The purpose of this paper is to derive modeling equations for debris flows on real
terrain. Thus, we use curvilinear coordinates adapted to the topography as introduced, e.g., by
Bouchut and Westdickenberg, and develop depth-averaged models of gravity-driven saturated mixtures of solid grains
and pore fluid on an arbitrary rigid basal surface. First, by only specifying the interaction force and
ordering approximations in terms of an aspect ratio between a typical length perpendicular to the
topography, and a typical length parallel to the topography, we derive the governing equations for
the shallow flow of a binary mixture, driven by gravitational force. In doing so, the non-uniformity
through the avalanche depth of the constituent velocities and of the solid volume fraction is accounted
for by coefficients of Boussinesq type. Then, the material behaviour peculiarities of both constituents
properly enter the theory. One constituent is a granular solid. For its stresses we propose three
models, one of them of Mohr-Coulomb type. The other constituent is a Newtonian/non-Newtonian
fluid with small viscosity, obeying a viscous bottom friction condition. The final governing equations
for the shallow flow of the mixture, incorporating the constitutive assumptions, are deduced, and the
limiting equilibrium is then investigated.