Wilcock and Crowe

Wilcock (2001) and Wilcock and Crowe (2003) developed a surface-based bed-load transport equation for graded beds with sand and gravel. The transport potential of the Wilcock and Crowe (WC) formula is given by

1)	$\begin{array}{l}\displaystyle q_{bk}^{}=\frac{u_{}^{3}W_{k}^{*}}{R_{k}g}\end{array}$

where

$\begin{array}{l}{q_{bk}}^*\end{array}$ = fractional bed-load sediment transport potential [L²/T]

$\begin{array}{l}u_{*}\end{array}$ = bed shear velocity [L/T]

$\begin{array}{l}{W_{k}}*\end{array}$ = transport function [-]

$\begin{array}{l}R_{k}\end{array}$ = $\begin{array}{l}ρ_{sk}/ρ_w−1\end{array}$ = submerged specific gravity of grain class [-]

$\begin{array}{l}ρ_{sk}\end{array}$ = sediment grain density [M/L³]

$\begin{array}{l}ρ_{w}\end{array}$ = water density [M/L³]

$\begin{array}{l}g\end{array}$ = gravitational constant [L/T^2]

The transport function is given by

2)	$\begin{array}{l}W_{k}^{*}=\begin{cases} 0.002\phi ^{7.5}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\,\,\phi < 1.35\\ 14\left(1-\frac{0.894}{\phi ^{0.5}}\right)^{4.5}\,\,\,\mathrm{for}\,\,\phi \geq 1.35 \end{cases}\right.\end{array}$

where

$\begin{array}{l}ϕ\end{array}$ = $\begin{array}{l}τ_b/τ_{r, k}\end{array}$ = sediment mobility [-]

$\begin{array}{l}τ_{b}\end{array}$ = bed shear stress [-]

$\begin{array}{l}τ_{r, k}\end{array}$ = reference shear stress