system appendpdf cover-forpdf - university of …...draft 1 gene-expression programming to predict...
Post on 17-Apr-2020
6 Views
Preview:
TRANSCRIPT
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: er.arpanpradhan@gmail.com (author for correspondence) 5
KISHANJIT K KHATUA, Associate Professor, Department of Civil Engineering, National 6
Institute of Technology Rourkela, Rourkela, India, 7
Email: kkkhatua@nitrkl.ac.in 8
9
Gene-Expression Programming to Predict Manning’s n in Meandering Flows 10
11
Page 1 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
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
Page 2 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
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
Page 3 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
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
Page 4 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
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
Page 5 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
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
Page 6 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
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
Page 7 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
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
Page 8 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
( )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
Page 9 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
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
Page 10 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
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
Page 11 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
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
Page 12 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
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
Page 13 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
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
Page 14 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
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
Page 15 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
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
References 375
Azamathulla, H. Md., Ahmad, Z. & Ghani, A. Ab. 2013. An Expert System for Predicting 376
Manning’s roughness coefficient in open channels by using Gene Expression 377
Programming. Neural Comput & Applic., 23:1343-1349. doi:10.1007/s00521-012-378
1078-z. 379
Dash, S. & Khatua, K. K. 2016. Sinuosity Dependency on Stage Discharge in Meandering 380
Channels. J. Irrig. Drain Eng., 04016030. 10.1061/(ASCE)IR.1943-4774.0001037 381
Ervine, D. A., Willetts, B. B., Sellin, R. H. J., & Lorena, M. 1993. Factors affecting 382
conveyance in meandering compound flows. J. Hydr. Engrg., 119(12), 1383–1399. 383
http://dx.doi.org/10.1061/(ASCE)0733-9429(1993)119:12(1383) 384
Fagan, S. D. 2001. Channel and floodplain characteristics of Cooper Creek, Central Australia. 385
Ph.D. thesis, University of Wollongong, Australia. 386
Ferreira C. 2001. Gene expression programming: A new adaptive algorithm for solving 387
problems. Complex Syst. 13(2), 87-129. 388
Gandomi, A. H., Alavi, A. H, Mirzahosseini, M. R. & Moqhadas, N. J. 2011. Nonlinear 389
genetic-based models for prediction of flow number of asphalt mixtures. J. Mater. Civ. 390
Eng., 23(3), 248-263. 391
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
Page 16 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
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
Page 17 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
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
Page 18 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
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
Page 19 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
( )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
Page 20 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
434
Figure 1 Planform of Experimental Procedure 435
436
437
Figure 2 Series I 438
439
440
Figure 3 Series II 441
442
Page 21 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
443
Figure 4 Series III 444
Page 22 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
Dependence of roughness coefficient on the various influencing parameters
Page 23 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
Coefficient of determination for training and validation data sets
Page 24 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
Percentage of error for different data series by the presumed models
Page 25 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
Percentage of error for US Army (1956)
Page 26 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
Percentage of error for FCF Phase B (1990-1991)
Page 27 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
Percentage of error for Willetts-Hardwick (1993) and Shiono-Al-Knight (1999)
Page 28 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
Percentage of error for NITR (2008, 2013) and present study (2017)
Page 29 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
Morphological Map of River Watawarra
103x134mm (96 x 96 DPI)
Page 30 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
Surveyed Sections of River Watawarra
103x83mm (96 x 96 DPI)
Page 31 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
Draft
Percentage of error for different models in river Watawarra
Page 32 of 32
https://mc06.manuscriptcentral.com/cjce-pubs
Canadian Journal of Civil Engineering
top related