Let R be a Noetherian ring, and I and J be two ideals of R. Let S be a Serre subcategory of the category of R-modules satisfying the condition CI and M be a ZD-module. As a generalization of the S-depth(I, M) and depth(I, J, M), the S-depth of (I, J) on M is defined as $$S{\rm{ - depth}}\left( {I,J,M} \right) = \inf \left\{ {S{\rm{ - depth}}\left( {\mathfrak{a},M} \right): \mathfrak{a}\in \widetilde{W}\left( {I,J} \right)} \right\}$$ , and some properties of this concept are investigated. The relations between S-depth(I, J, M) and H (M) are studied, and it is proved that S-depth(I, J, M) = inf{i: H (M) ∉ S}, where S is a Serre subcategory closed under taking injective hulls. Some conditions are provided that local cohomology modules with respect to a pair of ideals coincide with ordinary local cohomology modules under these conditions. Let SuppRH (M) be a finite subset of Max(R) for all i < t, where M is an arbitrary R-module and t is an integer. It is shown that there are distinct maximal ideals $$\mathfrak{m}_1,\mathfrak{m}_2,...\mathfrak{m}_k\in{W(I,J)}$$ such that $$H_{I,J}^i\left( M \right) \cong H_{\mathfrak{m}{_1}}^i\left( M \right) \oplus H_{\mathfrak{m}{_2}}^i\left( M \right) \oplus \ldots \oplus H_{\mathfrak{m}_{_k}}^i\left( M \right)$$ for all i < t.