Application of inverse routing methods to euphrates river iraq

International Journal of Civil Engineering and TechnologyENGINEERING – 6308
INTERNATIONAL JOURNAL OF CIVIL (IJCIET), ISSN 0976 AND
(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 1, January- February (2013), © IAEME
TECHNOLOGY (IJCIET)
ISSN 0976 – 6308 (Print)
ISSN 0976 – 6316(Online)
Volume 4, Issue 1, January- February (2013), pp. 97-109
IJCIET
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APPLICATION OF INVERSE ROUTING METHODS TO EUPHRATES
RIVER (IRAQ)

Mustafa Hamid Abdulwahid
Department of Civil Engineering,
University of Babylon, Iraq,
mstfee@gmail.com
Kadhim Naief Kadhim
Department of Civil Engineering,
University of Babylon, Iraq,
Altaee_Kadhim@yahoo.com

ABSTRACT

The problem of inverse flood routing is considered in the paper. The study deals with
Euphrates River in Iraq. To solve the inverse problem two approaches are applied, based on
the Dynamic wave equations of Saint Venant and the storage equation using Muskingum
Cunge Method, First the downstream hydrograph routed inversely to compute the upstream
hydrograph, Five models have been made for inverse Muskingum Cunge Method (CPMC,
VPMC3-1, VPMC3-2, VPMC4-1, VPMC4-2) and two models for inverse finite difference
scheme of Saint Venant Equations (explicit and implicit). Then the sensitivity analysis
studied for each parameter in each method to find out the effect of these parameters on the
solution and study the stability and accuracy of the result (inflow hydrograph) with each
parameter value change. Finally, the computed inflow hydrograph compared with the
observed inflow hydrograph and analyze the results accuracy for each inverse routing method
to evaluate the most accurate one. The results showed that the schemes generally provided
reasonable results in comparison with the observed hydrograph and hydraulic results for
inverse implicit scheme of Saint Venant equations are more accuracy and less oscillation than
that obtained by the inverse explicit scheme and these hydraulic results are more accuracy
than hydrologic results using Muskingum Cunge Method.

Keywords: CPMC, VPMC3-1, VPMC3-2, VPMC4-1, VPMC4-2

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International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308

I. INTRODUCTION

Euphrates is important river in Iraq, flowing from Turkish mountains and pour in Shatt
Alarab southern Iraq which withdrawing water from the pool upstream of the AL Hindya
Barrage. The river is surrounded by floodplains and considered as an important region of the
Iraqi agricultural incoming. The inverse flood routing appears to be a useful for water
management in Euphrates plains. The considered reach of length about 59.450 km between Al
Hindya Barrage ( 32o 43’ 40" N - 44o 16’ 04" E ) and Al Kifil gauging station ( 32o 12’ 53" N
- 44o 21’ 43" E ), where the river branches into Al Kuffa and Al Abbasia branches while there
is no branches within the studied river reach.

II. INVERSE FLOOD ROUTING USING SAINT VENANT EQUATIONS

The basic formulation of unsteady one dimensional flow in open channels is due to
Saint Venant (1871). It can be written in the following form:
(1) Continuity equation.

(2) Momentum equation.

where: t = time, x = longitudinal coordinate, y = channel height, A = cross-sectional flow
area, Q = flow discharge, B = channel width at the water surface, g = gravitational
acceleration, qL = lateral inflow, Sf = friction slope.

Figure( 1) . Euphrates river considered reach [ Google Map ].

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The equations (1, 2) are solved over a channel reach of length L for increasing time t, and
they are carried out in the following domain: 0 ≤ x ≤ L and t ≥ 0, Figure (1)

Figure 1 Direction of integration of the Saint Venant equations while (a) is conventional
solution and (b) is inverse solution [Szymkiewicz. (2010)].

The following auxiliary conditions should be imposed at the boundaries of the solution
domain (Szymkiewicz, 2010):
− Initial conditions: Q (x,t) = Qinflow(t) and h(x,t) = hinflow(t)
for x=L and 0 ≤ t ≤ T;
− Boundary conditions: Q(t,x) = Qo(x) or h(t,x) = ho(x)
for t=0 and 0 ≤ x ≤ L, Q(t,x) = QT(x) or h(t,x) = hT(x) for t=T and 0 ≤ x ≤ L.
where Q(x,t), h(x,t), Qo(t), QL(t), ho(t) and hL(t) are known functions imposed, respectively,
at the channel ends. The Saint Venant partial differential equations could be solved
numerically by finite difference formulation. The derivatives and functions in Saint Venant
Equation are approximated at point (P) which can moves inside the mesh to transform the
system of Saint Venant Equation to a system of non-linear 4 algebraic equations. Two
weighting factors must be specified, i.e., (θ) for the time step and (ψ) for the distance step as
shown in Figure (2). [Dooge and Bruen,(2005)].

Figure (3): Grid point applied in weighted four point scheme. [Dooge and Bruen,(2005)].
The spatial derivatives of the function ( ) at point (P) becomes [Szymkiewicz, (2008)]:

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The derivatives and functions approximated at point P transform the system of equations (1
and 2) to the following system of non-linear algebraic equations [Alattar M. H. (2012)]:

where i is index of cross section, j is index of time level, and ψ and θ are the weightings
parameters ranging from 0 to 1.

III. INVERSE FLOOD ROUTING USING MUSKINGUM CUNGE METHOD

The main features of flood wave motion are translation and attenuation; both are
nonlinear features resulting in nonlinear deformation. The linear convection – diffusion
equation, scheme stated in terms of the discharge [Miller and Cunge, (1975)]:

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where: Q = discharge , A = flow area, t = time, x = distance along the channel, y = depth of
flow, qL = lateral inflow per unit of channel length, the kinematic flood wave celerity ( ) and
the hydraulic diffusivity (µ) are expressed as follows:

where Bo is the top width of the water surface for wide channel, and is the reference flow
depend on its computation method, i.e. (CPMC) and (VPCM). an improved version of the
Muskingum - Cunge method is due to Ponce and Yevjevich, (1978).

where : i = index of cross section, j = index of time level, QL = total volume discharge, =
lateral flow discharge with :

where X is dimensionless weighting factor, ranging from 0.0 to 0.5 and the ( C ) value is the
Courant number, i.e., the ratio of wave celerity ( ck ) to grid celerity ( x / t) :

The Muskingum - Cunge equation is therefore considered an approximation of the
convective diffusion Equation (8). As a result, the parameters K and X are expressed as
follows [Ponce and Yevjevich, (1978)]:

From equation (11) Koussis et al.,(2012) develop the explicit reverse routing scheme. Figure
(3).

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The bi’s are related to the ais of equation (12) and written as follows:

Figure (3). Reverse flood routing in a stream reach of length L [Koussis et al.,(2012)].

The values of Qo and ck in equation (15) is obtained as follows [Samimi et al., (2009)] :
1) In Constant Parameters Muskingum Cunge Method (CPMC) ck and Qo are kept
unchanged during the entire computation. In other words they were computed based on a
constant “reference discharge”. Ponce, (1983) proposed reference discharge:

Where is the base flow taken from the inflow hydrograph.
2) In Variable Parameters Muskingum Cunge Method (VPMC) Ponce et al.,(1978) claimed
that variable routing parameters could more properly reflect the nonlinear nature of flood wave
than the constant parameters do. However, the variable parameter schemes induce volume loss
during the computation. Ponce et al. (1994) argued that the volume loss is the price that should
be paid to model the nonlinear characteristic of flood wave.
In the variable parameters Muskingum - Cunge method (VPMC), the computation of
routing parameters is based on a reference discharge that is determined within each
computational cell. Therefore, the routing parameters will vary from a time step to another
according to the nonlinear variations of the discharge. The reference discharge is determined
by averaging the discharge of the nodes of each computational cell.
Basically, two schemes were proposed for determining the reference discharge in each
computational cell. The first scheme uses known discharge values of three nodes of the cell,
which is referred to three points averaging scheme. The second one includes the unknown
discharge value of the forth node, that is (i+1, j+1), in the computation of the reference
discharge. This scheme is usually referred to as four points averaging scheme and clearly

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requires trial and error procedures to determine the reference discharge. Each of the averaging
schemes encompasses three computational techniques which were presented as follows [Tang
et al., (1999)]:

and ( m = 1.5 ) for wide channel [U.S. Army Corps of
Engineers, (1994)].

This method is clearly requires trial and error procedures to determine the reference discharge
( Qo ). Where the value of Qo (4) and ck (4) is obtained by one of the numerical iteration
techniques.

In the present study only five methods (CPMC, VPMC3-1, VPMC3-2, VPMC4-1, and
VPMC4-2) have been applied to estimate the reference discharge and flood wave celerity that
needed to estimate the parameters of Muskingum Cunge inverse routing Method.
The values of t and x are chosen for accuracy and stability. First, t should be evaluated by
looking at the following three criteria and selecting the smallest value: (1) the user defined
computation interval, (2) the time of rise of the inflow hydrograph divided by 20 ( Tr / 20 ),
and (3) the travel time through the channel reach [Ponce, (1983)].
For very small value of x, the value of in equation (15) may be greater than (0.5) leading to
negative value of (X) so x is selected as [Ponce, (1994)]:

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To satisfy ( 0 < X < 0.5) and preserve consistency and stability in the method , where the
solution will be more accurate when X 0.5 [Koussis et al.,(2012)].

IV. INVERSE ROUTING RESULTS

Five models have been made for inverse Muskingum Cunge Method (CPMC,
VPMC3-1, VPMC3-2, VPMC4-1, VPMC4-2) and two models for inverse finite difference
scheme of Saint Venant Equation (explicit and implicit) used to route the flood wave in
Euphrates river (28/11/2011 – 13/12/2011). Table (1) shows the actual and computed
hydrographs. the time intervals of 12 hour is used to satisfy the limitations of Muskingum
Cunge Method and for more smooth curvature line of the hydrographs that calculated in
different methods and the average bed slope founded to be ( So = 4.3 cm / km ), For inverse
explicit finite difference scheme of Saint Venant equations the weight factors were ( ¥=5 Ø=0)
this founded to gave the most accurate results and these values of weight factors are
compatible with the [Chevereau, (1991)] study and for Inverse implicit scheme the weight
factors were ( ¥=0.5 Ø=0). Figure (4)

Figures (5) and (6) shows model stability analysis with change of the weight factors
for explicit finite difference scheme of Saint Venant equations, The results show that
the time weight factor is very sensitive for model stability and only model of (Ø=0) is a
stable model, but less sensitive for space weight factor.

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The Channel bed slope is one of important parameters effects on river routing results
[U.S. Army Corps of Engineers, (1994)] , Figure (7) and Figure (8) shows that the
model is stabile with steep slope and the stability decreasing with mild slope with the
same distance interval ( i.e. x = 59450 m ) in inverse constant parameter Muskingum
Cunge scheme and the model stability decrease with less distance intervals, The model
with slope of ( 0.000023 ) and less is instable with dispersion numerical solution with
the same parameters of this study case. Note that the Courant number not depend on
slope and remain (C = 0.352) for all models for different slope values, While the
models with different space intervals both of Courant number and weight factor
changed.

Table (1). Comparison of computed results of inverse flood routing by various methods.

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Figure (5). Computed upstream discharge hydrograph in explicit 12 hour interval scheme in different time weight
factor.

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Figure (8). Computed upstream discharge hydrograph in inverse Muskingum Cunge
scheme with different model space intervals with 12 hour time interval and (Slope = 0.0001 ).

For one part model the space interval is ( x = 59450 m ); Courant number ( C = 0.352 )
and ( X = 0.387 ), The two parts model ( x = 30000 m, 29450 m ); ( C = 0.697 ) and ( X =
0.275 ) and the three parts model ( x = 20000 m, 20000 m, 19450 m ); ( C = 0.697 ) and ( X =
0.275 ).

V. ACCURACY ANALYSIS FOR INVERSE ROUTING RESULTS

According to the inverse routing results in table (1) the actual observed upstream
hydrograph is appointed for the error analysis by evaluate the deviation ratio of the results for
each model in different analytic methods with the actual observed inflow. the error percentage
is define as equation [Szymkiewicz, (2010)]:

Where the QObserved inflow in is upstream station and QComputed is the computed inflow
by one of the applied inverse routing methods. The values of the mean errors are listed in
Table (2) which summarizes the considerations of each method. Knowing that the percentage
of the accuracy was determined as :

The root mean square of error method (RMSE) equation (30) computes the magnitude of error
in the computed hydrographs [Schulze et al., (1995)].

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Where: I-SV is the inverse implicit routing scheme of Saint Venant equations, E-SV is the
inverse explicit routing scheme of Saint Venant equations, CPMC is the constant parameters
estimation for inverse hydrologic routing of Muskingum Cunge method, VPMC is the variable
parameters estimation for inverse hydrologic routing of Muskingum Cunge method.
The errors of the measuring field data effect on the accuracy analysis for comparison of these
actual measured field data with calculated results in different inverse routing methods, these
errors may align against or toward the one of the applied reverse routing methods and the
accuracy analysis will be less adequate. in addition to that the hydraulic and hydrologic
inverse routing methods have many approximations and assumptions for its equations that is
cause the deviation of the measured results of discharge and stage hydrograph compared with
the observed data.
The many field data approximation also decreases the models accuracy like neglecting of the
many channel geometric properties had an indirectly little effecting on the inverse river
routing i.e. the temperature, meandering and existing of the sapling and grass and neglecting
of the small lateral flow i.e. rainfall and seepage.

VI. CONCLUSION

For the hydraulic and hydrologic inverse routing methods, The inverse hydraulic
methods represented by (explicit and implicit schemes) considered the hydraulic properties
with the actual cross sections along the reach with more details than the other methods, this
reflect that the inverse hydraulic method give more accurate results than those at the inverse
Muskingum Cunge methods and the implicit scheme of hydraulic method for Saint Venant
equations gave more adequate results than the explicit scheme, where the implicit scheme
accuracy was ( 96.96% ) and for explicit ( 96.22% ). Channel bed slope is a sensitive
parameter effects on river routing results, this conclusion agree with [U.S. Army Corps of
Engineers, (1994)]. The present study of Muskingum Cunge Methods shows that the model is
stabile with steep slope and the stability decreasing with mild slope with the same distance
interval (i.e. the x = 59450 m), and the model stability decrease with less distance intervals,
The models with slope of (0.000023) and less are instable with dispersion numerical solution

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with the same parameters of this study case. The results show that the time weight factor of
finite difference explicit scheme of Saint Venant equations is very sensitive for model stability
and only model of (Ø=0) is a stable model , this compatible with [Liu et al, (1992)] as he
concluded that the inverse computations produced better results with (Ø=0). While the model
is less sensitive for space weight factor (¥) and the model remain stable for wide range of
space weight factor values. For implicit scheme the value of time weight factor (¥=5) and (Ø=
0.5) is founded that it gave the more stable results.

VII. REFERENCES

For the hydraulic and hydrologic inverse routing methods, The inverse hydraulic methods
represented by (explicit and implicit schemes) considered the hydraulic properties with the
actual cross sections along the reach with more details than the other
[1] Alattar M. H. (2012) :” Application of Inverse Routing Methods on Sement of Euphrates
River“. M.Sc. thesis, College of Engineering, University of Babylon, Iraq.
[2] Chevereau, G. (1991):” Contribution à l'étude de la régulation dans les systèmes
hydrauliques à surface libre”. Thèse de Doctorat de l'Institut National Polytechnic de
Grenoble, 122 p.
[3] Dooge, C.I. and Breuen, M. (2005): "Problems in reverse routing". Center of Water
Resources Research, University College of Dublin, Earlsfort Terrace, Dublin, Ireland.
[4] Koussis, A. D. (2012): " Reverse flood routing with the inverted Muskingum storage
routing scheme", Nat. Hazards Earth Syst. Sci., 12, 217–227.
[5] Miller W.A. and Cunge J.A. (1975):” Simplified Equations of Unsteady Flow in Open
Channels”. Water Resources Publications, Fort Collins, Colorado, 5: Pages 183 -257.
[6] Ponce, V. M. & Yevjevich, V. (1978) : ” Muskingum - Cunge method with variable
parameters”. Hydraulic. Div. ASCE, 104 (12), 1663-1667.
[7] Ponce, V. M. (1983) :” Accuracy of physically based coefficient methods of flood
routing”. Tech. Report SDSU Civil Engineering Series No. 83150, San Diego State
University, San Diego, California, Ponce, V.M. (1994):” Engineering Hydrology”, Principles
and Practice, Prentice Hall, New Jersey.
[8] Saint Venant, B. de (1871): “Théorie de liquids non-permanent des eaux avec application
aux cave des rivieres et à l’introduction des marées dans leur lit (Theory of the unsteady
move-ment of water with application to river floods and the introduction of tidal floods in their
river beds)”, Acad. Sci. Paris, Comptes Rendus 73, pp. 148-154 and 33, pp. 237-249.
[9] Samimi S.V. , Rad R. A. , Ghanizadeh, F. (2009): ” Polycyclic aromatic hydrocarbon
contamination levels in collected samples from vicinity of a highway ” . J. of Environ. Health
Sci. & Eng., 6:47-52. Iran.
[10] Schulze, E. D.ct al. (1995): ”Above-ground biom ass and nitrogen nutrition in a chronos
equence of pristine Lahurian Larix stands in Eastern Siberia”. Canadian Journal of Forest
Research 25: 943—960.
[11] Szymkiewicz, R. (2010) :”Numerical Modeling in Open Channel Hydraulics “,
University of Technology, Gdansk, Poland.
[12] Tang, X., Knight, D. W., SAMUELS, P. G., (1999):”Volume conservation in Variable
Parameter Muskingum - Cunge Method”, J. Hydraulic Eng. (ASCE), 125(6), pp. 610–620.
[13] U.S. Army Corps of Engineers (1994):” FLOOD-RUNOFF ANALYSIS”
DEPARTMENT OF THE ARMY EM U.S. Army Corps of Engineers Washington, DC
20314-1000 USA.

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