A multiscale restriction-smoothed basis method for compressible black-oil models

Simulation problems encountered in reservoir management are often computationally expensive because of the complex fluid physics for multiphase flow and the large number of grid cells required to honor geological heterogeneity. Multiscale methods have been proposed as a computationally inexpensive alternative to traditional fine-scale solvers for computing conservative approximations of the pressure and velocity fields on high-resolution geo-cellular models. Although a wide variety of such multiscale methods have been discussed in the literature, these methods have not yet seen widespread use in industry. One reason may be that no method has been presented so far that handles the combination of realistic flow physics and industry-standard grid formats in their full complexity. Herein, we present a multiscale method that handles both the most wide-spread type of flow physics (black-oil type models) and standard grid formats like corner-point, stair-stepped, PEBI, as well as general unstructured, polyhedral grids. Our approach is based on a finite-volume formulation in which the basis functions are constructed using restricted smoothing to effectively capture the local features of the permeability. The method can also easily be formulated for other types of flow models, provided one has a reliable (iterative) solution strategy that computes flow and transport in separate steps. The proposed method is implemented as open-source software and validated on a number of two and three-phase test cases with significant compressibility and gas dissolution. The test cases include both synthetic models and models of real fields with complex wells, faults, and inactive and degenerate cells. Through a prescribed tolerance, the solver can be set to either converge to a sequential or the fully implicit solution, in both cases with a significant speedup compared to a fine-scale multigrid solver. Altogether, this ensures that one can easily and systematically trade accuracy for efficiency, or vice versa.