We describe the exchange of fluid between a porous medium saturated with a viscous fluid coupled to the Stokes flow in a thin adjacent open channel. The asymptotic analysis of this system faces two big difficulties: the difference of scales at which Stokes and Darcy's law are valid and the mixed formulation of the system. This gives rise to two intermediate problems comparing Darcy's flow in both regions; with high permeability in the thin one. The first problem will be formulated in the standard minimization form and the second problem will follow the approach in mixed formulation. The asymptotic analysis will be presented in both cases as well as the convergence of the solutions.