**Abstract** : We derive high order homogenized models for the incompressible
Stokes system in a cubic domain filled with periodic obstacles. These models have
the potential to unify the three classical limit problems (namely the
``unchanged' Stokes system, the Brinkman model, and the Darcy's law)
corresponding to various asymptotic regimes of the ratio $\eta\equiv
a_{\epsilon}/\epsilon$ between the radius $a_{\epsilon}$ of the holes and the
size $\epsilon$ of the periodic cell. What is more, a novel, rather surprising
feature of our higher order effective equations is the occurrence of odd order
differential operators when the obstacles are not symmetric. Our derivation
relies on the method of two-scale power series expansions and on the existence of
a ``criminal' ansatz, which allows to reconstruct the oscillating velocity and
pressure $(\u_{\epsilon},p_{\epsilon})$ as a linear combination of the
derivatives of their formal average $(\u_{\epsilon}^{*},p_{\epsilon}^{*})$
weighted by suitable corrector tensors. The formal average
$(\u_\epsilon^{*},p_{\epsilon}^{*})$ is itself the solution to a formal, infinite
order homogenized equation, whose truncation at any finite order is in general
ill-posed. Inspired by the variational truncation method of
\cite{smyshlyaev2000rigorous,cherednichenko2016full}, we derive, for any
$K\in\N$, a well-posed model of order $2K+2$ which yields approximations of the
original solutions with an error of order $O(\epsilon^{K+3})$ in the $L^{2}$
norm. Furthermore, the error improves up to the order $O(\epsilon^{2K+4})$ if a
slight modification of this model remains well-posed. Finally, we find
asymptotics of all homogenized tensors in the low volume fraction limit
$\eta\rightarrow 0$ and in dimension $d\>3$. This allows us to obtain that our
effective equations converge coefficient-wise to either of the Brinkman or Darcy
regimes which arise when $\eta$ is respectively equivalent, or greater than
the critical scaling $\eta_{\mathrm{crit}}\sim\epsilon^{2/(d-2)}$