I am trying to calculate some intersection numbers and would appreciate help on the following problem:

Consider two divisors $D_1$ and $D_2$. Blowing up their intersection yields $\varphi^{*}(D_i) = \widetilde{D_i} + m_i E$, $\varphi$ is the blowup morphism and E is the exceptional Divisor. I hope there are no mistakes so far.

Now I am interested in calculating intersection numbers that would look like $\varphi^*(D_i) . \alpha$, where $\alpha$ is a cycle which is contained in the exceptional locus. My first intuition was to apply the projection formula, but on second thought I do not think this is the right approach, as $\varphi_*(\alpha)$ would have lower dimension that $\alpha$. Any thoughts on a good way to calculate an intersection number like this?