An analytical membrane-polarization model to predict the complex conductivity signature of immiscible liquid hydrocarbon contaminants
An analytical membrane-polarization model is developed to predict the frequency-dependent complex conductivity of hydrocarbon-contaminated sediments. In the absence of hydrocarbon contaminants, the effect of membrane polarization can be approximated using a recently developed analytical model, which describes the pore space as a sequence of two cylindrical pores of different lengths and radii. Different cation and anion concentrations in the electrical double layer at the pore wall lead to an ion-selective behaviour causing the membrane-polarization effect. This model can readily be adjusted to account for a wetting liquid hydrocarbon covering the pore wall. To model the effect of non-wetting hydrocarbon, we extend the analytical model by introducing a second cylindrical body into the cylindrical pores that represents a discrete droplet of the contaminant phase. In order to account for the high electrical resistivity of liquid hydrocarbons, the corresponding volumes are assumed to be electrically insulating. Because most liquid hydrocarbon surfaces are negatively charged when in contact with an electrolyte, they are covered by a second electrical double layer, which can easily be incorporated into the analytical model. We use our extended model to study the effect of varying saturations of wetting and non-wetting hydrocarbon on the complex electrical conductivity of the pore system. Our results predict that conductivity magnitude and conductivity phase generally decrease with hydrocarbon saturation. However, if the surface potential at the surface of non-wetting hydrocarbon droplets is larger than the one at the pore wall, we can observe an increase in the conductivity magnitude with the hydrocarbon saturation and a slight increase in the conductivity phase at intermediate hydrocarbon concentrations. This finding is particularly interesting as it offers a possible explanation for the relation between complex conductivity and hydrocarbon saturation observed in different field and laboratory experiments.