For each $i$, $E_{i}\cap({\mathcal{X}}\setminus{\mathcal{X}}_{\mathrm{snc}})$ contains finitely many prime divisors $F_{ik}$ of $E_{i}$. At the generic point of $F_{ik}$, ${\mathcal{X}}$ has cyclic quotient singularities, and

$$B_{E_{i}}=\sum_{j\neq i}E_{j}\cap E_{i}+\sum_{k}\left(1-\frac{1}{m_{ik}}\right)F_{ik}$$

with $m_{ik}$ the order of the corresponding cyclic groups, cf. [Kol13, 3.36.3].