Continuum percolation theory for transport properties in porous media

It is well-known that transport properties vanish according to power laws at a percolation threshold. In systems described by continuum percolation, theoretical results can give non-universal powers. Porous media have been argued to be systems, which typify such non-universal percolation results. It is shown here that the suite of properties, pressure saturation relationships, electrical conductivity, air permeability, hydraulic conductivity, and solute diffusion are perfectly described by using critical path analysis (from percolation theory) far from the percolation threshold and using universal scaling of transport properties near the percolation threshold. The appropriate description of most porous media is a truncated random fractal, with typical pore size ranges up to two orders of magnitude. It is shown that in the extreme condition of an infinite range of pore sizes, the present theoretical treatment generates the appropriate non-universal percolation exponents from theory. However, an infinite range of pore sizes corresponds to a porosity of 1, which would require in an arbitrary volume an infinite solid surface area with zero solid volume fraction. For a number of reasons this is not a realistic model of a porous medium. Thus the argument that typical natural porous media form a realistic basis to generate non-universal exponents from percolation theory is not supported.