Multidimensional upwind schemes and higher resolution methods for three-component two-phase systems including gravity driven flow in porous media on unstructured grids - Michael G. Edwards (Computer Methods in Applied Mechanics and Engineering, 2015)
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 multidimensional upwind schemes are presented for essentially hyperbolic three-component two-phase flow systems of conservation laws in porous media including counter current gravity flow on unstructured grids. The multidimensional methods employ cell-based tracing, which involves tracing characteristic wave directions over each control-volume subquadrant. The multidimensional methods reduce crosswind diffusion inherent in standard methods for convective flow approximation in porous media. The schemes are coupled with continuous Darcy-flux approximations resulting from the elliptic pressure equation on unstructured grids.Characteristic upwind approximations are proposed and compared with the classical upstream weighting schemes for cases including gravity segregated flow. When dealing with systems of hyperbolic equations, upwind characteristic wave decomposition is used for wave tracing. The multidimensional upwind cell-based tracing formulations are designed for unstructured grids (and include structured grids by default) and are stable subject to conditions on the tracing direction and CFL number and satisfy a local maximum principle that ensures solutions are free of spurious oscillations.Benefits of the resulting schemes are demonstrated for two-phase flow and a three-component two-phase flow system including gravity segregated flow. The multidimensional cell based schemes are shown to reduce crosswind diffusion induced by standard upwind methods, and prove to be particularly effective when flow is strongly non-aligned with the grid, leading to improved resolution of numerical saturation and concentration fronts. Extension of higher order schemes to a three-component two-phase flow systems of conservation laws on unstructured grids is also presented, which provides a significant improvement in flow resolution for the system cases. Comparison is drawn between the methods.