Four phenomena contribute to wind-induced sand movement and eventually to the formation and evolution of dunes: erosion from the sand bed, transport by the wind, sedimentation due to gravity, and sand grain slides occurring when the slope of the accumulated sand exceeds a critical angle of repose. In particular, erosion, sedimentation, and the formation of such small avalanches determine the evolution of the free-boundary over which wind blows and transports the sand. The need to couple the multiphase turbulent fluid-dynamics with the dynamics occurring at the sand surface requires to deduce mathematical models for such phenomena that are able to describe the evolution of the surface in an accurate way, but that is at the same time computationally fast. Starting from this need, the aim of this talk is to propose a new mathematical model based on using classical continuum mechanics tools under the assumptions that the thickness of the creep layer is small and that the grains in it move in the direction of the steepest descent with a speed that is determined by several constitutive closures (Coulomb-like or pseudo-plastic fluids). The mathematical models deduced as degenerate parabolic equations for the height of the sand pile. In spite of their simplicity, all the models reply many well known behaviours characterizing the evolution of sand piles, such as the non-uniqueness of static configurations in subcritical conditions and the link between critical stationary configurations and the eikonal equation, and behave well in all tested set-ups. The only slight difference among them lies in the temporal evolution of the interface.