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There have been many attempts over the years to relate the fluid permeability k of a porous material to some relevant microstructurally-defined length scale. This is done since the units of the permeability k, as defined by Darcy's law [11],
are length squared, so that the square root of k defines some sort of "typical" or "average" pore
diameter that characterizes the material. In eq. (3),
P is the applied pressure difference,
L is the thickness of the porous sample, 
is the fluid viscosity, and q is the volumetric flow
per unit cross-sectional sample area.
One way to define a length scale that could be appropriate for permeability is by the hydraulic radius, which is essentially the ratio of volume to surface area of a porous material. The permeability k then goes like the hydraulic radius squared. There are many equations that attempt to predict permeability based on this approach, which are all called Kozeny equations [11]. The hydraulic radius approach seems to work fairly well for simple porous materials, like packed powders, but not for more complicated porous materials like rocks [11].
Katz and Thompson have recently derived, using percolation theory, a prediction for the ratio of permeability to electrical conductivity, as given in eq. ( 2) [3,4]. This equation has no adjustable parameters, and relies on a mercury porosimetry measurement to define dc, their prediction of the length scale appropriate for permeability. The quantity dc is defined the following way. If the pore space is sequentially built up starting with the largest pores and working down, then dc is the diameter of the pore that just completes the first continuous pathway through the material. This pathway then consists only of pores with d greater than or equal to dc. The physical justification for this procedure is that if one were to pick out the subset of the pore space that contributes most to permeability, it would be the subset of pores that not only were the largest, but which also formed a continuous pathway through the pore space. Katz and Thompson found experimentally that dc corresponded closely to the inflection point on the cumulative intrusion curve [3,4].
Using this basic physical idea, Katz and Thompson then used methods from percolation theory
[7] to form lower bounds involving dc for both the permeability and electrical conductivity of
the porous material. Unknown constants arise in this approach, which are fortunately the same
for the two transport coeffficients. By maximizing these lower bound functions, so as to get the
best possible values, and taking the ratio of the results, which gets rid of the unknown constants,
they derived eq. (2). Since for rocks, the porous materials of interest for Katz and Thompson, dc
and 
are easily measurable
quantities, eq. (2) is thought of as a prediction for k in terms
of pertinent microstructural parameters [3,4]. The agreement with experimental results on
rocks is impressive, as noted above, but the criticism of using pore diameters as measured by
mercury porosimetry under the assumption of circular pore geometry must still be addressed.