Chow group of a specific scheme related to the localization of the affine line at two points.

Question

Given a scheme $X$, let $S=\varprojlim\limits_{U\subset X\times \mathbb{A}^1}U$ where $U$ is a Zariski open that contains both of $X\times\{1\}$ and $X\times\{0\}$. What does the Chow group $CH^i(S)$ look like? Is it $CH^i(X)\oplus CH^i(X)$?