Department of Civil Engineering, Indian Institute of Technology Guwahati, Guwahati, IndiaDepartment of Civil Engineering, RGM College of Engineering and Technology, Nandyal 518501, Andhra Pradesh, IndiaDepartment of Civil Engineering, Indian Institute of Science, Bangalore, India
r t i c l e i n f o
rticle history:eceived 4 November 2009eceived in revised form 17 July 2010ccepted 13 September 2010
eywords:
a b s t r a c t
Seepage through sand bed channels in a downward direction (suction) reduces the stability of particlesand initiates the sand movement. Incipient motion of sand bed channel with seepage cannot be designedby using the conventional approach. Metamodeling techniques, which employ a non-linear pattern anal-ysis between input and output parameters and solely based on the experimental observations, can beused to model such phenomena. Traditional approach to find non-dimensional parameters has not been
used in the present work. Parameters, which can influence the incipient motion with seepage, have beenidentified and non-dimensionalized in the present work. Non-dimensional stream power concept hasbeen used to describe the process. By using these non-dimensional parameters; present work describesa radial basis function (RBF) metamodel for prediction of incipient motion condition affected by seepage.The coefficient of determination, R2 of the model is 0.99. Thus, it can be said that model predicts thephenomena very well. With the help of the metamodel, design curves have been presented for designing
en it
uction the sand bed channel wh
. Introduction
In channel flow, incipient motion condition for a sediment bedefers to the beginning of movement of bed particles that previouslyere at rest. Incipient motion of streambeds is a fundamental pro-
ess with applications to a wide variety of research problems, suchs paleohydraulic reconstructions, placer formation, canal design,ushing flows and assessment of aquatic habitat (Buffington andontgomery, 1997). Incipient motion has been studied exten-
ively over the past years following the work of Shields (1936),ho presented a semi-empirical approach to incipient motion
Taylor and Vanoni, 1972; Lavelle and Mofjeld, 1987; Buffingtonnd Montgomery, 1997; Marsh et al., 2004; Mario and Lenzi, 2007).hields (1936) has applied dimensional analysis and obtained thehreshold shear stress at which the individual particles on a sed-mentary bed, comprising nearly spherical shaped and uniformediments, are on the verge of motion by a unidirectional streamow. Here, it may be worth to mention that Shields diagram does
ot include the effect the seepage on incipient motion.
Seepage through boundaries of alluvial canals, rivers andtreams is a common occurrence due to porosity of the granularaterial as well due to level difference between ground water and
surface water in the canal. Seepage losses from alluvial canals havebeen estimated to range from 15 to 45% of total inflow (Van der Leenet al., 1990). Clearly there is a continuing need to study or analyzeseepage phenomena undergoing in the alluvial canals (Hotchkiss etal., 2001). The presence of seepage affects the hydrodynamic char-acteristics of the channel. A study of the effect of seepage flows onthe detachment of particles from the bed or incipient motion is ofgreat interest, since this problem is related to the solution of impor-tant practical engineering problems. For instance, groundwatermovement plays an exceptionally important role when construct-ing hydraulic structures, particularly dams; these problems areurgent when solving problems of the stability of dams and canalslopes.
With the level of the free surface in a channel being differentfrom the adjoining groundwater table, two typical seepage flows(bed suction and bed injection) may occur through the channelboundary (Rao and Sitaram, 1999; Yan et al., 2008). In the case ofbed suction or downward seepage, water seeps out of the channel;while with bed injection or upward seepage, the channel receivesadditional water. Various researchers have studied the effect of bedsuction and injection on incipient motion (Harrison, 1968; Willetsand Drossos, 1975; Richardson et al., 1985; Cheng and Chiew, 1999;
Owoputi and Stolte, 2001; Dey and Zanke, 2004). However, it isvery interesting to note that the design methods are not available,which considers the seepage effects. Present work considers thedownward seepage in devising the design procedure for channel, aspublished studies show that channel bed stability is greatly affected
Used for incipient motiond* dimensionless particle diameterd16, d, d84 diameters of particles at 16%, 50% and 84% finer by
weightB width of the channelQ discharge in the channelqs seepage dischargeRe Reynolds numberRs vsy/�Sf energy slopeSo bed slopeSw water surface slopeu average velocity;y flow depth;vs seepage velocity;� shear stress�s sediment particle specific weight� specific weight of fluid� kinematics viscosity of the fluid�bo bed shear stress without seepage�bs bed shear stress with seepage�co critical shear stress without seepage�cs critical shear stress with seepage
Used for metamodelJ hidden layerX input vector� center˛ learning rate� width
iC
topwhefkea
ofmpsttepgafadi
Z RBF functions typeR2 correlation coefficient
n the case of downward seepage (Rao and Sitaram, 1999; Chen andhiew, 2004).
Channel seepage is hard to analyze and model using conven-ional techniques due to its nonlinearity, rapid change in amountf seepage along the channels and complexity. In the case of com-lex systems, where mathematics is too complex and process is notell defined, computer based simulation models impart a greatelp (Kleijnen, 1987; Sztipanovits, 1998). These simulation mod-ls are called metamodel. This has opened up new opportunitiesor modeling processes about which either the level of availablenowledge is too limited to put the relevant information in a math-matical framework or too little data is available for calibrating anppropriate model.
A metamodel, first proposed by Blanning (1975), is a modelf a simulation model that serves as a surrogate or substituteor the more complex and computationally expensive simulation
odel. The overlying basis of the metamodel is that if the com-uter simulation model is a realistic representation of the realystem, then the metamodel will yield on adequate determina-ion of the optimum conditions of the real systems. Metamodelingechnique is presently being utilized in almost all branches of sci-nce as an alternative and complementary to the more traditionalhysically-based modeling system. In metamodeling approach,enerally a non-linear parametric function approximator is usednd the coefficients of the function decomposition are obtained
rom input–output data pairs, a specified topology and system-tic learning rules. Once trained, the model becomes a parametricescription of the function being approximated. The goal of learn-
ng from examples is to find the general rule that created the
s in Agriculture 74 (2010) 321–328
specific examples, and this is achieved by trying out different modeltopology and related parameters. Metamodeling techniques havebeen used to study several hydrologic and hydraulic phenomenaincluding water quality, stream flows, rainfall, runoff, sedimenttransport and to infill missing data (Tatang et al., 1997; Haas, 1998;Govindaraju, 2000; Riddle et al., 2004; Caamano et al., 2006; Rao etal., 2007; Chang et al., 2008; Yuhong and Wenxin, 2009).
Many types of metamodels have been proposed over the last25 years including polynomial regression, neural networks, radialbasis functions, and splines. Barton (1993) provides a survey anddiscussion of the key properties of several types of metamodels.Polynomial regression is the most common type of metamodel;however, in general, such models are not suitable for fitting com-plex surfaces (Barton, 1993). Franke (1982) found radial basisfunctions are superior to all other metamodeling techniques basedon empirical comparison. Jin et al. (2001) also compared differenttypes of metamodel and showed that radial basis function meta-model was the best for both large-scale and small-scale engineeringproblems. Radial basis functions (RBF) have been extensively usedfor interpolation, regression and classification due to their uni-versal approximation properties and simple parameter estimation(Hardy, 1971; Dyn et al., 1986; Powell, 1987; Broomhead and Lowe,1988).
This paper attempts the application of the metamodel techniqueinto the sediment transport problem. The objectives of this studyare to develop an RBF metamodel for designing the incipient motionof sand bed channels affected by bed suction and to demonstratethe practical applicability.
2. Experimentation
In sediment transport experiments, scale effects must be con-sidered, especially for sand bed associated experiments. The mosteffective solution is to conduct experiments on multiple flumeswith a scale large enough to neglect scale effect problems. In thepresent work, experiments are conducted in two types of tiltinglaboratory flumes. The idea of doing experiments on two differentflumes is to limit the scale effect of hydraulic and geometric param-eters from the design curves and generate the comprehensive setof experimental observations. These experimental observations arebeing used to develop the metamodel. The geometric dimensionof the first flume is 360 cm (length), 15.75 cm (width) and 20 cm(deep). A schematic diagram of the flume with seepage arrange-ment is shown in Fig. 1. Sand bed thickness of 5 cm was maintainedin this flume. Second flume has a 23 cm thick sand bed in a straight,rectangular, smooth, rigid-walled flume 61.5 cm wide and 1416 cmlong (excluding a length of 560 cm for entry and stilling arrange-ments), facilitating the application of uniform seepage in eitherdirection perpendicular to the bed over a length of 1275 cm.
Sand bed is laid on a perforated sheet at an elevated level fromthe channel bottom covered with a fine wire mesh (to prevent thesand falling through) to facilitate the seepage flow through the sandbed. The space between the perforated sheet and the channel bot-tom acted as a pressure chamber to allow seepage flow through thesand bed either in a downward or an upward direction by creating apressure difference lower or higher, respectively, than the channelflow. Seepage lengths of 240 cm and 1275 cm have been adoptedin the smaller and larger flumes, respectively. These lengths havebeen chosen to make the observations free from the upstream anddownstream condition.
2.1. Sand sizes for experiments
Six different sizes, i.e., d = 0.58, 0.65, 0.8, 1, 1.3 and 3 mm wereused as bed material for seepage studies. All sizes have fairly uni-
B. Kumar et al. / Computers and Electronics in Agriculture 74 (2010) 321–328 323
inal se
ftt
2
wAwatwscuaonwcTmautoatss
l
osmsP
The hydrodynamic system of the channel with seepage can bedescribed by the following components of the phenomenon:
f (g, �, �s, �, d, y, Sf , u, vs) = 0 (4)
Fig. 1. Schematic longitud
orm material with gradation coefficient � = 0.5(d84/d + d/d16) is inhe range of 1.08–1.3 (where d16, d and d84 are the sizes pertainingo 16, 50 and 84 per cent finer respectively).
.2. Procedure and measurements
Initially, the sand bed was made plane for all the experimentsith a required bed slope, So. Then inflow discharge, Q was allowed.fter reaching stable conditions, slowly seepage flow, qs (suction)as allowed. With the applied suction, sand bed has been made
t incipient motion condition. A tailgate at the downstream end ofhe channel was used to adjust the flow depth. Pressure tapingsere provided at some sections inside the sand bed to measure the
eepage gradients and to verify the uniformity of seepage flow. Theriterion for incipient motion developed by Yalin (1976) has beensed in the present case. Before and after the application of seep-ge, the water surface elevations were measured with an accuracyf ±0.015 mm of water head at regular intervals along the chan-el by using a digital micro manometer in order to determine theater surface slope, Sw. Flow depths, y, along the central line of the
hannel were measured at regular intervals using a point gauge.he amount of Q and qs were measured through calibrated orificeeters. The values of Sf were calculated based on the values of So
nd Sw, since Sf = Sw − So. By knowing y and width of the channel,(average velocity) and vs (seepage velocity) is calculated. Thus,
he basic variables Sf, Q, qs, and y for each particle size (d) werebtained in every experimental run and presented in Table 1. Raond Sitaram (1999) suggested that the critical shear stress for par-icle entrainment due to suction �cs is related to both the bed sheartress without seepage �bo and the critical shear stress withouteepage �co as:
n(
�bo
�co
)= −0.2525
(�cs
�co
)−2.917for (�bo/�co) < 1 (1)
Here �bo is bed shear stress (=�RbSf, Rb is the hydraulic radiusf the bed after smooth wall correction), �co is the Shields’ criticalhear stress and �cs is the critical shear stress at pseudo incipientotion after application of seepage. The value of Shields’ critical
hear stress has been calculated by empirical formula given by
aphitis (2001):
�co
(�s − �)d= 0.273
1 + 1.2d∗ + 0.046(1 − 0.576 e−0.05d∗) (2)
ction of the tilting flume.
Williams (1970) method (given below) has been adopted for thesmooth wall correction:
�bo
�ySf= b2
b2 + 0.055y(3)
where b is the channel width and measured in meters. Presentexperimental observations have been also processed through Eq.(1) and plotted in Fig. 2. As shown in Fig. 2, all the data points fallin Eq. (1).
3. Dimensional analysis of the system
Fig. 2. Validations of the present observations.
324 B. Kumar et al. / Computers and Electronics in Agriculture 74 (2010) 321–328
that Zj has an appreciable value only when the distance ||X − �j||is smaller than the width �j. The value of width has a profoundeffect upon the accuracy of the system. For hidden layer neuronswhose centres are widely separated from others, width must belarge enough to cover the gap, whereas, those in the centre of a
here � and � are specific weight and kinematic viscosity of theuid respectively; �s is specific weight of the sediment particles andis the gravitational acceleration. A stable relationship between
ediment transport and flow can at best only be expected in a sit-ation where the mechanisms controlling sediment transport areependent only on the rate of flow of water in the channel and seep-ge occurring through the channel. Thus, it is felt that the ‘streamower concept’ is more appropriate for describing seepage induced
ncipient motion in alluvial channel. Stream power is the energyvailable to transport sediment (Velikanov, 1954; Bagnold, 1966).iven basic knowledge of a stream cross section (depth of flow,verage velocity and average bed slope) it is possible to computehe stream power per unit boundary area for a range of flows. Inunctional form, stream power can be expressed as follows (Petitt al., 2005):
tream power = QSf ≈ �csu (5)
Here it can be assumed that critical shear with seepage is respon-ible for the sediment transport.
Eq. (5) can be non-dimensionalized in a following way:
tream power = �csu
(�s − �)�(6)
Particle size, d can be non-dimensionalized as:
∗ = d
((�s − �)g
��2
)1/3
(7)
Reynolds number (Re = uy/�) is very essential parameter toescribe any fluid flow phenomena. In order to incorporate theeepage term, vs has been non-dimensionalized as:
s = vsy
�(8)
Critical shear stress under no seepage (�co) condition can beon-dimensionalized by using the critical shear stress with seepage�cs). The non-dimensional form of shear stresses �cs/�co must alsoe taken as important parameter in the analysis of incipient motionffected with seepage. So the system characteristics as described inq. (4) can be changed into a system of non-dimensional numbers:[
uy
�, d
((�s − �)g
��2
)1/3
,�csu
(�s − �)�,
vsy
�,
�cs
�co
](9)
It is hypothesized here that the non-dimensional number asiven in Eq. (9) can represent the hydrodynamic characteristicsf the channel affected with seepage. For the design purpose, its assumed that:) [ ( )1/3
]
�csu
(�s − �)�,
�cs
�co= f
uy
�, d
(�s − �)g��2
,vsy
�(10)
The purpose of re-writing Eq. (9) as Eq. (10) is to identify thenput and output of the system.
The objective of this study is to construct a RBF metamodel thatapproximates an unknown input–output mapping on the basis ofgiven experimental observations. The goal is to develop a meta-model, which will capture the underlying relationship betweenoutput and input patterns. The RBF network has a feed forwardstructure consisting of a single hidden layer of J locally tuned units,which are fully interconnected to an output layer of L linear units.All hidden units simultaneously receive the n-dimensional real val-ued input vector X (Fig. 3).
Hidden-unit output Zj is obtained by closeness of the input Xto an n-dimensional parameter vector �j associated with the jthhidden unit (Poggio and Girosi, 1990). The response characteristicsof the jth hidden unit (j = 1, 2, . . ., J) is assumed as:
Zj = K
(∥∥X − �j
∥∥�2
j
)(11)
where K is a strictly positive radially symmetric function (kernel)with a unique maximum at its ‘centre’ �j and which drops offrapidly to zero away from the centre. The parameter �j is the widthof the receptive field in the input space from unit j. This implies
Fig. 3. Structure of a RBF network.
tronics in Agriculture 74 (2010) 321–328 325
crtY
Y
dmf
Z
cwuw�t
�
wRuatuswbbibestvpmwe
teitFtiaunpn
4
(nd
B. Kumar et al. / Computers and Elec
luster must have a small width if the shape of the cluster is to beepresented accurately (Hertz et al., 1991). Given an input vector X,he output of the RBF network is the L-dimensional activity vector, whose lth component (l = 1, 2, . . ., L) is given by:
l =J∑
j=1
wljZi(X) (12)
There are several common types of functions used for the hid-en units, for example, the Gaussian, the multiquadric, the inverseultiquadric and the Cauchy. In the present work, Gaussian basis
unction has been assumed for the hidden unit, which is given as:
j = exp
(−∥∥X − �j
∥∥2
2�2j
)(13)
Training of the RBF neural network involved two critical pro-esses. First, the centers of each of the J Gaussian basis functionsere fixed to represent the density function of the input spacesing a dynamic ‘k means clustering algorithm’ (Bian, 1988). Thisas accomplished by first initializing the set of Gaussian centersj to random values. Then, for any arbitrary input vector X(t) in the
raining set, the closest Gaussian centre, �j, is modified as:
newj = �old
j + ˛(X(t) − �oldj ) (14)
here ˛ is a learning rate that decreases over time. This phase ofBF network training places the weights of the radial basis functionnits in only those regions of the input space where significant datare present. The parameter �j is set for each Gaussian unit to equalhe average distance to the two closest neighboring Gaussian basisnits. If �1 and �2 represent the two closest weight centers to Gaus-ian unit j, the intention was to size this parameter so that thereere no gaps between basis functions and only minimal overlap
etween adjacent basis functions were allowed. After the Gaussianasis centers were fixed, the second step of the RBF network train-
ng process was to determine the weight vector W which wouldest approximate the limited sample data X, thus leading to a lin-ar optimization problem that could be solved by ordinary leastquares method. Entire modelling work has been done by using theoolbox of MATLAB® 7. The total numbers of experimental obser-ations at incipient condition are 51. Input patterns and the outputarameters of the system has been shown in Eq. (10). The best fitodel is a radial basis function network using a Gaussian kernelith 11 centers and a global width of 0.008. The model ouput and
xperimental point have been plotted in Fig. 4a and b.The R2, also called multiple correlation or the coefficient of mul-
iple determination, is the percent of the variance in the dependentxplained uniquely or jointly by the independents. R2 can also benterpreted as the proportionate reduction in error in estimatinghe dependent when knowing the independents. It can be seen fromig. 4 that the value of R2 is very high. RBF networks have advan-ages and disadvantages over Multi Layer Perceptron (MLP), whichs feed forward neural network trained with the standard backprop-gation algorithm. RBF networks can model any non-linear functionsing a single hidden layer, which removes design-decisions aboutumbers of layers. RBF networks are fast and do not suffer fromroblems such as local minima which plague MLP training tech-iques.
.1. Design curves
Design curves correlating dimensionless number as given in Eq.10) have been developed through the RBF model. Alluvial chan-el design with seepage needs five basic design variables, namely,, y, Q, qs and friction/energy slope Sf. It is assumed here that one
Fig. 4. (a) Modelling result for �cs/�co. (b) Modelling result for �csu/(� s − �)�.
of the geometric variables from d or y is known. With known val-ues of d or y and one more design variables, other parameters canbe ascertained by design curves developed in the present work.Table 2 shows the possible design problems of alluvial channel.Known variables are marked with a tick (
√) and the unknown vari-
ables with a question mark (?). However, one should note that unitweights of both fluid and sediment material �s and � respectively,kinematic viscosity, � of the fluid and channel width B are assumedto be known in all types of problems.
Fig. 5a shows the relationship of �cs/�co and Re of a particulardiameter sediment particle. As it can be seen from Fig. 5a that thereis an increment of �cs/�co with increasing Re. An increase in Re willresult in increment of u and y. As �co is constant for a particulardiameter, �cs will increase as shown in Fig. 5a. �co is calculated fromShields diagram and it is high in the case of low values of d*. That iswhy �cs/�co is lower at low values of d*. Relationship of �cs/�co andRs is non-linear as shown in Fig. 5b. At constant Q, y decreases withan increase in Rs. This means lower values of �cs/�co at high d*. Itcan be also seen from Fig. 5b that there is an increase in �cs/�co withRs and then it decreases with Rs. Rs is controlled by vs and y. If thereis an increase in Q, y will increase because of slow rate of seepage.This will cause in increment in �cs. Once channel stabilizes, seepageflow is more and it decreases the flow depth. Same pattern can bealso seen in Fig. 5c, which relates the stream power and Re with Rs.Fig. 5d shows the relationship of stream power and Rs with d*. Highvalues of stream power is required for higher values of d* as shownin Fig. 5d. As bed suction destabilizes the bed, less stream power isneeded to maintain the channel at incipient condition as shown inFig. 5d. In Table 2, the design problems of I–VI are straightforwardin nature and whereas the design problem VII–IX requires iteration.Design problems of I and VII are explained here.
Problem No I: Known variables: y, d and SfUnknown variables: Q and qs
Solution: As d is known, Shields shear stress �co and d* can beestimated directly. �cs can be calculated as �ySf. So the dimension-
326 B. Kumar et al. / Computers and Electronics in Agriculture 74 (2010) 321–328
Fig. 5. (a) Design curve relating �cs/�co, Re and d*. (b) Design curve relating �cs/�co, Rs and d*. (c) Design curve relating �csu/(� s − �)�, Re and Rs . (d) Design curve relating�csu/(� s − �)�, Rs and d*.
B. Kumar et al. / Computers and Electronics in Agriculture 74 (2010) 321–328 327
Table 2The various problems of stable channel design.
Type of designproblem
Design variables
dParticle size
yFlow depth
Sf
Channel slopeQWater discharge
qs
Seepage velocityKnown dimensionless parameters
I√ √ √
? ? d* and �cs/�co
II√ √
?√
? d*and ReIII
√ √? ?
√d* and Rs
IV√
?√ √
? d* and SPV ?
√ √ √? SP and Re
VI ?√
?√ √
Re and Rs
VII√
?√
?√
d* and iterative solution (assume y)√?
N
lvikTed
R�kadtpu
5
idCobdtsbtsstmspobdwvddiwvo
VIII√
? ?IX ?
√ √
ote: The variables of � , � s , � and B are assumed to be known.
ess term �cs/�co and d* are known in this problem. Based on thealues of �cs/�co and d*, Re can be estimated from Fig. 5a. By know-ng Re, u can be estimated with known values of y and �. Withnown values of �cs/�co and d*, Rs can be calculated from Fig. 5b.hus the value of qs is known. In this problem, cross check of thestimated values of Q and qs can be performed by using Fig. 5c and.
Problem No VII: Known variables: d, Sf and qs
Unknown variables: y and QSolution: As d is known, �co and d* can be calculated. The value of
s can be estimated by assuming y. Using Fig. 5b and d, the values ofcs/�co and �csu/(�s − �)� can be estimated respectively. Now withnown values of �cs/�co, y can be estimated. The value of Re canlso be estimated from Fig. 5a with known values of �cs/�co and*. Substituting the estimated value of y in Re and �csu/(�s − �)�,he value of u can be compared. If difference is there, the wholerocedure can be repeated with new values of y until the values ofmatches.
. Conclusion
Seepage consideration in sediment transport problem is anmmensely complex process and the expressing it through aeterministic mathematical framework may not be possible yet.onventional Shields approach cannot be used in the predictionf incipient motion when seepage loss is happening in the sanded channel. Therefore, a departure from the traditional parametricesign approach is required. In parallel with research into sedimentransport has been the emergence of new modeling paradigmsuch as metamodeling techniques. Metamodeling techniques cane used to learn complex relationships from a set of experimen-al observations. Present work uses a RBF metamodel to design theand bed channel affected by seepage. Results from the presenttudy clearly show that RBF metamodel can be successfully appliedo analyze channel seepage by using key input variables. In the
etamodeling process, non-dimensional numbers governing theeepage phenomena in sand bed channel have been used so that theresent work can easily integrate into the field application. Basedn the metamodel, design charts in terms of non-dimensional num-er have been developed for the field application. Methodology toesign different kinds of problem such as determining Q and qs
ith known values of d, y and Sf, determining Q and Sf with knownalues of d, y and qs, determining qs and Sf with known values of, y and Q, determining qs and y with known values of d, Sf and Q,
etermining qs and d with known values of y, Sf and Q, determin-
ng d and Sf with known values of y, Q and qs, determining Q and yith known values of d, Sf and qs, determining y and Sf with known
alues of d, Q and qs and determining Q and d with known valuesf y, Sf and qs has been presented for the field application.
√d* and iterative solution (assume y)√Rs and iterative solution (assume d)
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