De nombreux travaux visent à caractériser l'influence des hétérogénéités sur le transport de soluté<br />dans les sous-sols. La dispersion anormale observée à l'échelle de l'aquifère est en partie attribuée aux<br />effets du non-équilibre, comme l'échange de masse entre des régions présentant un contraste de<br />perméabilité élevé.<br />En présence de non-équilibre à grande échelle, le transport miscible ne peut plus être décrit par une<br />équation classique de convection-dispersion. Une méthode de changement d'échelle doit permettre de<br />prendre en compte les hétérogénéités, et donner une représentation macroscopique du transport.<br />Différentes techniques peuvent être utilisées, la méthode de prise de moyenne volumique à grande échelle<br />est employée ici. Cette méthode calcule les équations de transport et les propriétés effectives associées<br />par un processus de moyenne spatiale sur les équations correspondant à l'échelle inférieure. Au travers de<br />trois problèmes de fermeture, une expression explicite des propriétés à grande échelle est proposée. Le<br />modèle obtenu peut être vu comme une extension des modèles à double-porosité, capable de représenter<br />la plupart des comportements anormaux. Différents modèles à une équation sont ensuite dérivés et<br />comparés entre eux (comportement asymptotique, hypothèse d'équilibre local, cas de non-équilibre).<br />Une procédure numérique générale est mise en place afin de résoudre les problèmes de fermeture,<br />et ainsi calculer les coefficients de transport macroscopiques. Afin de valider le modèle à deux équations,<br />les prédictions théoriques sont comparées aux expériences numériques réalisées sur des milieux stratifiés<br />et nodulaires. Nous explorons enfin la possibilité d'utiliser une approche à deux équations en relation avec<br />une définition géo-statistique des hétérogénéités. Des systèmes stratifiés aléatoires et des milieux<br />aléatoires bi-dimensionnels sont étudiés, un bon accord est obtenu entre l'approche théorique et les<br />résultats expérimentaux.

Since the past decade, use of electro-osmotic flow (EOF) as an alternative flow mechanism in microdevices is becoming more popular due to its less bulky and low maintenance system design. However, one of the biggest shortcomings for its usage in mainstream applications is that it requires the concerned liquid to be electrically conductive. One idea can be to use the flow of conductive fluids to transport non-conductive liquids passively via interfacial shear transfer. Such an idea can has numerous applications in a wide range of fields like bio-chemical processing (e.g. lab-on-a-chip reactors, mixers, etc.), to oil extraction from porous rock formations. One of the significant characteristics of micro-scale flows is high surface to volume ratio, which significantly highlights the role of multi-phase interfaces in such dynamics. The presence of a fluid-fluid interface in an EOF necessitates the characterization of the parameters responsible for hydrodynamic instability of such systems. The present work focuses on the role of steady and time-dependent electric stress (Maxwell stress), capillary force and disjoining pressure on fluid-fluid interfacial instability. A linear stability analysis of interfacial perturbation was performed for a thin film of electrolyte under DC and AC electric fields. Through long wave asymptotic analysis of the Orr-Sommerfeld equations, parametric stability thresholds of a thin aqueous film explored. Further, a set of experiments were performed in order to characterize the EOF in a rectangular microchannel. With the help of a Particle Tracking Velocimetry analysis, velocity distributions were obtained which agreed well to the theoretical values. This was further used to estimate PDMS zeta potential, which was found to be within the reported values in the existing literature. Liquid-liquid interfacial deformation was also explored under a time-periodic EOF and a wide range of the magnitudes of capillary force, and diffusive and convective transport.

L’approximation dite « anélastique » permet de filtrer les ondes acoustiques grâce à un développement asymptotique deséquations de Navier-Stokes, réduisant ainsi le pas en temps moyen, lors de la simulation numérique du développement d’instabilités hydrodynamiques. Ainsi, les équations anélastiques sont établies pour un mélange de deux fluides pour l’instabilité de Rayleigh-Taylor. La stabilité linéaire de l’écoulement est étudiée pour la première fois pour des fluides parfaits, par la méthode des modes normaux, dans le cadre de l’approximation anélastique. Le problème de Stokes issu des équations de Navier-Stokes sans les termes non linéaires (une partie de la poussée d’Archiméde est prise en compte) est défini ; l’éllipticité est démontrée, l’étude des modes propres et l’invariance liée à la pression sont détaillés. La méthode d’Uzawa est étendue à l’anélastique en mettant en évidence le découplage des vitesses en 3D, le cas particulier k = 0 et les modes parasites de pression. Le passage au multidomaine a permis d’établir les conditions de raccord (raccord Co de la pression sans condition aux limites physiques). Les algorithmes et l’implantation dans le code AMENOPHIS sont validés par les comparaisons de l’opérateur d’Uzawa développé en Fortran et à l’aide de Mathematica. De plus des résultats numériques ont été comparés à une expérience avec des fluides incompressibles. Finalement, une étude des solutions numériques obtenues avec les options anélastique et compressible a été menée. L’étude de l’influence de la stratification initiale des deux fluides sur le développement de l’instabilité de Rayleigh-Taylor est amorcée. / The « anelastic » approximation allows us to filter the acoustic waves thanks to an asymptotic development of the Navier-Stokes equations, so increasing the averaged time step, during the numerical simulation of hydrodynamic instabilitiesdevelopment. So, the anelastic equations for a two fluid mixture in case of Rayleigh-Taylor instability are established.The linear stability of Rayleigh-Taylor flow is studied, for the first time, for perfect fluids in the anelastic approximation.We define the Stokes problem resulting from Navier-Stokes equations without the non linear terms (a part of the buoyancyis considered) ; the ellipticity is demonstrated, the eigenmodes and the invariance related to the pressure are detailed.The Uzawa’s method is extended to the anelastic approximation and shows the decoupling speeds in 3D, the particular casek = 0 and the spurius modes of pressure. Passing to multidomain allowed to establish the transmission conditions.The algorithms and the implementation in the existing program are validated by comparing the Uzawa’s operator inFortran and Mathematica langages, to an experiment with incompressible fluids and results from anelastic and compressiblenumerical simulations. The study of the influence of the initial stratification of both fluids on the development of the Rayleigh-Taylor instability is initiated.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / O desenvolvimento de software livre de Jacobiana para a resolução de problemas formulados por equações diferenciais parciais não-lineares é de interesse crescente para simular processos práticos de engenharia. Este trabalho utiliza o chamado algoritmo espectral livre de derivada para equações não-lineares na simulação de fluxos em meios porosos. O modelo aqui considerado é aquele empregado para descrever o deslocamento do fluido compressível miscível em meios porosos com fontes e sumidouros, onde a densidade da mistura de fluidos varia exponencialmente com a pressão. O algoritmo espectral utilizado é um método moderno para a solução de sistemas não-lineares de grande porte, o que não resolve sistemas lineares, nem usa qualquer informação explícita associados com a matriz Jacobiana, sendo uma abordagem livre de Jacobiana. Problemas bidimensionais são apresentados, juntamente com os resultados numéricos comparando o algoritmo espectral com um método de Newton inexato livre de Jacobiana. Os resultados deste trabalho mostram que este algoritmo espectral moderno é um método confiável e eficiente para a simulação de escoamentos compressíveis em meios porosos. / The development of Jacobian-free software for solving problems formulated by nonlinear partial differential equations is of increasing interest to simulate practical engineering processes. This work uses the so-called derivative-free spectral algorithm for nonlinear equations in the simulation of flows in porous media. The model considered here is the one employed to describe the displacement of miscible compressible fluid in porous media with point sources and sinks, where the density of the fluid mixture varies exponentially with the pressure. The spectral algorithm used is a modern method for solving large-scale nonlinear systems, which does not solve linear systems, nor use any explicit information associated with the Jacobin matrix, being a Jacobian-free approach. Two dimensional problems are presented, along with numerical results comparing the spectral algorithm to a well-developed Jacobian-free inexact Newton method. The results of this paper show that this modern spectral algorithm is a reliable and efficient method for simulation of compressible flows in porous media.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / O desenvolvimento de software livre de Jacobiana para a resolução de problemas formulados por equações diferenciais parciais não-lineares é de interesse crescente para simular processos práticos de engenharia. Este trabalho utiliza o chamado algoritmo espectral livre de derivada para equações não-lineares na simulação de fluxos em meios porosos. O modelo aqui considerado é aquele empregado para descrever o deslocamento do fluido compressível miscível em meios porosos com fontes e sumidouros, onde a densidade da mistura de fluidos varia exponencialmente com a pressão. O algoritmo espectral utilizado é um método moderno para a solução de sistemas não-lineares de grande porte, o que não resolve sistemas lineares, nem usa qualquer informação explícita associados com a matriz Jacobiana, sendo uma abordagem livre de Jacobiana. Problemas bidimensionais são apresentados, juntamente com os resultados numéricos comparando o algoritmo espectral com um método de Newton inexato livre de Jacobiana. Os resultados deste trabalho mostram que este algoritmo espectral moderno é um método confiável e eficiente para a simulação de escoamentos compressíveis em meios porosos. / The development of Jacobian-free software for solving problems formulated by nonlinear partial differential equations is of increasing interest to simulate practical engineering processes. This work uses the so-called derivative-free spectral algorithm for nonlinear equations in the simulation of flows in porous media. The model considered here is the one employed to describe the displacement of miscible compressible fluid in porous media with point sources and sinks, where the density of the fluid mixture varies exponentially with the pressure. The spectral algorithm used is a modern method for solving large-scale nonlinear systems, which does not solve linear systems, nor use any explicit information associated with the Jacobin matrix, being a Jacobian-free approach. Two dimensional problems are presented, along with numerical results comparing the spectral algorithm to a well-developed Jacobian-free inexact Newton method. The results of this paper show that this modern spectral algorithm is a reliable and efficient method for simulation of compressible flows in porous media.

[pt] Umas das etapas mais complexas e críticas durante a construção de poços de petróleo é o processo de cimentação primária, definido como: O processo de instalação de cimento no anular entre o revestimento e a formação exposta ao poço [1]. Ela fornece isolamento zonal permanente para evitar contaminação ou migração de fluidos indesejáveis no anular, protege o revestimento da ocorrência de corrosão e fornece estabilidade hidráulica e mecânica para o revestimento ao longo da vida produtiva do poço de petróleo. Neste processo, ainda na etapa de perfuração, irregularidades na seção transversal (washouts) podem ser geradas como resultado de diversos colapsos parciais da seção do poço aberto em decorrência da presença de rochas pouco consolidadas da formação. Uma operação de cimentação primária bem sucedida dependerá se o sistema de fluidos espaçadores e a pasta de cimento deslocam de forma adequada e completa o fluido de perfuração do anular e washouts. Motivados por esse problema industrial, a presente dissertação usa um simulador numérico DNS (Direct Numerical Simulation) com o objetivo de analisar sistematicamente o comportamento hidrodinâmico e calcular a eficiência do deslocamento entre dois fluidos newtonianos miscíveis através de um anular contendo uma expansão seguida de uma contração abrupta. Investigamos como diferentes viscosidades e densidades dos fluidos, um injetado e outro deslocado, a miscibilidade entre eles, a taxa de injeção e a dimensão do washout retangular afetam o escoamento bifásico. Consideramos uma geometria axissimétrica durante processos de deslocamentos verticais, e as equações governantes são resolvidas em coordenadas cilíndricas, permitindo investigar diferentes aberturas anulares. Devido à miscibilidade entre os fluidos, nossos resultados preveem eficiências de deslocamento altíssimas, próximas a 100 Fluidos não-newtonianos são usualmente utilizados durante o processo industrial de cimentação primária de poços de petróleo. Contudo, as altas eficiências de deslocamentos encontradas em nossos resultados motivam estudos futuros sobre a influência da miscibilidade em deslocamentos de fluidos complexos. Estes resultados também motivam o uso de fluidos espaçadores para tentar controlar as propriedades de interface. Desta forma, é possível que a utilização de sistemas que se aproximem das condições reológicas e hidrodinâmicas de deslocamento entre fluidos newtonianos miscíveis poderá contribuir para um aumento da eficiência de deslocamento e, consequentemente, otimizar o processo de deslocamento de sistemas de fluidos, tendo em vista melhorias na integridade da cimentação de poços de petróleo. / [en] One of the most complex and critical stages during the construction ofoil wells is the primary cementing process, defined as the process of installingcement in the annulus between the casing and the exposed formation to thewell [1]. Primary cementing provides permanent zonal isolation to preventcontamination or migration of unwanted fluids in the annulus, protects thecasing from corrosion, and provides hydraulic and mechanical stability forthe casing throughout the productive life of the oil well. In this process,during the drilling stage, irregularities in the cross-section (washouts) can begenerated because of various partial collapses of the open wellbore section dueto the presence of poorly consolidated rocks in the formation. A successfulprimary cementing operation will depend on whether the spacer fluid systemand cement slurry adequately and completely displace the drilling fluid fromthe annulus and washouts.Motivated by this industrial problem, the present dissertation uses aDirect Numerical Simulation (DNS) numerical simulator to systematically analyze the hydrodynamic behavior and calculate the displacement efficiency between two miscible newtonian fluids through an annulus containing an expansion followed by an abrupt contraction. We investigate how different viscositiesand densities of the injected and displaced fluids, their miscibility, injectionrate, and the dimension of the rectangular washout affect the two-phase flow.We consider an axisymmetric geometry during vertical displacement processes,and the governing equations are solved in cylindrical coordinates, allowing theinvestigation of different annular clearances. Due to the miscibility betweenthe fluids, our results predict very high displacement efficiencies, close to 100Non-newtonian fluids are commonly used during the industrial processof primary cementing of oil wells. However, the high displacement efficienciesfound in our results motivate further studies on the influence of miscibilityin displacements of complex fluids. These results also encourage the use ofspacer fluids attempting to control the interfacial properties. Therefore, the useof systems that approximate the rheological and hydrodynamic conditions ofdisplacement between miscible newtonian fluids may contribute to an increasein displacement efficiency and, consequently, optimize the displacement processof fluid systems, aiming at improvements in the integrity of well cementing.