 
Summary: HOMOGENIZATION OF A DARCYSTOKES 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 BeaversJosephSaffman 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 twoscale 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 twoscale
convergence theory needed for our bimodal medium, including a twoscale convergence result on the
DarcyStokes 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 BeaversJoseph slip, and
(2) flow that alternates from vug to matrix behaves as if the vugs have infinite permeability.
Key words. Homogenization, twoscale convergence, DarcyStokes system, vuggy porous media,
BeaversJoseph 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
