Estimation of Open Channel Roughness by using Gradual Varied Flow Profiles

Open channel flow parameter estimation is an inverse problem, which involves the prediction of a function within a domain, given an error criterion with respect to a set of observed data. Various numerical methods have been developed to estimate open channel flow parameters. For this study, Genetic Algorithm optimization technique is selected. Because of its inherent characteristics, Genetic Algorithm optimization technique avoids the subjectivity, long computation time and ill-posedness often associated with conventional optimization techniques. An accurate estimation of roughness coefficients is of vital importance in any open channel flow study. In flood routing in natural rivers, most channels have compound sections and the roughness values in main channel and flood plains are usually different. In order to have more accurate results, the roughness of main channel and flood plains should be considered separately. It is possible to identify the values of roughness using optimization methods. However, studies on the inverse problem of estimating roughness values in compound channels are still limited. The present study involves estimation of open channel flow parameters having different bed materials invoking data of Gradual Varied Flow (GVF). Use of GVF data facilitates estimation of flow parameters. The necessary data base was generated by conducting laboratory experiments in Hydraulics Lab of civil Engineering at IIT Roorkee. In the present study, the efficacy of the Genetic Algorithm (GA) optimization technique is assessed in estimation of open channel flow parameters from the collected experimental data. Computer codes are developed to obtain optimal flow parameters Optimization Technique. Applicability, Adequacy & robustness of the developed code are tested using sets of theoretical data generated by experimental work. Estimation of Manning’s Roughness coefficient from the collected experimental work data by using Manning’s equation & GVF equation were made. The model is designed to arrive at such values of the decision variables that permit minimized mismatch between the observed & the computed GVF profiles. A simulation model was developed to compute GVF depths at preselected discrete sections for given downstream head and discharge rate. This model is linked to an optimizer to estimate optimal value of decision variables. The proposed model is employed to a set of laboratory data for three bed materials (i.e, d50=20mm, d50=6mm and lined concrete). Application of proposed model reveals that optimal value of fitting parameter ranges from 1.42 to 1.48 as the material gets finer. This value differs from the currently documented value i.e. 1.5. The optimal estimates of Manning’s n of three different bed conditions of experimental channel appear to be higher than the corresponding reported / Strickler’s’ estimates.


General
Nowadays, models are decision-making tools but their reliability depends considerably on the choice of these parameters. In practice, their values are obtained through tedious trial-and-error procedures mainly involving visual comparisons. There is clearly scope for improvement by applying automatic optimization methods for the introduction of objectivity and efficiency into the procedure. In general, parameters in mathematical models applied in the field of hydraulic engineering can be categorized into physical parameters and empirical parameters. Physical parameters describe physical properties and features of materials, e.g., the density of fluid. They usually are constants and probably have a set of independent state equations connected with some physical variables. Empirical parameters are based on mathematical models without definite and complete physical concept. Due to complexities of physical processes, the exact values of empirical parameters, such as Manning's n in shallow water equations, are often uncertain. These kinds of empirical parameters are widely used in process modeling. As an empirical parameter, Manning's n actually includes the components of surface friction resistance, form resistance, wave resistance, and resistance due to flow unsteadiness. Many empirical formulations for estimating the n value in practical problems have been suggested in the past (Urquhart 1975).
Roughness and flow estimation from given flow profiles for rivers having multiple estuaries is an important problem. The estuaries may be interconnected and form closed-loop channel networks.
The resulting distributed parameter system may hinder the roughness and flow estimation because of the gradually varied flow effects. This paper documents the development of optimization models for parameter estimation in closedloop channel networks. The parameter estimation procedure determines a set of unknown parameter values by minimizing the difference between the model-predicted water surface profile and observed values. The approach is known as inverse solution of the open-channel flow problem, and it stands as an independent area of research in many fields of engineering applications. For example, in the groundwater literature, Willis and Yeh (1987) reported a large number of inverse problem studies and in openchannel hydraulics, Yeh and Becker (1973) and Khatibi et al. (1997) reported the solution of inverse problems for transient flow in a single channel. The present study considers steady-state flow in closed-loop channel networks. Two approaches can be used for solving inverse problems. The first is an iterative solution in which a numerical model is used to compute the water surface profiles for a given set of input parameter values and the computed water surface profile is then compared with the known water surface profiles for minimum error. Such a procedure can be computationally demanding. The present study uses the second approach, which combines the numerical procedure

Channel Roughness
Channel roughness can be defined as the resistance offered to flow mainly by the bed friction and bed forms. In this study, channel roughness coefficients were identified as parameters by using an automatic optimization method. Where, d 50 is the size of particles in meters which are 50 percent finer. For mixtures of bed material, the above equation is modified as: Where, d 90 is the size of particle in meters and present the particle size in which 90 of the particle is finer than d 90 (K. Subramanya, 2012).
However, the actual n of a channel may be larger than the strickler's estimate because of additional roughness due to bed forms…etc. The other approach is to monitor the normal depth corresponding to several discharge rates in the channel and to compute n by regressing the Manning equation. This approach provides a composite value of n accounting for all sources of roughness and may work well provided several observations corresponding several discharge rates are available. We can estimate composite roughness (nc) by using the following equation.
Where, n c is composite roughness coefficient; N is total number of segment of wetted perimeter; n i is roughness coefficient of i th segment; p i is wetted perimeter of i th segment and is fitting parameter.
However, the underlying assumption of the flow being uniform may not hold. Further, in a typical natural channel with variable roughness along its wetted perimeter, Manning's n may vary with flow depth. The approach of using the uniform flow data does not account for this variability of n. For the present study, d 50 = 6mm particle size and d 50 = 20mm particle size are used.

Brief Review of Literature
Parameter identification techniques have been widely used in the field of hydrology, meteorology, and oceanography. The issue of parameter identification based on the optimal control theories in oceanography can be traced from the early work of Bennett and McIntosh (1982) and Prevost and Salmon (1986). Panchang and O'Brien (1989) carried out early an adjoint parameter identification for bottom drag coefficient in a tidal channel. Das and Lardner (1991) estimated the bottom friction and water depth in a twodimensional tidal flow. Yeh and Sun (1990) presented an adjoint sensitivity analysis for a groundwater system and identified the parameters in a leaky aquifer system. The identifications of parameters in some cases are hard to achieve due to ill-posedness in the inverse problems. Chavent (1974) noted instability and nonuniqueness of identified parameters in the distributed system. Due to the instability, some minimization procedures will lead to serious errors in the identified parameters and make the identification process unstable. In the case of nonuniqueness, the identified parameters will differ according to the initial estimations of the parameters, and not converge to their optimal (or ''true'') values. Yeh (1986) and Navon (1998)  to identification, which addresses the question of whether it is at all possible to obtain a unique solution of the inverse problem for unknown parameters. Although there are a lot of identification procedures available for estimating parameters in mathematical models, none of them can automatically guarantee stability and uniqueness in the parameter identifications in diverse engineering problems. It is therefore vital to confirm the performance of these procedures to find stable ones that can warrant obtaining the optimal solutions. For the present study, channel roughness is identified by using optimization technique.
Optimization techniques were successfully used by Becker andYeh (1972, 1972a), Fread and Smith (1978) and Wormleaton and Karmegam (1984) to identify parameters for regular prismatic channels having simple cross-sections. These researchers used the same optimization algorithm (the so-called "Influence Coefficient" Algorithm) which, mathematically, is closely related to both quasi linearization and the gradient method.

Objectives
The present study involves estimation of Manning roughness n of a channel having different channel bed materials invoking data of gradual varied flow (GVF). Use of GVF data facilitates estimation of depth dependent Manning's roughness n. the necessary data base was generated by conducting laboratory experiments. The overlying objective is fulfilled through the accomplishment of sub objectives listed below. 1) To identify open channel flow parameters by using Genetic Algorithm optimization Technique 2) To generate and monitor gradually varied flow profiles corresponding to different bed materials, discharge and ponded depths. 3) Invoking the observed data of the GVF profiles and the linked simulation optimization approach to estimates Manning's n corresponding to different channel bed materials in the experimental channel, and hence to calibrate the following composite roughness equation.
Where, n c is composite roughness coefficient; N is total number of segment of wetted perimeter; n i is roughness coefficient of i th segment; p i is wetted perimeter of i th segment and is fitting parameter.

Methodology
This study was carried out to identify open channel flow parameters by using Genetic Algorithm optimization technique. Manning's roughness coefficient and other parameters are estimated for different bed materials used ( d 50 =6mm and 20mm grain size and Lined concrete bed materials). Also, GVF flow profile is identified. Crank-Nicolson method is used to solve the governing differential equation.
Parameter optimization technique is used to find the optimal value of coefficient roughness for three different bed materials. Estimation of roughness coefficient is based on Manning's equation for estimation of manning roughness coefficient and corresponding manning roughness parameters. This estimation invokes the data of observed GVF profiles and such accounts for different bed materials with the flow depth. Experimental works is done to several sets of data monitored in Hydraulics Laboratory of Civil Engineering Department

General
Resistance to flow depends up on shape, size and density of object placed as an obstacle to offer resistance (Christodoulou, 2014). Effective roughness during overland flow having shallow depth depends on the fraction of flow depth over roughness element height. Also, an inverse relationship is supposed to exit between Manning's n /friction factor f and Froude number (Barros and Colello, 2001). Cylindrical roughness in open channel has been studied and shows that resistance to flow depends up on depth of flow and stem characteristics (Stone et al., 2002). It has been found that spacing as well as size of stripped bed roughness highly influences the flow over a rectangular channel as a free fall (Guo et al., 2008). Gravel bed roughness is conventionally given by grain size. A new approach was adopted to measure the resistance of flow due to gravel bed by a new parameter based a characteristics of elevation field (Qin and Leung Ng, 2012). Experiments for computation of Manning's n where conducted on twenty-one steep slope streams in Colorado which suggested following relationships of manning's n i.e. value of n decreases with increase in depth of flow, n decreases with decrease in gradient and streams are found to be in supercritical condition which corrected the prior conception of subcritical condition (Jarret, 1984). A numerical method was proposed in order to evaluate Manning's n of shallow water flows (Ding et al., 2004) and an equation is developed to predict Manning's n for high gradient streams (Jarret, 1984). It has been found that Manning's equation is effectively applicable in furrow irrigation problems and Manning's n varies for low discharge rates but attain a constant value at high discharge rates (Esfandiari and Maheshwari, 1998). Nonetheless, computation of Manning's roughness coefficient for border irrigation was also described (Li and Zhang, 2001). The behavior of Manning's n was studied for erodible and nonerodible soils and it has been found that n value is a function of gradient for erodible soils only (Hessel, 2003). Variation of Manning's n with space and time for cropped furrow as well as bare furrow was studied (Maillapalli et al., 2008). A potentially applicable model was developed for estimation of channel roughness which uses embeds finite difference approximation of governing equation to an optimizer having inputs as depth of flow and rates varying with time and space (Ramesh et al.,2000). Apart from experimental data and numerical models, n is also estimated by using software based simulation models. HEC-RAS model is used for roughness coefficient for Hilla River (Hameed and Ali, 2012).

Composite Roughness
Multi roughness channels are not uncommon in field application of open channel flow hydraulics. Due to different roughness of wetted perimeter the overall roughness of the channel is given by composite roughness. Composite roughness comprises of individual roughness effect of channel cross section. Seventeen different equations based on several assumptions along with six different techniques to sub divide the channel cross section were given by numerous investigators (Yen, 2002). The credibility of these equations would be assessed by employing experimental data. An effective methodology has been proposed to evaluate optimal design of cross sectional area of a channel having composite roughness using Manning's roughness equation (Das, 2000).

Introduction
In this chapter, water surface flow profiles corresponding to specific discharges, bed material and ponded depth have been obtained through experimentation. This chapter includes a detailed description of experimental setup, adopted procedures and the observations with range of data obtained for different flow conditions. The experiments for the investigation were carried out in Hydraulics Laboratory of Civil Engineering Department. IIT-Roorkee, India

Flume
A rectangular tilting flume of length 30m, width 0.205m and height 0.50m was used (fig 3.1). The bed of the flume was made up of lined concrete and the other two sides were made up of glass and GI sheet. Discharge was released through an inlet pipe of 0.010m diameter into the flume. The entrance of the channel was provided with flow suppressors in order to make the flow stable. In order to maintain desired depth of water at the downstream of the channel, a tail gate was fitted at the end of the channel. Water discharging from the tail gate, passed to the sump which was circulated again through a 15hp centrifugal pump for further experimentation.

Experimental Procedures
The experiments were conducting by adopting the following steps as mentioned below:-

Slope Measurement
All the sets of experiment were performed on a particular slope of the channel. The slope was measured by using two steel containers connected with a long rubber tube. Both the containers were placed on the channel bed separated by the rubber tube along the length of the channel. One of the containers placed at higher elevation was filled with water Where, H 2 and H 1 is the depth of water in second and first container respectively after equilibrium is established and L is the distance between the containers.

Sieve Analysis
Sieve analysis was performed to determine the particle size of the material used to create artificial bed roughness.
Results of sieve analysis were plotted to investigate the particle size of the bed material used in the present study. Experiments were conducted on two different bed materials. First on one rough bed condition having gravel as a bed particle size d 50 =20mm, d 50 =6mm and then on the smooth condition having lined concrete as bed material.

Calibration of orifice meter
Orifice meter was provided in the inlet pipe for the measurement of discharge. Orifice plate was made up of GI sheet having diameter of 0.06m and the diameter of inlet pipe was 0.10m. Ultrasonic flow meter was used for the calibration of coefficient of discharge of orifice meter. Different discharges were noted corresponding to varying head. This result was plotted and the best fitted line was used (Fig. 3.6).

Figure 3: Calibration curve
Cd was calibrated as 0.66. after calibration of Cd of orifice meter, the discharge in the channel was computed by using the following equation.
Where, a o is area of orifice plate; g= acceleration due to gravity and h= height of water column.

Measurement of water surface profiles
1) Water was released into the rectangular flume by opening the valve of inlet pipe.
2) The desired depth of flow was maintained at the downstream end by operating sluice gate provided at the end of the channel. The depth of water was measured using pointer gauge. 3) After a while when the flow become steady in the channel and the desired depth was maintained at the downstream end, the water surface profile was being measured. 4) Starting from the maintained depth at the downstream end (0.00m), the water surface profile is measured towards upstream at ten (21)

Model Formulation
The present study involves generation of the GVF data that are subsequently employed for estimation of Manning's n corresponding to different segment of wetted perimeter ( Where, n c is composite roughness coefficient; N is total number of segment of wetted perimeter; n i is roughness coefficient of i th segment; p i is wetted perimeter of i th segment and is fitting parameter. The monitored GVF data comprises the depths measured at finite number of discrete locations corresponding to several downstream heads and discharge rates (Fig. 4.3). The data base is thus enumerated as follow: Where M 1 is the number of discrete sections at which the GVF depths are measured; M 2 is number of discharge rates for which the GVF depths are measured and M 3 is number of downstream heads maintained at the tail gate during the measurement of GVF depths.

Figure 5: Measured GVF data
Invoking this data base, the problem of roughness estimation is posed as the following optimization problem.

Decision Variables: ( ,……N) and
Objective function: The objective function is designed to arrive at such values of decision variables that permit minimized mismatch between the observed and the computed GVF profiles. The mismatch is qualified in terms of the following data observed during the experiments.
Where, are simulated and experimentally measured depth at discrete section, discharge rate and downstream head maintained at the tail gate respectively.

Constraint:
Where, upper limit of and lower limit of .

Optimization
The following problem was solved three times corresponding to different bed conditions i.e. d 50 =20mm, d 50 =6mm and lined concrete as bed materials.

Decision Variables:
; and Objective Function: Where, and are simulated and experimentally measured depth at discrete section, discharge rate and downstream head respectively; is a subset of the locations where the observed depth is larger than 1.01 x normal depth; is the weight assigned to the mismatch at location. In the present study the weights are assigned to index the length discretized by the discrete sections. Thus ( ) is defined as follows: i) Following six constraints were assigned to impose upper and lower limits of the segment roughness coefficients The adopted values of the limits are given in Table 2  ii) Following three constraints were assigned to ensure realistic relative roughness of the three roughness coefficients.
iii) Following constraints was assigned to impose upper and limits of fitting parameters ( ).
Since the reported value of 1.5, a range of 1 to 2 was prescribed.
Linked simulation optimization approach is used to estimate the optimal values of the parameters for three bed conditions i.e. d 50 =20mm, d 50 =6mm and lined concrete as bed materials and their corresponding GVF profiles were simulated.

Optimal values
Optimal values of decision variables and their corresponding minimized objective function value for different bed materials are mentioned in Table 3.

Optimal reproduction of GVF profiles
Computed GVF profiles corresponding to the optimal parameter values and the variation of composite roughness are in the following figures. The profile is plotted for three different bed materials corresponding to discharge rates and water depth.

Estimated parameters
The bed roughness ( ) varies from 0.026 to 0.033 as bed material /condition changes from lined concrete to gravel (d 50 =20mm). The corresponding reported/ Strickler's estimates are given in Table 4 by using equation 2. It may be seen that optimal roughness estimates are higher than Strickler's estimates. The roughness coefficient of glass and GI sheet sides as optimized for various runs are presented in Table 5. The estimated roughness coefficients satisfy the known inequality ( < ) and are higher than the tabulated values. This establishes the credibility of the proposed model. The optimal value of (fitting parameter) ranges from 1.42 to 1.48, which differs from the reported value i.e. 1.5. The optimal value of increases as the bed materials get finer.

Reproduction of observed profile
Computed GVF profiles corresponding to the optimal parameter values match quite well with corresponding observed profiles.

Variability of composite roughness
It can be observed that composite roughness reduces with increase in flow depth. Apparently because of increase in weightage of side resistance, the value of composite roughness increase.

Conclusion
This study was carried out to identify open channel flow parameters. Manning's roughness coefficient and other parameters are estimated for different bed materials used ( An optimization method is applied to identify the parameters based on Manning formula for estimation of manning roughness coefficient and corresponding manning roughness parameters. This estimation invokes the data of observed GVF profiles and such accounts for different bed materials with the flow depth.
Experimental works is done to several sets of data monitored in Hydraulics Laboratory of Civil Engineering Department. The application led to the following conclusions; i) The GVF profile computed on the basis of estimated parameters match quite closely with the corresponding observed profiles. ii) Strickler's formula under estimate the roughness due to the bed material. iii) The following commonly used formula is calibrated for Manning coefficient estimation iv) The currently documented value of is 1.5. However, the present work reveals that it varies from 1.43 to 1.47.
The value of generally decreases as the bed material gets coarser.