On the Equivalence of Forchheimer and Inertial Terms in Fluid Dynamics in Porous Media

Resumo

With the advance of numerical methods and the capacity of computers to solve problems of high
computational cost, the dynamics of fluids in porous media started to give up with the simplifying hypotheses and
started to not only solve the complete equations, but also allowed to incorporate coupled phenomena such as
deformation of porous media, temperature, chemical reaction and electromagnetism. The first steps towards a more
rigorous study from the mathematical point of view, necessarily involve the progressive incorporation of more
rigorous equations as a state of the art in flow problems. The transition from Darcy’s law to Brinkman’s equations
and even the direct numerical simulations (DNS) of the Navier-Stokes equations on a pore scale, started to take
place in the current methods of predicting the hydrogeological behavior of materials. This work seeks, in a succinct
and humble way, to highlight and propose an analysis of the possible equivalence between the Forchheimer term
to simulate the nonlinear effects on the flow in porous media and the natural convective term of the Eulerian
formulation of the conservation of the of momentum in fluids. Despite the equivalence being affirmed in some
works, there seems to be a lack in literature concerning rigorous demonstrations of this affirmation. This paper
uses tools such as the topology of metric spaces and the weak formulation in finite dimensional spaces in the
analysis of the differential equations that govern the problem and seeks to be as clear as possible so that a reader
not used to the mathematics involved can understand, at least, the central idea.