Electric conductivities of a cracked solid

Calculations on the basis of the self-consistent approximation are used to study the effects of randomly distributed elliptical cracks and of non-randomly distributed circular cracks, either dry or saturated by a highly conductive material phase, on the electric conductivities of a cracked body. Analytic and numeric results are given for two special non-random distributions. In the first, the cracks are assumed randomly distributed in planes parallel to a given plane. In the second, the crack normals are randomly distributed in parallel planes. The results of the theoretical calculations indicate that the magnitudes of the crack induced variations of the dry cracked rock depend upon a crack density parameter ? rather than upon the crack porosity. Here, ? is defined as $$\varepsilon = \frac{{2N}}{\pi }< \frac{{A^2 }}{P} > $$ whereN is the average number of cracks per unit volume, andA andP are the crack area and perimeter respectively. (For circular cracks of radiusa, ?=N〈a³〉.) Although a straightforward relationship does connect ? with the porosity, it may be more meaningful for laboratory experiments to concentrate upon measuring crack-induced variations as functions of crack density rather than of porosity. For saturated cracked rocks, the results of the calculations indicate that, in addition to ?, variations in conductivity depend also upon a saturation parameter Ω, which relates crack aspect ratio α to matrix and fluid conductivities σ and σ_F $$\Omega = \frac{{{\sigma \mathord{\left/ {\vphantom {\sigma {\sigma _F }}} \right. \kern-\nulldelimiterspace} {\sigma _F }}}}{\alpha }.$$