Sequential Methods for Coupled Geomechanics and Multiphase Flow

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5.18 Pressure history at the observation well (2,5) for Case 5.6 with different deviation factors when 0.95 × K 1D dr , 1.05 × K1D dr , and K3D dr are taken for Kest dr . 162 6.1 The region for the linear theory. . . . . . . . . . . . . . . . . . . . . . . . . 170 6.2 Water injection and oil production. Left: coupled multiphase flow and geomechanics in a 1D poroelastic medium with overburden. Right: coupled multiphase flow and geomechanics in a 2D poroelastic medium with overburden and side burden . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 6.3 Top: pressure history at the monitoring well during simulation. Bottom: water saturation profile after simulation. Pd = P/Po,i. . . . . . . . . . . . . 197 6.4 Convergence behaviors during iterations at td = 0.08. Top: pressure. Bot- tom: water saturation. · is the L 2 norm. . . . . . . . . . . . . . . . . . . . 199 6.5 Pressure (top) and water saturation (bottom) distributions after simulation. 201 6.6 Top: pressure history at the monitoring well during simulation. Bottom: water saturation profile at the bottom layer after simulation. . . . . . . . . 202 6.7 Convergence behaviors during iterations at the first time step. Top: pressure. Bottom: water saturation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 6.8 Pressure history at the monitoring well during simulation. Top: the drained and fixed-strain splits. Bottom: the undrained and fixed-stress splits. The drained and fixed-strain splits become unstable due to water injection whereas the undrained and fixed-stress splits yield stability with good accuracy. . . 206 A.1 Left: the Terzaghi problem. Right: the two dimensional plain strain strip footing consolidation problem. . . . . . . . . . . . . . . . . . . . . . . . . . . 217 A.2 Case A.1 Stability of pressure at early time for Terzaghi’s problem. Top: pressure distributions at early and late time by the nodal based finite element method, where linear interpolation is used for pressure and displacement (Wan, 2002). Bottom: pressure distributions at early and late time by the combined finite-volume and finite-element approach. . . . . . . . . . . . . . 218 xxiv
A.3 Case A.2 Dimensionless pressure (top) and vertical displacement distributions (bottom) at early time, td = 3.7 × 10 −19 . Note that subsidence is positive. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 A.4 Dimensionless vertical-displacement distribution of the top layer at early time, td = 3.7 × 10 −19 . The left part shows large subsidence, and the subsidence decreases along the right direction. . . . . . . . . . . . . . . . . . . . 222 A.5 Case A.2 Convergence of pressure (top) and displacement (bottom). First- order accuracy in time is observed for both pressure and displacement. . . . 223 B.1 Typical shapes of capillary pressure regarding wetting and non-wetting phases (e.g., Sw and Sg, respectively). pco = po − pw, and pcg = pg − po. The capillary pressure curves are drawn based on the work of Lenhard and Parker (1987) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 xxv
Page 1 and 2: SEQUENTIAL METHODS FOR COUPLED GEOM
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48 CHAPTER 3. STABILITY OF THE DRAI
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82 CHAPTER 4. CONVERGENCE OF THE DR
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118 CHAPTER 5. FIXED-STRAIN AND FIX
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164 CHAPTER 6. COUPLED MULTIPHASE F
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208 CHAPTER 7. CONCLUSIONS 7.2 Coup
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210 CHAPTER 7. CONCLUSIONS 7.2.3 Ac
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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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