Sequential Methods for Coupled Geomechanics and Multiphase Flow

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126 CHAPTER 5. FIXED-STRAIN AND FIXED-STRESS SPLITS From Equation 5.28, the fixed-strain split does not inherit the contractivity relative to the norm · N at the stage of A p sn. As a result, the fixed-strain split is not contractive with respect to the full problem. When solving the mechanical problem after the flow problem, A u sn, we obtain d dχ 2 N dt = = Ω Ω dσ ′ : ˙ dεe − dκ · ˙ dξ + 1 dσ ′ : dεdΩ ˙ − dpd ˙p dΩ M dσ ′ : dεp ˙ + dκ · dξ ˙ dΩ Ω D d p≥0 (from d ˙p = 0) = −D d p ≤ 0, (5.29) dσ Ω ′ : dεdΩ ˙ = 0 from Equation 5.27 . 5.3.2 Contractivity of the fixed-stress split Again we consider two arbitrary initial conditions and study the contractivity of the fixed- stress split. In the fixed-stress split, the original operator A is decomposed as follows: ⎡ ⎣ dun dpn ⎤ ⎦ Ap ss ⎡ −→ ⎣ du∗ dpn+1 ⎤ ⎦ Au ss −→ ⎡ ⎣ dun+1 ⎤ dpn+1 ⎦, where ⎧ ⎪⎨ ⎪⎩ A p ss : ˙ dm + Div dv = 0, δdσv ˙ = 0, A u ss : Div dσ = 0, dp = 0, ⇒ Div dσ ′ = 0, (5.30) which has homogeneous boundary conditions with no source terms. Note that δ ˙ dσv = 0 is equivalent to δ ˙ dσ = 0 in A p ss, since pressure variations affect the volumetric stress, not the shear stress. Using Equation 5.3, the initial conditions of A p ss in Equation 5.30 become Div d ˙σt=0 = 0, Div dσ(t) t=0 = 0. (5.31) First, we show the contractivity of the fixed-stress split when solving the flow problem A p ss. In A p ss of Equation 5.30, δ ˙ dσ = 0 yields ˙ dσ(t) − ˙ dσt=0 = 0. Combining this result
5.4. DISCRETE STABILITY OF THE NONLINEAR PROBLEM 127 with the initial condition in Equation 5.311, we have Div ˙ dσ(t) = Div ˙ dσt=0 = 0. (5.32) Since the divergence operator Div is not a function of time under the infinitesimal transformation, Div ˙ dσ(t) = ∂t Div dσ(t). Then Equations 5.31 and 5.32 lead to Div dσ(t) = Div dσ(t) t=0 = 0. (5.33) Equation 5.33 and the homogeneous boundary condition yield The contractivity of the fixed-stress split is shown as d dχ 2 N dt = Ω Ω dσ : ˙ dε + dp dσ : ˙ dεdΩ = 0. (5.34) ρf,0 dm ˙ dΩ − dσ ′ : dεp ˙ + dκ · dξ ˙ dΩ Ω D d p≥0 = − dv · k Ω −1 µdvdΩ − D d p ≤ 0 (from Equation 5.34). (5.35) Thus, the fixed-stress scheme is contractive relative to the norm · N when solving the flow problem. Since the mechanical problem in the fixed-stress split, A u ss, is the same as that in the fixed-strain split, A u sn, the fixed-stress split satisfies contractivity in the norm · N when solving the mechanical problem as indicated by Equation 5.29. 5.4 Discrete Stability of the Nonlinear Problem The fixed-strain split is not contractive and cannot provide unconditional stability. Thus, we only investigate what time integration scheme can provide unconditional stability for the fixed-stress split. Let (u n , p n , ξ n ) and (ũ n , ˜p n , ˜ ξ n ) be two arbitrary solutions at time tn, yielding (σ ′n , m n , κ n , ε n p) and (˜σ ′n , ˜m n , ˜κ n , ˜εp n ), respectively.
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SEQUENTIAL METHODS FOR COUPLED GEOM
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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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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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176 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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