A combination of the Hashin-Shtrikman bounds aimed at modelling electrical conductivity and permittivity of variably saturated porous media
In this paper we propose a novel theoretical model for the dielectric response of variably saturated porous media. A model is constructed for fully saturated porous media as a combination of the well established Hashin and Shtrikman  bounds and Archie’s  law, the latter usually adopted for electrical conductivity of multiphase porous media. One of the key advantages of the new constitutive model is that it explains both electrical conductivity and permittivity using the same parameterization, so that the measurements made on one such physical quantity can be used to predict the other. The model for partially saturated media is derived as an extension of the fully saturated model, where the permittivity of the pore space is in turn obtained as a combination of the permittivity of the aqueous and non aqueous phases. Both fully saturated and unsaturated models are successfully tested against both synthetic data from pore-scale digital modeling and laboratory data from the literature, proving effective for a wide range of different porous media. As opposite to most of existing constitutive laws for the water saturation – dielectric constant relationship, model parameters are related to the micro-geometrical and chemical-physical properties of the porous medium.