Darcy's Law from the Stokes Equation: New Perspective on the Upscaling Problem

The theoretical development of Darcy's Law from the consideration of the pore-scale Stokes flow within the porous matrix is a canonical problem in upscaling theory. Although many successful derivations of Darcy's Law exist, there is still much that can be learned by reexamination of the derivation, particularly in regard to understanding the assumptions and restrictions that have been made in the development. In this work, a derivation of Darcy's Law is discussed in the context of integral solutions to the Stokes equation. In particular, the focus is on (1) a careful formulation of the microscale hydrodynamic problem, especially with regard to boundary conditions; (2) the development of the averaged equations and the associated closure that is required for predicting the hydraulic conductivity from the pore structure; and, (3) a presentation of the solutions for the hydraulic conductivity explicitly in terms of an integral solution to the Stokes equation. The integral solution presented relies on well-developed boundary integral methods (e.g., Pozrikids, 2011; Hapel and Brenner, 1965; Ladyzhenskaya, 1963). Although these solutions have been known for decades, they have been under-utilized in the development of Darcy's Law from the Stokes equation. This approach leads to some nonconventional terms appearing in the macroscale momentum balance, and is thus not exactly consistent with Darcy's Law. The importance and role of these nonconventional terms will be discussed.