Dielectric and acoustic response of rocks

Abstract

A self-similar model is developed to explain the conductivity, dielectric, and acoustic response of rocks. The model explicitly takes into account the correlations that lead to a zero percolation threshold, and gives the d.c. conductivity of rock (r(0) - σ_w(0)_θ ^m, with σ_w(0) the d.c. conductivity of the brine, θ the porosity. The exponent m depends on the grain shape. Using well-known results, it is shown that a small concentration, η, of plate-like grains of aspect ratio 8,8 < < 1, can give rise to a large (≈10⁴) low-frequency dielectric constant ε_s, and explain its salinity and frequency dependences. The case 8<-q, η-0, corresponds to the well-known Maxwell-Wagner effect where a divergent e8 is accompanied with a conductivity threshold σ(0)-0. The case δ-0,,η-0,δ>,η gives rise to a new result, previously overlooked, with divergent es and non-zero σ(0). The velocity v of Biot second wave at high frequency and the fourth sound wave in ⁴He saturated plug are given by v - v_F/n, VF being the velocity in the free fluid, n²=θσ_w,(0)/σ(0) (exact) =θ^1-m (self-similar model).