Percolative transport in fractal porous media

Abstract

Application of continuum percolation theory to a fractal pore space model yields results for the constitutive relationships for unsaturated flow in agreement with experiment. This application also unites understanding in that the same dry end moisture content, θt=0.039SAvol0.52 as a function of the surface area to volume ratio, is shown to be associated with the deviation of experimental water retention from fractal scaling as well as with the vanishing of the diffusion constant. Substituted into a critical path analysis (based on continuum percolation theory) for the dependence of the unsaturated hydraulic conductivity, K(θ), on moisture, the same value of θt produces excellent agreement with experimental data (y=1.0015x−0.0065, R2=0.96), with y experiment, x theory and using no adjustable parameters. Though critical path analysis is based on percolation theory, the result obtained for K(θ) is more closely tied to the fractal characteristics of the medium, and the dependence is referred to as a fractal scaling of the hydraulic conductivity. In all three properties, the interpretation of θt is the same; it represents the minimum value for which a continuous interconnected path of capillary flow is possible, making it the critical volume fraction for percolation. This identification means that the low moisture content deviation from fractal predictions in h(θ) does not conflict with fractal models of the pore space, as the deviation is due to dynamics rather than to structure. Critical path analysis does not yield percolation scaling, in which K vanishes as a power of (θ−θt). However, it is shown here that the data for K(θ) and h(θ) are consistent with an interpretation in which the fractal scaling of K at large moisture contents crosses over to a percolation scaling at a moisture content slightly above θt.