Sediment Transport Rate Calculator

Bedload transport is a threshold process, which is why this calculation proceeds in two stages rather than one. The first stage compares the tractive force the flow exerts on the bed, tau = rho g d S, against the submerged weight of a representative grain. Normalizing one by the other gives the dimensionless Shields parameter theta = tau / ((rho_s - rho_w) g D). With quartz sediment at 2650 kg/m3 in water at 1000 kg/m3, the submerged specific gravity is 1.65, and grains begin to move once theta rises past a critical value near 0.047 for fully turbulent flow over coarse beds. Below that threshold the transport rate is genuinely zero, not merely small. The second stage assumes transport scales with how far the flow exceeds threshold. Meyer-Peter and Muller expressed this as a dimensionless rate phi = 8 (theta - theta_c)^1.5, converted to volume per unit width by multiplying by sqrt(1.65 g D^3). The 1.5 power means transport is extremely sensitive to depth and slope: a 20 percent rise in shear stress can double the rate. Practical limits matter. The formula was fitted to near-uniform sediment between roughly 0.4 and 30 mm and predicts bedload only, so it will badly understate total load in sand-bed rivers where suspension dominates. It also uses total bed shear, whereas dunes and bars can absorb half or more of that stress as form drag that never reaches individual grains, and in mixed-size beds hiding and exposure shift the effective threshold for each fraction.

The threshold half of this calculation traces to Albert Shields, whose 1936 doctoral work at the Prussian research institute for hydraulic engineering and shipbuilding in Berlin used flume experiments with grains of differing density to collapse incipient motion data onto a single dimensionless curve plotted against grain Reynolds number. That diagram gave sedimentology its first genuinely general entrainment criterion and is still reproduced in modern textbooks, though later workers noted his critical values were partly extrapolated and have proposed adjustments. The transport half comes from Eugen Meyer-Peter and Robert Muller at the hydraulics laboratory of ETH Zurich, who published their bedload formula in 1948 at the International Association for Hydraulic Research congress in Stockholm. It rested on decades of Swiss flume runs with gravel of varied size and density, undertaken in part because Alpine rivers and their reservoirs posed acute sedimentation problems. Their excess-shear form proved durable enough to remain a default in engineering practice. Parallel approaches followed different logic: Hans Albert Einstein published a probabilistic bedload function as a USDA technical bulletin in 1950, treating grain motion as a statistical exchange between bed and flow, while Engelund and Hansen offered a total-load relation for sand beds in 1967. More recent surface-based formulations, notably Wilcock and Crowe in 2003, explicitly handle sand-gravel mixtures that the original single-diameter formulas were never built for.