Suppose $X/\mathbb C$ is a projective $\mathbb Q$-factorial variety with wild singularities. Let $N$ be a nef $\mathbb R$-Cartier divisor. Then is it possible that there are infinitely many curves $C_i \subset X$, such that $C_i \cdot N>0$ are infinitely close to $0$?

Notice that if $N$ is a $\mathbb Q$-Cartier divisor, then if $C \cdot N>0$, it must satisfy $C \cdot N > \frac 1 m$ for some $m$ such that $mN$ is Cartier. Also, if $N$ is an ample $\mathbb R$-Cartier divisor, then it can be written (in the ample cone) as $\sum r_i A_i$ with $r_i \in \mathbb R_{>0}$ and $A_i$ $\mathbb Q$-Cartier ample divisor, hence the statement does not hold either.

I am trying to work with Kodaira/Nagata example of a surface with infinitely $(-1)$-curves [Hartshorne's book V Ex 4.15(e)], but I have no idea of finding such $N$...