A self-similar model for sedimentary rocks with application to the dielectric constant of fused glass beads

Abstract

We develop a theory for dielectric response of water-saturated rocks based on a realistic model of the pore space. The absence of a percolation threshold manifest in Archie’s law, porecasts, electron-micrographs, and general theories of formation of detrital sedimentary rocks indicates that the pore spaces within such rocks remain interconnected to very low values of the porosity ϕ. In the simplest geometric model for which the conducting paths remain interconnected, each grain is envisioned to be coated with water. The dielectric constant of the assembly of water-coated grains is obtained by a self-consistent effective medium theory. In the dc limit, this gives Maxwell's relation for conductivity σ of the rock σ = 2σωφ/(3 − φ), where σω is the conductivity of water. In order to include the local environmental effects around a grain, a self-similar model is generated by envisioning that each rock grain itself is coated with a skin made of other coated spheres; the coating at each level consists of other coated spheres. The self-consistent complex dielectric constant ɛ* is given in this model in terms of that of water ɛω* and of rock ɛm*, by [(ɛm*−ɛ*)/(ɛm*−ɛω*)][ɛω*/ɛ*]1/3=φ for spherical particles. This gives, in the dc limit, σ = σωφ3/2. For nonspherical particles, the exponent m in Archie's law σ = σωφm is greater than 3/2 for the plate-like grains or cylinders with axis perpendicular to the external field and smaller than 3/2 for plates or cylindrical particles with axis parallel to the external field. Artificial rocks with a wide range of porosities were made from glass beads. We present data on the glass bead rocks for dc conductivity and the dielectric constant at 1.1 GHz. The data follow the conductivity and the dielectric responses given by the self-similar model. The present theory fails to explain the salinity dependence of ɛ* at lower frequencies.