Effective Conductivity of an Isotropic Heterogeneous Medium of Lognormal Conductivity Distribution

The study aims at deriving the effective conductivity Kef of a three-dimensional heterogeneous medium whose local conductivity K(x) is a stationary and isotropic random space function, of lognormal distribution and finite integral scale IY . We adopt a model of spherical inclusions of different K, of lognormal pdf, that we coin as a multi-indicator structure. The inclusions are inserted at random in an unbounded matrix of conductivity K0 within a sphere Ω, of radius R0, and they occupy a volume fraction n. Uniform flow of flux U∞ prevails at infinity. The effective conductivity is defined as the equivalent one of the sphere Ω, under the limits n→ 1 and R0/IY →∞. Following a qualitative argument, we derive an exact expression of Kef by computing it at the dilute limit n → 0. It turns out that Kef is given by the well-known self-consistent or effective medium argument. The above result is validated by accurate numerical simulations for σY ≤ 10 and for spheres of uniform radia. By using a face centered cubic lattice arrangement, the values of the volume fraction are in the interval 0 < n < 0.7. The simulations are carried out by the means of an analytic element procedure. To exchange space and ensemble averages, a large number N = 10000 of inclusions is used for most simulations. We surmise that the self-consistent model is an exact one for this type of medium, that is different from the multi-Gaussian one.