The Saint-Venant equations are widely used to model real river flows. Solving these equations requires the use of sophisticated partial differential equation (PDE) solvers. In this paper, it is proposed that, under different conditions on data availability, the solution of the Saint-Venant equations can be represented by feasibility models that depend on a moderate number of parameters. In particular, a new non-rectangular model for the shape of cross sections is proposed. This method can be used under different assumptions on the knowledge of the variables influencing the evolution of the system.