Kozeny–Carman relations and image processing methods for estimating Darcy’s constant

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Abstract
A natural connection is demonstrated between Kozeny–Carman relations for porous media and the image processing techniques which have recently been applied to the problem of estimating the parameters in such relations. It is shown that the term in the Kozeny–Carman relation related to the specific surface area is best estimated from a smoothed version of the actual material surface. To measure this image specific surface, the magnification of a cross section of the porous material should be chosen so that a typical correlation length for the sample corresponds to a distance comparable to 100 discrete picture elements. Under these conditions, the assumptions typically made in the derivation of a Kozeny–Carman relation are entirely compatible with the resolution constraints imposed by digitizing the image. Thus, although the measured image specific surface may be considerably smaller in magnitude than the true specific surface area of the material (due to resolution constraints), this smaller value is nevertheless the required input to the Kozeny–Carman relation. The argument is based on a known comparison theorem relating the permeabilities of two porous materials which differ only by the addition (without rearrangement) of the solid to the more porous material.