q-Families of CVD(MPFA) Schemes on General Elements: Numerical Convergence and the Maximum Principle

In this paper, families of flux-continuous, locally conservative, finite-volume schemes are presented for solving the general geometry-permeability tensor pressure equation on structured and unstructured grids in two and three dimensions. The schemes are applicable to the general tensor pressure equation with discontinuous coefficients and remove the O(1) errors introduced by standard reservoir simulation (two-point flux) schemes when applied to full anisotropic permeability tensor flow approximation (Edwards and Rogers in Multigrids Methods, vol. 1, pp. 190–200, 1993; Edwards and Rogers in Proceedings: 4th European Conference on the Mathematics of Oil Recovery, 1994; Edwards and Rogers in Comput. Geom. 2:259–290, 1998). Full tensors arise when the local orientation of the grid is non-aligned with the principal axes of the tensor field. Full tensors may also arise when fine scale permeability distributions are upscaled to obtain gridblock-scale permeability distributions. In general full tensors arise when using any structured or unstructured grid type that departs from K-orthogonality.