CONFINING
CHALK
AND 13.8 MPa PORE PRESSURES
........
I
0.1
, ,,,l.I
,
, , ..... t
1
10
........
T
100
........
1,000
PERMEABILITY (10.6 micrometers 2 )
Fig. 4. a as a function of permeability. The experimental procedure and statistical analysis used here can provide accurate data on a. The 95% confidence limits on a, based on the least-squares fit of the response surface for each sample, are _+0.02 for both Ekofisk chalks and the Albuskjell chalk, _+0.03 for the Austin chalk, and _+0.06 for the Saratoga limestone over the range of experimental conditions. DISCUSSION AND CONCLUSIONS
Ip
0.2 ALPHA AT 34.5 MPa CONFINING
AUSTIN CHALK
0 0
J
f
0.05
0.1
AND 13.8 MPa PORE PRESSURES I J I I
0.15
0.2
0.25
POROSITY (fraction)
Fig. 3. a as a function of porosity.
0.3
I
0.35
0.4
These results show that the value of a derived from equation (2) can be considerably in error. Measured values of a are 0.1-0.2 greater than the predicted value of equation (2) for all samples tested, giving a 15-30% discrepancy. For three of the five rocks, the simple assumption of a=l would provide a better estimate than equation (2).
~7.!
ROCK MECHANICS IN THE 1990s
The error in the theory is not surprising, considering the severe assumptions that were required. These assumptions include linear, homogeneous and isotropic behavior, and fully interconnected pore space. Nevertheless, for the carbonate rocks studied here, many of these assumptions are acceptable. For example, these rocks are primarily a single component, many are homogeneous, some exhibit very linear behavior, and many have nearly fully interconnected pore spaces, as measured by the unjacketed bulk modulus relative to pure calcite's bulk modulus. The primary assumption that cannot be met in most of these rocks is isotropy, and this is likely to be the primary source of model error. Carroll [6] provides a theoretical derivation for an anisotropic material, although the equations are quite complex, even for the simplest case of transverse anisotropy. Such a model should be considered in the future. The unloading data provide the best estimate of a because relaxation microcracks [9] are least active in unloading. During loading conditions, relaxation microcracks that are already open require considerably greater pressures to reclose. In this study, the values of a in loading are slightly lower than the unloading values. This behavior suggests that the in situ value of a may be somewhat greater than these lab measurements show, as decreasing microcrack activity appears to increase a. It is impossible to entirely eliminate the effects of relaxation microcracks in lab tests, but they can be minimized by seasoning the sample and using unloading data. While this study includes only five different rock types and no universal relationship can be established, there does appear to be a trend of decreasing a with decreasing permeability and porosity. This is not unexpected since decreasing permeability and porosity indicate an increasing volume of grain material relative to pore space, which will increase K in Equation (2). Finally, the importance of a correct formulation of the effective-stress law is best demonstrated by noting some applications. Any calculations involving a deformation of the in situ rock mass require some formulation of the effective-stress law. Example applications include drilling, in situ stress measurements, weUbore stability, sand control, reservoir compaction and subsidence, hydraulic fracturing, and any production influenced by reservoir geomechanics.
Acknowledgment- The authors would like to acknowledge the assistance of Phillips Petroleum Co. and Oryx Energy Co.' in providing both core and technical information. Interactions with both of these companies have been invaluable. The Project was conducted under funds received from the US Department of Energy's oil research program through the Bartlesville Project Office under contract DE-ACIM-76DP000789. Special thanks go to our project managers, Fred Burtch and Bob Lemmon, Results from the Oryx core were conducted as part of a project under DOE's Oil Recovery Technology Parmership.
REFERENCES 1. Robin, P-Y. F. Note on effective pressure. J. Geophys. Res. 78, 2434-2437 (1973). 2. Biot, M. A. General theory of three-dimensional consolidation. J. of Appl. Phys, 12, 155-164 (1941). 3. Bier, M. A. and Willis, D. G. The elastic coefficients of the theory of consolidation. J. of Appi. Mech. 24, 594-601 (1957). 4. Rice, J. R. and Cleary, M. P. Some basic stress diffusion solutions for fluid-saturated elastic porous media with compressible constituents. Rev. Geophys. and Space Phys. 14, 227-241 (1976). 5. Nur, A. and Byerlee, J. D. An effective stress law for elastic deformation of rocks with fluids. J. Geophys. Res. 76, 6414-6419 (1971). 6. Carroll, M. M. An effective stress law for anisotropie elastic deformation. J. Geophys. Res. 84, 7510-7512 (1979). 7. Berryman, J.G. Effectivestress for transport properties of inhomogeneous porous rock. J. Geophys. Res.97, 1740% 17424 (1992). 8. Warpinski,, N.R. and Teufel, L.W. Determination of the effective-stress law for permeability and deformation in low-permeability rocks. SPE Formation Evaluation. 7, 123131 (1992). 9. Teufel, L.W. and Warpinski, N.R: Laboratory determination of effective-stress laws for deformation and permeability of chalk. Third North Sea Chalk Syrup. Copenhagen, Denmark (1990). 10. Box, G.E.P. and Draper, N.R. Empirical Model Building and Response Surfacesl John Wiley & Sons. N.Y. (1987) 11. Bernabe, Y. The effective pressure law for permeability in Chelmsford granite and Barre granite. Int. J. Rock Mech.. Min. Sci. & Geomech. Abstr. 23, 267-275 (1986).
I)148-9062,'93 $6.00 + 0.00 Pergamon Press Ltd
Laboratory Measurements of the Effective-Stress Law for Carbonate Rocks under Deformation N.R. WARPINSKI* L.W. TEUFEL*
INTRODUCTION The behavior of rocks under the combined effects of confining stress and pore pressure is an important issue for any in situ petroleum process. In order to simplify the difficulties in dealing with two independent parameters, it is customary to introduce an effective-stress law which relates a net, or effective, stress to some combination of confining stress and pore pressure. This law is usually given as Creff= or- ap,
(1)
where cr is the stress, p is the pore pressure, and a is a parameter or function that may depend upon both stress and pressure. Assuming that the effective-stress law is known, then material behavior can be measured at one pore pressure (often zero) and subsequently predicted for any other pore pressure. The major difficulty is obtaining an accurate model for the effective-stress law, since rocks are often heterogeneous, nonlinear and anisotropic. Robin [1] has demonstrated that the effective-stress law is dependent upon the particular property or process that is active (e.g., failure, deformation, permeability), so that different laws will be required for each. In this study, deformation of rocks is of concern, a subject that is often designated poroelasticity. Important developments in poroelasticity include works by Biot [2], Biot and Willis [3] and Rice and Cleary [4], to name a few. An effective-stress-law model for deformation has been developed by Nur and Byerlee [5], although this model requires an isotropic, homogeneous, single-component material with fully interconnected pore space to be valid. For this model, ~ is given as a:l-K/Ks
(2)
,
where K is the drained frame bulk modulus of the rock and K s is the unjacketed bulk modulus. Carroll [6] has developed a model for anisotropic materials, but the *Sandia National Laboratories, PO Albuquerque, New Mexico, USA 87185
Box
5800,
formulation is highly complex requiring several material constants. Berryman [7] has developed equation (2) for multi-component rocks. An approach often taken for deformation behavior is to assume the Nut and Byerlee model is correct and to determine both K and Ks in the laboratory, thus arriving at a calculated value of a. The primary uncertainty with this approach is an unknown accuracy and/or range of validity of this model for real rocks. This study presents some initial laboratory data that quantifies the error found in using the Nur and Byerlee model. Since it is also likely that poroelastic behavior will be a strong function of the rock internal structure, different types of rocks (e.g., sandstones compared to limestones) are likely to exhibit distinct effective-stress laws and any variability in the poroelastic rock behavior will be difficult to understand if disparate rock types are compared. To minimize this complication, the present study deals only with carbonate rocks (chalks and limestones only). EXPERIMENTAL PROCEDURE The experimental procedure 18.9] consists of measuring the volumetric-strain response of a rock subjected to a compression composed of varying hydrostatic confining-stress and pore-pressure levels. A 2.5 cm diameter core plug is instrumented with axial and circumferential strain gages, inserted in a neoprene jacket, and emplaced in an overburden coreholder that has been modified to allow feed-throughs for the straingage wires. Samples are loaded and unloaded several times ("seasoning") with hydrostatic confining stress alone until rock behavior is stabilized (i.e., repeatable results are achieved on successive cycles), and the effects of relaxation microcracks are diminished [8]. The samples are then saturated with nitrogen and measurements are made at constant pore pressure through both loading and unloading cycles in the confining stress. Pore pressures are then changed and data are again taken through loading and unloading cycles in the confining stress [8].
1169
1170
ROCK MECHANICS IN THE 1990s
The analysis procedure [8,91 uses a statistical technique whereby an empirical response surface of volumetric strain is mathematically generated [10]. To do this, the volumetric strains are (1) power-law transformed and (2) fit in a least-squares sense by a quadratic surface in both tr and p [8]. The resultant mathematical form of the response surface is given by 0~
:
x 2+xzcr+xsp+x4~
+ % o p + x6p 2 ,
(3)
where 0 is the volumetric strain and the xi are constant parameters to be fit. The power law employed (the value of 2, a constant) is determined through a maximumlikelihood approach [10]. Given this empirical function, of the form O(tr,p), Bernabe's formula for the tangent a [ 11] can be employed as
i!i ......
~O
re . ' r 0.1 .. ~o.o o>
0
",s .~
1o <~.w 10
20
30
40
50
60
STRESS (MPo) Fig. I. Exampleresponsecurve for 24% porosity Ekofisk chalk
in unloading. (4)
Additionally, standard statistical techniques can be used to determine confidence limits, regression coefficients, and other indicators of the quality of the fit and the variability of the data. The attractive feature of this technique is that there is no a priori need to assume a model for the rock behavior, as is the case when a mechanistic model is fit to the data. This avoids any possibility of biasing the fit. Rock types tested included samples from the Albuskjell and Ekofisk reservoirs in the North Sea, the Austin chalk in Texas, and the Saratoga limestone in Louisiana. Table I gives a list of relevant properties, including both nominal porosity and permeability values and values at 34.5 MPa confining stress and 13.8 MPa pore pressure which will be used later. RESULTS The initial result of the analysis is a response surface that shows the model fit relative to the data. An example of a response surface, for a 24% Ekofisk chalk sample, is shown in Figure 1. This rock shows some nonlinear behavior, yet the response surface provides an excellent representation of the data. Lines on the response surface are at constant pore pressure (along which the data were taken) and constant volumetric strain. Plots such as these are obtained for both loading and unloading cycles (there can be a difference in the two cycles) for all samples.
1,101 I .o51
,oo I o.gs
~,~
÷o
~ o.9ol
< o.ss] 0-801 o.Ts 1 o.7o/ o
i0
20
30
40
50
60
STRESS (IdPo)
Fig.2. Example a behavior for 24% porosity Ekofisk chalk in unloading. The response surface in Figure 1 is differentiated to yield the oc functionality shown in Figure 2. In this example, a has a value near 0.87 at low stress and pore pressure, increasing to near 1.0 at high stress levels. There is only a slight dependence on pore pressure (for this sample) and it is small enough to be statistically insignificant. The theoretical data, shown for zero pore pressure, were obtained by measuring unjacketed volumetric strain as a function of confining stress (Ks), determining K from the measured data of Figure 1, and applying equation (2). Ks was found to be highly linear, and the values are given in Table 1. The theoretical
Table 1 Rock sample information Sample Ekofisk Ekofisk Albuskjell Austin SaraU~a
Porosity (fraction) 0.36 0.24 0.15 0.04 0.11
Porosity at stress (fraction) 0.35 0.24 0.15 0.04 0.11
Permeability
Permeabitit~ at stress
1.OxlO'3
1.OxlO~3
43.0
1.lxlO"4 5.5x10"6
1, lxlO"4 5.4x10"6
55.0 63.0
Ks
(QP,,),
1.5xlO "8
-
64.8
5.0xlO"5
5.0xlO"5
63.4
ROCK MECHANICS IN THE 1990s
1171
Table 2 Response surface parameters Ekofisk (0.36) 4) .08136 0.04676 -0.04682 -4.399E-5 1.371E-4 -5.345E-5 2.6 27.6 27.6 +p
Parameters x1 x2 x3 x4 x5 x6 2 Maximum p (MPa) Maximum tr (MPa)
Ekofisk (0.24) -0.02162 0.009113 0.007861 4.619E-6 -4.005E-5 3.315E-5 1.35 27.6 55.2
value of a is accurate at low stress levels, but at high stress levels it is in error by 25%. The results would be similar for any other pore pressure. The same procedure was carried out for each of the rock samples in Table 1. The response-surface parameters (xi, 2,) are listed in Table 2, and they can be used to reproduce the functional form of a, for these test conditions. However, care must be taken not to extrapolate outside the data domain given in the table (the maximum p and or). Since the measurement of a is a time-consuming, difficult process, it would be helpful if results from a few carbonate samples could be used to estimate a for other carbonates. If the matrix rock material is the same for all samples (calcite, in this case), then it is reasonable to assume that the pore and grain structures are the primary influence on the effective-stress law. Such structures would likely represent themselves in both the porosity and the permeability of the material, and cross plots of a as a function of these parameters could show behavior trends. Figure 3 shows the measured value of a at a selected condition (34.5 MPa stress, 13.8 MPa pore pressure) as a function of porosity. The unloading data are presented here because they are considered to be the most accurate measure of a (as discussed later). The value given by equation (2) (theory) is also shown. The theoretical value is about 15-30% lower than the measured value. 1
SARATOGA
0.8
24%
0.6
38%
ALPHA 0.4
Albuskjell Chalk 0.00545 0007987 4).006882 -8.656E-6 8.181E-6 5.703E-6 1.0 27.6 55.2
Saratoga Limestone -0.01205 0.0047144 -0.004 184 1.776E-6 -1.616E-5 1.643E-5 1.22 27.6 55.2
Austin Chalk 4).00301 0.001862 4).0005416 8.686E-7 -1.802E-6 -2.004E-6 1.04 27.6 55.2
Figure 4 shows the measured and theoretical values of a as a function of permeability. Although this plot is a log scale, the measured data show a more consistent monotonic relationship between a and permeability than between ~ and porosity (Figure 3). The results in Figures 3 and 4 are only valid for one confining-stress and porepressure condition and there is some variability of a with cr and p. However, the results shown here are representative of the general behavior throughout the data domain studied in these experiments.
0.8
0.6 ALPHA 0.4
ALBUSKJELL
0.2
0 0.01
AUSTIN
ALPHA AT 34.5 MPa CONFINING
CHALK
AND 13.8 MPa PORE PRESSURES
........
I
0.1
, ,,,l.I
,
, , ..... t
1
10
........
T
100
........
1,000
PERMEABILITY (10.6 micrometers 2 )
Fig. 4. a as a function of permeability. The experimental procedure and statistical analysis used here can provide accurate data on a. The 95% confidence limits on a, based on the least-squares fit of the response surface for each sample, are _+0.02 for both Ekofisk chalks and the Albuskjell chalk, _+0.03 for the Austin chalk, and _+0.06 for the Saratoga limestone over the range of experimental conditions. DISCUSSION AND CONCLUSIONS
Ip
0.2 ALPHA AT 34.5 MPa CONFINING
AUSTIN CHALK
0 0
J
f
0.05
0.1
AND 13.8 MPa PORE PRESSURES I J I I
0.15
0.2
0.25
POROSITY (fraction)
Fig. 3. a as a function of porosity.
0.3
I
0.35
0.4
These results show that the value of a derived from equation (2) can be considerably in error. Measured values of a are 0.1-0.2 greater than the predicted value of equation (2) for all samples tested, giving a 15-30% discrepancy. For three of the five rocks, the simple assumption of a=l would provide a better estimate than equation (2).
~7.!
ROCK MECHANICS IN THE 1990s
The error in the theory is not surprising, considering the severe assumptions that were required. These assumptions include linear, homogeneous and isotropic behavior, and fully interconnected pore space. Nevertheless, for the carbonate rocks studied here, many of these assumptions are acceptable. For example, these rocks are primarily a single component, many are homogeneous, some exhibit very linear behavior, and many have nearly fully interconnected pore spaces, as measured by the unjacketed bulk modulus relative to pure calcite's bulk modulus. The primary assumption that cannot be met in most of these rocks is isotropy, and this is likely to be the primary source of model error. Carroll [6] provides a theoretical derivation for an anisotropic material, although the equations are quite complex, even for the simplest case of transverse anisotropy. Such a model should be considered in the future. The unloading data provide the best estimate of a because relaxation microcracks [9] are least active in unloading. During loading conditions, relaxation microcracks that are already open require considerably greater pressures to reclose. In this study, the values of a in loading are slightly lower than the unloading values. This behavior suggests that the in situ value of a may be somewhat greater than these lab measurements show, as decreasing microcrack activity appears to increase a. It is impossible to entirely eliminate the effects of relaxation microcracks in lab tests, but they can be minimized by seasoning the sample and using unloading data. While this study includes only five different rock types and no universal relationship can be established, there does appear to be a trend of decreasing a with decreasing permeability and porosity. This is not unexpected since decreasing permeability and porosity indicate an increasing volume of grain material relative to pore space, which will increase K in Equation (2). Finally, the importance of a correct formulation of the effective-stress law is best demonstrated by noting some applications. Any calculations involving a deformation of the in situ rock mass require some formulation of the effective-stress law. Example applications include drilling, in situ stress measurements, weUbore stability, sand control, reservoir compaction and subsidence, hydraulic fracturing, and any production influenced by reservoir geomechanics.
Acknowledgment- The authors would like to acknowledge the assistance of Phillips Petroleum Co. and Oryx Energy Co.' in providing both core and technical information. Interactions with both of these companies have been invaluable. The Project was conducted under funds received from the US Department of Energy's oil research program through the Bartlesville Project Office under contract DE-ACIM-76DP000789. Special thanks go to our project managers, Fred Burtch and Bob Lemmon, Results from the Oryx core were conducted as part of a project under DOE's Oil Recovery Technology Parmership.
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