Higher order cell-based multidimensional upwind schemes for flow in porous media on unstructured grids

摘要

Standard reservoir simulation schemes employ single-point upstream weighting for convective flux approximation. These schemes introduce both coordinate-line numerical diffusion and cross-wind diffusion into the solution that is grid and geometry dependent. New locally conservative cell-based multi-dimensional upwind schemes and higher-order cell-based multi-dimensional upwind schemes that reduce both directional and cross-wind diffusion are presented for convective flow approximation. The new higher-order 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 information obtained by upstream tracing along multi-dimensional flow paths. 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 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 effects, leading to improved resolution of concentration and saturation fronts.