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 isotropic random space function of lognormal distribution and finite integral scale I-Y. 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 K-0 within a sphere Omega, of radius R-0, and they occupy a volume fraction n. Uniform flow of flux U-infinity prevails at infinity. The effective conductivity is defined as the equivalent one of the sphere Omega, under the limits n --> 1 and R-0/I-Y --> infinity. 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 sigma(Y)(2) less than or equal to 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.