Let $f:U \to S$ be a flat morphism. Let us say that $f$ is ** flatly compactifiable** if there exists a proper morphism $\bar{f}:X \to S$ and a closed subscheme $Z \subset X$ such that

1) $U = X \setminus Z$ and $f = \bar{f}_{|U}$;

2) $\bar{f}$ is proper;

3) BOTH $X$ and $Z$ are flat over $S$.

My question is whether this notion already appeared in the literature and what is the correct name for it?