  A two-layer shallow water type model is proposed to describe bedload sediment
transport. The upper layer is filled by water and the lower one by sediment.
The key point falls on the definition of the friction laws between the two
layers, which are a generalization of those introduced in Fern\'andez-Nieto et
al. (ESAIM: M2AN, 51:115-145, 2017). This definition allows to apply properly
the two-layer shallow water model for the case of intense and slow bedload
sediment transport. Moreover, we prove that the two-layer model converges to a
Saint-Venant-Exner system (SVE) including gravitational effects when the ratio
between the hydrodynamic and morphodynamic time scales is small. The SVE with
gravitational effects is a degenerated nonlinear parabolic system. This means
that its numerical approximation is very expensive from a computational point
of view, see for example T. Morales de Luna et al. (J. Sci. Comp., 48(1):
258-273, 2011). In this work, gravitational effects are introduced into the
two-layer system without such extra computational cost. Finally, we also
consider a generalization of the model that includes a non-hydrostatic pressure
correction for the fluid layer and the boundary condition at the sediment
surface. Numerical tests show that the model provides promising results and
behave well in low transport rate regimes as well as in many other situations.
