Severe accidents arising from the fusion of a nuclear reactor core must be anticipated to enhance the efficiency of their mitigation. Such accidents have occurred at TMI-2 (1979) and Fukushima (2011). Following a loss of coolant accident, core heating and oxidation of the fuel cladding followed by reflooding (injection of water) may lead to the collapse of fuel rods and formation of porous debris bed in the core. Steam produced upon reflooding may activate the exothermic oxidation of Zircaloy leading to partial melting of materials. Such evolution generates zones with reduced porosity limiting coolant penetration and/or impermeable blocked zones. In this situation, the efficiency of injecting water into the core to stop the progress of degradation and prevent the reactor core melting may be significantly reduced. In this scope, IRSN launched PEARL program to investigate the thermal hydraulics of reflooding of hot debris beds surrounded by a more permeable zone simulating the presence of intact or less damaged zones in the core. The PEARL experiments were modeled and simulated using ICARE/CATHARE code to assess the evolution of a bottom reflooding of a superheated debris bed surrounded by a bypass of larger permeability. The thermal hydraulics of the quenching process has been analyzed and the effect of each of the initial conditions on the reflooding behavior was assessed. The effect of pressure was investigated and related to the entrainment of injected water at quench front level into the bypass. An analytical model was then developed to investigate thoroughly the reflooding of a superheated heterogeneous porous medium, composed of two layers of contrasting permeability and porosity, and to describe the water entrainment in the bypass. This model computes the main variables characterizing the reflooding process such as quench front velocity, water-to-steam conversion ratio, and the flow rate of water entrained in the bypass. It provides good qualitative and quantitative results for the two-phase flow redistribution as compared to experimental results. This model has several advantages. It is written in a rather general form including the Forchheimer correction terms and non-zero cross-terms in the generalized Darcy-Forchheimer momentum equation. Variations of proposed momentum equations including changes in correlations andinterfacial friction laws can be tested easily and efficiently. Comparison of the calculations against experimental results indicated that it is necessary to include an interfacial friction law to obtain good predictions. This model allows performing fast evaluations of the efficiency of cooling bycomputing the fraction of the injected flow rate that participates in cooling. Upscaling to the reactor scale is straightforward and calculations were performed to assess the impact of geometric parameters of the debris bed (particle size, porosity, dimensions) as well as thermal hydraulic conditions (temperature, pressure, injection flow rate) on the reflooding process. Thus the model is very useful to estimate the total quenching time and the maximum temperature that could be reached by the hot debris bed at large scales. This allows assessing the probability of a successful quenching of a hot debris bed formed during a hypothetical accidental scenario.

Ce travail de thèse poursuit l’objectif d’une simulation numérique multiéchelle d’un assemblage de réacteur à neutrons rapides. Malgré la puissance croissante des ordinateurs, la CFD fine complète d’un tel système demeure extrêmement coûteuse dans un contexte de recherche et développement. Nous proposons alors, une fois déterminé le comportement thermohydraulique moyen de l’assemblage, de reconstruire localement l’information aux fines échelles, l’ensemble de la démarche requérant un temps de calcul bien moindre qu’une simulation de la totalité de la structure. La description à l’échelle moyenne est obtenue soit par le formalisme de prise de moyenne volumique en milieu poreux, soit via une approche alternative historiquement développée pour les assemblages de RNR-Na. Elle fournit des informations utilisées comme contraintes d’un sous-problème de raffinement d’échelle, par l’intermédiaire d’une technique de pénalisation des équations de conservation locales. Ce sous-problème exploite le caractère périodique de la structure en s’appuyant sur des conditions aux limites de périodicité des champs recherchés ou de leur déviation spatiale. Après validation des méthodologies sur des applications modèles, nous entreprenons leur mise en oeuvre sur des configurations « industrielles » qui démontrent la viabilité de cette approche multiéchelle. / The present work is devoted to a multiscale numerical simulation of an assembly of fast neutron reactor. In spite of the rapid growth of the computer power, the fine complete CFD of a such system remains out of reach in a context of research and development. After the determination of the thermalhydraulic behaviour of the assembly at the macroscopic scale, we propose to carry out a local reconstruction of the fine scale information. The complete approach will require a much lower CPU time than the CFD of the entire structure. The macroscale description is obtained using either the volume averaging formalism in porous media, or an alternative modeling historically developed for the study of fast neutron reactor assemblies. It provides some information used as constraint of a downscaling problem, through a penalization technique of the local conservation equations. This problem lean on the periodic nature of the structure by integrating periodic boundary conditions for the required microscale fields or their spatial deviation. After validating the methodologies on some model applications, we undertake to perform them on “industrial” configurations which demonstrate the viability of this multiscale approach.

Dans cette thèse, deux problèmes complètement indépendants sont étudiés. Le premier sujet porte sur l'analyse mathématique d'un modèle simple de délamination de films minces et d'apparition de cloques. Le second sujet est une étude fine de l'équation des milieux poreux motivée par des problèmes d'intrusion saline. Dans le premier sujet de ce travail, nous considérons un modèle variationnel simple unidimensionnel décrivant la délamination de films minces sous l'effet d'un refroidissement. Nous caractérisons les minimiseurs globaux qui correspondent à trois types de films: non-délaminés, partiellement délaminés (appelés cloques) ou bien complètement délaminés. Deux paramètres jouent un rôle crucial dans un tel phénomène : la longueur du film et la température. Dans le plan de phase de ces deux derniers, nous classifions l'ensemble des minimiseurs. Comme conséquence de notre analyse, nous identifions explicitement les cloques les plus petites pour ce modèle. Dans le deuxième sujet, nous répondons d'abord à une question ouverte depuis longtemps concernant l'existence de nouvelles contractions pour l'équation de type milieux poreux. Pour m>0, nous nous sommes intéressés à des solutions positives de l'équation suivanteU_t=Delta U^m. Pour 0<m<2, nous présentons une nouvelle famille de contractions pour cette équation en toute dimension, ce qui induit une extension des propriétés de la contraction L^1. Notre contraction peut être considérée comme étant la quatrième contraction connue pour cette équation. Même pour le cas m=1, notre approche aboutit à des nouveaux résultats pour l'équation de la chaleur standard. Dans un second temps, nous avons traité le même problème mais en utilisant une approche différentielle différente basée sur les distances géodésiques. Cette approche originale et générale sert à fabriquer des familles de contractions pour des équations aux dérivées parielles non linéaires, d'évolutions ou stationnaires. Nous présentons dans ce cadre diverses applications. En particulier, nous traitons à nouveau l'équation des milieux poreux et l'équation doublement non linéaire / In this thesis, we study two completely independent problems. The first one focuses on a simple mathematical model of thin films delamination and blistering analysis. In the second one, we are interested in the study of the porous medium equation motivated by seawater intrusion problems. In the first part of this work, we consider a simple one-dimensional variational model, describing the delamination of thin films under cooling. We characterize the global minimizers, which correspond to films of three possible types : non delaminated, partially delaminated (called blisters), or fully delaminated. Two parameters play an important role : the length of the film and the cooling parameter. In the phase plane of those two parameters, we classify all the minimizers. As a consequence of our analysis, we identify explicitly the smallest possible blisters for this model. In the second part, we answer a long standing open question about the existence of new contractions for porous medium type equations. For m>0, we consider nonnegative solutions U(t,x) of the following equationU_t=Delta U^m.For 0<m<2, we present a new family of contractions for this equation in any dimension, which extends the L^1 contraction properties. Our contraction can be seen as the fourth known contraction for this equation. Even for the case m=1, our approach leads to new results for the standard heat equation. A second work focuses on the same problem but using a differential approach based on geodesic distances. This original and general method is used to produce families of contractions for nonlinear partial differential equations, of evolution or stationary type. We present in this part various applications of this original method. In particular, we are concerned with the porous medium and the doubly nonlinear equations

In this dissertation, theoretical, computational, and experimental methodologies are introduced to determine the rate-dependent material properties of the brain tissue. Experiments have shown that the brain tissue is significantly rate-dependent. To examine the range of strain rates at which trauma might happen, a validated finite element (FE) human head model was initially employed to examine the biomechanics and dynamic behavior of the head and brain under impact and blast loads. The strain rates to cause traumatic brain injury (TBI) were found to be in the range of 36 to 241 1/s, under these types of loadings. These findings provided a good estimation prior to exploring the required experiments for characterizing the brain tissue. The brain samples were tested by employing unconfined compression tests at three different deformation rates of 10 (n= 10 brain samples), 100 (n=8), and 1000 mm/sec (n=12). It was found that the tissue exhibited a significant rate-dependent behavior with various compression rates. Two different material characterization approaches were proposed to evaluate the rate-dependent mechanical responses of the brain. In the first approach, based on the parallel rheological framework, a single-phase viscoelastic model which captures the key aspects of the rate-dependency in large strain behavior was introduced. The extracted material parameters showed an excellent constitutive representation of tissue response in comparison with the experimental test results (R^2=0.999). The obtained material parameters were employed in the FE simulations of the brain tissue and successfully verified by the experimental results. In the second approach, the brain tissue is modeled as a biphasic continuum, consisting of a compressible solid matrix fully saturated with an incompressible interstitial fluid. The governing equations based on conservation of mass and momentum are used to describe the solid-fluid interactions. This viscoelastic biphasic model can effectively estimate the rate-dependent tissue deformations, the hydrostatic pressure as well as fluid diffusion through the tissue. Although both single-phasic, as well as bi-phasic models, can successfully capture the key aspects of the rate-dependency in large strain deformation, it was shown the biphasic model can demystify more phenomenological behavior of this tissue that could not be perceived with yet established, single-phasic approaches.

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In this thesis we design and study a particle method that can be used to numericallyapproximate solutions to the quadratic porous medium equation. The idea consists offirst approximating the porous medium equation using a non-local transport equation,to which we approximate the solution with a particle method. We prove that theparticle method converges, in a suitable norm, to the solution to the non-localtransport equation. We provide numerical simulations to illustrate this convergenceand estimate the order of convergence. In particular, we use the particle method toapproximate the Barenblatt solutions to the quadratic porous medium equation. Theanalysis of the partial differential equations is to a large extent carried out in the senseof integrable functions, while the analysis of the particle method relies on a dualityapproach on the space of finite signed Radon measures. / Vi konstruerar och undersöker en partikelmetod som kan användas för att lösaden kvadratiska porös-medium-ekvationen numeriskt. Huvudidén är att förstapproximera ekvationen med en icke-lokal transportekvation, som vi sedan lösernumeriskt med en partikelmetod.Vi bevisar att partikelmetoden konvergerar, i en passande norm, till lösningen tillden icke-lokala transport-ekvationen. Vi presenterar numeriska simulationer föratt illustera denna konvergens och estimera hur snabb konvergensen är. För attgöra detta försöker vi använda partikelmetoden för att approximera Barenblattslösningar till den kvadratiska porös-medium-ekvationen. Vår analys av de partielladifferentialekvationerna görs till stor del i rummet av Lebesgue-integrerbarafunktioner, medan vår analys av partikelmetoden är baserad på att se rummet avändliga Radon-mått som ett underrum till ett dualrum.

[pt] A primeira parte da dissertação é dedicada ao estudo do modelo em meios porosos em contato com reservatórios e à obtenção, heurística, da equação hidrodinâmica para esse modelo, com o intuito de iniciar o estudo do limite hidrodinâmico que garante que a evolução da densidade de partículas desse modelo é descrita pela solução fraca da equação hidrodinâmica, nomeadamente, a equação em meios porosos com condições de Dirichlet. A segunda parte da dissertação é dedicada ao estudo do método da representação matricial, a chamada matriz ansatz, que será utilizado para caracterizar as medidas estacionárias de sistemas de partículas fora do equilíbrio. Usaremos o processo de exclusão simples simétrico como motivação para apresentar as técnicas utilizadas nesse método. Munido dessas técnicas conseguimos obter pela primeira vez a função de correlação de segunda ordem para o processo de exclusão simples simétrico em contato com reservatórios lentos, e além disso, conseguimos obter informação do estado estacionário do modelo em meios porosos em contato com reservatórios. / [en] The first part of the dissertation is dedicated to the study of the porous medium model in contact with reservoirs and to, heuristically, obtain the hydrodynamic equation for this model, with the pursuit of starting the study of the hydrodynamic limit which guarantees that the evolution of the density of particles of this model is described by the weak solution of the hydrodynamic equation, namely, the porous medium equation with Dirichlet boundary conditions. The second part of the dissertation is dedicated to the study of the matrix representation method, the so-called matrix ansatz, which will be used to characterize the stationary measures of particle systems out of equilibrium. For warming up, we will use the symmetric simple exclusion process as a toy model to present the techniques used in this method. With those techniques, for the first time we obtained the second order correlation function for the symmetric simple exclusion process in contact with slow reservoirs, and in addition, we were able to obtain information about the steady state of the porous medium model in contact with reservoirs.

Groundwater inflow is a challenge in construction of tunnels in fractured bedrocks since it affects the safety function of tunnels and leads to potential problems in the surrounding environment, such as subsidence, dropdown of the groundwater table. Quantification of groundwater inflow into the tunnel is also important for design of grouting in the construction of the tunnel. Modelling groundwater flow in fractured bedrocks currently remains a challenge. Commonly used groundwater models are based on continuum assumptions and they do not consider realistic structures of discrete fractures, which leads to high potential uncertainty in prediction of tunnel groundwater inflow. This thesis focuses on prediction of tunnel groundwater inflow, using a discreet fracture-matrix (DFM) model. The DFM model is evaluated and compared with the conventional continuum model based on Darcy’s law. This DFM model considers, in particular, multi-scale heterogeneity, e.g. fracture networks and variable fracture aperture structures. Applying this DFM model, the impact of variable fracture aperture structures on tunnel inflow is investigated through stochastic analysis. The results show that under the same boundary conditions, the traditional continuum model overestimates the inflow compared to the DFM model. The difference in equivalent permeability is 2 to 3 orders of magnitude. The sensitivity analysis shows that the discreet fracture model is sensitive to the variability of fracture aperture. The estimated equivalent permeability values by discreet fracture modelling is in the order of 5×10-6 to 1×10-7 m/s for a fracture density of 1.2 fractures per square meter. This study demonstrates that the DFM represents the more realistic features of fractured rock structures, which is useful and can be used to predict groundwater inflow in fractured rock tunnels. / Grundvatteninflöde är en utmaning vid byggnation av tunnlar i sprucken berggrund eftersom det påverkar tunnlarnas säkerhetsfunktion och leder till potentiella problem i den omgivande miljön, såsom sättningar och Grundvattennivåsänkning. Kvantifiering av grundvatteninflöde till tunneln är också viktig för utformning av injektering i tätning? byggandet av tunneln. Att modellera grundvattenflödet i sprucken berggrund är för närvarande en utmaning. Grundvattenmodeller man normalt använder är baserade på kontinuumantaganden, och de tar inte hänsyn till realistiska strukturer av diskreta sprickor, vilket leder till hög potentiell osäkerhet i uppskattning av tunnelgrundvatteninflöde. Denna avhandling fokuserar på förutsägelse av tunnelinläckage, med hjälp av en diskret sprickmatris (DFM) modell. DFM-modellen utvärderas och jämförs med den konventionella kontinuummodell vilken är baserad på Darcys lag. Denna DFM-modell tar särskilt hänsyn till multi-skala heterogenitet, till exempel spricknätverk och variabla dubbelkolla. Genom att tillämpa denna DFM-modell undersöks effekten av strukturer med variabel spricköppning på grundvatteninflödet genom stokastisk analys. Resultaten visar att under samma randvillkor överskattar den traditionella kontinuummodellen inflödet jämfört med DFM-modellen. Skillnaden i ekvivalent permeabilitet är 2 till 3 storleksordningar. Känslighetsanalysen visar att den diskreta sprickmodellen är känslig möt variationen i spricköppningen. De uppskattade ekvivalenta permeabilitetsvärdena med diskret sprickmodellering är i storleksordningen 5x10-6 till 1x10-7 m/s för en spricktäthet på 1,2 sprickor per kvadratmeter. Denna studie visar att DFM representerar de mer realistiska egenskaperna hos sprickiga bergstrukturer, vilket är användbart och kan användas för att uppskatta grundvatteninflöde i sprickiga bergtunnlar.