The Savage-Hutter and the Hadeler-Kuttler equations can be used to describe the movement of granular matter. The former, in its standard formulation, neglects erosion/deposition phenomena. The latter captures several relevant features of granular dynamics but, due to its very simple structure, is not equipped with the conservation of energy. The models presented in this talk aim at keeping the good qualitative features of the Hadeler-Kuttler model, while complying with the basic physical principles. These models ask for new results, both at the analytical and at the numerical levels.