By G. Ghantmakher
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Extra info for Green's Functions and Condensed Matter
By contrast, the periodic homogenization technique provides a rational means to derive Darcy’s law together with the macroscopic permeability tensor as a function of the geometry of the pore space. 9), and the solid matrix has been assumed to be rigid. The focus of this and the next section is to generalize the micromechanics derivation beyond these particular situations, while remaining within the framework of the periodic homogenization technique;4 that is, the porous 4 The theory presented below is inspired by the contributions of Ene and Sanchez-Palencia  and Auriault ,.
4, an upper bound will be developed. 31). 47) For a given v′ ∈ H, let δv = v′ − v. 31). 29). Ordered Relations of Permeability It is instructive to investigate how the permeability is affected by the magnitude of the porosity. That is, two porous media that differ in porosity have different permeabilities. The question is: in which sense and order? It is intuitively expected that the permeability of a porous medium with a smaller porosity is smaller than one with a greater porosity. This result is established in the following, under some conditions.
91), the pressure and the advective velocity have to be scaled. More precisely, considering an advective ﬂow driven by a pressure gradient which is characterized by the pressure scale P and the macroscopic length L, the advective velocity is scaled so as to balance the pressure gradient P/L. 12), and we assume that the characteristic length of the spatial variations of the velocity ﬁeld is the pore size d = a . 96), all p ∗( j) (resp. v∗( j) ) are expected to be of the same magnitude. 95). 97) Generalization: Periodic Homogenization Based on Double-Scale Expansion 47 In other words, p ∗(0) depends only on the macroscopic variable X and does not ﬂuctuate within the cell.