In this dissertation, we focus on the uncertainty quantification problems where the goal is to sample the porous media properties given integrated responses. We first introduce a reduced order model using the level set method to characterize the channelized features of permeability fields. The sampling process is completed under Bayesian framework. We hence study the regularity of posterior distributions with respect to the prior measures. The stochastic flow equations that contain both spatial and random components must be resolved in order to sample the porous media properties. Some type of upscaling or multiscale technique is needed when solving the flow and transport through heterogeneous porous media. We propose ensemble-level multiscale finite element method and ensemble-level preconditioner technique for solving the stochastic flow equations, when the permeability fields have certain topology features. These methods can be used to accelerate the forward computations in the sampling processes. Additionally, we develop analysis-of-variance-based mixed multiscale finite element method as well as a novel adaptive version. These methods are used to study the forward uncertainty propagation of input random fields. The computational cost is saved since the high dimensional problem is decomposed into lower dimensional problems. We also work on developing efficient advanced Markov Chain Monte Carlo methods. Algorithms are proposed based on the multi-stage Markov Chain Monte Carlo and Stochastic Approximation Monte Carlo methods. The new methods have the ability to search the whole sample space for optimizations. Analysis and detailed numerical results are presented for applications of all the above methods.

Le cœur d'un réacteur nucléaire est un milieu très hétérogène encombré de nombreux obstacles solides et les phénomènes thermohydrauliques à l'échelle macroscopique sont directement impactés par les phénomènes locaux. Toutefois les ressources informatiques actuelles ne suffisent pas à effectuer des simulations numériques directes d'un cœur complet avec la précision souhaitée. Cette thèse est consacré au développement de méthodes d'éléments finis multi-échelles (MsFEMs) pour simuler les écoulements incompressibles dans un milieu hétérogène avec un coût de calcul raisonnable. Les équations de Navier-Stokes sont approchées sur un maillage grossier par une méthode de Galerkin stabilisé, dans laquelle les fonctions de base sont solutions de problèmes locaux sur des maillages fins prenant précisément en compte la géométrie locale. Ces problèmes locaux sont définis par les équations de Stokes ou d'Oseen avec des conditions aux limites ou des termes sources appropriés. On propose plusieurs méthodes pour améliorer la précision des MsFEMs, en enrichissant l'espace des fonctions de base locales. Notamment, on propose des MsFEMs d'ordre élevée dans lesquelles ces conditions aux limites et termes sources sont choisis dans des espaces de polynômes dont on peut faire varier le degré. Les simulations numériques montrent que les MsFEMs d'ordre élevés améliorent significativement la précision de la solution. Une chaîne de simulation multi-échelle est construite pour simuler des écoulements dans des milieux hétérogènes de dimension deux et trois. / The nuclear reactor core is a highly heterogeneous medium crowded with numerous solid obstacles and macroscopic thermohydraulic phenomena are directly affected by localized phenomena. However, modern computing resources are not powerful enough to carry out direct numerical simulations of the full core with the desired accuracy. This thesis is devoted to the development of Multiscale Finite Element Methods (MsFEMs) to simulate incompressible flows in heterogeneous media with reasonable computational costs. Navier-Stokes equations are approximated on the coarse mesh by a stabilized Galerkin method, where basis functions are solutions of local problems on fine meshes by taking precisely local geometries into account. Local problems are defined by Stokes or Oseen equations with appropriate boundary conditions and source terms. We propose several methods to improve the accuracy of MsFEMs, by enriching the approximation space of basis functions. In particular, we propose high-order MsFEMs where boundary conditions and source terms are chosen in spaces of polynomials whose degrees can vary. Numerical simulations show that high-order MsFEMs improve significantly the accuracy of the solution. A multiscale simulation chain is constructed to simulate successfully flows in two- and three-dimensional heterogeneous media.