Relative cone of curves and extremal contractions of a successive blowup

Let $X$ be a normal variety, and let $\pi \text{\hspace{0.17em}⁣}:\tilde{X}\to X$ be the successive blowup along subvarieties $Z_1,\dots ,Z_n\subseteq X$ of codimension at least two that have simple normal crossings and satisfy $Z_h\supseteq ̸Z_i$ whenever $h<I$. We prove that the relative cone of curves $\stackrel{}{\mathrm{NE}}(\tilde{X}\mathrm{/}X)$ is generated by the classes of finitely many elementary curves, and that every face admits a contraction over $X$. We describe the exceptional loci of extremal ray contractions, and prove that every small extremal ray contraction admits a $D$-flip for every $\mathbb{R}$-Cartier divisor $D$ negative on the corresponding ray.