A full tensor velocity field occurs in reservoir simulation whenever the computational grid is non-aligned with the tensor principal axes. Three contributing factors that can give rise to a full tensor pressure equation in reservoir simulation are anisotropic media, upscaling (with cross-flow effects) and non K-orthogonal structured and unstructured grids.

The introduction of a full tensor typically increases the support of the standard scheme on a logically rectangular grid from 7 to 27 nodes in 3-D, which can increase simulation costs by over 100% in 3-D.

A major assumption in many simulators is that the pressure equation always has a diagonal tensor. The design and efficiency of such codes is intrinsically linked to the diagonal tensor assumption. However, when employing general non K-orthogonal grids with corner point geometry the diagonal tensor assumption leads to an O(1) error in velocity field.

The focus of this paper is on the development of finite volume schemes that maintain consistent full tensor flux approximations while retaining standard diagonal tensor Jacobian matrix inversion in three dimensions on structured and unstructured grids. Three dimensional Hexahedral and Tetrahedral grid examples are included which demonstrate the benefits of the method.

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