Grid Optimization for Improved Monotonicity of MPFA Solutions on Unstructured Grids

Abstract

Unstructured grids are useful for resolving key geological or flow features in reservoir simulation. Accurate finite volume discretization of the reservoir flow equations on such grids generally requires the use of multipoint flux approximation (MPFA). MPFA can be applied to heterogeneous, anisotropic systems on generally unstructured grids, though it may suffer from loss of monotonicity of the inverse of the resulting linear operator at moderate to high permeability anisotropy ratios. As a consequence, the resulting pressure solution can show errors in the form of spurious oscillations. In recent work, we developed new algorithms for 2D and 3D systems that address this loss of monotonicity from a grid optimization perspective. Given an underlying fine-scale heterogeneous permeability field, the approaches can be generalized (via iteration) to couple permeability upscaling with the unstructured grid optimization. In our previous work, solutions of the single-phase incompressible pressure equation were considered for 2D and 3D models. In this paper we describe our grid optimization procedures, consider new example cases, and apply the approach to a two-phase flow problem. We demonstrate that the overall procedure is capable of providing accurate solutions (for both single-phase and two-phase flows) that are free of spurious pressure oscillations. An upscaling example demonstrates results in reasonably close agreement with the corresponding fine-grid solution. Although not presented here, the method can be coupled with a flow-based grid generation procedure to provide an overall gridding capability that resolves key flow regions while minimizing or eliminating spurious pressure oscillations.