A family of flux-continuous, locally conservative, finite-volume schemes has been developed for solving the general geometry-permeability tensor pressure equation on structured and unstructured grids and are control-volume distributed [1,2]. 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 tensor flow approximation. The family of flux-continuous schemes is quantified by a quadrature parameterization and has been tested for ranges of quadrature points. Specific points have been observed to yield improved convergence for the family of flux-continuous schemes for structured and unstructured grids in two dimensions .
This paper presents a complete extension of the family of control-volume distributed (CVD) multi-point flux approximation (MPFA) flux-continuous schemes for general three dimensional grids comprised of any element type, hexahedra, tetrahedra, prisms and pyramid elements. Discretization principles are presented for each element. The pyramid element is shown to be a special case with unique construction of the continuity conditions. The Darcy flux approximations are applied to a range of test cases that verify consistency of the schemes. Convergence tests of the three-dimensional families of schemes are presented, with emphasis on use of quadrature parameterization. Monotonicity issues are also discussed and tests performed confirm optimal monotonicity of the schemes as determined by an M-matrix analysis .