Effective conductivity of an isotropic heterogeneous medium of lognormal conductivity distribution

The study aims at deriving the effective conductivity K_ef of a three-dimensional heterogeneous medium whose local conductivity K(x) is a stationary and isotropicrandom 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 K_ef by computing it at the dilute limit n → 0. It turns out that K_ef is given by the well-known self-consistent or effective medium argument. The above result is validated by accurate numerical simulations for σ² _Y ≤ 10 and for spheres of uniform radii. By using a faced-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.