Brauer groups of conic bundles over elliptic curves

We study the Brauer groups of regular conic bundles over elliptic curves defined over a number field $k$. We explicitly compute the Brauer group of the conic bundle when the singular fibres lie above $k$-points that are divisible by $2$ in $E(k)$, and the corresponding ramification fields are isomorphic. We apply the result to compute the Brauer group of a class of surfaces analogous to that of Ch\^{a}telet surfaces. We investigate Brauer-Manin obstructions to weak approximation coming from the real places on such surfaces.