Effective conductivity of an isotropic heterogeneous medium of lognormal conductivity distribution

Igor Jankovic, Aldo Fiori, Gedeon Dagan

Research output: Contribution to journalArticlepeer-review


The study aims at deriving the effective conductivity Kef 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 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 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.

Original languageEnglish
Pages (from-to)40-56
Number of pages17
JournalMultiscale Modeling and Simulation
Issue number1
StatePublished - 2003


  • Effective conductivity
  • Heterogeneity
  • Porous media
  • Self-consistent model


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