Sequential Methods for Coupled Geomechanics and Multiphase Flow

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

74 CHAPTER 3. STABILITY OF THE DRAINED AND UNDRAINED SPLITS for the modified Cam-clay model is shown in the top picture of Figure 3.19 (refer to Borja and Lee (1990); Borja (1991) for more details). Case 3.4—1D fluid injection and production in elastoplasticity The constrained modulus is Kdr = 1 GPa and the Biot modulus is M = 83.3 MPa. The hardening moduli are unspecified for the numerical tests. The other data and geometry of the domain are the same as in Case 3.2. The coupling strength τ is 0.083 when the me- chanical problem is elastic. An iterative staggered method is used, whereby. Each iteration involves solving two problems in sequence, such that each problem is solved implicitly. So, for a given time step, a single-pass strategy would entail solving the first sub-problem implicitly (subject to a given tolerance), updating the appropriate terms to set up the second problem, and finally solving the second problem implicitly. We then move to the next time step. In the iterative approach, the implicit-implicit solution sequence is repeated until convergence. Figure 3.17 shows the numerical results for Case 3.4. On the top figure, the yield stress and hardening modulus are σ ′ Y = 1.5 MPa and H = 250 MPa, respectively. This condition yields a coupling strength less than one for plasticity, τ = 0.417. The drained split is stable in the elastic regime, since the coupling strength is less than one. After the simulation enters the plastic regime, at about td = 0.05, the drained split is still stable, shown on the top of Figure 3.17, because the coupling strength is still less than one. The case that the yield stress and hardening modulus are σ ′ Y = 1.5 MPa and H = 25 MPa, respectively, is shown in Figure 3.17 (bottom). These conditions yield a coupling strength greater than one for plasticity, τ = 3.41. The bottom figure shows that the drained split is unstable in the plastic regime because the coupling strength is higher than one. This supports the a-priori stability estimates from the Von Neumann method. Moreover, the instability of the drained split cannot be fixed by tuning the time step size, since the stability limit is independent of the time step size. However, the undrained split is stable regardless of the coupling strength, as shown in Figure 3.17.
3.7. NUMERICAL EXAMPLES 75 Dimensionless pressure(P/P i ) Dimensionless pressure(P/P i ) 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 Case 3.4. τ 0.083 −> 0.417 Drained Undrained Fully Coupled 0.6 0 0.05 0.1 0.15 0.2 Dimensionless time (pore volume produced) 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 Case 3.4. τ 0.083 −> 3.41 Drained Undrained Fully Coupled 0.6 0 0.05 0.1 0.15 0.2 Dimensionless time (pore volume produced) Figure 3.17: 1D problem with injection and production with the isotropic hardening (Case 3.4). Top: the drained split is stable in plastic regime, where τ = 0.417. Bottom: the drained split is unstable in the plastic regime, where τ = 3.41.
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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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124 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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