application of inverse routing methods to euphrates river iraq

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

(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 1, January- February (2013), © IAEME

97

APPLICATION OF INVERSE ROUTING METHODS TO EUPHRATES

RIVER (IRAQ)

Mustafa Hamid Abdulwahid

Department of Civil Engineering, University of Babylon, Iraq,

[email protected]

Kadhim Naief Kadhim Department of Civil Engineering,

University of Babylon, Iraq, [email protected]

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

INTERNATIONAL JOURNAL OF CIVIL ENGINEERING AND

TECHNOLOGY (IJCIET)

ISSN 0976 – 6308 (Print)

ISSN 0976 – 6316(Online)

Volume 4, Issue 1, January- February (2013), pp. 97-109 © IAEME: www.iaeme.com/ijciet.asp

Journal Impact Factor (2012): 3.1861 (Calculated by GISI)

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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 ais 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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