Summary In this paper, we present efficient multiscale preconditioners for solving multiphase compressible flow in highly heterogeneous porous media. The proposed method employs finite volume discretization for spatial conservation and a backward Euler scheme for time integration, ensuring stability and local conservation properties. The resulting nonlinear system is addressed through two strategies: (1) a fully coupled approach using Newton’s method with constrained pressure residual (CPR) preconditioning and (2) a decoupled sequential fully implicit (SFI) approach that iteratively solves pressure and saturation subproblems with inner Newton iterations. In both strategies, solving the pressure equation remains the primary computational bottleneck due to its elliptic nature and the challenges posed by highly heterogeneous permeability fields. To address this, we develop multiscale-based two-grid and three-grid preconditioners to accelerate the pressure solve. These preconditioners are built upon local spectral basis functions that capture fine-scale heterogeneities and enable efficient coarse-grid corrections. This work is, to the best of our knowledge, the first to extend general multiscale (GMs) basis-based preconditioning—previously restricted to incompressible single- and two-phase problems—to the challenging setting of compressible multiphase (black-oil) flow. We demonstrate the effective integration of these multiscale preconditioners within both CPR and SFI frameworks. Their efficiency and practical applicability are further highlighted through tests on 3D and irregular industrial reservoir models. Numerical experiments on benchmark problems demonstrate that the proposed method is highly competitive with the algebraic multigrid (AMG) method.
Shubin Fu (Thu,) studied this question.