An algebraic multiscale solver for the simulation of two-phase flow in heterogeneous and anisotropic porous media using general unstructured grids (AMS-U)

Abstract

We introduce a new multiscale finite volume framework for the simulation of multi-phase flow in heterogenous and anisotropic porous media on quite general unstructured grids that enable geophysical grid defined properties to be used directly on a high definition grid.A novel background grid strategy is presented, where an auxiliary mesh aids the creation of multiscale primal and dual coarse grids. Due to the unstructured grid connectivity, the dual grids do not retain the natural structured grid decomposition. This may induce errors in the computed basis functions, and contribute to loss of mass conservation. A generalization of the Algebraic Multiscale Solver (AMS) is proposed to prevent the basis function induced leakage outside the support region enabling coupling with a Control Volume Distributed MultiPoint Flux Approximation (CVD-MPFA), ensuring consistency on non k-orthogonal unstructured grids.We present three different problems in which the accuracy of the framework is validated by comparing the multiscale solver with the direct simulation on the fine-scale. The results obtained show that the method can produce well resolved solutions for two-phase flow in highly heterogeneous and anisotropic porous media using general unstructured grids on all scales. As a consequence, the method captures the most important flow features that result on high-resolution geophysical models while providing suitable discretization for complex geological formations found in current petroleum reservoir problems.The novelty of this scheme involves three components which are combined to enable consistent multiscale simulation on unstructured grids: i) a new approach to primal and dual coarse grid generation, ii) a novel technique to prevent basis function induced leakage and iii) the coupling of the Algebraic Multiscale Solver (AMS) with a Multipoint Flux approximation with a Diamond Stencil (MPFA-D).