  We prove the global well-posedness to the 2D Oldroyd-B type models with $\nu
\Lambda^{2\alpha}u$ and $\eta\Lambda^{2\beta}\tau$ satisfying $(i)\ \alpha>1,
\eta=0$ or $(ii)\ \alpha=1,\ \beta>0$. By establishing the gradient estimate of
$u$, $\tau$ and $L^\infty$ bound of ${\rm curl u+\Lambda^{-2}curldiv \tau}$,
Elgidi-Rousset (Commun. Pure Appl. Math. online, 2015) obtained the global
well-posedness for the case $\nu=0$, $\beta=1$. However, for the cases $(i)$
and $(ii)$, it is difficult to improve the regularity of $u$ and $\tau$
directly, especially when $\alpha\rightarrow 1^{+}$ in case $(i)$ and
$\beta\rightarrow 0^{+}$ in case $(ii)$. To overcome this difficulty, we
exploit a new structure of the equations coming from the dissipation and
coupled term. Then we prove the global well-posedness to these cases by energy
method which brings us closer to the more interesting case $\alpha=1$,
$\eta=0$.
