Rock flow simulation by high-order quasi-characteristics scheme

A pure second-order scheme of quasi-characteristics based on a pyramidal stencil is applied to the numerical modelling of non-stationary two-phase flows through porous media with the essentially heterogeneous properties. In contrast to well-known other high-resolution schemes with monotone properties, this scheme preserves a second-order approximation in regions, where discontinuities of solutions arise, as well as monotone properties of numerical solutions in those regions despite of well-known Godunov theorem. It is possible because the scheme under consideration is defined on a non-fixed stencil and is a combination of two high-order approximation scheme solutions with different dispersion properties. A special criterion according to which, one or another admissible solution is chosen, plays a key role in this scheme. A simple criterion with local character suitable for parallel computations is proposed. Some numerical results showing the efficiency of present approach in computations of two-phase flows through porous media with strongly discontinuous penetration coefficients are presented.