Total-load

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)	$\begin{array}{l}\displaystyle \frac{\partial }{\partial t}\left(\frac{h C_{tk}}{\beta_{tk}}\right) + \nabla \cdot \left( h \boldsymbol{U} C_{tk} \right) = \nabla \cdot \left(\varepsilon_{thk} h \nabla C_{tk} \right) + E_{tk} - D_{tk}\end{array}$

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 vector [L/T]
$\begin{array}{l}\boldsymbol{U}_{tk}\end{array}$ : total-load correction velocity [L/T]
$\begin{array}{l}h\end{array}$ : water depth [L]
$\begin{array}{l}ε_{thk }\end{array}$ : total-load diffusion (mixing) coefficient corresponding to the k^th grain class [L²/T]
$\begin{array}{l}E_{tk}\end{array}$ : total-load erosion rate [M/L²/T]
$\begin{array}{l}D_{tk}\end{array}$ : total-load deposition rate [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.

Fraction of Suspended Sediments

The fraction of suspended sediments is defined as:

2)	$\begin{array}{l}\displaystyle r_{sk} = \frac{q_{sk}}{q_{tk}}\end{array}$

where

$\begin{array}{l}r_{sk}\end{array}$ : fraction of suspended sediments [-]
$\begin{array}{l}q_{sk}\end{array}$ : suspended-load transport rate [M/L/T]
$\begin{array}{l}q_{tk}\end{array}$ : total-load transport rate [M/L/T]

The fraction of suspended sediments is needed in the total-load transport model in order to separate the contributions from bed- and suspend-load to various parameters including the horizontal mixing coefficient and advection coefficients. The parameter is needed in order to close the system of equations. The fraction of suspended sediments is approximated by the transport mode parameter which is the ration of the suspended-load to total-load transport potential rates:

3)	$\begin{array}{l}\displaystyle r_{sk}\approx f_{sk} = \frac{q_{sk}^}{q_{tk}^}\end{array}$

where

$\begin{array}{l}f_{sk}\end{array}$ : transport mode parameter [-]
$\begin{array}{l}q_{sk}^*\end{array}$ : suspended-load transport capacity [M/L/T]
$\begin{array}{l}q_{tk}^*\end{array}$ : total-load transport capacity [M/L/T]

There methods available to estimate the transport mode parameter:

1. Transport capacity method (Wu 2007)
2. Rouse parameter method of Greimann et al. (2008)
3. van Rijn (1984)

Figure 1. Representative transport mode parameter curves as a function of the Rouse number with $\begin{array}{l}d\end{array}$ = 1 mm, $\begin{array}{l}\rho_{w}\end{array}$ = 1000 kg/m³, and $\begin{array}{l}\rho_{s}\end{array}$ = 2650 kg/m³.

Transport Capacity Method

This is the simplest and preferred method.

4)	$\begin{array}{l}\displaystyle f_{sk} =\frac{q_{sk}^}{q_{tk}^}\end{array}$

The default option is the transport capacity method. If the total-load transport capacity formula may be written in terms of bed and suspended capacities, the transport capacity method is utilized. This is the most consistent approach with the transport formulas. However, if the transport formula is a bed-material formula such as the Laursen formula, another approach must be used.

Greimann et al.

The transport mode parameter proposed by Greimann et al. (2008) is as follows

5)	$\begin{array}{l}\displaystyle f_{sk} = \min[1, ~2.5\exp(−r_k)]\end{array}$

where

$\begin{array}{l}r_{k}\end{array}$ : Rouse parameter [-]
$\begin{array}{l}ω_{sk}\end{array}$ : sediment settling velocity [L/T]
$\begin{array}{l}u_{*}\end{array}$ : bed shear velocity [L/T]

van Rijn

An alternative formulation for estimating the fraction of suspended sediments was given by van Rijn (1984)

6)	$\begin{array}{l}\displaystyle f_{sk} = 0.25 + 0.325 \ln(u_*/ \omega_{sk})\end{array}$

where

$\begin{array}{l}ω_{sk}\end{array}$ : sediment settling velocity [L/T]
$\begin{array}{l}u_{*}\end{array}$ : bed shear velocity [L/T]

Total-load Correction Factor

The total-load correction advection coefficient, β_tk, accounts for the vertical distribution of the suspended sediment concentration and velocity profiles, and the generally slower bed-load velocity compared to the depth-averaged current velocity (see figure below) (Wu 2007).

Figure 2.9: Schematic of sediment and current velocity profiles.

The total-load correction factor is given by

7)	$\begin{array}{l}\displaystyle \beta_{tk} = \frac{1}{r_{sk}/\beta_{sk} + (1-r_{sk})/\beta_{bk}}\end{array}$

where

$\begin{array}{l}r_{sk}\end{array}$ : fraction of suspended-load [-]
$\begin{array}{l}\beta_{sk}\end{array}$ : suspended-load correction factor [-]
$\begin{array}{l}\beta_{bk}\end{array}$ : bed-load correction factor [-]

It is noted that the sediment transport is assumed to be in the same direction as the flow. Therefore, the influence of the bed-slope on the bedload is not included here. This effect is included separately in the model as an additional term in the bed-change equation. The user is given several options on how to compute the total-load correction factor including specifying a constant or computing it from the bed- and suspended-load correction factors which also may be computed with one of several methods or specified as a constant.

Total-load Horizontal Diffusion Coefficient

The total-load horizontal mixing/diffusion coefficient is determined as

8)	$\begin{array}{l}\displaystyle \varepsilon_{thk} = r_{sk} \varepsilon_{shk} + (1 − r_{sk}) \varepsilon_{bhk} \quad \text{for } k = 1,\hdots, N\end{array}$

where

$\begin{array}{l}r_{sk}\end{array}$ : fraction of suspended-load [-]
$\begin{array}{l}ε_{shk}\end{array}$ : suspended-load horizontal mixing coefficient [L²/T]
$\begin{array}{l}ε_{bhk}\end{array}$ : bed-load horizontal mixing coefficient [L²/T]

The calculation of the suspended- and bed-load mixing coefficients is described in their respective sections.