Since 1996, super-critical CO$_2$ has been injected at a rate of $\sim$0.85~Mt~yr$^{-1}$ into a pristine, saline aquifer at the Sleipner carbon capture and storage project. A suite of time-lapse, three-dimensional seismic reflection surveys have been acquired over the injection site. This suite includes a pre-injection survey acquired in 1994 and seven post-injection surveys acquired between 1999 and 2010. Nine consistently bright reflections within the reservoir, mapped on all post-injection surveys, are interpreted to be thin layers of CO$_2$ trapped beneath mudstone horizons. The areal extents of these CO$_2$ layers are observed to either increase or remain constant with time. However, volume flux of CO$_2$ into these layers has proven difficult to measure accurately. In addition, the complex planform of the shallowest layer, Layer 9, has proven challenging to explain using reservoir simulations. In this dissertation, the spatial distribution of CO$_2$ in Layer~9 is measured in three dimensions using a combination of seismic reflection amplitudes and changes in two-way travel time between time-lapse seismic reflection surveys. The CO$_2$ volume in this layer is shown to be growing at an increasing rate through time. To investigate CO$_2$ flow within Layer~9, a numerical gravity current model that accounts for topographic gradients is developed. This vertically-integrated model is computationally efficient, allowing it to be inverted to find reservoir properties that minimise differences between measured and modelled CO$_2$ distributions. The best-fitting reservoir permeability agrees with measured values from nearby wells. Rapid northward migration of CO$_2$ in Layer~9 is explained by a high permeability channel, inferred from spectral decomposition of the seismic reflection surveys. This numerical model is found to be capable of forecasting CO$_2$ flow by comparing models calibrated on early seismic reflection surveys to observed CO$_2$ distributions from later surveys. Numerical and analytical models are then used to assess the effect of the proximity of an impermeable base on the flow of a buoyant fluid, motivated by the variable thickness of the uppermost reservoir. Spatial gradients in the confinement of the reservoir are found to direct the flow of CO$_2$ when the current is of comparable thickness to the reservoir. Finally, CO$_2$ volume in the second shallowest layer, Layer~8, is measured using structural analysis and numerical modelling. CO$_2$ in Layer~8 is estimated to have reached the spill point of its structural trap by 2010. CO$_2$ flux into the upper two layers is now $\sim$40\% of total CO$_2$ flux injected at the base of the reservoir, and is increasing with time. This estimate is supported by observations of decreasing areal growth rate of the lower layers. The uppermost layers are therefore expected to contribute significantly to the total reservoir storage capacity in the future. CO$_2$ flow within Layer~9 beyond 2010 is forecast to be predominantly directed towards a topographic dome located $\sim$3~km north of the injection point. This dissertation shows that advances in determining the spatial distribution and flow of CO$_2$ in the sub-surface can be made by a combination of careful seismic interpretation and numerical flow modelling.

This investigation completed the verification of a three-dimensional resin transfer molding/resin film infusion (RTM/RFI) process simulation model. The model incorporates resin flow through an anisotropic carbon fiber preform, cure kinetics of the resin, and heat transfer within the preform/tool assembly. The computer model can predict the flow front location, resin pressure distribution, and thermal profiles in the modeled part. The formulation for the flow model is given using the finite element/control volume (FE/CV) technique based on Darcy's Law of creeping flow through a porous media. The FE/CV technique is a numerically efficient method for finding the flow front location and the fluid pressure. The heat transfer model is based on the three-dimensional, transient heat conduction equation, including heat generation. Boundary conditions include specified temperature and convection. The code was designed with a modular approach so the flow and/or the thermal module may be turned on or off as desired. Both models are solved sequentially in a quasi-steady state fashion. A mesh refinement study was completed on a one-element thick model to determine the recommended size of elements that would result in a converged model for a typical RFI analysis. Guidelines are established for checking the convergence of a model, and the recommended element sizes are listed. Several experiments were conducted and computer simulations of the experiments were run to verify the simulation model. Isothermal, non-reacting flow in a T-stiffened section was simulated to verify the flow module. Predicted infiltration times were within 12-20 percent of measured times. The predicted pressures were approximately 50 percent of the measured pressures. A study was performed to attempt to explain the difference in pressures. Non-isothermal experiments with a reactive resin were modeled to verify the thermal module and the resin model. Two panels were manufactured using the RFI process. One was a stepped panel and the other was a panel with two `T' stiffeners. The difference between the predicted infiltration times and the experimental times was 4 to 23 percent. / Master of Science

[pt] Este estudo procurou observar e quantificar os parâmetros operacionais que governam as propriedades permoporosas da torta de filtração, formada após o escoamento de uma solução de glicerina com uma determinada concentração de sólidos. A formação de um reboco externo de baixa permeabilidade é um dos fatores mais importantes para minimizar da invasão do filtrado de fluido na rocha reservatório. A contaminação do reservatório pelo filtrado do fluido pode trazer vários problemas operacionais, que serão discutidos nesta dissertação. A eficiência do sistema de fluidos em minimizar a invasão é normalmente avaliada através de ensaios padrão de filtração estática. Neste trabalho dois objetivos centrais são definidos: Identificar os parâmetros operacionais que governam as propriedades permoporosas do reboco externo através de ensaios de filtração estática e disponibilizar uma metodologia para a avaliação da invasão do filtrado do fluido de perfuração na geometria poço-reservatório (escoamento radial), a partir de ensaios de laboratório de filtração estática (escoamento linear). Os resultados indicam que a solução da lei Darcy para o problema de filtração com formação de torta incompressível mostrou - se adequada para grande maioria dos ensaios experimentais com solução de glicerina contendo agentes obturantes. O mesmo não se verificou para ensaios com solução de goma xantana como meio contínuo. Os resultados experimentais obtidos mostraram também que, para uma mesma solução de glicerina contendo agente obturante, os valores de permeabilidade da torta de filtração obtidos na geometria linear e na geometria radial são semelhantes. Desta forma, pode - se validar a metodologia de previsão do grau da invasão de fluidos de perfuração na rocha reservatório (configuração radial) a partir de ensaios convencionais de laboratório (configuração linear). / [en] This work deals with the understanding of the major operational parameters governing filter cake building drilling fluids invasion through reservoir rocks. The ability of the fluid system to prevent invasion is normally evaluated by standardized static filtration experiments. In these tests, the fluid is pressurized through a filter paper or into a consolidated inert porous medium. The volume which crosses the porous core is monitored along the time. Darcy flow modeling of non-compressible cakes proved to reproduce adequately the filtration of a Newtonian fluid + particulate system through ceramic and sinterized steel disks. Pressure differential, particle size and shape proved to be relevant parameters affecting filter cake permeability and porosity. The present study proposes, through the coupling of a linear filtration formulation (lab configuration) and a radial single phase formulation (wellbore vicinity), to predict fluid invasion depth of fluid filtrate in the reservoir rock. Modeling is validated with linear and radial lab tests. The proposed methodology is a requirement for optimum drilling fluid design to be used in the drilling of reservoir sections in both exploratory and development wells.

[pt] O escoamento de emulsões é encontrado em diversos processos de recuperação e produção de petróleo. O escoamento de emulsões em meios porosos depende de diversos parâmetros como a relação do tamanho das gotas ao tamanho dos poros, a razão de viscosidades, a vazão volumétrica e o efeito destes parâmetros ainda não é bem compreendido. Uma análise detalhada na escala microscópica dos fenômenos envolvidos se faz essencial para a melhora do entendimento completo do escoamento de emulsões em um reservatório. Isto permitiria o desenvolvimento de melhores modelos de simulação para o escoamento multifísico em meios porosos. Neste trabalho, o escoamento de emulsões óleo-água através de um capilar com garganta foi estudado através de experimentos e teoria. A análise experimental consistiu da visualização sob um microscópio do escoamento e da medição da queda de pressão em função da vazão para diferentes emulsões. A análise teórica englobou o estudo do escoamento em regime permanente de uma gota de óleo imersa em água através de um capilar e o estudo do escoamento transiente da mesma gota através de um capilar com uma garganta. Os resultados mostram que os modelos de escoamento de emulsões em meios porosos não devem ser baseados em propriedades macroscópicas da emulsão quando o tamanho das gotas da fase dispersa for da mesma ordem de grandeza do tamanho dos poros. Neste caso, a queda de pressão é função da tensão interfacial, a razão de viscosidades, a vazão e a razão entre o tamanho das gotas e o diâmetro do poro. Os resultados apresentados neste trabalho podem ser usados no projeto de emulsões apropriadas para controle de mobilidade em operações de recuperação avançada através de injeção de emulsões. / [en] Flow of emulsions is found in many petroleum recovery and production processes and it is often referred to in the context of tertiary oil recovery. The characteristics of emulsion flow in porous media depend on several parameters such as medium drop size to pore size ratio, viscosity ratio, flow rate and the effect of these parameters is far from being entirely understood. A detailed analysis at a microscopic scale of the flow is essential to improve the understanding of flow of an emulsion in a reservoir. This would lead to the development of better simulation models, henceforth increasing the predictability capability of reservoir simulators for enhanced oil recovery applications. In this work, flow of oil-water emulsions through constricted capillaries, used as model for the geometry inside a porous media, is studied experimentally and theoretically. The experimental approach consisted of measuring pressure drop response as a function of flow rate for different emulsions and visualizing the flow under an optical microscope to understand the phenomena involved. The theoretical approach is divided in two parts. First, the immiscible steady flow of a infinite single drop suspended in an less viscous fluid through a capillary was analyzed by solving the Navier- Stokes equations with the appropriate boundary conditions for free-surface flow. The second part of the theoretical analysis consisted of solving the transient flow of a drop suspended in a less viscous fluid through a capillary with a constriction. It is shown the effect of capillary number and viscosity ratio over the main responses of the flow. The results show that models of emulsion flow in a porous media cannot be based on the macroscopic properties of the emulsion when the drop diameter is of the same order of magnitude as the pore throat diameter. In this case flow rate-pressure drop is a strong function of the interfacial tension, viscosity ratio, flow rate and drop to pore size ratio. The results can be used to design appropriate emulsions to control the water mobility during EOR operations by emulsion injection.

Submitted by Maria Cristina (library@lncc.br) on 2015-04-07T18:12:16Z No. of bitstreams: 1 tese_Yoisell-Rodriguez-Nunez[versao-final].pdf: 18140163 bytes, checksum: c8fa79a9fb3973c70ccce56923608a18 (MD5) / Approved for entry into archive by Maria Cristina (library@lncc.br) on 2015-04-07T18:12:32Z (GMT) No. of bitstreams: 1 tese_Yoisell-Rodriguez-Nunez[versao-final].pdf: 18140163 bytes, checksum: c8fa79a9fb3973c70ccce56923608a18 (MD5) / Made available in DSpace on 2015-04-07T18:12:43Z (GMT). No. of bitstreams: 1 tese_Yoisell-Rodriguez-Nunez[versao-final].pdf: 18140163 bytes, checksum: c8fa79a9fb3973c70ccce56923608a18 (MD5) Previous issue date: 2014-09-04 / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (Capes) / Fundação de Apoio ao Desenvolvimento da Computação Científica (FACC) / The numerical simulation of incompressible miscible displacements in porous media has obtained significant progress in the last decades being an useful tool in several areas of interest. In particular, in the oil industry the use of numerical simulation allows to obtain qualitative and quantitative data that may provide a better understanding of the physical and chemical processes that occur, for example, in petroleum reservoirs. The system of partial differential equations governing miscible displacement consists of an elliptic subsystem stemming from conservation of mass, Darcy’s law and a transport equation expressing the conservation of the injected fluid (concentration). Although the concentration is the variable of greatest interest, the calculation of the velocity field requires special attention since it is responsible for transporting the mixture and, therefore, its production. Besides, it has a strong influence on the stability and accuracy of the transport equation when we are dealing with adverse mobility ratios, that is, the solvent is less viscous than the resident oil. These facts motivate us to seek for an efficient and accurate numerical method for the calculation of the velocity field (Darcy’s system) in order to reduce inaccuracies in calculating the concentration. In this thesis we propose a Stabilized Dual Hybrid Mixed (SDHM) method to approximate the Darcy subsystem. It is shown that this methodology is stable with usual finite element approximations, such as Lagrangian polynomial approximations, where all variables can be interpolated by equal-order functions. Furthermore, the SDHM method is conservative with appropriate parameters choices, and more accurate and robust when compared to the Galerkin method and postprocessing techniques. In order to verify the efficiency of the SDHM method, computer simulations are presented for the recovery processes of reservoirs for patterns flow problems in homogeneous and heterogeneous porous media, such as tracer injection and continuous injection. The SDHM method is employed together with a combination of the SUPG method in the spatial discretization, and an implicit finite difference scheme in the time for the concentration approximation of the transport equation. To verify the proposed methodology a semi-analytical approach is also employed which combines the SDHM approximation for velocity field with the concentration calculated analytically by the streamline method. The results obtained with the proposed formulation showed to be efficient, accurate and free of spurious oscillations even for highly heterogeneous scenarios, where we consider random permeability and adverse mobility ratios. / A simulação numérica de deslocamentos miscíveis incompressíveis em meios porosos obteve avanços significativos nas últimas décadas, sendo uma ferramenta útil em várias áreas de interesse. Em particular, na indústria do petróleo, a utilização de simulação numérica permite obter dados qualitativos e quantitativos que podem proporcionar uma melhor compreensão dos processos físicos e químicos que ocorrem, por exemplo, em reservatórios de petróleo. O sistema de equações diferenciais parciais que regem o deslocamento miscível consiste em um subsistema elíptico, decorrente de conservação de massa, a lei Darcy, e uma equação de transporte, que expressa a conservação do fluido injetado(concentração). Embora a concentração é a variável de maior interesse, o cálculo do campo de velocidade requer uma atenção especial, uma vez que é responsável pelo transporte da mistura e, por conseguinte, a produção da mesma. Além disso,o campo de velocidades tem uma forte influência sobre a estabilidade e a precisão da equação de transporte quando estamos lidando com razões de mobilidade adversas. Esses fatos nos motivam a procurar um método numérico eficiente e preciso para o cálculo do campo de velocidade (subsistema de Darcy), a fim de reduzir as viii imprecisões no cálculo da concentração. Nesta tese propomos um método Misto Híbrido Dual Estabilizado (MHDE) para aproximar o subsistema de Darcy. Mostra-se que essa metodologia é estável com aproximações de elementos finitos usuais, tais como aproximações polinomiais de Lagrange, onde todas as variáveis podem ser interpoladas por funções de igual ordem. Além disso, o método MHDE é conservativo, para escolhas adequadas dos parâmetros de estabilização, e mais preciso e robusto em comparação ao método de Galerkin e técnicas de pós-processamento. Com o intuito de verificar a eficiência do método MHDE, são apresentadas simulações computacionais para os processos de recuperação de reservatórios, para problemas padrões de escoamentos em meios porosos homogêneos e heterogêneos, tais como: injeção de traçadores e injeção continua. É empregado o método MHDE juntamente com uma combinação do método SUPG, na discretização espacial, e um esquema implícito de diferenças finitas no tempo, para a aproximação da concentração da equação de transporte. Outra abordagem, denominada semianalítica,utilizada na validação da metodologia proposta, consiste na combinação do campo de velocidades obtidos pelo método MHDE com a concentração calculada analiticamente pelo método das linhas de fluxo. Os resultados obtidos com a formulação proposta mostram-se eficientes, acurados e livres de oscilações espúrias até para os cenários altamente heterogêneos, onde consideramos permeabilidades aleatórias e razões de mobilidades adversas.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-05T19:26:54Z No. of bitstreams: 1 JULIANA ARAGÃO DE ARAÚJO - DISSERTAÇÃO PPGMAT 2005..pdf: 3880310 bytes, checksum: 796ad7c042587fcea49b37b64d7b2fc1 (MD5) / Made available in DSpace on 2018-07-05T19:26:54Z (GMT). No. of bitstreams: 1 JULIANA ARAGÃO DE ARAÚJO - DISSERTAÇÃO PPGMAT 2005..pdf: 3880310 bytes, checksum: 796ad7c042587fcea49b37b64d7b2fc1 (MD5) Previous issue date: 2005-05 / Neste trabalho é apresentada a solução do problema de Riemann associado a um sistema de leis de conservação. Este sistema é proveniente de um escoamento bifásico unidimensional em meios porosos e considera os efeitos de histerese nas curvas de permeabilidade das fases. A principal contribuição deste trabalho é que a solução do problema de Riemann é obtida para um modelo que leva em conta a histerese nas duas fases e que considera as curvas de embebição e de drenagem distintas sempre que haja uma reversão de regime de drenagem para embebição e vice-versa. Os resultados obtidos aqui ampliam aqueles obtidos para um modelo mais simplificado em que a histerese era considerada apenas numa das fases e as curvas de permeabilidade eram tomadas coincidentes após urna segunda reversão. Uma vez obtida a solução do problema de Riemann, base para a construção de simuladores numéricos de alta precisão, é feita uma comparação entre esta solução e aquela obtida anteriormente, para os mesmo dados iniciais, mostrando que não só as velocidades de ondas podem ser distintas, mas também as próprias sequências de ondas que formam tais soluções. / In this work we prasent the Riemann solution for a system of conservation laws associatcd to an unidimensional two-ph&sc fiow in a porous media taking into account the hysteresis effects on the permeability curves. Our main contribution in this work is that the solution of the Riemann problem is obtained for a model that takes into account the hysteresis in both wetting and non-wetting phases and considers the scanning curves of embebition and drainagc distincts whenever there is a reversion of regime. The results obtained here improve those obtained for a simplified model where hysteresis is considered only on the non-wetting phase and the scanning curves coincide aftcr a second reversion of regime. Once obtained the solution of the Riemann problem, which is basic for the construction of high aceurate numeric simulators, we compare this solution and that one already obtained, for the same initial data. showing that not only the speeds of waves can be distinct, but also the sequences of waves in such Solutions.

La compréhension des écoulements biphasiques dans des milieux hétérogènes est déterminante pour un certain nombre de processus industriels. Le plus marquant étant la récupération assistée du pétrole piégé dans des roches poreuses,qui est un moteur pour la recherche dans ce domaine depuis plusieurs décennies.Pourtant, si l’imbibition spontanée de liquides dans un milieu poreux est relativement bien comprises, l’imbibition forcée d’un liquide dans un milieu poreux rempli d’un liquide plus visqueux n’a pas encore été décrite quantitativement. En combinant des expériences de microfluidique et d’imagerie confocale, nous avons étudié la morphologie et la dynamique d’imbibition forcée dans des milieux poreux réguliers. Nous avons mis en évidence la présence de quatre régimes d’imbibition,provenant de dynamiques d’invasion différentes à l’échelle du pore et ayant une signature claire sur les observables macroscopiques. L’étude de la dynamique locale associée à la microscopie confocale, qui donne accès à des informations tridimensionnelles,nous a permis de montrer que les transitions entre ces régimes étaient la conséquence de deux transitions de mouillage et d’une instabilité d’interface.De plus, contrairement à la majorité des études menées jusqu’à présent, nous nous sommes intéressés à l’évolution de ces motifs d’imbibition soumis à une injection continue sur des temps longs devant le temps de percolation. Cela a permis de mettre en évidence le vieillissement de ces motifs qui est le résultat d’un mûrissement d’origine capillaire. / Understanding two-phase flow in heterogeneous media is of great importance for a number of industrial processes. One of the most prominent examples is enhanced oil recovery which has driven fundamental and applied research in this fieldfor decades. However our understanding has remained extremely unbalanced. The case of spontaneous imbibition of a fluid in a porous medium is fairly well understood,whereas quantitative descriptions of forced imbibition in the presence of anunfavorable viscosity ratio is still lacking. Combining large-scale observations and confocal imaging of microfluidic experiments, we studied the morphology and dynamicsof forced imbibition in homogeneous porous media. We identify four classesof three-dimensional patterns resulting from different dynamics at the pore-scale, and having a clear signature on the macroscopic observables. By means of confocalmicroscopy allowing us to visualize three dimensional features of the local dynamics,we show that the transitions between the four imbibition scenarios result from two dynamical wetting transitions and one interfacial instability. In addition,unlike previous studies, we investigate the evolution of those patterns undercontinuous injection over long time scales. We evidence their aging according to acapillary-coarsening process.

Orientador: Caio Gorla Nogueira / Resumo: Este trabalho apresenta um conjunto de técnicas estatísticas básicas aplicadas à modelagem de problemas de fluxo em meios porosos fraturados e de estabilidade de taludes, com o objetivo de identificar as variáveis explicativas mais influentes sobre a variabilidade das variáveis resposta. Diferentes planejamentos de experimentos foram utilizados para possibilitar a construção de metamodelos polinomiais representativos dos fenômenos estudados. Uma modificação do planejamento do tipo Box-Behnken é apresentada e foi proposta pelo autor para analisar problemas que envolvem elevado número de variáveis explicativas (e.g. 30). Os metamodelos, obtidos via método dos mínimos quadrados, são também chamados de superfícies de resposta ou modelos de regressão e são indispensáveis à verificação da sensibilidade das variáveis explicativas. O conjunto de técnicas mostrou- se muito eficaz na identificação das variáveis explicativas que provocaram efeitos mais significativos sobre a variável resposta. Evidenciou-se também, por meio dos exemplos de estabilidade de taludes tratados, a possibilidade de se quantificar incertezas com o uso de metamodelos suficientemente adequados, uma opção que pode ser bastante útil no processo de quantificação de incertezas de problemas que não possuem soluções analíticas simples. / Abstract: This paper presents a set of basic statistical techniques applied to the modeling of flow problems in fractured porous and slope stability media, aiming to identify the most influential explanatory variables on the response variables variability. Different designs of experiments were used to enable the construction of polynomial metamodels representative of the studied phenomena. A Box-Behnken type design modification is presented and was proposed by the author to analyze problems involving high number of explanatory variables (e.g. 30). The metamodels, obtained by the least squares method, are also called response surfaces or regression models and are indispensable for verifying the sensitivity of the explanatory variables. The set of techniques was very effective in identifying the explanatory variables that had the most significant effects on the response variable. It was also evidenced, through the examples of stability of treated slopes, the possibility of quantifying uncertainties using sufficiently adequate metamodels, an option that can be very useful in the process of quantifying uncertainty of problems that do not have simple analytical solutions. / Mestre

Nous construisons une méthode numérique fiable pour simuler un écoulement dans un milieu poreux modélisé par une équation elliptique. La simulation est rendue difficile par les hétérogénéités du milieu, la taille et la géométrie complexe du domaine de calcul. Un maillage d'hexaèdres réguliers ne permet pas de représenter fidèlement les couches géologiques du domaine. Par conséquent, nous sommes amenés à travailler avec un maillage de cubes déformés. Il existe différentes méthodes de volumes finis ou d'éléments finis qui résolvent ce problème avec plus ou moins de succès. Pour la méthode que nous proposons, nous nous imposons d'avoir seulement un degré de liberté par maille pour la pression et un degré de liberté par face pour la vitesse de Darcy, pour rester au plus près des habitudes des codes industriels. Comme les méthodes d'éléments finis mixtes standards ne convergent pas, notre méthode est basée sur un élément fini mixte composite. En deux dimensions, une maille polygonale est découpée en triangles en ajoutant un point au barycentre des sommets, et une expression explicite des fonctions de base a pu être obtenue. En dimension 3, la méthode s'étend naturellement au cas d'une maille pyramidale. Dans le cas d'un hexaèdre ou d'un cube déformé quelconque, la maille est divisée en 24 tétraèdres en ajoutant un point au barycentre des sommets et en divisant les faces en 4 triangles. Les fonctions de base de l'élément sont alors construites en résolvant un problème discret. Les méthodes proposées ont été analysées théoriquement et complétées par des estimateurs a posteriori. Elles ont été expérimentées sur des exemples académiques et réalistes en utilisant le calcul parallèle. / We develop a reliable numerical method to approximate a flow in a porous media, modeled by an elliptic equation. The simulation is made difficult because of the strong heterogeneities of the medium, the size together with complex geometry of the domain. A regular hexahedral mesh does not allow to describe accurately the geological layers of the domain. Consequently, this leads us to work with a mesh made of deformed cubes. There exists several methods of type finite volumes or finite elements which solve this issue. For our method, we wish to have only one degree of freedom per element for the pressure and one degree of freedom per face for the Darcy velocity, to stay as close to the habits of industrial software. Since standard mixed finite element methods does not converge, our method is based on composite mixed finite element. In two dimensions, a polygonal mesh is split into triangles by adding a node to the vertices's barycenter, and explicit formulation of the basis functions was obtained. In dimension 3, the method extend naturally to the case of pyramidal mesh. In the case of a hexahedron or a deformed cube, the element is divided into 24 tetrahedra by adding a node to the vertices's barycenter and splitting the faces into 4 triangles. The basis functions are then built by solving a discrete problem. The proposed methods have been theoretically analyzed and completed by a posteriori estimators. They have been tested on academical and realistic examples by using parallel computation.

Experimental data convincingly show that viscosity of a fluid may change significantly with pressure. This observation leads to various generalizations of well-known models, like Darcy's law, Stokes' law or the Navier-Stokes equations, among others. This thesis investigates three such models in a series of three published papers. Their unifying topic is development of existence theory and finding a weak solution to systems of partial differential equations stemming from the considered models.