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 impermeablebed. 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.