Consider a $\textbf{flat}$ surjective map $f: X \rightarrow \mathbb{A}^1$. The general fibers $F_{\epsilon}$ are canonically isomorphic, and the special fiber $F_0$ above $0 \in \mathbb{A}^1$ is not isomorphic to the general fibers.

Given a closed subscheme $B \subset X$, we define its special fiber limit $\widetilde{B}$ to be the intersection of the special fiber $F_0$ with the closure of $B \times \mathbb{A}^1 \backslash \{ 0 \}$ in $X$, $F_0 \cap \overline{B \times \mathbb{A}^1 \backslash \{ 0 \}}$.

Let $B_1$ and $B_2$ be two closed subschemes of $X$. When does the limit of their intersection equal to the intersection of their limits, i.e. $\widetilde{B_1} \cap \widetilde{B_2} = \widetilde{B_1 \cap B_2}$? Is it enough for the dimension of $\widetilde{B_1} \cap \widetilde{B_2}$ to be equal to the dimension of $B_1 \cap B_2$? Any relevant comments and references are welcome. Thanks!