Non-linear flux-splitting schemes with imposed discrete maximum principle for elliptic equations with highly anisotropic coefficients

Families of flux-continuous, locally conservative, finite-volume schemes have been developed for solving the general tensor pressure equation of petroleum reservoir simulation on structured and unstructured grids. The schemes are applicable to diagonal and full tensor pressure equation with generally discontinuous coefficients and remove the O(1) errors introduced by standard reservoir simulation schemes when applied to full tensor flow approximation. The family of flux-continuous schemes is quantified by a quadrature parameterization. Improved convergence using the quadrature parameterization has been established for the family of flux-continuous schemes. When applied to strongly anisotropic full-tensor permeability fields the schemes can fail to satisfy a maximum principle (as with other FEM and finite-volume methods) and result in spurious oscillations in the numerical pressure solution. This paper presents new non-linear flux-splitting techniques that are designed to compute solutions that are free of spurious oscillations. Results are presented for a series of test-cases with strong full-tensor anisotropy ratios. In all cases the non-linear flux-splitting methods yield pressure solutions that are free of spurious oscillations. Copyright (C) 2010 John Wiley & Sons, Ltd.