Proceedings of the Seventh International Conference on Hydroscience and Engineering, Philadelphia, PA, September 2006.  http://hdl.handle.net/1860/732Details are given herein of the development and application of three-dimensional layer
integrated numerical model to predict cohesive sediment transport in estuarine waters. The Finite
Volume Method on the staggered grid system was deployed to discretize governing differential
equation which consists the mass balance equation for suspended sediments. Three dimensional
advection–diffusion equation was solved by the use of splitting algorithm which divided the
equation into two-dimensional horizontal and one dimensional vertical part. The layer integrated
two-dimensional advection-diffusion equation was solved horizontally using modified ULTIMATE
QUICKEST scheme for the advection acceleration while in the vertical part the QUICKEST scheme
was deployed. Verification of model was carried out by comparison of model results and analytical
solutions. Comparison of these two-set of results represent a reasonable degree of similarity. Model
application was carried out by simulating the cohesive sediment transport in Boushehr Bay. Results
of developed numerical model were compared with measurements which represents the applicability
of the model in real case studies

Hessaroeyeh, Mahnaz Ghaeini

Kolahdoozan, Morteza

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Proceedings of the 7th International Conference on HydroScience and Engineering 
Philadelphia, USA September 10-13, 2006 (ICHE 2006) 
ISBN: 0977447405 
 
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College of Engineering 
 
 
 
 
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The 7th Int.  Conf. on Hydroscience and Engineering (ICHE-2006),Sep 10 –Sep 13,Philadelphia,USA 1 
 
 
 
 
 
 
THREE DIMENSIONAL MODELLING OF COHESIVE SEDIMENT TRANSPORT IN 
ESTUARINE WATERS 
 
 
Morteza Kolahdoozan1 and Mahnaz Ghaeini Hessaroeyeh2  
 
 
ABSTRACT 
 
Details are given herein of the development and application of three-dimensional layer 
integrated numerical model to predict cohesive sediment transport in estuarine waters. The Finite 
Volume Method on the staggered grid system was deployed to discretize governing differential 
equation which consists of the mass balance equation for suspended sediments. Three dimensional 
advection–diffusion equation was solved by the use of splitting algorithm which divided the 
equation into two-dimensional horizontal and one dimensional vertical part. Two dimensional 
horizontal part was solved using modified ULTIMATE QUICKEST scheme for the advection 
acceleration and for the vertical part, the QUICKEST scheme was deployed. Verification of model 
was carried out by comparison of model results and analytical solutions. Comparison of these two-
set of results represents a reasonable degree of similarity. Model application was carried out by 
simulating the cohesive sediment transport in Boushehr Bay. Results of developed numerical model 
were compared with measurements which represents the applicability of the model in real case 
studies. 
 
 
1. INTRODUCTION 
 
Cohesive sediment transport can cause siltation in waterways and harbour docks. 
Cohesive sediments absorbed contaminated materials, industrial effluents, heavy metals, and 
toxic contaminant. It re-suspend by strong tidal currents, short wave action and dredging 
operations. It can impact on the ecological balance of shallow waters in near shore areas 
(Wu et. al, 1998). Therefore prediction of cohesive sediment transport process in estuarine 
and coastal waters is very important for environmental engineers as wall as hydraulic 
engineers. In recent years there have been a number of studies, aimed at developing 
numerical models to predict cohesive sediment transport in coastal and estuarine waters. For 
example, semi three dimensional model of Christodoulou (Murakam et. al, 1995); three 
dimensional mathematical model of Nicholson et al. (1986) and three dimensional modeling 
of Kazaue (Murakam et. al, 1995); Wu et al. (1998); Cancino et al. (1999); Pandoe (2004) 
and Uchiyama (2005).  This study describes the development of a three-dimensional layer-
                                                 
1 Assistant Professor, Department of Civil and Environmental Engineering, AmirKabir University, Tehran, IRAN 
(mklhdzan@aut.ac.ir) 
2 Ph.D student, Department of Civil and Environmental Engineering, AmirKabir University, Tehran, IRAN 
(mghaeini@cic.aut.ac.ir) 
  2 
integrated model has been developed. In this model the method of solving equation is the 
same as the method used by Wu et al. in horizontal part, however in vertical part the 
advection term was treated with QUICKEST scheme which has the third order accuracy in 
time and space. 
 
 
2. GOVERNING MATHEMATICAL EQUATIONS 
 
As the solution accuracy of advection–diffusion equation was very important in terms 
of prediction of cohesive sediment transport different schemes have been proposed for 
numerical modeling. Leonard (1979) developed QUICK and QUICKEST scheme for 
modelling advection terms. These schemes are based on the quadratic upstream interpolation 
(Leonard, 1979). As the advective acceleration is dominant in estuarine waters one needs to 
use an accurate scheme for current modeling. QUICKEST scheme still suffers from 
numerical oscillations near discontinuities, therefore the universal limiter ULTIMATE has 
been used in conjunction with it (Leonard, 1998). In this study three dimensional advection-
diffusion equation has been deployed. Cohesive sediment transport in coastal waters can be 
described by unsteady three dimensional advection-diffusion equation, which as follows: 
( ) ( ) ( ) rzsysxss sz
s
zy
s
yx
s
x
wws
z
vs
y
us
xt
s =∂
∂ε∂
∂−∂
∂ε∂
∂−∂
∂ε∂
∂−−∂
∂+∂
∂+∂
∂+∂
∂ )()()()( ,,,  (1) 
Where:  = sediment concentration; ,  and w = are fluid velocity components in s u v x , y  
and  directions respectively; = sediment settling velocity, z sw xs,ε , ys,ε , zs,ε  = the turbulent 
diffusion coefficient in x , y  and  directions respectively and  = source or sink term. z rs
In estuarine and coastal waters, the horizontal length scale is much larger than the 
vertical scale, therefore a splitting algorithm has been deployed to divide three dimensional 
transport equation into two equations (i.e. horizontal part and vertical part) (Wu et. al, 1998). 
                                                     (2) ( ) 0)()( , =∂
∂ε∂
∂−−∂
∂+∂
∂
z
s
z
wws
zt
s
zss
( ) ( ) rysxs sy
s
yx
s
x
vs
y
us
xt
s =∂
∂ε∂
∂−∂
∂ε∂
∂−∂
∂+∂
∂+∂
∂ )()( ,,                                  (3) 
Integration of horizontal part over each layer gives (Wu and Falconer, 2000):  
( ) ( ) arysxs ssy
sz
yyx
sz
xz
sv
y
su
xt
s +=∂
∂εΔ∂
∂
Δ−∂
∂εΔ∂
∂
Δ−∂
∂+∂
∂+∂
∂ )(1)(1 ,,                   (4) 
⎪⎪
⎪⎪
⎩
⎪⎪
⎪⎪
⎨
⎧
Δ+⎟⎟⎠
⎞
⎜⎜⎝
⎛
∂
∂+∂
∂
−Δ+⎟⎟⎠
⎞
⎜⎜⎝
⎛
∂
∂+∂
∂
Δ−⎟⎟⎠
⎞
⎜⎜⎝
⎛
∂
∂+∂
∂
=
−
+−
+
2/1
2/12/1
2/1
)(
k
kk
k
a
w
z
s
y
v
x
us
ww
z
s
y
v
x
us
w
z
s
y
v
x
us
s                                              (5) 
 
 
 
 
  3 
3. NUMERICAL MODEL 
 
3.1 Discretisation 
 
In present model the Finite Volume Method on the staggered grid system were 
deployed to discretize governing differential equation which consists mass balance equation 
for suspended sediment. Three dimensional advection-diffusion equation were solved by the 
use of splitting algorithm, which divided the equation into two dimensional horizontal and 
one dimensional vertical parts. As advection is dominant in estuarine waters, attempt has 
been made to use more accurate methods in discretizing these terms. The layer integrated 
two dimensional advection-diffusion equation was solved horizontally using modified 
ULTIMATE QUICKEST scheme for advective acceleration while for diffusive part a 
central differencing scheme was deployed. The two dimensional layer integrated equation 
discretized over the control volume,  which is shown in Figure 1.  
 
 
     N     
      n     
W w   p e E 
            
      s     
      S     
Figure 1. Control volume of two dimensional equation 
 
n
a
n
r
n
sys
n
nys
n
wxs
n
exssysnynwxwexe
n
p
n
p
StSt
y
sz
y
sz
yz
t
x
sz
x
sz
xz
tSCSCSCSCSS
Δ+Δ+⎥⎦
⎤⎢⎣
⎡
∂
∂εΔ−∂
∂εΔΔΔ
Δ+
⎥⎦
⎤⎢⎣
⎡
∂
∂εΔ−∂
∂εΔΔΔ
Δ+−−−−=+
)()(
)()()()(
,,
,,
1
           (6) 
x
tUC exe Δ
Δ=  ,  , C  , C                             (7) 
x
tUC wxw Δ
Δ=
x
tVn
yn Δ
Δ=
x
tVs
ys Δ
Δ=
 
Where  = average cell face values over the time increment Δ ,  = average of  over the 
control volume.  = average of s  over the control volume. The superscript n  is an index 
for time. the subscript  denotes the northern cell face. The cell face gradients in equation 
(6) are approximated by central differencing and the averaged cell face values , ,  
and  are estimated using two dimensional modified ULTIMATE QUICKEST scheme. A 
sketch of layers in the x-z plane is illustrated in Figure 2. The one dimensional advection-
diffusion in vertical part is discretized over non-uniform grid. It was solved implicitly using 
QUICKEST scheme (third order in time and space), and for diffusive part a central 
differencing scheme was deployed. Finally the three diagonal matrix is obtained in vertical 
part and then concentration is estimated using Thomas algorithm.  
S t rS rs
aS a
n
wS eS sS
nS
  4 
 
Figure 2. The sketch of layers in x-z direction 
H 
I, k-1 
I+1, k I-1, k 
I, k+1 
I , k 
Z 
 
3.2 Boundary Conditions 
 
At the inflow open boundaries concentration were prescribed, while at the outflow open 
boundary the zero gradient open boundary condition has been deployed. The free surface 
and bed boundary condition can be described by equations (8) and (9) respectively. 
( ) 0, =∂
∂ε−−
z
ssww zss                                                         (8) 
⎪⎪
⎪
⎩
⎪⎪
⎪
⎨
⎧
τττ=∂
∂ε−−
τ≥τ=∂
∂ε−−
τ≤τ=∂
∂ε−−
ebdz,ss
eberoz,ss
dbdepz,ss
z
ssw
q
z
ssw
q
z
ssw
ππ0
                               (9) 
Where  =  bed shear stress,  =critical shear stress for deposition,  =critical shear stress 
for erosion,  and  = deposition and erosion rate at bed. 
bτ dτ eτ
depq eroq
 
3.3 Exerting Cohesive Sediment Property 
 
In present model the sediment settling velocity is divided into three parts according to 
classification of Mehta based on suspended sediment concentration (Abbott and Price, 
1994). 
⎪⎪
⎪
⎩
⎪⎪
⎪
⎨
⎧
>−−
<<
<υ
−
=
10))(1(
103.0
3.0
18
)1(
66.4
220
3/4
1
2
ssskw
ssk
s
gDsg
w
s
s
s                                       (10) 
Where = sediment concentration,  =settling velocity, s sw sg  =specific density, υ  
=kinematic viscosity, g  =gravity acceleration,  =floc diameter,  =an empirical 
coefficient, = the value of  at concentration ,  inverse of the concentration at 
which 0  and 
sD 1k
0sw sw 2s 2k
=sw lit
grs 102 = . 
  5 
The Krone (Van Rijn, 1993) deposition rate model was used for numerical modeling 
purposes expressed by: 
)1(
d
b
bsdep swq τ
τ                                                      (11) −−=
Where  = settling velocity,  = cohesive sediment concentration near the bed, sw bs bτ  = the 
bed shear stress,  = critical shear stress for deposition. dτ
Erosion rate can be estimated from equation (12) for consolidated bed proposed by 
Partheniades (Abbott and Price, 1994). 
⎟⎟⎠
⎞
⎜⎜⎝
⎛
τ
τ−τ=
e
ebME                                                         (12) 
Where E  = erosion rate,  = the bed shear stress, bτ eτ  =critical shear stress for erosion and 
M  =constant erosion coefficient. 
 
 
4. VERIFICATION OF MODEL 
 
Numerical test has been carried out to study the performance of the model. First, to test 
the performance of the present model the advection of the square wave was studied in a one 
dimensional uniform flow. The initial condition was represented as follows: 
15.0,05.0,10
1,00
1
)0,(
21
21
21
==≤≤
⎩⎨
⎧
≤≤≤
≤≤==
xxx
xxxx
xxx
txs π                              (13) 
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 0.2 0.4 0.6 0.8 1x
s
Ultimate Quickest
Analytical solution
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 0.2 0.4 0.6 0.8 1x
s
Quickest
Analytical solution
 
Figure 3 The result of present model 
 
Figure 3 represents the comparison of QUICKEST and ULTIMATE QUICKEST scheme 
with analytical solution proposed by Wu and Falconer (2000). The results of ULTIMATE 
QUICKEST scheme show no oscillations. Also it predicts well in the region of 
discontinuities and omitted the negative concentration. Furthermore, QUICKEST scheme 
overestimated the peak concentration about 10%.  
Secondly, the simulation of advection-diffusion of square column concentration distribution 
in a two dimensional flow has been carried out. Equation (14) shows the initial condition of 
the test. 
  6 
1.0,1.0,0),,(
0
,1
)0,,( 2121
===
⎩⎨
⎧ ≤≤≤≤=
yxtyxs
yyyxxx
yxs
                             (14) 
Figure 4 shows the result of the model at the end of simulation time (T=0.6). A comparison 
of the present result and Wu et al. analytical solution is shown in Figure 5 at the section 
Y=0.7. Comparing these two set of results represents the reasonable degree of similarity. 
 
Fig.4. Numerical results of advection-diffusion of a square column 
-0.5
0.0
0.5
1.0
0 0.2 0.4 0.6 0.8 1x
s
Model Result
Analytical Solution
 
Fig.5.Comparision of numerical results and exact solution at section y=0.7. 
 
 
5. MODEL APPLICATION TO BOUSHEHR BAY 
 
 
To study the performance of the model in real cases, model application was carried out 
by simulating the cohesive sediment transport in Boushehr Bay, north of Persian Gulf, Iran 
coastline. The harbour approach channel suffers heavily from sediment depositions which 
mostly consist of cohesive sediments. Measurements were shown that the nature of the flow 
in this area is three dimensional (Water Research Center, 2003). Initial data that is used in 
present model obtained from the project of modelling cohesive sediment transport in 
Boushehr Bay. In that project different scenario of deposition has been studied. Finally the 
transport and settling of bed cohesive sediments that is influenced by tidal flows are 
identified as the most important phenomena of deposition (Water Research Center, 2003). 
Overall modelling area is shown in Figure 6. 
  7 
 
Figure 6. Physical domain of Boushehr Bay 
Bushehr  
B4
 
Physical domain has three open boundaries; two open velocity boundaries (north and west 
boundaries) and one water elevation boundary (south boundary). Boundary conditions are 
obtained from the intermediate model (Water Research Center, 2003). The area is covered 
by 300*300 m grid size in horizontal and at most six layers in the vertical part.  
The simulation time of the modelling is for the period that shows the properties of flow in 
that area. Flow calibration is performed with the use of field data of Transit and Iransiam 
stations which are shown in figure 7. Turbulent diffusion coefficients vary with flow 
properties. Different values of critical shear stress for deposition and erosion in the proposed 
range was tested. Finally, the value of critical shear stress for erosion and deposition are 
equal to 0.085 and 0.008  is considered according to comparison of the model 
results and field measurements. Figure 8 shows the variation of concentration, tidal level and 
current speed for Transit station. Initial concentration of all points was set to zero. Sediment 
concentration increased with time due to the erosion of the bed material. There is a lag 
between peak of concentration and velocity. The comparison of model result and field data 
is shown in Figure 9 for Transit and B4 stations which represents the applicability of the 
present model. 
)/( 2mN
 
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Flood Transit              Ebb Transit Flood Iransiam Ebb Iransiam
C
ur
re
nt
 S
pe
ed
(m
/s
)
Measurement
Model
 
Figure 7 Calibration results for hydrodynamic model 
 
  8 
0
2
4
6
8
10
12
14
16
0 0.5 1 1.5 2 2.5time (day)
tidal level (m)
Current speed (m/s)
concentration*100 (gr/m3)
 
Figure 8 variation of concentration, tidal level and current speed for Transit station 
. 
0
200
400
600
800
1000
1200
1400
1600
Transit Flood Transit Ebb B4 Flood B4 Ebb
C
on
ce
nt
ra
tio
n 
(g
r/m
3)
Measurement
Model
 
Figure 9 comparison of model result and field data for Transit and B4 stations 
 
 
6. CONCLUSIONS 
 
Cohesive sediment transport can be described by the time dependent advection-
diffusion equation. A three dimensional layer integrated model has been developed for 
predicting cohesive sediment concentration. In the current study the high resolution methods 
with third order accuracy both in time and space has been used for advective terms. Model 
verification has been made by comparison of the model results with analytical solution. 
Comparison of two set of results represent a reasonable degree of similarity. Then model 
was applied to a real case study, Boushehr Bay, north of Persian Gulf, Iran coastline. Results 
of developed numerical model were compared with measurements which represents the 
applicability of the model for real cases. 
 
 
REFERENCES 
 
Abbot, M.B. and Price, W.A., (1994), Coastal estuarine and harbour engineers, Reference book, 
E&Fr Spon Pulishing 
  9 
Cancino, L. and Neves,R., (1999), "Hydrodynamic and sediment suspension modeling in estuarine 
systems", J.Marine System, No. 22, pp. 105-116. 
Leonard, B.P., (1979), "A stable and accurate convective modeling procedure based on quadratic 
upstream interpolation",No. 19, pp. 59-98. 
Leonard, B.P., (1998), "The ultimate conservative difference scheme applied to unsteady one- 
dimensional advection", Computers Methods in Applied Mechanics and Engineering, No. 88, 
pp. 17-74. 
Murakam K. and Tsuruya,H. and Irie,I., (1995), " A three- dimensional model of sediment transport 
with consendration of fluid mud", Coastal And Estuarine Morfodynamics,3(3B7) 
Nicholson, J. and O'connor,A.B., (1986), "Cohesive sediment transport model", Journal of 
Hydraulic Engineering, Vol. 112, No. 7,pp. 621-640. 
Pandoe,W. and Edge,L., (2004), "cohesive sediment transport in the hydrodynamic –baroclinic 
circulation model, study case for idealize tidal inlet" 
Uchiyama, Y., (2005), "Modeling three- dimensional cohesive sediment transport and associated 
morphological variation in estuarine intertidal mudflats" report of the port and airport institute, 
Japan 
Van Rijn, L., (1993), Principal of sediment transport in rivers, estuarine and coastal seas, Aqua 
Publication 
Water Research Center, (2003), The Report of Cohesive Sediment Modeling of Boushehr Bay, north 
of Persian Gulf, Iran 
Wu, Y. and Falconer, R.A., (2000), "A mass conservative 3-D numerical model for predicting solute 
fluxes in estuarine waters", Advances in water resources, No. 23, pp. 531-543. 
Wu, Y., Falconer, A.R. and Uncles,R.J., (1998), "Modeling of water flows and cohesive sediment 
fluxes in Humber estuary, UK", Marine Pollution Bulletin, No. 37, pp. 182-189. 
 


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Three dimensional modelling of cohesive sediment transport in estuarine waters

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