In this paper a system of depth-integrated equations for over-saturated debris flows on three-dimensional topography is derived. The lower layer is a saturated mixture of density preserving solid and fluid constituents, where the pore fluid is in excess, so that an upper fluid layer develops above the mixture layer. At the layer interface fluid mass exchange may exist and for this a parameterization is needed. The emphasis is on the description of the influence on the flow by the curvature of the basal surface, and not on proposing rheological models of the avalanching mass. To this end, a coordinate system fitted to the topography has been used to properly account for the geometry of the basal surface. Thus, the modeling equations have been written in terms of these coordinates, and then simplified by using (1) the depth-averaging technique and (2) ordering approximations in terms of an aspect ratio ϵ which accounts for the scale of the flowing mass. The ensuing equations have been complemented by closure relations, but any other such relations can be postulated. For a shallow two-layer debris with clean water in the upper layer, flowing on a slightly curved surface, the equilibrium free surface is shown to be horizonta
