UC Berkeley

Abstract Fractured rocks play a crucial role in myriad natural and engineered systems, including Earth's critical zone, oil/gas/geothermal reservoirs, and geological CO 2 /H 2 /waste storage systems. While complex electrical conductivity is extensively used to estimate the pore and grain sizes of conventional porous rocks and soils, it is rarely used to predict the aperture size of fractured rocks and this remains poorly understood. Here, integrating theory, simulations, and experiments, we show that under external fields, fractured rocks follow the fracture‐and‐matrix coupling to make the bulk complex conductivity non‐linear with respect to water conductivity. We find that the relaxation time and quadrature conductivity for porous media do not apply to fractured rocks, but, instead, reasonably accurate predictions of aperture size can be made based on the true formation factor. This study unravels the fundamental mechanism governing conduction and polarization of fractured rocks and paves the way for the non‐invasive investigation of global fractured rocks. Plain Language Summary In fractured rocks, most water flow and solute transport occur in fractures and hence depend on the aperture of those fractures. While the complex electrical conductivity is widely used to infer the pore and grain size of porous rocks and soils, equivalent methods for fractured rocks are limited by theoretical and experimental challenges. Specifically, the applicability of Archie's law that relates electrical conductivity merely to water conductivity remains debated in this context, and experimental measurements often face difficulties due to the small phases in impedance from the dominant conduction in fractures. To bridge this knowledge gap, we systematically investigate the complex conductivity of fractured rocks by integrating theory, simulations, and experiments. For the first time, we demonstrate that the underlying conduction and polarization mechanism of fractured rocks to be the fracture‐and‐matrix coupling. We then employ a non‐linear model to account for this coupling to obtain the true formation factor, and finally predict the mean aperture size with the cementation exponent and the fracture shape. The methodology used in this study may be extended to large scales to facilitate the quantitative non‐invasive investigation of fractured rocks in various natural and engineered systems. Key Points Electrical conduction and polarization of fractured rocks are affected by the coupling between the fracture and matrix Fracture‐and‐matrix coupling leads to non‐linear complex conductivity, which are more pronounced for low‐salinity or high‐polarizable matrix Reasonably accurate aperture size predictions can be made based on the true formation factor, cementation exponent, and fracture shape