Total-load Transport Equation

The total-load sediment transport is the sum of all particles transported. The total-load may be divided into bed and suspended loads as a function of the transport mode. The total-load transport equation may be written as

1)	$\begin{array}{l}\displaystyle \frac{\partial }{\partial t}\left(\frac{hC_{tk}}{\beta _{tk}}\right)+\nabla \cdot (h\boldsymbol{U}C_{tk})=\nabla \cdot \left(\varepsilon _{tk}h\nabla C_{tk}\right)+E_{tk}^{HF}-D_{tk}^{HF}+S_{tk}\end{array}$

where

$\begin{array}{l}C_{tk}\end{array}$ = total-load sediment concentration of the kth grain class [M/L³]

$\begin{array}{l}β_{tk}\end{array}$ = total-load correction factor for the kth grain class

$\begin{array}{l}U\end{array}$ = depth-averaged current velocity in j^th - direction [L/T]

$\begin{array}{l}h\end{array}$ = water depth [L]

$\begin{array}{l}ε_{tk }\end{array}$ = total-load diffusion (mixing) coefficient corresponding to the k^th grain class}

$\begin{array}{l}{E_{tk}}^{HF}\end{array}$ = total-load erosion rate in hydraulic flow [M/L²/T]

$\begin{array}{l}{D_{tk}}^{HF }\end{array}$ = \text{total-load deposition rate in hydraulic flow }[M/L²/T]

$\begin{array}{l}S_{tk }\end{array}$ = total-load source/sink term [M/L²/T]

The main advantage for solving the total-load transport formula instead of separate bed- and suspended-load transport equations is the reduced computational costs since it requires one less transport equation solution and also simplifies the bed change and sorting computations.

It is noted that the velocity-weighted concentration definition is utilized. This definition results in the total-load correction factor appearing in the temporal term whereas the depth-averaged concentration definition results in an advection coefficient in the of the advection terms. Experience has shown that when simulating bedload dominate transport, the advection coefficient can have sharp spatial variations which may not be consistent with the transport capacity leading to unrealistic results. The load correction factor in the temporal on the other hand is extremely well behaved even when its value varies significantly in space and/or time.

The above formulation is utilized for both cohesive and noncohesive sediments. Erosion is computed differently for cohesive and noncohesive sediment grain classes depending on the grain size and the bed composition. Erosion is also computed differently for the hydraulically wet and dry portions of the domain. The hydraulically wet portion is the region which is submerged by water and erosion is primarily due to bottom shear stresses. The hydraulically dry portion is the region where the erosion is primarily due to precipitation splash and surface runoff in the form of sheet flow. The source/sink term S_tk includes boundary conditions including surface runoff S_tk^SR.