  We consider the flow of a generalized Newtonian fluid through a thin porous
medium of thickness $\epsilon$, perforated by periodically distributed solid
cylinders of size $\epsilon$. We assume that the fluid is described by the 3D
incompressible Stokes system, with a non-linear viscosity following the Carreau
law of flow index $1<r<+\infty$, and scaled by a factor $\epsilon^{\gamma}$,
where $\gamma\in \mathbb{R}$. Generalizing (Anguiano et al., Q. J. Mech. Math.,
75(1), 2022, 1-27), where the particular case $r<2$ and $\gamma=1$ was
addressed, we perform a new and complete study on the asymptotic behaviour of
the fluid as $\epsilon$ goes to zero. Depending on $\gamma$ and the flow index
$r$, using homogenization techniques, we derive and rigorously justify
different effective linear and non-linear lower-dimensional Darcy's laws.
Finally, using a finite element method, we study numerically the influence of
the rheological parameters of the fluid and of the shape of the solid obstacles
on the behaviour of the effective systems.
