HOMOGENIZATION OF A DARCY-STOKES SYSTEM MODELING VUGGY POROUS MEDIA TODD ARBOGAST * AND HEATHER L. LEHR † Abstract. We derive a macroscopic model for single phase, incompressible, viscous fluid flow in a porous medium with small cavities called vugs. We model the vuggy medium on the microscopic scale using Stokes equations within the vugular inclusions, Darcy’s law within the porous rock, and a Beavers-Joseph-Saffman boundary condition on the interface between the two regions. We assume periodicity of the medium, and obtain uniform energy estimates independent of the period. Through a two-scale homogenization limit as the period tends to zero, we obtain a macroscopic Darcy’s law governing the medium on larger scales. We also develop some needed generalizations of the two-scale convergence theory needed for our bi-modal medium, including a two-scale convergence result on the Darcy-Stokes interface. The macroscopic Darcy permeability is computable from the solution of a cell problem. An analytic solution to this problem in a simple geometry suggests that: (1) flow along vug channels is primarily Poiseuille with a small perturbation related to the Beavers-Joseph slip, and (2) flow that alternates from vug to matrix behaves as if the vugs have infinite permeability. Key words. Homogenization, two-scale convergence, Darcy-Stokes system, vuggy porous media, Beavers-Joseph boundary condition 1. Introduction. A vug is a cavity in a porous medium that is relatively larger than the intergranular pore space. Vugular inclusions are especially common in car- bonate rocks, and are endemic to many of the world’s groundwater aquifers and petroleum reservoirs. Although small, vugs can significantly increase both the effec- tive porosity and permeability of the medium. We consider in this paper a porous medium with many small vugs scattered throughout its extent. It is well established, both empirically and theoretically, that Darcy’s law governs fluid flow in a porous medium on scales above the pore diameter [6, 23, 27, 29]. Since the flow is expected to have a relatively low Reynolds number, the Stokes equations should adequately model fluid flow in the vugs. In 1967, Beavers and Joseph [7] determined experimentally that a free fluid in contact with a porous medium flows faster than a fluid in contact with a completely solid surface. Although thin boundary layers arise in both cases, the latter case is generally modeled by assuming that all components of the velocity vanish at the solid contact surface. In the former case, the experiments of Beavers and Joseph demonstrate that the tangential velocity of the fluid cannot vanish. They proposed to account for this slippage by imposing a boundary condition of the form ∂U s ∂y = α √ K (U s − U d ) , where ∂/∂y is the normal derivative, U s is the tangential component of the Stokes velocity, U d is the tangential component of the Darcy velocity, K is the permeability of the porous medium, and α is the dimensionless Beavers-Joseph slippage coefficient. Saffman [24] justified this law theoretically, and showed that the term involving U d could be dropped (see also [15, 16]). Jones [18] reinterpreted this law so that it applies * Department of Mathematics, C1200; and Institute for Computational Engineering and Sciences, C0200; The University of Texas at Austin; Austin, TX 78712 (arbogast@ices.utexas.edu). † Department of Mathematics, C1200; The University of Texas at Austin; Austin, TX 78712; current address: Department of Mathematics, The Ohio State University, 231 W. 18th Avenue, Columbus, OH 43210 (heather@math.ohio-state.edu). 1