[pt] A teoria da homogeneização de equações diferenciais tornou-se um campo aberto de pesquisa em diversas áreas das ciências exatas e mostrou-se ser uma poderosa ferramenta para a compreensão do comportamento global de materiais heterogêneos. Apesar de ser conhecido que a dedução da lei de Darcy através das equações de Navier-Stokes já é tema debatido há décadas muitas questões continuam em aberto, principalmente a respeito de condições de contorno mais complexas, casos envolvendo fluxos multifásicos e técnicas de homogeneização numérica. Sabe-se que a lei de Darcy se apresenta sob forma de uma relação linear apenas para um intervalo de gradiente hidráulico e que este intervalo se sobrepõe ao intervalo de fluxo laminar do fluido através dos vazios do solo.Se propõe neste trabalho, então, compreender a perda de linearidade na lei de Darcy, a partir da teoria da homogeneização, modificando e explorando os resultados obtidos anteriormente na literatura. / [en] The theory of homogenization of differential equations has become an open field of research in several areas of the exact sciences and has proved to be a powerful tool for understanding the global behavior of heterogeneous materials. Despite knowing that the deduction of Darcy s law through the Navier-Stokes equations has been debated for decades, many questions remain open, mainly regarding more complex boundary cenditions, cases involving multiphase flows and the numerical homogenization techniques. It is known that Darcy s law is presented in the form of a linear relationship only for a range of hydraulic gradient that overlaps the range of laminar flow of fluids through soil voids. Therefore, it is proposed in this work to understand the loss of linearity in Darcy s law, based on the theory of homogenization, modifying and exploring the limit results obtained in the literature.

Cette thèse porte sur la modélisation de l’écoulement et du transport en milieu poreux ;nous effectuons des simulations numériques et démontrons des résultats de convergence d’algorithmes.Au Chapitre 1, nous appliquons des méthodes de volumes finis pour la simulation d’écoulements à densité variable en milieu poreux ; il vient à résoudre une équation de convection diffusion parabolique pour la concentration couplée à une équation elliptique en pression.Nous nous appuyons sur la méthode des volumes finis standard pour le calcul des solutions de deux problèmes spécifiques : une interface en rotation entre eau salée et eau douce et le problème de Henry. Nous appliquons ensuite la méthode de volumes finis généralisés SUSHI pour la simulation des mêmes problèmes ainsi que celle d’un problème de bassin salé en dimension trois d’espace. Nous nous appuyons sur des maillages adaptatifs, basés sur des éléments de volume carrés ou cubiques.Au Chapitre 2, nous nous appuyons de nouveau sur la méthode de volumes finis généralisés SUSHI pour la discrétisation de l’équation de Richards, une équation elliptique parabolique pour le calcul d’écoulements en milieu poreux. Le terme de diffusion peut être anisotrope et hétérogène. Cette classe de méthodes localement conservatrices s’applique àune grande variété de mailles polyédriques non structurées qui peuvent ne pas se raccorder.La discrétisation en temps est totalement implicite. Nous obtenons un résultat de convergence basé sur des estimations a priori et sur l’application du théorème de compacité de Fréchet-Kolmogorov. Nous présentons aussi des tests numériques.Au Chapitre 3, nous discrétisons le problème de Signorini par un schéma de type gradient,qui s’écrit à l’aide d’une formulation variationnelle discrète et est basé sur des approximations indépendantes des fonctions et des gradients. On montre l’existence et l’unicité de la solution discrète ainsi que sa convergence vers la solution faible du problème continu. Nous présentons ensuite un schéma numérique basé sur la méthode SUSHI.Au Chapitre 4, nous appliquons un schéma semi-implicite en temps combiné avec la méthode SUSHI pour la résolution numérique d’un problème d’écoulements à densité variable ;il s’agit de résoudre des équations paraboliques de convection-diffusion pour la densité de soluté et le transport de la température ainsi que pour la pression. Nous simulons l’avance d’un front d’eau douce assez chaude et le transport de chaleur dans un aquifère captif qui est initialement chargé d’eau froide salée. Nous utilisons des maillages adaptatifs, basés sur des éléments de volume carrés. / This thesis bears on the modelling of groundwater flow and transport in porous media; we perform numerical simulations by means of finite volume methods and prove convergence results. In Chapter 1, we first apply a semi-implicit standard finite volume method and then the generalized finite volume method SUSHI for the numerical simulation of density driven flows in porous media; we solve a nonlinear convection-diffusion parabolic equation for the concentration coupled with an elliptic equation for the pressure. We apply the standard finite volume method to compute the solutions of a problem involving a rotating interface between salt and fresh water and of Henry's problem. We then apply the SUSHI scheme to the same problems as well as to a three dimensional saltpool problem. We use adaptive meshes, based upon square volume elements in space dimension two and cubic volume elements in space dimension three. In Chapter 2, we apply the generalized finite volume method SUSHI to the discretization of Richards equation, an elliptic-parabolic equation modeling groundwater flow, where the diffusion term can be anisotropic and heterogeneous. This class of locally conservative methods can be applied to a wide range of unstructured possibly non-matching polyhedral meshes in arbitrary space dimension. As is needed for Richards equation, the time discretization is fully implicit. We obtain a convergence result based upon a priori estimates and the application of the Fréchet-Kolmogorov compactness theorem. We implement the scheme and present numerical tests. In Chapter 3, we study a gradient scheme for the Signorini problem. Gradient schemes are nonconforming methods written in discrete variational formulation which are based on independent approximations of the functions and the gradients. We prove the existence and uniqueness of the discrete solution as well as its convergence to the weak solution of the Signorini problem. Finally we introduce a numerical scheme based upon the SUSHI discretization and present numerical results. In Chapter 4, we apply a semi-implicit scheme in time together with a generalized finite volume method for the numerical solution of density driven flows in porous media; it comes to solve nonlinear convection-diffusion parabolic equations for the solute and temperature transport as well as for the pressure. We compute the solutions for a specific problem which describes the advance of a warm fresh water front coupled to heat transfer in a confined aquifer which is initially charged with cold salt water. We use adaptive meshes, based upon square volume elements in space dimension two.

[pt] A Equação da Difusividade Hidráulica Não-Linear (EDHN) modela o escoamento monofásico de fluidos em meios porosos levando em conta a variação das propriedades da rocha e do fluido presente no interior de seus poros. Normalmente, a solução adimensional da linha-fonte pD(rD, tD) para escoamento de líquidos é encontrada por meio do uso da transformada de Laplace ou transformação de Boltzmann, o qual, o perfil transiente de pressões em coordenadas cartesianas é descrito pela função erro complementar erfc(xD, yD, tD) e, em coordenadas cilíndricas pela função integral exponencial Ei(rD, tD). Este trabalho propõe a solução analítica pelo método de expansão assíntotica de primeira ordem em séries, para solução de alguns problemas de escoamento de petróleo em meios porosos com permeabilidade dependente da pressão de poros e termo fonte. A solução geral será implementada no software Matlab (marca registrada) e a calibração do modelo matemático será realizada comparandose a solução obtida neste trabalho com a solução calculada por meio de um simulador de fluxo óleo em meios porosos denominado IMEX (marca registrada) , amplamente usado na indústria de petróleo e em pesquisas científicas e que usa o método de diferenças finitas. A solução geral da equação diferencial é dada pela soma da solução para escoamento de líquidos com permeabilidade constante e o termo de primeira ordem da expansão assintótica, composto pela não linearidade devido à variação de permeabilidade. O efeito da variação instantânea de permeabilidade em função da pressão de poros é claramente demonstrado nos gráficos diagnósticos e especializados apresentados. / [en] The Nonlinear Hydraulic Diffusivity Equation (NHDE) models the singlephase flow of fluids in porous media considering the variation in the properties of the rock and the fluid present inside its pores. Normally, the dimensionless linear solution for the flow of oil is performed using the Laplace and Fourier transform or Boltzmann transformation and provides the unsteady pressure profile in Cartesian coordinates given by complementary error function erfc(xD, yD, tD) and in cylindrical coordinates described by the exponential integral function Ei(rD, tD). This work develops a new analytical model based on an integro-differential solution to predict the formation mechanical damage caused by the permeability loss during the well-reservoir life-cycle for several oil flow problems. The appropriate Green s function (GF) to solve NHDE for each well-reservoir setting approached in this thesis is used. The general solution is implemented in the Matlab (trademark) and the mathematical model calibration will be carried out by comparing the solution obtained in this work to the porous media finite difference oil flow simulator named IMEX (trademark). The general solution of the NHDE is computed by the sum of the linear solution (constant permeability) and the first order term of the asymptotic series expansion, composed of the nonlinear effect of the permeability loss. The instantaneous permeability loss effect is clearly noticed in the diagnostic and specialized plots.