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
(v~) = ___.K. (V(pz) z _ pzg) _ ~. (v~).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.
Nomenclature,A~,, AZ~ Ap b B dp g I e~ L Lo m M nBo.
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