Computing qualitatively correct approximations of partial differential equations in porous media transport phenomena

Abstract

The qualification of this proposal is scientific research in applied mathematics. The focus will be the construction of a new class of well-balanced scheme and its application to understanding nonlinear and unconventional models of partialdifferential equations (PDEs) governing multiphase flow in porous media. In some recent research programs (2010-2014), E. Abreu with some research colleagues developed a new numerical scheme for computing qualitative correct approximate numerical solutions for three-phase flows in multidimensional porous media made of several rock types characterized by spatial multiscale discontinuities. This procedure is based on an operator splitting strategy, which in turn leads to three distinct subproblems, being computed separately and sequentially, as follows convection, diffusion and pressure-velocity equations. The above mentioned method is the first multidimensional scheme in the literature able to show strong numerical evidence of existence and structurally stable nonclassical waves for three-phase flow with or without the gravity effect under excitations imposed by heterogeneity of porous media systems. Based on this new approach, this project aims to investigate and to develop a new class of well balancing scheme respecting the local equilibria associated with the convection (hyperbolic PDE), the diffusion (parabolic PDE) and the pressure-velocity (elliptic PDE) problems induced by flow functions, which in turn exhibit various types of discontinuities upon their arguments. The purpose of the project is two-fold:( 1 ) a qualitative study of solutions of nonlinear and unconventional models of governing PDEs for multiphase flow problems in porous media and ( 2 ) improve the accuracy of entropic solutions given by the new well-balanced scheme without increasing too much the computational cost. Unconventional two-phase and three-phase PDE transport models, relevant in applications, will be investigated. (AU)