Keywords: full pressure continuity, finite volume scheme, control volume distributed.

A new family of flux-continuous control-volume distributed (CVD) finite volume schemes are presented for the general full-tensor pressure equation arising in porous media. In addition to being flux-continuous, the new locally conservative schemes are designed such that pressure is continuous over the entire control-volume surface. Comparisons are made with previous full tensor formulations [1,2], that have point-wise continuity in pressure. The new formulation has improved regularity with respect to quadrature point where flux is defined.

The key feature of the family of new schemes is the full pressure continuity over each subcell. In standard CVD(MPFA) schemes [1,2], only four pressure values on the surface of the subcells are introduced, thus the pressure support is only part of the subcell. In order to impose full pressure continuity, an additional auxiliary pressure is introduced, i.e. the common vertex of the four subcells in the dual cell. Now in local system we have five auxiliary surface pressure values (unknown),thus an extra condition is required to complete the system. We adopt the similar technique to [3], viz. divergence free condition for the auxiliary control volume (dual control volume). But in our methods the size of dual control volume is not fixed as in [3].

The flexibility of the new family of schemes lies in the selection of quadrature point q as well as the size coefficient c of dual control. For example, when c=1, the dual control volume coincide with the dual cell; while c=0, the dual control volume shrinks to the common vertex of the four subcells. Proper selection of these two parameters will enhance the performance of the new schemes. The performance of standard CVD regarding quadrature q is studied, demonstrating the benefit of a family of schemes.

The schemes are tested on a range of cases involving discontinuous coefficients and distorted grids. Comparisons of convergence rate demonstrate that super-convergence is obtained for certain quadrature points. The new schemes prove to be robust for the cases tested including strongly anisotropic full tensor fields. While such cases can induce oscillations when using the previous CVD(MPFA) formulations or indeed other standard formulations, the new full pressure support scheme is able to remove or minimize oscillatory behaviour. The quasi-monotonicity property of the new schemes is studied in [4]. This formulation is also being extended to unstructured grids as well as in 3D.

1

M.G. Edwards, C.F. Rogers, "Finite volume discretization with imposed flux continuity for the general tensor pressure equation", Computational Geoscience, 2, 259-290, 1998. doi:10.1023/A:1011510505406

2

M.G. Edwards, "Unstructured,control volume distributed,full-tensor finite-volume schemes with flow based grids", Computational Geoscience, 6, 433-452, 2002. doi:10.1023/A:1021243231313

3

P.I. Crumpton, G.J. Shaw, A.F. Ware, "Discretization and multigrid solution of elliptic equations with mixed derivative terms and strongly discontinuous coefficients", Journal of Computational Physics, 116, 343-358, 1995. doi:10.1006/jcph.1995.1032

4

M.G. Edwards, H. Zheng, "A Quasi-Monotonic Family of Continuous Darcy-Flux CVD(MPFA) Finite Volume Schemes with Full Pressure Support", submitted, 2007.

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