  For an absolutely unramified field extension $L/\mathbb{Q}_p$ with imperfect
residue field, we develop the notion of Wach modules in the setting of
$(\varphi,\Gamma)$-modules for $L$. Our main result establishes a categorical
equivalence between lattices inside crystalline representations of the absolute
Galois group of $L$ and integral Wach modules. Moreover, we provide a direct
relation between a rational Wach module equipped with the Nygaard filtration
and the filtered $\varphi$-module of its associated crystalline representation
in the imperfect residue field case.
