A Multiscale Control Volume framework using the Multiscale Restriction Smooth Basis and a non-orthodox Multi-Point Flux Approximation for the simulation of two-phase flows on truly unstructured grids

In these days, the advances in the fields of geology have led to geocellular models containing up to hundreds millions cells. Meanwhile, the standard petroleum reservoir simulations are limited to models with only a fraction of this amount. Thus, the multiples simulation of multiphase flow in heterogeneous and anisotropic media on the geological scale are simply impractical. To overcome this problem Multiscale Finite Volume (MsFV) uses operators to project the fine-scale system of equations onto a coarse-scale space originated from a lower-resolution grid. The resulting coarse system is solved, and by using the multiscale operator projected back onto the higher-resolution grid. Notwithstanding, the MsFV is not capable of capturing geologic properties on unstructured grids, as in its classical form, it relies on a Two Point Flux Approximation (TPFA) for fluxes, which is only consistent for k-orthogonal grids. Furthermore, the algorithm used by the MsFV to generate the geometric entities, such as, coarse-scale and dual meshes and its centers are not valid for general unstructured grids. The Multiscale Restricted Smoothed Basis (MsRSB) method expands this framework by creating the algorithms capable of generating these structures and by proposing an iterative multiscale operator capable of handling simulation on unstructured coarse grids. However, it fails to produce consistent solutions on fine-scale unstructured grids and for arbitrary permeability tensors, as it uses the TPFA method. In this article, we couple a Multi-Point Flux Approximation with a Diamond stencil to the MsRSB, to extend its use to general unstructured grids. We call our framework the “Multiscale Control-Volume” (MsCV) method. Moreover, we bring the iterative Dirichlet pressure correction scheme and the upscaling flux correction to the unstructured multiscale context. On the first examples, we study the impact of these corrections on the simulation of one-phase flow using the SPE Model 2 benchmark. In addition, we show the behavior of the MsCV framework on the simulation of two-phase flows on general unstructured grids and highly heterogeneous and isotropic permeability fields.