Double-families of Quasi-Positive Flux-continuous Finite-volume Schemes on Structured and Unstructured Grids

This paper focuses on flux-continuous pressure equation approximation for strongly anisotropic media. Previous work on families of flux continuous schemes for solving the general geometry-permeability tensor pressure equation has focused on single-parameter families e.g. [1]. These schemes have been shown to remove the O(1) errors introduced by standard two-point flux reservoir simulation schemes when applied to full tensor flow approximation. Improved convergence of the schemes has also been established for specific quadrature points [2]. However these schemes have conditional M-matrices depending on the strength of the cross-terms [1]. When applied to cases involving full tensors arising from strongly anisotropic media, these schemes can fail to satisfy the maximum principle. Loss of solution monotonicity then occurs at high anisotropy ratios, causing spurious oscillations in the numerical pressure solution. New double-family flux-continuous locally conservative schemes are presented for the general geometry-permeability tensor pressure equation. The new double-family formulation is shown to expand on the current single-parameter range of existing schemes that have an M-matrix. While it is shown that a double family formulation does not lead to an unconditional M-matrix scheme, an analysis is presented that classifies the sensitivity of the new schemes with respect to monotonicity and double-family quadrature. Convergence performance with respect to a range of double-family quadrature points is presented for some well known test cases.