Sequential Methods for Coupled Geomechanics and Multiphase Flow

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82 CHAPTER 4. CONVERGENCE OF THE DRAINED AND UNDRAINED SPLITS where ets is the error between the true solution and the numerical solutions from the sequential method, and · is an appropriate norm (e.g., L 2 norm). e n+1 fs = xn+1 f − xn+1,niter s is the error between the fully coupled and sequential methods, n is the time step, and niter is the iteration number within a time step, and x t f = [ut , p t ]. The subscripts f and s (i.e., (·)f and (·)s) denote the fully coupled and sequential methods, respectively, the subscript t (i.e., (·)t) denotes the true solution. If efs is O(∆t m ), where m > 0, the numerical scheme is convergent (Strikwerda, 2004). If efs is O(1), the numerical scheme is not convergent. In Appendix B.2, we show that x n+1 − xn+1 has O(∆t) with the backward Euler time discretization. t f For the fully coupled method, the algebraic form of the coupled problem is written as ⎡ ⎤⎡ K −Lt ⎣ ⎦⎣ L F A u ⎤n+1 ⎡ ⎤⎡ 0 0 ⎦ − ⎣ ⎦⎣ p L Q B u ⎤n ⎡ ⎦ = ⎣ p fu ⎤n+1 ⎦ , fp ⎡ f x = ⎣ u ⎤ ⎦, F = Q + ∆tT, p (4.2) where K is the stiffness matrix with the drained moduli, and F is composed of Q and T (Lewis and Schrefler, 1998). Q is the fluid compressibility matrix, which includes the Biot modulus, and T is the transmissibility matrix of the flow problem. L is associated with the coupling (Biot) coefficient. u and p are the displacement and pressure, respectively. The drained split decomposes the matrix A into ⎡ ⎤⎡ K 0 ⎣ ⎦⎣ L F u ⎤ ⎦ p ⎡ ⎤⎡ 0 0 − ⎣ ⎦⎣ L Q u ⎤ ⎦ p n+1,k+1 n ⎡ ⎤⎡ 0 Lt − ⎣ ⎦⎣ 0 0 u ⎤ ⎦ p ⎡ ⎤ = ⎣ fu fp ⎦ n+1 n+1,k , (4.3) where k is the iteration index. From Equations 4.2 and 4.3, the errors of pressure and
4.1. ERROR ESTIMATION 83 displacement between the fully coupled method and drained split are written as ⎡ ⎣ efs,u where efs = efs,p ⎤ ⎦ n+1,k+1 ⎡ ⎤ ⎡ ⎤⎡ K 0 = ⎣ ⎦ L F ⎜ 0 Lt ⎝⎣ ⎦⎣ 0 0 efs,u ⎤ ⎦ efs,p ⎡ K = ⎣ −1 −F 0 −1LK−1 F−1 ⎤⎛⎡ ⎤⎡ ⎦⎜ 0 Lt ⎝⎣ ⎦ 0 0 ⎡ ⎤⎡ 0 0 + ⎣ ⎦⎣ L Q efs,u ⎤n⎞ ⎦ ⎠ efs,p ⎡ ⎤⎡ ⎤ = −1 ⎛ ⎣ 0 K−1Lt 0 −F−1LK−1Lt ⎡ ⎦ D ⎣ efs,u efs,p e n+1,niter fs ⎤ ⎦, and e n fs ⎣ efs,u efs,p ⎦ n+1,k n+1,k ⎣ efs,u ⎡ ⎤ ⎡ 0 0 + ⎣ ⎦ L Q efs,p ⎡ + ⎣ = en,niter fs . Then e n+1,niter fs = D niter niter n+1,0 e + l=1 ⎤ ⎦ D l−1 He n,niter fs = D niter n+1 (xf − xnf ) + (Dniter niter + where en+1,0 is written as (x n+1 f −xn f )+(xn f −xn,niter s in terms of the initial error e 0,0 fs , Dniter , and S as e n+1,niter fs = n l=0 D niter S l (x n+1−l f n+1,k 0 0 F −1 L F −1 Q ⎣ efs,u efs,p ⎤⎡ ⎦ H can be written as l=1 S ⎤ ⎦ ⎣ efs,u efs,p D l−1 H) e n,niter fs , n ⎞ ⎟ ⎠ ⎤ ⎦ n , (4.4) (4.5) ). By recursion, e n+1,niter is expressed − x n−l f )+S n+1 e 0,0 fs . (4.6)
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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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30 CHAPTER 3. STABILITY OF THE DRAI
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132 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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