Fluid filtration through the media with random sets of crack-like inclusions

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Fluid filtration through homogeneous permeable media containing random sets of thin crack-like inclusions is considered. One of the characteristic sizes of such inclusions is much smaller than the others, and filtration coefficients of the inclusion material are much larger than the coefficients of the host medium. First, a homogeneous medium with an isolated crack-like inclusion subjected to a constant external fluid pressure gradient is considered (the one-particle problem). Using the method of matching of asymptotic expansions, the problem is reduced to a 2D-integral equation for the fluid flux along the middle surface of the inclusion. Analytical solutions of this equation for thin inclusions with elliptical middle surfaces are presented. Then, the solution of the one-particle problem is used in the framework of the self-consistent effective field method for calculation of effective filtration coefficients of the medium with crack-like inclusions. The method takes into account statistical characteristics of random sets of the inclusions. Detailed equations for the tensors of effective filtration coefficients of the heterogeneous media are presented for inclusions of the same orientations and for a homogeneous distribution of the inclusion over the orientations. Predictions of the method are compared with numerical solutions of the homogenization problem presented in the literature.