Resolução numérica de equações diferenciais parciais hiperbólicas não lineares: um estudo visando a recuperação de petróleo

• Autor(a) principal: Barbosa, Nelson Machado
• Data de Publicação: 2010
• Tipo de documento: Dissertação
• Idioma: por
• Título da fonte: Biblioteca Digital de Teses e Dissertações da UERJ
• Texto Completo: http://www.bdtd.uerj.br/handle/1/13828
• topic: Secondary recovery of oil - Mathematical models; Differential equations, Hyperbolic - Numerical solutions; Burgers equation; Conservation laws (Mathematics); Nonlinear hyperbolic equations; Burgers and Buckley-Leverett problems; LWLF-k Composite Scheme; Recuperação secundária do petróleo Modelos matemáticos; Equações diferenciais hiperbólicas Soluções numéricas; Burgers, Equação de; Lei da conservação (Matemática); Equações hiperbólicas não lineares; Problemas de Burgers e Buckley-Leverett; Método composto LWLF-k; CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA::MATEMATICA APLICADA

The secondary recovery of petroleum is usually performed with injection of water through an oil reservoir to keep the oil pressure for the exploration. In order to make the exploration profitable, the extraction cost must be less than the financial return, which means that the simulation of the exploration process is extremely relevant. In this work, the Burgers- and- Buckley-Leverett problems are studied seeking a two-phase displacement in porous media. The flow is considered incompressible and capillary effects are ignored. In order to analyze the problem, it was necessary to use the theory of conservation law in a spatial variable. Weak solutions, which can be understood as shock or rarefaction waves, are studied with the entropy condition, so that only the physically correct solutions are considered. Some classical numerical methods, which can be seen as conservative schemes of three points, are studied, among them the Lax-Friedrichs (LF) and Lax-Wendroff (LW) methods. A composite scheme, called LWLF-k, is tested using LF and LW methods, being respected the CFL condition, with satisfactory results. In order to validate the numerical schemes, we consider analytical solutions of the Burgers-and-Buckley-Leverett equations. Was also made a comparison with TVD s methods with flux limiters, obtaining satisfactory results. We emphasize that to the best of our knowledge, the LWLF-4 scheme has never been used to solve the Buckley-Leverett equation.