Homogenization of a degenerate triple porosity model with thin fissures

Abstract

We consider the problem of modelling the flow of a slightly compressible fluid in a periodic fractured medium assuming that the fissures are thin with respect to the block size. As a starting point we used a formulation applied to a system comprising a fractured porous medium made of blocks and fractures separated by a thin layer which is considered as an interface. The inter-relationship between these three characteristics comprise the triple porosity model. The microscopic model consists of the usual equation describing Darcy flow with the permeability being highly discontinuous. Over the matrix domain, the permeability is scaled by $(\varepsilon \delta)^2$, where $\varepsilon$ is the size of a typical porous block, with $\delta$ representing the relative size of the fracture. We then consider a model with Robin type transmission conditions: a jump of the density across the interface block-fracture is taken into account and proportional to the flux by the mean of a function $(\varepsilon\delta)^{-\gamma}$, where $\gamma$ is a parameter. Using two-scale convergence, we get homogenized models which govern the global behaviour of the flow as $\varepsilon$ and $\delta$ tend to zero. The resulting homogenized problem is a dual-porosity type model that contains a term representing memory effects for $\gamma\le 1$, and it is a single porosity model with effective coefficients for $\gamma >1$.