Sequential Methods for Coupled Geomechanics and Multiphase Flow

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148 CHAPTER 5. FIXED-STRAIN AND FIXED-STRESS SPLITS Dimensionless pressure (P/P i ) Dimensionless pressure (P/P i ) 1 0.95 0.9 0.85 0.8 0.75 0.7 Case 5.1. The Terzaghi problem, τ=16.67 Fixed stress 1iter Fully coupled Undrained 10iter Undrained 1iter 0.65 0 0.02 0.04 0.06 0.08 1 0.95 0.9 0.85 0.8 0.75 0.7 Dimensionless time (t d =4c v t/(Lz) 2 ) Case 5.2. τ=12.12 Fixed stress 1iter Fully coupled Undrained 10ter Undrained 1iter 0.65 0 0.05 0.1 0.15 0.2 0.25 Dimensionless time (pore volume produced) Figure 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.
5.6. NUMERICAL EXAMPLES 149 and 5.10). This behavior is the same as the drained split. However, the solutions from the fixed-stress split are stable regardless of the coupling strength, and they match the analytical solutions. Case 5.4—Fluid production scenario in 2D with elastoplasticity We consider a nonlinear poroelastoplastic problem using the staggered method, where the numerical stability is compared with a-priori stability estimates of the fixed-strain and fixed- stress splits. The modified Cam-clay model is used for plastic modeling (Borja and Lee, 1990; Borja, 1991). We adopt the associative plasticity formulation (Coussy, 1995; Simo, 1991; Simo and Hughes, 1998) for Case 5.4. We use the backward Euler time discretization. The input values for Case 5.4 are the same as Case 3.5. In Case 5.4, compaction occurs because of production. As a result, subsidence occurs and the fluid pressure decreases. Figure 5.11 shows that the fixed-strain split becomes unstable when it enters the plastic regime, even though it is stable in the elastic regime. This is because plasticity increases the coupling strength beyond unity. The fixed-stress split is, however, stable in the plastic regime. Around td = 0.018, we observe some roughness in the pressure solution because we treat the relaxation term b 2 /Kdr explicitly. The solution from fixed-stress split is slightly different from the fully coupled solution, since one iteration is performed. But, when two iterations are taken, the solutions by the fixed-stress split match those from the fully coupled method, as shown in Figure 5.11. That figure also shows that plasticity can slow down the rate of decline in the reservoir pressure. Staggered Newton schemes for Case 5.4 We apply a staggered Newton scheme to Case 5.4. Figure 5.12 shows that the fixed-strain split is stable during the early-time elastic regime, which has a weak coupling strength (τ < 1). However, when plasticity is reached, the solution by the fixed-strain method is no longer stable because the coupling strength increases beyond unity. The bottom of Figure 5.11 shows the variation of the coupling strength during the simulation. It is clear that when the coupling strength increases beyond unity, the solution by the fixed-strain
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SEQUENTIAL METHODS FOR COUPLED GEOM
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I certify that I have read this dis
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splits, are, at best, conditionally
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deeply. There was deep discussion o
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3 Stability of the Drained and Undr
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6.2.3 Undrained split . . . . . . .
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3.7 Left: the Terzaghi problem in 1
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4.1 The distribution of the error a
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5.3 Case 5.2 (1D injection-producti
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5.18 Pressure history at the observ
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2 CHAPTER 1. INTRODUCTION Reservoir
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4 CHAPTER 1. INTRODUCTION problem (
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6 CHAPTER 1. INTRODUCTION undrained
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8 CHAPTER 1. INTRODUCTION where A c
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10 CHAPTER 1. INTRODUCTION Even whe
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12 CHAPTER 1. INTRODUCTION
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14 CHAPTER 2. FORMULATION full form
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16 CHAPTER 2. FORMULATION dE dt =
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18 CHAPTER 2. FORMULATION where the
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20 CHAPTER 2. FORMULATION coupling
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22 CHAPTER 2. FORMULATION where Div
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24 CHAPTER 2. FORMULATION between g
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26 CHAPTER 2. FORMULATION
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28 CHAPTER 3. STABILITY OF THE DRAI
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82 CHAPTER 4. CONVERGENCE OF THE DR
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198 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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