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

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.