Numerical analysis and mathematical modeling of non-linear PDEs applied to fluid dynamics in porous media

Abstract

This project focuses to scientific research in mathematical modeling and numerical analysis of partial differential equations in fluid dynamics in heterogeneous porous media. Recent results in numerical mathematics obtained by Prof. Eduardo Abreu will be used for a in deep understanding of nonlinear dynamics interaction of fluids (e.g., liquid and gas phases) in porous media. Mathematically, the key challenge lies in understanding the nonlinear mapping that takes the micro-physics (centimeter scale) to the macro-physics (field scale of tens of meters), which is more appropriate for a description of the nature of problems in fluid dynamics in porous media in real -- field -- scale.This research proposal integrates scientific research to understand the mathematics of multiphase flow in porous media by means of modeling, numerical analysis and computer simulation. Mathematically, the comprehensive effects of nonlinear dynamics of the flow of liquid and gas phases in porous media is scientifically relevant, for instance, in the context of mechanisms for capturing CO2 from the atmosphere and its injection, by means of different techniques in deep reservoirs, such as those in pre-salt layer spread along the Brazilian coast (e.g., in the Santos Basin at São Paulo).On off-shore, deep saline reservoirs are natural candidates to store CO2. Moreover, the mathematical results obtained can be applied in the production of oil and natural gas. These problems are modeled by nonlinear differential partial equations, typically stiff in their nature, and therefore is not always possible to develop a general non-local analytic theory to such equations. Thus the mathematical modeling and numerical analysis are an alternative to mathematical understanding of nonlinear dynamics of these PDEs.This research program provides an challenging, high-level, mathematical problem for qualified students. The training of qualified students is also included in this proposal. A doctoral thesis (in progress) under direction of Professor E. Abreu will be related to this proposal. (AU)