New analytical and numerical solutions are developed to both the kinematic approximation to the St Venant equations and the Hairsine-Rose (HR) soil erosion model in order to gain a better physical understanding of soil erosion and sediment transport in shallow overland flow. The HR model is unique amongst physically based erosion models in that it is the only one that: considers the entire distribution of the soil s sediment size classes, considers the development of a layer of deposited non-cohesive sediment having different characteristics to the original underlying cohesive soil and considers separately the erosion processes of rainfall detachment, runoff entrainment and gravitational deposition. The method of characteristics and the method of lines were used to develop both the analytical and numerical solutions respectively. These solutions were obtained for boundary and initial conditions typical of those used in laboratory flume experiments along with physically realistic constant and time dependent excess rainfall rates. Depending on the boundary and initial conditions, interesting new solutions of the kinematic wave equation containing expansion waves, travelling shocks as well as solutions which split into an upslope and downslope drying profiles were found. Numerical solutions of the HR model were applied to the experimental flume data of Polyakov and Nearing (2003) obtained under flow conditions which periodically cycled between net erosion and net deposition conditions. While excellent agreement was found with suspended sediment data, the analysis suggested that an additional transport mechanisms, traditionally not included in soil erosion models, was occurring. While the inclusion of bed-load transport improved the ii overall model prediction, it was still not sufficient. Subsequent asymptotic analysis then showed that the interaction of the flow with an evolving bed morphology was in fact far more important than bed load transport. A very interesting finding from this work showed that the traditional criterion of validating sediment transport model based solely on suspended sediment data was not sufficient as reliable predictions could be obtained even when important transport mechanisms were neglected. Experimental plots of sediment discharge or suspended sediment concentration against water discharge in overland flow have been shown to contain significant hysteresis between the falling and rising limbs of the discharge hydrograph. In the final Chapter, the numerical solution developed for the complete system of soil erosion and kinematic flow was used to show that it was possible for the HR model to simulate three of the four hysteresis loops identified in the literature. Counter clock-wise loops, clock-wise loops and figure 8 loops could all be produced as a result of starting with different initial conditions, being mi(x; 0) = 0, mi(x; 0) = pimt and mi(x; 0) = 0:5pimt respectively. This is the first time that these types of hysteresis loops have been produced by any erosion model. The generation of these hysteresis loops are physically explainable in terms of sediment availability and is consistent with data obtained on the field scale.

[pt] O estudo de fenômenos transitórios em sistemas hidráulicos está presente em diversos ramos da engenharia, encontra-se na área de hidrelétricas e na indústria do petróleo. O presente trabalho apresenta uma análise comparativa deste fenômeno utilizando três abordagens de modelagem distintas: o clássico método das características, a técnica de grafos de ligação e o tratamento através do fluxo de potência. No caso do método das características emprega-se o procedimento proposto por Streeter. Na modelagem através de grafo de ligação adota-se a analogia generalizada entre os sistemas fluido, mecânico e elétrico. Na representação pelo fluxo de potência acopla-se módulos previamente desenvolvidos considerando as relações de causa e efeito entre os componentes de um sistema. Todos os métodos foram implementados em Matlab e foram testados e comparados a partir de simulações em dois sistemas simples compostos de um reservatório, linha fluida e válvula. Os resultados obtidos através da metodologia de fluxo de potência aproximam-se dos encontrados pelas abordagens tradicionais. / [en] The study of transitory phenomena in hydraulic systems is present in several branches of the engineering, in the hydroelectric power stations area and in the petroleum industry. The present work introduce a comparative analysis of this phenomenon using three approaches of distinct modeling: the classic characteristics method, the technique of bond graphs and the treatment through the power flow. In the case of characteristics method is used the procedure proposed by Streeter. In the modeling through bond graphs is adopted the generalized analogy among the fluid, mechanic and electric systems. In the representation by power flow modules previously developed are coupled considering the cause and effect relationships between components of one system. All methods were implemented in Matlab and were tested and compared from simulations in two simple systems composed of a reservoir, fluid lines and valve. The results obtained through the methodology of power flow are near of the discoveries by the traditional approaches.

Apresentamos alguns métodos numéricos para a resolução das equações hiperbólicas que regem problemas de transporte de cargas elétricas em isolantes, aplicando-os a quatro problemas específicos: injeção de corrente por um contato ôhmico em uma amostra com voltagem constante aplicada; transporte de um pulso de cargas através de uma amostra em circuito aberto; transporte de um pulso de cargas através de uma amostra submetida a uma diferença de potencial constante (tempo de vôo); e, finalmente, descarga termo-estimulada em circuito aberto. Empregamos, basicamente, dois tipos de métodos: características e diferenças finitas. Concluímos que, quando as descontinuidades são importantes, é mais conveniente usar o método das características; porém, quando não houver descontinuidades ou se estas não forem importantes, alguns métodos de diferenças finitas podem ser utilizados com boa precisão e menores tempos de computação do que aqueles gastos pelos métodos das características. / Numeral methods for solving partial differential equations of the hiperbolic type, governing some problems of transport of electric charge in dielectrics are presented and then applied to four specific problems: injection of charge via an ohmic contact into a sample with a constant applied voltage; transport of a pulse of charge through a sample in the open circuit mode; transport of a pulse of charge through a sample subjected to a constant voltage; and finally, thermally stimulated discharge in open circuit. Essentially two kinds of methods are employed: the method of characteristics and finite-difference methods. It is concluded that when discontinuities are important, the method of characteristics is the most convenient; otherwise, appropriate finite-difference schemes can be used with sufficient precision and less time expenses in computers.

Apresentamos alguns métodos numéricos para a resolução das equações hiperbólicas que regem problemas de transporte de cargas elétricas em isolantes, aplicando-os a quatro problemas específicos: injeção de corrente por um contato ôhmico em uma amostra com voltagem constante aplicada; transporte de um pulso de cargas através de uma amostra em circuito aberto; transporte de um pulso de cargas através de uma amostra submetida a uma diferença de potencial constante (tempo de vôo); e, finalmente, descarga termo-estimulada em circuito aberto. Empregamos, basicamente, dois tipos de métodos: características e diferenças finitas. Concluímos que, quando as descontinuidades são importantes, é mais conveniente usar o método das características; porém, quando não houver descontinuidades ou se estas não forem importantes, alguns métodos de diferenças finitas podem ser utilizados com boa precisão e menores tempos de computação do que aqueles gastos pelos métodos das características. / Numeral methods for solving partial differential equations of the hiperbolic type, governing some problems of transport of electric charge in dielectrics are presented and then applied to four specific problems: injection of charge via an ohmic contact into a sample with a constant applied voltage; transport of a pulse of charge through a sample in the open circuit mode; transport of a pulse of charge through a sample subjected to a constant voltage; and finally, thermally stimulated discharge in open circuit. Essentially two kinds of methods are employed: the method of characteristics and finite-difference methods. It is concluded that when discontinuities are important, the method of characteristics is the most convenient; otherwise, appropriate finite-difference schemes can be used with sufficient precision and less time expenses in computers.

[pt] O estudo de controle de poço é fundamental para determinar as principais variáveis envolvidas nas operações de perfuração e construção de um poço. A análise dessas operações visa estabelecer parâmetros importantes que devem ser acompanhados, a fim de evitar um influxo de fluidos da formação para o poço (kick). Para que as operações de perfuração sejam sempre seguras, é necessário que a pressão hidrostática no fundo do poço seja superior a pressão de poros da formação. Este trabalho apresenta uma modelagem matemática, desenvolvida para simular um kick de água e óleo da formação que adentra um poço terrestre vertical e, assim, prever o comportamento da pressão dentro da coluna de produção e no espaço anular durante uma situação de controle de poço. A operação de controle é composta pelo influxo do kick, fechamento do poço e expulsão do fluido invasor. O caso de kick de gás não foi abordado devido à sua complexidade. O modelo estudado foi simulado no Visual Basic do Excel com emprego de uma formulação do método das características com diferenças finitas. Foram implementadas considerações sobre o efeito da geometria do poço, perdas de carga por fricção, expansão do influxo e modelo de escoamento bifásico. / [en] The study of well control is critical to determine the main variables involved in well drilling and construction operations. The analysis of these operations aims to establish important parameters that must be followed in order to avoid an influx of fluids from the formation into the well (kick). In order to guarantee safety during drilling operations, the hydrostatic pressure at the bottom of the well must be greater than the formation pore pressure. This work presents a mathematical modeling, developed to simulate a water and oil kick from the formation that enters a vertical onshore well and, thus, predict the pressure behavior inside the production string and the annular space during a well control situation. The control operation is composed of the kick inflow, well closure and expulsion of the invading fluid. The gas kick case was not assessed because of its complexity. The studied model has been simulated in Visual Basics, in Excel, using the characteristics method with finite difference formulation. Considerations were taken about the effect of well geometry, friction loss, influx expansion and two phase flow model.

La présente thèse est divisée en deux parties. La première partie concerne l'analyse mathématique et la contrôlabilité exacte à zéro pour une catégorie de systèmes structurés décrivant la dynamique d'une population d'insectes. La seconde partie est consacrée à l'étude de la stabilité de la conductivité d'un système de réaction diffusion modélisant l'activité électrique du coeur.Dans le chapitre 2, on considère que la population d'adultes se diffuse dans la vignoble,la fonction de la croissance des individus à chaque stade dépend des variations climatiques et de la variété des raisins. En utilisant la méthode de point fixe, on obtient l'existence et l'unicité des solutions du modèle. On démontre ensuite l'existence d'un attracteur global pour le système dynamique. Enfin, on utilise la théorie des opérateurs compacts et le théorème de point fixe de Krasnoselskii pour prouver l'existence des états stationnaires.Dans le chapitre 3, on traite le problème de contrôlabilité exacte du modèle de Lobesia Botrana, lorsque la fonction de croissance est égale à 1. On suppose que les quatre sous-catégories de ce système sont dans une phase statique. On obtient que la population d'oeufs peut être contrôlée à zéro. Ce résultat est basé sur des estimations à priori combinées avec un théorème de point fixe.Lorsque les papillons adultes se dispersent spatialement, on introduit un contrôle sur la population d'oeufs, de larves et de femelles dans une petite région du vignoble. On montre alors la contrôlabilité exacte à zéro pour les femelles.Dans la deuxième partie de cette thèse, on analyse la stabilité des coefficients de diffusion d'un système parabolique qui modélise l'activité électrique du coeur. On établit une estimation de Carleman pour le système de réaction-diffusion. En combinant cette estimation avec des estimations d'énergie avec poids on obtient le résultat de stabilité. / This thesis is divided into two parts.One is mainly devoted to make a qualitative analysis and exact null controlfor a class of structured population dynamical systems, and the other concernsstability of conductivities in an inverse problem of a reaction-diffusion systemarising in electrocardiology.In the first part, we study the dynamics ofEuropean grape moth, which has caused serious damages on thevineyards in Europe,North Africa, and even some Asian countries.To model this grapevine insect, physiologically structured multistage population systems are proposed.These systemshave nonlocal boundary conditions arising in nonlocal transition processes in ecosystem.We consider the questions of spatial spread of the populationunder physiological age and stage structures,and show global dynamical properties for the model.Furthermore, we investigate the control problem for this Lobesia botrana modelwhen the growth function is equal to $1$.For the case that four subclasses of this system are all in static station,we conclude that the population of eggs can be controlled to zero at acertain moment by acting on eggs.While the adult moths can disperse,we describe a control by a removal of egg and larvapopulation, and also on female moths in a small region of the vineyard.Then the null controllability for female mothsin a nonempty open sub-domain at a given time is obtained.In the second part, a reaction-diffusion system approximating a parabolic-elliptic systemwas proposed tomodel electrical activity in the heart. We are interested inthe stability analysis of an inverse problem for this model.Then we use the method of Carleman estimates and certain weight energyestimatesfor the identification of diffusion coefficients for the parabolicsystem to draw the conclusion.