Higher Order Multidimensional Wave Oriented Upwind Schemes for Flow in Porous Media on Unstructured Grids

摘要

Standard reservoir simulation schemes employ single-point upstream weighting for convective flux approxi- mation. These schemes introduce both coordinate-line numerical diffusion and cross-wind diffusion into the solution that is grid and geometry dependent. New locally conservative higher-order multi-dimensional upwind schemes that minimize both directional and cross-wind diffusion are presented for convective flow approximation. The new schemes are comprised of two steps; (a) Higher-order approximation that corrects the directional diffusion of the approximation. (b) Truly multi-dimensional upwind approximation, which involves flux approximation using upwind infor- mation obtained by upstream tracing along wave-vector paths where wave information travels in multiple dimensions. This approximation reduces cross-wind diffusion. Conditions on tracing direction and CFL number lead to a local maximum principle that ensures stable solutions free of spurious oscillations. The schemes are coupled with full-tensor Darcy flux approximations. Benefits of the resulting schemes are demonstrated for classical convective test cases in reservoir simula- tion 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 formu- lations are shown to effectively reduce numerical cross-wind diffusion effects, leading to improved resolution of concentration and saturation fronts.