  Given a family of graphs $\mathcal{F}$, a graph $G$ is said to be
$\mathcal{F}$-saturated if $G$ does not contain a copy of $F$ as a subgraph for
any $F\in\mathcal{F}$ but the addition of any edge $e\notin E(G)$ creates at
least one copy of some $F\in\mathcal{F}$ within $G$. The minimum size of an
$\mathcal{F}$-saturated graph on $n$ vertices are called the saturation number,
denoted by $\sat(n, \mathcal{F})$. Let $\mathcal{C}_{\ge r}$ be the family of
cycles of length at least $r$. Ferrara et al. (2012) gave lower and upper
bounds of $\sat(n, C_{\ge r})$ and determined the exact values of $\sat(n,
C_{\ge r})$ for $3\le r\le 5$. In this paper, we determine the exact value of
$\sat(n,\mathcal{C}_{\ge r})$ for $r=6$ and $28\le \frac{n}2\le r\le n$ and
give new upper and lower bounds for the other cases.
