The influence of the fractal dimension on the complex conductivity of porous materials

SUMMARY The bulk component of the electrical conductivity of a porous material is related to the (connected) porosity and saturation by power-law functions defining the first and second Archie's laws. Recently, it was shown that for porous materials with fractal characteristics, the power-law exponent of Archie's law could be related to the fractal dimension of such materials. Similarly, the real and imaginary parts of the complex-valued surface conductivity are not just proportional to the specific surface area and saturation of the material but to power-law functions of these properties defining two additional ‘interfacial’ Archie's laws, which are called the third (saturated case) and fourth (unsaturated case) Archie's laws in this paper. These new laws have been poorly recognized and studied so far. A number of porous materials and especially clay-rich media are multiscale materials characterized by broad distributions of particle and pore sizes. We extend Archie's laws concept to describe the complex conductivity of such materials. We use both numerical simulations in fractal porous materials as well as published experimental data sets to propose a unified physical interpretation of the exponents entering the four Archie's type power-law relationships, which offer an updated complex conductivity model for natural porous media.