Sequential Methods for Coupled Geomechanics and Multiphase Flow

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186 CHAPTER 6. COUPLED MULTIPHASE FLOW AND GEOMECHANICS multiphase flow and mechanics. 6.4.4 Contractivity of the fixed-stress split From Equation 6.10, the fixed-stress split has the form ⎡ ⎣ dun dp n J ⎤ ⎦ Ap,m ss −→ ⎡ ⎣ du∗ dp n+1 J ⎤ ⎦ Au,m ss −→ ⎡ ⎣ dun+1 dp n+1 J ⎤ ⎦, where ⎧ A ⎪⎨ p,m ss : dmJ ˙ + Div dwJ = 0, δ ˙ dσv = 0, ⎪⎩ A u,m ss : Div dσ = 0, dpJ = 0 ⇒ Div dσ ′ = 0, (6.85) which has homogeneous boundary conditions. Note that the maximum plastic dissipation is applied implicitly to multiphase flow, just as it does for single-phase flow. The constraint δ ˙ dσv = 0 yields δ ˙ dσ = 0 because the flow problem A p,m ss does not affect the shear stress field. When we solve the flow problem, A p,m ss , the initial conditions for the stress field (i.e., Div ˙ dσt=0 = 0 and Div dσt=0 = 0) and δ ˙ dσ = 0 yield Then we obtain which is the same as the fixed-stress split for single-phase flow. as Ω Div dσ = 0. (6.86) dσ : ˙ dεdΩ = 0, (6.87) Hence, when solving the flow problem, we show the contractivity of the fixed-stress split
6.4. STABILITY ANALYSIS VIA THE ENERGY METHOD 187 d dχm 2 Nm dt = Ω dσ : ˙ dε + dp ρ J dmJ ˙ dΩ − dσ ′ : dεp ˙ + dκ · dξ ˙ dΩ Ω D d p = − dvJ · k Ω −1 p,JK µdvK dΩ − D d p ≤ 0. (6.88) When solving the mechanical problem A u,m ss , we show the contractivity of the fixed-stress split as d dχm 2 Nm dt = Ω dσ : ˙ dε + dp ρ J dmJ ˙ dΩ − dσ ′ : dεp ˙ + dκ · dξ ˙ dΩ Ω D d p = dσ Ω ′ : dεdΩ ˙ d − Dp = −D d p ≤ 0, (6.89) dσ Ω ′ : dεdΩ ˙ = 0 from Equation 6.852 Therefore, the fixed-stress split holds the contractivity property relative to the norms · Nm and · Tm . 6.4.5 B-stability of the sequential methods Since the undrained and fixed-stress splits are contractive, we study the two sequential schemes for discrete stability, B-stability relative to the norms · Nm or · Tm , which is expressed as dχm n+1 ≤ dχm Nm n , Nm dζm n+1 ≤ dζm Tm n . (6.90) Tm We use the generalized midpoint rule for time discretization. Thus, the return mapping .
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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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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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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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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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