Higher Resolution Hybrid Unstructured Spectral Finite-volume Methods For Flow In Porous Media

Abstract

Summary Novel higher resolution hybrid spectral finite-volume methods are presented and coupled with the control-volume distributed multi-point flux approximation (CVD-MPFA) for flow in porous media including fractures and gravity. The spectral-volume is the primal cell which is sub-divided into sub-cell control-volumes, the sub-cells are then used to develop an efficient strategy for reconstruction of both a higher resolution approximation of the convective transport flux and Darcy flux approximation on sub-cell interfaces. The new method extends the first order structured grid hybrid method of [Lee et al] to a higher resolution spectral formulation on unstructured grids. The novel hybrid spectral-volume method presented involves reconstruction of both the convection variable(s) and the Darcy-flux. First the pressure equation is solved on a primal coarse grid using the CVD-MPFA method. Each coarse primal cell is subdivided into sub-cells following the spectral-volume method. Darcy-fluxes are then reconstructed on the sub-cell control-volume faces from using the course primal cell solution. For the convective transport equation approximation, each sub-cell is assigned a degree of freedom for saturation and/or concentration and a higher resolution hybrid sub-cell control-volume transport approximation is efficiently reconstructed over each spectral-volume. The fine scale saturation field is then updated via the new hybrid reconstructed finite volume approximations over the sub-cell control-volumes. Performance comparisons are presented for tracer and two-phase flow problems, including fracture and gravity problems on unstructured meshes. Comparisons between the standard lower-order and higher resolution hybrid CVD-MPFA methods and the new higher resolution hybrid spectral-volume method demonstrate both the improved resolution of flow fields achieved by the standard higher resolution method, and the extra fine flow field resolution achieved by the spectral-volume method, while the spectral-volume method is also shown to be considerably more efficient.