Let $X$ be a variety and $i:Y\hookrightarrow X$ be a regularly embedded closed subvariety, and write $\tilde X$ for $Bl_Y X$ the blow-up of $X$ along $Y$. Let $A^*(X)$ refer to the Fulton-Macpherson operational Chow groups of $X$.
If one assumes $i^*:A^*(X)\rightarrow A^*(Y)$ is surjective, there is a result of Keel (1992, Trans. Amer. Math. Soc., the appendix) that describes $A^*(\tilde X)$ in terms of $A^*(X)$ and the Chern classes $c_i(N_Y X)$ of the normal bundle to $Y$ in $X$. In more detail, $A^*(\tilde X)$ is isomorphic to $A^*(X)[T]/(P(T),(T\cdot ker(i^*)))$, where $P(T)\in A^*(X)[T]$ is any polynomial with constant term $[Y]$ and whose restriction to $A^*(Y)$ is the Chern polynomial of the normal bundle $N_Y X$.
If $Y$ is not regularly embedded, then instead of a normal bundle there is only the normal cone $C_Y X$. Cones do not have Chern classes, but they have Segre classes, and so I am wondering:
Has anyone developed a description of $A^*(\tilde X)$ in terms of $A^*(X)$ and the Segre classes $s_i(C_Y X)$? Even special cases (e.g. toric varieties) are welcome.