gene-expression programming for sediment transport in...

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Gene-expression programming for sediment transport in sewer pipe systems

Aminuddin Ab. Ghani1

and H. Md. Azamathulla, M.ASCE2

________________________________________________________________

Abstract

Old sewerage systems were designed based on empirical rules to minimize sediment

problems and a list of codes for self-cleansing sewers. These codes were applicable to non-

cohesive sediments (typically storm sewers). This study presents gene-expression

programming (GEP), which is an extension of genetic programming (GP), as an alternative

approach to modeling the functional relationships of sediment transport in sewer pipe

systems. A functional relation has been developed using GEP. The proposed relationship can

be applied to different boundaries with partial flow. The proposed GEP approach gives

satisfactory results (r2=0.97 and MSE=0.0054) compared to existing predictor.

Key Words: Sewers, Sediment transport, Part-full flow, Gene-expression programming,

Regression analysis.

________________________________________________________________________________________________________________

1Professor, River Engineering and Urban Drainage Research Centre (REDAC), Universiti

Sains Malaysia, Engineering Campus, Seri Ampangan, 14300 Nibong Tebal, Pulau Pinang,

Malaysia. Email: [email protected]

2Senior Lecturer, REDAC, Universiti Sains Malaysia, Engineering Campus, Seri Ampangan,

14300 Nibong Tebal, Pulau Pinang, Malaysia; Email: [email protected],

[email protected] (author for correspondence)

Journal of Pipeline Systems Engineering and Practice. Submitted August 3, 2010; accepted December 8, 2010; posted ahead of print December 10, 2010. doi:10.1061/(ASCE)PS.1949-1204.0000076

Copyright 2010 by the American Society of Civil Engineers

mailto:[email protected]

mailto:[email protected]

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Introduction

In sewer networks, the deposition of solids occurs occasionally due to the intermittent nature

of flow (Nalluri et al., 1994). The longer the deposits remain in the sewer the more likely that

the sediment properties change and can eventually become consolidated or cemented,

especially during dry weather flows. Such permanent deposits in pipe inverts will change the

nature of the velocity and boundary shear distributions, which affects the sediment carrying

capacity and hydraulic resistance of sewers. Previous work on sediment transport with no

deposition includes that by Novak and Nalluri (1975, 1984), May et al. (1996), Mayerle et al.

(1991) and Vongvisessomjai et al., 2010.

Sewer system designs must satisfy two major criteria: high flow and low flow criteria. During

high flows, sewer systems must convey the design discharge. For low flows, sewers should

be free from sediment deposit as much as possible. Traditionally, a fixed minimum flow

velocity for non-deposition, such as 0.6 m/s (ASCE 1970) is used as a low flow criterion.

This criterion may be inadequate because the loading and sediment characteristics vary

considerably under different environmental conditions (Vongvisessomjai et al., 2010).

The present study investigates the hydraulic characteristics of the flow in channels with a

circular cross-section with different bed roughness and their effects on sediment transport

capacity (Figure 1) (Nalluri and Ab. Ghani, 1996). The pipe channel can be represented by a

trapezoidal section, especially at low depths; at high depths, the cross-section is influenced by

the „crowing‟ effect of the pipe, through the changes in velocity and shear distributions due to

changes in cross-sectional shapes (Nalluri et al., 1994).



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Extensive experimental work (Ab. Ghani 1993; Nalluri and Ab. Ghani 1994a, b) on bed load

transport of non-cohesive sediments with no deposition (Fig. 2) was carried out in pipe

channels with diameters of 154 mm, 305 mm, and 450 mm covering a wide range of flow

depths, sediment sizes and three different bed roughness values, as shown in Table 1. The

limiting sediment concentrations Cv(=Qs/Q) for the „no deposition‟ criterion with uniform

flow conditions were established for several flow depths (yo) over each bed roughness.

Multiple Linear Regression - Clean Pipes

The sediment transport rate in channels with a circular cross section or pipe channels depends

on many factors, such as flow depth (yo), bed slope (So), sediment size (d50), density of

sediment (ρs), kinematic viscosity (ν) and density (ρ ) of fluid, friction factor (λ), pipe

diameter (D), and gravitational constant (g).

For the case of bed load transport with no deposition, Ab. Ghani (1993) suggested the

following equation to describe bed load transport with a limiting velocity for no deposition

(clean pipes):

21.053.05021.009.0 )(08.3

svgr

s

R

dCD

dg

V

(1)

where

Vs = Limiting velocity

Dgr = Dimensionless grain size = 3

12 )/)1(( SsdDgr

R = Hydraulic radius



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d50 = Sediment size

∆ = Relative density of sediment in water = 1)(

s

λs = Overall friction factor

98.002.001.013.1 cvgrs CD

λc = Clear water friction factor of the channel

Ab. Ghani (1993) has shown that this simple regression equation can provide a good

estimation of sediment transport in clean pipes. The above equation yielded r2= 0.95 and a

discrepancy ratio of 1.0. Additional data from the work by Mayerle (1988), May et al. 1996)

and Loveless (1991) were also included in the derivation of Equation 1 (Fig. 3). The Q-So-D

plot (Figure 1), where Q is the flow discharge, shows the sediment transporting capacity for

different pipe sizes based on Equation 1.

Although a number of successful modeling attempts have been reported by Dogan et al.

2007; Kisi et al. 2008; Azamathulla et al. (2009, 2010); Azamathulla and Ab. Ghani (2010,

2011) and Dogan et al. (2009), a wider application of theoretical models is restricted by their

heavy demand in terms of computing capacity and time. Alternatively, soft computing

techniques, such as artificial neural networks (ANNs), evolutionary computation (EC), fuzzy

logic (FL), and genetic programming, have been successfully applied in water engineering

problems in last the two decades. Thus, the present study attempts a new soft computing



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technique, GEP, to obtain a new sediment transport equation for bed load transport in pipes

with no deposition

Overview of GEP

GEP, which is an extension of GP (Koza, 1992), is a search technique that involves computer

programs (e.g., mathematical expressions, decision trees, polynomial constructs, and logical

expressions). GEP computer programs are all encoded in linear chromosomes, which are then

expressed or translated into expression trees (ETs). ETs are sophisticated computer programs

that have usually evolved to solve a particular problem and are selected according to their

fitness at solving that problem.

GEP is a full-fledged genotype/phenotype system, with the genotype totally separated

from the phenotype, whereas in GP, genotype and phenotype are mixed together in a simple

replicator system. As a result, the full-fledged genotype/phenotype system of GEP surpasses

the old GP system by a factor of 100-60,000 (Ferreira 2001a, b).

Initially, the chromosomes of each individual in the population are generated randomly.

Then, the chromosomes are expressed, and each individual is evaluated based on a fitness

function and selected to reproduce with modification, leaving progeny with new traits. The

individuals in the new generation are, in their turn, subjected to some developmental

processes, such as expression of the genomes, confrontation of the selection environment,

and reproduction with modification. These processes are repeated for a predefined number of

generations or until a solution is achieved (Ferreira 2001a, b). The functionality of each

genetic operator included in GEP system has been explained by Guven and Aytek (2009).



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Derivation of Froude Number based on GEP

In this section, the sediment load is modeled using the GEP approach. Initially, the “training

set” is selected from the entire data set, and the rest is used as the “testing set”. Once the

training set is selected, one could say that the learning environment of the system is defined.

The modeling also includes five major steps to prepare to use GEP. The first is to choose the

fitness function. For this problem, the fitness, fi, of an individual program, i, is measured by:

tC

jjjii TCMf

1),(

(2)

where M is the range of selection, C(i,j) is the value returned by the individual chromosome i

for fitness case j (out of Ct fitness cases) and Tj is the target value for fitness case j. If |C(i,j) -

Tj| (the precision) ≦ 0.01, then the precision is 0, and fi = fmax = CtM. In this case, M = 100 is

used; therefore, fmax = 1000. The advantage of this kind of fitness function is that the system

can find the optimal solution by itself.

Secondly, the set of terminals T and the set of functions F are chosen to create the

chromosomes. In this problem, the terminal set consists of single independent variable, i.e., T

= {h}. The choice of the appropriate function set is not so clear; however, a good guess is

helpful if it includes all the necessary functions. In this study, four basic arithmetic operators

(+, -, *, /) and some basic mathematical functions (√) are utilized.

The third major step is to choose the chromosomal architecture, i.e., the length of the

head and the number of genes. We initially used single gene and two head lengths and

increased the number of genes and heads one at a time during each run while we monitored

the training and testing performances of each model. We observed that more than two genes

more and a head length greater than 8 did not significantly improve the training and testing



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performance of GEP models. Thus, the head length, lh = 8, and two genes per chromosome

are employed for each GEP model in this study.

The fourth major step is to choose the linking function. In this study, addition and

multiplication operators are used as linking functions, and it is observed that linking the sub-

ETs by addition gives better fitness (Eq. 2) values. The fifth and final step is to choose the set

of genetic operators that cause variation and their rates. A combination of all genetic

operators (mutation, transposition and crossover) is used for this purpose (Table 2).

Table 3 compares the GEP model with one of the independent parameters removed in

each case and any independent parameter from the input set that yielded larger RMSE and

lower R2 values also removed. These five independent parameters affect Fr =

dg

Vs

; thus, the

functional relationship given in Eq. (1) is used for the GEP model in this study. The GEP

approach resulted in a highly nonlinear relationship between dg

Vs

and the input

parameters, and the GEP model had the highest accuracy and the lowest error (Table 3).

The GEP model was calibrated with 220 input-target pairs of collected data by Ab. Ghani

(1993) and Vongvisessomjai et al., 2010. Among the 220 data sets, 55 (25%) were reserved

for validation (testing), and the remaining 165 sets were used to calibrate the GP model.

The best individual in each generation has 30 chromosomes has and a fitness 780.5

fordg

Vs

. The explicit formulations of GEP for

dg

Vs

are given in Eq. (3), and the

corresponding expression trees are shown in Fig. 4.



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8

)34.8(

)(91.5

91.5

014.0411.0

41.0

014.0 sd

RsDs

D

C

d

Rd

R

dg

Vgr

s

gr

sVS

s

s

(3)

Equation 3 is further simplified as:

d

RDss

D

C

d

Rdg

Vgr

sgr

V

s 23

34.8014.0

191.5

)(

41.0425.1

(4)

Results and discussion of GEP

The performance of the GEP model was compared with the sediment transport (Eq. 1) by Ab.

Ghani (1993). Overall, particularly for laboratory measurements, the GEP models give better

predictions than the existing models. The GEP model produced the lowest errors (r2=0.97,

MAE=0.02456 and MSE=0.0054) for the training data (Fig. 5) and for the test data (r2=0.94,

MAE=0.06566 and MSE=0.0056) (Fig. 6).

The most significant advantage of the proposed GEP compared to classical regression

analysis based models (traditional equations) is that it is capable of mapping the data into a

high dimensional feature space, where a variety of methods (described in the previous

section) are used to find relations in the data. Because the mapping is quite general, the

relations are very general. We used JEdit open source software in this study (JEdit).



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Conclusions

Sediment transport in pipes is a complex phenomenon. The nature and motivation of

traditional models differ significantly. These approaches are normally able to make

predictions within about one order-of-magnitude of the actual measurements.. A single bed

load transport model for storm sewers is given by equation (4). To overcome the complexity

and uncertainty associated with bed load estimation, this study demonstrates that a GEP

model can be applied for accurate prediction. The GEP model was able to successfully

predict bed load transport in storm sewers. The high value of the coefficient of determination

(r2=0.97; MSE = 0.0054) implies that the GEP model provides an excellent fit for the

measured data. These results suggest that the proposed GEP model is robust and useful for

practitioners.

Notation

d50 = Sediment size

g = gravitational acceleration

yo = uniform flow depth

D = Pipe diameter

Dgr = Dimensionless grain size = 3

12 )/)1(( SsdDgr

Fr = Froude Number = dg

Vs

R = Hydraulic radius



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Vs = Limiting velocity

r2 = coefficient of determination

MAE = mean average error

∆ = Relative density of sediment in water =

λs = Overall friction factor

λc = Clear water friction factor of the channel

= fluid density

s = buoyant sediment density

References

Ab. Ghani, A., (1993). Sediment transport in sewers. PhD. Thesis, University of Newcastle

upon Tyne, UK.

American Society of Civil Engineers (ASCE)(1970). Water pollution control federation,

Design and construction of sanitary and storm sewers. American Society of Civil

Engineers Manuals and Reports on Engineering Practices, No. 37.

Azamathulla, H. Md, Ab. Ghani, A. Zakaria, N. A. and Guven, A., (2010). Genetic

programming to predict bridge pier scour. ASCE Journal of Hydraulic Engineering, 2010;

136(3),165-169.

Azamathullah, H.Md., Chang, C.K. Ab Ghani, A., Ariffin, J., Zakaria, N.A., and Abu Hasan,

Z., (2009). An ANFIS-Based Approach for Predicting the Bed Load for Moderately-

Sized Rivers. Journal of Hydro-Environment Research, IAHR, Vol. 3, No. 1, pp. 35-44.

Azamathulla, H. MD. and Ab. Ghani, A., (2010). Genetic programming to predict river

pipeline scour, ASCE, Journal of Pipeline system and Engineering Practice. 1(3)127-132.



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Azamathulla, H. MD. and Ab. Ghani, A., (2011). An ANFIS-based approach for predicting

the scour depth at culvert outlet, ASCE, Journal of Pipeline system and Engineering

Practice. (In press).

Bhattacharya, B., Price, R.K., and Solomatine, D,P., (2007). Machine Learning Approach to

Modeling Sediment Transport, ASCE Journal of Hydraulic Engineering, 2007, 133(4),

440-450.

Department of Irrigation and Drainage Malaysia or DID., (2009). Study on River Sand

Mining Capacity in Malaysia.

Dogan, E., Yuksel, I. and Kisi, O., (2007). Estimation of Sediment Concentration Obtained

by Experimental Study Using Artificial Neural Networks, Environmental Fluid

Mechanics, 7, 271-288.

Dogan, E., Tripathi, S., Lyn, D. A and Govindaraju, R. S., (2009). From flumes to rivers: Can

sediment transport in natural alluvial channels be predicted from observations at the

laboratory scale?, Water Resour. Res., 114, W08433, doi:10.1029/2008WR007637.

Engelund F. and Hansen.E., (1967). A monograph on sediment transport in alluvial streams,

Teknisk Forlag, Copenhagen, Denmark.

Ferreira, C., (2001a). Gene expression programming in problem solving, 6th

Online World

Conference on Soft Computing in Industrial Applications (invited tutorial).

Ferreira, C. (2001b). Gene expression programming: A new adaptive algorithm for solving

problems. Complex Systems, 13 (2), 87–129.

Guven, A. and Gunal, M., (2008a). Prediction of scour downstream of grade-control

structures using neural networks.ASCE Journal of Hydraulic Engineering, 134(11),

1656-1660.



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Guven, A. and Gunal, M., (2008b). Genetic programming for prediction of local scour

downstream of grade-control structures. ASCE Journal of Irrigation and Drainage

Engineering, 134(2), 241-249.

Guven, A. and Aytek, A.. (2009). A new approach for stage-discharge relationship: Gene-

Expression Programming, J. Hydrologic Engineering, 14(8),812-820.

Holland, J. H. (1975). Adaptation in natural and artificial system. University of Michigan

Press, Ann Arbor, Mich.

JEdit, http://sourceforge.net/projects/jedit/

Kisi, O, Yuksel, I. and Dogan, E., (2008). Modelling daily suspended sediment of rivers in

Turkey using several data driven techniques, Hydrol. Sci. J., 53(6), 1270-1285.

Koza, J.R., (1999). Genetic Programming: On the Programming of Computers by means of

Natural Selection. The MIT Press, Cambridge, MA.

Keijzer, M., and Babovic, V., (2002). Declarative and preferential bias in GP-based scientific

discovery. Genetic Programming and Evolvable Machines, 1(3), 41-79.

Kizhisseri, A.S., Simmonds, D., Rafiq, Y., and Borthwick, M., (2005). An evolutionary

computation approach to sediment transport modeling. In: Fifth International Conference

on Coastal Dynamics, April 4–8, 2005, Barcelona, Spain.

Loveless, J. H.,( 1991). Sediment transport in rigid boundary channels with particular

reference to the condition of incipeint deposition. PhD thesis , Univeristy of London,

U.K.

Mayerle, R., (1988). Sediment transport in rigid boundary channel. PhD Thesis, University of

Newcastle upon Tyne, UK.

Mayerle, R., Nalluri, C., and Novak, P., (1991). Sediment transport in rigid bed conveyances.

Journal of Hydraulic Research, 29 (4), 475–496.



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May, R.W.P., Ackers, J.C., Butler, D. and John, S., (1996). Development of design

methodology for self-cleansing sewers. Water Science and Technology, 33 (9), 195–205.

Nalluri, C., Ghani, A.A., and El-Zaemey, A.K.S., (1994). Sediment transport over deposited

beds in sewers. Water Science and Technology, 29 (1–2), 125–133.

Nalluri, C. & Ab. Ghani, A., (1994). Sediment Transport in Sewers and Design Implications.

National Conference on Hydraulic Engineering; Hydraulic Division, ASCE, Buffalo,

USA, Vol. 2, pp. 933-938, 1 - 5 August.

Nalluri, C. & Ab. Ghani, A., (1994). Sediment Transport in Sewers with and without

Deposited Beds. 9th Congress of Asia-Pacific Division of IAHR, Singapore, Vol. 2, pp.

27-34, 24 - 26 August.

Nalluri, C. and Ab. Ghani, A., (1996). Design Options For Self-Cleansing Storm Sewers.

Journal of Water Science and Technology, IWA, Vol. 33, No. 9, pp. 215-220.

Oltean, M., and Groşan, C,, (2003). A comparison of several linear genetic programming

techniques. Complex Systems, 14(1), 1-29.

Solomatine, D.P. and Otsfeld, A., (2008). Data-driven modelling: some past experiences and

new approaches. Journal of Hydroinformatics, 2008, 10, No. 1, 3-22.

Vongvisessomjai, N., Tingsanchali, T. and Babel, M. S., (2010). Non-deposition design

criteria for sewers with part-full flow, Urban Water Journal, 7(1), 61-77.



14

Table 1: Range of Experimental Data in Clean Pipes

Author D

(mm)

K0

(mm) D50(mm) Yo/D V (m/s)

Cv

(ppm

Ab. Ghani (1993)

154 0.0 0.93-5.7 0.15-0.76 0.24-0.86 38-

1450

305 0.0-1.34 0.46-8.4 0.2-0.8 0.5-1.2 1-1280

450 0.15 0.72 0.5-0.76 0.5-1.22 2-37

Vongvisessomjai et al. (2010) 100 -

150 0.0

0.20-

0.43 0.2-0.9

0.24 –

1.5

4-

1453

Loveless (1991) 88

220

0.0

0.3

0.45-1.3

0.45-6.0

0.16-0.6

0.14-0.42

0.63-1.09

0.24-0.86

337-

2010

74-

1289

Mayerle (1988) 152 0.0 0.5-8.74 0.2-0.76 0.38-1.4 20-

1280

May et al. (1996) 158

300

0.0

0.15 0.640.72

0.37-0.76

0.38-0.76

0.49-1.09

0.5-1.5

11-507

1-44

Accepted Manuscript Not Copyedited



15

Table 2: Parameters of the optimized GEP model

Parameter Description of parameter Setting of parameter

P1 Function set +, -, *, /, √

P2 mutation rate % 30

P3 Inversion rate % 30

P4

One point and two point

recombination rate respectively %

30,30

P5 Gene recombination rate 95

P6 Gene transportation rate 0.1




16

Table 3. Sensitivity Analysis for Independent Parameters for the Testing

Set Model MSE MAE

r2

),,,( 50svgr

s

R

dCDf

dg

V

0.0056 0.65 0.94

),,( 50sv

s

R

dCf

dg

V

0.099 0.88 0.86

),,( 50sgr

s

R

dDf

dg

V

0.094 0.95 0.79

),,( svgrs CDfdg

V

0.109 0.85 0.76

),,( 50

R

dCDf

dg

Vvgr

s

0.38 0.91 0.78




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List of Figures:

Fig. 1: Q-S0-D plot: clean pipe (Ab. Ghani, 1993)

Fig. 2: Cross section of clean pipes (Ab. Ghani, 1993)

Fig. 3: Limiting velocity criteria for clean pipes (Ab Ghani, 1993)

Fig. 4: Expression Tree (ET) for GEP formulation

Fig. 5: Observed versus predicted sediment load by GEP for partially full pipe flow –

Training data

Fig. 6: Observed versus predicted sediment load by GEP for partially full pipe flow – Test

data



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