Theoretical and numerical modeling of electric conductance in porous media

The spotlight is on the electric conductance of 2D and 3D media with perfectly conducting or insulating inclusions. Using the known formulas for incremental conductance in introduction of a small inclusion in a medium, the differential equations are obtained for the conductance of a medium with the non-small concentration of inclusions. The relations behave exponentially and contain no percolation thresholds. In 2D case, the relations between the conductance and concentration of circle and square inclusions are obtained. In terms of a 3D medium, the spherical inclusions are analyzed, which makes it possible to model porous rocks. The results were tested in COMSOL Multiphysics by means of construction of single dimension and unitary conductance samples added then with inclusions. The tests aimed to examine the conformity between the experimental and theoretical relations of conductivity and concentration of inclusions in the two-component media, as well as to check the agreement with the known predictions for two models–effective model approximation and Rayleigh model with inclusions located at the periodic lattice sites. The accomplished numerical experiments exhibit a larger advantage of the proposed formulas over the formulas derived using the method of effective medium. The Rayleigh model and the proposed model agree well with the numerical experimental data up to the inclusion concentration of ~0.3, i.e. within the whole range of values when fracturing of a sample is yet far. In 3D case, the proposed model is more accurate than the Rayleigh model. Moreover, the proposed model is much simpler in terms of mathematical notations.

Sizin P. E. Theoretical and numerical modeling of electric conductance in porous media. MIAB. Mining Inf. Anal. Bull. 2023;(5):43-56. [In Russ]. DOI: 10.25018/0236_ 1493_2023_5_0_43.

P.E. Sizin, Cand. Sci. (Phys. Mathem.), Assistant Professor, Institute of Basic Education, National University of Science and Technology «MISiS», 119049, Moscow, Russia, e-mail: mstranger@list.ru, ORCID ID: 0000-0001-8156-4972.

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