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

Abstract

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

Combined estimation of effective electrical conductivity and permittivity for soil monitoring

“…The petrophysical constitutive model of Brovelli and Cassiani [2010b] was developed by combining variational bounds for the transport properties of granular composites [ Hashin and Shtrikman , 1962; Milton , 1981] and Archie 's [1942] law, a constitutive model commonly used to parameterize electrical conductivity of variably saturated porous media. The model was consistently developed for two‐ and three‐phase materials on the basis of the same basic assumptions.…”

“…On the contrary, electrical conductivity is often influenced by the conductance arising from the excess of charge in the vicinity of the negatively charged solid surfaces of silica, clays, and organic matter [ Brovelli et al , 2005; Revil and Glover , 1997; Revil et al , 1998]. The additional contribution of the charged interface is often converted into an equivalent grain (volumetric) conductivity [ Bussian , 1983; De Lima and Sharma , 1990], and therefore analogous boundary value problems can describe both bulk permittivity and electrical conductivity [ Brovelli and Cassiani , 2010b; Brovelli et al , 2005; Linde et al , 2006]. Although developed using different approaches, including empirical or semiempirical considerations [ Archie , 1942; Clavier et al , 1984; Waxman and Smits , 1968], effective medium theories [ Bussian , 1983; Miller , 1969; Sen et al , 1981], and volume‐averaging algorithms [ Linde et al , 2006; Pride , 1994], most of the petrophysical equations suited to model both DC electrical conductivity and high‐frequency permittivity utilize a parameterization that is compatible with that introduced by Archie [1942].…”

Assessment of local hydraulic properties from electrical resistivity tomography monitoring of a three‐dimensional synthetic tracer test experiment

“…In other words, identification of many parameters using relatively little, somewhat noisy information that is collected only at the domain boundary clearly represents a difficult challenge [e.g., Menke , 1984]. In addition, suitable constitutive models linking geophysical and hydrological parameters are needed [e.g., Brovelli and Cassiani , 2010, 2011], often to be calibrated on a site‐by‐site basis. These limitations can make geophysical methods more of a qualitative tool, unless information concerning the relevant hydrological processes involved are properly taken into account.…”

1. Joint Inversion of Crosshole GPR and Seismic Traveltime Data

“…A key step in all interpretations is to know what field-scale petrophysical relationships should be used to transform the models into geologic or hydrogeologic properties. It is helpful to use petrophysical relationships that share similar parameterizations and assumptions about the pore structure for all model types, as done here for the HS lower bounds, using a weighted average of the HS bounds (Brovelli and Cassiani, 2010) or volume averaging . Even if the joint inversion improves resolution, similar resolution-dependent petrophysical relationships, as for individual inversions, remain.…”

“…The lower and upper bounds for dielectric permittivity in saturated media are (e.g., Hashin and Shtrikman, 1962;Brovelli and Cassiani, 2010) …”