Higher Dimensional Upwind Schemes for Flow in Porous Media on Unstructured Grids

Abstract

Standard reservoir simulation schemes employ single-point upstream weighting for approximation of the convective fluxes when multiple phases or components are present. These schemes rely upon upwind information that is determined according to the grid geometry. As a consequence directional diffusion is introduced into the solution that is grid dependent. The effect can be particularly important for cases where the flow is across grid coordinate lines and is known as cross-wind diffusion. Novel higher dimensional upwind schemes that minimize cross-wind diffusion are presented for convective flow approximation on structured and unstructured grids. The schemes are free of spurious oscillations and remain locally conservative. The higher dimensional schemes are coupled with full-tensor Darcy flux approximations. Benefits of the resulting schemes are demonstrated for classical convective test cases in reservoir simulation including cases with full tensor permeability fields, where the methods prove to be particularly effective. The test cases involve a range of unstructured grids with variations in orientation and permeability that lead to flow fields that are poorly resolved by standard simulation methods. The higher dimensional formulations are shown to effectively reduce numerical cross-wind diffusion effect, leading to improved resolution of concentration and saturation fronts. TECHNICAL CONTRIBUTIONS Locally conservative unstructured multi-dimensional schemes coupled with full-tensor Darcy flux approximations are presented for reservoir simulation. The multi-dimensional schemes are developed for general unstructured grids. The schemes are proven to be positive subject to conditions on the tracing direction and permit higher CFL conditions than standard schemes. Comparisons with single point upstream weighting scheme are made on a range of unstructured grids for different grid orientation and aspect ratios in cases involving full tensor coefficient velocity fields. The numerical results demonstrate the benefits of the higher dimensional schemes both in terms of improved front resolution and significant reduction in cross-wind diffusion.