We investigate the $N_f=2$ Schwinger model with the massive staggered fermions in the presence of a $2\pi$ periodic $\theta$ term, using the Grassmann tensor renormalization group.
Thanks to the Grassmann tensor network formulation, there is no difficulty in dealing with the massive staggered fermions.
We study the $\theta$ dependence of the free energy in the thermodynamic limit.
Our calculation provides consistent results with the analytical solution in the large mass limit.
The results also suggest that the $N_f=2$ Schwinger model on a lattice has a different phase structure from that described by the continuum theory.