The numerical discretization of the reservoir flow equations is a challenging problem on three-dimensional unstructured grids and in the presence of full-tensor permeabilities. Finite volume methods based on a two-point flux approximation (TPFA) do not properly account for grid distortion or permeability anisotropy that is misaligned with the grid. Over the past decade, multipoint flux approximation (MPFA) methods have been devised to address these issues. Recent research has shown, however, that MPFA methods may lead to unphysical nonmonotonic solutions for high anisotropy ratios or strongly distorted grids.

In this paper, we work in the mathematical framework of mixed finite element methods (MFEM). We study a low- and high-order accurate MFEM, based on the lowest order Raviart-Thomas space (RT0) and the Brezzi-Douglas-Marini space of order one (BDM1). On tetrahedrons, the BDM1 space can be localized into a compact finite volume method through the use of an appropriate numerical quadrature. However, this is not possible for hexahedrons.

The main result of this paper is the development of a new mixed finite element method for 3D hexahedral grids, which leads to a multipoint flux approximation finite volume method through the use of a vertex-based numerical quadrature. This is possible because the new velocity space is designed so that the degrees of freedom are the normal component of the velocity field at the vertices of each face. This establishes for the first time a direct link between MFEM and MPFA methods on 3D hexahedral grids.

We compare the performance of classical (RT0 and BDM1) mixed finite element methods with the newly developed method on a number of test cases. We conclude that high-order methods lead to higher-quality velocity fields that are less sensitive to the choice of the grid, and that they have better monotonicity properties for highly anisotropic full-tensor permeability fields. While the added benefit of the new space is that it can be approximated by a cell-centered finite volume scheme, our numerical experiments indicate that the accuracy of the scheme is reduced.

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