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Méthode multi-échelle pour la simulation d'écoulements miscibles en milieux poreux

Abstract : This work deals with the study and the implementation of a multiscale finite element method for the simulation of miscible flows in porous media. The definition of the multiscale basis functions is based on the idea introduced by F. Ouaki. The novelty of this work lies in the combination of this multiscale approach with Discontinuous Galerkin methods (DG) so that these new finite elements can be used on nonconforming meshes composed of cells with various shapes. We first recall the basics of DG methods and their application to the discretisation of a convection-diffusion equation that arises in the flow problem considered in this work. After establishing the existence and uniqueness of a solution to the continuous problem, we prove again the convergence of DG methods towards this solution by establishing an a priori error estimate. We then introduce the nonconforming multiscale finite element method and explain how it can be implemented for this convection-diffusion problem. Assuming that the boundary conditions and the parameters of the problem are periodic, we prove a new a priori error estimate for this method. In a second part, we consider the whole flow problem where the equation, studied in the first part of that work, is coupled and simultaneously solved with Darcy equation. We introduce various synthetic test cases which are close to flow problems encountered in geosciences and compare the solutions obtained with both DG methods, namely the classical method based on the use of a single mesh and the one studied here. For the resolution of the cell problems, we propose new boundary conditions which, compared to classical linear conditions, allow us to better reproduce the variations of the solutions on the interfaces of the coarse mesh. The results of these tests show that the multiscale method enables us to calculate solutions which are close to the ones obtained withDG methods on a single mesh and also enables us to reduce significantly the size of the linear system that has to be solved at each time step.
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Aboubacar Konaté. Méthode multi-échelle pour la simulation d'écoulements miscibles en milieux poreux. Physique mathématique [math-ph]. Université Pierre et Marie Curie - Paris VI, 2017. Français. ⟨NNT : 2017PA066006⟩. ⟨tel-01558994⟩

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