Poromechanics V © ASCE 2013 2423 Poroelasticity of High Porosity Chalk under Depletion Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. K.A. Andreassen1 and I.L. Fabricius1 1 Department of Civil Engineering, Technical University of Denmark, Brovej – Building 118, 2800 Kgs. Lyngby, Denmark; PH (+45) 45251781; email: kall@byg.dtu.dk ABSTRACT The theory of poroelasticity for the elastic region below pore collapse by means of three different loading paths gives the possibility to compare the static and dynamically determined Biot coefficient for a set of experimental data with uniaxial loading on outcrop chalk performed with different levels of pore pressure. The chalk is oilsaturated Lixhe chalk from a quarry near Liège, Belgium, with a general porosity of 45%. Additionally, we compare the theoretical lateral stress to the experimentally determined lateral stress at the onset of pore collapse. The static Biot coefficient based on mechanical test results is found to be lower than the pretest dynamic Biot coefficient determined from elastic wave propagation for the loading path and with less deviation under depletion. The calculated lateral stress is lower than the experimentally measured lateral stress depending on loading path. An explanation to this behaviour is pore pressure build up. INTRODUCTION Knowledge of the Biot coefficient is important for prediction of basin or reservoir subsidence especially for depletion under hydrocarbon production. As the depth for newly discovered reservoirs continually increases so will the pressures involved increase. Consequently, there is a large effect by not applying the correct Biot coefficient due to these higher overburden and pore pressure. There exist several studies of Biot’s coefficient (Engstrøm, 1992; Fabre and Gustkiewicz, 1997; Ojala and Fjær, 2007; Fabricius et al., 2010; Alam et al., 2012) but most are for hydrostatic conditions as described by Biot (1941). In this paper we present a combined poroelastic analysis of laboratory tests on outcrop chalk from the Pasachalk project (Schroeder et al. 2000; 2003). A number of tests are under reservoir conditions with elevated pore pressures and have not been treated by use of poroelastic theory previously. METHODS We apply theory of poroelasticity stated with stresses and strains positive for compression (Biot, 1941; Zimmerman, 2000a): = 2 + � + + � + , Poromechanics V Poromechanics V © ASCE 2013 2424 = 2 + � + + � + , (1) Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. = 2 + � + + � + , = 2 , = 2 , = 2 , where εii denotes the principal strain, εij denotes the shear strain, σii denotes the principal stress, σij denotes the shear stress, G is the shear modulus, λ is Lamé’s parameter, Pp is the pore pressure and α is Biot’s coefficient. For hydrostatic stresses it is well known that Biot’s coefficient is = 1 − dry , min valid for isotropic, monomineralitic materials. On the other hand, uniaxial strain, ∆εxx = ∆εyy = 0, is generally thought to resemble reservoir conditions, Addis (1997), and therefore often relevant in petroleum engineering. Thus, following the derivation of Alam et al. (2012), poroelasticity for uniaxial strain conditions results in the relationship: (2) = + , with σa the applied axial stress, εa the resulting axial strain, M the compressional modulus (often termed the oedometer or p-wave modulus) valid for ∆εxx = ∆εyy = 0, and n the effective stress coefficient for uniaxial strain conditions. The term Mεa is the effective stress experienced by the soil skeleton. Alam et al. (2012) show that n is identical to α, which is in accordance with the derivation by Zimmerman (2000a). By differentiation of eq. (2) � � = 1 , � � = 1− , with σd the difference in stress, σd = σa – Pp. This yields =1− � � � � (3) . Differentiating eq. (2) with respect to σa gives =− � � � � (4) . Poromechanics V Poromechanics V © ASCE 2013 2425 From eq. (1) with the conditions of uniaxial strain (∆εxx = ∆εyy = 0) the lateral stress is � 1− = � 1−2 � 1− +� (5) , Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. as stated by Zimmerman (2000b) and Engelder and Fischer (1994). Finally, differentiating eq. (5) yields � � 1−2 � 1− =� ⇒ =� 1− � � � 1−2 (6) , used by Addis (1997) and Goulty (2003) among others for reservoir depletion or passive basin response. Three equations above determine individually the Biot coefficient under uniaxial strain conditions. In a laboratory setting these correspond to the following: • Eq. (3) with first changing pore pressure with the difference of stress constant followed by a phase with changing differential stress with constant pore pressure, • Eq. (4) for changing pore pressure and total axial stress constant followed by a phase with changing total axial stress with constant pore pressure, • Eq. (6) for simultaneous change in lateral stress and pore pressure with the total axial stress held constant. These phases are all relevant for the determination of the static Biot coefficient for the mechanical test results reported in the Pasachalk project (Schroeder et al., 2000; 2003). From elastic wave propagation we find the dynamic Biot coefficient by use of the isoframe model (Fabricius et al., 2007; 2010) as the Pasachalk data set contains dry and saturated p-wave measurements but not s-wave measurements and thus does not facilitate direct determination of Biot’s coefficient. The isoframe model is based on modified upper Hashin-Shtrikman bounds and the isoframe values show how large part of the solid constitutes the load bearing skeleton, = 4 3 4 3 �min + min �fl (min −sat )+�min + min �sat (fl −min ) 4 3 4 3 ��sat + min �min (fl −min )+�min + min �fl (min −sat )�(1−) , (7) with Kmin the mineral bulk modulus, Gmin the mineral shear modulus, Kfl the fluid bulk modulus, Ksat the saturated bulk modulus, and φ the porosity. It can be visualised as a modelled mixture of relatively large hollow spheres and smaller solid spheres. For a homogenous Reuss model the bulk modulus of a suspension of particles is = +(1−)(1−) (1−) + fl min (8) . The Modified Upper Hashin–Shtrikman bound for the bulk modulus gives, Poromechanics V Poromechanics V © ASCE 2013 HS+ =� 2426 4 3 sus + min + (1−)(1−) Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. together with the shear modulus HS+ = � + (1−)(1−) 4 3 sus + min + + (1−) (1−) −1 min + 4 3 min + min � � −1 − , = Additionally, the p-wave modulus is defined as 4 3 (9) − min , min 9min +8min � +2 � 6 min min . (10) 4 3 HS+ = HS+ + HS+ , and by back-calculation of MHS+ to equal the measured p-wave modulus by change of IF in Equations (7–10) the result is the Biot coefficient from the dry bulk modulus 4 min min (1−) +4 min min ) − 3 min (1−) dry = ( RESULTS , dry =1− min . (11) The Pasachalk project (Schroeder et al., 2000; 2003) covers aside from a high number of tests on water- or oil-saturated chalk also a number of tests on oil-saturated chalk under reservoir conditions. Figure 1 shows an example of the loading path for one of these. The loading consists of a hydrostatic uploading including increasing pore pressure. This is followed by a deviatoric phase – actually termed triaxial here and in the Pasachalk project – where the lateral stress is constant while increasing the axial stress. Finally comes a depletion phase with constant total axial stress while decreasing the pore pressure under uniaxial strain conditions. The yield point is the onset of acceleration of deformation or the end of the linear elastic behaviour, which is after the start of the depletion phase. Therefore, we apply the theory of poroelasticity to obtain Biot’s coefficient and the resulting lateral stress. Using Eq. (3) for the loading phase and Eq. (4) for the depletion phase determines the Biot coefficient for the example in Figure 1. Using Eq. (5) gives the lateral stress from the total axial stress with the pore pressure, Poisson’s ratio and the Biot coefficient. We include the tests numbered 177, 182, 189, 182, and 190 for comparison of the poroelastic behaviour with and without raised pore pressure. Pretesting measurements of the saturated and dry p-wave exist and Equations (7–11) form the basis for calculation of the Biot coefficient. These calculations use a fluid bulk modulus for soltrol of 1.292 GPa valid for 20°C and determined by use of the procedure listed in Mavko et al. (2009) from the stated density of 0.783 g/cm2 in the Pasachalk reports. Poromechanics V Poromechanics V © ASCE 2013 2427 Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. (a) (b) Figure 1. Sample of testing procedure. Test 198 with (a) time evolution of the stress and strain and (b) effective stress or pore pressure versus axial strain. The effective stress is based on the dynamic Biot coefficient of 0.887, see Table 1. Biot’s coefficient – dynamical versus statical. The above procedures lead to the results listed in the Table 1. Two tests had the onset of pore collapse before the actual uniaxial strain loading phase started therefore the lateral stress calculation is not applied. Table 1. Liège chalk test results with sound velocity measurements. Loading paths are stated as HxTy: Hydrostatic to x MPa, triaxial (deviatoric) to y MPa. All are then uniaxially loaded (∆εxx = ∆εyy = 0) to beyond yield. Initial Dyn. α, Dyn. ν, Load Pp before before Static α, Static α, Test path (MPa) Porosity testing testing loading depletion 177 H7 0 0.450 0.888 0.257 181 H7 0 0.453 0.895 0.271 189 H7 0 0.448 0.892 0.271 165 H7T12 38 Yield onset before uniaxial strain loading phase 166 H7T12 38 0.459 0.891 0.267 0.74 0.77 167 H7T12 38 0.454 0.905 0.271 0.78 0.87 172 H7T12 0 0.434 0.887 0.270 173 H7T12 0 0.448 0.882 0.255 170 H7T14.5 38 0.441 0.889 0.273 0.70 0.78 198 H7T17 33 0.436 0.887 0.267 0.74 0.83 141 H7T17 38 0.453 0.903 0.271 – 0.84 142 H7T17 38 0.442 0.890 0.270 – 0.88 174 H7T17 0 Yield onset before uniaxial strain loading phase 182 H12 0 0.438 0.884 0.268 190 H12 0 0.448 0.892 0.270 196 H12T17 33 0.445 0.890 0.263 0.75 0.88 197 H5T18 32 0.453 0.904 0.276 0.71 0.83 Poromechanics V Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. Poromechanics V © ASCE 2013 2428 Lateral stress. The loading paths contain hydrostatic and deviatoric loading phases followed by a uniaxial strain phase. Therefore computation of the part of the total axial stress contribution to the lateral stress from the uniaxial strain phase is achieved by subtraction of both of these levels of stress. Following this, the hydrostatic contribution is added to the total stress value while the deviatoric phase do not contain strain restrictions and consequently do not lead to an increase in lateral stress. Table 2 lists the results. All tests have a loading rate of 4·10–3 MPa/s except tests 189 and 190, which are slower with a loading rate of 1·10–3 MPa/s. Table 2. Liège chalk test results with lateral stress measurements. Loading paths are stated as HxTy: Hydrostatic to x MPa, triaxial (deviatoric) to y MPa. All are then uniaxially loaded (∆εxx = ∆εyy = 0) to beyond yield. Total lateral stress calculated using dynamic properties, α and ν, listed in Table 1. At yield onset, At yield onset, Load Initial Pp measured stresses σr calc. from σa Test path (MPa) Porosity [σa, σr, Pp] (MPa) (MPa) 177 H7 0 0.450 15.4 7.5 0 9.9 181 H7 0 0.453 24.0 8.5 0 13.3 189 H7 0 0.448 19.2 7.2 0 11.6 165 H7T12 38 Yield onset before uniaxial strain loading phase 166 H7T12 38 0.459 49.8 42.5 26.8 37.0 167 H7T12 38 0.454 49.8 42.8 28.8 38.5 172 H7T12 0 0.434 20.3 6.5 0 10.1 173 H7T12 0 0.448 19.0 6.7 0 9.4 170 H7T14.5 38 0.441 47.5 39.1 28.2 33.6 198 H7T17 33 0.436 50.0 38.9 25.8 34.6 141 H7T17 38 0.453 50.0 43.7 31.5 38.2 142 H7T17 38 0.442 49.9 41.4 31.0 37.9 174 H7T17 0 Yield onset before uniaxial strain loading phase 182 H12 0 0.438 25.2 12.6 0 16.9 190 H12 0 0.448 18.1 12.2 0 14.3 196 H12T17 33 0.445 50.2 42.8 27.5 40.7 197 H5T18 32 0.453 50.0 35.9 24.8 32.0 DISCUSSION The static and dynamic Biot coefficients are not expected to be equal due to the strain amplitude is smaller for the elastic wave propagation than for the mechanical strain resulting from the loading (Fjær et al., 2008). Others find that there is agreement within the level of estimated error (Ojala and Fjær, 2007; Alam et al., 2012). The Biot coefficient is also expected to decrease with applied stress level and as the dynamic Biot coefficient is determined before the test it should give higher values compared to Poromechanics V Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. Poromechanics V © ASCE 2013 2429 what could be expected during testing (Ojala and Fjær, 2007; Alam et al., 2012) although hysteresis is anticipated, i.e. the loading path influences the Biot coefficient. The results in Table 1 indicate that hysteresis is present for the Liège Lixhe chalk and that the dynamic Biot coefficient is higher than the static Biot coefficient. This is most pronounced for the loading phase and less for the depletion phase. For the lateral stress calculation the results in Table 2 show higher calculated lateral stress compared with the measured stress for tests with zero pore pressure while lower calculated lateral stress for tests at higher pore pressures. To compare the effect of applying static Poisson’s ratio and static Biot coefficient instead of the dynamic parameters we find the average for the difference between the measured and calculated total lateral stress. The Poisson’s ratio used is a general value of 0.18 valid for oil-saturated Liège Lixhe specimens (Schroeder et al., 2000). The average difference goes from 3.5 MPa to 1.4 MPa for the zero pore pressure tests. On the contrary, for the elevated pore pressure tests by applying static parameters the average difference goes from 3.7 MPa to 6.9 MPa. Therefore, the static parameters seem to yield the best calculation of the lateral stress based on the zero pore pressure tests while the dynamic parameters are expressing the higher pore pressure tests well. An important reason for the discrepancy for the higher pore pressure tests could be the high rate by which the pore pressure is depleted. This raises concerns that the pore pressure is not fully dissipated and the pressure build up contributes to a higher measurement of the lateral stress. The lower onset of yield of tests 189 and 190 compared with the other tests verify this. Another aspect is time dependent effects. The higher pore pressure tests are in principle performed with a much slower loading rate for the first part of the test (hydrostatic loading to 2 MPa, with ramp in the pore pressure) and as such this could be seen as a consolidation phase changing the material properties. Additionally, the Poisson’s ratio is a mean value and therefore not specific for each test specimen in Table 2. By obtaining shear wave velocities this could be further investigated. CONCLUSION The theory of poroelasticity applied for chalk at reservoir conditions gives consistent results for the static and dynamic Biot coefficient. Hysteresis exists between loading and depletion phases. The theoretical lateral stress is compared with the measured stress giving small differences. Both of these effects could be attributed to the high depletion rate raising concerns that the pore pressure was not left enough time to dissipate. REFERENCES Addis, M.A. (1997). “Stress-depletion response of reservoirs.” Proceedings – SPE Annual Technical Conference and Exhibition, Oct. 5–8, San Antonio, Texas, 55–65. Alam, M.M., Fabricius, I.L. and Christensen, H.F. (2012). “Static and dynamic effective stress coefficient of chalk.” Geophysics, 77, No. 2, L1–L11. Poromechanics V Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. Poromechanics V © ASCE 2013 2430 Biot, M.A. (1941). “General theory for three-dimensional consolidation.” Journal of Applied Physics, 12, 155–164. Engelder, T., and Fischer, M.P. (1994). “Influence of poroelastic behaviour on the magnitude of minimum horizontal stress, Sh, in overpressured parts of sedimentary basins.” Geology, 22, 949–952. Engstrøm, F. (1992). “Rock mechanical properties of Danish North Sea chalk.” 4th North Sea Chalk Symposium. Sept. 21–23, Deauville, France. Fabre, D. and Gustkiewicz, J. (1997). “Poroelastic properties of limestones and sandstones under hydrostatic conditions.” International Journal of Rock Mechanics and Mining Sciences, 34, 127-134. Fabricius, I.L., Høier, Ch., Japsen, P., and Korsbech, U. 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Proceedings of the 1st Canada-US Rock Mechanics Symposium, May 27–31, Vancouver, Canada, 813–819. Schroeder, Ch., Illing, P., Charlier, R., Collin, F., Delage, P., Cui, Y-J., De Gennaro, V., De Leebeeck, A., Keül, P., and A.-P. Bois (2000). Mechanical behaviour of partially and multiphase saturated chalks fluid-skeleton interaction. Part 1. Pasachalk. European Joule III contract JOF3CT970033. Schroeder, Ch., Illing, P., Charlier, R., Collin, F., Delage, P., Cui, Y-J., De Gennaro, V., De Leebeeck, A., Keül, P., and A.-P. Bois (2003). Mechanical behaviour of partially and multiphase saturated chalks fluid-skeleton interaction. Part 2. Final Technical Report. European Joule III contract ENK6-2000-00089. Zimmerman, R.W. (2000a). “Pore Compressibility under Uniaxial Strain.” Proc. 6th Int. Symp. on Land Subsidence, Sep. 25–29, Ravenna, Italy. 57–65. Zimmerman, R.W. 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