base change of line bundles

Question

Under what kind of assumptions on $X$ and $k$ (here $X$ is a scheme over $k$ and $\bar{k}$ the algebraic closure) is the map $\mathcal{L}\mapsto \mathcal{L}\otimes_k\bar{k}$ from $\mathrm{Pic}(X)$ to $\mathrm{Pic}(X\otimes_k\bar{k})$ injective?