Sequential Methods for Coupled Geomechanics and Multiphase Flow

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

52 CHAPTER 3. STABILITY OF THE DRAINED AND UNDRAINED SPLITS the drained split can be expressed as ⎡ ⎣ dun dp n ⎤ ⎦ Au dr −→ ⎡ ⎣ dun+1 dp n ⎤ ⎦ Ap dr −→ ⎡ ⎣ dun+1 ⎤ dp n+1 ⎦, where ⎧ ⎪⎨ ⎪⎩ A u dr : Div dσ = 0, δ(dp) = 0, A p dr : ˙ dm + Div dw = 0, dε ˙ = 0, dεp ˙ = 0, dξ ˙ = 0, (3.86) which has homogeneous boundary conditions with no source terms. Since ˙ε, εp, ˙ and ˙ ξ are prescribed, they are not affected by the perturbation of the initial condition, yielding dε ˙ = 0, dεp ˙ = 0, and ˙ dξ = 0. When we solve the mechanical problem Au dr split, we have, dΨ d dt = = = Ω Ω dσ : ˙ dε + dp ρf,0 dm ˙ dΩ − dσ ′ : dεp ˙ + κ · ˙ dξ dΩ Ω D d p dσ : dε ˙ dp + b ρf,0 ˙ dεv dΩ − D d p (from δ(dp) = 0) dp by the drained b Ω ρf,0 ˙ dεvdΩ − D d p ≤ 0 (from Div dσ = 0) . (3.87) Equation 3.87 proves that the drained split is not contractive when we solve the mechanical problem. This non-contractivity of the drained split has been pointed out by Armero and Simo (1992) and Armero (1999). Then, when we solve the flow problem A p dr by the drained split, we have dΨ d dt = Ω dσ : dε ˙ dp + dm ˙ dΩ − dσ ρf,0 Ω ′ : dεp ˙ + dκ · ˙ dξ dΩ = −dp Div(dv)dΩ .. . dε ˙ = 0, dεp ˙ = 0, dξ ˙ = 0 Ω = − dv · µk −1 dvdΩ ≤ 0. (3.88) Ω Dp
3.6. DISCRETE STABILITY OF THE NONLINEAR PROBLEM 53 ⎡ For the undrained split, we have ⎣ dun dp n ⎤ −→ ⎡ dp ∗ ⎤ −→ ⎡ ⎦ Au ud ⎣ dun+1 ⎦ Ap ud ⎣ dun+1 ⎤ dp n+1 ⎦, where ⎧ ⎪⎨ ⎪⎩ A u ud : Div dσ = 0, δdm = 0, A p ud : ˙ dm + Div dw = 0, dε ˙ = 0, dεp ˙ = 0, dξ ˙ = 0, (3.89) which has homogeneous boundary conditions with no source terms. When we solve the mechanical problem A u ud dΨ d dt = Ω by the undrained split, we have dσ : ˙ dε + dp ρf,0 dm ˙ dΩ − dσ ′ : dεp ˙ + dκ · ˙ dξ dΩ Ω D d p≥0 = dσ : ˙ d dεdΩ − Dp Ω = −D d p ≤ 0. (3.90) Note that the operator for flow in the undrained split is the same as that in the drained split, A p ud is the same as Ap dr . Thus, the undrained split satisfies the contractivity condition for both mechanics and flow. 3.6 Discrete Stability of the Nonlinear Problem As shown in the previous section (Armero, 1999), the undrained split honors the dissipative properties of the coupled problem. But, we need to find which algorithm (i.e. time inte- gration technique) provides unconditional stability - so called B-stability. By B-stability we mean the discrete counterpart of contractivity (Simo, 1991; Simo and Govindjee, 1991; Simo and Hughes, 1998), i.e., dχ n+1 N ≤ dχ n N ∀n. (3.91)
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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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208 CHAPTER 7. CONCLUSIONS 7.2 Coup
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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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