Brahim Amaziane - Academia.edu (original) (raw)

Papers by Brahim Amaziane

Nonlinear Analysis-real World Applications, Aug 1, 2016

Abstract The paper is devoted to the homogenization of immiscible compressible two-phase two-comp... more Abstract The paper is devoted to the homogenization of immiscible compressible two-phase two-component flow in heterogeneous porous media. We consider liquid and gas phases, two-component (water and hydrogen) flow in a porous reservoir with periodic microstructure, modeling the hydrogen migration through engineered and geological barriers for a deep repository for radioactive waste. Phase exchange, capillary effects included by the Darcy–Muskat law and Fickian diffusion are taken into account. The hydrogen in the gas phase is supposed compressible and could be dissolved into the water obeying the Henry law. The flow is then described by the conservation of the mass for each component. The microscopic model is written in terms of the phase formulation, i.e. the liquid saturation phase and the gas pressure phase are primary unknowns. This formulation leads to a coupled system consisting of a nonlinear parabolic equation for the gas pressure and a nonlinear degenerate parabolic diffusion–convection equation for the liquid saturation, subject to appropriate boundary and initial conditions. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Under some realistic assumptions on the data, 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 the two-scale convergence.

Mathematical Methods in The Applied Sciences, May 8, 2015

This paper presents a study of immiscible compressible two-phase, such as water and gas, flow thr... more This paper presents a study of immiscible compressible two-phase, such as water and gas, flow through double porosity media. The microscopic model consists of the usual equations derived from the mass conservation laws of both fluids, along with the standard Darcy-Muskat law relating the velocities to the pressure gradients and gravitational effects. The problem is written in terms of the phase formulation, that is, where the phase pressures and the phase saturations are primary unknowns. The fractured medium consists of periodically repeating homogeneous blocks and fractures, where the absolute permeability of the medium becomes discontinuous. Consequently, the model involves highly oscillatory characteristics. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. We obtain the convergence of the solutions, and a macroscopic model of the problem is constructed using the notion of two-scale convergence combined with the dilatation technique.

Nonlinearity, Nov 14, 2006

The goal of the paper is to study the asymptotic behaviour of solutions to a high contrast quasil... more The goal of the paper is to study the asymptotic behaviour of solutions to a high contrast quasilinear equation of the form −div (|∇u ε | p−2 ∇u ε) + G ε (x)|u ε | p−2 u ε = f (x) in , where ⊂ R n with n 2, 1 < p n, and the coefficient G ε (x) is assumed to blow up as ε → 0 on a set of N ε isolated inclusions of asymptotically small measure. Here N ε −→ +∞ as ε → 0. It is shown that the asymptotic behaviour, as ε → 0, of the solution u ε 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 ε (x) in the original problem. This result is then illustrated with a periodic example and a nonperiodic one.

Discrete and Continuous Dynamical Systems-series B, 2018

This paper presents a study of immiscible incompressible twophase flow through fractured porous m... more This paper presents a study of immiscible incompressible twophase flow through fractured porous media. The results obtained earlier in the pioneer work by A. Bourgeat, S. Luckhaus, A. Mikelić (1996) and L. M. Yeh (2006) are revisited. The main goal is to incorporate some of the most recent improvements in the convergence of the solutions in the homogenization of such models. The microscopic model consists of the usual equations derived from the mass conservation of both fluids along with the Darcy-Muskat law. The problem is written in terms of the phase formulation, i.e. the saturation of one phase and the pressure of the second phase are primary unknowns. We will consider a domain made up of several zones with different characteristics: porosity, absolute permeability, relative permeabilities and capillary pressure curves. The fractured medium consists of periodically repeating homogeneous blocks and fractures, the permeability being highly discontinuous. Over the matrix domain, the permeability is scaled by ε θ , where ε is the size of a typical porous block and θ > 0 is a parameter. The model involves highly oscillatory characteristics and internal nonlinear interface conditions. Under some realistic assumptions on the data, the convergence of the solutions, and the macroscopic models corresponding to various range of contrast are constructed using the two-scale convergence method combined with the dilation technique. The results improve upon previously derived effective models to highly heterogeneous porous media with discontinuous capillary pressures.

Multiscale Modeling & Simulation, 2010

This paper is devoted to the homogenization of a coupled system of diffusionconvection equations ... more This paper is devoted to the homogenization of a coupled system of diffusionconvection 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.

Comptes Rendus Mecanique, Mar 1, 2009

We study the asymptotic behaviour, as ε → 0, of u ε solutions to a nonlinear elliptic equation wi... more We study the asymptotic behaviour, as ε → 0, of u ε solutions to a nonlinear elliptic equation with nonstandard growth condition in domains containing a grid-type microstructure F ε that is concentrated in an arbitrary small neighborhood of a given hypersurface Γ. We assume that u ε = A ε on ∂F ε , where A ε is an unknown constant. The macroscopic equation and a nonlocal transmission condition on Γ are obtained by the variational homogenization technique in the framework of Sobolev spaces with variables exponents. This result is then illustrated by a periodic example.

Networks and Heterogeneous Media, 2017

The paper deals with a degenerate model of immiscible compressible two-phase flow in heterogeneou... more The paper deals with a degenerate model of immiscible compressible two-phase flow in heterogeneous porous media. We consider liquid and gas phases (water and hydrogen) flow in a porous reservoir, modeling the hydrogen migration through engineered and geological barriers for a deep repository for radioactive waste. The gas phase is supposed compressible and obeying the ideal gas law. The flow is then described by the conservation of the mass for each phase. The model is written in terms of the phase formulation, i.e. the liquid saturation phase and the gas pressure phase are primary unknowns. This formulation leads to a coupled system consisting of a nonlinear degenerate parabolic equation for the gas pressure and a nonlinear degenerate parabolic diffusion-convection equation for the liquid saturation, subject to appropriate boundary and initial conditions. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. The aim of this paper is to extend our previous results to the case of an ideal gas. In this case a new degeneracy appears in the pressure equation. With the help of an appropriate regularization we show the existence of a weak solution to the studied system. We also consider the corresponding nonlinear homogenization problem and provide a rigorous mathematical derivation of the upscaled model by means of the two-scale convergence.

Applied Mathematics Letters, Nov 1, 2006

We study the asymptotic behaviour of the solution of a reaction-diffusion equation in a ε-periodi... more We study the asymptotic behaviour of the solution of a reaction-diffusion equation in a ε-periodic partially fractured medium with Robin interface conditions. We consider a model where the solution has a jump of order ε −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.

Advances in Mathematical Physics, 2016

We consider two-phase flow in a heterogeneous porous medium with highly permeable fractures and l... more We consider two-phase flow in a heterogeneous porous medium with highly permeable fractures and low permeable periodic blocks. The flow in the blocks is assumed to be in local capillary disequilibrium and described by Barenblatt's relaxation relationships for the relative permeability and capillary pressure. It is shown that the homogenization of such equations leads to a new macroscopic model that includes two kinds of long-memory effects: the mass transfer between the blocks and fractures and the memory caused by the microscopic Barenblatt disequilibrium. We have obtained a general relationship for the double nonequilibrium capillary pressure which represents great interest for applications. Due to the nonlinear coupling and the nonlocality in time, the macroscopic model remains incompletely homogenized in general case. The completely homogenized model was obtained for two different regimes. The first case corresponds to a linearized flow in the blocks. In the second case, we assume a low contrast in the block-fracture permeability. Numerical results for the two-dimensional problem are presented for two test cases to demonstrate the effectiveness of the methodology.

Applicable Analysis, Jul 1, 2013

A model for immiscible compressible two-phase flow in heterogeneous porous media is considered. S... more A model for immiscible compressible two-phase flow in heterogeneous porous media is considered. Such models appear in gas migration through engineered and geological barriers for a deep repository for radioactive waste. The main feature of this model is the introduction of a new global pressure and it is fully equivalent to the original equations. The resulting equations are written in a fractional flow formulation and lead to a coupled degenerate system which consists of a nonlinear parabolic (the global pressure) equation and a nonlinear diffusion–convection one (the saturation) equation with rapidly oscillating porosity function and absolute permeability tensor. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Under some realistic assumptions on the data, we obtain a nonlinear homogenized problem with effective coefficients which are computed via a cell problem and give a rigorous mathematical derivation of the upscaled model by means of two-scale convergence.

European Journal of Applied Mathematics, Jun 30, 2005

We consider the problem of modelling the flow of a slightly compressible fluid in a periodic frac... more 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. 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 (εδ) 2 , where ε is the size of a typical porous block, δ representing the relative size of the fracture. We consider a nonstandard interface condition: a jump of the pressure is taken into account and proportional to the flux by mean of a function (εδ) −γ where γ is a parameter. Using two-scale convergence, we get homogenized models which govern the global behaviour of the flow as ε and δ tend to zero. The resulting homogenized equation is of dual-porosity type model that contains a term representing memory effects for γ ≤ 1, and it is a simpleporosity model with effective coefficients for γ > 1.

Physical Review E, Jan 9, 2012

For two-phase flow in porous media, the natural medium heterogeneity necessarily gives rise to ca... more For two-phase flow in porous media, the natural medium heterogeneity necessarily gives rise to capillary nonequilibrium effects. The relaxation to the equilibrium is a slow process which should be introduced in macroscopic flow models. Many nonequilibrium models are based on a phenomenological approach. At the same time there exists a rigorous mathematical way to develop the nonequilibrium equations. Its formalism, developed by Bourgeat and Panfilov [Computational Geosciences 2, 191 (1998)], is based on the homogenization of the microscale flow equations over medium heterogeneities. In contrast with the mentioned paper, in which the case of a sufficiently fast relaxation was analyzed, we consider the case of long relaxation, which leads to the appearance of long-term memory on the macroscale. Due to coupling between the nonlinearity and nonlocality in time, the macroscopic model remains, however, incompletely homogenized, in the general case. At the same time, frequently only the relationship for the nonequilibrium capillary pressure is of interest for applications. In the present paper, we obtain such an exact relationship in two different independent forms for the case of long-term memory. This relationship is more general than that obtained by Bourgeat and Panfilov. In addition, we prove the comparison theorem which determines the upper and lower bounds for the macroscopic model. These bounds represent linear flow models, which are completely homogenized. The results obtained are illustrated by numerical simulations.

HAL (Le Centre pour la Communication Scientifique Directe), 2009

arXiv (Cornell University), Mar 3, 2005

Homogenization method is used to analyse the equivalent behavior of transient flow of a passive s... more Homogenization method is used to analyse the equivalent behavior of transient flow of a passive solute through highly heterogeneous porous media. The flow is governed by a coupled system which includes an elliptic equation and a linear convection-diffusion concentration equation with a diffusion term small with respect to the convection, i.e. a high Peclet number is considered. Asymptotic expansions lead to the definition of a macroscale transport model. Numerical computations to obtain the effective hydraulic conductivity and the macrodiffusivity tensor have been carried out via finite volume methods. Numerical experiments related to the simulations of solute transport have been performed comparing the heterogeneous medium to the corresponding effective medium. The results of the simulations are compared in terms of spatial moments, L2 errors and concentration contours. Results obtained from the simulations using the model obtained by the homogenization method show good agreement with the heterogeneous simulations.

arXiv (Cornell University), May 19, 2016

This paper presents a study of immiscible incompressible two-phase flow through fractured porous ... more This paper presents a study of immiscible incompressible two-phase flow through fractured porous media. The results obtained earlier in the pioneer work by A. Bourgeat, S. Luckhaus, A. Mikelić (1996) and L. M. Yeh (2006) are revisited. The main goal is to incorporate some of the most recent improvements in the convergence of the solutions in the homogenization of such models. The microscopic model consists of the usual equations derived from the mass conservation of both fluids along with the Darcy-Muskat law. The problem is written in terms of the phase formulation, i.e. the saturation of one phase and the pressure of the second phase are primary unknowns. We will consider a domain made up of several zones with different characteristics: porosity, absolute permeability, relative permeabilities and capillary pressure curves. The fractured medium consists of periodically repeating homogeneous blocks and fractures, the permeability being highly discontinuous. Over the matrix domain, the permeability is scaled by ε θ , where ε is the size of a typical porous block and θ > 0 is a parameter. The model involves highly oscillatory characteristics and internal nonlinear interface conditions. Under some realistic assumptions on the data, the convergence of the solutions, and the macroscopic models corresponding to various range of contrast are constructed using the two-scale convergence method combined with the dilation technique. The results improve upon previously derived effective models to highly heterogeneous porous media with discontinuous capillary pressures.

Mathematical Methods in The Applied Sciences, 2005

We study the asymptotic behaviour of the solution of a stationary quasilinear elliptic problem po... more We study the asymptotic behaviour of the solution of a stationary quasilinear elliptic problem posed in a domain Ω(ε) of asymptotically degenerating measure, i.e. meas Ω(ε) → 0 as ε → 0, where ε is the parameter that characterizes the scale of the microstructure. We obtain the convergence of the solution and the homogenized model of the problem is constructed using the notion of convergence in domains of degenerating measure. Proofs are given using the method of local characteristics of the medium Ω(ε) associated with our problem in a variational form. Copyright © 2005 John Wiley & Sons, Ltd.

A mathematically rigorous method of homogenization is presented and used to analyze the equivalen... more A mathematically rigorous method of homogenization is presented and used to analyze the equivalent behavior of transient flow of two incompressible fluids through heterogeneous media. Asymptotic qpansions and H-convergence lead to the definition of a global or effective model of an equivalent homogeneous reservoir. Numerical computations to obtain the homogenized coefficients of the entire reservoir have been carried out via a finite element method. Numerical experiments involving the simulation of incompressible two-phase flow have been performed for each heterogeneous medium and for the homogenized medium as well as for other averaging methods. The results of the simulations are compared in terms of the transient saturation contours, production curves, and pressure distributions. Results obtained from the simulations with the homogenization method presented show good agreement with the heterogeneous simulations.

HAL (Le Centre pour la Communication Scientifique Directe), 2010