Let $X$ be a normal projective rational surface over $\mathbb{C}$ with finitely generated divisor class group $\text{Cl}(X)$. Consider the exact sequence $$0 \rightarrow \text{Pic}(X) \rightarrow \text{Cl}(X) \rightarrow \bigoplus_{x \in X^{\text{sing}}} \text{Cl}(\mathcal{O}_{X,x})$$ and the injections $\text{Cl}(\mathcal{O}_{X,x}) \hookrightarrow \text{Cl}(\hat{\mathcal{O}}_{X,x})$ where $\hat{\mathcal{O}}_{X,x}$ is the completion of the local ring of $x$.

What criteria are there for the right arrow to be surjective? Or generally, can you say anything specific about the image of $\text{Cl}(X)$?

My situation is pretty specific so I am willing to assume further that $\text{Pic}(X)$ is free and $[\text{Cl}(X) : \text{Pic}(X)] < \infty$ if that helps.