Sequential Methods for Coupled Geomechanics and Multiphase Flow

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5.3 Case 5.2 (1D injection–production problem). Evolution of the dimensionless pressure as a function of dimensionless time (pore volume produced). Shown are the results for the fully coupled method, the fixed-strain, and fixed-stress splits. Top: coupling strength τ = 0.83. Bottom: coupling strength τ = 1.21. 142 5.4 Case 5.2 (1D injection–production problem). Results are shown for the fixed- strain split, with two different coupling strengths: τ = 0.95 (top), and τ = 1.05 (bottom), and two very different time step sizes. The stability of the fixed-strain split is independent of time step size. The fixed-strain split is stable if τ < 1 and unstable if τ > 1, regardless of time step size. . . . . . . 144 5.5 1D problem with injection and production (Case 5.2). Kdr = 1 GPa. α = 0.5 for both flow and mechanics is considered. Top: τ = 0.83. Bottom: τ = 1.11. 145 5.6 Stability behaviors of different η’s and τ’s. Top: pressure history at the observation well when τ = 3.33. Bottom: pressure history at the observation well when τ = ∞. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5.7 Behavior of the undrained and fixed-stress splits for cases with high coupling strength. Top: Case 5.1 with τ = 16.67. Bottom: Case 5.2 with τ = 12.12. In both cases, the fixed-stress split requires one single iteration per time step to match the fully coupled solution, while many iterations are required for the undrained split. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.8 Case 5.3 (the Mandel problem). Evolution of the pressure at the observation well as a function of dimensionless time. Shown are the results for the fully coupled, fixed-strain, and fixed-stress split methods. Top: τ = 0.90. Bottom: τ = 1.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.9 Case 5.3 (the Mandel problem). Evolution of the dimensionless vertical displacement on the top as a function of dimensionless time. Shown are the results for the fully coupled, fixed-strain, and fixed-stress split methods. Top: τ = 0.90. Bottom: τ = 1.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 xxii
5.10 Case 5.3 (the Mandel problem). Evolution of the dimensionless horizontal displacement on the right side as a function of dimensionless time. Shown are the results for the fully coupled, fixed-strain, and fixed-stress split methods. Top: τ = 0.90. Bottom: τ = 1.10. . . . . . . . . . . . . . . . . . . . . . . . . 152 5.11 2D problem with a production well (Case 5.4). Top: the history of pressure. Bottom: the history of the coupling strength. FSN and FSS indicate the fixed-strain and fixed-stress splits, respectively. The model enters the plastic regime at td ≈ 0.018. Beyond this point, the fixed-strain split becomes unstable.153 5.12 Case 5.4 (2D production problem in elastoplastic regime). Shown is the evolution of the dimensionless pressure at the observation well against dimensionless time (pore volume produced). Once the model enters the plastic regime, the fixed-strain split becomes unstable and fails to produce a solution at all. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 5.13 Convergence analysis of Case 5.1 on pressure (top) and displacement (bottom). ∆td = 4cv∆t/Lz 2 . FC, FSS, and FSN indicate the fully coupled, fixed-stress, and fixed-strain split methods, respectively. The fixed-strain split shows zeroth-order accuracy. But, the fixed-stress split yields first-order accuracy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.14 Non-convergence of the fixed-strain split with one iteration for Case 5.1: pressure (top) and displacement (bottom) . . . . . . . . . . . . . . . . . . . 156 5.15 Convergence of the fixed-stress split with one iteration for Case 5.1: pressure (top) and displacement (bottom) . . . . . . . . . . . . . . . . . . . . . . . . 157 5.16 Two test cases to investigate the effect of different η’s. Case 5.5: consolidation problem with constrained boundary (left picture). Case 5.6: consolidation problem with unconstrained boundary (right picture). . . . . . . . . . . . . 159 5.17 Convergence of the fixed-stress split for Case 5.5 when K est dr = K3D dr . Top: convergence of pressure. Bottom: pressure distribution along the vertical axis (X-Y section). ∆td = 4cv∆t/LxLz. . . . . . . . . . . . . . . . . . . . . 161 xxiii
Page 1 and 2: SEQUENTIAL METHODS FOR COUPLED GEOM
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164 CHAPTER 6. COUPLED MULTIPHASE F
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212 CHAPTER 7. CONCLUSIONS
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214 APPENDIX A. STABILITY IN SPACE
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226 APPENDIX B. MISCELLANEOUS DERIV
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236 APPENDIX C. CONSTITUTIVE RELATI
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244 BIBLIOGRAPHY Mech Eng 122: 145-
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246 BIBLIOGRAPHY Aug. Murad M.A. an
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248 BIBLIOGRAPHY ˇZeniˇsek A. 198
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Sequential Methods for Coupled Geomechanics and Multiphase Flow

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