Computational Geosciences 1 (1997) 289–315 289 Control-volume mixed finite element methods Z. Cai a , J.E. Jones b , S.F. McCormick c and T.F. Russell d a Center for Applied Mathematics, Purdue University, West Lafayette, IN 47907-1395, USA b Lawrence Livermore National Laboratory, MS L-316, P.O. Box 808, Livermore, CA 94551-0808, USA c Department of Applied Mathematics, University of Colorado at Boulder, Boulder, CO 80309-0526, USA d Department of Mathematics, University of Colorado at Denver, Campus Box 170, Denver, CO 80217-3364, USA Received 22 October 1996; revised 19 September 1997 A key ingredient in simulation of flow in porous media is accurate determination of the velocities that drive the flow. Large-scale irregularities of the geology (faults, fractures, and layers) suggest the use of irregular grids in simulation. This paper presents a control-volume mixed finite element method that provides a simple, systematic, easily implemented pro- cedure for obtaining accurate velocity approximations on irregular (i.e., distorted logically rectangular) block-centered quadrilateral grids. The control-volume formulation of Darcy’s law can be viewed as a discretization into element-sized “tanks” with imposed pressures at the ends, giving a local discrete Darcy law analogous to the block-by-block conservation in the usual mixed discretization of the mass-conservation equation. Numerical results in two dimensions show second-order convergence in the velocity, even with discontinuous anisotropic permeability on an irregular grid. The method extends readily to three dimen- sions. Keywords: control-volume method, mixed method, local mass conservation, local Darcy law, block-centered grid, distorted grid, anisotropy, heterogeneity 1. Introduction As techniques of reservoir description become more sophisticated, it becomes increasingly important to model flows of reservoir fluids accurately. In particular, it is desirable to accurately represent large-scale irregularities of reservoir geology in mod- els. The control-volume mixed finite element method allows the use of irregular grids while maintaining many of the familiar properties of block-centered finite difference methods for rectangular grids. For example, it preserves the notion of block-by-block material balance, with physical interblock-flow terms. It also yields an analogue of the local discrete Darcy law (relating a combination of fluxes to a pressure drop between blocks) on a block-sized “tank” between two pressure nodes. The method can be ap- plied to any “logically rectangular” grid of irregular quadrilaterals in two dimensions, or analogous hexalaterals in three dimensions. In two dimensions, “logically rectan- gular” means that each grid block can be assigned an index (i, j ) such that it shares an  Baltzer Science Publishers BV