Sediment Transport Functions

Because different sediment transport functions were developed under different conditions, a wide range of results can be expected from one function to the other. Therefore it is important to verify the accuracy of sediment prediction to an appreciable amount of measured data from either the study stream or a stream with similar characteristics. It is very important to understand the processes used in the development of the functions in order to be confident of its applicability to a given stream.

Typically, sediment transport functions predict rates of sediment transport from a given set of steady-state hydraulic parameters and sediment properties. Some functions compute bed-load transport, and some compute bed-material load, which is the total load minus the wash load (total transport of particles found in the bed). In sand-bed streams with high transport rates, it is common for the suspended load to be orders of magnitude higher than that found in gravel-bed or cobbled streams. It is therefore important to use a transport predictor that includes suspended sediment for such a case.

The following sediment transport functions are available in HEC-RAS:

Ackers-White
Engelund-Hansen
Laursen
Meyer-Peter Müller
Toffaleti
Yang

These functions were selected based on their validity and collective range of applicability. All of these functions, except for Meyer-Peter Müller, are compared extensively by Yang and Schenggan (1991) over a wide range of sediment and hydraulic conditions. Results varied, depending on the conditions applied. The Meyer-Peter Müller and the bed-load portion of the Toffaleti function were compared with each other by Amin and Murphy (1981). They concluded that Toffaleti bed-load procedure was sufficiently accurate for their test stream, whereby, Meyer-Peter Müller was not useful for sand-bed channels at or near incipient motion. The ranges of input parameters used in the development of each function are shown in Table 12-7. Where available, these ranges are taken from those presented in the SAM package user's manual (Waterways Experiment Station, 1998) and are based on the developer's stated ranges when presented in their original papers. The ranges provided for Engelund and Hansen are taken from the database (Guy, et al, 1966) primarily used in that function's development. The parameter ranges presented are not limiting, in that frequently a sediment transport function will perform well outside the listed range. For example, Engelund-Hansen was developed with flume research only, and has been historically applied successfully outside its development range. The parameter ranges are presented as a guideline only.

A short description of the development and applicability of each function follows. It is strongly recommended that a review of the respective author's initial presentation of their function be undertaken prior to its use, as well as a review of "comparison" papers such as those referenced in the preceding paragraph. References are included in "References". Sample solutions for the following sediment transport methods are presented in "Sediment Transport Functions – Sample Calculations".

Table 12-8 Range of input values for sediment transport functions (Sam User's Manual, 1998)

Function	d	d_m	s	V	D	S	W	T
Ackers-White (flume)	0.04 - 7.0	NA	1.0 - 2.7	0.07 - 7.1	0.01 - 1.4	0.00006 - 0.037	0.23 - 4.0	46 - 89
Englund-Hansen (flume)	NA	0.19 - 0.93	NA	0.65 – 6.34	0.19 – 1.33	0.000055 – 0.019	NA	45 - 93
Laursen ( field)	NA	0.08– 0.7	NA	0.068 – 7.8	0.67 – 54	0.0000021 – 0.0018	63 – 3640	32 - 93
Laursen (flume)	NA	0.011 -29	NA	0.7 - 9.4	0.03 – 3.6	0.00025 – 0.025	0.25 – 6.6	46 - 83
Meyer-Peter Muller (flume)	0.4 – 29	NA	1.25 – 4.0	1.2 – 9.4	0.03 – 3.9	0.0004 – 0.02	0.5 – 6.6	NA
Tofaletti ( field)	0.062 – 4.0	0.095 – 0.76	NA	0.7 - 7.8	0.07 – 56.7 (R)	0.000002 – 0.0011	63 - 3640	32 – 93
Tofaletti (flume)	0.062 – 4.0	0.45 – 0.91	NA	0.7 - 6.3	0.07 – 1.1 (R)	0.00014 – 0.019	0.8 – 8	40 - 93
Yang (field-sand)	0.15 – 1.7	NA	NA	0.8 - 6.4	0.04 – 50	0.000043 – 0.028	0.44 – 1750	32 - 94
Yang (field-gravel)	2.5 – 7.0	NA	NA	1.4 - 5.1	0.08 – 0.72	0.0012 – 0.029	0.44 – 1750	32 - 94

Where:

$\begin{array}{l}d\end{array}$ = Overall particle diameter, mm
$\begin{array}{l}d_m\end{array}$ = Median particle diameter, mm
$\begin{array}{l}s\end{array}$ = Sediment specific gravity
$\begin{array}{l}V\end{array}$ = Average channel velocity, fps
$\begin{array}{l}D\end{array}$ = Channel depth, ft
$\begin{array}{l}S\end{array}$ = Energy gradient
$\begin{array}{l}W\end{array}$ = Channel width, ft
$\begin{array}{l}T\end{array}$ = Water temperature, ^oF
$\begin{array}{l}(R)\end{array}$ = Hydraulic Radius, ft
$\begin{array}{l}NA\end{array}$ = Data not available

Ackers-White: The Ackers-White transport function is a total load function developed under the assumption that fine sediment transport is best related to the turbulent fluctuations in the water column and coarse sediment transport is best related to the net grain shear with the mean velocity used as the representative variable. The transport function was developed in terms of particle size, mobility, and transport.

A dimensionless size parameter is used to distinguish between the fine, transitionary, and coarse sediment sizes. Under typical conditions, fine sediments are silts less than 0.04 mm, and coarse sediments are sands greater than 2.5 mm. Since the relationships developed by Ackers-White are applicable only to non-cohesive sands greater than 0.04 mm, only transitionary and coarse sediments apply. Original experiments were conducted with coarse grains up to 4 mm, however the applicability range was extended to 7 mm.

This function is based on over 1000 flume experiments using uniform or near-uniform sediments with flume depths up to 0.4 m. A range of bed configurations was used, including plane, rippled, and dune forms, however the equations do not apply to upper phase transport (e.g. anti-dunes) with Froude numbers in excess of 0.8.

The general transport equation for the Ackers-White function for a single grain size is represented by:

1)	$\begin{array}{l}\displaystyle \displaystyle X= \frac{G_{gr}sd_s}{D \left( \frac{u_*}{V} \right) ^n}\end{array}$

and

2)	$\begin{array}{l}\displaystyle \displaystyle G_{gr} = C \left( \frac{F_{gr}}{A} - 1 \right)\end{array}$

Symbol	Description	Units
$\begin{array}{l}X\end{array}$	Sediment concentration, in parts per part
$\begin{array}{l}G_{gr}\end{array}$	Sediment transport parameter
$\begin{array}{l}s\end{array}$	Specific gravity of sediments
$\begin{array}{l}d_s\end{array}$	Mean particle diameter
$\begin{array}{l}D\end{array}$	Effective depth
$\begin{array}{l}u_*\end{array}$	Shear velocity
$\begin{array}{l}V\end{array}$	Average channel velocity
$\begin{array}{l}n\end{array}$	Transition exponent, depending on sediment size
$\begin{array}{l}C\end{array}$	Coefficient
$\begin{array}{l}F_{gr}\end{array}$	Sediment mobility parameter
$\begin{array}{l}A\end{array}$	Critical sediment mobility parameter

A hiding adjustment factor was developed for the Ackers-White method by Profitt and Sutherland (1983), and is included in RAS as an option. The hiding factor is an adjustment to include the effects of a masking of the fluid properties felt by smaller particles due to shielding by larger particles. This is typically a factor when the gradation has a relatively large range of particle sizes and would tend to reduce the rate of sediment transport in the smaller grade classes.

Engelund-Hansen: The Engelund-Hansen function is a total load predictor which gives adequate results for sandy rivers with substantial suspended load. It is based on flume data with sediment sizes between 0.19 and 0.93 mm. It has been extensively tested, and found to be fairly consistent with field data.

The general transport equation for the Engelund-Hansen function is represented by:

3)	$\begin{array}{l}\displaystyle \displaystyle g_s = 0.05 \gamma _s V^2 \sqrt{\frac{d_{50}}{g \left( \frac{\gamma _s}{\gamma} -1 \right)}} \left[ \frac{\tau _0}{(\gamma _s -\gamma ) d_{50}} \right] ^{3/2}\end{array}$

Symbol	Description	Units
$\begin{array}{l}g_s\end{array}$	Unit sediment transport
$\begin{array}{l}\gamma\end{array}$	Unit wt of water
$\begin{array}{l}\gamma _s\end{array}$	Unit wt of solid particles
$\begin{array}{l}V\end{array}$	Average channel velocity
$\begin{array}{l}\tau _0\end{array}$	Bed level shear stress
$\begin{array}{l}d_{50}\end{array}$	Particle size of which 50% is smaller

Laursen: The Laursen method is a total sediment load predictor, derived from a combination of qualitative analysis, original experiments, and supplementary data. Transport of sediments is primarily defined based on the hydraulic characteristics of mean channel velocity, depth of flow, energy gradient, and on the sediment characteristics of gradation and fall velocity. Contributions by Copeland (Copeland, 1989) extend the range of applicability to gravel-sized sediments. The range of applicability is 0.011 to 29 mm, median particle diameter.
The general transport equation for the Laursen (Copeland) function for a single grain size is represented by:

4)	$\begin{array}{l}\displaystyle \displaystyle c_m =0.01 \gamma \left( \frac{d_s}{D} \right) ^{7/6} \left( \frac{\tau _0}{\tau _c} -1 \right) f \left( \frac{u_*}{\omega} \right)\end{array}$

Symbol	Description	Units
$\begin{array}{l}c_m\end{array}$	Sediment discharge concentration, in weight/volume
$\begin{array}{l}G\end{array}$	Unit weight of water
$\begin{array}{l}d_s\end{array}$	Mean particle diameter
$\begin{array}{l}D\end{array}$	Effective depth of flow
$\begin{array}{l}\tau _0\end{array}$	Bed shear stress due to grain resistance
$\begin{array}{l}\tau _c\end{array}$	Critical bed shear stress
$\begin{array}{l}\displaystyle f \left( \frac{u_*}{\omega} \right)\end{array}$	Function of the ratio of shear velocity to fall velocity as defined in Laursen's Figure 14 (Laursen, 1958).

Meyer-Peter Müller: The Meyer-Peter Müller bed load transport function is based primarily on experimental data and has been extensively tested and used for rivers with relatively coarse sediment. The transport rate is proportional to the difference between the mean shear stress acting on the grain and the critical shear stress.
Applicable particle sizes range from 0.4 to 29 mm with a sediment specific gravity range of 1.25 to in excess of 4.0. This method can be used for well-graded sediments and flow conditions that produce other-than-plane bed forms. The Darcy-Weisbach friction factor is used to define bed resistance. Results may be questionable near the threshold of incipient motion for sand bed channels as demonstrated by Amin and Murphy (1981).

The general transport equation for the Meyer-Peter Müller function is represented by:

5)	$\begin{array}{l}\displaystyle \displaystyle \left( \frac{k_r}{k'_r} \right) ^{3/2} \gamma RS=0.047 (\gamma _s - \gamma ) d_m +0.25 \left( \frac{\gamma}{g} \right) ^{1/3} \left( \frac{\gamma _s - \gamma}{\gamma _s} \right) ^{2/3} g^{2/3}_s\end{array}$

Symbol	Description	Units
$\begin{array}{l}g_s\end{array}$	Unit sediment transport rate in weight/time/unit width
$\begin{array}{l}k_r\end{array}$	A roughness coefficient
$\begin{array}{l}k_r'\end{array}$	A roughness coefficient based on grains
$\begin{array}{l}\gamma\end{array}$	Unit weight of water
$\begin{array}{l}\gamma _s\end{array}$	Unit weight of the sediment
$\begin{array}{l}g\end{array}$	Acceleration of gravity
$\begin{array}{l}d_m\end{array}$	Median particle diameter
$\begin{array}{l}R\end{array}$	Hydraulic radius
$\begin{array}{l}S\end{array}$	Energy gradient

Toffaleti: The Toffaleti method is a modified-Einstein total load function that breaks the suspended load distribution into vertical zones, replicating two-dimensional sediment movement. Four zones are used to define the sediment distribution. They are the upper zone, the middle zone, the lower zone and the bed zone. Sediment transport is calculated independently for each zone and the summed to arrive at total sediment transport.

This method was developed using an exhaustive collection of both flume and field data. The flume experiments used sediment particles with mean diameters ranging from 0.3 to 0.93 mm, however successful applications of the Toffaleti method suggests that mean particle diameters as low as 0.095 mm are acceptable.

The general transport equations for the Toffaleti function for a single grain size is represented by:

6)	$\begin{array}{l}\displaystyle \displaystyle g_{ssL} =M \frac{\left( \frac{R}{11.24} \right) ^{1+n_{\nu} -0.756z} - (2d_m)^{1+n_{\nu} -0.756z}}{1+n_{\nu} -0.756z} \;(lower \;zone)\end{array}$

7)	$\begin{array}{l}\displaystyle \displaystyle g_{ssM} =M \frac{\left( \frac{R}{11.24} \right) ^{0.244z} \left[ \left( \frac{R}{2.5} \right) ^{1+n_{\nu} -z} - \left( \frac{R}{11.24} \right) ^{1+n_{\nu} -z} \right]}{1+n_{\nu} -z} \;(middle \;zone)\end{array}$

8)	$\begin{array}{l}\displaystyle \displaystyle g_{ssU} =M \frac{\left( \frac{R}{11.24} \right) ^{0.244z} \left( \frac{R}{2.5} \right) ^{0.5z} \left[ R^{1+n_{\nu} -z} - \left( \frac{R}{2.5} \right) ^{1+n_{\nu} -z} \right]}{1+n_{\nu} -1.5z} \;(upper \;zone)\end{array}$

9)	$\begin{array}{l}\displaystyle g_{sb} = M(2d_m) ^{1+n_{\nu} -0.756z} \;(bed \;zone)\end{array}$

10)	$\begin{array}{l}\displaystyle M=43.2C_L(1+n_{\nu} )VR^{0.756z-n_{\nu}}\end{array}$

11)	$\begin{array}{l}\displaystyle g_s = g_{ssL} + g_{ssM} + g_{ssU} + g_{sb}\end{array}$

Symbol	Description	Units
$\begin{array}{l}g_{ssL}\end{array}$	Suspended sediment transport in the lower zone	tons/day/ft
$\begin{array}{l}g_{ssM}\end{array}$	Suspended sediment transport in the middle zone	tons/day/ft
$\begin{array}{l}g_{ssU}\end{array}$	Suspended sediment transport in the upper zone	tons/day/ft
$\begin{array}{l}g_{sb}\end{array}$	Bed load sediment transport	tons/day/ft
$\begin{array}{l}g_s\end{array}$	Total sediment transport	tons/day/ft
$\begin{array}{l}M\end{array}$	Sediment concentration parameter
$\begin{array}{l}C_L\end{array}$	Sediment concentration in the lower zone
$\begin{array}{l}R\end{array}$	Hydraulic radius
$\begin{array}{l}d_m\end{array}$	Median particle diameter
$\begin{array}{l}z\end{array}$	Exponent describing the relationship between the sediment and hydraulic characteristics
$\begin{array}{l}n_{\nu}\end{array}$	Temperature exponent

Yang: Yang's method (1973) is developed under the premise that unit stream power is the dominant factor in the determination of total sediment concentration. The research is supported by data obtained in both flume experiments and field data under a wide range conditions found in alluvial channels. Principally, the sediment size range is between 0.062 and 7.0 mm with total sediment concentration ranging from 10 ppm to 585,000 ppm. Channel widths range from 0.44 to1746 ft, depths from 0.037 to 49.4 ft, water temperature from 0 $\begin{array}{l}^{\circ}\end{array}$ to 34.3 $\begin{array}{l}^{\circ}\end{array}$ Celsius, average channel velocity from 0.75 to 6.45 fps, and slopes from 0.000043 to 0.029.

Yang (1984) expanded the applicability of his function to include gravel-sized sediments. The general transport equations for sand and gravel using the Yang function for a single grain size is represented by:

12)	$\begin{array}{l}\displaystyle logC_t =5.435-0.286log \frac{\omega d_m}{\nu} -0.457log \frac{u_}{\omega} + \left( 1.799 - 0.409log \frac{\omega d_m}{\nu} -0.314log \frac{u_}{\omega} \right) log \left( \frac{VS}{\omega} - \frac{V_{cr}S}{\omega} \right) \;for \;sand \;d_m \;< \;2mm\end{array}$

13)

$\begin{array}{l}\displaystyle logC_t =6.681-0.633log \frac{\omega d_m}{\nu} -4.816log \frac{u_*}{\omega} + \left( 2.784 - 0.305log \frac{\omega d_m}{\nu} -0.282log \frac{u_*}{\omega} \right) log \left( \frac{VS}{\omega} - \frac{V_{cr}S}{\omega} \right) \;for \;gravel \;d_m \; \geq \;2mm\end{array}$

Symbol	Description	Units
$\begin{array}{l}C_t\end{array}$	Total sediment concentration
$\begin{array}{l}\omega\end{array}$	Particle fall velocity
$\begin{array}{l}d_m\end{array}$	Median particle diameter
$\begin{array}{l}\nu\end{array}$	Kinematic viscosity
$\begin{array}{l}u_*\end{array}$	Shear velocity
$\begin{array}{l}V\end{array}$	Average channel velocity
$\begin{array}{l}S\end{array}$	Energy gradient