Classification of conic bundles on a rational elliptic surface in any characteristic

Let X be a rational elliptic surface with elliptic fibration \(\pi {:}\,X\rightarrow {\mathbb {P}}^1\) $\pi :X\to {\mathbb{P}}^1$ over an algebraically closed field k of any characteristic. Given a conic bundle \(\varphi {:}\,X\rightarrow {\mathbb {P}}^1\) $\phi :X\to {\mathbb{P}}^1$ we classify all possible fibers of \(\varphi \) $\phi$ and study the interplay between singular fibers of \(\pi \) $\pi$ and \(\varphi \) $\phi$ .