Mathematical Treatment of Saturated Macroscopic Flow in Heterogeneous Porous Medium: Evaluating Darcy’s Law

Abstract

We present a rigorous mathematical treatment of water flow in saturated heterogeneous porous media based on the classical Navier-Stokes formulation that includes vorticity in a heterogeneous porous media. We used the mathematical approach proposed in 1855 by James Clark Maxwell. We show that flow in heterogeneous media results in a flow field described by a heterogeneous complex lamellar vector field with rotational flows, compared to the homogeneous lamellar flow field that results from Darcy’s law. This analysis shows that Darcy’s Law does not accurately describe flow in a heterogeneous porous medium and we encourage precise laboratory experiments to determine under what conditions these issues are important. We publish this work to encourage others to perform numerical and laboratory experiments to determine the circumstances in which this derivation is applicable, and in which the complications can be disregarded.