  In this work we present a deduction of the Saint-Venant-Exner model through
an asymptotic analysis of the Navier-Stokes equations. A multi-scale analysis
is performed in order to take into account that the velocity of the sediment
layer is smaller than the one of the fluid layer. This leads us to consider a
shallow water type system for the fluid layer and a lubrication Reynolds
equation for the sediment one. This deduction provides some improvements with
respect to the classical Saint-Venant-Exner model: (i) the deduced model has an
associated energy. Moreover, it allows us to explain why classical models do
not have an associated energy and how to modify them in order to recover a
model with this property. (ii) The model incorporates naturally a necessary
modification that must be taken into account in order to be applied to
arbitrarily sloping beds. Furthermore, we show that this modification is
different of the ones considered classically, and that it coincides with a
classical one only if the solution has a constant free surface. (iii) The
deduced solid transport discharge naturally depends on the thickness of the
moving sediment layer, what allows to ensure sediment mass conservation.
Moreover, we include a simplified version of the model for the case of
quasi-stationary regimes. Some of these simplified models correspond to the
generalization of classical ones such as Meyer-Peter$\&$M\"uller and
Ashida-Michiue models. Three numerical tests are presented to study the
evolution of a dune for several definition of the repose angle, to see the
influence of the proposed definition of the effective shear stress in
comparison with the classical one, and by comparing with experimental data.
