An extension of Darcy's law to non-Stokes flow in porous media

Abstract

A momentum equation for Newtonian-fluid flow in homogeneous and isotropic porous media is derived on the basis of momentum conservation over a representative elementary volume. The resultant equation can be expressed in the dimensionless form as Fo −1 ∂ u ∗ ∂t ∗ +Re K 1/2 K 1/2 l u ∗·∇ ∗ u ∗+(1+Re p) u ∗=−∇ ∗ p ∗+ f ∗, where u ∗, t ∗, p ∗ and f ∗ are, respectively, the dimensionless superficial velocity, time, pressure, and body force, Fo≡ νt c / K is a Fourier number, Re K 1/2 ≡ U c K 1/2/ ν is a Reynolds number based on the permeability K and a macroscopic characteristic velocity U c , Re p ≡ U p l p / ν is a Reynolds number based on the microscopic characteristic length l p and characteristic velocity U p , ν is the kinematic viscosity of the fluid, t c is a characteristic time, and l is the length scale of the representative elementary volume. Under the conditions that t c ≫ K/ ν (i.e., Fo≫1) and l/ K 1/2≫ Re K 1/2 , the unsteady and convection terms become negligible in comparison to the viscous term. These conditions are generally met for fluid flow in natural porous media. The reduced form of the equation may be considered as a generalized form of Darcy's law which applies to both Stokes and non-Stokes flows in porous media. The pressure drop vs. the mass–flux relationship predicted from this study agrees reasonably well with the experimental data reported in the literature.