Universal scaling of the formation factor in porous media derived by combining percolation and effective medium theories

Abstract

The porosity dependence of the formation factor for geologic media is examined from the perspective of universal scaling laws from percolation and effective medium theories. Over much of the range of observed porosity, the expected percolation scaling is observed, but the values of the numerical prefactor do not conform to the simple predictions from percolation theory. Combining effective medium and percolation theories produces a numerical prefactor whose value depends on both the threshold porosity and the porosity above which the formation factor crosses from percolation to effective medium scaling. This change allows extraction of a numerical value of the prefactor, which is reasonably close to experimental values. Subsequent evaluation of the porosity dependence of the formation factor shows that difficulties in prior comparisons of theory and experiment are largely removed when percolation scaling is allowed to transition to effective medium scaling far above the percolation threshold.

Relating Topological and Electrical Properties of Fractured Porous Media: Insights into the Characterization of Rock Fracturing

“…As before, we use the two Archie's law formulations F = φ −m and F = τ /φ. In addition, we consider the percolation-theory model F = (φ − φ c ) −t (e.g., [31,52,53]), in order to evaluate whether the notion of a critical porosity φ c helps to interpret data corresponding to low-porosity fractured porous media with sparse fracture networks. Note that in this case, the estimated value of φ c is restricted to lie between 0 and the minimum porosity of the considered fracture family.…”

“…This has been the objective of numerous laboratory experiments (e.g., [22][23][24][25]) and modeling approaches (e.g., [26][27][28][29][30]) conducted for a wide variety of porous and fractured rocks. Whereas these studies defined and characterized the corresponding petrophysical relationships for porous media (e.g., [16,31,32]), some experimental results obtained for fractured-rock samples remain unexplained because the impact of the fracture-network properties on the resistivity response is unclear (e.g., [25,[33][34][35]). From a numerical modeling standpoint, multiple-porosity model formulations show how the presence of vugs and fractures impacts the effective electrical properties of rocks (e.g., [14,26,36]).…”

Physical Explanation of Archie's Porosity Exponent in Granular Materials: A Process‐Based, Pore‐Scale Numerical Study

“…Clair et al, 2015). To link the measured resistivity to other properties of interest, considerable efforts have been made to develop constitutive models for geological materials (e.g., Bussian, 1983;Ghanbarian et al, 2014;Sen et al, 1981). The most widely used model is probably Archie's law (Archie, 1942), which relates the conductivity σ of rock to the porosity ϕ by…”

Universal scaling of the formation factor in clays: Example from the Nankai Trough

“…Nonetheless, analyses presented by Ghanbarian‐Alavijeh and Hunt [], Ghanbarian et al []. Ghanbarian and Hunt [], Hunt et al [], Ghanbarian et al [], and Ghanbarian et al [] indicate that universal scaling from percolation theory is valid over a broader region above the percolation threshold in natural porous media. In the effective medium approach, a relatively disordered porous medium is replaced by a uniform one.…”

“…In general, the percolation theory universal scaling is theoretically valid when porosity is close to the percolation threshold, while the effective medium universal scaling is appropriate for porosity much greater than the percolation threshold. The porosity ϕ x that separates the percolation and effective medium domains varies; the fraction of conducting bonds at which crossover occurs as reported by Kirkpatrick [] and Kiefer et al [] for numerical simulations of resistor networks corresponds to a value of roughly 0.75, but values between 0.26 and 1 have been determined for natural porous media, and there is no evidence that there is a universal value [ Ghanbarian et al , ]. The crossover porosity ϕ x might be above the porosity of the medium (see, e.g., Ghanbarian et al []) or below it [ Ghanbarian et al , ].…”