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HOMOGENIZATION OF A DARCY-STOKES SYSTEM MODELING VUGGY POROUS MEDIA
 

Summary: 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

  

Source: Arbogast, Todd - Center for Subsurface Modeling & Department of Mathematics, University of Texas at Austin

 

Collections: Mathematics; Geosciences