Sequential methods for coupled geomechanics and multiphase flow.

摘要

We study sequential solution methods for coupled multiphase flow and geomechanics. Sequential methods are desirable from a software development perspective. If the sequential solution strategies have stability and convergence properties that are close to those of the fully coupled approach, they can be very competitive for solving problems of practical interest. This is because of the high investment cost associated with developing unified flow-mechanics simulators and the high computational cost of the fully coupled (i.e., simultaneous solution) method. In these sequential-implicit solution strategies, the flow and mechanics problems are solved in sequence. Implicit time discretization is used when solving each of the flow and mechanics problems. The specific details of the form of the coupling scheme and how the two problems of flow and mechanics communicate play important roles in the viability of the coupling strategy. Here, four sequential coupling methods are considered in great detail: drained, undrained, fixed-strain, and fixed-stress splits. For space discretization, we employ a finite-volume method for flow and a finite-element approach for mechanics. This space discretization yields stable solutions at early time and allows for using existing flow and mechanical simulators. The drained and undrained splits solve the mechanical problem first, whereas the fixed-strain and fixed-stress splits solve the flow problem first. The stability and convergence properties for single-phase flow are analyzed for the four sequential-implicit methods. The Von Neumann and energy methods are used to analyze the stability of the linear and nonlinear problems, respectively. The derived stability estimates indicate that the drained and fixed-strain splits, which are the obvious splits, are, at best, conditionally stable. Moreover, their stability limit depends on the coupling strength only and is independent of time step size. On the other hand, the derived a-priori estimates indicates that the undrained and fixed-stress splits are unconditionally stable regardless of the coupling strength. All the results have been verified by performing numerical simulations for several test cases.;To analyze the convergence rates of the various coupling algorithms, we use matrix and spectral methods. The drained and fixed-strain splits can suffer from non-convergence, even when they are stable. On the other hand, the undrained split yields first-order accuracy in time for a compressible fluid, but it exhibits slow convergence rates for high coupling strength and suffers from non-convergence for purely incompressible systems (solid grains and fluid). The fixed-stress split shows first-order accuracy in time regardless of the fluid type and coupling strength, and it yields a less stiff mechanical problem. Furthermore, the fixed-stress split requires only a few iterations to converge, even for very difficult problems with strong coupling.;The stability and convergence behaviors of the four sequential methods for coupled multiphase flow and geomechanics are also analyzed using spectral and energy methods. The formulation for the flow part can be either fully implicit, or IMPES (IMplicit Pressure, Explicit Saturations). The derived a-priori estimates for the four sequential methods are similar to their single-phase counterparts. That is, the undrained and fixed-stress splits show unconditional stability, and the fixed-stress split exhibits faster convergence rates compared with the other sequential methods. Therefore, we strongly recommend the fixed-stress split with backward Euler time integration, a finite-volume scheme for flow, and a finite-clement discretization for mechanics.