Logically Rectangular Mixed Methods for Darcy Flow on General Geometry

Abstract

ABSTRACT We consider an expanded mixed finite element formulation (cell centered finite differences) for Darcy flow with a tensor absolute permeability. The reservoir can be geometrically general with internal features, but. the computational domain is rectangular. The method is defined on a curvilinear grid that need not, be orthogonal, obtained by mapping the rectangular, computational grid. The original flow problem becomes a similar problem with a modified permeability on the computational grid. Quadrature rules turn the mixed method into a cell-centered finite difference method with a. 9 point stencil in 2-D and 19 in 3-D. As shown by theory and experiment, if the modified permeability on the computational domain is smooth, then the convergence rate is optimal and both pressure and velocity are superconvergent at certain points. If not, Lagrange multiplier pressures can be introduced on boundaries of elements so that optimal convergence is retained. This modification presents only small changes in the solution process; in fact, the same parallel domain decomposition algorithms can be applied with little or no change to the code if the modified permeability is smooth over the subdomains. This Lagrange multiplier procedure can be. used to extend the difference scheme to multi-block domains, and to give, a coupling with unstructured grids. In all cases, the mixed formulation is locally conservative. Computational results illustrate the advantage and convergence of this method.