Darcy-Forchheimer gravity currents in porous media

We theoretically and experimentally study gravitycurrents of a Newtonian fluid advancingin atwo-dimensional, infinite and saturated porous domain over a horizontal impermeable bed. The driving force is due to the density difference between the denser flowing fluid andthe lighter, immobile ambient fluid. The current is taken to be in the Darcy-Forchheimerregime, where a term quadratic in the seepage velocity accounts for inertial contributionsto the resistance. The volume of fluid of the current varies as a function of time as similar to T gamma, where the exponentparameterizes the case of constant volume subject to dambreak (gamma=0), of constant (gamma=1), waning (gamma<1)and waxing inflow rate (gamma>1). Thenonlinear governing equations, developed within the lubrication theory, admit self-similarsolutions for some combinations of the parameters involved and for two limiting conditionsof low and high local Forchheimer number, a dimensionless quantity involving the localslope of the current profile. Another parameterNexpresses the relative importance of thenonlinear term in Darcy-Forchheimer's law; values ofNin practical applications may varyin a large interval around unity, e.g.N is an element of[10-5,102]; in our experiments,N is an element of[2.8,64].Sixteen experiments with three different grain sizes of the porous medium and differentinflow rates corroborate the theory: the experimental nose speed and current profiles arein good agreement with the theory. Moreover, the asymptotic behaviour of the self-similarsolutions is in excellent agreement with the numerical results of the direct integration ofthe full problem, confirming the validity of a relatively simpleone-dimensional model.