In this paper, we deal with the study of intermediate domains between a domain $R$ and a domain $T$ such that $T$ is an intersection of localizations of $R$, namely the pair $\left(R,T\right)$. More precisely, we study the pair $\left(R,R_d\right)$ and the pair $\left(R,\tilde{R}\right)$, where $R_d=\cap \left\{R_M\mid M\in \text{Max}\left(R\right),htM=dimR\right\}$ and $\tilde{R}=\cap \left\{R_M\mid M\in \text{Max}\left(R\right),htM\ge 2\right\}$. We prove that, if $R$ is a Jaffard domain, then $\left(R,R_d\left[n\right]\right)$ is a Jaffard pair, which generalize [5, Théorème 1.9]. We also show that if $R$ is an $S$-domain, then $\left(R,\tilde{R}\right)$ is a residually algebraic pair (that is for each intermediate domain $S$ between $R$ and $\tilde{R}$, if $Q$ is a prime ideal of $S$...