Sequential Methods for Coupled Geomechanics and Multiphase Flow

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

162 CHAPTER 5. FIXED-STRAIN AND FIXED-STRESS SPLITS Dimensionless pressure P d =(P−P i )/P i 0.8 0.7 0.6 0.5 0.4 0.3 Pressure history at (2,5), τ=∞, α=1.0 K est =1.05× K1D dr dr K est =0.95× K1D dr dr K est dr =K3D dr FC 0 0.2 0.4 0.6 0.8 1 1.2 x 10 −4 0.2 Dimensionless time t =4c t/Lz/Lx d v Figure 5.18: Pressure history at the observation well (2,5) for Case 5.6 with different devi- ation factors when 0.95 × K 1D dr , 1.05 × K1D dr , and K3D dr are taken for Kest dr .
Chapter 6 Coupled Multiphase Flow and Geomechanics 6.1 Sequential Methods for Multiphase Flow Here, we analyze the stability and convergence of sequential implicit methods for multiphase flow. Let us denote by A m the original operator of coupled multiphase flow and mechanics, shown in Equations 2.1 and 2.2 in Chapter 2. The superscript m in the operators, (·) m , means multiphase. Using the fully coupled method, the discrete approximation of A m is written as: ⎡ ⎤ ⎡ ⎣ un pn ⎦ J Am fc −→ ⎣ un+1 p n+1 J ⎤ ⎦, where A m fc : ⎧ ⎪⎨ Div σ + ρbg = 0, ⎪⎩ ˙ mJ + Div wJ = (ρf)J, (6.1) where we solve the coupled problem simultaneously using the Newton-Raphson method. Then the fully coupled method leads to the following system of Equations: ⎡ ⎣ Km −LT ⎤⎡ m ⎦⎣ Lm F m δu ⎤ ⎦ δpJ Jfc,m 163 n+1,k = − ⎡ ⎣ Ru R p J ⎤ ⎦ n+1,k , (6.2)
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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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100 CHAPTER 4. CONVERGENCE OF THE D
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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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