Sequential Methods for Coupled Geomechanics and Multiphase Flow

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

120 CHAPTER 5. FIXED-STRAIN AND FIXED-STRESS SPLITS linear poroelasticity only). Other values of cp could be used in order to enhance the stability and convergence of sequential schemes, and this has been studied, in the context of linear poroelasticity, by Bevillon and Masson (2000), and Mainguy and Longuemare (2002). In order to properly account for geomechanical effects, a correction needs to be included as a source term. This source term is known as porosity correction, ˙ Φ (Settari and Mourits, 1998; Mainguy and Longuemare, 2002) and takes the following two equivalent expressions: ˙Φ = φ0cp + φ0 − b ˙p − b ˙εv (from Equation 3.10), (5.5) Ks = φ0cp + φ0 − Ks b 1 ˙p − − Kdr Kdr 1 ˙σv (from Equation 3.12). (5.6) Ks Even though the pore-volume compressibility has been recognized as a stabilization term, a complete stability analysis and comparison study of sequential methods including plasticity is lacking. In the next section, we analyze the stability and accuracy of the two sequential implicit methods presented here. 5.2 Stability Analysis for Linear Poroelasticity We use the Von Neumann method to analyze the stability of the fixed-strain and fixed-stress splits. 5.2.1 Fixed-strain split Using the generalized mid-point rule, we can express the total stress σh and velocity Vh in the discretized space as σ n+α h V n+α h = (1 − α)σ n h = (1 − α)V n h + ασn+1 h , (5.7) + αV n+1 h . (5.8) Figure 3.3 shows the one dimensional spatial discretization. The fixed-strain split freezes
5.2. STABILITY ANALYSIS FOR LINEAR POROELASTICITY 121 the variation of the strain rate, that is Full discretization in one dimension yields h ∆P M n j ∆t −( Kdr + bh ∆t (− h Un+α j− 3 2 ∆U n−1 j− 1 2 − 2 Kdr ∆ε n = ∆ε n−1 . (5.9) − ∆U n−1 j+ 1 2 ) − h kp P µh n+α n+α j−1 − 2Pj h Un+α j− 1 2 + Kdr h Un+α j+ 1 2 ) − b(P n+α j−1 + P n+α j+1 = 0, (5.10) − P n+α j ) = 0. (5.11) Introducing solutions of the form U n j = γn e i(j)θ Û and P n j = γn e i(j)θ ˆ P, where γ is the amplification factor, e (·) = exp(·), i = √ −1, and θ ∈ [−π, π], we have ⎡ ⎣ Un j P n j ⎤ ⎦ = γ n ⎡ ⎣ ei(j)θ Û e i(j)θ ˆ P Substituting Equation 5.12 into Equations 5.10 and 5.11, we obtain Gsn ⎤ ⎦. (5.12) ⎡ ⎣ 1 kp∆t θ M h(γ − 1)γ + µh 2((1 − α) + αγ)γ(1 − cos θ) b(γ − 1)2i sin 2 b2i sin θ ⎤⎡ ⎦⎣ Kdr 2 h 2(1 − cos θ) ˆ ⎤ ⎡ P ⎦ = ⎣ Û 0 ⎤ ⎦. 0 (5.13) Since the matrix needs to be singular, detGsn = 0. Then the characteristic equation is obtained and can be written as F α sn(γ) = Kdr kp∆t + Kdr α2(1 − cos θ) γ M µh2 2 + − Kdr kp∆t + Kdr (1 − α)2(1 − cos θ) + b2 γ − b M µh2 2 = 0. (5.14)
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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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28 CHAPTER 3. STABILITY OF THE DRAI
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170 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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