Percolation approach to the problem of hydraulic conductivity in porous media

While percolation theory has been studied extensively in the field of physics, and the literature devoted to the subject is vast, little use of its results has been made to date in the field of hydrology. In the present study, we carry out Monte Carlo computer simulations on a percolating model representative of a porous medium. The model considers intersecting conducting permeable spheres (or circles, in two dimensions), which are randomly distributed in space. Three cases are considered: (1) All intersections have the same hydraulic conductivity, (2) The individual hydraulic conductivities are drawn from a lognormal distribution, and (3) The hydraulic conductivities are determined by the degree of overlap of the intersecting spheres. It is found that the critical behaviour of the hydraulic conductivity of the system, K, follows a power-law dependence defined by K ∞ (N/N_c-1)^x, where N is the total number of spheres in the domain, N_c is the critical number of spheres for the onset of percolation, and x is an exponent which depends on the dimensionality and the case. All three cases yield a value of x≈1.2±0.1 in the two-dimensional system, while x≈1.9±0.1 is found in the three-dimensional system for only the first two cases. In the third case, x≈2.3±0.1. These results are in agreement with the most recent predictions of the theory of percolation in the continuum. We can thus see, that percolation theory provides useful predictions as to the structural parameters which determine hydrological transport processes.