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Universal Frequency-Dependent Permeability of Heterogeneous Porous Media: Effective–Medium Approximation and Critical-Path Analysis

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Abstract

Experiments, numerical simulations, and analytical models for simple models of porous media, such as a single pore and spatially-periodic models, have provided evidence that the dynamic, frequency-dependent permeability of porous media, when rescaled by its static value, may follow a universal function of the suitably-rescaled frequency, independent of the morphology of the pore space. No approach has, however, been developed to prove or refute the universality for a general model of a heterogeneous porous medium. We propose two approaches to analyze the problem. One is based on a dynamic effective-medium approximation (EMA) for d-dimensional networks of interconnected pores as the model of porous media, characterized by a pore-size or pore-conductance distribution. The EMA is accurate when the heterogeneity of the pore space is not very strong. The second approach is based on the critical-path analyzis that provides accurate estimates of the permeability when the pore space is highly heterogeneous. We show that both approaches predict that the rescaled frequency-dependent permeability is a universal function of the rescaled frequency. Thus, the two approaches together strongly support the universality of the rescaled dynamic permeability in any porous medium. The implications for the frequency-dependent electrical conductivity, the formation factor, and the diffusion and dispersion coefficients of porous media are also discussed.

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Acknowledgements

The author would like to thank Behzad Ghanbarian for his help in deriving Eq. (16), and for bringing to his attention the experimental data for frequency-dependent electrical conductivity of porous media mentioned above. This work was supported in part by the National Science Foundation Grant CBET 2000966.

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    Muhammad Sahimi

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Sahimi, M. Universal Frequency-Dependent Permeability of Heterogeneous Porous Media: Effective–Medium Approximation and Critical-Path Analysis. Transp Porous Med 144, 759–773 (2022). https://doi.org/10.1007/s11242-022-01839-8

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