TY - JOUR
AU - Moosavi,, M
AB - Abstract For the characterization of the mechanical behavior of porous media in elastic conditions, the theory of poroelasticity is used. The number of poroelastic coefficients is greater in elastic conditions because of the complexity of porous media. The laboratory measurement of poroelastic coefficients needs a system that can control and measure the variables of poroelasticity. In this paper, experimental measurements of these coefficients are presented. Laboratory tests are performed using a system designed by the authors. Laboratory hydrostatic tests are performed on cylindrical samples in drained, pore pressure loading, undrained and dry conditions. Compressibilities (bulk and pore compressibility), effective stress and Skempton coefficients are measured by these tests. Samples are made of a composition (sand and cement) and are made by a compaction process synthetically. Calibration tests are performed for the setup to identify possible errors in the system and to correct the results of the main tests. This is done by performing similar compressibility tests at each stress level on a cylindrical steel sample (5.47 mm in diameter) with a longitudinal hole along it (hollow cylinder). A steel sample is used to assume an incompressible sample. The results of the tests are compared with the theory of poroelasticity and the obtained graphs and their errors are analyzed. This study shows that the results of the drained and pore pressure loading tests are compatible with poroelastic formulation, while the undrained results have errors because of extra fluid volume in the pore pressure system and calibration difficulties. poroelasticity, experimental measurement, synthetic sandstone, drained, pore pressure loading, undrained, compressibility, effective stress coefficient, Skempton or build up pressure coefficient 1. Introduction Poroelasticity theory accounts reasonably well for a small deformation of fluid-saturated porous solids (Gueguen et al2004). The mechanical behavior of porous rocks is different from that of continuum media due to pores, their geometry, interstitial fluids and pore pressure conditions. The relation between stress and pore pressure rather than strain and fluid conditions in the medium are the equations of poroelasticity (Wang 2000, Detournay and Cheng 1993, Jaeger et al2007, Gueguen et al2004, Zimmerman 1991, Hart and Wang 1995). Some of the poroelastic coefficients are the compressibilities, effective stress coefficients, pore pressure buildup and Skempton coefficient. (Wang 2000, Jaeger et al2007, Zimmerman 1991, Detournay and Cheng 1993, Hart and Wang 1995). These coefficients have a critical role in the study of mechanical behavior of reservoir rock and in petroleum engineering issues such as reservoir subsidence and reservoir production calculations. Furthermore, these coefficients are important in the analysis of the poromechanical behavior of porous materials (Coussy 2004, Gueguen et al2004). Thus a poroelastic measurement system is necessary for measuring these coefficients and analyzing the mechanical behavior of reservoir rocks. In fact, by measuring poroelastic variables such as stress, strain, pore pressure and the increment of fluid content, and using the constitutive relations among them, the poroelastic coefficients are obtainable. This system could reduce many ambiguities and problems in petroleum-related rock mechanics (Zimmerman 1991). Many rock poroelastic measurement systems have been introduced in the literature, such as Fatt (1958), Greenwald (1980), Zimmerman (1984), Hart (2000), Lockner and Stanchits (2002), Sulem and Ouffroukh (2006), Jalalh (2006) and Andersen (1988) from the 1950s until the present day. Systems have been designed differently; some of them, such as Fatt (1958), Greenwald (1980) and Zimmerman (1991), could measure coefficients only in hydrostatic terms, and some, such as Andersen (1988) and Hart (2000), could apply other non-hydrostatic stress fields such as the uniaxial strain condition that is the predominant stress field in reservoirs (Andersen 1988). Also some systems were designed for drained, pore pressure loading tests (Zimmerman 1984, Andersen 1988, Greenwald 1980) or the undrained measurement of coefficients (Lockner and Stanchits 2002, Hart 2000). In this study, tests are done by a rock poroelastic measurement system which was designed by the authors. The system is able to be applied in drained, pore pressure loading and undrained conditions. The tests are hydrostatic; however the system could work in triaxial stress fields. Synthetic samples are made to have uniform properties and this enables performance of destructive tests. So, it is possible to control the properties of rock, and we can even make an extreme condition that hardly exists in nature. For the controlled properties, it becomes easier to understand the behavior of each property in nature. In addition, natural cores are very limited in number and are not numerous enough to perform a good number of experimental tests. The composition and production process of synthetic sandstone are determined by mineralogy and geological studies (David et al1998, Heath 1965, Saidi et al2005). In our research, a number of sandstone samples are made by the compaction and cementation process. In this paper, the theory of poroelasticity and its mathematic formulations are first briefly explained. In the next part, the production process of synthetic samples is presented and the measurement system and its components are illustrated. Following this, hydrostatic tests in drained, pore pressure loading, undrained and dry conditions are explained completely. Finally, the results and graphs obtained using the poroelastic formulations are discussed and interpreted. The main goal of this paper is to perform a series of hydrostatic tests that can be done by the designed system to compare the results of the tests to the ones predicted by the theory of poroelasticity and to discuss the possible sources of error for each test. A procedure has also been discussed to make synthetic sandstone samples easily. The results of drained and pore pressure loading tests are compatible with each other and poroelastic theory, while undrained results have errors because of extra fluid volume in the pore pressure system. The undrained coefficient could be obtainable by the poroelastic formulation helping drained and pore pressure loading tests indirectly. So it is possible to calculate the error of the system for each coefficient. 2. Review of poroelasticity Poroelasticity is about the elasticity of a fluid-infiltrated porous solid; a body composed of two phases (solid and fluid). In other word, it is the study of the elastic deformation mechanism considering pore fluid conditions (Cui 1995). This theory was introduced by Biot (1941), an American–Belgian engineer, assuming compressible components, and saturated and continuous solid and fluid phases, so that the porous medium is composed of a solid skeleton that is considered as a permeable phase including saturated pores (Detournay and Cheng 1993). Porosity is defined as the ratio of the volume of the pore space (Vp) to the total volume (Vb) in the actual (deformed) state (Ghabezloo and Sulem 2008). The total volume is sum of porous space (Vp) and solid volume (Vs), as in (2): φ=VpVb1 Vb=Vp+Vs.2 In poroelasticity, two scalar variables are added to the variables of classic elasticity: the increment of fluid content (ζ) and pore pressure (p). All the poroelastic variables are defined as their mean on the represented element volume (REV) (Jaeger et al2007, Detournay and Cheng 1993, Paterson and Wong 2005). Changes in fluid contents are caused by two events: first, changes in pore volume due to stress and pore pressure variations, and second, fluid volume changes due to its compressibility. So the expression of this variable is obtained by (3), where Kf is the fluid bulk modulus (Detournay and Cheng 1993): ζ=φΔVpVp+pKf.3 This variable is unitless, and negative values represent the exit of fluid from rock or REV (Wang 2000). Considering it in the principal and isotropic stress field, two main linear equations are obtained (Wang 2000): ɛb=-1KPc+αKp.4 Parameter εb represents the volumetric strain and Pc is the confining stress. The strain in expansion and stress in compression are positive. Stress and pore pressure are independent variables while strain and the increment of fluid content are dependents. These relationships can be written in three other modes, based on which variable is independent or dependent (Wang 2000): ɛb=-PcKu+Bζ5 Pc=-Kɛb+αp6 Pc=-Kuɛb+(KuB)ζ.7 Using (4), (5), (6) and (7) with restrictions to one of the independent variable, the poroelastic coefficients are defined as below (Wang 2000): 1K=-δɛbδPcδp=08 1Ku=-δɛbδPcδζ=09 α=δζδɛbδp=010 B=δpδPcδζ=011 νu=3ν+αB(1-2ν)3-αB(1-2ν).12 The parameter 1/K is drained bulk compressibility, 1/Ku is undrained bulk compressibility, α is the effective stress coefficient, B is the buildup pore pressure or Skempton coefficient and ν, νu are the drained and undrained Poisson ratios, respectively. Compressibility is one of the mechanical properties of rock that represents the volume changes versus stress changes. In porous media, the term ‘volume’ could mean bulk volume (Vb), pore volume (Vp) or solid volume (Vs), and the term ‘stress’ could mean confining pressure (Pc), pore pressure (P) or differential pressure (PcP = Pd). So there are some compressibilities for porous rocks. Equation (13) defines four rock compressibilities that are shown as below (Wang 2000): ɛb=δVbVb=-1KPd-1Ks′p.13 The parameter 1/Kp is the drained pore compressibility, which is defined as (Wang 2000) 1Kp=δVpVp1dPddp=0.14 Assuming a homogenous and isotropic solid phase, the parameters 1/Ks′, 1/Kφ and 1/Ks are equal to the compressibility of the solid phase (Cr = 1/Ks) (Zimmerman 1991, Wang 2000): Cr=1Ks=1Ks′=1Kφ.15 In this equation, 1/Ks′ and 1/Kφ are the unjacketed bulk and unjacketed pore compressibilities. The solid compressibility according to poroelastic theory is expressed by the equation below. This equation is obtained, while solid volume changes and pore relative changes are considered as dependent variables in (13) (Detournay and Cheng 1993): Cr=-δVsVs(1-φ)/dPc.16 Using (13) and the effective stress concept, the bulk and pore effective stress coefficients are obtained as below (Wang 2000, Detournay and Cheng 1993): δVbVb=-1K(Pc-αp)17 δVpVp=-1KpPc-βp.18 Based on the theory of poroelasticity, the below relation is established (Wang 2000): φ≤α≤β≤1.19 If in (13) the differential pressure (Pd) is replaced with the confining pressure (Pc) the Zimmerman equations are obtained (Zimmerman et al1986, Zimmerman 1991): ɛb=δVbVb=-CbcPc+Cbpp.20 Every coefficient has two subscripts. The first subscript represents the volume: a subscript b shows the bulk and a subscript p shows the pore volume; and the second subscript represents the stress: a subscript c shows the confining stress and a subscript p shows the pore pressure. So Cbc is compressibility related to bulk volume changes due to confining pressure changes in constant pore pressure (Jaeger et al2007, Zimmerman 1991). Using (13), (20) and the definition of differential pressure, Cbc = 1/K, Cpc = 1/Kp and also Cbp and Cpp are expressed as below (Zimmerman 1991, Wang 2000): Cbp=δVbVb1dpdPc=021 Cpp=δVpVp1dpdPc=0.22 Using (13), (20) and Betti's reciprocal theorem of elasticity, the follow relations result (Zimmerman 1991): Cr=1Ks=Cbc-Cbp23 Cr=1Ks=Cpc-Cpp24 Cbp=φCpc.25 The compressibility factors in undrained condition are obtained as below, using (26), which represents the undrained condition (Jaeger et al2007): δVpVp=δVfVf=Cfdp26 Cbu=Cbc-CbpCpcCpp+Cf27 Cpu=Cpc1+(Cpp/Cf).28 Buildup pore pressure or the Skempton coefficient according to poroelasticity is expressed in (29) (Jaeger et al2007): B=Cpp+CrCpp+Cf.29 3. Sample preparation The mineral compositions of synthetic sandstone samples and their bulk moduli are shown in table 1. The size of grains is in range of 0.2–0.5 mm. Table 1. Mineral composition of synthetic sandstones. Minerals . Volume percentage (%) . Bulk modulus (10E4 MPa) . SiO2 97.60 3.707 Al2O3  0.48 6.305 Fe2O3  0.12 9.814 CaO  0.18 6.726 MgO  0.08 8.736 Na2O  0.63 6.305 K2O  0.14 6.305 Minerals . Volume percentage (%) . Bulk modulus (10E4 MPa) . SiO2 97.60 3.707 Al2O3  0.48 6.305 Fe2O3  0.12 9.814 CaO  0.18 6.726 MgO  0.08 8.736 Na2O  0.63 6.305 K2O  0.14 6.305 Open in new tab Table 1. Mineral composition of synthetic sandstones. Minerals . Volume percentage (%) . Bulk modulus (10E4 MPa) . SiO2 97.60 3.707 Al2O3  0.48 6.305 Fe2O3  0.12 9.814 CaO  0.18 6.726 MgO  0.08 8.736 Na2O  0.63 6.305 K2O  0.14 6.305 Minerals . Volume percentage (%) . Bulk modulus (10E4 MPa) . SiO2 97.60 3.707 Al2O3  0.48 6.305 Fe2O3  0.12 9.814 CaO  0.18 6.726 MgO  0.08 8.736 Na2O  0.63 6.305 K2O  0.14 6.305 Open in new tab This composition is combined with 35% cement and some water, and then it is poured into polyethylene cylindrical casts which are 55 mm in diameter (NX standard size) and have heights that are 2–3 times longer. The composition is poured into the cast stage by stage. At each stage, the composition is compacted under 5.2 MPa stress for 10 s. The process continues until the cast is filled, as is shown in figure 1(a). The percentage of cement, water, the amount of stress and the loading duration time are determined by trial and error to obtain a synthetic sample with porosity and strength in a range of sandstone reservoir rocks. After 24 h of filling the casts, the samples are separated from the casts and are kept in water at 80 °C for 24 h. It should be mentioned that the cement hydration at high temperatures is done quickly. After this step, the samples and the water container are taken out of the oven and are kept at room temperature for 24 h. In the next step, the saturated weight is measured and then samples are put in the oven at 70–80 °C to dry, for 24 h (figure 1(b)). After this step, the samples are pulled out and their dry weight and bulk volume are measured. So it is possible to calculate the samples' porosity, which is in the range of 22.5–23.5%. Synthetic sandstone samples are shown in figure 1(c). One of these samples is tested by uniaxial compression to obtain some elastic coefficients and uniaxial strength (figure 1(d)). The results of this test are given in table 2. Figure 1. Open in new tabDownload slide (a) Filled casts of synthetic sandstones; (b) the drying of synthetic samples in the oven; (c) synthetic samples; and (d) uniaxial strength test by the MTS system. Figure 1. Open in new tabDownload slide (a) Filled casts of synthetic sandstones; (b) the drying of synthetic samples in the oven; (c) synthetic samples; and (d) uniaxial strength test by the MTS system. Table 2. Results of uniaxial strength test. Sample no . Young modulus (GPa) . Poisson ratio . Uniaxial strength (MPa) . Bulk compressibility (MPa-1) . 1-A 17.11 0.09 21.36 1.417 × 10-4 Sample no . Young modulus (GPa) . Poisson ratio . Uniaxial strength (MPa) . Bulk compressibility (MPa-1) . 1-A 17.11 0.09 21.36 1.417 × 10-4 Open in new tab Table 2. Results of uniaxial strength test. Sample no . Young modulus (GPa) . Poisson ratio . Uniaxial strength (MPa) . Bulk compressibility (MPa-1) . 1-A 17.11 0.09 21.36 1.417 × 10-4 Sample no . Young modulus (GPa) . Poisson ratio . Uniaxial strength (MPa) . Bulk compressibility (MPa-1) . 1-A 17.11 0.09 21.36 1.417 × 10-4 Open in new tab It should be mentioned that the samples must be prepared before every test. The end of samples should be smooth, and their height should be same for simplicity. It is possible to calculate the solid phase compressibility of the samples using the well-known Voigt–Reuss–Hill method. However, the solid phase compressibility can be tested experimentally through the so-called unjacketed test. In this test, the rock is pressurized by a fluid which is allowed to seep into its pores, so that the pore and confining pressures are equal, with the assumption that the mineral phase effectively behaves like an isotropic elastic medium. In this paper, the solid phase compressibility is obtained by the Voigt–Reuss–Hill method because it is fairly easy to use when the mineralogical composition of the rock is known (Zimmerman 1991). In the Voigt–Reuss–Hill method, the compressibilities (Ci) and volume fractions (Xi) of the minerals' compositions are used (Zimmerman 1991). The volume fraction of mineral composition in table 1 should be corrected because of the 35% cement which is added. The bulk modulus of cement is 1.52E4 MPa (Haecker et al2005). So the solid phase compressibility based on the Reuss equation is C¯ Reuss =-δɛδP=∑χiCi=4.003E-5 MPa -1.30 This coefficient is calculated by the Voigt method, as follows: C¯ Voigt =-δɛδP=∑(χi/Ci)-1=3.379E-5 MPa -1.31 And the solid phase compressibility is obtained by averaging the two above values using the Voigt–Reuss–Hill method: C¯ Voigt - Reuss - Hill =C¯ Reuss +C¯ Voigt 2=3.684E-5 MPa -1.32 As mentioned above, this coefficient is obtainable by tests, and so it is possible to check the laboratory data with this empirical method. This comparison will be shown and discussed in the next sections. 4. Poroelastic measurement system The designed rock poroelastic measurement system is manual and can record data in automatic and continuous conditions. This system is able to measure the coefficients in hydrostatic, non-hydrostatic, drained, pore pressure loading, undrained and unjacketed conditions. The system is comprised of three main parts—the main body (A), the confining and pore systems (B)—and others instruments of connectors (C), gauges (E) and pumps (D). To clarify the description, each part is named by capital letters in the text and figures. As shown in figure 2(a), the main body (A) is a Hoek cell (54.7 mm in diameter), and its capacity is 70 MPa. This part is the main part of system and is placed in a loading frame. The sample is located in the cell and then confining stresses and pore pressure are applied by the confining and pore systems (B). In hydrostatic mode, confining and axial stress are applied simultaneously and equally by the confining system. The different components of the main body are illustrated in figure 2(a). The confining and pore systems (B) are similar, and consist of two jacks with 1/144 ratios of cross sections. These parts and their components are shown in figure 2(b). The role of the confining and pore systems is the measurement and control of the stresses and strains to high precision. The main part is connected to the confining and pore systems by some tubes and connectors (C). In figure 3, the hydrostatic mode of the system and its circuit is illustrated. All the connectors, valves, tubes and couplings (C) are the high-pressure capacity type (up to 700 bars or 10 000 Psi), and they are designed in such a way as not to leak at high pressure. It is obviously very important in poroelastic tests that volume change is measured accurately, as any leakage may cause a considerable source of error in the measuring pressure data. Loading tools (D) such as pumps and jacks are also the high-pressure capacity type and they apply primary stresses to the rock. After applying the primary stresses, these tools are disconnected at the connectors (C), and the control and measurement of variables is done by the confining and pore systems (B). Recordings of stresses and displacements are made by pressure and displacement gauges (E), which are installed on main body (A) and confining and pore pressure systems (B). The accuracy of the pressure gauges is 1 MPa. The strains are calculated from the injected fluid volume by confining and pore pressure systems (B), and the recorded displacement is calculated by gauges (E). These data are calibrated by calibration data and then pore and bulk strains are obtained. The cylinder diameters of the confining and pore systems are 10 mm and the precision of displacement gauges are 0.01 mm, so the accuracy of the system in strain measurement is calculated as follows: Accuracy =πD22×0.01=0.785 mm 3=7.85×10-4 cc .33 Figure 2. Open in new tabDownload slide Main body in hydrostatic mode (a), and confining and pore systems (b). Main body (A): (A1) body, (A2) sleeve, (A3) upper cap, (A4) piston cap, (A5) hydrostatic loading cap, (A6) lower cap, and (A7) hydrostatic cap. Confining and pore systems (B): (B1) small jack's cylinder, (B2) small jack's piston, (B3) big nut, (B4) big jack's cylinder, (B5) big jack's piston, (B6) end cap, (B7) fixed cap, (B8) sealing screw, (B9) transmission loading road, (B10) threated road, (B11) fixed base, and (B12) handle. Figure 2. Open in new tabDownload slide Main body in hydrostatic mode (a), and confining and pore systems (b). Main body (A): (A1) body, (A2) sleeve, (A3) upper cap, (A4) piston cap, (A5) hydrostatic loading cap, (A6) lower cap, and (A7) hydrostatic cap. Confining and pore systems (B): (B1) small jack's cylinder, (B2) small jack's piston, (B3) big nut, (B4) big jack's cylinder, (B5) big jack's piston, (B6) end cap, (B7) fixed cap, (B8) sealing screw, (B9) transmission loading road, (B10) threated road, (B11) fixed base, and (B12) handle. Figure 3. Open in new tabDownload slide Poroelastic measurement system in hydrostatic mode. Figure 3. Open in new tabDownload slide Poroelastic measurement system in hydrostatic mode. 5. Experimental procedure In this paper, the results of drained, pore pressure loading, undrained and dry hydrostatic tests on synthetic sandstone samples are presented. Calibration tests are done on an NX cylindrical steel sample which has a longitudinal hole with a 10 mm diameter. Assuming an incompressible steel sample, the goal of the calibration tests is to obtain the extra volume changes of the fluid, tubes and connectors at each stress level. So the results of the main tests after calibration correction are presented. Drained, pore pressure loading and undrained calibration tests are done at different stress levels on a steel sample enabled evaluation of the extra volume changes due to the compressibility of the pores and confining fluid, tube and connectors. The used pore fluid is hydraulic oil with compressibility of 6.425 E-4 Mpa-1 (6.425 E-10 Pa-1). Calibration for drained and pore pressure loading tests is easier than for undrained conditions because the results of undrained tests are very sensitive to extra fluid volume in the pore system. In addition, the calibration procedure for very small amounts of differential stress is difficult to perform and may encounter some errors. It is noted that all of the tests are done with two loading–unloading cycles, and the first cycle is for removing the non-elastic deformation of the rock. The recording of stresses and displacements is done in the second cycle. The equation of compressibility (8) has a vertical asymptote mathematically. So the reference points and stress paths are selected carefully as it is possible to compare the results of different tests at the same differential stresses. The rate of loading and unloading is nearly 1.2 MPa per minute for all tests. Note that because of the measuring of elastic coefficients, the stress should not exceed the elastic behavior of rock. This could be checked by the shape of the curves and the Mohr–Coulomb envelope of synthetic sandstone. In figure 4, the failure envelope is presented and the stress path should be below the envelope in each test. This curve is obtained with a multi-step triaxial test by MTS. Figure 4. Open in new tabDownload slide Failure envelope of synthetic sandstones. Figure 4. Open in new tabDownload slide Failure envelope of synthetic sandstones. 6. Experimental results In the below subsections, drained, pore pressure loading, undrained and dry hydrostatic tests on synthetic samples are explained. In each part, first, the procedure, assumptions and specific boundary conditions are expressed, and then related results and discussion for each test are explained. 6.1. Drained test The compressibility coefficients (Cbc, Cpc), effective stress coefficients (α, β) and solid phase compressibility are obtained by this test. The drained condition is defined as there being change in the pore pressure while the confining pressure is variable. The reference point is within 5 MPa of the confining pressure and 2 MPa of the pore pressure. The confining pressure is increased by 3 MPa in each step until it gets to 40 MPa, while pore pressure is constant during the test. In figure 5(a), the stress paths for confining and pore pressure are shown. The first cycle is for removing the non-elastic deformation of rock. At each step, stresses and displacements are recorded by the related gauges, and the bulk and pore volume changes are obtained after correction by the calibration data of the confining and pore systems. In figure 5(b) the curves of bulk and pore volume strains are shown. These curves are plotted by dividing the bulk and pore volume changes in bulk and pore volume at a reference point. The slope of the curves is the compressibility of the rock. By increasing the differential pressure, the slopes of the curves, or the compressibility, are decreasing, until it gets to a constant value. In figure 5(c), the drained bulk compressibility factor (Cbc) is plotted by (8), and its amount is 2.261E-4 MPa-1 (2.261E-10 Pa-1). In figure 5(d), the drained pore compressibility (Cpc) is plotted by (14), and its amount is 8.412E-4 MPa-1. This graph is also obtained by (25) and (23). The two curves are matched exactly and it can be verified for calculation. In figure 5(e), the effective stress coefficients versus differential stresses are plotted. The Biot coefficient is obtained by (17) and another effective stress coefficient related to pores is calculated by (18). According to these graphs, equation (19) is confirmed. In figure 5(f), the graph of solid phase compressibility is shown based on (16). The amount of this coefficient (3.626E-5 MPa-1) is comparable with the amount calculated by the Voigt–Reuss–Hill method or equation (32). This comparison confirms that the difference between these two values is negligible and is about 2%. It should be noted that in figures 5(d) and (e), the data of Cr for figure 5(f) are used. Figure 5. Open in new tabDownload slide Results of the drained hydrostatic test: (a) stress path; (b) bulk and pore strain; (c) bulk compressibility (Cbc)—its amount is limited to 2.261E-4 MPa-1 asymptotically; (d) pore compressibility (Cpc)—its amount is limited to 8.412E-4 MPa-1 asymptotically; (e) effective stress coefficients; (f) rock matrix or solid phase compressibility (Cr)—its amount is limited to 3.626E-5 MPa-1 asymptotically. Figure 5. Open in new tabDownload slide Results of the drained hydrostatic test: (a) stress path; (b) bulk and pore strain; (c) bulk compressibility (Cbc)—its amount is limited to 2.261E-4 MPa-1 asymptotically; (d) pore compressibility (Cpc)—its amount is limited to 8.412E-4 MPa-1 asymptotically; (e) effective stress coefficients; (f) rock matrix or solid phase compressibility (Cr)—its amount is limited to 3.626E-5 MPa-1 asymptotically. The rock matrix compressibility is slightly pressure dependent, but in this figure there are variable amounts, particularly at small amounts of differential stresses. That is because of the natural behavior of rocks and the existence of very thin crack-like voids. Also, as noted in section 4, the equation of compressibility has a vertical asymptote mathematically, and the amount of compressibility is higher at differential pressures near to zero. In addition, the data at these amounts of differential stresses are not reliable because of calibration errors. So, the amount of compressibility at a small differential pressure is not very operational. However the data at higher differential stresses according to figure 5(f) are nearly constant, and are compatible with the Voigt–Reuss–Hill method. 6.2. Pore pressure loading test The compressibility coefficients (Cbp, Cpp) and the solid phase compressibility (Cr) are obtained by this test. The pore pressure loading condition is defined by there being no change in confining pressure while the pore pressure is variable. It should be mentioned that in this test, the amount of confining pressure is higher than the pore pressure during the test. In this test, the confining pressure is increased to 38 MPa while the pore pressure is constant at 2 MPa, after the first cycle for removing non-elastic deformation. Then the pore pressure is increased to 35 MPa in a constant confining pressure at 38 MPa. This is the reference point for this test and its values are used in the calculation. Then the pore pressure decreases step by step in 3 MPa intervals while the confining stress is constant, and in each step the data (stresses and displacements) are recorded. In figure 6(a), the stress paths for the confining and pore pressure are shown. The bulk and pore volume changes, which are recorded by gauges, are obtained after correction with the calibration data of the confining and pore systems. There are two calibration curves for the pore and bulk systems, which are used for the pore and bulk coefficients, respectively. The calibration of the pore system in this test is more important than the confining system, and is needed to accurately consider the system compliance at each stress level. The strains are calculated by dividing the bulk and pore volume changes in the bulk and pore volume values at a reference point. In figure 6(b), the graphs of bulk and pore strain are shown. Bulk compressibility (Cbp) versus differential stress based on (21) is illustrated in figure 6(c). The value of Cbp is limited to 2.013E-4 MPa-1 asymptotically. According to (23), the difference between Cbc and Cbp should be nearly the solid phase compressibility (Cr) value. This compressibility is about 2.466E-5 MPa-1 and is compatible with empirical (32) and experimental values of (Cr) in a drained test. Graphs of pore compressibility (Cpp) versus differential stress based on (22) are illustrated in figure 6(d). The value of Cpp is limited to 8.122E-4 MPa-1 asymptotically. According to (24), the difference between Cpc and Cpp should be nearly the solid phase compressibility (Cr), and the results show that this discrepancy is about 2.9E-5 MPa-1 and is compatible with empirical (Voigt–Reuss–Hill method) and experimental (drained test) values of Cr. The rock matrix compressibility is obtainable by an empirical method (Voigt–Reuss–Hill) and indirectly by an experimental method (the drained pore pressure loading test). The value obtained by the empirical method is a constant, and values from the experimental method were read from rock matrix compressibility curves at maximum differential stresses. In figure 6(e), the solid phase compressibility is shown based on the data from pore pressure loading and drained tests. According to this graph, the Cr value for the pore pressure loading test data is 4.44E-5 MPa-1, which has a nearly negligible error compared with (32) and the results of the drained hydrostatic test. This value could not be read from the graph because of scale limitations, so it was read from the recording data. According to this figure, like the drained test data, the amounts of rock matrix or solid compressibility are nearly constant at higher differential stresses and the data at small differential stresses are more variable and are almost unreliable because of the existence of very thin crack-like voids, mathematical vertical asymptotes and calibration errors. As can be seen in figure 6(e), the point at the smallest differential stress is unreasonable. Figure 6. Open in new tabDownload slide Results of the pore pressure loading test: (a) stress path; (b) bulk and pore strain; (c) bulk compressibility (Cbp)—its amount is limited to 2.013E-4 MPa-1 asymptotically; (d) pore compressibility (Cpp)—its amount is limited to 8.122E-4 MPa-1asymptotically; (e) rock matrix or solid phase compressibility (Cr)—its amount is limited to 3.626E-5 MPa-1in the drained test and 4.44E-5 MPa-1 in the pore pressure loading test asymptotically. Figure 6. Open in new tabDownload slide Results of the pore pressure loading test: (a) stress path; (b) bulk and pore strain; (c) bulk compressibility (Cbp)—its amount is limited to 2.013E-4 MPa-1 asymptotically; (d) pore compressibility (Cpp)—its amount is limited to 8.122E-4 MPa-1asymptotically; (e) rock matrix or solid phase compressibility (Cr)—its amount is limited to 3.626E-5 MPa-1in the drained test and 4.44E-5 MPa-1 in the pore pressure loading test asymptotically. 6.3. Undrained test Compressibility coefficients (Cbu, Cpu) and the solid phase compressibility are obtained by this test. The undrained condition is defined by there being no change in the fluid content of the rock. At this test, the confining pressure is variable and the pore pressure will increase due to the boundary limitations of the pore fluid. According to figure 7(a), the reference point is 3 MPa of confining pressure and 2 MPa of pore pressure. The first cycle is for removing of the non-elastic deformation of rock. The confining pressure is increased in 3 MPa steps and the volume changes of the confining system and pore pressure in each step are recorded. Measurement systems for the undrained test should be designed so precisely that the extra volume of the pore system, including the tube and connector volume, is negligible compared to the rock pore volume (Lockner and Beeler 2003). So the pore pressure changes are nearly all due to the deformation of the pores, and the measurement of the coefficients is more accurate. The extra volume of the pore system is nearly 25% of the rock pore volume. So it is proved that the error in the undrained test result is unavoidable. Figure 7. Open in new tabDownload slide Results of the undrained tests. Figure 7. Open in new tabDownload slide Results of the undrained tests. Below, the results of this test and the undrained coefficients are presented. These coefficients are obtained by the drained and pore pressure loading tests indirectly in addition to the direct undrained test. So it is possible to compare the direct and indirect value and obtain the error of the system in an undrained condition. In figure 7(b), the buildup pore pressure or Skempton coefficient is plotted against the differential stress. The measured Skempton coefficient is drawn directly based on (11). This coefficient is obtained by (29), once with data from the drained test and once with data from the pore pressure loading test. The fluid compressibility is considered to be a constant value and its amount is 6.425E-4 MPa-1 (4.43E-6 1/psi). According to this graph, the curves of Skempton are nearly the same and the Skempton coefficient is not sensitive to errors due to extra pore volume. In figure 7(c), the measured undrained bulk compressibility is plotted using calibrated recording volume-changes data based on (9) directly. The actual value is obtained by the drained test result based on (27) indirectly. The graph shows that there is a difference between the actual and directing measured values. The measured pore volume changes are gained from the pore pressure changes considering the fluid compressibility and calibration test results. At first, the sum of the system and rock volume changes are obtained by pore pressure changes and fluid compressibility based on (26), and then these data are corrected by calibration tests related to pore system. So it is possible to plot the graph of Cpu directly. The actual curve of Cpu is obtained by the drained test result based on (28) indirectly. In figures 7(b)–(d), there is a difference between the actual and measured undrained coefficients. The difference is considerable for Cpu, and this confirms the sensitivity of Cpu to pore pressure changes and the extra fluid volume in the pore system. But the difference between the measured and actual Cbu and the Skempton coefficient is not influenced by extra fluid volume, such Cpu, in the system. It should be noted that the calibration procedure for the pore system and the measuring of Cpu in the undrained test is difficult, and that the best way to do this is to decrease the extra pore volume in the pore system. So, the existence of errors in this test is unavoidable because of the extra pore volume in the pore system. 6.4. Hydrostatic dry test In this test a dry sample, like the sample which is tested by MTS, is put in the system. This test is done by confining the system and there is no fluid in the pore volume. In the graph in figure 8, the dry bulk compressibility factor is illustrated. According to the figure, the amount of this factor is 1.45E-4 MPa-1 (1.001E-6 1/Psi), which is nearly compatible with the results of the MTS test (table 2). But the results of the MTS and hydrostatic dry tests are lower than Cbc in the drained test, when these values should be higher because of the lack of pore pressure limitation. This event may be related to synthetic samples and cement limitations. In actual rock reservoir samples the role of cement (amount and place) is very important in rock behavior (Gueguen et al2004). In these synthetic samples during the saturating process, an amount of cement exits from the pore rock. So this can be a justification for the lower bulk compressibility in the dry samples than in the saturated samples. In figure 8, a power law regression equation and R-square value are shown. The R-square value shows that there is a good match between the equation and test data. Figure 8. Open in new tabDownload slide Bulk compressibility from the dry test. Figure 8. Open in new tabDownload slide Bulk compressibility from the dry test. 7. Discussion and summary In this paper, the drained, pore pressure loading, undrained and dry tests on synthetics sandstone samples in hydrostatic conditions have been explained by a designed rock compressibility measurement system. Poroelastic coefficients such as compressibility (bulk, pore and matrix), effective stress coefficients and the Skempton factor are measured by these tests. The main goal of this paper is to design and manufacture a test system by which a series of tests can be done and the results compared to those predicted by the theory of poroelasticity, and by comparing both results, to discuss the possible sources of error for each test. A procedure has also been discussed to easily make synthetic sandstone samples. Drained, pore pressure loading and undrained calibration tests are done on an incompressible NX cylindrical steel sample with a longitudinal hole. At each test, calibration curves are obtained for the confining and pore pressure systems in each stress level. So it is possible to evaluate the extra volume changes due to the compressibility of the pore and confining fluid, tubes and connectors. Calibration tests for drained and pore pressure loading tests are easier than undrained tests, because the results of undrained tests are very sensitive to extra fluid volume in the pore pressure system. In this paper, solid phase compressibility is obtained by the Voigt–Reuss–Hill method because it is fairly easy when the mineralogical composition of the rock is known. However, this coefficient is measured indirectly by drained and pore pressure loading tests based on a poroelastic formulation. These test results are nearly compatible with the Voigt–Reuss–Hill method. According to figures 5(f) and 6(e), the amount of rock matrix or solid compressibility is nearly constant at higher differential stresses and the data at small differential stresses are more variable and almost unreliable because of the existence of very thin crack-like voids, the mathematical vertical asymptotes of curves and calibration errors. The results show that drained and pore loading pressure-test data are compatible with each other and a poroelastic formulation, but in undrained tests there is some error between the actual and measured values because of the extra volume in pore pressure system and calibration difficulties. The error in the Cpu coefficient is more than in other coefficients. This is due to the sensitivity of Cpu to extra fluid volume in the pore system and system compressibility. The calibration procedure for the pore system and the measuring of Cpu in the undrained test is difficult, and the best way to make the coefficient more accurate is to decrease the extra pore volume in the pore system. The lower bulk compressibility in the dry samples than in the saturated samples is due to an exit of cement from the pores during the saturation of samples and the synthetic sample limitations. 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TI - Experimental measurement of compressibility coefficients of synthetic sandstone in hydrostatic conditions
JF - Journal of Geophysics and Engineering
DO - 10.1088/1742-2132/10/5/055002
DA - 2013-10-01
UR - https://www.deepdyve.com/lp/oxford-university-press/experimental-measurement-of-compressibility-coefficients-of-synthetic-Hjems3uSAY
VL - 10
IS - 5
DP - DeepDyve
ER -