  Pseudo-parabolic equations have been used to model unsaturated fluid flow in
porous media. In this paper it is shown how a pseudo-parabolic equation can be
upscaled when using a spatio-temporal decomposition employed in the
Peszyn'ska-Showalter-Yi paper [8]. The spatial-temporal decomposition
transforms the pseudo-parabolic equation into a system containing an elliptic
partial differential equation and a temporal ordinary differential equation. To
strengthen our argument, the pseudo-parabolic equation has been given
advection/convection/drift terms. The upscaling is done with the technique of
periodic homogenization via two-scale convergence. The well-posedness of the
extended pseudo-parabolic equation is shown as well. Moreover, we argue that
under certain conditions, a non-local-in-time term arises from the elimination
of an unknown.
