Multidimensional and Higher Order Edge Based Upwind Schemes for Hyperbolic Systems Including Gravity in Porous Media on Structured and Unstructured Grids

Standard reservoir simulation schemes employ single-point upstream weighting for approximation of the convective fluxes when multiple phases or components are present. These schemes introduce both coordinate-line numerical diffusion and crosswind diffusion into the solution that is grid and geometry dependent. Families of locally conservative higher-order multidimensional upwind schemes are presented for convective flow approximation in porous media that reduce both directional and crosswind diffusion. The schemes are coupled with full-tensor Darcy flux approximations and handle general flow conditions including counter current gravity driven flows and systems of hyperbolic equations. Characteristic vector upwind approximations are proposed and compared with the simulation standard single-point upstream weighting schemes. When dealing with systems of hyperbolic equations, upwind characteristic wave decomposition is used in combination with different limiting strategies involving conservative, primitive and characteristic variables. Alternate wave vector tracing approximations are proposed based on phase velocities and characteristic velocities and comparisons are presented. The higher order multidimensional schemes are designed for general grids. Conditions for a local discrete maximum principle are presented that ensure solutions are free of spurious oscillations. Benefits of the resulting schemes are demonstrated for gravity segregated flow and polymer flood systems. The test cases involve a range of unstructured grid types with variations in orientation and permeability that lead to flow fields that are poorly resolved by standard simulation methods. The higher order multidimensional formulations are shown to effectively reduce numerical diffusion, leading to improved resolution of concentration and saturation fronts.