We first examine Lp-viscosity solutions to fully nonlinear elliptic equations with bounded measurable ingredients. By considering p0 < p < d, we
focus on gradient-regularity estimates stemming from nonlinear potentials.

We find conditions for local Lipschitz-continuity of the solutions and continuity of the gradient. We survey recent breakthroughs in regularity theory
arising from (nonlinear) potential estimates.

Our findings follow from – and are inspired by – fundamental facts in the theory of Lp-viscosity solutions, and results in the work of Panagiota Daskalopoulos, Tuomo Kuusi and Giuseppe Mingione (DKM2014). In the second part we prove partial regularity of weakly stationary weighted harmonic maps with free boundary data on a cone.

As a starting point we take a look at the interior partial regularity theory for intrinsic energy minimising fractional harmonic maps from Euclidean space into smooth compact Riemannian manifolds for fractional powers strictly between zero and one.

Intrinsic fractional harmonic maps can be extended to weighted harmonic maps, so we prove partial regularity for locally minimising harmonic maps with (partially) free boundary data on half-spaces, fractional harmonic maps then inherit this regularity.

O contato com modelos geométricos tais como planos, esferas ou quádricas é uma importante ferramenta para entender a geometria diferencial de uma superfície. Nesta tese, estudamos o contato de superfícies com quádricas, e mais particularmente, com as quádricas de Moutard. Estendemos os resultados conhecidos para o caso de pontos parabólicos e curvas flecnodais em superfícies genéricas. Consideramos também o contato de hipersuperfícies com hiperquádricas de Moutard.

A microtomografia computadorizada de uma amostra de rocha possibilita uma caracterização do meio poroso e pode ser utilizada para estimar propriedades da rocha em macroescala, isto é, em escala de reservatório. Métodos baseados em mapas de distâncias e em algoritmos de erosão são as principais abordagens utilizadas para extração de uma rede de poros e gargantas a partir de imagens microtomográficas de rocha. Este trabalho propõe um método híbrido para a construção da rede, de modo que, durante o processo de modelagem na escala de poros, obtemos um esqueleto do espaço poroso por meio de um algoritmo de erosão e utilizamos um mapa de distâncias para construir uma rede de poros e gargantas. A determinação dos poros e gargantas a partir do esqueleto adota uma abordagem não-determinística possibilitando a geração de múltiplas redes com configurações distintas a partir de um mesmo esqueleto. Avaliamos a variabilidade dos cenários gerados e comparamos as estimativas para as propriedades petrofísicas com as obtidas pelo método de Bolas Máximas por meio dos resultados de uma simulação de fluxo monofásica na rede.

Injectivity test is a conventional technique in reservoir engineering used for oil recovery and formation evaluation. Typically, water is injected to displace the existing oil by increasing the pressure in the pores.
In this test, the pressure response measurement provides valuable information about the reservoir parameters, including permeability data.

Therefore researchers aim to develop mathematical equations that could accurately model pressure response during these tests for reservoir management and maintenance prediction purposes.

This work introduces a new analytical solution for injectivity test analysis. The solution combines the pressure-pressure convolution technique with a two-zone radial model. It allows the evaluation of the injectivity test without precise flow rate data, as the pressure-pressure convolution exclusively uses the pressure data acquired at different positions in the reservoir.

The reservoir model comprises an injector well in the inner zone of the reservoir and an observation well in the outer zone for measuring pressure response.

The proposed solution was validated by comparing the analytical results with those obtained from a commercial simulator with finite differences

Neste trabalho aplicamos o método Lattice Boltzmann (LBM) para simular os processos de reações químicas que ocorrem na interação entre o fluido e a fase sólida, modificando o meio poroso. Para isso apresentaremos como o método LBM aborda a simulação do escoamento de fluido em um meio poroso irregular para os casos de um ou mais fluidos incluindo o processo de dissolução química. A partir dos processos anteriores, propomos uma modificação onde a dissolução possa ocorrer como uma característica do fluído que interage com a fase sólida. Ao abordar a dissolução como característica da interação do fluido com a fase sólida, é possível ter uma maior compreensão de como o fluido pode modificar a geometria do meio poroso e impactar nas mudanças de fluxo. A proposta de modificação foi avaliada em alguns casos em que o fluxo no meio poroso é bem definido: O canal aberto, canal com cilindros e em um meio poroso de geometria complexa. A proposta foi estendida para a simulação em um meio poroso 3D onde analisamos como a dissolução foi impactada pela presença de formas externas como a gravidade

We study several models of base dynamics and linear cocycles over such systems. We establish effective rates of convergence of the Birkhoff averages oftoral translations. We derive large deviations estimates for mixed Markov-quasiperiodic dynamics. We prove continuity properties of the Lyapunov exponents of linear cocycles over Markov shifts. Besides their intrinsic interest,these results prepare the ground for a larger project concerned with the study of linear cocycles over mixed Markov-quasiperiodic base dynamics. As crucial steps in this study, we obtain a version of Kifer’s non random filtration and an upper large deviations estimate for such systems.

Physics Informed Machine Learning is the strategy of developing a neural network with physical constraints, commonly expressed in partial differential equations (PDEs) and their initial and boundary conditions. In this approach, the main idea is to incorporate underlying physical laws expressed in these PDEs as prior information for the neural network. In this work, we investigate the applicability of this technique to the direct problem of two-phase fluid transport in porous media, particularly in the context of gas injection in an oil reservoir, whose physical constraints are described using nonlinear first-order hyperbolic PDEs, subject to specific initial and boundary conditions. Initially, we develop the equations governing the problem without considering the fluid volume change factor to study the convergence of the solutions to these PDEs. Based on the obtained results, we introduce the volume change equations to capture the gas phase's behavior better. The fractional flux functions used in our examples were chosen as non-convex to include shock and refraction phenomena in the solutions. We also incorporate a diffusive factor, transforming the hyperbolic PDEs into parabolic ones. Through this approach, the neural network could learn consistent approximate solutions. Consequently, this effect smoothens the solution curves at the points of shock.

Uma reta bissetora divide uma região convexa do plano em duas partes com áreas iguais. É natural estudar a envoltória destas linhas bissetora, que em geral apresentam singularidades. O caso de polígonos é particularmente interessante, pois existem diversas noções distintas de envoltórias discretas. Nesta dissertação, nós estudamos três tipos diferentes de envoltórias discretas de retas bissetoras e as conexões entre elas.

Esta dissertação versa sobre a teoria das soluções de viscosidade para equações diferenciais parciais elípticas completamente não-lineares com ingredientes mensuráveis. Nosso principal objetivo é demonstrar o Princípio do Máximo de Aleksandrov- Bakelman-Pucci neste contexto.

We consider, in this dissertation, the question of determining if a graph has a property P, such as "G is triangle-free" or "G is 4-colorable". In particular, we consider for which properties P there exists a random algorithm with constant error probabilities that accept graphs that satisfy P and reject graphs that are epsilon-far from any graph that satisfies it. If, in addition, the algorithm has complexity independent of the size of the graph, the property is called testable. We will discuss the results of Alon, Fischer, Newman and Shapira that obtained a combinatorial characterization of testable graph properties, solving an open problem raised in 1996. This characterization informally says that a graph property P is testable if and only if testing P can be reduced to testing the property of satisfying one of finitely many Szemerédi-partitions.

Precisely forecasting oil field performance is essential in oil reservoir planning and management. Nevertheless, forecasting oil production is a complex nonlinear problem due to all geophysical and petrophysical properties that may result in different effects with a bit of change. Thus, all decisions to be made during an exploitation project must consider different efficient algorithms to simulate data, providing robust scenarios to lead to the best deductions. To reduce the uncertainty in the simulation process, recent studies have efficiently introduced machine learning algorithms for solving reservoir engineering problems since they can extract the maximum information from the dataset. This thesis proposes using two machine learning techniques to predict the daily oil production of an offshore reservoir. Initially, the oil rate production is considered a time series and is pre-processed and restructured to fit a supervised learning problem. The Random Forest model is used to forecast a one-time step, which is an extension of decision tree learning, widely used in regression and classification problems for supervised machine learning. Regardless, the restrictions of this approach lead us to a more robust model, the LSTM RNN's, which are proposed by several studies as a suitable deep learning technique for time series modeling. Various configurations of LSTM RNN's were constructed to implement single-step and multi-step oil rate forecasting and down-hole pressure was incorporated to the inputs. For testing the robustness of the proposed models, we use four different datasets, three of them synthetically generated and one from a public real dataset, the Volve oil field, as a case study to conduct the experiments. The results indicate that the Random Forest model could sufficiently estimate the one-time step of the oil field production, and LSTM could handle more inputs and adequately estimate multiple-time steps of oil production.

Este trabalho analisa um mercado de eletricidade em que os geradores declaram funções de custo quadráticas para o operador da rede e também suas disponibilidades máximas de produção. O operador, então, determina as quantidades a serem produzidas por cada gerador de modo a atender a uma demanda inelástica, ao menor custo possível. Estabelecem-se alguns resultados que permitem computar os equilíbrios de Nash deste modelo e descrevem-se algumas de suas propriedades, tais como condições de existência.

Oil and gas reservoir simulation is a common demand in petroleum engineering, and research, which may have a high computational cost, solving a mathematical numeric problem, or high computational time. Moreover, several reservoir characterization methods require multiple iterations, resulting in many simulations to obtain a reasonable characterization. It is also possible to mention ensemble-based methods, such as the EnKF and the ES-MDA, which demand lots of simulation runs to provide the output result. As a result, reservoir simulation might be a complex subject to deal with when working with reservoir characterization. The use of machine learning has been increasing in the energy industry. It can improve the accuracy of reservoir predictions, optimize production strategies, and many other applications. The complexity and uncertainty of reservoir models pose significant challenges to traditional modeling approaches, making machine learning an attractive solution. Aiming to reduce reservoir simulation's complexities, this work investigates using a machine-learning model as an alternative to conventional simulators. The Random Forest regressor model is experimented with to reproduce pressure response solutions for multi-region radial composite reservoirs. An analytical approach is employed to create the training dataset in the following procedure: the permeability is sorted using a specific distribution, and the output is generated using the analytical solution. Through experimentation and analysis, this work aims to advance our understanding of using machine learning in reservoir simulation for the energy industry.