A depth-averaged two-phase model for debris flows over erodible beds

Mass exchange between debris flow and the bed plays a vital role in debris flow dynamics. Here a depth-averaged two-phase model is proposed for debris flows over erodible beds. Compared to previous depth-averaged two-phase models, the present model features a physical step forward by explicitly incorporating the mass exchange between the flow and the bed. A widely used closure model in fluvial hydraulics is employed to estimate the mass exchange between the debris flow and the bed, and an existing relationship for bed entrainment rate is introduced for comparison. Also, two distinct closure models for the bed shear stresses are evaluated. One uses the Coulomb friction law and Manning's equation to determine the solid and fluid resistances respectively, while the other employs an analytically derived formula for the solid phase and the mixing length approach for the fluid phase. A well-balanced numerical algorithm is applied to solve the governing equations of the model. The present model is first shown to reproduce average sediment concentrations in steady and uniform debris flows over saturated bed as compared to an existing formula underpinned by experimental datasets. Then, it is demonstrated to perform rather well as compared to the full set of USGS large-scale experimental debris flows over erodible beds, in producing debris flow depth, front location and bed deformation. The effects of initial conditions on debris flow mass and momentum gain are resolved by the present model, which explicitly demonstrates the roles of the wetness, porosity and volume of bed sediments in affecting the flow. By virtue of extended modeling cases, the present model produces debris flow efficiency that, as revealed by existing observations and empirical relations, increases with initial volume, which is enhanced by mass gain from the bed.