Formulações numéricas conservativas para aproximação de modelos hiperbólicos com termos de fonte e problemas de transporte relacionados

Orientador: Eduardo Cardoso de AbreuTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: O objetivo desta tese é desenvolver, pelo menos no aspecto formal, algoritmos construtivos e bem-balanceados para a aproximação de classes específicas de modelos diferenciais. Nossas principais aplicações consistem em equações de água rasa e problemas de convecção-difusão no contexto de fenômenos de transporte, relacionados a problemas de pressão capilar descontínua em meios porosos. O foco principal é desenvolver sob o framework Lagrangian-Euleriano um esquema simples e eficiente para, em nível discreto, levar em conta o delicado equilíbrio entre as aproximações numéricas não lineares do fluxo hiperbólico e o termo fonte, e entre o fluxo hiperbólico e o operador difusivo. Os esquemas numéricos são propostos para ser independentes de estruturas particulares das funções de fluxo. Apresentamos diferentes abordagens que selecionam a solução entrópica qualitativamente correta, amparados por um grande conjunto de experimentos numéricos representativosAbstract: The purpose of this thesis is to develop, at least formally by construction, conservative methods for approximating specific classes of differential models. Our major applications consist in shallow water equations and nonstandard convection-diffusion problems in the context of transport phenomena, related to discontinuous capillary pressure problems in porous media. The main focus in this work is to develop under the Lagrangian-Eulerian framework a simple and efficient scheme to, on the discrete level, account for the delicate nonlinear balance between the numerical approximations of the hyperbolic flux and source term, and between the hyperbolic flux and the diffusion operator. The proposed numerical schemes are aimed to be independent of particular structures of the flux functions. We present different approaches that select the qualitatively correct entropy solution, supported by a large set of representative numerical experimentsDoutoradoMatematica AplicadaDoutor em Matemática Aplicada165564/2014-8CNPQCAPE

A numerical study of two-phase flow with dynamic capillary pressure using an adaptive moving mesh method

Motivated by observations of saturation overshoot, this paper investigates numerical modeling of two-phase flow incorporating dynamic capillary pressure. The effects of the dynamic capillary coefficient, the infiltrating flux rate and the initial and boundary values are systematically studied using a travelling wave ansatz and efficient numerical methods. The travelling wave solutions may exhibit monotonic, non-monotonic or plateau-shaped behaviour. Special attention is paid to the non-monotonic profiles. The travelling wave results are confirmed by numerically solving the partial differential equation using an accurate adaptive moving mesh solver. Comparisons between the computed solutions using the Brooks-Corey model and the laboratory measurements of saturation overshoot verify the effectiveness of our approach

Improved simulation of naturally fractured reservoirs using unstructured grids and multi-rate dual-porosity models

Naturally Fractured Reservoirs (NFR) hold about half of the world’s remaining oil reserves and are typically very heterogeneous. NFR are also important for many other subsurface engineering applications including (nuclear) waste storage, CO2 sequestration, groundwater aquifers, and geothermal energy extraction. They contain faults, fracture corridors, large fractures but also many small-scale fractures as well as a heterogeneous rock matrix. Multi-phase flow in NFR is strongly influenced by this multi-scale heterogeneity. Therefore, accurate conceptual models that reliably quantify fluid flow in NFR are needed. In this thesis, three important contributions are made towards an improved simulation of multi-phase flow processes in NFR. First, the Implicit Pressure Implicit Saturation (IMPIS) method using unstructured grids was implemented to numerically simulate two-phase flow in a Discrete Fracture and Matrix (DFM) model. Second, a Multi-Rate Dual-Porosity (MRDP) model was developed including fracture-matrix transfer functions that are based on analytical solutions for spontaneous imbibition and gravity drainage. Finally, the two approaches were combined to a DFM-MRDP model. This model represents the multi-scale heterogeneity inherent to NFR more accurately by resolving fluid-flow processes in large-scale fractures directly using the DFM model while accounting for complex matrix heterogeneities when modelling fluid exchange between small-scale fractures and rock matrix using the MRDP model