  The asymptotic analysis of a Darcy-Stokes system modeling the fluid exchange
between a narrow channel (Stokes) and a porous medium (Darcy) coupled through a
$ C^{2} $ curved interface, is presented. The channel is a cylindrical domain
between the interface ($ \Gamma $) and a parallel translation of it ($ \Gamma +
\epsilon \, \boldsymbol{\widehat{e}}_{N} $). The introduction of a change
variable to fix the domain's geometry and the introduction of two systems of
coordinates: the Cartesian and a local one (consistent with the geometry of the
surface), permit to find a Darcy-Brinkman lower dimensional coupled system as
the limiting form, when the width of the channel tends to zero ($ \epsilon
\rightarrow 0 $).
