Positive solutions of the equation \(h_t =k(h^2)_{xx}\) are studied in the domain \(\{x>x_0 (t)\), \(t>0\}\), with the boundary conditions \(h=0\), \(mk(h^2)_x =q(t)\) at the free boundary \(x=x_0 (t)\). This boundary value problem is proposed as a model for floating of a porous stratum with forced draining. Here, \(h\) is the fluid level, \(m\) and \(k\) are the porosity and permeability of the stratum. The derivation of the model is obtained through a passage to the limit in the problem of ground water filtration in a two-layered setting where a porous medium with constant permeability sits on a thin layer of elastic porous medium whose permeability changes abruptly under external pressure from very high values for small preflowing back from the stratum. The solutions with compact support in space with a right boundary point \(x=x_f (t)\), where \(h=0\), describe the floating of the stratum. Some self-similar solutions are found indicating that the following effects occur: infinite extension of the fluid mound in infinite time, localization of the fluid mound in infinite time, and depletion (extinction) of the fluid mound in finite time.