A family of MPFA finite-volume schemes with full pressure support for the general tensor pressure equation on cell-centered triangular grids

  • Authors:

  • Helmer A. Friis;Michael G. Edwards

  • Affiliations:

  • International Research Institute of Stavanger, P.O. Box 8046, 4068 Stavanger, Norway and Department of Informatics, University of Bergen, P.O. Box 7800, N-5020 Bergen, Norway;Civil and Computational Engineering Centre, School of Engineering, Swansea University, Swansea SA2 8PP, Wales, United Kingdom

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2011

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Abstract

A new family of cell-centered finite-volume schemes is presented for solving the general full-tensor pressure equation of subsurface flow in porous media on arbitary unstructured triangulations. The new schemes are flux continuous and have full pressure support (FPS) over each subcell with continuous pressure imposed across each control-volume sub-interface, in contrast to earlier formulations. The earlier methods are point-wise continuous in pressure and flux with triangle-pressure-support (TPS) which leads to a more limited quadrature range. An M-matrix analysis identifies bounding limits for the schemes to posses a local discrete maximum principle. Conditions for the schemes to be positive definite are also derived. A range of computational examples are presented for unstructured triangular grids, including highly irregular grids, and the new FPS schemes are compared against the earlier pointwise continuous TPS formulations. The earlier pointwise TPS methods can induce strong spurious oscillations for problems involving strong full-tensor anisotropy where the M-matrix conditions are violated, and can lead to decoupled solutions in such cases. Unstructured cell-centered decoupling is investigated. In contrast to TPS, the new FPS formulation leads to well resolved solutions that are essentially free of spurious oscillations. A substantial degree of improved convergence behavior, for both pressure and velocity, is also observed in all convergence tests. This is particularly important for problems involving high anisotropy ratios. Also the new formulation proves to be highly beneficial for an upscaling example, where enhancement of convergence is highly significant for certain quadrature points, clearly demonstrating further advantages of the new formulation.