Let $X$ be a smooth projective variety over $\mathbb{C}$. Let $D$ be a Cartier divisor on $X$. If $C$ is a curve on $X$ such that the intersection $C\cdot D <0$, we have $C \subseteq \mathbf{B}(D)$, where $$\mathbf{B}(D):= \bigcap_{m \in \mathbb{N}, F\in |mD|}F$$ is the stable base locus of $D$.

I want to know if there is any result related to the inverse direct. To be precise, I want to know if the followings are true or false:

(1)For a general point of $\mathbf{B}(D)$, is there a curve $C$ passing through that point with $C\cdot D<0$?

(2)The union of all curves with negative intersections with D has the same dimension as $\mathbf{B}(D)$, that is $$\dim \mathbf{B}(D) = \dim \bigcup_{C, C\cdot D < 0}{C}\qquad?$$