Consider a smooth minimal $3$-fold conic bundle $f:X\rightarrow\mathbb{P}^2$. Then $X$ has Picard rank two and consequently also the vector space of $1$-cycles is $2$-dimensional. Then the cone of effective curves of $X$ has two generators. On the other hand if $F = F_1 + F_2$ is a reducible fibers it seems that $F_1,F_2$ should be generators, and there should be another generator coming from a curve $C\subset X$ which is not contracted by $f$. What am I missing here? What are the generators of the cone of curves of $X$?

Thank you.