A Fractional Differential Model for the Electrical Conductivity of Clay Rocks

Abstract

Summary A micro-macro approach is proposed to model the effects of surface phenomena occurring at the pore fluid/solid interface on the electrical conductivity of clay rocks. This new model is a generalization of the so-called differential scheme (DS) used to infer the effective properties of a composite made up of several materials. It is named the fractional differential scheme and is based on a fractional integral formulation of the DS for a porous medium considered as a two-component composite. The formulation of the fractional DS introduces two parameters: a cementation exponent m and a fractional order α. The fractional order α is assumed to account for the amplitude of the so-called surface conduction on clay minerals. Both parameters m and α are inverted from a set of electrical conductivity measurements obtained on clay rocks. The inversion results demonstrate that the fractional DS model is able to capture the dependence of the cation concentration on the effective electrical conductivity of the clay rocks under study. Our results also show that the fractional order α can be considered an indirect indicator of the amplitude of physico-chemical interactions between hydrated cations and swelling clay minerals occurring at the pore fluid/solid interface.