A combination of the Hashin-Shtrikman bounds aimed at modelling electrical conductivity and permittivity of variably saturated porous media

P>In this paper, we propose a novel theoretical model for the dielectric response of variably saturated porous media. The model is first constructed for fully saturated systems as a combination of the well-established Hashin and Shtrikman bounds and Archie's first law. One of the key advantages of the new constitutive model is that it explains both electrical conductivity-when surface conductivity is small and negligible-and permittivity using the same parametrization. The model for partially saturated media is derived as an extension of the fully saturated model, where the permittivity of the pore space is obtained as a combination of the permittivity of the aqueous and non-aqueous phases. Model parameters have a well-defined physical meaning, can be independently measured, and can be used to characterize the pore-scale geometrical features of the medium. Both the fully and the partially saturated models are successfully tested against measured values of relative permittivity for a wide range of porous media and saturating fluids. The model is also compared against existing models using the same parametrization, showing better agreement with the data when all the parameters are independently estimated. An example is also presented to demonstrate how the model can be used to predict the relative permittivity when only electrical conductivity is measured, or vice versa.