Sequential Methods for Coupled Geomechanics and Multiphase Flow

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$Drilling-induced tensile wall-fractures - Stanford University$

3 Stability of the Drained and Undrained Splits 27 3.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3 Operator Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3.1 Fully coupled method . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3.2 Drained split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.3.3 Undrained split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4 Stability Analysis for Linear Poroelasticity . . . . . . . . . . . . . . . . . . . 37 3.4.1 Drained split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.4.2 Undrained split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.5 Contractivity of the Nonlinear Continuum Problem . . . . . . . . . . . . . . 46 3.6 Discrete Stability of the Nonlinear Problem . . . . . . . . . . . . . . . . . . 53 3.7 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.7.1 Numerical results for elasticity . . . . . . . . . . . . . . . . . . . . . 58 3.7.2 Numerical results for elastoplasticity . . . . . . . . . . . . . . . . . . 69 4 Convergence of the Drained and Undrained Splits 81 4.1 Error Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.1.1 Error estimation of the drained split by matrix algebra . . . . . . . . 81 4.1.2 Error estimation of the undrained split by matrix algebra . . . . . . 85 4.2 Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.2.1 Error amplification of the drained split . . . . . . . . . . . . . . . . . 88 4.2.2 Comparison with the coupled flow and dynamics . . . . . . . . . . . 90 4.2.3 Error amplification of the undrained split . . . . . . . . . . . . . . . 94 4.3 Convergence Rate with Full Iterations . . . . . . . . . . . . . . . . . . . . . 95 4.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.4.1 Convergence behavior for a fixed number of iterations . . . . . . . . 96 4.4.2 Convergence rate with full iterations . . . . . . . . . . . . . . . . . . 110 x
5 Fixed-strain and Fixed-stress Splits 117 5.1 Operator Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.1.1 Fixed-strain split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.1.2 Fixed-stress split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.1.3 Conventional reservoir simulation . . . . . . . . . . . . . . . . . . . . 119 5.2 Stability Analysis for Linear Poroelasticity . . . . . . . . . . . . . . . . . . . 120 5.2.1 Fixed-strain split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.2.2 Fixed-stress split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.3 Contractivity of the Nonlinear Continuum Problem . . . . . . . . . . . . . . 125 5.3.1 Non-contractivity of the fixed-strain split . . . . . . . . . . . . . . . 125 5.3.2 Contractivity of the fixed-stress split . . . . . . . . . . . . . . . . . . 126 5.4 Discrete Stability of the Nonlinear Problem . . . . . . . . . . . . . . . . . . 127 5.5 Error Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 5.5.1 Error amplification of the fixed-strain split . . . . . . . . . . . . . . 134 5.5.2 Error amplification of the fixed-stress split . . . . . . . . . . . . . . . 136 5.6 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.6.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.6.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 5.6.3 Effect of the deviation factor η . . . . . . . . . . . . . . . . . . . . . 158 6 Coupled Multiphase Flow and Geomechanics 163 6.1 Sequential Methods for Multiphase Flow . . . . . . . . . . . . . . . . . . . . 163 6.1.1 Drained split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 6.1.2 Undrained split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 6.1.3 Fixed-strain split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 6.1.4 Fixed-stress split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 6.2 Staggered Newton schemes for Multiphase Flow . . . . . . . . . . . . . . . . 167 6.2.1 Convergence of FIM . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 6.2.2 Drained split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 xi
Page 1 and 2: SEQUENTIAL METHODS FOR COUPLED GEOM
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Page 28 and 29: 2 CHAPTER 1. INTRODUCTION Reservoir
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34 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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