L. Pankratov

HOMOGENIZATION OF A SINGLE PHASE FLOW THROUGH A POROUS MEDIUM IN A THIN LAYER

The paper deals with homogenization of stationary and non-stationary high contrast periodic double porosity type problem stated in a porous medium containing a 2D or 3D thin layer. We consider two different types of high contrast medium. The medium of the first type is characterized by the asymptotically vanishing volume fraction of fractures (highly permeable part). The medium of the second type has uniformly positive volume fraction of fracture part. In both cases we construct the homogenized models and prove the convergence results. The techniques used in this work are based on a special version of the two-scale convergence method adapted to thin structures. The resulting homogenized problems are dual-porosity type models that contain terms representing memory effects.

On the homogenization of some double-porosity models with periodic thin structures

Models describing global behaviour of incompressible flow in fractured media are discussed. A fractured medium is regarded as a porous medium consisting of two superimposed continua, a connected fracture system and a system of disjoint matrix blocks. We derive global behaviour of fractured media versus different parameters such as the fracture thickness, the size of blocks and the ratio of the block permeability and the permeability of fissures. The homogenization results are obtained by means of the convergence in domains of asymptotically vanishing measure.

Upscaling of an immiscible non-equilibrium two-phase flow in double porosity media

We study an immiscible incompressible two-phase, such as water–oil flow through a -periodic double porosity media. The mesoscopic model consists of equations derived from the mass conservation laws of both fluids, along with a generalized Darcy law in the framework of Kondaurov’s non-equilibrium flow model. The mobility functions as well as the capillary pressure function for each component of the porous medium are the functions of the saturation and an additional non-equilibrium parameter, which satisfies a kinetic equation coming from the definition of the Helmholtz free energy. The fractured medium consists of periodically repeating homogeneous blocks and fractures, where the permeability ratio of matrix blocks to fracture planes is of order . Using the method of two-scale asymptotic expansions, we derive the macroscopic model of the flow which is written in terms of the homogenized phase pressures, saturation, and the non-equilibrium parameter. For small relaxation times, we compare our model with the global models obtained earlier by H. Salimi and J. Bruining. We show the novelty of our macroscopic double porosity flow model.

A fully homogenized model for incompressible two-phase flow in double porosity media

In this paper, we discuss a model describing the global behavior of the two-phase incompressible flow in fractured porous media. The fractured medium is regarded as a porous medium consisting of two superimposed continua, a connected fracture system, which is assumed to be thin of order , where being the relative fracture thickness, and an –periodic system of disjoint matrix blocks. We derive the global behavior of the fractured medium by passing to the limit as , taking into account that the permeability of the blocks is proportional to , while the permeability of the fractures is of order one and obtain the corresponding global –model, i.e. the homogenized model with the coefficients depending on the small parameter . In the –model, we linearize the cell problem in the matrix block and then by letting , we obtain the macroscopic model which does not depend on and , and is fully homogenized in the sense that all the coefficients are calculated in terms of given data and do not depend on the additional coupling or cell problems.

New non-equilibrium matrix imbibition equation for double porosity model

Abstract The paper deals with the global Kondaurov double porosity model describing a non-equilibrium two-phase immiscible flow in fractured-porous reservoirs when non-equilibrium phenomena occur in the matrix blocks, only. In a mathematically rigorous way, we show that the homogenized model can be represented by usual equations of two-phase incompressible immiscible flow, except for the addition of two source terms calculated by a solution to a local problem being a boundary value problem for a non-equilibrium imbibition equation given in terms of the real saturation and a non-equilibrium parameter.

Homogenization of a reaction-diffusion equation with Robin interface conditions

Abstract We study the asymptotic behaviour of the solution of a reaction–diffusion equation in a e -periodic partially fractured medium with Robin interface conditions. We consider a model where the solution has a jump of order e − 1 with respect to the flux which is continuous at the interface. The macroscopic model consists of two semi-linear parabolic equations with a linear exchange term.

Homogenization of Kondaurov’s non-equilibrium two-phase flow in double porosity media

ABSTRACT We consider a two-phase incompressible non-equilibrium flow in fractured porous media in the framework of Kondaurov’s model, wherein the mobilities and capillary pressure depend both on the real saturation and a non-equilibrium parameter satisfying a kinetic equation. The medium is made of two superimposed continua, a connected fracture system, which is assumed to be thin of order , where is the relative fracture thickness and an -periodic system of disjoint cubic matrix blocks. We derive the global behavior of the model by passing to the limit as , assuming that the block permeability is proportional to , while the fracture permeability is of order one, and obtain the global –model. In the -model we linearize the cell problem in the matrix block and letting , obtain a macroscopic non-equilibrium fully homogenized model, i.e. the model which does not depend on the additional coupling. The numerical tests show that for sufficiently small, the exact global -model can be replaced by the fully homogenized one without significant loss of accuracy.

Homogenization of quasilinear elliptic equations with non-uniformly bounded coefficients

The goal of the paper is to study the asymptotic behaviour of solutions to a high contrast quasilinear equation of the form −div (|∇u e | p−2 ∇u e ) + G e (x)|u e | p−2 u e = f( x) in �, where � ⊂ R n with n 2, 1 <p n, and the coefficient G e (x) is assumed to blow up as e → 0 on a set of Ne isolated inclusions of asymptotically small measure. Here Ne −→ +∞ as e → 0. It is shown that the asymptotic behaviour, as e → 0, of the solution u e is described in terms of a homogenized quasilinear equation of the form −div (|∇u| p−2 ∇u) + B(x)|u| p−2 u = f( x) in �, where the coefficient B(x) is calculated as a local energy characteristic of the microstructure associated with the potential G e (x) in the original problem. This