Sequential Methods for Coupled Geomechanics and Multiphase Flow

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6.2.3 Undrained split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 6.2.4 Fixed-strain split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 6.2.5 Fixed-stress split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 6.3 Staggered Newton scheme with IMPES . . . . . . . . . . . . . . . . . . . . 177 6.3.1 Drained split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 6.3.2 Undrained split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 6.3.3 Fixed-strain split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 6.3.4 Fixed-stress split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 6.4 Stability Analysis via the Energy Method . . . . . . . . . . . . . . . . . . . 181 6.4.1 Constitutive relations for multiphase flow and elastoplasticity . . . . 181 6.4.2 Contractivity of multiphase flow . . . . . . . . . . . . . . . . . . . . 182 6.4.3 Contractivity of the undrained split . . . . . . . . . . . . . . . . . . 185 6.4.4 Contractivity of the fixed-stress split . . . . . . . . . . . . . . . . . . 186 6.4.5 B-stability of the sequential methods . . . . . . . . . . . . . . . . . . 187 6.4.6 B-stability of the undrained split . . . . . . . . . . . . . . . . . . . . 188 6.4.7 B-stability of the fixed-stress split . . . . . . . . . . . . . . . . . . . 191 6.5 Numerical Examples with the staggered Newton methods . . . . . . . . . . 195 6.5.1 Case 6.1—Water injection and oil production in the 1D poroelasticity 196 6.5.2 Case 6.2—Water injection and oil production in 2D with poroelasticity200 6.6 Numerical Examples with the staggered method . . . . . . . . . . . . . . . 205 7 Conclusions 207 7.1 Formulation of Coupled Flow and Mechanics . . . . . . . . . . . . . . . . . 207 7.2 Coupled Mechanics and Single-Phase flow . . . . . . . . . . . . . . . . . . . 208 7.2.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 7.2.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 7.2.3 Accuracy and Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . 210 7.3 Coupled Mechanics and Multiphase Flow . . . . . . . . . . . . . . . . . . . 210 7.4 Finite-Volume and Finite-Element Methods . . . . . . . . . . . . . . . . . . 211 xii
7.5 Recommendation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 A Stability in Space 213 A.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 A.2 Finite-Volume and Finite-Element Methods . . . . . . . . . . . . . . . . . . 216 A.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 A.3.1 Case A.1—Terzaghi’s problem . . . . . . . . . . . . . . . . . . . . . 217 A.3.2 Case A.2—Two dimensional plain strain strip footing consolidation problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 B Miscellaneous Derivations 225 B.1 Positive-Definite Biot’s Moduli, M . . . . . . . . . . . . . . . . . . . . . . . 225 B.2 Accuracy of the Fully Coupled Method . . . . . . . . . . . . . . . . . . . . . 227 B.3 Drained Split in Coupled Flow and Dynamics . . . . . . . . . . . . . . . . . 228 B.4 Non-Contractivity with Different Constitutive Relations . . . . . . . . . . . 228 B.5 δφ for single-phase flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 C Constitutive Relations in Reservoir Simulation 235 C.1 po, pw, pg formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 C.2 po, sw, sg formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 xiii
Page 1 and 2: SEQUENTIAL METHODS FOR COUPLED GEOM
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Page 6 and 7: splits, are, at best, conditionally
Page 8 and 9: deeply. There was deep discussion o
Page 10 and 11: 3 Stability of the Drained and Undr
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Page 28 and 29: 2 CHAPTER 1. INTRODUCTION Reservoir
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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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