Shallow two-component gravity-driven flows with vertical variation

Gravity-driven geophysical mass flows often consist of a heterogeneous fluid-solid mixture. The complex interplay between the components leads to phenomena such as lateral levee formation in avalanches, or a granular front and an excess fluid pore pressure in debris flows. These effects are very important for predicting runout and the forces on structures, yet they are only partially represented in simplified shallow flow theories, since rearrangement of the mixture composition perpendicular to the main flow direction is neglected. In realistic flows, however, rheological properties and effective basal drag may depend strongly on the relative concentration of the components. We address this problem and present a depth-averaged model for shallow mixtures that explicitly allows for rearrangement in this direction. In particular we consider a fluid-solid mixture that experiences bulk horizontal motion, as well as internal sedimentation and resuspension of the particles, and therefore resembles the case of a debris flow. Starting from general mixture theory we derive bulk balance laws and an evolution equation for the particle concentration. Depth-integration yields a shallow mixture flow model in terms of bulk mass, depth-averaged particle concentration, the particle vertical centre of mass and the depth-averaged velocity. This new equation in this model for the particle vertical centre of mass is derived by taking the first moment, with respect to the vertical coordinate, of the particle mass conservation equation. Our approach does not make the Boussinesq approximation and results in additional terms coupling the momentum flux to the vertical centre of mass. The system is hyperbolic and reduces to the shallow-water equations in the homogeneous limit of a pure fluid or perfect mixing. We highlight the effects of sedimentation on resuspension and finally present a simple friction feedback which qualitatively resembles a large-scale experimental debris flow data set acquired at the Illgraben, Switzerland.