A fractional differential scheme for the effective transport properties of multiscale reactive porous media: Applications to clayey geomaterials

A new homogenization scheme is proposed to model the effects of surface phenomena occurring at the pore fluid/solid interface on the effective transport properties of multiscale and reactive geosystems, e.g., clayey geomaterials. This new scheme 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 two key elements: (i) the concept of realizability of the DS itself and (ii) 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 alpha. The cementation exponent m is related to the microstructure of the material. The fractional order alpha accounts for the amplitude of the "surface" transport of ions resulting from the physico-chemical interactions between hydrated cations and swelling clay minerals. Both parameters m and alpha are inverted from two sets of data: electrical conductivity measurements obtained on clayey rocks and effective diffusion coefficient measurements acquired from a Na-montmorillonite geomaterial. The inversion results demonstrate that the fractional DS model is able to capture the dependence of the cation concentration on the effective transport properties of the clayey materials under study. Our results also show that the fractional order alpha 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.