Estimation of velocity profile through mud depth is a long-standing and essential problem in debris-flow dynamics. Until now, various velocity profiles have been proposed based on the regression of experimental measurements, but these are often limited by the observation conditions, such as the number of the configured sensors. Therefore, the resulting linear velocity profiles exhibit limitations in reproducing the nonlinear behavior and its temporal variation during the debris-flow process. In this study, we present a novel approach to explore debris-flow velocity profile in detail upon our previous 3D-HBP-SPH numerical model, i.e., the three-dimensional Smoothed Particle Hydrodynamic model incorporating with the Herschel-Bulkley-Papanastasiou rheology. Specifically, we propose a stratification statistical algorithm for interpreting the details of SPH particles, which enables the recording of temporal velocities of debris flow at different mud depths. To regress the velocity profile, we introduce a logarithmic-based nonlinear function with two empirical parameters, that controlling the shape of velocity profile and concerning its temporal evolution. We verify the proposed velocity profile and explore its sensitivity using 34 sets of velocity data from three individual flume experiments in previous literatures. Our results demonstrate that the proposed temporal-varying and depth-nonlinear velocity profile outperforms the previous ones.

The computational analysis of debris-flow dynamics and its impact on the structure, i.e., sabo dam, is a long-standing problem for hazard prevention. It is a complex problem that involves fluid-solid coupling and large deformation process of sabo dam, for which three-dimensional numerical simulation remains a scientific challenge until now. The smooth particle hydrodynamics (SPH) and discrete element method (DEM) coupling model can enable the numerical simulation for the large deformation failure of sabo dam under debris-flow impact. For this purpose, built upon our previous Herschel-Bulkley-Papanastasiou (HBP) rheology-based 3D SPH model, the impact forces posed by debris-flow particles acting on the sabo dam are obtained. The sabo dam is modeled by a series of particles with relatively fixed positions in order to generate blocks for simulating their large deformation by DEM, wherein a nonlinear elastic-plastic bond model with a pre-defined bond strength degradation coefficient between DEM blocks is incorporated. To verify the effectiveness of the proposed 3D SPH-DEM numerical coupling model, a simple pier failure case under debris-flow impact is simulated in prior, and the 2010 Yohutagawa debris-flow event, at Amami Oshima Island in Japan is selected as a case study, in which sabo dam with different bond strength degradation coefficients are tested. Results show that the proposed 3D SPH-DEM numerical model well simulates the fluid-solid coupling phenomenon and is able to explore the large deformation of the sabo dam with different strengths under debris-flow impact.