Sequential Methods for Coupled Geomechanics and Multiphase Flow

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

8 CHAPTER 1. INTRODUCTION where A can be linear, or nonlinear, y is a solution vector, and ˙ () is the time derivative, we solve two subproblems in sequence as follows: ˙y(t) = A1y(t) problem 1 and ˙y(t) = A2y(t) . problem 2 (1.2) In this case, even when one iteration is performed (i.e., a staggered method), the sequential method from Equation 1.2 is still convergent by Lie’s formula (Chorin et al., 1978; Lapidus, 1981). When the operator splitting of Equation 1.2 is applied to the coupled heat flow and mechanical-dynamics, convergence with first-order accuracy in time is obtained, as long as the algorithm is stable (Armero and Simo, 1992). In general, however, sequential methods do not guarantee convergence for a fixed number of iterations, even when they are numerically stable (Turska et al., 1994). Operator splitting of coupled flow and geomechanics can be written as: 0 = A1y(t) mechanics and 0 = A2(y(t), ˙y(t)) , flow (1.3) which is different from the sequential method of Equation 1.2, where the mechanical problem is elliptic. Thus, Lie’s formula cannot be applied to Equation 1.3, and it is not clear whether sequential methods are convergent for a fixed iteration number, even though they may be stable. Armero (1999) mentions that the undrained split may suffer from accuracy issues for strongly coupled problems. The convergence of sequential methods with a fixed iteration number is important, since a fixed iteration number is typically required in order to save computational resources. Vijalapura and Govindjee (2005) propose a hybrid scheme of staggered and fully coupled methods, in which the time step size is controlled for accuracy based on the assumption that refining the time step size improves the accuracy of the staggered method. Further- more, Vijalapura et al. (2005) investigated the order of accuracy in the index-1 differential algebraic equations because the mechanical problem in Equation 1.3 can be viewed as an
1.5. FINITE-VOLUME AND FINITE-ELEMENT METHODS 9 algebraic constraint (Brenan et al., 1996). They claimed that the differential algebraic equations have first-order accuracy when the first step in the algebraic equation is redundant. However, this is not applicable to the coupled flow and quasi-static mechanics because the mechanical problem is not redundant for the first time step. For example, consolidation problems are often driven by instant loading in the mechanical problem during the first time step. Rather, the work of Vijalapura et al. (2005) supports the fact that typical sequential methods can provide zeroth order accuracy in time, when the first step in the algebraic equation is not redundant. 1.5 Finite-Volume and Finite-Element Methods Numerical experiments are performed using the finite-volume and finite-element methods (the FVM and FEM) for flow and mechanics, respectively. The benefits from this discretization strategy are as follows. 1. The choice for the discretization and the primary unknowns are practical and natural in order to make use of a reservoir simulator and a geomechanics simulator in a sequential fashion. 2. Reservoir simulation with the finite-volume method yields local mass conservation, while that with the finite-element method is not locally conservative. 3. This mixed-space discretization strategy avoids spatial oscillations at early time, which are typically encountered in consolidation problems. There are two reasons for the early time spatial oscillations in the nodal based finite- element methods. One is the violation of the LBB condition (Fortin and Brezzi, 1991) and the other is the discontinuity of pressure at the drainage boundary. At early time, the coupled problem converges to the undrained mechanical response. The LBB condition is required to solve the incompressible mechanical problem (Fortin and Brezzi, 1991), and it must be satisfied when the fluid and solid grains are incompressible (Murad and Loula, 1992, 1994; Truty and Zimmermann, 2006; White and Borja, 2008).
Page 1 and 2: SEQUENTIAL METHODS FOR COUPLED GEOM
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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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