Nonlinear domain decomposition scheme for sequential fully implicit formulation of compositional multiphase flow

摘要

New sequential fully implicit (SFI) methods for compositional flow simulation have been recently investigated. These SFI schemes decouple the fully coupled problem into separate pressure and transport problems and have convergence properties comparable with those of the fully implicit (FI) method. The pressure system is a parabolic problem with fixed overall compositions and the transport system is a hyperbolic problem with fixed pressure and total velocity. We discuss some aspects of how to design optimal SFI schemes for compositional flow with general equation-of-states by localizing the computations. The different systems are solved sequentially and the fully implicit solution is recovered by controlling the a posteriori splitting errors due to the choice of decoupling. When the parabolic and the hyperbolic operators are separated, it is possible to design nonlinear domain decomposition schemes taking the advantage of the specific properties of each operator. Usually, for reservoir simulation models, most of the reservoir is converged with SFI methods in one outer iteration. However, in some localized regions with strong coupling between the pressure and the compositions, the SFI algorithms may need several outer iterations. Here, we propose a domain decomposition method based on a predictor-corrector strategy. As a first step, the nonlinear parabolic pressure equation is solved on the whole domain with the Multiscale Restriction-Smooth Basis (MsRSB) method used as a linear domain decomposition solver. In a second step, the compositions system is solved. At the end of this first outer iteration, most of the reservoir is converged. Based on a posteriori splitting errors of the SFI scheme in volume and velocity, we define local regions where additional global outer iterations would be required in the conventional SFI scheme. We then fix Dirichlet boundary conditions for the pressure and the compositions and solve local problems in these non converged regions. After convergence of these smaller nonlinear problems, if the boundary conditions are changed by the updated regions, the global pressure problem is revisited. An additional post-processing of local transport iterations makes sure mass is conserved everywhere. The resulting algorithm converges to the same solution as the FI solver, with all simultaneous updates to composition and pressure in localized regions. We demonstrate the robustness of this nonlinear domain decomposition algorithm across a wide parameter range. Realistic compositional models with gas and water injection are presented and discussed.