Sequential Methods for Coupled Geomechanics and Multiphase Flow

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30 CHAPTER 3. STABILITY OF THE DRAINED AND UNDRAINED SPLITS The boundary conditions for the mechanical problem are u = ū (prescribed displacement) on Γu and σ · n = ¯t (prescribed traction) on Γσ. Again, we assume Γu ∩ Γσ = ∅, and Γu ∪ Γσ = ∂Ω. The initial displacements and strains are, by definition, equal to zero. The initial con- dition of the coupled problem is p|t=0 = p0 and σ|t=0 = σ0. The initial stress field must satisfy mechanical equilibrium. 3.2 Discretization The finite-volume method is widely used by the reservoir simulation community (Aziz and Settari, 1979), whereas most numerical models in geotechnical engineering and thermo- mechanics use nodal-based finite-element discretizations (Zienkiewicz et al., 1988; Lewis and Sukirman, 1993; Lewis and Schrefler, 1998; Armero and Simo, 1992; Armero, 1999; Wan et al., 2003; White and Borja, 2008). In the finite-volume method for the flow problem, the pressure is located at the cell (grid block) center. In the nodal-based finite-element method for the mechanical problem, the displacement vector is located at vertices (Hughes, 1987). This space discretization has the following characteristics: local mass conservation at the element level, a continuous displacement field, which allows for tracking the deforma- tion, and convergent approximations with the lowest order discretization (Jha and Juanes, 2007). Since we assume slightly compressible fluid flow, the given space discretization pro- vides a stable pressure field at early time (Phillips and Wheeler, 2007a,b), while for nodal based finite element methods, spurious pressure oscillations are identified for the equal-order approximations of pressure and displacement (e.g., piecewise continuous interpolation) and incompressible fluid flow (Vermeer and Verruijt, 1981; Murad and Loula, 1992, 1994; White and Borja, 2008). Stabilization techniques to deal with spurious pressure oscillations have been studied by several authors (Murad and Loula, 1992, 1994; Wan, 2002; Wan et al., 2003; Truty and Zimmermann, 2006; White and Borja, 2008). Refer to Appendix A. for further discussion on the finite-volume and finite-element methods. Let the domain be partitioned into nonoverlapping elements (grid blocks), Ω = ∪ ne j=1 Ωj,
3.2. DISCRETIZATION 31 where ne is the number of elements. Let Q ⊂ L 2 (Ω) and U ⊂ (H 1 (Ω)) d (where d = 2, 3 is the number of space dimensions), be the functional spaces of the solution for pressure, p, and displacements, u. Let Q0 and U0 be the corresponding spaces for the test functions ϕ and η, for flow and mechanics, respectively (Jha and Juanes, 2007). Let Qh, Qh,0, Uh and Uh,0 be the corresponding finite-dimensional subspaces. Then the discrete approximation of the weak form of the governing equations 3.1 and 3.9 becomes: find (uh, ph) ∈ Uh × Qh such that Ω Grad s ηh : σh dΩ = ηh · ρbg dΩ + Ω 1 dmh dΩ + ϕh Div vh dΩ = dt Ω ρf,0 ϕh Ω The pressure and displacement fields are approximated as follows: ph = uh = nelem j=1 nnode b=1 Γσ Ω η h · ¯t dΓ ∀η h ∈ Uh,0, (3.14) ϕhf dΩ, ∀ϕh ∈ Qh,0. (3.15) ϕjPj, (3.16) ηbUb, (3.17) where nnode is the number of nodes, Pj are the element pressures, and Ub are the displacement vectors at the element nodes (vertices). Figure 3.1: Element shape functions for displacement (left) and pressure (right) in 2-D We restrict our analysis to pressure shape functions that are piecewise constant, so
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SEQUENTIAL METHODS FOR COUPLED GEOM
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80 CHAPTER 3. STABILITY OF THE DRAI
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82 CHAPTER 4. CONVERGENCE OF THE DR
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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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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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