Homogenization of a Darcy–Stokes system modeling vuggy porous media

• Published in: Computational geosciences 2006-09, Vol.10 (3), p.291-302
• Main Authors: Arbogast, Todd, Lehr, Heather L
• Format: Article
• Language: English
• Subjects: Boundary conditions; Darcys law; Fluid dynamics; Fluid flow; Homogenization; Permeability; Porous media
• ISSN: 1420-0597; 1573-1499
• DOI: 10.1007/s10596-006-9024-8

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 bimodal 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.[PUBLICATION ABSTRACT]