A Simplex Solid–Fluid Model for Debris Flows over Erodible Beds with Multi-Size Sediments

Debris flows are composed of solid grains and fluid, in which the grains span a range of size, and the interstitial fluid is viscous. Erosion and deposition processes have significant impacts on post-event morphology, and their mechanisms are closely related with the grains and the viscosity of the interstitial fluid within the flow body. In the present study, we present a two-phase erodible model, extended from Wong et al., (2024) in which mono-grain-size is assumed, for modeling heterogeneous grain-fluid mixtures composed of multiple grain sizes and a viscous interstitial fluid. That is, the solid phase within the flow body is supposed to consist of grains of various sizes. In this simplex approach, the effects of grain size are explicitly incorporated into the erosion-deposition processes. The erosion rate is proportional to the shear stress and follows the Shields parameter (Shields, 1936) for the threshold magnitude, while deposition is assumed to be induced by settling speed and to follow the regressed Hjulström-Sundborg diagram (Hjulström, 1935). Because both the Shields parameter and the settling speed depend on grain size and fluid viscosity, the resulting entrainment or deposit patterns vary with the grain-size composition of the flow body. For example, sediments of smaller size are entrained first and settle latter, whereas larger grains tend to deposit at earlier stage. The key features of this simplex approach will be demonstrated through numerical investigations on  flows in chutes with simple geometry, as well as through an application to a back-calculation of a large-scale historical event.