Appearance of the nonlinearity from the nonlocality in diffusion through multiscale fractured porous media

We shall consider diffusion or single-phase flow in a multiscale porous medium which represents an infinite set of self-similar double-porosity media. At each scale, the medium consists of a highly permeable network of connected channels and low-permeable blocks. The characteristic scale of heterogeneity is ε at the highest level of hierarchy, wherein ε is a small parameter. The ratio between the channel and block permeability at each scale is ε ². The process analyzed is described using a diffusion equation with an oscillating multiscale diffusion parameter. The macroscale behavior is of interest. The transition to the macroscale is performed by means of the two-scale homogenization procedure. One step of averaging at each level of hierarchy leads to the appearance of the memory terms in the averaged equation. The successive averaging steps lead to progressive memory accumulation, so at each step of averaging, the macroscale model changes its type, and even the result of the second step is unknown a priori. The objective was to determine the macroscopic limit model for the infinite number of scales. By the method of induction, we obtained the macroscale model for an arbitrary number of scales and its limit for the infinite hierarchy. The limit model represents the system of two equations with memory terms. The kernel of the memory operator is the solution of a nonlinear integro-differential equation. Its solution is obtained through Laplace transform.