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q-Families of CVD(MPFA) Schemes on General Elements: Numerical Convergence and the Maximum Principle

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Abstract

In this paper, families of flux-continuous, locally conservative, finite-volume schemes are presented for solving the general geometry-permeability tensor pressure equation on structured and unstructured grids in two and three dimensions. 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 anisotropic permeability tensor flow approximation (Edwards and Rogers in Multigrids Methods, vol. 1, pp. 190–200, 1993; Edwards and Rogers in Proceedings: 4th European Conference on the Mathematics of Oil Recovery, 1994; Edwards and Rogers in Comput. Geom. 2:259–290, 1998). Full tensors arise when the local orientation of the grid is non-aligned with the principal axes of the tensor field. Full tensors may also arise when fine scale permeability distributions are upscaled to obtain gridblock-scale permeability distributions. In general full tensors arise when using any structured or unstructured grid type that departs from K-orthogonality.

A family of schemes is quantified by a quadrature parametrization q, where the position of continuity defines the quadrature and hence the family.

This paper presents complete extensions of the q-families of control-volume distributed (CVD) multi-point flux approximation (MPFA) flux-continuous schemes for general three dimensional grids comprised of classical element types, hexahedra, tetrahedra, prisms and pyramids. 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.

A series of numerical test cases are presented and numerical convergence studies are conducted for the q-families of schemes, using different types of two and three dimensional structured and unstructured grids. Use of quadrature parametrization is investigated and specific quadrature points are observed to yield improved convergence for the families of flux-continuous schemes on structured and unstructured grids in two and three dimensions.

When applying the CVD(MPFA) schemes to strongly anisotropic heterogeneous media they can fail to satisfy a maximum principle as with other finite element and finite-volume methods, (M-matrix conditions are given in (Edwards and Rogers in Proceedings of the 4th European Conference on the Mathematics of Oil Recovery, 1994; Edwards and Rogers in Comput. Geom. 2:259–290, 1998; Edwards in 13th SPE Reservoir Simulation Symposium, pp. 553–562, 1995; Edwards et al. Quasi-montonic variable support (q1,q2) families of continuous darcy-flux cvd(mpfa) finite volume schemes, 2006; Edwards and Zheng in J. Comput. Phys. 227:9333–9364, 2008) and for high (full-tensor) anisotropy ratios they can yield numerical pressure solutions with spurious oscillations. In this work methods for obtaining optimal discretization with minimal spurious oscillations are also investigated. New flux-splitting techniques based on (Edwards in J. Comput. Phys. 160:1–28, 2000) are also developed so as to impose a discrete maximum principle, the new methods prove to be effective for problems with high anisotropy ratios (Pal and Edwards in Proceedings of the 10th European Conference on the Mathematics of Oil Recovery, 2006; Pal and Edwards in Proceedings of the ECOMASS CFD-2006 Conference, 2006). In all cases the resulting numerical pressure solutions are free of spurious oscillations.

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Authors and Affiliations

  1. Shell Exploration and Production, Kessler Park 1, 2288GS, Rijswijk, The Netherlands

    Mayur Pal

  2. Civil and Computational Engineering Centre, Swansea University, Wales, UK

    Michael G. Edwards

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  1. Mayur Pal
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  2. Michael G. Edwards
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Correspondence to Mayur Pal.

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This work was done when the first author was a PhD Student at Swansea University, UK (2004-2007).

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Pal, M., Edwards, M.G. q-Families of CVD(MPFA) Schemes on General Elements: Numerical Convergence and the Maximum Principle. Arch Computat Methods Eng 17, 137–189 (2010). https://doi.org/10.1007/s11831-010-9043-4

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