The river silting process is complex and represents one of the greatest challenges faced in the future. In this paper, we propose an asymptotic approach for describing silting of rivers to study asymptotic solutions governing a coupled (hydro-sedimentary) model. We focus our paper on the inner and outer asymptotic expansions of solutions. The proposed approach is based on the hypothesis that rivers silting can be explain by an asymptotic behavior near shock curves of the governing equations of sediment deposits. We look for an inner and outer asymptotic expansion which gives us expressions of the solution in the neighborhood of the shock zone. Once both asymptotic expansions found , we give matching property of both expansions.