Unification — removing equations and updating the solution

Question

This question is concerned with the first order unification. Suppose I have a set $D$ of equations and a solution to these equations. Let this solution be a set $S$ of substitutions. Now, suppose I remove some equations from $D$. Is there an algorithm that just uses these removed equations and quickly updates $D$, or the unification needs to be done from scratch?

(In my application, I tend to have a lot of equations, so that is why I am interested in a solution that does not recompute everything.)