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Numerical resolution of algebraic systems with complementarity conditions. Application to the thermodynamics of compositional multiphase mixtures

Abstract : In reservoir simulators, it is usually delicate to take into account the laws of thermodynamic equilibrium for multiphase hydrocarbon mixtures. The difficulty lies in handling the appearance and disappearance of phases for different species. The traditional dynamic approach, known as variable switching, consists in considering only the unknowns and equations of the present phases. It is cumbersome and costly, insofar as "switching" occurs constantly, even from one Newton iteration to another. An alternative approach, called unified formulation, allows a fixed set of unknowns and equations to be maintained during the calculations. From a theoretical point of view, this is an major advance. On the practical level, because of the nonsmoothness of the complementarity conditions involved in the new formulation, the discretized equations have to be solved by the semi-smooth Newton-min method, whose behavior is often pathological. In order to fully exploit the interest of the unified approach, this thesis aims at circumventing this numerical obstacle by means of more robust resolution algorithms, with a better convergence. To this end, we draw inspiration from the methods that have proven their worth in constrained optimization and we try to transpose them to general systems. This gives rise to interior-point methods, of which we propose a nonparametric version called NPIPM. The results appear to be superior to those of Newton-min. Another contribution of this doctoral work is the understanding and (partial) resolution of another obstruction to the proper functioning of the unified formulation, hitherto unidentified in the literature. This is the limitation of the domain of definition of Gibbs' functions associated with cubic equations of state. To remedy the possible non-existence of a system solution, we advocate a natural extension of Gibbs' functions.
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Submitted on : Tuesday, October 13, 2020 - 11:41:47 AM
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Duc Thach Son Vu. Numerical resolution of algebraic systems with complementarity conditions. Application to the thermodynamics of compositional multiphase mixtures. Numerical Analysis [math.NA]. Université Paris-Saclay, 2020. English. ⟨tel-02965421⟩

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