Etude théorique et numérique de couplages entre écoulements et déformations mécaniques dans l'extraction d'hydrocarubres

Abstract : )The purpose of this thesis is to study theoretical and numerical aspects of some coupled models between a flow in porous medium and the mechanical deformation of the ground. We consider two coupled models on the one hand (1) between a linear compressible flow and the mechanical deformation of the ground, and on the other hand (2) between a nonlinear two-phase flow and the mechanical deformation of the medium. For the first model (1), we show the existence and the uniqueness of the weak solution by means of the Galerkin method. The second model (2) is strongly coupled and involves a parabolic degenerate equation. In order to show the existence of a weak solution, we consider a sequence of related uniformly parabolic problems and apply the Schauder fixed point theorem to show that they possess a classical solution. We then prove the relative compactness of sequences of solutions by means of the Fréchet-Kolmogorov theorem which yields the convergence of a subsequence to a weak solution of the parabolic system. The second part is devoted to the numerical study; we compare two coupling algorithms for the coupled models. The first one, which is used by engineers, is based upon computations by means of a fixed point method; the second one, which we propose, relies on a preconditionned conjugate gradient approach. We show that the second algorithm is more robust than the first one.
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Fatima Zahra Daim. Etude théorique et numérique de couplages entre écoulements et déformations mécaniques dans l'extraction d'hydrocarubres. Mathématiques [math]. Université Paris Sud - Paris XI, 2004. Français. ⟨tel-00007909v2⟩

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