Sequential Methods for Coupled Geomechanics and Multiphase Flow

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184 CHAPTER 6. COUPLED MULTIPHASE FLOW AND GEOMECHANICS d dχm 2 Nm dt = ∂dχm 2 = = 2 ∂dχm ∂dχm Nm : dεe ˙ Nm + · dξ ˙ + ∂dεe ∂dξ 2 Nm dmJ,e ˙ ∂dmJ,e dσ Ω ′ dme : dεe ˙ − − bJdεe,v MJKbK ρ J ˙ dεe,v dme dme ˙ −dκ · dξ ˙ + − bJdεe,v MJK dΩ ρ J ρ K dσ : dε ˙ dp + dmJ ˙ dΩ − dσ ρ ′ : dεp ˙ + dκ · dξ ˙ dΩ Ω = dσ : dε ˙ − dpJ Div(dvJ) Ω = − .. . dσ : dεdΩ ˙ = 0 from Equation 6.781 J Ω D d p dΩ − D d p (from Equation 6.782) dvJ · µk Ω −1 p,JKdvKdΩ − D d p ≤ 0, (6.79) Ω where D d p ≥ 0 for maximum plastic dissipation, and the divergence theorem and Darcy’s law are applied to the last equation. By Darcy’s law in Chapter 2, vJ = −kp,JK GradpK, vJ,i ∈ H (div,Ω), (6.80) where vJ,i is a component of vJ. Note that kp,JK includes viscosity effects, shown in Equation 2.11. Thus, Equation 6.79 yields χm(t) − ˜χm(t) Nm ≤ χm0 − ˜χm0 Nm , ζm(t) − ˜ ζm(t) Tm ≤ ζm0 − ˜ ζm0 Tm , (6.81) where (·) 0 indicates a quantity at the initial (reference) condition. Therefore, coupled multiphase flow and geomechanics are contractive relative to the norms · Nm and · Tm . Since the drained and fixed-strain splits are not contractive in single-phase flow, we exclude them for the investigation of contractivity properties for multiphase flow.
6.4. STABILITY ANALYSIS VIA THE ENERGY METHOD 185 6.4.3 Contractivity of the undrained split Similar to Equation 6.78 of the fully coupled method, Equation 6.4 gives ⎡ ⎣ dun ⎤ ⎡ dpn ⎦ J Au,m ud −→ ⎣ dun+1 dp∗ ⎦ J Ap,m ud −→ ⎣ dun+1 dp n+1 J ⎤ ⎡ ⎤ ⎦, where ⎧ A ⎪⎨ u,m ud : Div dσ = 0, δdmJ = 0, A p,m ud : ˙ dmJ + Div dwJ = 0, ⎪⎩ dε ˙ = 0, dεp ˙ = 0, dξ ˙ = 0, (6.82) where maximum plastic dissipation is assumed for elastoplasticity. We have homogeneous boundary conditions. When solving the mechanical problem A u,m ud obtain d dχm 2 Nm dt = Ω dσ : ˙ dε + dp ρ J dmJ ˙ dΩ − of Equation 6.82, we dσ ′ : dεp ˙ + dκ · dξ ˙ dΩ Ω D d p = dσ : ˙ d dεdΩ − Dp Ω = −D d p ≤ 0 (from Equation 6.821), (6.83) which shows the contractivity property relative to the norm · Nm . Then when solving the flow problem A p,m ud d dχm 2 Nm dt = = = − Ω dσ : ˙ dε + dp ρ J −dpJ Div(dvJ)dΩ Ω of Equation 6.82, dmJ ˙ dΩ − dσ ′ : dεp ˙ + dκ · dξ ˙ dΩ Ω D d p .. . dε ˙ = 0, dεp ˙ = 0, dξ ˙ = 0 dvJ · k Ω −1 p,JKdvKdΩ ≤ 0, (6.84) which shows the contractivity property relative to the norm · Nm . Therefore, the undrained split holds the contractivity property relative to the norms · Nm and · Tm for coupled
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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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2 CHAPTER 1. INTRODUCTION Reservoir
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6 CHAPTER 1. INTRODUCTION undrained
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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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234 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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