## Abstract

The paper deals with the effective conductivity tensor K_{ef} of anisotropic randommedia subject to mean uniform flux. The hydraulic conductivity K field is modeled as a collection of spheroidal disjoint inclusions of different, isotropic and independent Y = ln K; the latter is a random variable with given distribution of variance σ^{2}_{y}. Inclusions are embedded in homogeneous background of anisotropic conductivity K _{0}. The Kfield is anisotropic, characterized by the anisotropy ratio f , ratio of the vertical and horizontal integral scales of K.We derive K _{ef} by accurate numerical simulations; the numericalmodel for anisotropic media is presented here for the first time, and it generalizes a previously developed model for isotropic formations [I. Jankovic, A. Fiori, and G. Dagan, Multiscale Model. Simul., 1 (2003), pp. 40-56]. The numerical model is capable of solving complex threedimensional flow problems with high accuracy for any configuration of the spheroidal inclusions and arbitrary K distribution. The numerically derived K_{ef} for a normal Y is compared with its prediction by (i) the self-consistent solution K_{sc}, (ii) the first-order approximation in σ^{2}_{y} , and (iii) the exponential conjecture [L. J. Gelhar and C. L. AxnessWater. Resour. Res., 19 (1983), pp. 161-180]. It is found that the self-consistent solution K _{sc} is very accurate for a broad range of the values of the parameters σ^{2}_{y} ,f and for the densest inclusions packing. In contrast, the first-order solution strongly deviates from K_{ef} for large σ^{2}_{y} , as expected, and the exponential conjecture is generally unable to correctly represent the effective conductivity. The numerical solution for the potential is expressed as an infinite series of spheroidal harmonics, attached to the interior and exterior of each inclusion, which accounts for the nonlinear interaction between neighboring inclusions.

Original language | English |
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Pages (from-to) | 933-954 |

Number of pages | 22 |

Journal | Multiscale Modeling and Simulation |

Volume | 9 |

Issue number | 3 |

DOIs | |

State | Published - 2011 |

## Keywords

- Anisotropic formations
- Effective conductivity
- Effective medium approximation
- Self-consistent approximation