Leonid Pankratov

Homogenization of Immiscible Compressible Two-Phase Flow in Porous Media: Application to Gas Migration in a Nuclear Waste Repository

This paper is devoted to the homogenization of a coupled system of diffusion-convection equations in a domain with periodic microstructure, modeling the flow and transport of immiscible compressible, such as water-gas, fluids through porous media. The problem is formulated in terms of a nonlinear parabolic equation for the nonwetting phase pressure and a nonlinear degenerate parabolic diffusion-convection equation for the wetting saturation phase with rapidly oscillating porosity function and absolute permeability tensor. We obtain a nonlinear homogenized problem with effective coefficients which are computed via a cell problem. We rigorously justify this homogenization process for the problem by using two-scale convergence. In order to pass to the limit in nonlinear terms, we also obtain compactness results which are nontrivial due to the degeneracy of the system.

Nonlinear “double porosity” type model

We consider a variational problem inf u∈H 1(�) � {a e |∇u e | m + g|u e | m − mf e u e } dx in a bounded domain � = F (e) ∪ M (e) with a microstructure F (e) which is not in general periodic; a e = a e (x) is of order 1 in F (e) and sup x∈M(e) a e (x) → 0a se → 0. A homogenized model is constructed. To cite this article: L. Pankratov, A. Piatnitski, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 435-440.  2002 Academie des sciences/Editions scientifiques et medicales Elsevier SAS

Homogenization of a degenerate triple porosity model with thin fissures

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

Homogenized model of reaction-diffusion in a porous medium

(\varepsilon \delta)^2

Homogenization of attractors for semilinear parabolic equations in domains with spherical traps

, where

Γ-convergence of nonlinear functionals in thin reticulated structures

\varepsilon

Homogenization of immiscible compressible two-phase flow in highly heterogeneous porous media with discontinuous capillary pressures

is the size of a typical porous block, with

Homogenization of one phase flow in a highly heterogeneous porous medium including a thin layer

\delta

Homogenization of variational functionals with nonstandard growth in perforated domains

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

Homogenization of Parabolic Equations in Domains of Degenerating Measure

(\varepsilon\delta)^{-\gamma}