Comportamento das cascatas de energia em hidrodinâmica para a compreensão do fluxo de petróleo em reservatórios

Abstract

When we study ows in porous media, we see that there are a number of open problems involving the mathematics of hydrodynamic equations. Among these challenges, is the problem of the Euler equations in 3-D for incompressible uids, where it is not known if there are singularities in nite time. Problems of a similar nature are currently studied using other hydrodynamic equations, such as the Navier-Stokes equations or even simpler equations aiming to reduce the mathematical complexities in seeking to gain a better understanding. A key aspect to problems involving these equations is that the kinetic energy present in large spatial scales can be redistributed to smaller spatial scales. If this is repeated on smaller and smaller scales and ever faster times, the process can even result in nite time singularity. Thus the premise of this work is to study the process of cascades of energy using the Fourier transform. We study the non-viscous Burger equation in 1-D, which allows us to investigate in detail the formation of the power-law tail in the energy spectrum, which is associated with singularities.