New insights on steady, non-linear flow in porous media

Abstract

We consider two problems of nonlinear flow in porous media: 1) a derivation of a cubic weak inertia correction of Darcy's law which is valid for any matrix anisotropy, and 2) a description of flow by the weak inertia equation for low Reynolds numbers and a Forchheimer equation for high Reynolds number laminar flow. Recent homogenization studies show that the weak inertia correction to Darcy's law is not a square term in velocity, as it is in the Forchheimer equation, but instead a cubic term in velocity. By imposing that the pressure loss is invariant under flow reversion, it has been shown that the weak inertia equation is valid even for anisotropic media. We show, by using the homogenization technique, that the weak inertia equation is valid for any anisotropic matrix symmetry without imposing a reversed flow symmetry. For the second problem, we reexamine published data. We find that the description 2) applies well. A spline may be applied in the crossover regime.