Multi-dimensional higher resolution methods for flow in porous media. - Mohamed Sadok Lamine (2009)
Currently standard first order single-point upstream weighting methods are employed in reservoir simulation for integrating the essentially hyperbolic system components. These methods introduce both coordinate-line numerical diffusion (even in 1-D) and cross-wind diffusion into the solution that is grid and geometry dependent. These effects are particularly important when steep fronts and shocks are present and for cases where flow is across grid coordinate lines. In this thesis, families of novel edge-based and cell-based truly multidimensional upwind formulations that upwind in the direction of the wave paths in order to minimise crosswind diffusion are presented for hyperbolic conservation laws on structured and unstructured triangular and quadrilateral grids in two dimensions. Higher resolution as well as higher order multidimensional formulations are also developed for general structured and unstructured grids. The schemes are coupled with existing consistent and efficient continuous CVD (MPFA) Darcy flux approximations. They are formulated using an IMPES (Implicit in Pressure Explicit in Saturation) strategy for solving the coupled elliptic (pressure) and hyperbolic (saturation) system of equations governing the multi-phase multi-component flow in porous media. The new methods are compared with single point upstream weighting for two-phase and three-component two-phase flow problems. The tests arc conducted on both structured and unstructured grids and involve full-tensor coefficient velocity fields in homogeneous and heterogeneous domains. The comparisons demonstrate the benefits of multidimensional and higher order multidimensional schemes in terms of improved front resolution together with significant reduction in cross-wind diffusion.