Sequential Methods for Coupled Geomechanics and Multiphase Flow

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

32 CHAPTER 3. STABILITY OF THE DRAINED AND UNDRAINED SPLITS that ϕj takes a constant value of 1 over element j and 0 at all other elements (Figure 3.1 right plot). Therefore, Equation 3.15 can be interpreted as a mass conservation statement, element-by-element. The second term can be integrated by parts to arrive at the sum of integral fluxes, Vh,ij, between element i and its adjacent elements j: ϕi Div vh dΩ = − Ω ∂Ωi vh · ni dΓ = − nface j=1 Γij nface vh · ni dΓ = − Vh,ij. (3.18) The inter-element flux can be evaluated using a two-point or a multipoint flux approximation (Aavatsmark, 2002). The interpolation functions for the displacement vectors are the usual C 0 -continuous isoparametric functions, such that ηb takes a value of 1 at node b, and 0 at all other nodes (the left of Figure 3.1). Inserting the interpolation from Equations 3.16–3.17, and testing Equations 3.14–3.15 against each individual shape function, the semi-discrete finite- element/finite-volume equations can be written as Ωi 1 dPi M dt Ω B T a σh dΩ = dΩ + Ωi b dεv dt Ω ηaρbg dΩ + j=1 Γσ nface dΩ − Vh,ij = Ωi j=1 ηa ¯t dΓ ∀a = 1, . . .,nnode, (3.19) f dΩ, ∀i = 1, . . .,nelem. (3.20) The matrix Ba is the linearized strain operator, which in 2D takes the form ⎡ ⎤ ∂xηa ⎢ Ba = ⎢ 0 ⎣ 0 ⎥ ∂yηa⎥ . ⎦ (3.21) ∂yηa ∂xηa The stress and strain tensors are expressed in compact engineering notation (Hughes, 1987). For example, in 2D, ⎡ ⎢ σh = ⎢ ⎣ σh,xx σh,yy σh,xy ⎤ ⎡ ⎥ ⎦ , εh ⎢ = ⎢ ⎣ εh,xx εh,yy 2εh,xy ⎤ ⎥ ⎥. (3.22) ⎦
3.3. OPERATOR SPLITTING 33 The stress–strain relation for linear poroelasticity takes the form: σh = σ ′ h − bph1, δσ ′ h = Dpsδεh, (3.23) where σ ′ is the effective stress tensor, and Dps is the elasticity matrix which, for 2D plane strain conditions, can be written as Dps = E(1 − ν) (1 + ν)(1 − 2ν) ⎡ 1 ⎢ ν ⎣ ν 1−ν ν 1−ν ν 1−ν 1 1−ν ν 1−ν ν 1−ν where E is the Young modulus, and ν is the Poisson ratio. 1 ⎤ ⎥ , (3.24) ⎦ The coupled equations that describe quasi-static poromechanics form an elliptic–parabolic system. A fully discrete system of equations can be obtained by further discretizing in time the mass accumulation term in Equations 3.19–3.20. We use the generalized mid-point rule for time discretization. 3.3 Operator Splitting The fully coupled method solves the equations of flow and mechanics simultaneously and obtains a converged solution through iteration, typically using the Newton-Raphson method (Lewis and Sukirman, 1993; Wan et al., 2003; Jha and Juanes, 2007; Jean et al., 2007). The fully coupled method is unconditionally stable, but requires high computational cost and the development of a unified simulator for flow and mechanics. It is thus desirable to develop solution methods that have stability properties comparable to the fully coupled approach, but that are computationally more efficient and easier to implement. Sequential solution methods offer such a possibility. With sequential methods, we can employ separate flow and mechanics simulators. There are two representative sequential strategies, in which one solves the mechanical problem first. One is the drained split shown in the left diagram of Figure 3.2. The other
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SEQUENTIAL METHODS FOR COUPLED GEOM
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splits, are, at best, conditionally
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118 CHAPTER 5. FIXED-STRAIN AND FIX
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164 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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