# Homogenization and Numerical Modeling of Flow and Transport in Heterogeneous Porous Media. Applications to Energy and Environmental Studies.

Abstract : Single and multiphase flow and transport processes in heterogeneous porous media are involved in a wide variety of engineering applications, such as oil recovery, radioactive nuclear waste and groundwater remediation. In this work, we study flow and transport through heterogeneous porous media using homogenization methods and numerical modeling. We are concerned with approximating effective coefficients for such problems using conforming, mixed finite elements and finite volume methods and their implementation. Numerical methods have been developed for the simulation of miscible or immiscible two-phase flow in heterogeneous porous media. Three topics are investigated. The first one deals with the homogenization of single and multiphase flow in porous media. The convergence results are obtained by mean of the two-scale convergence and/or L-convergence. Each homogenization method leads to the definition of a global or effective model of a homogeneous medium defined by the computed effective coefficients. Homogenization methods allow the determination of these effective coefficients from knowledge of the geometrical structure of a basic cell and its heterogeneities by solving appropriate local problems. The technique is based on numeric. We assume that data given on a fine grid fully represents the important physical scales and that a practical computational grid must be somewhat coarser. In the homogenization methods described and implemented in this work we use conforming, mixed finite elements and FV methods to compute approximate solutions of the local problems used in the calculation of the effective coefficients. We have developed a user friendly computational tool, Homogenizer++, for the computation of effective parameters. The platform Homogenizer++ is based on Object Oriented Programming approach. The second topic concerns numerical methods for miscible or immiscible two-phase flow in porous media. The mathematical models used to describe these fluid flow processes are coupled system of partial differential equations. The miscible model under consideration includes an elliptic pressure-velocity equation coupled to a linear convection-diffusion-reaction concentration equation. While the immiscible model is described by an elliptic pressure-velocity equation coupled to a nonlinear, degenerate, convection-diffusion saturation equation. For each model, it is shown that the scheme satisfies a discrete maximum principle. We derive $L^\infty$ and BV estimates under an appropriate CFL condition. Then we prove the convergence of the approximate solutions to a weak solution of the coupled system. Several numerical simulations prove the efficiency of these schemes. A posteriori error estimation for a finite volume scheme on anisotropic meshes for Darcy equation has been derived. The scheme is a analyzed theoretically and numerically. Numerical simulations underline the applicability of the scheme in adaptive computations. Finally, we present some numerical methods applied to groundwater flow problems. These include a meshless method, genetic algorithms, and stochastic boundary element algorithms. A meshless method based on radial basis functions is coupled with genetic algorithms for parameter identification to a diffusion equation with some specific boundary conditions describing the groundwater fluctuation in a leaky confined aquifer system near open tidal water. Then a stochastic boundary element method coupled to a genetic algorithm is employed for the optimization of groundwater pumping in coastal aquifers under the threat of saltwater intrusion. Numerical examples of deterministic and stochastic problems are provided to demonstrate the feasibility of the proposed schemes.
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Habilitation à diriger des recherches
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https://tel.archives-ouvertes.fr/tel-00010339
Contributor : Brahim Amaziane <>
Submitted on : Friday, September 30, 2005 - 9:16:08 AM
Last modification on : Thursday, March 5, 2020 - 7:20:40 PM
Long-term archiving on: : Friday, April 2, 2010 - 10:48:40 PM

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• HAL Id : tel-00010339, version 1

### Citation

Brahim Amaziane. Homogenization and Numerical Modeling of Flow and Transport in Heterogeneous Porous Media. Applications to Energy and Environmental Studies.. Mathematics [math]. Université de Pau et des Pays de l'Adour, 2005. ⟨tel-00010339⟩

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