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In this work we present a deduction of the Saint-Venant-Exner model throughan asymptotic analysis of the Navier-Stokes equations. A multi-scale analysisis performed in order to take into account that the velocity of the sedimentlayer is smaller than the one of the fluid layer. This leads us to consider ashallow water type system for the fluid layer and a lubrication Reynoldsequation for the sediment one. This deduction provides some improvements withrespect to the classical Saint-Venant-Exner model: (i) the deduced model has anassociated energy. Moreover, it allows us to explain why classical models donot have an associated energy and how to modify them in order to recover amodel with this property. (ii) The model incorporates naturally a necessarymodification that must be taken into account in order to be applied toarbitrarily sloping beds. Furthermore, we show that this modification isdifferent of the ones considered classically, and that it coincides with aclassical one only if the solution has a constant free surface. (iii) Thededuced solid transport discharge naturally depends on the thickness of themoving sediment layer, what allows to ensure sediment mass conservation.Moreover, we include a simplified version of the model for the case ofquasi-stationary regimes. Some of these simplified models correspond to thegeneralization of classical ones such as Meyer-Peter$\&$M\"uller andAshida-Michiue models. Three numerical tests are presented to study theevolution of a dune for several definition of the repose angle, to see theinfluence of the proposed definition of the effective shear stress incomparison with the classical one, and by comparing with experimental data.

Formal deduction of the Saint-Venant-Exner model including arbitrarily sloping sediment beds and associated energy