We consider the flow of generalized Newtonian fluid through a thin porous media Ωε of thickness ε which is perforated by periodically distribute solid cylinders of size ε. Generalizing (Anguiano et al., Q. J. Mech. Math., 75(1), 2022, 1-27) we do a new and complete study on the fluid described by the 3D incompressible Stokes system with non-linear viscosity, following the Carreau law of flow index 1 < r < +∞, which is scaled by a factor ε^γ , where γ ∈ R. Depending of γ and the flow index r, using homogenization techniques, we derive different effective linear and non-linear lower-dimensional Darcy's laws.