Continuum percolation theory and Archie's law

Abstract

[1] Results for the unsaturated hydraulic conductivity, K, are adapted to generate the electrical conductivity, s, as a function of moisture content, q. The result for s, evaluated q = f, (f the porosity), yields s / f m , known as Archie's Law. This result is based on the fact that s must vanish as the critical moisture content for percolation, q t , is approached, and on the apparent proportionality of q t to f, at least in coarser porous media (with insignificant clay content). In three dimensions the power is expected to be 1.88, but in two dimensions, approximately 1.35. The former result is compatible with data summarized in Thompson et al., giving m = 1.86 ± 0.19, while the latter appears compatible with two-dimensional simulations by Kuentz et al., who found m = 1.28 ± 0.07. The same analysis demonstrates that K cannot be written as a power of the porosity.

Analytical modeling of one-dimensional diffusion in layered systems with position-dependent diffusion coefficients

“…The similar power-law function relationship D=D 0 / e c eff was also observed by other authors in various fields, such as gas diffusion in variable saturated porous materials [17,51,24,28], electron diffusion on porous film [6,19], and simulation studies [14]. Attempts have been made to explain Archie's law using percolation models [24,23]. In general, there is a broad range of observed c, which depends on the dimensionality, pore throat size distributions, and degree of anisotropy of the percolation system [3].…”

“…The similar power-law function relationship D=D 0 / e c eff was also observed by other authors in various fields, such as gas diffusion in variable saturated porous materials [17,51,24,28], electron diffusion on porous film [6,19], and simulation studies [14]. Attempts have been made to explain Archie's law using percolation models [24,23].…”

Geophysical Properties of the Near Surface Earth: Electrical Properties

“…At low porosities, there may be a fraction of the porosity f o that does not take part in conduction, which may be taken into consideration using a percolation limit approach (Bernabé and Bruderer, 1998;Hunt, 2004;Kirkpatrick, 1973;Seager and Pike, 1974;Shah and Yortsos, 1996): At low porosities, there may be a fraction of the porosity f o that does not take part in conduction, which may be taken into consideration using a percolation limit approach (Bernabé and Bruderer, 1998;Hunt, 2004;Kirkpatrick, 1973;Seager and Pike, 1974;Shah and Yortsos, 1996):…”

Relevance of percolation theory to power‐law behavior of dynamic processes including transport in disordered media

“…Thus, when the electrical conductivity is a power of the saturation (fraction of pore space occupied by water), this result is usually due to the relevance of percolation theory to the conductivity rather than rooted in the pore size distribution [103]. But not all properties have the same simple behavior [103].…”

“…Thus, when the electrical conductivity is a power of the saturation (fraction of pore space occupied by water), this result is usually due to the relevance of percolation theory to the conductivity rather than rooted in the pore size distribution [103]. But not all properties have the same simple behavior [103]. In general, it is recognized that the two main influences on flow and transport properties of porous media are the pore size distribution and the connectivity of dominant pathways for fluid (or solute) motion.…”