Unified analytical model and theoretical revision of the Hjulström diagram

The classical Hjulström diagram is an empirical rule that qualitatively describes the relationship between flow velocity, particle size, and erosion–transport–deposition. It has long lacked a unified theoretical foundation, which limits the quantitative interpretation of nonlinear variations in these processes. To address this, we establish a unified analytical model for the critical mean flow velocities that govern erosion, transport, and deposition. The model is based on Yang Meiqing’s formula for critical bed shear stress, applicable to the full range of particle sizes, and the Ferguson–Church unified settling velocity formula. It incorporates the logarithmic velocity distribution law and uses the Rouse number as the critical deposition criterion. Theoretically, it demonstrates the U-shaped characteristic of the erosion zone, the “wide for fine and narrow for coarse” feature of the transport zone, and the convex curve of the deposition zone in double-logarithmic coordinates, thereby theoretically revising the traditional understanding in the classical Hjulström diagram that approximates the deposition boundary as a straight line. The correspondence between the unified analytical model and the classical Shields and Rouse theory, which can be regarded as a special case of this unified model under certain conditions, is further analyzed. By explicitly introducing the factors of riverbed inclination angle and water depth, the application scope of the Hjulström diagram is expanded, providing a reliable theoretical basis for quantitatively studying the erosion–transport–deposition relationship of river particles.