Studying the suspended sediments concentration (SSC) is of critical interest for water management. However, acquiring information on suspended sediment concentration (SSC) through direct in situ measurements has always been a challenging task due to technical difficulties especially in flows where the character of suspended materials changes frequently. Ideally, researchers would like to be able to measure the suspended-sediment concentration at all points in a given river. Regarding the cost in terms of material and labor time, they simplify the task to the measurement at all points in a cross section, the measurement along one vertical or even the measurement at one point. Each time the procedure includes a smaller portion of the river, spatial error becomes grater. Regarding all these difficulties, numerical modeling is used instead of or coupled with in situ measurement to optimize the cost of studying suspended sediments concentration. Various numerical and analytical formulas of different order of complexity are used to represent SSC, and knowing the complexity of transport in rivers, it is still impossible to have a universal profile.  In this context, we are interested into establishing an innovative model to predict SSC in river flows. The model suggested in the present study is obtained by combining the properties of the sediment diffusivity coefficients of the parabolic constant model and of the model presented by Itakura and Kishi (1980). The model established is validated with a range of experimental data

CHIBAN, Samia

GHENAIM, Abdellah

POULET, Jean-Bernard

SABAT, Mira

TERFOUS, Abdelali

I-Revues

21ème Congrès de Mécanique                                                                    Bordeaux, 26 au 30 août 2013     
	   1	  
Prediction of suspended sediment concentrations in river 
flows 
A. TERFOUS, M. SABAT, A. GHENAIM, J. B. POULET, S. CHIBAN 
INSA Graduate School of Engineering, 24 Boulevard de la Victoire, Strasbourg France 67084 
Icube UMR 7357 – Laboratoire des Sciences de l’Ingénieur, de l’Informatique et de l’Imagerie   
Résumé : 
Nous proposons un modèle pour calculer la concentration des sédiments transportés en suspension 
dans un écoulement à surface libre en régime uniforme quand la concentration moyenne est connue. 
Le coefficient de diffusion obtenu par l’application de la correction d’Itakura et Kishi à un profil 
parabolique constant est implémenté dans une forme simplifiée de l’équation de convection diffusion. 
Le modèle proposé est validé par des résultats de mesures expérimentales. 
Abstract : 
This paper suggests a method for profiling suspended sediments concentration (SSC) through the 
water column for uniform open channel flows when the mean value of concentration is known. A new 
sediment diffusivity coefficient is defined by applying the Itakura and Kishi correction method to the 
well known parabolic constant profile. This coefficient is implemented into a simplified version of the 
convection diffusion equation. The proposed profile is validated through comparison with 
experimental data and shows an enhanced prediction of SSC near the free surface. 
Mots clefs: open channel flows, suspended sediment concentration, vertical distribution,   
sediment diffusivity. 
1  Introduction 
Various numerical and analytical formulas of different order of complexity are used to represent 
suspended sediments concentration (SSC), and knowing the complexity of transport in rivers, it is still 
impossible to have a universal profile. To state some examples, for steady problems, many analytical 
formulas are proposed [1, 2, 3]. For unsteady flows uniform in width and flow directions, modelers 
used one-dimensional vertical simulation with finite differences schemes [4].  For unsteady flows 
uniform in the width direction, two dimensional simulations are used along with a finite differences 
method [5], finite volumes method [6]. For non-uniform and non-steady flow conditions, full 3D 
models are developed using finite differences [7], using finite volumes CFD softwares (sediment 
simulation in intakes with multiblock option SSIM), finite elements (Fluent). 
In this context, we are interested into establishing an innovative model to predict SSC in open channel 
flows. In a previous work [8] we studied the main parameters influencing the suspended sediment 
concentration distribution and we compared the simplest method in order to combine them later on 
into new hybrid ones that have some of the properties of the parent models and their own originality. 
The model suggested in the present study is obtained by combining the properties of the sediment 
diffusivity coefficients of the parabolic constant model [9] and of the model presented by Itakura and 
Kishi [2]. 
2 Transport equation 
We consider the vertical distribution of suspended sediment concentration in a steady and uniform 
flow and under equilibrium conditions. For small concentrations, the convection diffusion equation 
21ème Congrès de Mécanique                                                                    Bordeaux, 26 au 30 août 2013     
	   2	  
giving the transport of suspended sediment concentration in the vertical direction can be obtained by 
equating the downward fluxes caused by gravity and the upward fluxes caused by turbulence : 
 
s 0sdc w cdzε + =  
(1) 
   
2.1 Sediment diffusivity coefficient 
In order to keep the non zero property of the sediment diffusivity at the free surface of an open 
channel flow, we use the following sediment diffusivity model : 
 
 ( )
( )( )
1
* 2
s 1
* 2
0.25 1 0.5
1 1 0.5
u h z h for z h
u h z h z h z h for z h
κ ϕ
ε
κ ϕ
−
−
⎧ + >⎪
= ⎨
− + ≤⎪⎩
 
(2a) 
 
(2b) 
Where 2 h Lϕ α= , L being the Monin Obukhov length scale given by:  
 
3
*
( 1)1 sg s w c
L u
κ −
=  
(3) 
7α ≅  is the Monin Obukhov coefficient [2], c is the given mean concentration. 
 
The graph of this function is shown in figure 1 along with the profiles of Rouse [1], Kerssens et al. [9] 
and Itakura and Kishi [2].  
 
FIG. 1 – Sediment diffusivity coefficients: parabolic [1], parabolic constant [9], Itakura and Kishi  [2]  
and present study. 
2.2 Suspended sediment concentration profile 
The vertical suspended sediment distribution is given by the Rouse profile (Eq. 4): 
 ( )
( )
1Z
a
a h zc
c z h a
⎡ ⎤−
= ⎢ ⎥
−⎢ ⎥⎣ ⎦
 
(4) 
 
With :  1
*
swZ
uκ
=
 
0	  0,1	  0,2	  
0,3	  0,4	  0,5	  
0,6	  0,7	  0,8	  
0,9	  1	  
0	   0,005	   0,01	   0,015	   0,02	   0,025	   0,03	  
z/
h	  
εs	  (m2/s)	  
parabolic	  	  parabolic	  constant	  	  Itakura	  	  present	  study	  
21ème Congrès de Mécanique                                                                    Bordeaux, 26 au 30 août 2013     
	   3	  
Kerssens et al. [9] introduced the parabolic constant concentration profile with momentum diffusivity 
equal to the sediment diffusivity: 
 ( )
( )
1
1
1
0.5
1exp 4 0.5
2
Z
Z
a
a h z
for z h
z h ac
c a zZ for z h
h a h
⎧⎡ ⎤−
⎪ ≤⎢ ⎥
−⎪⎢ ⎥⎣ ⎦= ⎨
⎪ ⎛ ⎞⎡ ⎤ ⎛ ⎞− − >⎪ ⎜ ⎟⎜ ⎟⎢ ⎥−⎣ ⎦ ⎝ ⎠⎝ ⎠⎩
 
 
(5a) 
 
(5b) 
Itakura and Kishi [2] introduced a correction factor on the parabolic diffusion coefficient based on the 
Monin Obukhov length scale to better account for open channel flows with suspended sediment. They 
considered the sediment diffusivity and the momentum diffusivity to be equal: 
 ( ) 121 Z
a
c a h z
c z h a
ϕ+⎡ ⎤−⎛ ⎞= ⎢ ⎥⎜ ⎟−⎝ ⎠⎢ ⎥⎣ ⎦
 
 
(6) 
 In the present study we apply thee modifications of Itakura and Kishi [9] on the parabolic constant 
profile of Kerssens et al. [9]. 
We develop the vertical suspended sediment concentration profiles using Eqs. (2) and we validate it 
by comparison with some experimental data. 
The reference concentration is considered as the concentration at the lowest measurement point a. The 
reference concentration and the reference elevation are considered for each experiment separately. 
We consider two domains of study: 0.5z h ≤ and 0.5z h >  
2.2.1    Sediment concentration profile in the region z/h ≤ 0.5 
In this region the sediment diffusivity is the one of Itakura and Kishi  
         ( )( )
1
s * 21 1u z z h z hε κ ϕ
−
= − +                                                                                   (7)           
and Eq. (2b). 
After separating variables, the convection diffusion equation (Eq. (1)) reduces to: 
 ( ) ( )2 2
1
*
1 1
(1 ) (1 )
sw z h z hdc dz Z dz
c u z z h z z h
ϕ ϕ
κ
− + +
= = −
− −
 
(7) 
By using partial fractions: 
( )2 21 11 1
(1 ) (1 / )
z h
z z h z h z h
ϕ ϕ+ +
= +
− −
 
Then: 
2
1
11 1
(1 / )
dc Z dz
c z h z h
ϕ⎛ ⎞+
= − +⎜ ⎟−⎝ ⎠
 
Integrating between the reference elevation “a” and “z” an arbitrarily chosen position in the lower half 
of the flow: 
 ( )21
1ln ln
a
c z h aZ
c a h z
ϕ+⎡ ⎤−⎛ ⎞= − ⎢ ⎥⎜ ⎟−⎝ ⎠⎢ ⎥⎣ ⎦
 
(8) 
Or also: 
21ème Congrès de Mécanique                                                                    Bordeaux, 26 au 30 août 2013     
	   4	  
 ( ) 121 Z
a
c a h z
c z h a
ϕ+⎡ ⎤−⎛ ⎞= ⎢ ⎥⎜ ⎟−⎝ ⎠⎢ ⎥⎣ ⎦
 
(9) 
2.2.2    Sediment concentration profile in the region z/h > 0.5 
In this region the sediment diffusivity is given by Eq. (2a). 
Implementing this equation in Eq. (1), the following differential equation is obtained: 
 ( ) ( )2 1 2
*
4 1
4 1s
w z h Zdc dz z h dz
c u h h
ϕ
ϕ
κ
− +
= = − +  
(10) 
For all height h/2 < z < h, we get the concentration by integrating: 
 
( ) ( )1 2
/ 2
ln 4 2 1 2
2h
Zc z h z h
c h h
ϕ⎡ ⎤= − − + +⎢ ⎥⎣ ⎦
 
(11) 
Or also: 
 
( ) ( )21
/ 2
exp 4 1 2 1 1 2
2h
c Z z h z h
c
ϕ⎧ ⎫⎡ ⎤= − − + +⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
 
(12) 
 
/ 2hc  is the sediment concentration at height h/2. It can be calculated from Eq. (9): 
 
( ) ( )
( )( )
1
2
1
2
2
1
/ 2 1
2 / 2 2
Z
Z
h a a
haa hc c c
h h a h a
ϕ
ϕ
ϕ
+
+
⎡ ⎤
⎡ ⎤⎛ ⎞ ⎢ ⎥= =⎢ ⎥⎜ ⎟ ⎢ ⎥−⎝ ⎠ −⎢ ⎥⎣ ⎦ ⎢ ⎥⎣ ⎦
 
 
(13) 
 
Then the sediment concentration for z/h > 0.5 is given by: 
 
 
( )
( )( )
( ) ( )
1
2
2
2
11
2 exp 4 1 2 1 1 2
2
Z
a
hac Z z h z h
c h a
ϕ
ϕ
ϕ
+
⎡ ⎤
⎧ ⎫⎡ ⎤⎢ ⎥= − − + +⎨ ⎬⎢ ⎥⎢ ⎥ ⎣ ⎦− ⎩ ⎭⎢ ⎥⎣ ⎦
 
 
(14) 
 
3 Validation and discussion 
The obtained concentration model was compared to the data of Itakura and Kishi [2 ] and Vanoni and 
Brooks [10]. It shows good agreement with these experimental data. 
Fig. 2 and Fig. 3 show the parabolic profile [1], the parabolic constant profile [9], Itakura and Kishi 
[2] profile and the profile of the present study given by Eq. (9) and Eq. (14) along with the above 
mentioned experimental data. We can remark in all these figures that the Itakura and Kishi and present 
study models are in better agreement with data then the parabolic and parabolic constant profiles. 
21ème Congrès de Mécanique                                                                    Bordeaux, 26 au 30 août 2013     
	   5	  
  
FIG. 2. The concentration profiles compared  FIG. 3. The concentration profiles compared to the  
             to the data of Itakura and Kishi (1980)   data of Vanoni and Brooks (1957). 
Both Itakura and Kishi profile and ours show the same level of agreement with Itakura and Kishi’s 
measurements.  
These measurements are concentrated in the lower half of the water height were the profile of the 
present study coincides with the one of Itakura and Kishi. 
4 Conclusions 
Knowing the difficulty of measuring the distribution of suspended sediment concentration over the 
water depth for uniform open channel flows and the non existence of a universal numerical profile, we 
suggest a method for predicting concentrations when the mean concentration is known.  
The model proposed is obtained by applying the Itakura and Kishi correction method to the parabolic 
constant profile. The profile is based on a sediment diffusivity coefficient that increases in the lower 
half of the height and decreases in the upper half without reaching zero at the free surface which is 
physically more realistic. It represents smaller values than the parabolic constant one at the free 
surface. This profile is validated with a range of experimental data.  
References 
[1]  Rouse H., Modern conception of the mechanics of fluid turbulence. Trans. ASCE, 102, 463-543, 
1937. 
[2]  Itakura T., Kishi T., Open channel flow with suspended sediments. J. Hydr. Div. ASCE, 106 
(HY8), 1325 – 1343, 1980. 
[3]  Umeyama M., Vertical distribution of suspended sediment in uniform open – channel flow. J. 
Hydr. Eng. ASCE, 118(6), 936 – 941, 1992. 
[4]  Yu D., TIAN C., Vertical distribution of suspended sediment at the Yangtze river estuary. In: 
Proceedings of the International conference on estuaries and coasts, Hangzhou, China, 214-220, 
2003. 
[5]  Zhang J., Liu, H., A vertical 2-D numerical simulation of suspended sediment transport. J. of  
hydrodynamics, 19(2), 217-224, 2007. 
[6]  Peña E., Fe J., Sánchez-Tembleque F., Puertas J., A 2D numerical model using finite volume 
method for sediment transport in rivers. Proceedings of International Conference on Fluvial 
Hydraulics, Louvainla-Neuve, Belgium, 693–698, 2002. 
[7]  Lin B., Namin M. M., Modelling suspended sediment transport using an integrated numerical and 
ANNs model. Journal of Hydr. Research, 43(3), 302-310, 2005. 
0,01	  
0,1	  
1	  
1E-­‐09	   0,0000001	  0,00001	   0,001	   0,1	  
z/
h	  
c/ca	  
Itakura	  data	  	  parabolic	  	  parabolic	  constant	  	  Itakura	  pro@ile	  	  present	  study	  
0,01	  
0,1	  
1	  
1E-­‐09	   1E-­‐07	   1E-­‐05	   0,001	   0,1	  
z/
h	  
c/ca	  
Vanoni	  and	  Brooks	  data	  	  parabolic	  	  parabolic	  constant	  	  Itakura	  pro@ile	  present	  study	  
21ème Congrès de Mécanique                                                                    Bordeaux, 26 au 30 août 2013     
	   6	  
[8]  Sabat M., Terfous A., Ghenaim A., Poulet J. B., The wicked problem of suspended sediment 
profile: a choice criterion. Hydrocomplexity: New Tools for Solving Wicked Water Problem 
IAHS, 338, 251- 252, 2010. 
[9]  Kerssens P., Prins A., Van Rijn L. C., Model for suspended sediment transport. J. Hydr., ASCE, 
105(HY5), 461 – 476, 1979. 
[10]  Vanoni V. A., Brooks N. H., Laboratory studies of the roughness and suspended load of alluvial 
streams. Ed. U.S. Army Engineer Division, Missouri River Division Collection, 1957. 
Notations 
a = water height corresponding to the reference concentration (m) 
c = volumetric concentration (0≤ ! ≤ 1) 
c = mean concentration (-) 
ac = reference concentration (-) 
g = gravity acceleration (m2/s) 
h = water height(m) 
L = Monin Obukhov length scale (m) 
s = sediments relative density (-) 
*u = bed shear velocity (m/s) 
sw = particle fall velocity (m/s) 
z = vertical coordinate (m) 
1Z = Rouse suspension number (-) 
α =Monin Obukhov coefficient (-) 
mε = momentum diffusivity coefficient (m2/s) 
sε = sediment diffusivity coefficient (m2/s) 
2ϕ = coefficient (-) 
κ =Von Karman constant = 0.4 (-) 
 
	  


Prediction of suspended sediment concentrations in river flows

Abstract

Similar works

Full text

Available Versions

I-Revues