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Méthodes Numériques pour la Simulation des Ecoulements Miscibles en Milieux Poreux Hétérogènes

Abstract : In this thesis, we are interested in numerical methods for a model of incompressible and miscible flows having application in hydrogeology and oil engineering. We study and analyze a numerical scheme combining a mixed finite element method (MFE) and a finite volumes method (FV) to discretize the coupled system between an elliptic equation (pressure-velocity) and a convection-diffusion-reaction equation (concentration). The FV scheme considered is "vertex centered" type semi-implicit in time: explicit for the convection and implicit for the diffusion. We use a Godunov scheme to approach the convectif term and a P1 finite element approximation for the diffusion term. We prove that the FV scheme is L≂ and BV stable and satisfy the discrete maximum the principle under a suitable CFL condition. Then, we show the convergence of the approximate solution obtained by the combined scheme MFE-FV towards the solution of the coupled problem. The proof of convergence is done in several steps: first we deduce strong convergence of the approximate solution in L2(Q), using L≂ stability, BV estimates and a compactness argument. In the second step we study the decoupled MFE scheme, by giving a convergence result for the pressure and velocity. In the final step, the process of convergence of the approximate solution of the combined scheme MFE-FV towards the exact solution is obtained by passing in the limit and uniqueness of the solution of the continuous problem. Academic and realistic numeric simulations for two-dimensional problems confirm the stability and the effectiveness of the combined scheme. Finally, We analyze a residual error estimator for a convection-diffusion-reaction equation discretized by a semi-implicit finite volume. We introduce two kinds of indicators. The first is local in time and space and constitutes an effective tool for the adaptation of the grid to each time step. The second is total in space but local in time and can be used for the adaptation in time. The error estimators with respect to both time and space yield global upper and local lower bounds on the error measured in the energy norm. Numerical results of adaptations of grid are presented and show the effectiveness of the method. The software part of this work concerns two shutters. The first allowed to carry out an IMPES simulator, MFlow, written in C++, for the simulation of the system of miscible flows considered in this thesis. The second shutter relates to the collaboration with a group of researchers for the development of the Homogenizer++ platform realized within the framework of the GDR MoMaS (http://momas.univ-lyon1.fr/).
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https://tel.archives-ouvertes.fr/tel-00009683
Contributor : Mustapha El Ossmani <>
Submitted on : Tuesday, July 5, 2005 - 6:26:25 PM
Last modification on : Tuesday, February 2, 2021 - 2:54:04 PM
Long-term archiving on: : Friday, September 14, 2012 - 1:35:36 PM

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Mustapha El Ossmani. Méthodes Numériques pour la Simulation des Ecoulements Miscibles en Milieux Poreux Hétérogènes. Mathématiques [math]. Université de Pau et des Pays de l'Adour, 2005. Français. ⟨tel-00009683⟩

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