The Forchheimer equation: A theoretical development

In this paper we illustrate how the method of volume averaging can be used to derive Darcy's law with the Forchheimer correction for homogeneous porous media. Beginning with the Navier-Stokes equations, we find the volume averaged momentum equation to be given by $$\langle v_\beta \rangle = - \frac{K}{{\mu _\beta }} \cdot (\nabla \langle p_\beta \rangle ^\beta - \rho _\beta g) - F\cdot \langle v_\beta \rangle .$$ The Darcy's law permeability tensor, K, and the Forchheimer correction tensor, F, are determined by closure problems that must be solved using a spatially periodic model of a porous medium. When the Reynolds number is small compared to one, the closure problem can be used to prove that F is a linear function of the velocity, and order of magnitude analysis suggests that this linear dependence may persist for a wide range of Reynolds numbers.