The theory of homogenization allows to find for a given system of partial differential equations governing a model with a very complicated internal struc- ture a system governing a model without this structure, whose solution is in a certain sense an approximation of the solution of the original problem. In this thesis, methods of the theory of homogenization are applied to three sys- tems of partial differential equations. The first one governs a flow of a class of non-Newtonian fluid through a porous medium. The second system is utilized for modeling of a flow of a fluid through an electric field wherein the viscosity depends significantly on the intensity of the electric field. For the third system is considered an elliptic operator having growth and coercivity indicated by a general anisotropic inhomogeneous N-function. 1

No description available.

[pt] A injeção de emulsão em meio poroso como método de recuperação avançada pode se tornar realidade na operação de campos de petróleo devido à maior rigidez no descarte de água produzida e aos potenciais ganhos na produção de óleo. Para entender o comportamento macroscópico desta técnica de EOR, é necessário compreender o fenômeno microscópico. Com este objetivo, fez-se a modelagem numérica do escoamento de uma gota imersa em fase contínua escoando em capilar restrito utilizando o método de Lattice-Boltzmann. Este método foi escolhido devido à sua facilidade de ser aplicado em geometrias complexas de rocha e fluido e ao bom compromisso na representação dos fenômenos de microescala. Para entender a influência de cada parâmetro, foram realizadas diversas simulações em domínio tridimensional, alterando a velocidade do fluxo, a razão de viscosidades dos fluidos, a relação entre os diâmetros da gota e do tubo e a magnitude da tensão interfacial. Observou-se que a passagem da gota pela restrição causa uma redução na mobilidade do escoamento, representada por um aumento na perda de carga, pela conjunção dos efeitos viscosos e capilares. Obteve-se correlação negativa do fator redutor de mobilidade 𝑓 com a razão de viscosidade e com o tamanho da gota, assim como fora determinado numericamente por Roca-Reyes (2011) com o método level-set. Foi notada uma pequena sensibilidade de 𝑓 ao número de capilaridade, assim como estabelecido experimentalmente por Robles-Castillo (2011). Verificou-se a importância de se determinar o conjunto adequado de parâmetros do sistema para ter sucesso na implantação de injeção de emulsões. / [en] Emulsion injection in porous medium as an Enhanced Oil Recovery method can turn out to be reality in the operation of onshore and offshore fields, due to increasing rigidity in the disposal of produced water and also due to the potential additional oil production. In order to understand macroscopic behavior of this EOR method, it is necessary to understand the microscopic phenomenon. With this objective, it was performed the numerical modeling of the flow of a droplet immerse in continuous phase through a constricted capillary using the Lattice-Boltzmann method. This method was chosen due to its simplicity on being applied to complex rock geometries and multiphasic flow and due to its good commitment in representing microscopic phenomena. Focusing on understanding the influence of each parameter on flow behavior, several simulation studies were performed altering flow velocity, viscosity ratio, ratio between droplet s and pipe s diameter and interfacial tension. A reduction in mobility is observed as the droplet passes through the restriction caused by the conjunction of viscous and capillary effects. A negative correlation of mobility reduction factor 𝑓 in relation to the viscosity ratio and to droplet size was noticed, as it had been observed before by Roca-Reyes (2011) in a numerical implementation of level-set method. Weak correlation between 𝑓 and capillary number was determined, as in previous experimental essay performed by Robles-Castillo (2011). In this study, it was verified the importance of determining the appropriate set of system parameters, in order to achieve success in the implementation of emulsion injection.

Extremely small pore size, low porosity, and ultra-low permeability are among the characteristics of shale rocks. In tight shale reservoirs, the nano-confinement effects that include large gas-oil capillary pressure and critical property shifts could alter the phase behaviors, thereby affecting the oil or gas production. In this research, two in-house simulation models, i.e., a compositionally extended black-oil model and a fully composition model are developed to examine the nano-pore confinement effects on convective and diffusive multicomponent multiphase fluid transport. Meanwhile, the effect of nano-confinement and rock intrinsic properties (porosity and tortuosity factor) on predicting effective diffusion coefficient are investigated. First, a previously developed compositionally extended black-oil simulation approach is modified, and extended, to include the effect of large gas-oil capillary pressure for modeling first contact miscible (FCM), and immiscible gas injection. The simulation methodology is applied to gas flooding in both high and very low permeability reservoirs. For a high permeability conventional reservoir, simulations use a five-spot pattern with different reservoir pressures to mimic both FCM and immiscible displacements. For a tight oil-rich reservoir, primary depletion and huff-n-puff gas injection are simulated including the effect of large gas-oil capillary pressure in flow and in flash calculation on recovery estimations. A dynamic gas-oil relative permeability correlation that accounts for the compositional changes owing to the produced gas injection is introduced and applied to correct for changes in interfacial tension (IFT), and its effect on oil recovery is examined. The results show that the simple modified black-oil approach can model well both immiscible and miscible floods, as long as the minimum miscibility pressure (MMP) is matched. It provides a fast and robust alternative for large-scale reservoir simulation with the purpose of flaring/venting reduction through reinjecting the produced gas into the reservoir for EOR. Molecular diffusion plays an important role in oil and gas migration in tight shale formations. However, there are insufficient reference data in the literature to specify the diffusion coefficients within porous media. Another objective of this research is to estimate the diffusion coefficients of shale gas, shale condensate, and shale oil at reservoir conditions with CO2 injection for EOR/EGR. The large nano-confinement effects including large gas-oil capillary pressure and critical property shifts could alter the phase behaviors. This study estimates the diffusivities of shale fluids in nanometer-scale shale rock from two perspectives: 1) examining the shift of diffusivity caused by nanopore confinement effects from phase change (phase composition and fluid property) perspective, and 2) calculating the effective diffusion coefficient in porous media by incorporating rock intrinsic properties (porosity and tortuosity factor). The tortuosity is obtained by using tortuosity-porosity relations as well as the measured tortuosity of shale from 3D imaging techniques. The results indicated that nano-confinement effects could affect the diffusion coefficient through altering the phase properties, such as phase compositions and densities. Compared to bulk phase diffusivity, the effective diffusion coefficient in porous shale rock is reduced by 102 to 104 times as porosity decreases from 0.1 to 0.03. Finally, a fully compositional model is developed, which enables us to process multi-component multi-phase fluid flow in shale nano-porous media. The validation results for primary depletion, water injection, and gas injection show a good match with the results of a commercial software (CMG, GEM). The nano-confinement effects (capillary pressure effect and critical property shifts) are incorporated in the flash calculation and flow equations, and their effects on Bakken oil production and Marcellus shale gas production are examined. The results show that including oil-gas capillary pressure effect could increase the oil production but decrease the gas production. Inclusion of critical property shift could increase the oil production but decrease the gas production very slightly. The effect of molecular diffusion on Bakken oil and Marcellus shale gas production are also examined. The effect of diffusion coefficient calculated by using Sigmund correlation is negligible on the production from both Bakken oil and Marcellus shale gas huff-n-puff. Noticeable increase in oil and gas production happens only after the diffusion coefficient is multiplied by 10 or 100 times. / Doctor of Philosophy / Shale reservoir is one type of unconventional reservoir and it has extremely small pore size, low porosity, and ultra-low permeability. In tight shale reservoirs, the pore size is in nanometer scale and the oil-gas capillary pressure reaches hundreds of psi. In addition, the critical properties (such as critical pressure and critical temperature) of hydrocarbon components will be altered in those nano-sized pores. In this research, two in-house reservoir simulation models, i.e., a compositionally extended black-oil model and a fully composition model are developed to examine the nano-pore confinement effects on convective and diffusive multicomponent multiphase fluid transport. The large nano-confinement effects (large gas-oil capillary pressure and critical property shifts) on oil or gas production behaviors will be investigated. Meanwhile, the nano-confinement effects and rock intrinsic properties (porosity and tortuosity factor) on predicting effective diffusion coefficient are also studied.

Cette thèse est consacrée à l'étude de méthodes numériques pour la simulation des écoulements polyphasiques en milieu poreux, en vue de leur application à des problèmes d'ingénierie pétrolière ou environnementale. Nous présentons une formulation générique du modèle d'écoulements à nombre quelconque de composants présents dans un nombre quelconque de phases. Dans notre approche l'approximation des flux diffusifs (issus, par exemple, de la loi de Darcy) s'appuie sur de nouveaux schémas, appelés schémas gradient, qui ont plusieurs avantages sur les schémas industriels standard : ces derniers, qui sont des schémas volumes finis multi-points centrés aux mailles, ne sont généralement pas symétriques et convergent difficilement sur des cas à forts rapports d'anisotropie. Nous montrons en revanche que les schémas gradient conduisent naturellement à des approximations symétriques et convergentes. Parmi cette classe de schémas, nous étudions plus particulièrement le schéma "VAG" qui fait intervenir des inconnues au centre des mailles et aux sommets du maillage. Ce schéma conduit à la définition de flux entre le centre d'une maille et ses sommets, qui sont utilisés pour généraliser la méthode "VAG" au contexte polyphasique. Des tests numériques montrent alors que ce schéma est robuste, et conduit à un très bon compromis précision/coût, ce qui en fait un candidat idoine pour les applications industrielles. Nous présentons notamment un cas test, basé sur des observations de terrains, d'injection et de dissolution de CO2 dans la région proche d'un puits foré dans un aquifère salin. Nous montrons alors que le schéma numérique permet de simuler l'assèchement et la précipitation de minéral observée en pratique. Un chapitre de la thèse est enfin consacré à l'étude pratique et théorique d'une méthode numérique générique pour contrôler l'effet d'axe lors de l'utilisation de schémas industriels / This thesis is focused on numerical methods dedicated to the simulation of multiphase flow in porous media, involved in petroleum or environmental engineering. We present a generic formulation of the flow model which is able to take into account any number of components within any number of phases. In our approach the approximation of the diffusive fluxes (mainly resulting from Darcy's law) is based on new schemes, called gradient schemes, which show several advantages over the standard industrial numerical schemes : these schemes, which belong to the class of the cell-centred MultiPoint Flux Approximation finite volume schemes, are not symmetric and may lead to difficulties of convergence in the case of high anisotropy ratios. We indeed show that gradient schemes are naturally providing symmetric and convergent approximations. We particularly study the Vertex Approximate Gradient scheme (called the VAG scheme), whose discrete unknowns are the values at the cell centres and at the vertices of the mesh. This scheme implies the definition of fluxes between the centre of a given cell and its vertices, used for the extension of the scheme to multiphase flow. Numerical tests show the robustness and the accuracy of the method for a low computational cost, which enables the use of the VAG scheme in an industrial framework. A test case, based on experimental data of injection and dissolution of CO2 in the near-well region within a saline aquifer, shows the aptitude of the scheme for reproducing drying and salt precipitation, which are practically observed. Finally, a chapter of the thesis is devoted to the theoretical and practical study of a general numerical method for controlling Grid Orientation Effect in industrial simulators

Neste trabalho é apresentada uma nova modelagem matemática para a descrição do escoamento de um líquido incompressível através de um meio poroso rígido homogêneo e isotrópico, a partir do ponto de vista da Teoria Contínua de Misturas. O fenômeno é tratado como o movimento de uma mistura composta por três constituintes contínuos: o primeiro representando a matriz porosa, o segundo representando o líquido e o terceiro representando um gás de baixíssima densidade. O modelo proposto possibilita uma descrição matemática realista do fenômeno de transição insaturado/saturado a partir de uma combinação entre um sistema de equações diferenciais parciais e uma desigualdade. A desigualdade representa uma limitação geométrica oriunda da incompressibilidade do líquido e da rigidez do meio poroso. Alguns casos particulares são simulados e os resultados comparados com resultados clássicos, mostrando as consequências de não levar em conta as restrições inerentes ao problema. / This work is concerned with a new mathematical modelling for describing the flow of an incompressible fluid (a liquid) through a rigid, homogeneous and isotropic porous medium, from a Continuum Mixtures point of view. The phenomenon is regarded as the motion of a mixture composed by three overlaping continuous constituents: the first one representing the porous matrix, the second one representing the liquid and the third one representing a (very) low density gas. The proposed mathematical modelling allows a realistic mathematical description for the unsaturated/saturated transition process by means of a combination between a system of partial differential equations and an inequality. This inequality represents a geometrical constraint arising from the liquid incompressibility merged with the porous matrix rigidity. The simulation of some interesting particular cases is carried out presenting a comparison between the obtained results and the classical ones, showing the consequences of disregarding the constraints associated to the phenomenon.

Neste trabalho é apresentada uma nova modelagem matemática para a descrição do escoamento de um líquido incompressível através de um meio poroso rígido homogêneo e isotrópico, a partir do ponto de vista da Teoria Contínua de Misturas. O fenômeno é tratado como o movimento de uma mistura composta por três constituintes contínuos: o primeiro representando a matriz porosa, o segundo representando o líquido e o terceiro representando um gás de baixíssima densidade. O modelo proposto possibilita uma descrição matemática realista do fenômeno de transição insaturado/saturado a partir de uma combinação entre um sistema de equações diferenciais parciais e uma desigualdade. A desigualdade representa uma limitação geométrica oriunda da incompressibilidade do líquido e da rigidez do meio poroso. Alguns casos particulares são simulados e os resultados comparados com resultados clássicos, mostrando as consequências de não levar em conta as restrições inerentes ao problema. / This work is concerned with a new mathematical modelling for describing the flow of an incompressible fluid (a liquid) through a rigid, homogeneous and isotropic porous medium, from a Continuum Mixtures point of view. The phenomenon is regarded as the motion of a mixture composed by three overlaping continuous constituents: the first one representing the porous matrix, the second one representing the liquid and the third one representing a (very) low density gas. The proposed mathematical modelling allows a realistic mathematical description for the unsaturated/saturated transition process by means of a combination between a system of partial differential equations and an inequality. This inequality represents a geometrical constraint arising from the liquid incompressibility merged with the porous matrix rigidity. The simulation of some interesting particular cases is carried out presenting a comparison between the obtained results and the classical ones, showing the consequences of disregarding the constraints associated to the phenomenon.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-24T14:45:11Z No. of bitstreams: 1 LUCIANO MARTINS BARROS - DISSERTAÇÃO PPGMAT 2010..pdf: 2620845 bytes, checksum: dc95731aa66d4dab123e738a1dc6b49c (MD5) / Made available in DSpace on 2018-07-24T14:45:11Z (GMT). No. of bitstreams: 1 LUCIANO MARTINS BARROS - DISSERTAÇÃO PPGMAT 2010..pdf: 2620845 bytes, checksum: dc95731aa66d4dab123e738a1dc6b49c (MD5) Previous issue date: 2010-03 / Neste trabalho obtivemos uma solução do problema de Riemann associado a um sistema de duas leis de conservação proveniente da modelagem matemática de um escoamento trifásico num meio poroso. Consideramos o caso de um reservatório petrolífero contendo inicialmente uma mistura arbitrária do tipo gás/óleo à ser deslocada pela injeção de uma mistura do tipo água/gás, também arbitrária. Usando uma combinação de métodos analíticos e computacionais determinamos a geometria das chamadas curvas de onda sob a condição de entropia de viscosidade, com matriz de viscosidade sendo a identidade. Determinamos todas as possíveis sequências de ondas que descrevem o escoamento para cada par de misturas de injeção e de produção representandoosdadosdeRiemann. Mostramosqueparadadosdeproduçãorepresentando uma mistura próxima de óleo puro, ou de gás puro, apenas duas ondas estão presentes no escoamento, independentemente da mistura de injeção. No entanto, para dados de produção representando uma melhor proporção gás/óleo mostramos a existência de uma faixa de dados de injeção para a qual três ondas estão presentes no escoamento, uma delas sendo uma onda de choque transicional. / In this work we describe a Riemann solution for a system of two conservation laws modeling a three-phase flow in a porous media. We consider the case where a petroleum reservoir is initially filled with an arbitrary gas/oil mixture to be displaced by the injection of a gas/water mixture, also arbitrary. By using a combination of analytical and computational methods we obtain the geometry of the so called wave curves under the viscous profile entropy condition, with the viscosity matrix as the identity. We determine all wave sequences describing the flow for each pair of injection andproductionmixtures,representingtheRiemanndata. Weshowthatforproduction mixture data close to pure oil, or pure gas, only two waves are present in the flow independentlyontheinjectionmixture. Nevertheless, forproductiondatarepresenting a more proportional gas/oil mixture we show the existence of a injection data range for which three waves are present in the flow, one of them being a transitional shock wave.

Le but de cette thèse est de faire l'étude de méthodes de volumes finis pour des équations aux dérivées partielles déterministes et stochastiques; nous effectuons des simulations numériques et démontrons des résultats de convergence d'algorithmes.Au Chapitre 1, nous appliquons un schéma semi-implicite en temps combiné avec la méthode de volumes finis généralisés SUSHI 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 présentons ensuite une méthode de simulation numérique pour un problème d'écoulements à densité variable couplé à un transfert de chaleur.Au Chapitre 2, nous effectuons une étude numérique de l'équation de Burgers non visqueuse en dimension un d'espace, avec des conditions aux limites périodiques, un terme source stochastique de moyenne spatiale nulle et une condition initiale déterministe. Nous utilisons un schéma de volumes finis combinant une intégration en temps de type Euler-Maruyama avec le flux numérique de Godunov. Nous effectuons des simulations par la méthode de Monte-Carlo et analysons les résultats pour différentes régularités du terme source. Il apparaît que la moyenne empirique des réalisations converge vers la moyenne en espace de la condition initiale déterministe quand t → ∞. Par ailleurs, la variance empirique converge elle aussi en temps long, vers une valeur qui dépend de la régularité et de l'amplitude du terme stochastique.Au Chapitre 3, nous démontrons la convergence d'une méthode de volumes finis pour une loi de conservation du premier ordre avec une fonction de flux monotone et un terme source multiplicatif faisant intervenir un processus Q-Wiener. Le terme de convection est discrétisé à l'aide d'un schéma amont. Nous présentons des estimations a priori pour la solution discrète dont en particulier une estimation de type BV faible. A l'aide d'une interpolation en temps, nous démontrons deux inégalité entropiques vérifiées par la solution discrète, ce qui nous permet de prouver que la solution discrète converge selon une sous-suite vers une solution stochastique faible entropique à valeurs mesures de la loi de conservation.Au Chapitre 4, nous obtenons des résultats similaires à ceux du Chapitre 3 dans le cas où la fonction flux n'est pas monotone; le terme de convection est discrétisé à l'aide d'un schéma monotone. / This thesis bears on numerical methods for deterministic and stochastic partial differential equations; we perform numerical simulations by means of finite volume methods and prove convergence results.In Chapter 1, we apply a semi-implicit time scheme together with the generalized finite volume method SUSHI for the numerical simulation of density driven flows in porous media; it amounts to solve a nonlinear convection-diffusion parabolic equation for the concentration coupled with an elliptic equation for the pressure. We then propose a numerical scheme to simulate density driven flows in porous media coupled to heat transfer. We use adaptive meshes, based upon square or cubic volume elements.In Chapter 2, We perform Monte-Carlo simulations in the one-dimensional torus for the first order Burgers equation forced by a stochastic source term with zero spatial integral. We suppose that this source term is a white noise in time, and consider various regularities in space. We apply a finite volume scheme combining the Godunov numerical flux with the Euler-Maruyama integrator in time. It turns out that the empirical mean converges to the space-average of the deterministic initial condition as t → ∞. The empirical variance also stabilizes for large time, towards a limit which depends on the space regularity and on the intensity of the noise.In Chapter 3, we study a time explicit finite volume method with an upwind scheme for a first order conservation law with a monotone flux function and a multiplicative source term involving a Q-Wiener process. We present some a priori estimates including a weak BV estimate. After performing a time interpolation, we prove two entropy inequalities for the discrete solution and show that it converges up to a subsequence to a stochastic measure-valued entropy solution of the conservation law in the sense of Young measures.In Chapter 4, we obtain similar results as in Chapter 3, in the case that the flux function is non-monotone, and that the convection term is discretized by means of a monotone scheme.

[pt] Tradicionalmente, os testes de formação em poços de petróleo buscam caracterizar o campo de permeabilidades a partir da interpretação dos transientes de pressão (PTA) nos períodos de fluxo e estática baseados em modelos isotérmicos de escoamento em meios porosos. Com o avanço da instrumentação dos testes, registros mais precisos de temperatura passaram a estar disponíveis e fomentaram a pesquisa baseada em modelos não isotérmicos que possibilitaram a análise a partir dos transientes de temperatura (TTA). Além da caracterização de parâmetros do reservatório como permeabilidade e porosidade com a interpretação dos transientes de temperatura, os dados de pressão obtidos a partir de um modelo não isotérmico representa de forma mais fidedigna o fenômeno físico sobretudo quando os testes são submetidos a maiores diferenciais de pressão. Este trabalho consiste no desenvolvimento de um simulador para teste de formação que considera a modelagem não isotérmica de reservatório unidimensional radial acoplado a um poço produtor e na utilização deste simulador, associado a métodos de otimização multivariável, para resolução do problema inverso da caracterização de parâmetros do reservatório. Alguns métodos de otimização foram testados e o algoritmo do Simplex de Nelder-Mead apresentou melhor eficácia. Foram estabelecidos três tipos de problemas e utilizados em três casos hipotéticos considerando inclusive a imposição artificial de ruídos nos sinais de pressão e temperatura utilizados para resolução do problema inverso. / [en] Traditionally, oil well formation tests aim to characterize the reservoir permeability field from pressure transient analysis (PTA) of drawdown and build up based on isothermal flow models in porous media. With the advancement of well test instrumentation, more accurate temperature records became available and have encouraged researches based on non-isothermal models that made possible the temperature transient analysis (TTA). In addition to the characterization of reservoir parameters such as permeability and porosity by TTA, the pressure data obtained from a non-isothermal model represent better the physical phenomenon, especially when the tests are subjected to greater drawdowns. This work consists in the development of a simulator for formation test that considers non-isothermal modeling of a unidimensional radial reservoir coupled to a production well and in the use of this simulator, associated with multivariable optimization methods, to solve the inverse problem of reservoir parameters characterization. Some optimization methods were tested and the Nelder-Mead Simplex algorithm presented better efficiency. Three types of problems were established and used in three hypothetical cases, including artificially imposed noise in pressure and temperature signals used to solve the inverse problem.