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Draft

Gene-Expression Programming to Predict Manning’s n in

Meandering Flows

Journal: Canadian Journal of Civil Engineering

Manuscript ID cjce-2016-0569.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? :

N/A

Keyword: Conveyance Estimation, Meandering Channel, Manning's roughness coefficient, Gene-Expression Programming, rivers-lakes-est-& reserv < Hydrotechnical Eng.

https://mc06.manuscriptcentral.com/cjce-pubs

Canadian Journal of Civil Engineering

Draft

Gene-Expression Programming to Predict Manning’s n in Meandering 1

Flows 2

ARPAN PRADHAN, PhD Scholar, Department of Civil Engineering, National Institute of 3

Technology Rourkela, Rourkela, India, 4

Email: [email protected] (author for correspondence) 5

KISHANJIT K KHATUA, Associate Professor, Department of Civil Engineering, National 6

Institute of Technology Rourkela, Rourkela, India, 7

Email: [email protected] 8

9

Gene-Expression Programming to Predict Manning’s n in Meandering Flows 10

11

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Gene-Expression Programming to Predict Manning’s n in Meandering 12

Flows 13

ABSTRACT 14

Accurate prediction of Manning’s roughness coefficient is essential for the computation of conveyance 15

capacity in open channels. There are various factors affecting the roughness coefficient in a 16

meandering compound channel and not just the bed material. The factors, geometric as well as 17

hydraulic, are investigated and incorporated in the prediction of Manning’s n. In this study, a new and 18

accurate technique, gene-expression programming (GEP) is used to estimate Manning’s n. The 19

estimated value of Manning’s n is used in the evaluation of the conveyance capacity of meandering 20

compound channels. Existing methods on conveyance estimation are assessed in order to carry out a 21

comparison between them and the proposed GEP model. Results show that the discharge capacity 22

computed by the new model provides far better results than the traditional models. The developed GEP 23

model is validated with three individual sections of a natural river, signifying that the model can be 24

applied to field study of rivers, within the stated range of parameters. 25

Keywords: Manning’s roughness coefficient, meandering channel, conveyance estimation, 26

gene-expression programing. 27

1 Introduction 28

The stage or depth of water passing through a river is the simplest way to define its discharge. 29

Prediction of discharge is one of the important works in river flow analysis. Streams and 30

rivers, on the event of high rainfall, overtop their banks and cause damage to the overlying 31

floodplain areas. Reliable estimation of discharge capacity is essential for the design, 32

operation and maintenance of open channels, and more importantly, for flood forecasting. 33

Methods for assessing the discharge capacity of a meandering channel is therefore essential in 34

controlling floods and in designing artificial waterways. 35

Accurate prediction of roughness coefficient is also helpful in predicting discharge in 36

open channels. Manning’s, Chezy’s and Darcy-Weisbach’s, equations have been in use for 37

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obtaining discharge for uniform flows in simple channels but it fails to predict discharge for 38

compound channels let alone for meandering channels. These methods were typically 39

developed for simple channels to find the characteristic of the bed material, called the 40

roughness coefficient. The roughness coefficient in meandering channels depend not only on 41

the bed roughness but also on other geometric and hydraulic parameters. Therefore, an 42

attempt is made to develop a model for predicting the roughness coefficient with respect to 43

these parameters. 44

Manning's formula is primarily the most popular formula in open channel flow. 45

Proper care need to be undertaken for implementing Manning’s formula to non-uniform and 46

compound channels. Manning’s n is a roughness factor which measures n in terms of a 47

geometric measure of the boundary roughness, reflecting the actual or effective unevenness of 48

the boundary as suggested by Yen (1992) for simple uniform flows. In case of compound 49

meandering channels, Manning’s n is presumed to be a roughness coefficient, which is 50

affected not only by the boundary unevenness but also by the dynamic behaviour of the 51

channel. Computation of roughness coefficient is challenging due to the various hydraulic 52

complexities in an open channel. There are various methods for estimating the roughness 53

coefficient of a channel by use of tables, photographs and even equations. 54

While the conventional methods might be capable of providing adequate results in the 55

case of simple channels, it is known that the adequacy of these methods for compound section 56

let alone meandering channels is insufficient. Hence, soft computing techniques are highly 57

demanded for calculating Manning’s n. The main advantage of genetic programing over 58

regression and other soft computing techniques is the ability to generate a simplified 59

prediction equation without assuming a prior form of the existing relationship. Recently 60

Gandomi and Alavi (2011) developed a new strategy using multistage genetic programming 61

for nonlinear system modelling. 62

This paper presents a new set of experimental data, for channels having 63

heterogeneous roughness for the main channel and floodplains. Laboratory data sets for other 64

investigators have also been collected to find an improved model for predicting composite 65

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Manning’s roughness coefficient n by using gene-expression programming. The analysis 66

takes into account various geometric and hydraulic parameters such as, relative flow depth, β 67

= (H-h)/H i.e. the ratio of water depth over the floodplain to that of the overall depth of flow 68

in the main channel; width ratio, α = B/b i.e. the total floodplain width to the main channel 69

width; sinuosity = s i.e. the ratio of curve distance to the straight distance in the meandering 70

main channel; bed slope, So and relative roughness, γ = nfp/nmc i.e. the ratio of bed roughness 71

of floodplain to that of the main channel. Estimation of the roughness coefficient n for other 72

traditional models is also computed. Subsequently, discharge estimation is conducted for all 73

the undertaken models and respective error analysis is carried out to substantiate the strength 74

of the proposed model. 75

2 Experimental Investigation and other Sources of Data 76

A highly meandering channel of sinuosity 4.11 was built at National Institute of Technology, 77

Rourkela (NITR); over a 15m long flume of 4m width. The meandering main channel is a 78

sine-generated curve of crossover angle 110°. The 1:1 trapezoidal meandering channel of 79

bottom width 0.33m and 0.065m bank-full depth was constructed within a floodplain of 80

overall width 3.95m with a meandering wavelength of 3.6m. The overall bottom slope or the 81

valley slope of the meandering compound channel is 0.00165 in the downward direction. The 82

assembly for the experimental process is demonstrated in Fig. 1. Water from the underground 83

sump is pumped to the overhead tank from which a regulated flow is maintained into the 84

channel. A volumetric tank at the rear end of the channel is used for measurement of actual 85

discharge. The water is recirculated bank into the sump from the volumetric tank. 86

Analysis for any experimental channel is suitably achieved, if the flow is fully 87

developed and uniform. Due to high sinuosity (i.e. 4.11) of the channel, the meander 88

wavelength is 45.45 times to that of the width of the main channel. Therefore, to attain 89

uniform and developed flow; a quasi-uniform flow condition is achieved for each of the 90

stages of water depth by maintaining equal flow depths at the second, third and fourth bend 91

apex sections by regulating the tailgate. M1 surface profile occurs if the depth of flow in the 92

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downstream section is greater than the upstream one. In such a case, the tailgate is opened. In 93

the case, if the upstream section has a higher depth of flow, M2 profile is formed, in this case 94

the tailgate is closed. The procedure is repeated such that all the three gauges at the apex 95

sections provide equal depths of flow. 96

A set of three series experiments were carried out for the above mentioned channel 97

with differential roughness in the main channel and the floodplains. Series I consists of 98

smooth (Perspex sheet of n=0.01) main channel and floodplains. Series II has artificial grass 99

as the floodplain roughness with n= 0.018 and Series III has a uniform grade of 8.5mm 100

diameter gravels (n=0.014) fixed to the bottom of the channel bed to the Series II channel. 101

The Manning’s n value for all the three types of bed conditions is their respective base n 102

values, computed by investigating them independently on a straight rectangular channel. 103

Figure 2 to Fig. 4 represents the three different experimental series’. 104

Stage-discharge assessment were carried out at the third bend apex section and the 105

discharge values were recorded with the help of a rectangular notch arrangement at the 106

beginning of the channel. Prior calibration of the notch is conducted by calculating the actual 107

discharge by the volumetric tank. The summary of the experimental observation is shown in 108

Table 1, mentioned as NITR (2017) Series I, II and III. 109

Various aspects of meandering channels due to the effect of one or two parameters 110

have been studied by different investigators. Experimental process on meandering compound 111

channels, commenced by United States Army Corps of Engineers (1956) at Vicksburg; had 112

two basic trapezoidal channels of 0.305m and 0.610m main channel widths for 1:0.5 side 113

slopes. The overall floodplain width was varied to achieve various width ratios (α=B/b) for 114

different sinuosity. Three different combination of bed roughness’ for main channel and 115

floodplains was undertaken, represented as ', '', ''' against their series notations. The ratio of 116

roughness of floodplain to that of the main channel is termed as relative roughness and is 117

denoted by γ. A total of 44 such data series (i.e. II to XVI) with different combinations of 118

width ratio, relative roughness and sinuosity have been considered. 119

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Experimental investigations were carried out at the SERC Flood Channel Facility in 120

1990 and 1991 on large scale meandering channels in Phase B at Wallingford, UK, termed as 121

FCF B (1990-1991). The data sets were obtained from the website 122

http://www.birmingham.ac.uk/ and from different reports and articles such as James and 123

Wark (1992), Ervine, Willetts, Sellin and Lorena (1993), Greenhill and Sellin (1993). One set 124

of rigid trapezoidal channel of 60° cross-over angle (B21) and two sets of natural channels 125

with 60° (B26, B31, B32, B33 and B34) and 110° (B39, B43, B46, B47, B48) cross-over 126

angles have been investigated. Different type of block arrangements were introduced on the 127

floodplains in order to vary the roughness. The relative roughness for such channels has been 128

assumed 1, as the base value of Manning’s n for such blocks have not been provided. 129

Experimental data sets of Willetts and Hardwick (1993) conducted at the University 130

of Aberdeen (denoted as 101, 102 and 104) and Shiono, Al-Romaih and Knight (1999) have 131

also been considered where the effect of bed slope, So and sinuosity, s were examined. There 132

are 9 data sets of Shiono, Al-Romaih and Knight (1999), denoted as 1(a, b, c); 2(a, b); 4(a, b) 133

and 5(a, b). Investigations previously carried out at NITR i.e. Khatua (2008) (KII and KIII) 134

and Mohanty (2013) have also been taken into the analysis. The experimentations carried out 135

by Mohanty (2013) has the similar geometric features as the present experimental 136

observations with a sinuosity of 1.11. The extensive set of data series used, aids in analysing 137

the effect of various parameters that affect the roughness coefficient in a meandering 138

compound channel. 139

3 Development of Manning’s n Model 140

3.1 Factors Affecting Manning’s n 141

Experimental investigations by Ervine, Willetts, Sellin and Lorena (1993) and Dash 142

and Khatua (2016) proposed major factors influencing conveyance and roughness coefficient 143

in meandering compound channels. It is observed, that the roughness coefficient is dependent 144

on various factors such as width ratio, relative depth of flow, relative roughness of floodplain 145

to that of the main channel, valley slope and sinuosity of the meandering compound channel. 146

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Flow in open channels is characterized as subcritical and turbulent, hence the effect of Froude 147

number, Fr and Reynold’s number, Re is also taken into account. 148

Aforesaid, there are seven parameters which have been presumed to affect roughness 149

coefficient. While perceiving the effects of one parameter, the other depending parameters 150

need to be similar. In order to achieve the individual characteristics, the data sets undertaken 151

in the paper are utilized to select series’ which have similar geometric and hydraulic features 152

with one varying parameter. 153

Figure 5 exhibits the relationship of data sets where all but one of the characteristic 154

features, either geometric or hydraulic, influences the roughness coefficient of the channel. 155

Six of the parameters being same, the variation in the roughness coefficient in the graphical 156

representation with respect to the assumed parameter, indicates that the parameter has an 157

influence on the roughness coefficient. Figure 5 consists of seven insets for each of the 158

influencing factors. Only a few of the studied data sets are represented in Fig. 5 to illustrate 159

the dependence of roughness coefficient on these parameters. 160

Insets a, b and c show the variation of roughness coefficient with respect to relative 161

depth β, Reynold’s number Re and Froude number Fr respectively for one individual data 162

sets, implying that the other depending geometric parameters are invariable. Insets d, e, f and 163

g suggests the effects of width ratio α, relative roughness γ, bed slope So and sinuosity s on the 164

roughness coefficient of a channel. As direct influence of these parameters with respect to 165

Manning’s n is not plausible, there variation with respect to relative depth is considered. The 166

variation of different curves in these insets suggests that the varying parameter is an 167

influencing factor causing the change in the curves. The value of the varying parameter has 168

been mentioned against the data series reference in each inset. As all the other geometric and 169

hydraulic parameters are same, the difference in the variation curves is supposedly because of 170

the one dissimilar parameter. 171

3.2 Gene Expression Programming 172

Gene Expression Programming (GEP), suggested by Azamathulla, Ahmad and Ghani (2013) 173

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is a search technique which involves computer programs such as mathematical expressions, 174

decision trees, polynomial constructs, logical expressions etc. GEP was developed by Ferreira 175

(2001) on the basis of generation and evaluation of its suitability. First, the chromosomes are 176

generated randomly for each of the individuals in the population. Then the fitness of each of 177

these chromosomes is evaluated based on a fitness function, 178

( )∑=

−=N

j

jj YXf1

(1) 179

where Xj=value returned by the chromosome for the fitness case j and Yj=expected value for 180

the fitness case j. A fitness function provides a quantitative analysis of how close the model is 181

able to predict the expected value. The function f, in the Eq. (1) returns the summation of 182

error in the target value for which the root mean square error (RMSE) is calculated. Various 183

fitness functions are available, but in the current study, RMSE is considered for the 184

development of the GEP model. 185

The individuals are then subjected to modifications, and the process is repeated for a 186

predefined number of generations or until a desired solution is achieved. The chromosomes 187

could be unigenic (single gene) or multigenic with equal or unequal program lengths 188

consisting of variables and mathematical operators (function set). The mathematical operators 189

could be arithmetic (+, -, *, /) as well as functions (sin, cos, tan, log, sqrt, power, exp, etc). 190

GeneXproTools 5.0 is used for modelling the gene expression programming in this 191

study. GeneXproTools works with population of models which are selected according to their 192

respective fitness. The selected models are reproduced by introducing genetic variations by 193

using one or more genetic operators like mutation or recombination. Repetition of this process 194

for a certain number of generations, provide with a more improved model. 195

3.2.1 Development of GEP Model for Manning’s n 196

The following relationship describes Manning’s n or roughness coefficient as a function of 197

geometric and hydraulic factors as discussed earlier: 198

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( )FrsSfn o Re,,,,,, γβα= (2) 199

The model development in this study designates Manning’s roughness as the output 200

and the seven independent parameters in Eq. (2) as input. Four basic arithmetic operators (+, -201

, *, /) and some basic mathematical functions (sqrt, ex, ln) were used as function set in the 202

model development. A multigenic programming i.e. with 3 genes and addition as the linking 203

function is used. A large number of generations (5000) were tested. The functional set and 204

operational parameters used in the GEP modelling during this study are listed in Table 2. 205

The overall 477 data sets were randomly distributed as 70% for training and the rest 206

30% as testing data. The data sets need not be normalized in the analysis, as the modelling is 207

carried out by fitness function which generates an expression to calculate n from the 208

depending parameters. These parameters whether dimensional or non-dimensional can be 209

used directly in their usual form, i.e. the one used during the model generation, to calculate 210

Manning’s n. 211

The simplified analytical form of the proposed GEP model is expressed as 212

( ) ( )( )( )

oo

ooSS

s

Fr

SSn α

βγβ

ln54.2Re

264.117

89.886.12

−+

−

−−

+

−−−= (3) 213

3.2.2 Training and Testing of GEP Model 214

The execution of the GEP model in training and testing sets were validated in terms 215

of Coefficient of determination (R2), Average Error (AE), Mean Absolute Error (MAE), Root 216

Mean Square Error (RMSE) and Mean Absolute Percentage Error (MAPE). These were 217

computed as given in (4) to (8) to find the acceptability of each of the models with respect to 218

the data sets. 219

2

22

2

=

∑∑∑

yx

xyR (4) 220

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100∑

−=

X

YXAE (5) 221

−= ∑

p

YXMAE (6) 222

( ) 2

12

−= ∑

p

YXRMSE (7) 223

−

=∑

p

X

YX

MAPE

100

(8) 224

where ( )XXx −= ; ( )YYy −= ; X is the observed values; X is mean of X; Y is the 225

predicted value; Y is mean of Y; and p is the number of samples. 226

Influence of each individual parameter on Manning’s n was verified by developing 227

series of models in GeneXproTools 5.0, with one independent parameter removed in each 228

case. Table 3 indicates the error analysis of the developed GEP models for the training data 229

sets. It is observed that on excluding any one of the independent parameters, larger RMSE 230

and lower R2 values were generated. Thus indicating that each of the seven independent 231

parameters have significant effect on the roughness coefficient. Hence the functional 232

relationship demonstrated in Eq. (2) is used in this study. 233

Figure 6 represents the coefficient of determination for the training and validation 234

data sets for the developed GEP model in Eq. (3). The predicted value of Manning’s n can be 235

used to obtain the discharge capacity carried by a meandering compound channel. 236

3.3 Other Conveyance Prediction Methods 237

The proposed GEP model for predicting conveyance is checked and compared with the 238

following existing conveyance estimation methods: 239

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1. The linearized SCS method (LSCS) for inbank flows in meandering channels given 240

by James and Wark (1992) is derived for two ranges of sinuosity and is represented as, 241

>=′

<+=′

5.13.1

5.15.043.0

sforn

n

sforsn

n

(9) 242

where n is value of Manning coefficient due to friction loss and nʹ is the value of Manning 243

coefficient including bend losses. 244

2. Meander-belt method given by Greenhill and Sellin (1993) which suggested five 245

methods with different combination of division lines and bed slopes. The method with 246

inclined division lines was observed to give better results which is used in this study and is 247

represented as GH5. The discharge stimation is represented as, 248

fpfpfpmbmcmbmcmcmc ASRn

ASRn

ASRn

Q 21

32

21

32

21

32 111

++= (10) 249

where the subscripts mc represents main channel area; mb as meander-belt region; and fp is 250

the area outside the meander-belt. 251

3. Shiono, Al-Romaih and Knight (1999) carried out experimental investigation on 252

meandering channels by varying the bed slope, So for different sinuosity. Consequently, they 253

derived a model by dimensional analysis to illustrate that friction factor, f is mainly dependent 254

on sinuosity. The relationship is shown below, 255

( ) 21

10 fs = (11) 256

4 Results and Analysis 257

4.1 Discharge Prediction in Laboratory Channels 258

The predicted Manning’s n value by the GEP model is used in estimating the conveyance 259

capacity of a meandering compound channel. It is essential to note that the n value predicted, 260

has taken into account the various geometric and hydraulic aspects of a compound channel 261

and is different to that of the composite n computed backwards by the Manning’s equation 262

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from the actual discharge. The conveyance capacity for all the experimental channels are 263

estimated by the presumed models such as LSCS (1992), GH5 (1993) and Shiono-Al- Knight 264

(1999). 265

The data sets of individual investigators are coupled together to carry out an overall 266

error analysis. Such an analysis presumably provides with a general indication regarding the 267

suitability of different models. Different error analysis approaches such R2, MAE, RMSE and 268

MAPE have been carried out and illustrated in a tabular form as shown in Table 4. Figure 7 269

also shows the percentage of error for each of the data series using the different models. 270

It is observed that GEP provides with high R2 values, closer to 1 whereas for the other 271

methods, the coefficient of determination is lower. The developed GEP model also provides 272

lower values of MAE, RMSE and MAPE with respect to other models. Especially the mean 273

absolute percentage error provides less than 7% error whereas the error percentage in the 274

other models is quite high. 275

From Fig. 7 it is observed that the mean error for the GEP model gives best result 276

along with GH5, which in respect to other models gives lower mean percentage error values. 277

The lower error provided by GH5 with respect to other models is because the model was 278

specifically developed for meandering compound channels. On analysing Table 4, the R2 279

values for GH5 is also observed to be closer to 1, quite comparable to the GEP model. 280

However, on observing the other error analysis techniques, such as MAPE, the 281

average error for GH5 (in the case of FCF Phase B – Roughened) is as high as 50.28, 282

whereas the MAPE error by GEP is 6.6 (for the present study) at most. 283

It is important to mention that by categorizing the data sets according to the 284

investigators, it becomes inconclusive, as each individual researcher has carried out 285

experimental investigation on different types of meandering compound channels, by varying 286

different parameters. Hence associating all those experimental observations as a single set, 287

might provide with spurious results. Even the data sets are in different ranges i.e. some are 288

large scale channels while others being small scale. Therefore, comparing by the above 289

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method would give large differences among the data sets. Therefore, the best two models, i.e. 290

GH5 and GEP from the above analysis is further investigated by finding the percentage of 291

error in each individual data series. Figures 8 to 11 demonstrate the percentage of error for 292

every individual data series in US Army (1956); FCF Phase B (1990-1991); Willetts-293

Hardwick (1993) and Shiono-Al-Knight (1999); NITR (2008, 2013) and present study (2017) 294

respectively along with the values of standard deviation. 295

From the extensive error analysis for all the data sets, it clearly illustrated that the 296

proposed model provides better estimation of discharge in a meandering compound channel 297

as compared to the other models. 298

4.2 Incorporation of Discharge Prediction Methods to Natural Rivers 299

Any method must pass through the test of reasonably performing in genuine situations, i.e., 300

for field cases or rivers. Therefore, the method should be tested for its suitability to the field 301

data. The Watawarra channel in the Cooper Creek, Central Australia is selected as study area 302

for implementation and investigation of the different discharge prediction models. The 303

characteristics of the channel and floodplains in the Cooper Creek, Central Australia were 304

studied by Fagan (2001). The Watawarra channel occurs after the junction of the Cooper and 305

Wilson rivers and runs approximately 33km along the channel in a south/south-westerly 306

direction with an increasing channel sinuosity and decreasing overall width downstream. It 307

has a composite cross-section with a small, relatively narrow and deep channel inset into a 308

wider, deeper channel. The inset channel is approximately constant in size throughout the 309

length of the channel, which is considered as the main channel width in this paper. The 310

roughness throughout the cross-section is assumed to be same, hence the relative roughness is 311

taken as 1. 312

Fagan (2001) selected three cross-sections for the Watawarra channel with sufficient 313

length of reach for a meaningful measurement of sinuosity and other planform characteristics. 314

Other characteristics such as, no confluences or bifurcations and that the reach length was a 315

minimum 100 times to that of the channel width were also considered for validly treating the 316

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channel as homogenous. The morphological map of the Watawarra channel with sub-317

divisions and surveyed sections is shown in Figs.12 and 13. The channel reach and planform 318

characteristics for the three surveyed sections has been summarized in Table 5. 319

The different models analyzed in the previous section is applied to the river 320

Watawarra. Table 6 illustrates the values of R2, MAE, RMSE and MAPE whereas Fig. 14 321

shows the percentage of error by different models along with the standard deviation. The GEP 322

model is observed to successfully predict the discharge capacity for a natural section. It is 323

significant to mention that the model requires only the geometric features and the mean 324

velocity at a section to estimate the discharge capacity, much better than the other traditional 325

models. It is relevant to state that the proposed model is developed by undertaking seven 326

geometric and hydraulic parameters and can be appropriately applied to real cases within the 327

prescribed ranges. 328

5 Conclusions 329

A new and improved technique to predict discharge in a meandering compound channel is 330

proposed, based on gene-expression programming. Three new sets of experimental data (15 331

runs each), along with a wide range of data sets of other researchers (i.e. 477 runs in total) 332

with different channel parameters have been used in the development of the model. The data 333

sets used have width ratio in the range 6.79 up to 30 which are both small scale as well as 334

large scale data. The data sets have different slopes and sinuosity with homogenous as well as 335

heterogeneous roughness. It is pertinent to mention that the proposed GEP model is based on 336

laboratory data sets with dimensionless geometric parameters in the ranges; 6.79 ≤ α ≤ 30, 337

0.014 ≤ β ≤ 0.64, .0005 ≤ So ≤ 0.0053, 1.092 ≤ s ≤ 4.11 and 1 ≤ γ ≤ 2.92 338

A selected number of models for predicting roughness coefficient were studied to 339

estimate conveyance for compound meandering channels using the same data sets in order to 340

investigate the suitability of the various methods. It was observed that the developed model 341

provided with satisfactory result as compared to the other models in terms of R2, MAE, 342

RMSE and MAPE for groups of data series’. When observed in a more amplified approach, 343

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i.e. by considering the percentage of error along with the standard deviation for each 344

individual data set, the proposed model showed noticeably better results in the range of less 345

than 7% error, proving to be a quite advanced model with respect to the others. The 346

developed GEP model is also observed to predict well for natural rivers with an acceptable 347

range of about 10% average error 348

Notation 349

AT = Total Area 350

b = Main Channel Width 351

B = Total Floodplain Width 352

f = Darcy-Weisbach’s friction factor 353

Fr = Froude’s number 354

h = Height of Main Channel 355

H = Overall Depth of Flow 356

LW = Meander Wavelength 357

n = Manning’s roughness coefficient 358

Q = Discharge (m3s-1) 359

rc = Radius of Curvature 360

R = Hydraulic radius (m) 361

Re = Reynold’s number 362

s = Sinuosity 363

So = Bed slope 364

V = Mean velocity (m2s-1) 365

α = Width ratio 366

β = Relative flow depth 367

γ = Relative roughness 368

ν = Kinematic viscosity (m2s

-1) 369

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Acknowledgement 370

The authors wish to acknowledge thankfully the support received by the third author from 371

DST India, under grant no. SR/S3/MERC/066/2008 and SR/S3/MERC/0080/2012 for 372

conducting experimental research works. 373

374

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Greenhill, R. K., & Sellin, R. H. J. 1993. Development of a simple method to predict 392

discharges in compound meandering channels. Proc. Inst. of Civ. Engrs., Water, 393

Maritime and Energy, 101(1), 37–44. http://dx.doi.org/10.1680/iwtme.1993.22986 394

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James, C. S. 1994. Evaluation of methods for predicting bend loss in meandering channels. J. 395

Hydr. Engrg., 120(2), 245–253. http://dx.doi.org/10.1061/(ASCE)0733-396

9429(1994)120:2(245) 397

James, C. S., and Wark, J. B. 1992. Conveyance estimation for meandering channels. Report 398

SR 329. HR Wallingford, Wallingford, U.K. 399

Khatua, K. K. 2008. Interaction of flow and estimation of discharge in two stage meandering 400

compound channels. Ph.D. thesis, National Institute of Technology, Rourkela, India. 401

Mohanty, P. K. 2013. Flow analysis of compound channels with wide floodplains. Ph.D. 402

Thesis, National Institute of Technology, Rourkela, India. 403

Shiono, K., Al-Romaih, J. S., & Knight, D. W. 1999. Stage-discharge assessment in 404

compound meandering channels. J. Hydr. Engrg., 125(1), 66-77. 405

http://dx.doi.org/10.1061/(ASCE)0733-9429(1999)125:1(66) 406

U.S. Army Corps of Engineers. 1956. Hydraulic capacity of meandering channels in straight 407

floodways. Waterways Experiments Station, Vicksburg, MS 408

U.S. Department of Agriculture 1955. Engineering handbook: hydraulics. U.S. Department of 409

Agriculture, Soil Conservation Service, sec. 5. 410

U.S. Department of Agriculture 1963. Guide for selecting roughness coefficient n values for 411

channels. U.S. Department of Agriculture, Soil Conservation Service 412

U.S. Department of Transportation 1979 Design charts for open-channel flow: U.S. 413

Department of Transportation, Federal Highway Administration, Hydraulic Design 414

Series 3. 415

Willetts, B. B., and Hardwick, R. I. 1993. Stage dependency for overbank flow in meandering 416

channels. Proc. Inst. of Civ. Engrs., Water, Maritime and Energy, 101(1), 45-54. 417

http://dx.doi.org/10.1680/iwtme.1993.22989 418

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Yen, B. C. 1992. Dimensionally homogeneous Manning's formula. J. Hydr. Engrg., 118(9), 419

1326-1332. http://dx.doi.org/10.1061/(ASCE)0733-9429(1992)118:9(1326) 420

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Table 1 Experimental observations 421

Data Series Side

Slope α s So γ β Q

NITR

(2017)

Series I (5) 1V:1H 11.97 4.11 0.00165 1 0.235, 0.270, 0.297,

0.307, 0.350

0.028, 0.034, 0.040,

0.042, 0.052

Series II (5) 1V:1H 11.97 4.11 0.00165 1.8 0.414, 0.425, 0.458,

0.496, 0.519

0.031, 0.044, 0.058,

0.077, 0.100

Series III (5) 1V:1H 11.97 4.11 0.00165 1.286 0.356, 0.381, 0.430,

0.467, 0.476

0.041, 0.052, 0.066,

0.086, 0.090

422

423

Table 2 Functional set and Operational parameters used in GEP Model 424

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 50000

Constants per Gene 5

Data Type Integer

Crossover Frequency (%) 50

Block Mutation Rate (%) 30

Homologous Crossover (%) 95

425

Table 3 Sensitivity Analysis for different GEP models and Error analysis of ANN 426

Model R2 AE (%) RMSE MAPE

( )FrsSfn o Re,,,,,, γβα= 0.9958 0.35 0.00097 7.1281

( )Re,,,,, sSfn oγβα= 0.9246 0.98 0.00257 10.5824

( )FrsSfn o ,,,,, γβα= 0.8462 11.86 0.00573 19.2457

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( )FrSfn o Re,,,,, γβα= 0.6891 18.42 0.00874 20.6980

( )Frsfn Re,,,,, γβα= 0.8614 9.73 0.00368 18.4257

( )FrsSfn o Re,,,,,βα= 0.9356 2.36 0.00126 8.2365

( )FrsSfn o Re,,,,,γα= 0.8785 5.61 0.00325 15.4268

( )FrsSfn o Re,,,,,γβ= 0.8963 2.54 0.00286 10.6482

427

Table 4 Error Analysis of individual data series by presumed methods 428

LSCS

(1967)

GH5

(1992)

SAK

(1999)

GEP

Model

LSCS

(1967)

GH5

(1992)

SAK

(1999)

GEP

Model

LSCS

(1967)

GH5

(1992)

SAK

(1999)

GEP

Model

LSCS

(1967)

GH5

(1992)

SAK

(1999)

GEP

Model

US Army (1956) 0.8462 0.9795 0.8089 0.9996 0.0638 0.0226 0.0560 0.0038 0.0866 0.0310 0.0761 0.0057 56.2079 14.9539 48.0716 2.3362

FCF Phase B -

Smooth (1990-1991)0.9683 0.9789 0.9721 0.9997 0.0486 0.0447 0.0594 0.0170 0.0829 0.0678 0.0885 0.0297 54.0816 41.5404 36.5178 4.4101

FCF Phase B -

Roughened (1990-1991)0.8549 0.8451 0.8438 0.9996 0.1771 0.1252 0.1108 0.0148 0.2624 0.2096 0.1838 0.0236 65.1244 50.2860 39.1820 4.6944

Hardwick-Willetts

(1993)0.8700 0.9880 0.8600 0.9998 0.0008 0.0018 0.0020 0.0003 0.0011 0.0024 0.0027 0.0004 8.7226 17.9173 18.2020 5.2951

Shiono-Al-Knight (1999) 0.9234 0.9336 0.9000 0.9991 0.0030 0.0023 0.0020 0.0005 0.0044 0.0032 0.0033 0.0006 24.6897 17.8408 16.2784 5.0837

NITR (2008, 2013) 0.8200 0.9811 0.8900 0.9933 0.0017 0.0041 0.0059 0.0016 0.0027 0.0050 0.0075 0.0020 6.1642 14.4345 20.6460 5.7865

Present Study (2017) 0.5821 0.6894 0.8775 0.9994 0.0272 0.0155 0.0297 0.0032 0.0363 0.0199 0.0324 0.0033 48.4290 24.1828 53.0845 6.6055

R2 MAE RMSE MAPE

429

Table 5 Sectional Parameters of River Watawarra 430

Data Series b

(m)

B

(m)

h

(m)

H

(m)

LW

(m)

rc

(m) s So AT (m

2)

V

(m2s

-1)

Re Fr Q

(m3s

-1)

Watawarra

Channel

W1 11.3 46.7 0.47 1.6 433 76 1.78 1.5E-04 27.6 0.37 1.92E+05 0.154 10.2

W2 10.3 32.8 0.4 1.33 277 51 2.3 1.5E-04 16.4 0.33 2.30E+05 0.118 5.4

W3 7.3 19 0.4 0.73 215 37 2.96 1.5E-04 7.22 0.27 1.17E+05 0.123 2

431

Table 6 Error Analysis of River Watawarra 432

Methods R2 MAE RMSE MAPE

LSCS 0.9940 4.652 5.36555 79.0463

GH5 0.9944 4.66128 6.57643 57.3821

SAK 0.9959 1.26002 1.80736 20.2755

GEP Model 0.9992 0.99791 1.62506 11.2175

433

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434

Figure 1 Planform of Experimental Procedure 435

436

437

Figure 2 Series I 438

439

440

Figure 3 Series II 441

442

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443

Figure 4 Series III 444

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Dependence of roughness coefficient on the various influencing parameters

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Coefficient of determination for training and validation data sets

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Percentage of error for different data series by the presumed models

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Percentage of error for US Army (1956)

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Percentage of error for FCF Phase B (1990-1991)

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Percentage of error for Willetts-Hardwick (1993) and Shiono-Al-Knight (1999)

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Percentage of error for NITR (2008, 2013) and present study (2017)

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Morphological Map of River Watawarra

103x134mm (96 x 96 DPI)

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Surveyed Sections of River Watawarra

103x83mm (96 x 96 DPI)

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Percentage of error for different models in river Watawarra

Page 32 of 32



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