Canadian Journal of Civil Engineering. Gene-Expression Programming to Predict Manning s n in Meandering Flows

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1 Gene-Expression Programming to Predict Manning s n in Meandering Flows Journal: Canadian Journal of Civil Engineering Manuscript ID cjce r2 Manuscript Type: Article Date Submitted by the Author: 09-Nov-2017 Complete List of Authors: Pradhan, Arpan; National Institute of Technology Rourkela, Civil Engineering Khatua, Kishanjit; National Institute of Technology Rourkela, Civil Engineering Is the invited manuscript for consideration in a Special Issue? : Keyword: N/A Conveyance Estimation, Meandering Channel, Manning's roughness coefficient, Gene-Expression Programming, rivers-lakes-est-& reserv < Hydrotechnical Eng.

2 Page 1 of 32 Canadian Journal of Civil Engineering 1 2 Gene-Expression Programming to Predict Manning s n in Meandering Flows ARPAN PRADHAN, PhD Scholar, Department of Civil Engineering, National Institute of Technology Rourkela, Rourkela, India, er.arpanpradhan@gmail.com (author for correspondence) KISHANJIT K KHATUA, Associate Professor, Department of Civil Engineering, National Institute of Technology Rourkela, Rourkela, India, kkkhatua@nitrkl.ac.in 9 10 Gene-Expression Programming to Predict Manning s n in Meandering Flows 11

3 Page 2 of Gene-Expression Programming to Predict Manning s n in Meandering Flows 14 ABSTRACT Accurate prediction of Manning s roughness coefficient is essential for the computation of conveyance capacity in open channels. There are various factors affecting the roughness coefficient in a meandering compound channel and not just the bed material. The factors, geometric as well as hydraulic, are investigated and incorporated in the prediction of Manning s n. In this study, a new and accurate technique, gene-expression programming (GEP) is used to estimate Manning s n. The estimated value of Manning s n is used in the evaluation of the conveyance capacity of meandering compound channels. Existing methods on conveyance estimation are assessed in order to carry out a comparison between them and the proposed GEP model. Results show that the discharge capacity computed by the new model provides far better results than the traditional models. The developed GEP model is validated with three individual sections of a natural river, signifying that the model can be applied to field study of rivers, within the stated range of parameters Keywords: Manning s roughness coefficient, meandering channel, conveyance estimation, gene-expression programing Introduction The stage or depth of water passing through a river is the simplest way to define its discharge. Prediction of discharge is one of the important works in river flow analysis. Streams and rivers, on the event of high rainfall, overtop their banks and cause damage to the overlying floodplain areas. Reliable estimation of discharge capacity is essential for the design, operation and maintenance of open channels, and more importantly, for flood forecasting. Methods for assessing the discharge capacity of a meandering channel is therefore essential in controlling floods and in designing artificial waterways. Accurate prediction of roughness coefficient is also helpful in predicting discharge in open channels. Manning s, Chezy s and Darcy-Weisbach s, equations have been in use for

4 Page 3 of 32 Canadian Journal of Civil Engineering obtaining discharge for uniform flows in simple channels but it fails to predict discharge for compound channels let alone for meandering channels. These methods were typically developed for simple channels to find the characteristic of the bed material, called the roughness coefficient. The roughness coefficient in meandering channels depend not only on the bed roughness but also on other geometric and hydraulic parameters. Therefore, an attempt is made to develop a model for predicting the roughness coefficient with respect to these parameters. Manning's formula is primarily the most popular formula in open channel flow. Proper care need to be undertaken for implementing Manning s formula to non-uniform and compound channels. Manning s n is a roughness factor which measures n in terms of a geometric measure of the boundary roughness, reflecting the actual or effective unevenness of the boundary as suggested by Yen (1992) for simple uniform flows. In case of compound meandering channels, Manning s n is presumed to be a roughness coefficient, which is affected not only by the boundary unevenness but also by the dynamic behaviour of the channel. Computation of roughness coefficient is challenging due to the various hydraulic complexities in an open channel. There are various methods for estimating the roughness coefficient of a channel by use of tables, photographs and even equations. While the conventional methods might be capable of providing adequate results in the case of simple channels, it is known that the adequacy of these methods for compound section let alone meandering channels is insufficient. Hence, soft computing techniques are highly demanded for calculating Manning s n. The main advantage of genetic programing over regression and other soft computing techniques is the ability to generate a simplified prediction equation without assuming a prior form of the existing relationship. Recently Gandomi and Alavi (2011) developed a new strategy using multistage genetic programming for nonlinear system modelling. This paper presents a new set of experimental data, for channels having heterogeneous roughness for the main channel and floodplains. Laboratory data sets for other investigators have also been collected to find an improved model for predicting composite

5 Page 4 of Manning s roughness coefficient n by using gene-expression programming. The analysis takes into account various geometric and hydraulic parameters such as, relative flow depth, β = (H-h)/H i.e. the ratio of water depth over the floodplain to that of the overall depth of flow in the main channel; width ratio, α = B/b i.e. the total floodplain width to the main channel width; sinuosity = s i.e. the ratio of curve distance to the straight distance in the meandering main channel; bed slope, S o and relative roughness, γ = n fp /n mc i.e. the ratio of bed roughness of floodplain to that of the main channel. Estimation of the roughness coefficient n for other traditional models is also computed. Subsequently, discharge estimation is conducted for all the undertaken models and respective error analysis is carried out to substantiate the strength of the proposed model Experimental Investigation and other Sources of Data A highly meandering channel of sinuosity 4.11 was built at National Institute of Technology, Rourkela (NITR); over a 15m long flume of 4m width. The meandering main channel is a sine-generated curve of crossover angle 110. The 1:1 trapezoidal meandering channel of bottom width 0.33m and 0.065m bank-full depth was constructed within a floodplain of overall width 3.95m with a meandering wavelength of 3.6m. The overall bottom slope or the valley slope of the meandering compound channel is in the downward direction. The assembly for the experimental process is demonstrated in Fig. 1. Water from the underground sump is pumped to the overhead tank from which a regulated flow is maintained into the channel. A volumetric tank at the rear end of the channel is used for measurement of actual discharge. The water is recirculated bank into the sump from the volumetric tank. Analysis for any experimental channel is suitably achieved, if the flow is fully developed and uniform. Due to high sinuosity (i.e. 4.11) of the channel, the meander wavelength is times to that of the width of the main channel. Therefore, to attain uniform and developed flow; a quasi-uniform flow condition is achieved for each of the stages of water depth by maintaining equal flow depths at the second, third and fourth bend apex sections by regulating the tailgate. M1 surface profile occurs if the depth of flow in the

6 Page 5 of 32 Canadian Journal of Civil Engineering downstream section is greater than the upstream one. In such a case, the tailgate is opened. In the case, if the upstream section has a higher depth of flow, M2 profile is formed, in this case the tailgate is closed. The procedure is repeated such that all the three gauges at the apex sections provide equal depths of flow. A set of three series experiments were carried out for the above mentioned channel with differential roughness in the main channel and the floodplains. Series I consists of smooth (Perspex sheet of n=0.01) main channel and floodplains. Series II has artificial grass as the floodplain roughness with n= and Series III has a uniform grade of 8.5mm diameter gravels (n=0.014) fixed to the bottom of the channel bed to the Series II channel. The Manning s n value for all the three types of bed conditions is their respective base n values, computed by investigating them independently on a straight rectangular channel Figure 2 to Fig. 4 represents the three different experimental series. Stage-discharge assessment were carried out at the third bend apex section and the discharge values were recorded with the help of a rectangular notch arrangement at the beginning of the channel. Prior calibration of the notch is conducted by calculating the actual discharge by the volumetric tank. The summary of the experimental observation is shown in Table 1, mentioned as NITR (2017) Series I, II and III. Various aspects of meandering channels due to the effect of one or two parameters have been studied by different investigators. Experimental process on meandering compound channels, commenced by United States Army Corps of Engineers (1956) at Vicksburg; had two basic trapezoidal channels of 0.305m and 0.610m main channel widths for 1:0.5 side slopes. The overall floodplain width was varied to achieve various width ratios (α=b/b) for different sinuosity. Three different combination of bed roughness for main channel and floodplains was undertaken, represented as ','',''' against their series notations. The ratio of roughness of floodplain to that of the main channel is termed as relative roughness and is denoted by γ. A total of 44 such data series (i.e. II to XVI) with different combinations of width ratio, relative roughness and sinuosity have been considered.

7 Page 6 of Experimental investigations were carried out at the SERC Flood Channel Facility in 1990 and 1991 on large scale meandering channels in Phase B at Wallingford, UK, termed as FCF B ( ). The data sets were obtained from the website and from different reports and articles such as James and Wark (1992), Ervine, Willetts, Sellin and Lorena (1993), Greenhill and Sellin (1993). One set of rigid trapezoidal channel of 60 cross-over angle (B21) and two sets of natural channels with 60 (B26, B31, B32, B33 and B34) and 110 (B39, B43, B46, B47, B48) cross-over angles have been investigated. Different type of block arrangements were introduced on the floodplains in order to vary the roughness. The relative roughness for such channels has been assumed 1, as the base value of Manning s n for such blocks have not been provided. Experimental data sets of Willetts and Hardwick (1993) conducted at the University of Aberdeen (denoted as 101, 102 and 104) and Shiono, Al-Romaih and Knight (1999) have also been considered where the effect of bed slope, S o and sinuosity, s were examined. There are 9 data sets of Shiono, Al-Romaih and Knight (1999), denoted as 1(a, b, c); 2(a, b); 4(a, b) and 5(a, b). Investigations previously carried out at NITR i.e. Khatua (2008) (KII and KIII) and Mohanty (2013) have also been taken into the analysis. The experimentations carried out by Mohanty (2013) has the similar geometric features as the present experimental observations with a sinuosity of The extensive set of data series used, aids in analysing the effect of various parameters that affect the roughness coefficient in a meandering compound channel Development of Manning s n Model 3.1 Factors Affecting Manning s n Experimental investigations by Ervine, Willetts, Sellin and Lorena (1993) and Dash and Khatua (2016) proposed major factors influencing conveyance and roughness coefficient in meandering compound channels. It is observed, that the roughness coefficient is dependent on various factors such as width ratio, relative depth of flow, relative roughness of floodplain to that of the main channel, valley slope and sinuosity of the meandering compound channel.

8 Page 7 of 32 Canadian Journal of Civil Engineering Flow in open channels is characterized as subcritical and turbulent, hence the effect of Froude number, Fr and Reynold s number, Re is also taken into account. Aforesaid, there are seven parameters which have been presumed to affect roughness coefficient. While perceiving the effects of one parameter, the other depending parameters need to be similar. In order to achieve the individual characteristics, the data sets undertaken in the paper are utilized to select series which have similar geometric and hydraulic features with one varying parameter. Figure 5 exhibits the relationship of data sets where all but one of the characteristic features, either geometric or hydraulic, influences the roughness coefficient of the channel. Six of the parameters being same, the variation in the roughness coefficient in the graphical representation with respect to the assumed parameter, indicates that the parameter has an influence on the roughness coefficient. Figure 5 consists of seven insets for each of the influencing factors. Only a few of the studied data sets are represented in Fig. 5 to illustrate the dependence of roughness coefficient on these parameters. Insets a, b and c show the variation of roughness coefficient with respect to relative depth β, Reynold s number Re and Froude number Fr respectively for one individual data sets, implying that the other depending geometric parameters are invariable. Insets d, e, f and g suggests the effects of width ratio α, relative roughness γ, bed slope S o and sinuosity s on the roughness coefficient of a channel. As direct influence of these parameters with respect to Manning s n is not plausible, there variation with respect to relative depth is considered. The variation of different curves in these insets suggests that the varying parameter is an influencing factor causing the change in the curves. The value of the varying parameter has been mentioned against the data series reference in each inset. As all the other geometric and hydraulic parameters are same, the difference in the variation curves is supposedly because of the one dissimilar parameter Gene Expression Programming Gene Expression Programming (GEP), suggested by Azamathulla, Ahmad and Ghani (2013)

9 Page 8 of is a search technique which involves computer programs such as mathematical expressions, decision trees, polynomial constructs, logical expressions etc. GEP was developed by Ferreira (2001) on the basis of generation and evaluation of its suitability. First, the chromosomes are generated randomly for each of the individuals in the population. Then the fitness of each of these chromosomes is evaluated based on a fitness function, 179 f = N ( X j Y j ) j= 1 (1) where X j =value returned by the chromosome for the fitness case j and Y j =expected value for the fitness case j. A fitness function provides a quantitative analysis of how close the model is able to predict the expected value. The function f, in the Eq. (1) returns the summation of error in the target value for which the root mean square error (RMSE) is calculated. Various fitness functions are available, but in the current study, RMSE is considered for the development of the GEP model The individuals are then subjected to modifications, and the process is repeated for a predefined number of generations or until a desired solution is achieved. The chromosomes could be unigenic (single gene) or multigenic with equal or unequal program lengths consisting of variables and mathematical operators (function set). The mathematical operators could be arithmetic (+, -, *, /) as well as functions (sin, cos, tan, log, sqrt, power, exp, etc). GeneXproTools 5.0 is used for modelling the gene expression programming in this study. GeneXproTools works with population of models which are selected according to their respective fitness. The selected models are reproduced by introducing genetic variations by using one or more genetic operators like mutation or recombination. Repetition of this process for a certain number of generations, provide with a more improved model Development of GEP Model for Manning s n The following relationship describes Manning s n or roughness coefficient as a function of geometric and hydraulic factors as discussed earlier:

10 Page 9 of 32 Canadian Journal of Civil Engineering ( α, β, γ, S, s, Fr) n= f o Re, (2) The model development in this study designates Manning s roughness as the output and the seven independent parameters in Eq. (2) as input. Four basic arithmetic operators (+, -, *, /) and some basic mathematical functions (sqrt, e x, ln) were used as function set in the model development. A multigenic programming i.e. with 3 genes and addition as the linking function is used. A large number of generations (5000) were tested. The functional set and operational parameters used in the GEP modelling during this study are listed in Table 2. The overall 477 data sets were randomly distributed as 70% for training and the rest 30% as testing data. The data sets need not be normalized in the analysis, as the modelling is carried out by fitness function which generates an expression to calculate n from the depending parameters. These parameters whether dimensional or non-dimensional can be used directly in their usual form, i.e. the one used during the model generation, to calculate Manning s n. The simplified analytical form of the proposed GEP model is expressed as 213 n 1.86 S ( β S 8.89) 7 2 γ s + Re ( 2β) ( 2.54 S ln( α) S ) o o = + o o (3) Fr Training and Testing of GEP Model The execution of the GEP model in training and testing sets were validated in terms of Coefficient of determination (R 2 ), Average Error (AE), Mean Absolute Error (MAE), Root Mean Square Error (RMSE) and Mean Absolute Percentage Error (MAPE). These were computed as given in (4) to (8) to find the acceptability of each of the models with respect to the data sets. 220 xy x 2 2 R = (4) 2 2 y

11 Page 10 of X Y AE = 100 (5) X 222 = X Y MAE (6) p 223 ( X Y) RMSE = (7) p 224 MAPE= X Y 100 X p (8) 225 where x= ( X X) ; y ( Y Y) = ; X is the observed values; X is mean of X; Y is the predicted value; Y is mean of Y; and p is the number of samples. Influence of each individual parameter on Manning s n was verified by developing series of models in GeneXproTools 5.0, with one independent parameter removed in each case. Table 3 indicates the error analysis of the developed GEP models for the training data sets. It is observed that on excluding any one of the independent parameters, larger RMSE and lower R 2 values were generated. Thus indicating that each of the seven independent parameters have significant effect on the roughness coefficient. Hence the functional relationship demonstrated in Eq. (2) is used in this study. Figure 6 represents the coefficient of determination for the training and validation data sets for the developed GEP model in Eq. (3). The predicted value of Manning s n can be used to obtain the discharge capacity carried by a meandering compound channel Other Conveyance Prediction Methods The proposed GEP model for predicting conveyance is checked and compared with the following existing conveyance estimation methods:

12 Page 11 of 32 Canadian Journal of Civil Engineering The linearized SCS method (LSCS) for inbank flows in meandering channels given by James and Wark (1992) is derived for two ranges of sinuosity and is represented as, 242 n n = 0.43s+ 0.5 n = 1.3 n fors< 1.5 fors> 1.5 (9) where n is value of Manning coefficient due to friction loss and nʹ is the value of Manning coefficient including bend losses. 2. Meander-belt method given by Greenhill and Sellin (1993) which suggested five methods with different combination of division lines and bed slopes. The method with inclined division lines was observed to give better results which is used in this study and is represented as GH5. The discharge stimation is represented as, Q RmcSmcAmc + RmbSmcAmb + RfpSfp = Afp (10) n n n where the subscripts mc represents main channel area; mb as meander-belt region; and fp is the area outside the meander-belt. 3. Shiono, Al-Romaih and Knight (1999) carried out experimental investigation on meandering channels by varying the bed slope, S o for different sinuosity. Consequently, they derived a model by dimensional analysis to illustrate that friction factor, f is mainly dependent on sinuosity. The relationship is shown below, ( ) 2 s= 10 f (11) Results and Analysis 4.1 Discharge Prediction in Laboratory Channels The predicted Manning s n value by the GEP model is used in estimating the conveyance capacity of a meandering compound channel. It is essential to note that the n value predicted, has taken into account the various geometric and hydraulic aspects of a compound channel and is different to that of the composite n computed backwards by the Manning s equation

13 Page 12 of from the actual discharge. The conveyance capacity for all the experimental channels are estimated by the presumed models such as LSCS (1992), GH5 (1993) and Shiono-Al- Knight (1999). The data sets of individual investigators are coupled together to carry out an overall error analysis. Such an analysis presumably provides with a general indication regarding the suitability of different models. Different error analysis approaches such R 2, MAE, RMSE and MAPE have been carried out and illustrated in a tabular form as shown in Table 4. Figure 7 also shows the percentage of error for each of the data series using the different models. It is observed that GEP provides with high R 2 values, closer to 1 whereas for the other methods, the coefficient of determination is lower. The developed GEP model also provides lower values of MAE, RMSE and MAPE with respect to other models. Especially the mean absolute percentage error provides less than 7% error whereas the error percentage in the other models is quite high From Fig. 7 it is observed that the mean error for the GEP model gives best result along with GH5, which in respect to other models gives lower mean percentage error values. The lower error provided by GH5 with respect to other models is because the model was specifically developed for meandering compound channels. On analysing Table 4, the R 2 values for GH5 is also observed to be closer to 1, quite comparable to the GEP model. However, on observing the other error analysis techniques, such as MAPE, the average error for GH5 (in the case of FCF Phase B Roughened) is as high as 50.28, whereas the MAPE error by GEP is 6.6 (for the present study) at most. It is important to mention that by categorizing the data sets according to the investigators, it becomes inconclusive, as each individual researcher has carried out experimental investigation on different types of meandering compound channels, by varying different parameters. Hence associating all those experimental observations as a single set, might provide with spurious results. Even the data sets are in different ranges i.e. some are large scale channels while others being small scale. Therefore, comparing by the above

14 Page 13 of 32 Canadian Journal of Civil Engineering method would give large differences among the data sets. Therefore, the best two models, i.e. GH5 and GEP from the above analysis is further investigated by finding the percentage of error in each individual data series. Figures 8 to 11 demonstrate the percentage of error for every individual data series in US Army (1956); FCF Phase B ( ); Willetts- Hardwick (1993) and Shiono-Al-Knight (1999); NITR (2008, 2013) and present study (2017) respectively along with the values of standard deviation. From the extensive error analysis for all the data sets, it clearly illustrated that the proposed model provides better estimation of discharge in a meandering compound channel as compared to the other models Incorporation of Discharge Prediction Methods to Natural Rivers Any method must pass through the test of reasonably performing in genuine situations, i.e., for field cases or rivers. Therefore, the method should be tested for its suitability to the field data. The Watawarra channel in the Cooper Creek, Central Australia is selected as study area for implementation and investigation of the different discharge prediction models. The characteristics of the channel and floodplains in the Cooper Creek, Central Australia were studied by Fagan (2001). The Watawarra channel occurs after the junction of the Cooper and Wilson rivers and runs approximately 33km along the channel in a south/south-westerly direction with an increasing channel sinuosity and decreasing overall width downstream. It has a composite cross-section with a small, relatively narrow and deep channel inset into a wider, deeper channel. The inset channel is approximately constant in size throughout the length of the channel, which is considered as the main channel width in this paper. The roughness throughout the cross-section is assumed to be same, hence the relative roughness is taken as 1. Fagan (2001) selected three cross-sections for the Watawarra channel with sufficient length of reach for a meaningful measurement of sinuosity and other planform characteristics. Other characteristics such as, no confluences or bifurcations and that the reach length was a minimum 100 times to that of the channel width were also considered for validly treating the

15 Page 14 of channel as homogenous. The morphological map of the Watawarra channel with subdivisions and surveyed sections is shown in Figs.12 and 13. The channel reach and planform characteristics for the three surveyed sections has been summarized in Table 5. The different models analyzed in the previous section is applied to the river Watawarra. Table 6 illustrates the values of R 2, MAE, RMSE and MAPE whereas Fig. 14 shows the percentage of error by different models along with the standard deviation. The GEP model is observed to successfully predict the discharge capacity for a natural section. It is significant to mention that the model requires only the geometric features and the mean velocity at a section to estimate the discharge capacity, much better than the other traditional models. It is relevant to state that the proposed model is developed by undertaking seven geometric and hydraulic parameters and can be appropriately applied to real cases within the 328 prescribed ranges Conclusions A new and improved technique to predict discharge in a meandering compound channel is proposed, based on gene-expression programming. Three new sets of experimental data (15 runs each), along with a wide range of data sets of other researchers (i.e. 477 runs in total) with different channel parameters have been used in the development of the model. The data sets used have width ratio in the range 6.79 up to 30 which are both small scale as well as large scale data. The data sets have different slopes and sinuosity with homogenous as well as heterogeneous roughness. It is pertinent to mention that the proposed GEP model is based on laboratory data sets with dimensionless geometric parameters in the ranges; 6.79 α 30, β 0.64,.0005 S o , s 4.11 and 1 γ 2.92 A selected number of models for predicting roughness coefficient were studied to estimate conveyance for compound meandering channels using the same data sets in order to investigate the suitability of the various methods. It was observed that the developed model provided with satisfactory result as compared to the other models in terms of R 2, MAE, RMSE and MAPE for groups of data series. When observed in a more amplified approach,

16 Page 15 of 32 Canadian Journal of Civil Engineering i.e. by considering the percentage of error along with the standard deviation for each individual data set, the proposed model showed noticeably better results in the range of less than 7% error, proving to be a quite advanced model with respect to the others. The developed GEP model is also observed to predict well for natural rivers with an acceptable range of about 10% average error Notation A T = Total Area b = Main Channel Width B = Total Floodplain Width f = Darcy-Weisbach s friction factor Fr = Froude s number h = Height of Main Channel H = Overall Depth of Flow L W = Meander Wavelength n = Manning s roughness coefficient Q = Discharge (m 3 s -1 ) r c = Radius of Curvature R = Hydraulic radius (m) Re = Reynold s number s = Sinuosity S o = Bed slope V = Mean velocity (m 2 s -1 ) α = Width ratio β = Relative flow depth γ = Relative roughness ν = Kinematic viscosity (m 2 s -1 )

17 Page 16 of Acknowledgement The authors wish to acknowledge thankfully the support received by the third author from DST India, under grant no. SR/S3/MERC/066/2008 and SR/S3/MERC/0080/2012 for conducting experimental research works References Azamathulla, H. Md., Ahmad, Z. & Ghani, A. Ab An Expert System for Predicting Manning s roughness coefficient in open channels by using Gene Expression Programming. Neural Comput & Applic., 23: doi: /s z. 380 Dash, S. & Khatua, K. K Sinuosity Dependency on Stage Discharge in Meandering Channels. J. Irrig. Drain Eng., /(ASCE)IR Ervine, D. A., Willetts, B. B., Sellin, R. H. J., & Lorena, M Factors affecting conveyance in meandering compound flows. J. Hydr. Engrg., 119(12), Fagan, S. D Channel and floodplain characteristics of Cooper Creek, Central Australia. Ph.D. thesis, University of Wollongong, Australia. Ferreira C Gene expression programming: A new adaptive algorithm for solving problems. Complex Syst. 13(2), Gandomi, A. H., Alavi, A. H, Mirzahosseini, M. R. & Moqhadas, N. J Nonlinear genetic-based models for prediction of flow number of asphalt mixtures. J. Mater. Civ. Eng., 23(3), Greenhill, R. K., & Sellin, R. H. J Development of a simple method to predict discharges in compound meandering channels. Proc. Inst. of Civ. Engrs., Water, Maritime and Energy, 101(1),

18 Page 17 of 32 Canadian Journal of Civil Engineering James, C. S Evaluation of methods for predicting bend loss in meandering channels. J. Hydr. Engrg., 120(2), James, C. S., and Wark, J. B Conveyance estimation for meandering channels. Report SR 329. HR Wallingford, Wallingford, U.K. Khatua, K. K Interaction of flow and estimation of discharge in two stage meandering compound channels. Ph.D. thesis, National Institute of Technology, Rourkela, India. Mohanty, P. K Flow analysis of compound channels with wide floodplains. Ph.D. Thesis, National Institute of Technology, Rourkela, India. Shiono, K., Al-Romaih, J. S., & Knight, D. W Stage-discharge assessment in compound meandering channels. J. Hydr. Engrg., 125(1), U.S. Army Corps of Engineers Hydraulic capacity of meandering channels in straight floodways. Waterways Experiments Station, Vicksburg, MS U.S. Department of Agriculture Engineering handbook: hydraulics. U.S. Department of Agriculture, Soil Conservation Service, sec. 5. U.S. Department of Agriculture Guide for selecting roughness coefficient n values for channels. U.S. Department of Agriculture, Soil Conservation Service U.S. Department of Transportation 1979 Design charts for open-channel flow: U.S. Department of Transportation, Federal Highway Administration, Hydraulic Design Series 3. Willetts, B. B., and Hardwick, R. I Stage dependency for overbank flow in meandering channels. Proc. Inst. of Civ. Engrs., Water, Maritime and Energy, 101(1),

19 Page 18 of Yen, B. C Dimensionally homogeneous Manning's formula. J. Hydr. Engrg., 118(9),

20 Page 19 of 32 Canadian Journal of Civil Engineering 421 Table 1 Experimental observations Data Series NITR (2017) 422 Side Slope α s S o γ β Q Series I (5) 1V:1H Series II (5) 1V:1H Series III (5) 1V:1H , 0.270, 0.297, 0.307, , 0.425, 0.458, 0.496, , 0.381, 0.430, 0.467, , 0.034, 0.040, 0.042, , 0.044, 0.058, 0.077, , 0.052, 0.066, 0.086, Table 2 Functional set and Operational parameters used in GEP Model 425 Description of Parameter Parameter Setting Function Set +, -, *, /, sqrt, exp, ln Number of Chromosomes 30 Head Size 8 Number of Genes 3 Linking Function Addition Fitness Function RMSE Program Size 40 Number of Generations Constants per Gene 5 Data Type Integer Crossover Frequency (%) 50 Block Mutation Rate (%) 30 Homologous Crossover (%) Table 3 Sensitivity Analysis for different GEP models and Error analysis of ANN Model R 2 AE (%) RMSE MAPE ( α, β, γ, S, s, Fr) n= f o Re, ( α, β, γ, S, s,re) n= f ( α, β, γ, S, s Fr) o n= f o,

21 Page 20 of 32 ( α, β, γ, S, Fr) n= f Re, o ( α, β, γ, s, Re, Fr) n= f ( α,β, S, s, Fr) n= f o Re, ( α,γ, S, s, Fr) n= f o Re, ( β,γ, S, s, Fr) n= f o Re, Table 4 Error Analysis of individual data series by presumed methods LSCS (1967) GH5 (1992) R 2 MAE RMSE MAPE SAK (1999) GEP Model LSCS (1967) GH5 (1992) SAK (1999) US Army (1956) FCF Phase B Smooth ( ) FCF Phase B Roughened ( ) Hardwick-Willetts (1993) GEP Model LSCS (1967) GH5 (1992) SAK (1999) GEP Model LSCS (1967) GH5 (1992) SAK (1999) GEP Model Shiono-Al-Knight (1999) NITR (2008, 2013) Present Study (2017) Table 5 Sectional Parameters of River Watawarra Data Series b (m) B (m) h (m) H (m) L W (m) r c (m) s S o A T (m 2 ) V (m 2 s -1 ) Re Fr Q (m 3 s -1 ) Watawarra Channel W E E W E E W E E Table 6 Error Analysis of River Watawarra Methods R 2 MAE RMSE MAPE LSCS GH SAK GEP Model

22 Page 21 of 32 Canadian Journal of Civil Engineering Figure 1 Planform of Experimental Procedure Figure 2 Series I Figure 3 Series II 442

23 Page 22 of Figure 4 Series III

24 Page 23 of 32 Canadian Journal of Civil Engineering Dependence of roughness coefficient on the various influencing parameters

25 Page 24 of 32 Coefficient of determination for training and validation data sets

26 Page 25 of 32 Canadian Journal of Civil Engineering Percentage of error for different data series by the presumed models

27 Page 26 of 32 Percentage of error for US Army (1956)

28 Page 27 of 32 Canadian Journal of Civil Engineering Percentage of error for FCF Phase B ( )

29 Page 28 of 32 Percentage of error for Willetts-Hardwick (1993) and Shiono-Al-Knight (1999)

30 Page 29 of 32 Canadian Journal of Civil Engineering Percentage of error for NITR (2008, 2013) and present study (2017)

31 Page 30 of 32 Morphological Map of River Watawarra 103x134mm (96 x 96 DPI)

32 Page 31 of 32 Canadian Journal of Civil Engineering Surveyed Sections of River Watawarra 103x83mm (96 x 96 DPI)

33 Page 32 of 32 Percentage of error for different models in river Watawarra