The incipient motion safety factor of sediment considers the coupling between the surface and subsurface flows under multi-motion modes

•Beavers-Joseph boundary conditions realize the coupling of surface–subsurface flows.•Derived incipient motion velocity analytical solutions for multi-motion modes.•Discussed the impact of particle characteristics on sediment incipient motion. The incipient movement of sediment particles in natural rivers under the action of water scour has remained a key issue in river sediment movement research. In this study, we establish a mathematical coupling model between surface and subsurface flows for investigating the interface velocity. The Navier-Stokes equation and Darcy's law are employed to describe surface and subsurface water flows, respectively. The interface velocity U between surface and subsurface flows is derived by introducing the Beavers–Joseph boundary conditions. Furthermore, based on the force equilibrium of a single particle, combined with three incipient modes, namely, sliding, rolling, and saltation, analytical solutions are derived for the critical instantaneous bottom velocity u0, and mean incoming velocity. The derived analytical solutions exhibit a higher computational accuracy than existing flume test data and classical flow velocity model results, with relative errors less than 12 %. Furthermore, the incipient motion state of sediment particles can be directly assessed by defining the critical incipient safety factor for particle motion (i.e., k = U/u0). Sensitivity analysis of the safety factor for sediment particle motion reveals a nonlinear positive correlation with the slope angle, riverbed permeability, riverbed porosity, and hydraulic gradient, while indicating a linear positive correlation with the water depth. However, a nonlinear negative correlation is observed with the B-J coefficient, relative exposure degree, and sediment particle dry bulk density. The safety factor shows a rapid decrease followed by a stabilizing trend with increasing B-J coefficient. Additionally, the safety factor first increases and then decreases with increasing in sediment particle size.