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Effect of memory accumulation in three-scale fractured-porous media

Abstract : We consider the flow of low compressible liquid described by the diffusion equation, which is also identical to the conductive heat transport, in a triple porosity medium that consists of two hierarchical connected networks of thin fractures and isolated low permeable blocks between the fractures. We reveal a unique parameterization of fracture thicknesses and permeabilities that ensures the contribution of all three medium subdomains into the macroscopic behavior. Such a parameterization corresponds to a non-selfsimilar medium. In such a medium the main flow occurs through the large-scale fractures, the small fractures/fissures play the role of fluid sources for large fractures, while the porous blocks play the role of fluid sources for small fractures. The delay in flow between different sub-domains leads to the appearance of memory in the macroscopic model described by the integro-differential operators. The double delay between three scales leads to the effect of memory accumulation, in such a way that the memory kernel of the integral operators is it-self the solution of an integro-differential cell problem. For thin fractures, the memory kernels have a specific structure that corresponds to Abel’s type. The macroscopic model is obtained by means of the asymptotic two-scale homogenization applied sequentially twice. These two sequential steps are non-symmetrical. At the first scale one deals with the problem of flow in continuous network of thin fractures/fissures surrounded by blocks, but only the boundary layer in each block is perturbed. At the second step we homogenize the flow in the medium that consists of large connected fractures and averaged blocks. Each averaged block contains a lot of small fractures and small porous blocks. At this step, in contrast, the appearance of a boundary layer inside the averaged blocks is prohibited. This model is used to calculate flow around a producing well in an oil reservoir, in comparison with the double porosity medium. The qualitative difference is revealed.
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Submitted on : Friday, December 16, 2016 - 3:24:54 PM
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Mojdeh Rasoulzadeh, Mikhail Panfilov, Fikri Kuchuk. Effect of memory accumulation in three-scale fractured-porous media. International Journal of Heat and Mass Transfer, Elsevier, 2014, 76, pp.171 - 183. ⟨10.1016/j.ijheatmasstransfer.2014.04.029⟩. ⟨hal-01418283⟩



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