Numerical studies of transport in porous media

Williams has recently developed a new formulation for flow through porous media by using a modification of the kinetic equations of particle transport. We explore in this paper several numerical techniques for solution of this equation. The methods explored include the upwind, the Smolarkiewicz method [1983], and the NND (non oscillatory, no free parameters, dissipative) schemes of Zhang and Zhuang [1992]. The numerical methods were benchmarked against an exact two-directional solution (the forward-backward model) and it was found that of these finite difference methods, the NND schemes, developed to accommodate shock waves, reduce the inherent diffusion error the most while avoiding spurious oscillations. The numerical techniques were further explored for a multi-directional case for which exact solutions cannot be obtained, and it was again found that the NND schemes outperformed the other methods. (Author)