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The contributions of the current analytical models on the prediction of the stability of the slope subjected to slide head toppling failure mechanisms, have always focused on the idealized geometry comprising regular blocks dipping into the slope face. Besides, the influence of groundwater and stabilizations from the lowermost block of the slope have been overlooked in the available literature. In this article, the analytical solutions that incorporates the kinematic mechanisms of the jointed rock slope under the influence of groundwater and stabilizing the lowermost block subjected to slide head toppling are derived based on the limit equilibrium. Furthermore, a real slide head toppling failure case history was studied to illustrate the effectiveness of the presented analytical solutions. The investigation results indicate that the presence of groundwater in the jointed rock slope, lowers the distributions of the normal and shear forces thereby inducing slide head toppling. Reinforcing the lowermost block of the slope, enhances the distributions of the normal and shear forces thus improving the stability of the jointed rock slope. The study results depict that the presented analytical solutions can provide an accurate and efficient stability analyses of the jointed rock slope subjected to slide head toppling failure mechanisms considering the presence of groundwater and stabilization effects.

O escoamento de fluidos miscíveis em meios porosos é modelado matematicamente pelo acoplamento do problema de Darcy com o transporte. Essas equações apresentam uma forte relação de dependência principalmente em casos com razão de mobilidade adversa. Além da não-linearidade e do forte acoplamento entre as equações, há ainda dificuldades numéricas relacionadas à predominância dos efeitos convectivos, heterogeneidade dos meios porosos e compatibilidade dos espaços de aproximação. Neste sentido, propomos o estudo de métodos de elementos finitos híbridos estáveis e estabilizados capazes de superar as dificuldades típicas deste problema e, em alguns casos, assegurar a conservação local de massa. Os métodos ditos estáveis são caracterizados pelo uso dos espaços de aproximação de Raviart-Thomas, enquanto que os estabilizados incorporam resíduos de mínimos quadrados à formulação. Dessa forma, desenvolvemos métodos mistos híbridos para o problema de Darcy e para o problema do transporte, visando gerar formulações para a equação do transporte compatíveis com as formulações para o problema de Darcy, onde os métodos estáveis para Darcy são combinados aos estáveis para o transporte e o mesmo ocorre para os métodos estabilizados. Portanto para manter esta correspondência entre as formulações, a escolha para os multiplicadores de Lagrange, em ambas as abordagens, é de mesma natureza, associada ao traço da variável escalar. Os métodos estáveis e estabilizados para o problema do transporte são combinados a um esquema upwind, empregado para suavizar os efeitos do regime predominantemente convectivo, e discretizados no tempo por uma abordagem de segunda ordem baseada no método de Crank-Nicolson. Neste contexto, os métodos estáveis e estabilizados empregando multiplicadores contínuos e descontínuos são testados e validados, através de estudos de convergência, para o problema de Darcy e do transporte separadamente e de forma acoplada para problema Darcy-transporte. Além disso, diversos cenários são simulados em meios homogêneos e heterogêneos supondo razão de mobilidade unitária e adversa onde os métodos propostos demonstram a capacidade de capturar os fenômenos de geração de “dedos” e padrões fractais gerados por este tipo de abordagem, além de apresentar uma redução dos efeitos oscilatórios provenientes da convecção dominante que é inerente a essas aplicações. Miscible displacement in porous media is modeled mathematically by coupling the Darcy problem with transport. These equations are strongly linked, especially in cases with adverse mobility ratio. In addition to the nonlinearity and the strong coupling between the equations, there are still numerical difficulties related to the predominance of convective effects, heterogeneity of the porous media and the compatibility between approximation spaces. In this context, we propose the study of stable and stabilized hybrid finite element methods capable of overcoming the typical difficulties of this problem and, in some cases, ensuring local mass conservation. The so-called stable methods are characterized by the use of Raviart-Thomas approximation spaces, while the stabilized methods incorporate residual terms into the formulation. In this way, we developed mixed hybrid methods for the Darcy problem and for the transport problem, aiming to generate formulations for the transport equation compatible with the formulations for the Darcy problem, where the stable methods for Darcy are combined with the stable ones and the same goes for stabilized methods. Therefore, to maintain this correspondence between the formulations, the choice for Lagrange multipliers, in both approaches, is of the same nature, associated with the trace of the scalar variable. The stable and stabilized methods for the transport problem are combined with an upwind scheme, used to soften the effects of the predominantly convective regime, and discretized in time by a second-order approach based on the Crank-Nicolson method. In this context, the stable and stabilized methods employing continuous and discontinuous multipliers are tested and validated, through convergence studies, independently in order for the Darcy and transport problems and in a coupled way for the Darcy-transport problem. In addition, scenarios are simulated in homogeneous and heterogeneous environments assuming unitary and adverse mobility ratios where the proposed methods demonstrate an ability to capture the viscous fingers generation phenomena and fractal patterns generated by this type of approach, in addition to presenting a reduction in the oscillatory effects from the dominant convection that is inherent to these applications.

O desenvolvimento de técnicas adequadas para extração eficiente de óleo de reservatórios de petróleo passa pela simulação precisa de tais fenômenos, que é alcançada através do estudo de modelos matemáticos e métodos computacionais robustos, eficientes e precisos. Neste contexto, este trabalho visa o estudo de métodos numéricos para a simulação de escoamentos bifásicos em meios porosos heterogêneos. Para tanto, propomos uma abordagem numérica do tipo staggered para estes modelos, que se baseia na aproximação de forma desacoplada dos sistemas de equações diferenciais parciais referentes aos problemas de Darcy e da saturação das fases. Dessa forma, podem ser empregados métodos numéricos específicos para cada sistema, que melhor se adequem às suas caractrerísticas. Assim, propomos o estudo de métodos de elementos finitos mistos, estáveis e estabilizados, clássicos e híbridos e localmente conservativos para o cômputo da velocidade da mistura e de um método de volumes finitos não-oscilatório de alta ordem, baseado em esquemas centrais, para a equação hiperbólica não-linear que governa o transporte da saturação das fases. Resultados numéricos comprovam a flexibilidade, a taxa de convergência e o custo computacional dos métodos adotados, além de demonstrar a eficácia dos métodos quando aplicados a simulação de problemas associados a extração de petróleo em cenários fortemente heterogêneos. The development of techniques for efficient oil extraction from reservoirs passes through the simulation of such phenomena, which is achieved by the study of mathematical models and robust, precise and efficient computational methods. This dissertation studies methods for the simulation of two-phase flows in heterogeneous porous media. To do so, we propose a “staggered” numerical approach for the numerical methods, that is based on the approximation of uncoupled systems of differential equations related to Darcy’s problems and saturation of the phases. Then, appropriate methods for each system, that best suit its characteristics can be applied. Therefore, we propose studying locally conservative finite element methods, stable and stabilized, classical and hybrid to approximate the velocity field and a non-oscillatory high order finite volume method, based on central schemes, to approximate the non-linear hyperbolic equation that governs the transport of phases. Numerical results attest to the flexibility, convergence rate and computational cost of the adopted methods, and demonstrate the effectiveness of such methods when applied to oil extraction in various heterogeneous porous media scenarios.

It is known that the addition and elimination of suitable bubble functions can result in a stabilized scheme of the SUPG-type. Residual-Free Bubbles (RFB), in particular, can assure a quasi-optimal stabilized scheme, but they are difficult to compute in one dimension and nearly impossible to compute in 2 and 3 dimensions, unless in special limit cases. Strongly convection-dominated problems (without reaction terms) are one of these cases, where it is possible to find reasonably simple computable bubbles that provide a stabilizing effect as good as that of true RFB. Here, although in a one-dimensional framework, we analyze the case in which a non-negligible reaction term is present, and we provide a simple recipe for spotting a suitable bubble space (adding two bubbles per element) that provides a very good stabilizing effect. The method adapts very well to all regimes with continuous transitions from one regime to another. It is clear that the one-dimensional case, in itself, has no real interest. We believe, however, that the discussion can cast some light on the interaction between convection and reaction that could be useful in future works dealing with multidimensional, more realistic problems.