The vertical distribution of debris flow profile velocity is the key to studying debris flow, impulse and the sediment carrying process. At present, the linear distribution model based on flume test results cannot describe the vertical distribution of debris flow velocity effectively due to the limitation of measurement methods. In this paper, the smooth particle hydrodynamics (SPH) numerical model based on the Herschel–Bulkley–Papanastasiou (HBP) constitutive model is utilized to invert the three-dimensional dynamic process of debris flow based on a large-scale debris flow flume experiment. With a hierarchical statistical approach, a huge number of particle velocity data were analyzed and processed to obtain the vertical distribution law of velocity. We proposed a nonlinear vertical distribution model of debris flow velocity based on logarithm function accordingly. We also applied the proposed model to the existing debris flow entrainment estimation framework. A flume dam break test case was inverted to verify the performance of erosion calculations. The results show that the numerical simulation results of erosion depth are close to the experimental values. The error percentage of maximum erosion depth is 4.1%. The average error percentage of erosion depth simulation results is 15.5%.

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.