A Mixed Finite Volume Element Method for Flow Calculations in PorousMediaJim

SUMMARY A key ingredient in the simulation of ow in porous media is the accurate determination of the velocities that drive the ow. The large scale irregularities of the geology, such as faults, fractures, and layers suggest the use of irregular grids in the simulation. Work has been done in applying the nite volume element (FVE) methodology as developed by McCormick in conjunction with mixed methods which were developed by Raviart and Thomas. The resulting mixed nite volume element discretization scheme has the potential to generate more accurate solutions that standard approaches. The focus of this paper is on a multilevel algorithm for solving the discrete mixed FVE equations. The algorithm uses a standard cell centered nite diierence scheme as thècoarse' level and the more accurate mixed FVE scheme as thè`ne' level. The algorithm appears to have potential as a fast solver for large size simulations of ow in porous media. The Mixed Finite Volume Element Discretization In this rst section, we brieey introduce the mixed nite volume element (FVE) dis-cretization technique. We will not dwell too much on the details of the discretization itself as our focus here is on solving the discrete set of equations that the discretization produces; a detailed description of the discretization can be found in 7]. We begin by considering the following partial diierential equation deened on a domain in R 2 : (? r A(x)r(x) = f(x) x 2 ; r(x) = g(x) x 2 @: (1) Here we assume the diiusion coeecient A is diagonal, but values of the coeecients may jump orders of magnitude at material interfaces. In the context of reservoir simulation, this is the pressure equation for incompressible single-phase ow where is the pressure in the reservoir , and the boundary condition speciies the ux on @. As one of our goals for the new discretization is accurate approximations of