Conductivity and permeability of rocks

TLDR

Salt‑water‑saturated rock conductivity is modeled by a random resistance network with a zero percolation threshold. The study varies rock porosity via a random bond‑shrinkage mechanism, uses numerical and analytical calculations to recover Archie's law, and tests the model experimentally on artificial and natural rocks. The model predicts an Archie's exponent greater than one linked to pore‑size skewness, yields a permeability power law with exponent m′ = m(m+1) in 1D and 2m in higher dimensions, and experimental data confirm both the larger exponent for heterogeneous samples and the m′ = 2m relationship.

Abstract

The electrical conductivity of salt-water-saturated rocks is modeled by a random resistance network which has a zero percolation threshold. The porosity is varied by a random bond-shrinkage mechanism. Numerical and analytical calculations of the model in different dimensions show an Archie's-law behavior: ${\ensuremath{\sigma}}_{r}=a{\ensuremath{\sigma}}_{w}{\ensuremath{\varphi}}^{m}$, where $\ensuremath{\varphi}$ is the porosity of the rock, and ${\ensuremath{\sigma}}_{r}$ and ${\ensuremath{\sigma}}_{w}$ are the conductivities of the rock and water, respectively. We find that the Archie's exponent $m$ is always greater than unity and is related to the skewness of the "pore-size distribution" of the rock. Applying the same model to fluid-flow permeability (${k}_{r}$) gives ${k}_{r} \ensuremath{\propto}{\ensuremath{\varphi}}^{{m}^{\ensuremath{'}}}$, where ${m}^{\ensuremath{'}}=m(m+1)$ in one dimension, and ${m}^{\ensuremath{'}}=2m$ in higher dimensions. This power-law form is consistent with the well-known Kozeny equation and has been frequently suggested by empirical studies. Experimental tests of the model are performed on artificial rocks, made by fusing small glass beads, as well as real rocks. From resistivity measurements, we demonstrate that $m$ is larger in samples with a wider fluctuation of pore sizes, which is qualitatively consistent with the model. From fluid-flow experiments on fused glass beads, we find quantitative support for the ${m}^{\ensuremath{'}}=2m$ prediction.