The impact of instability appearance on the quadratic law for flow through porous media

Abstract

In the present paper, we attempt to explain the macroscopic flow law evolution in porous media according to the Reynolds number. A crenellated channel, considered as an element of such a medium, is used to perform numerical simulations in stationary and non-stationary cases. In the case of non-stationary laminar flows, we point out flow instabilities occurring in the channel at high Reynolds numbers and we focus on their influence on the macroscopic law. We qualitatively prove that they generate an additional quadratic contribution to Forchheimer's law. We use two methods to study this contribution: first, a periodic disturbance, for which the instabilities appearing at the beginning of disturbance become regular oscillations; then a pulse disturbance of the entry velocity field which enables us to link the additional quadratic contribution to the existence of an accumulation of fluid at low velocity in the channel.

Analytical Solution of Unsteady-state Forchheimer Flow Problem in an Infinite Reservoir: The Boltzmann Transform Approach

“…Lucas et al (2007), presented that quadratic deviation in Forchheimer law appear as a result of non-periodicity in porous media and fractures [6]. Several other authors have modelled the Forchheimer flow in porous media by employing the numerical method [7][8][9][10][11][12][13], and semi-analytical approach [14][15][16][17][18][19][20][21][22][23][24][25].…”