Etude d'estimations d'erreur a posteriori et d'adaptivité basée sur des critères d'arrêt et raffinement de maillages pour des problèmes d'écoulements multiphasiques et thermiques. Application aux procédés de récupération assistée d'huile

摘要

The goal of this thesis is the a posteriori error analysis and the conception of adaptive strategies based on stopping criteria and local mesh refinement. We treat a class of multidimensional degenerate parabolic equations which represent typical examples of industrial interest. In Chapter 1 we consider the time-dependent two-phase Stefan problem that models a phase change process governed by the Fourier law. We regularize the relation between the enthalpy and the temperature and we discretize the problem by the backward Euler temporal stepping method with a conforming spatial discretization, such as the finite element or the vertex-centered finite volume one. We prove un upper bound for the dual norm of the residual, the L2(0; T;H-1) error in the enthalpy, and L2(0; T;L2) error in the temperature, by fully computable error estimators. These estimators include: an estimator associated to the regularization error, an estimator associated to the linearization error, an estimator associated to the temporal error, and an estimator associated to the spatial error. Consequently, these estimators allow to formulate an adaptive resolution algorithm where the corresponding errors can be equilibrated. We also propose a strategy of local mesh refinement. Finally, we prove the efficiency of our a posteriori estimates. A numerical test illustrates the efficiency of our estimates and the performance of the adaptive algorithm. In particular, effectivity indices close to the optimal value of 1 are obtained. In Chapter 2 we derive a posteriori error estimates for the isothermal compositional model of the multiphase Darcy ow in porous media, consisting of a system of strongly coupled nonlinear unsteady partial differential and nonlinear algebraic equations. This model is discretized by a cell-centered finite volume scheme in space with the backward Euler temporal stepping. We establish an upper bound for a dual norm of the residual augmented by a nonconformity evaluation term by fully computable estimators. We focus in this chapter on the formulation of criteria for the iterative linearization (such as the Newton method) and iterative algebraic solvers (such as the GMRes method) that stop the iterations when the corresponding error components no longer affect the overall estimate significantly. We apply our analysis to several real-life reservoir engineering examples to confirm that significant computational gains (up to an order of magnitude in terms of the total number of algebraic solver iterations) can be achieved thanks to our adaptive stopping criteria, already on fixed meshes, and this without any noticeable loss of precision. In Chapter 3 we complete the model described in Chapter 2 by considering a nonisothermal condition for the flow in order to treat the general thermal multiphase compositional foow in porous media. For this problem, we derive fully computable a posteriori error estimates analogous to Chapter 2 for a dual norm of the residual supplemented by a nonconformity evaluation term. We then show how to estimate separately the space, time, linearization, and algebraic errors, giving the possibility to formulate adaptive stopping and balancing criteria. Specification of the abstract theory to the so-called dead oil model closes the chapter. The proof of efficiency of our a posteriori estimate is also provided. Finally, in Chapter 4 we consider the Steam-Assisted Gravity Drainage (SAGD) process, more precisely a thermal oil-recovery technique of the deal oil type with steam injection designed to increase the oil mobility. The main subjects of this chapter are to apply the a posteriori error analysis of Chapters 2 and 3, propose a simplification and a quadrature formula for an easy evaluation of the estimators, propose a space-time adaptive mesh reffinement algorithm, and illustrate by numerical results on real-life examples its performance. In particular, a signi cant gain in terms of the number of mesh cells is achieved on examples in 3 space dimensions.