Scanning electron microscope images of cross sections of several porous specimens have been digitized and analyzed using image processing techniques. The porosity and specific surface area may be estimated directly from measured two-point spatial correlation functions. The measured values of porosity and image specific surface were combined with known values of electrical formation factors to estimate fluid permeability using one version of the Kozeny-Carman empirical relation. For glass bead samples with measured permeability values in the range of a few darcies, our estimates agree well (±10–20%) with the measurements. For samples of Ironton-Galesville sandstone with a permeability in the range of hundreds of millidarcies, our best results agree with the laboratory measurements again within about 20%. For Berea sandstone with still lower permeability (tens of millidarcies), our predictions from the images agree within 10–30%. Best results for the sandstones were obtained by using the porosities obtained at magnifications of about 100× (since less resolution and better statistics are required) and the image specific surface obtained at magnifications of about 500× (since greater resolution is required).

In a recent article, Blair et al. [1996] developed an image processing method for characterizing the microstructure of rock and other porous materials. The method is based on the statistical properties of the microgeometry as observed in scanning electron micrograph (SEM) images. The statistical properties are evaluated through analysis of the spatial correlation function. Blair et al. [1996] proceed to estimate permeabilities of sandstones and porous glass, using the microstructural properties as input for a Kozeny-Carman type of equation. Their results are very good, although their permeability estimates are, in general, somewhat low. I think that their approach is very valuable, but I would like to point out that their analysis uses equations that hold in three dimensions. However, at a certain point in their analysis it is more appropriate to use the two-dimensional analogue of their equation, because SEM images are two-dimensional. When one performs the analysis with the appropriate two-dimensional equation, the results become even more impressive. Blair et al. [1996] define the two-point correlation function S2(r) for a given function f(x) as S2(r) = (f(x) f(x + (1) where the angle brackets denote volume averaging over all positions x. The two-point correlation function can be used to measure a number of quantities in a SEM image. A particularly interesting quantity is the specific surface area s, i.e., the surface area of the rock material per unit volume. The specific surface area is related to the derivative of S2(r) at zero distance. Berryman and Blair [1986] have shown that in three dimensions this relation is given by S[(O) =-s/4. (2) Blair et al. [1996] implicitly assume that this relation also holds in two dimensions. Unfortunately, it does not. In two dimensions it reads s[(o) = (3) This can be explained as follows. An intermediate result in the derivation is Berryman and Blair's [1986] equation (10): dS2(r) dr 6oV aft dxf(x) •rr (x + r). (4) Here fi is the direction of r in space and (1/6o) f d fi is the average over this direction. In three dimensions this average is Copyright 1997 by the American Geophysical Union. Paper number 97JB02373. 0148-0227/97/97JB-02373509.00 (l/4rr) fo 2• de f(• dO sin O. In two dimensions, however, the average over direction takes a different form, namely, (1/2 rr) fo 2• de. As a consequence, the proportionality factor in equation (2) changes. Performing the same steps as Berryman and Blair [1986], one arrives at equation (3). Blair et al. [1996] used equation (2) to measure the specific surface s in SEM images. They subsequently used a Kozeny-Carman equation for the permeability k, k = ½:/cFs:, (5) where F is the so-called formation and c is a constant, which is taken to be 2. In this comment I argue that it is better to use equation (3) to measure s, instead of equation (2), because one measures s in two-dimensional SEM images. When one then uses equation (5) to estimate permeabilities, only one assumption is necessary, which is that the two-dimensional s is a good...

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.

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