Grain shape effects on dielectric and electrical properties of rocks

Recently there has been a considerable interest in the effect of anisotropy in the grain shape in the electrical and dielectrical properties of rocks and other inhomogeneous media (Sen 1981a, b; Sen et al, 1981; Mendelson and Cohen, 1982; and Kenyon, 1983). In this note I point out that equation (34) of Mendelson and Cohen (MC) is incorrect. The dc limit of MC equation (34) for the conductivity of rock σ, in terms of porosity φ and water conductivity σw, gives (1)Inφ=m̂−1Inσσw, or (2)σ=σwφm^, where (3)m^−1=〈1+2(1−3L)2(5−3L)(1+3L)−3L(1−L)1+3L〉.L is the depolarization factor along the principal axis of spheroidal grain and 〈 〉 denotes an average over the distribution in L. This value of m^ is in disagreement with the correct value of m in equation (28) of MC [equation (6) below]. [When the sign mistakes in equations MC (33)–(34) are corrected, m^=〈3(1−L2)/(5−3L)〉−1. This agrees with equation (6) below for the case when L has a single value and averaging is redundant.] This inconsistency arises from an incorrect replacement of the inverse of an average in MC equation (33) by an average of inverses. The corrected form of MC equation (33) is (4)−dφ3φ=dεε(εm−ε)..1〈(1+3L)εm+(5−3L)ε[Lεm+(1−L)ε][(1−L)εm+(1+L)ε]〉, where ε and εm are the dielectric constants of the mixture and of the matrix, respectively. The dielectric constant ε = ε′ + σ/iωε0 is complex, εm is real, ε0 is the permittivity of vacuum, σ the conductivity, ω the angular frequency. The last factor in the right-hand side of the equation was replaced incorrectly by the average of the inverse, which is incorrect in general. Note that in the dc limit equation (4) above gives (5)mdφφ=dσ(0)σ(0), and, by integration, σ(0)=σw(0)φm, where σw(0) is the dc conductivity of water, σ(0) is the dc conductivity of formation, and (6)m=〈5−3L3(1−L2)〉.