Mathematical Modeling of water surface at Unsteady Flow in Al-Msharah River











This study is concerned with unsteady flow in open channels in boundary conditions that may exist in river or channel system. A software Hydrologic Engineering Center-River Analysis System (HEC-RAS) was used for a case study. The software contains a hydrodynamic model for simulating the unsteady flow in open channels based on a one-dimensional form of the Saint-Venant equations, by formulating a discrete form using the implicit finite difference scheme, then solving using the Newton-Raphson iteration procedure and the modified Gaussian elimination technique. The case study adopted with the help of HEC-RAS application was applied to a reach of Al Msharah River (channel) between AL Msharah Barrage (upstream)and Al Malah Bridge (Downstream), for a length of (49 Km) divided to (52) cross sections. Model runs and the comparison of results with actual field data indicate that the accuracy of the results obtained for(θ=1 and ∆t=5 minute) is quite acceptable, where θ is a weighting parameter and ∆t is the time interval. The paper covers the case of normal depth of flow and observed water stage data within the period ( from January 2006 to July 2006) . The results demonstrate that the area at Al Am'arah city at distance 17.5 Km from upstream(cross section 30)could be subjected to flooding at High Flow, therefore, it is recommended to adjust cross sections to prevent the flooding in this area. At last , Calibration of the hydrodynamic model is achieved in a study reach using the observed data (water stage) along AL Msharah River and show that a good agreement.












1-INTRODUCTION
The study of water surface of unsteady flow in natural channel is very important in water resources systems, design of hydraulic structures, analyses of river mechanics problems, the development of river control works and to the life and works of humans.Movement of a flood wave in a single channel or a network of channels is good example of unsteady non uniform flow.There are problems characterized by the time dependence of flow and cover a wide range of phenomena including surface runoff, tidal motions, reservoir regulation and flood movements.
Simulation of unsteady flows in open channels and shallow water bodies is an important, interesting, and difficult subject in hydraulic engineering, because many variables enter into the functional relationship and the differential equations cannot be integrated in closed forms except under very simplified conditions.The subject has gained an entirely new dimension since the advent of the high-speed digital computers which the numerical solution of unsteady flow equation attainable and practical especially in the one-dimensional flow field.As one of the several mathematical methods developed for unsteady flow computation, the method of characteristics has been studied and used among hydraulic engineers (Woolhiser & Morris, 1980).
In the present study the application of implicit finite difference scheme is simulated for unsteady flow in open channel(Al Mshara River) as case study were conducted.The Saint-Venant equations (a combination of both the continuity and momentum equations) are used to solve unsteady flow problems.

MATHEMATICAL MODEL
The law of conservation of mass (continuity equation for unsteady flow may) be established by considering the conservation of mass in an infinitesimal space between two channel sections in unsteady one dimensional flow the discharge changes with distance at a rate Q/ x, and the depth changes with time at a rate of y/ t.Referring to Figure (1).
The continuity equation of unsteady flow at the following form:- (Mohammed, A. Y., 1993) And the last form of the conservation of momentum equation should be: The tow equations are named Saint Venant equations have been used for one dimensional unsteady flow to calculate water surface profile in single channel and looped channel (networks).
The four-point implicit scheme is sought over a discrete rectangular mesh of points on the (distance, time) plane.The mesh points are determined by the intersection of straight lines of location and time.The location lines are drawn with spacing x i , which is parallel to t-axis.The t-axis may be used as the upstream channel boundary location, and the last line drawn parallel to the t-axis, to be designated the N th line, can be used to represent the downstream channel boundary location.The time lines are drawn parallel to the x-axis with spacing t j .Each point of the network is identified by two indices, a subscript (i) to designate the x-position of the point, and a superscript ( j ) to designate the t-value.Figure(1) shows the computational grid on the x-t plane to be used for the development of the numerical procedure.
Function K in the intervals i,i+1 and j, j+1 may be replaced by its weight average between these points.For the time interval x Distance Increments

Figure (1)Network of Points on the (x-t)Plane
And for the space interval The time and space derivatives of K become, ( ) ( ) Clearly ,whole families of finite difference schemes may be obtained by varying the parameters and , all such schemes, however, are four-point schemes.
When =1/2 ,the system of equation ( 3) to ( 6) becomes the Preissmann fourpoint scheme whereby the time derivative i: and the function K is expressed as Substitution of the finite difference approximations is defined by equations ( 6) to(8) for the derivative and non-derivative terms in the unsteady flow equations ( 1)and ( 2), the finite difference formulations are presented the continuity, F i , and momentum, G i, equations for the reach i.For the equations written in terms of the discharge Q and depthY, the formulations of these equations in Preissmann scheme ( = 1 / 2 ) are as follows : Multiplied by 1 j i j B t + , the continuity equation (1) becomes where the friction slope S f is computed from Manning 's equation for uniform flow:- Since R=A/P ,then: Equations ( 9) and ( 10)are nonlinear with respect to unknowns Y and Q at the points i and i+1 on the j+1 time line.All terms associated with the j th time line are known either from the initial conditions or previous computations.This system of nonlinear algebraic equations cannot be solved directly since there are four unknowns , Y and Q at points i and i+1 on the j+1 time line , and only two equations.However, if these equations are applied to the (N-1)reaches between the upstream and downstream boundaries, a total of (2N-2) equations with 2N unknowns is obtained (N denotes the total number of nodes or cross sections).The two supplementary equations needed to close the system are provided from upstream and downstream boundary conditions.The resulting system of 2N equations with 2N unknowns may be symbolized as follows: )

Downstream boundary condition
The system (12) of nonlinear algebraic equations is solved by using the Newton-Raphson iterative method.The Newton-Raphson technique is derived from Taylor series expansion of the nonlinear function in which all terms of second and higher order are neglected.It should be noted that although the system of equations ( 12) involves 2N unknowns, each equation contains a maximum of four unknowns.
This can be used to a great advantage in the computational schemes.

HEC-Ras Package
The HEC-RAS has the interactive program features, such as multi windows facilities, which can treat any river system configurations with the ability of storing these configurations.It can provide the user with a report showing the river system data and presenting the computation results by charts and/or tables.
There are five main steps in creating a hydraulic model with HEC-RAS (U. S.

A. Starting a NEW Project:
The first step is to establish the directory of working and enter a title for the new project.To start anew project, go to the File Menu on the Main HEC-RAS window and select New Project.
Water Resources, Center for Restoration of Iraqi Marshlands, 2006).In the present study two hypothetical flow cases are considered, wet (Peak Flood) and Normal flow.
Boundary geometry for the analysis of flow in a natural stream is specified in terms of cross sections and the reach lengths.Cross sections are located at intervals along a stream to characterize the flow carrying capacity of the stream and its adjacent flood plain.Figures(6)

Results And Discussions
Results of this study have been discussed for a normal depth, the normal depth has analyzed for most important flow parameters describing the flow characteristics of the study reach (between Al-Msharah bridge and Al-Malah bridge ) .
Computed water surface profile is shown in finger ( 10 ) .This may be attributed to the topography nature of the channel basin and longitudinal bank elevation was lower than water surface profile .Therefore cross sections should be adjusted to allow passing any flow , as well as changing the ground level with distance along the cross sections of the river .At last calibration of model computed with measured water surface profile is platted in fig ( 13) and show that a good agreement.

Conclusions
The most conclusion of this study show that the area at Am'arah city at distance 17.5 km from upstream (cross section 30) could be subjected to flooding at high flow, since its longitudinal bank elevation is lower than water surface, therefore, it is recommended to adjust cross sections to prevent the flooding in this area.
The computation needed data can be divided into two typies, Geometric Data and Unsteady Flow Data.The information needed to perform the mathematical model is shown in Figure(2).

Figure ( 2 )
Figure (2)Needed Information to perform the Mathematical Model to (9)demonstrate the importance of the cross sections geometry.

Figure ( 3 )
Figure (3) Location Map showing The Surreged Cross Sections For Al-Msharah River Between Al-Msharah Barrage And Al-Malah Bridge

Fig( 11 )
Fig( 11 ) show the stage decreased form upstream toward downstream, and fig(12 ) show the water volume profile where it decrease from Al-Msharah Barrage station towards Al-Malah Bridge station because of water consumption and water losses along the study reach .
Figure ( 4 ) The location of AsSanna'f Marsh and Al Msharah RiverOutlet into the marsh