Application of the Bergman–Milton theory of bounds to the permittivity of rocks

Abstract

The permittivity of brine-saturated porous rocks below 2 GHz varies with frequency in a way that is linked closely to the pore geometry. The theories of bounds of the permittivity by Bergman and by Milton, based on Bergman’s work on two-component composites, impose restrictions on this frequency dependence. We give a detailed and systematic analysis of these restrictions for this special case. We show to what extent the conductivity at low frequencies, combined with a measured value of the permittivity at an intermediate frequency, restricts the permittivity at all other frequencies. We establish a scaling law, according to which the permittivity depends on the brine conductivity σ2 and the frequency ω, only through the ratio σ2/ω, to a good approximation. We apply the analysis to data on sandstones by Poley, Nooteboom, and de Waal. As a further application of this theory, we derive bounds for the electrical or thermal conductivity of a two-component composite using the values that this same property would have if either of the components were an insulator.