Analytical model for the dielectric response of brine-saturated rocks

Rocks whose pore space is saturated with a conducting brine have a dielectric permittivity which increases significantly with decreasing frequency in the range 1–1000 MHz, and a conductivity which shows a corresponding decrease in the same range. Here a simple model is developed which accounts for this effect within the framework of an analytical representation proposed by Bergman. The model postulates a particular analytic form for the density of resonances characterizing the geometry of the pore structure. The parameters of the model are completely determined by information about the dc conductivity of the rock, plus two exact sum rules satisfied by any composite density function and an inequality which may be taken as an equality in the limit when the contact area between rock grains is small in comparison to the surface area of the grains. The resulting dielectric permittivity of the composite varies as ${\omega }^{\mathrm{-}b}$ at low frequencies, where b can be calculated in the model from measurements of the static conductivity of the rock. No microscopic derivation is given for the resulting composite dielectric permittivity in terms of any geometric model of the rock. Nevertheless, the model is shown to agree well with measurements of the permittivity and the conductivity of a variety of brine-saturated rocks over a broad range of frequencies.