In this work in collaboration with T. Iguchi, we show that the Saint-Venant equations in 2D with a partially submerged obstacle is well-posed. To do so, we show that the problem is equivalent to the usual Saint-Venant equations in an external domain, with additionnal non-standard boundary conditions because they are not local in space and time. These conditions do not fit into any category of dissipativity for which the hyperbolic theory is well posed, but we introduce a new class of well-posed hyperbolic boundary problems: that of weakly dissipative boundary conditions. We then show that our system belongs to this class and is therefore well posed.