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

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