Saturation Dependence of Transport in Porous Media Predicted by Percolation and Effective Medium Theories

Abstract

Accurate prediction of the saturation dependence of different modes of transport in porous media, such as those due to conductivity, air permeability, and diffusion, is of broad interest in engineering and natural resources management. Most current predictions use a "bundle of capillary tubes" concept, which, despite its widespread use, is a severely distorted idealization of natural porous media. In contrast, percolation theory provides a reliable and powerful means to model interconnectivity of disordered networks and porous materials. In this study, we invoke scaling concepts from percolation theory and effective medium theory to predict the saturation dependence of modes of transport — hydraulic and electrical conductivity, air permeability, and gas diffusion — in two disturbed soils. Universal scaling from percolation theory predicts the saturation dependence of air permeability and gas diffusion accurately, even when the percolation threshold for airflow is estimated from the porosity. We also find that the non-universal scaling obtained from the critical path analysis (CPA) of percolation theory can make excellent predictions of hydraulic and electrical conductivity under partially saturated conditions.

Electrical Conductivity of Partially Saturated Packings of Particles

“…(6), the effect of the PCD on σ is negligible. Ewing and Hunt (2006) and Ghanbarian et al ( , 2015a) emphasized also that one should expect the saturation dependence of σ to follow universality, if the medium is not too heterogeneous (i.e., if D < 2). Kogut and Straley (1979), Sahimi et al (1983a), and Feng et al (1987) all emphasized that the propensity of very small or zero conductance is the root cause of the non-universality Putting it another way, r min = 0 is the origin of non-universality (Sahimi et al 1983a).…”

“…That for all the packings D < 2 (Table 1) implies that in the present formulation the key parameter for the universality or non-universality of t is D : σ r follows the universal power law of percolation, if D < 2. Ghanbarian et al (2015a) discussed the special case β = φ using the CPA and pointed out that there exists no criterion to distinguish the crossover point separating the power law from the CPA when β = φ. They proposed combining the two power laws to model σ r over the entire range of saturation, if D ≥ 2.…”

“…Among them is the empirical Archie's law (Archie 1942), which is valid only for water-wet rocks deprived of microporosity or substantial clay-exchange cations (Toumelin and TorresVerdín 2008) and has been extensively used to determine σ . A recent review of the theoretical models of σ is given by Ghanbarian et al (2015a). Mualem and Friedman (1991) represented the pore space by a bundle of series-parallel capillary tubes (Mualem 1976) in order to predict σ .…”

The Evaluation of Electrical Impedance of Three-Dimensional Fractal Networks Embedded in a Cube

“…This phenomenon can be explained from the percolation of impedance response in terms of pore size. Ghanbarian et al 29 investigated the saturation dependence of transport and percolation in porous media by effective medium theory and considered that the percolation of impedance response was strongly associated with the pore size distribution. The large value of β indicates the large diameter of pore channels in the whole fractal pore network.…”

Fractal dimension of soil fragment mass-size distribution: A critical analysis

“…1). In each iteration, regions previously assigned to the fractal phase are randomly partitioned into P, S, and F; the fractional value assigned to each phase does not change across iterations, and P + S + F = 1 (Ghanbarian et al, 2015).…”