Homogenization theory and nonlinearities in Darcy's law

The theory of homogenization of differential equations has become an open field of research in several areas of the exact sciences and has proved to be a powerful tool for understanding the global behavior of heterogeneous materials. Despite knowing that the deduction of Darcy’s law through the Navier-Stokes equations has been debated for decades, many questions remain open, mainly regarding more complex boundary conditions, cases involving multiphase flows and the numerical homogenization techniques. It is known today that Darcy’s law is presented in the form of a linear relationship only for a range of hydraulic gradient and that this range overlaps the range of laminar flow through soil voids. Therefore, it is proposed in this work to understand the loss of linearity in Darcy’s law, based on the theory of homogenization, modifying and exploring the limit results obtained by Allaire in 1991.