Sedimentation in sewer pipes has a negative impact on the performance of sewerage systems. However, due to the complex nature of sedimentation, determining the governing equations is difficult and the results of the available classic models for computing bedload transport rate often differ from each other. This paper focuses on the capability of a support vector machine (SVM) as a meta-model approach for predicting bedload transport in pipes. The method was applied for the deposition and limit of deposition states of sediment transport. Two different scenarios were proposed: in Scenario 1, the input combinations were prepared using only hydraulic characteristics, on the other hand, Scenario 2 was built using both hydraulic and sediment characteristics as model inputs of bedload transport. A comparison between the SVM and the employed classic approaches in predicting sediment transport indicated the supreme performance of the SVM, in which more accurate results were obtained. Also it was found that for estimation of bedload transport in pipes, Scenario 2 led to a more valid outcome than Scenario 1. Based on the sensitivity analysis, parameters Frm and d50/y in the limit of deposition state and Frm in the deposition state had the more dominant role in prediction of bedload discharge in pipes than other parameters.

INTRODUCTION

The prediction of sediment load is an important issue in hydraulic engineering due to its importance in the design and management of water resources projects. The performance of sewerage systems can be affected by sediments and sedimentation in sewer pipes. These issues can significantly reduce the flow capacity of pipes by decreasing their cross-sectional area and increasing the overall hydraulic roughness, which in turn leads to blockage, surcharging, and local flooding. Therefore, accurate estimation of bedload discharge rates in pipes is an important issue. There are various sediment transport equations that have been developed to predict sediment transport. These proposed methods are based on statistical correlations, a combination of the theoretical models, logical assumptions and experimental information. However, due to the complex nature of sedimentation and some parameters being more influential than others in the sediment transport process, determining the governing equations is difficult. For simplifying the governing equations, which describe complex phenomena of the sediment transport process, numerous sediment transport research studies have been conducted in channels and flumes. Perrusquia (1991) performed transport experiments over continuous loose beds in pipes (154, 225 and 450 mm dia.) flowing part full. Dimensional analyses were used to derive the best-fit transport equations. May (2003) studied sediment transport in horizontal pipes and developed a design method. Based on laboratory experiments, Vongvisessomjai et al. (2010) studied the sediment transport for non-cohesive sediment in uniform flow at a no-deposition state. Ota & Perrusquıa (2013) studied sediment transport at the limit of the deposition state in sewers and a semi-empirical equation was developed for computing bed shear stress in pipes. Limited databases, untested model assumptions and a general lack of field data make the predictive accuracy of these models often questionable. Consequently, the application of many formulas is limited to special cases from their development, therefore the existing equations do not show uniform results under different conditions, and this issue causes uncertainty in the estimation of bedload transport. Therefore, it is extremely critical to utilize methods that are capable of predicting sediment transport within pipes under varied hydraulic conditions. The artificial intelligence approaches (e.g. artificial neural networks (ANNs), neuro-fuzzy models, genetic programming and support vector machine (SVM)) have been applied for assessing the accuracy of complex hydraulic and hydrologic phenomena in recent years, such as prediction of total bedload (Chang et al. 2012), prediction of suspended sediment concentration (Kisi & Shiri 2012), prediction of total bed material load (Roushangar et al. 2014) and estimation of the solid load discharge of an alluvial river (Roushangar & Alizadeh 2015).

The SVM method has been applied in modeling various components of water resources systems. Roushangar & Koosheh (2015) evaluated the genetic algorithm-support vector regression (GA-SVR) method for modeling bedload transport in gravel-bed rivers. Kisi (2012) compared least square support vector machine (LSSVM) models with ANNs and the sediment rating curve (SRC) in prediction of suspended sediment concentration. Azamathulla et al. (2010) used the SVM approach to predict sediment transport in rivers.

In the present study, the capability of the SVM approach was assessed for modeling bedload transport in pipes with different boundary conditions (i.e. deposition and limit of deposition states). The models were prepared under two scenarios with various input combinations (based on hydraulic characteristics and properties of the solid load) in order to find the most appropriate input combination for modeling bedload transport in pipes. Then, the accuracy of the best SVM model was compared with the existing classic bedload approaches.

MATERIALS AND METHODS

The data sets

In this study, the experimental data presented by Ghani (1993) and May et al. (1989) were employed for prediction goals, which were collected for pipes carrying storm water. Ghani (1993) conducted some tests under part-full flow conditions with sediments transported as bedload in rigid and loose boundaries. In all, 254 experiments on bedload transport of non-cohesive sediments in non-deposition state were carried out in sewer pipes with diameters of (D = 154, 305 and 450 mm) and length of 20.5 m, covering wide ranges of flow depths (0.15 < y0/D < 0.80), median diameter of particles (0.46 < d50 (mm) < 8.3), flow discharge (0.44 < Q(L/s) < 115.04) and different bed roughness values (0.0 < k0 (mm) < 1.34). Also, 43 data on transport over loose beds were collected in a 450 mm diameter channel with various bed thicknesses up to 23% of pipe diameter, sediment with d50 = 0.72 mm, flow depths (0.5 < y0/D < 0.75) and flow discharge (28.75 < Q(L/s) < 104.98).

May et al. (1989) conducted experiments on a pipe, sized 300 mm in diameter and 20 m in length. Thirty-eight tests under part-full flow conditions were carried out with non-cohesive sediment (d50 = 0.72 mm) and flow velocity 0.082 < V(m/s) < 1.5 in limit of deposition state. Thirty-five tests were also carried out with a small depth of sediment deposition state, with various bed thicknesses up to 16% of pipe diameter. Figure 1 shows the cross-sectional geometry of pipes in deposition and the limit of deposition states. The ranges of some parameters used in the tests are given in Table 1, in which S0, Qs and Re represent pipe slope, sediment discharge and flow Reynolds number, respectively.
Table 1

Range of data used in the tests

    Parameters
 
No. of 
Condition Researcher D (mm) V (m/s) y0/D d50 (mm) Cv (Qs/Q) Re (105S0 (10−2data 
Limit of deposition 
 Smooth bed Ghani 154 0.24–0.862 0.153–0.756 0.93–5.7 38–145 0.13–1.43 0.13–0.53 39 
305 0.395–1.2 0.210–0.8 0.46–8.30 1–1280 0.87–2.7 0.06–0.53 87 
450 0.502–1.2 0.50–0.75 0.72 2–37 1.04–4.6 0.04–0.31 27 
May et al. 300 0.082–1.5 0.37–0.75 0.72 0.31–443 0.75–6.5 0.14–0.56 38 
 Rough bed Ghani 305 Roughness 1 (k0 = 0.53 mm) 0.411–1 0.18–0.77 0.97–8.30 1–923 0.89–2.52 0.07–0.56 71 
305 Roughness 2 (k0 = 1.34 mm) 0.56–0.827 0.243–0.764 2.00–8.30 7–403 0.98–2.1 0.13–0.56 30 
Deposition 
 Separate dunes bedform Ghani 450 0.501–1.011 0.5–0.75 0.72 4–391 1.85–3.72 0.056–0.34 17 
May et al. 300 0.5–1.52 0.49–0.505 0.72 5.7–1165 1.2–3.9 0.031–0.75 30 
 Continuous loose bedform Ghani 450 0.492–1.332 0.5–0.75 0.72 21.1269 1.54–4.1 0.069–0.46 26 
May et al. 300 0.6–1.14 0.5 0.72 280–1186 1.5–2.4 0.29–.57 
    Parameters
 
No. of 
Condition Researcher D (mm) V (m/s) y0/D d50 (mm) Cv (Qs/Q) Re (105S0 (10−2data 
Limit of deposition 
 Smooth bed Ghani 154 0.24–0.862 0.153–0.756 0.93–5.7 38–145 0.13–1.43 0.13–0.53 39 
305 0.395–1.2 0.210–0.8 0.46–8.30 1–1280 0.87–2.7 0.06–0.53 87 
450 0.502–1.2 0.50–0.75 0.72 2–37 1.04–4.6 0.04–0.31 27 
May et al. 300 0.082–1.5 0.37–0.75 0.72 0.31–443 0.75–6.5 0.14–0.56 38 
 Rough bed Ghani 305 Roughness 1 (k0 = 0.53 mm) 0.411–1 0.18–0.77 0.97–8.30 1–923 0.89–2.52 0.07–0.56 71 
305 Roughness 2 (k0 = 1.34 mm) 0.56–0.827 0.243–0.764 2.00–8.30 7–403 0.98–2.1 0.13–0.56 30 
Deposition 
 Separate dunes bedform Ghani 450 0.501–1.011 0.5–0.75 0.72 4–391 1.85–3.72 0.056–0.34 17 
May et al. 300 0.5–1.52 0.49–0.505 0.72 5.7–1165 1.2–3.9 0.031–0.75 30 
 Continuous loose bedform Ghani 450 0.492–1.332 0.5–0.75 0.72 21.1269 1.54–4.1 0.069–0.46 26 
May et al. 300 0.6–1.14 0.5 0.72 280–1186 1.5–2.4 0.29–.57 
Figure 1

Cross-sectional geometry for pipes with limit of deposited (a) and deposited (b) beds.

Figure 1

Cross-sectional geometry for pipes with limit of deposited (a) and deposited (b) beds.

Support vector machine

The foundations of SVMs were developed by Vapnik (1995). The SVM is essentially used in information categorization and data set classification. This approach is known as structural risk minimization, which minimizes an upper bound on the expected risk, as opposed to the traditional empirical risk, which minimizes the error on the training data. It is this difference which equips SVM with a greater ability to generalize, which is the goal in statistical learning (Gunn 1998). An SVM constructs a hyper plane or set of hyper planes in a high or infinite dimensional space, in fact the SVM method is based on the concept of the optimal hyper plane, which separates samples of two classes by considering the widest gap between them (see Figure 2). SVM was originally developed for binary decision problems. This classification method has also been extended to solve prediction problems. SVR is an extension of SVM regression. The aim of SVR is to characterize a kind of function that has at most deviation from the actually obtained objectives for all training data and at the same time it would be as flat as possible. The SVR formulation is as follows: 
formula
1
where φ (x) denotes a nonlinear function in feature of input x, the vector w, is known as the weight factor and b is called the bias. These coefficients are predicted by minimizing the regularized risk function as expressed below: 
formula
2
where 
formula
3
Figure 2

Data classification and support vectors.

Figure 2

Data classification and support vectors.

The constant C is the cost factor and represents the trade-off between the weight factor and approximation error. is the radius of the tube within which the regression function must lie. The represents the loss function in which is the forecasted value and is the desired value in period i. The is the norm of the w vector and the term can be expressed in the form wT. w in which wT is the transpose form of the w vector. According to Equation (3), the loss will be zero if the forecasted value is within the -tube. However, if the value is out of the -tube then the loss is the absolute value, which is the difference between the forecasted value and . Since some data may not lie inside the -tube, the slack variables (ξ, ξ*) must be introduced. These variables represent the distance from the actual values to the corresponding boundary values of the -tube. Therefore, it is possible to transform Equation (2) into the objective function: 
formula
4
subject to: ti−wi φ (xi)−b≤ɛ + ξi, wi φ (xi) + b−ti ≤ ɛ + ξi*, ξi + ξi* ≥ 0.
Using Lagrangian multipliers in Equation (4), thus yields the dual Lagrangian form: 
formula
5
subject to: ∑j=1ni−αi*) = 0, 0 ≤ αi, αi* ≤ C i = 1, 2, … ,N where and are Lagrange multipliers and represents the Lagrange function. is a kernel function to yield the inner products in the feature space and . Different kernel functions have been used in SVR problems. The selection of kernel type which has direct impact on the successful training and classification precision is the most important step in the SVM. Variable parameters used with each kernel function which considerably affect the flexibility of function. Table 2 shows some of the kernel functions and their parameters.
Table 2

Kernel functions

Kernel type Function Kernel parameter 
Linear  – 
Polynomial  d 
RBF  γ 
Sigmoid  α,c 
Kernel type Function Kernel parameter 
Linear  – 
Polynomial  d 
RBF  γ 
Sigmoid  α,c 

Classical bedload approaches

So far, a variety of formulas have been developed to predict bedload transport, ranging from simple regressions to complex multi-parameter formulations. There are different concepts and approaches that are used in the derivation and extraction process of these formulas. The utilized semi-theoretical formulas in this study are as following:

Formula of Ackers

Ackers (1991) developed a bedload transport formula for sewer pipes by modifying Ackers & White's (1973) bedload transport model, which was based on stream power concept. The sediments used ranged from 0.04 to 28.65 mm. The Ackers function, which covers wide alluvial channels, can be written as: 
formula
6
 
formula
where V: flow velocity, the various coefficients Agr, m, C and n are empirically related to the dimensionless grain size Dgr, Cv: transport parameter (QS/Q), d50: median diameter of particles, g: acceleration due to gravity, K representing the incipient motion condition, J relating to sediment transport, d: particle size, A: cross sectional area, W0: sediment bed width, specific density parameter, ρs and ρ: sediment and flow density, : overall friction factor, R: hydraulic radius.

Formula of May

May (1993) derived a semi-empirical transport formula at the limit of deposition state in a concrete pipe (450 mm diameter), based on the shear stress acting on the surface layer of sediments. Two uniform sands (d50= 0.47 mm and 0.73 mm) with an average specific gravity of 2.64 were used and observed moving as bedload. The transport parameter defined as: 
formula
7
 
formula
where : transport parameter, D: pipe diameter, Wb: sediment bed width, Re: particle Reynolds number, θ: related transition factor, ν: kinematic viscosity.

Formula of Neilsen

Neilsen (1992) developed a bedload equation using wide range of laboratory data as follows: 
formula
8
 
formula
where qb: sediment transport rate, θ: Shields’ parameter, τ0: shear stress, Gs: relative density of sediment, Ds: diameter of particles and : specific weight of sediment and water.

Formula of Mayerle

Mayerle (1988) studied the limit of deposition at part-full flow conditions in a smooth pipe (152 mm diameter) and two smooth and rough rectangular channels (311. 5 mm and 462.3 mm wide). Transport function was fitted to the experimental data by the use of multiple regression analysis. The best-fit line was given by: 
formula
9

Formula of Laursen

Laursen (1956) summarized the results of four investigations which carried out with 51 mm and 152 mm diameter pipes and sands with sizes between 0.25 mm and 1.6 mm. Using dimensional analysis Laursen’ equation can be expressed as: 
formula
10

Performance criteria

Evaluating the performance of a model is commonly done using different statistical indexes. In this study, the performance of SVM models were evaluated using three statistical indexes: Determination Coefficient (DC), Correlation Coefficient (R) and Root Mean Square Errors (RMSE). The RMSE describes the average difference between predicted values and measured values, R provides information for linear dependence between observation and corresponding predicted values and DC is the coefficient used to point the relative assessment of the model performance in dimensionless measures. The smaller the RMSE and the more the DC and R, the higher the accuracy of the model will be.

Predicting the intended parameter using non-normalized data may lead to undesirable results, therefore in this study all utilized data were normalized before modeling. All input variables were scaled to fall in the range 0.1–1. The following equation was used to normalize the utilized data in this study: 
formula
11
where Xn, X, Xmax, Xmin, respectively, are: the normalized value of variable i, the original value, the maximum and minimum of variable i.

Simulation and models development

Input variables

Selection of various parameters as input combinations can affect the accuracy of the results throughout the modeling process. Therefore, appropriate selection of parameters is an important step during the modeling process. To determine the dominant input and output variables, an attempt was made to express the process through a set of dimensionless variables. The steady and uniform two-dimensional flow of water and sediments can be defined by the set of the following parameters (Yalin 1977): 
formula
From dimensional analysis, these parameters can be express as follows: 
formula
which namely are the particle mobility parameter θ = u*2/(s − 1)gd50; Particle Reynolds number Re* = u*d50/ν; depth particle size ratio R/d50; and the specific density parameter s = ρsw.
Experimental studies by Ghani (1993), May et al. (1989) and Vongvisessomjai et al. (2010) revealed that in addition to the mentioned parameters, sediment transport in pipes may also be affected by some other parameters. The other selected parameters used herein are as follows: 
formula
where Modified Froude number; Dimensionless particle number; = sediment width; = sediment bed depth.
It should be noted that the parameters Wb and ys were considered to reflect the influence of the movable bed width (deposition state) and depth on sediment discharge for pipes with loose beds. In this study, for predicting bedload transport, two scenarios with different input parameters were proposed for developing models for SVM. Figure 3 demonstrates the proposed scenarios. The developed models are represented in Table 3. For assessing the applicability of the proposed technique for a wider range of data by regarding the fact that there may be a lack of information about sediment transport in pipes, the data series were analyzed all together. This state was evaluated using some of the developed models from Scenario 2. For all cases, 75% of data were used for training the model and the remaining 25% of data were used to test the model.
Table 3

SVM developed models

Scenario 1
 
Scenario 2
 
Parameters of flow conditions
 
Parameters of flow conditions and sediment properties
 
All states
 
Limit of deposited bed (LDb) in smooth and rough beds
 
Deposited bed (Db) in separate dunes and continuous loose bedforms
 
Models Input variables Models Input variables Models Input variables 
(I) Fr LDb(I) λs, Frm Db(I) λs, Frm 
(II) Re LDb(II) Dgr, Frm Db(II) λs, Frm, Dgr 
(III) Fr, y0/D LDb(III) λs, Frm, Dgr Db(III) λs, Frm, wb/y0 
(IV) Re, y0/D LDb(IV) λs, Frm, y/d50 Db(IV) ys/D, Frm, wb/y0 
  LDb(V) λs Frm, Dgr, d50/D Db(V) ys/D, Frm, wb/y0, λs 
  LDb(VI) λs, Frm, Dgr, y/d50 Db(VI) λs, Frm, Dgr, wb/y0 
  LDb(VII) λs, Frm, Dgr, d50/y Db(VII) λs, Frm, Dgr, d50/y 
Scenario 1
 
Scenario 2
 
Parameters of flow conditions
 
Parameters of flow conditions and sediment properties
 
All states
 
Limit of deposited bed (LDb) in smooth and rough beds
 
Deposited bed (Db) in separate dunes and continuous loose bedforms
 
Models Input variables Models Input variables Models Input variables 
(I) Fr LDb(I) λs, Frm Db(I) λs, Frm 
(II) Re LDb(II) Dgr, Frm Db(II) λs, Frm, Dgr 
(III) Fr, y0/D LDb(III) λs, Frm, Dgr Db(III) λs, Frm, wb/y0 
(IV) Re, y0/D LDb(IV) λs, Frm, y/d50 Db(IV) ys/D, Frm, wb/y0 
  LDb(V) λs Frm, Dgr, d50/D Db(V) ys/D, Frm, wb/y0, λs 
  LDb(VI) λs, Frm, Dgr, y/d50 Db(VI) λs, Frm, Dgr, wb/y0 
  LDb(VII) λs, Frm, Dgr, d50/y Db(VII) λs, Frm, Dgr, d50/y 
Figure 3

Considered scenarios in this study.

Figure 3

Considered scenarios in this study.

SVM models development

To determine the best performance of SVM and selecting the best kernel function, different models were predicted via SVM using various kernels. Table 4 shows the results of statistical parameters of different kernels for model LDb (VII) of a smooth bed. The results of Table 6 revealed that using a model with a kernel function of RBF led to better prediction accuracy in comparison to the other kernes. Therefore, the RBF kernel was selected as a core tool of SVM, which was applied for the rest of the models. Implementation of SVM requires the selection of three parameters, which are the constant C, ε and the kernel parameter (γ), where γ is a constant parameter of the RBF kernel. In this study, optimization of these parameters has been performed by a systematic grid search of the parameters using cross-validation on the training set. In this grid search, a normal range of parameter settings is investigated. First, optimized values of C and ε for a specified γ were obtained and then γ was changed. Statistical parameters were used to find optimums. Figure 4 shows the statistics parameters via γ values to find the SVM optimums of the testing set for model (VII) of a smooth bed. In the same way, optimal parameters were obtained for all models.
Table 4

The statistical parameters of the SVM method with different kernel functions; model (VII) of a smooth bed

  Training
 
Testing
 
Kernel types DC RMSE DC RMSE 
Linear 0.821 0.607 0.14 0.755 0.565 0.152 
Polynomial 0.938 0.87 0.081 0.935 0.847 0.085 
RBF 0.973 0.946 0.0415 0.962 0.94 0.062 
Sigmoid 0.594 0.25 0.2 0.533 0.215 0.23 
  Training
 
Testing
 
Kernel types DC RMSE DC RMSE 
Linear 0.821 0.607 0.14 0.755 0.565 0.152 
Polynomial 0.938 0.87 0.081 0.935 0.847 0.085 
RBF 0.973 0.946 0.0415 0.962 0.94 0.062 
Sigmoid 0.594 0.25 0.2 0.533 0.215 0.23 

Note: The bold row shows the superior kernel type.

Figure 4

Statistics parameters via γ values to find the SVM optimums of the testing set for model (VII) of a smooth bed.

Figure 4

Statistics parameters via γ values to find the SVM optimums of the testing set for model (VII) of a smooth bed.

RESULTS AND DISCUSSION

Developed models based on hydraulic parameters (Scenario 1)

Scenario 1 which was developed based on flow characteristics (hydraulic features) is more easily applied via SVM than Scenario 2 because there are fewer variables to be dealt with. On the other hand, it would be more useful in some cases, since there might only be flow characteristics as available data without any sediment data. For assessing the modeling of bedload discharge in pipes using only hydraulic characteristics as inputs (Scenario 1), four models were developed for the limit of deposition (rigid boundary) and deposition states (loose boundary). The results of the SVM models are shown in Table 5 and Figure 5. From the obtained results of statistical parameters (RMSE, R, DC) it can be stated that the estimated and observed values of scenario 1 are not in good agreement. For this state, model (III) with parameters Fr and y0/D led to a more accurate outcome than the other models. However, according to the results, it could be deduced that models that use only hydraulic characteristics are not so accurate.
Table 5

Statistical parameters of the SVM models; Scenario 1

    Performance criteria
 
  Optimal parameters
 
Train
 
Test
 
Condition SVM models ɛ γ DC RMSE DC RMSE 
Limit of deposited bed Smooth bed          
(I) 0.1 0.83 0.69 0.089 0.751 0.52 0.11 
(II) 0.1 0.647 0.53 0.163 0.638 0.428 0.18 
(III) 5 0.1 5 0.854 0.728 0.085 0.778 0.61 0.1017 
(IV) 0.1 0.8 0.6 0.101 0.78 0.53 0.11 
Rough bed          
(I) 10 0.1 6.7 0.89 0.78 0.095 0.85 0.636 0.097 
(II) 0.001 0.691 0.584 0.161 0.652 0.546 0.171 
(III) 5 0.1 5 0.89 0.8 0.093 0.859 0.65 0.0937 
(IV) 0.1 0.88 0.78 0.096 0.86 0.64 0.0964 
Deposited bed Separate dunes bed          
(I) 10 0.1 0.828 0.721 0.117 0.81 0.57 0.136 
(II) 10 0.1 0.73 0.59 0.158 0.729 0.568 0.159 
(III) 10 0.1 2 0.841 0.725 0.109 0.82 0.69 0.131 
(IV) 10 0.1 10 0.829 0.723 0.115 0.814 0.572 0.134 
Continuous loose bed          
(I) 10 0.1 0.81 0.718 0.128 0.79 0.61 0.154 
(II) 10 0.1 0.5 0.69 0.58 0.165 0.651 0.58 0.182 
(III) 10 0.1 5 0.82 0.75 0.107 0.81 0.7 0.142 
(IV) 10 0.1 0.81 0.719 0.1279 0.791 0.62 0.155 
    Performance criteria
 
  Optimal parameters
 
Train
 
Test
 
Condition SVM models ɛ γ DC RMSE DC RMSE 
Limit of deposited bed Smooth bed          
(I) 0.1 0.83 0.69 0.089 0.751 0.52 0.11 
(II) 0.1 0.647 0.53 0.163 0.638 0.428 0.18 
(III) 5 0.1 5 0.854 0.728 0.085 0.778 0.61 0.1017 
(IV) 0.1 0.8 0.6 0.101 0.78 0.53 0.11 
Rough bed          
(I) 10 0.1 6.7 0.89 0.78 0.095 0.85 0.636 0.097 
(II) 0.001 0.691 0.584 0.161 0.652 0.546 0.171 
(III) 5 0.1 5 0.89 0.8 0.093 0.859 0.65 0.0937 
(IV) 0.1 0.88 0.78 0.096 0.86 0.64 0.0964 
Deposited bed Separate dunes bed          
(I) 10 0.1 0.828 0.721 0.117 0.81 0.57 0.136 
(II) 10 0.1 0.73 0.59 0.158 0.729 0.568 0.159 
(III) 10 0.1 2 0.841 0.725 0.109 0.82 0.69 0.131 
(IV) 10 0.1 10 0.829 0.723 0.115 0.814 0.572 0.134 
Continuous loose bed          
(I) 10 0.1 0.81 0.718 0.128 0.79 0.61 0.154 
(II) 10 0.1 0.5 0.69 0.58 0.165 0.651 0.58 0.182 
(III) 10 0.1 5 0.82 0.75 0.107 0.81 0.7 0.142 
(IV) 10 0.1 0.81 0.719 0.1279 0.791 0.62 0.155 

Note: The bold rows show the superior model in each state.

Figure 5

Comparison of observed and predicted sediment load for superior model.

Figure 5

Comparison of observed and predicted sediment load for superior model.

Developed models based on hydraulic parameters and sediment particle features (Scenario 2)

In Scenario 2, for estimating the bedload discharge in pipes with different boundary conditions, several models were developed according to the flow condition and particle features. Also, as a new data set, data for smooth and rough beds were used all together. The results obtained from the SVM models are indicated in Table 6 and Figures 6 and 7. The superior performance was obtained from the model LDb(VII), in which the smooth and rough beds and mixed data set in the limit of deposition state were used and the input parameters were λs, Frm, Dgr, and d50/y. Based on Table 6, using the mixed data set reduced the models' accuracy. Also, it was deduced that for estimating sediment discharge in pipes, using the relative flow depth and overall friction factor as input parameters improved the efficiency of the models. Based on the results of the deposition state demonstrated in Table 6, model Db(IV) with input parameters ys/D, Frm, and Wb/y0 in separate dunes and continuous loose bedforms was more accurate than the other models. It could be inferred that using parameters ys/D and Wb/y0 increased the accuracy of the models, which confirms the importance of the flow depth and width and the depth of the sediment bed in the bedload estimating process in pipes with deposited beds. Figures 6 and 7 show the verification between the measured and estimated values for the best proposed model for all data sets. According to the obtained results, Scenario 2, which took advantage of both flow and sediment parameters as input in modeling sediment discharge in pipes, performed more successfully than Scenario 1.
Table 6

The optimal parameters and statistical parameters of the SVM models; Scenario 2

    Performance criteria
 
  Optimal parameters
 
Train
 
Test
 
Condition SVM model ɛ γ DC RMSE DC RMSE 
Limit of deposited bed Smooth bed          
LDb(I) 10 0.1 0.53 0.37 0.188 0.507 0.27 0.199 
LDb(II) 10 0.1 10 0.748 0.553 0.145 0.729 0.528 0.148 
LDb(III) 4.001 0.101 0.88 0.85 0.0796 0.873 0.813 0.083 
LDb(IV) 0.02 10 0.89 0.887 0.0731 0.881 0.813 0.0812 
LDb(V) 9.001 0.201 0.913 0.89 0.0668 0.883 0.816 0.0809 
LDb(VI) 10 0.001 0.96 0.91 0.052 0.94 0.87 0.0662 
LDb(VII) 10 0.1 3 0.973 0.946 0.0415 0.962 0.94 0.062 
Rough bed          
LDb(I) 10 0.1 0.5 0.48 0.354 0.196 0.445 0.31 0.199 
LDb(II) 0.13 0.698 0.466 0.1542 0.659 0.438 0.157 
LDb(III) 0.001 0.875 0.821 0.0838 0.871 0.8 0.0918 
LDb(IV) 0.001 0.89 0.874 0.0761 0.879 0.805 0.082 
LDb(V) 0.001 0.5 0.912 0.873 0.076 0.88 0.81 0.0813 
LDb(VI) 0.001 0.953 0.913 0.0456 0.922 0.831 0.0761 
LDb(VII) 5 0.002 2 0.986 0.97 0.032 0.98 0.929 0.0472 
Mixed data          
LDb(III) 0.2 0.75 0.521 0.121 0.7 0.49 0.136 
LDb(IV) 10 0.1 10 0.84 0.78 0.0861 0.818 0.75 0.106 
LDb(V) 10 0.001 10 0.902 0.84 0.079 0.852 0.8 0.097 
LDb(VI) 10 0.1 10 0.92 0.85 0.073 0.876 0.821 0.089 
LDb(VII) 5 0.001 5 0.94 0.9 0.0678 0.93 0.89 0.0782 
Deposited bed Bed with separate dunes          
Db(I) 10 0.001 0.791 0.548 0.108 0.78 0.532 0.144 
Db(II) 10 0.001 0.918 0.89 0.059 0.9 0.849 0.062 
Db(III) 0.001 0.939 0.93 0.053 0.912 0.87 0.056 
Db(IV) 10 0.1 6 0.986 0.96 0.0318 0.981 0.951 0.033 
Db(V) 0.1 0.94 0.935 0.048 0.92 0.88 0.055 
Db(VI) 10 0.1 0.5 0.97 0.94 0.045 0.94 0.92 0.052 
Db(VII) 0.001 2.5 0.91 0.88 0.0599 0.9 0.83 0.64 
Continuous loose bed          
Db(I) 10 0.1 1.5 0.89 0.79 0.104 0.85 0.538 0.169 
Db(II) 10 0.1 0.914 0.88 0.09 0.89 0.759 0.122 
Db(III) 10 0.1 1.5 0.94 0.92 0.048 0.91 0.89 0.077 
Db(IV) 10 0.1 2.5 0.992 0.981 0.031 0.988 0.975 0.039 
Db(V) 10 0.1 0.945 0.93 0.046 0.931 0.92 0.065 
Db(VI) 10 0.1 0.5 0.97 0.94 0.044 0.96 0.93 0.052 
Db(VII) 10 0.1 0.5 0.92 0.9 0.05 0.89 0.88 0.079 
    Performance criteria
 
  Optimal parameters
 
Train
 
Test
 
Condition SVM model ɛ γ DC RMSE DC RMSE 
Limit of deposited bed Smooth bed          
LDb(I) 10 0.1 0.53 0.37 0.188 0.507 0.27 0.199 
LDb(II) 10 0.1 10 0.748 0.553 0.145 0.729 0.528 0.148 
LDb(III) 4.001 0.101 0.88 0.85 0.0796 0.873 0.813 0.083 
LDb(IV) 0.02 10 0.89 0.887 0.0731 0.881 0.813 0.0812 
LDb(V) 9.001 0.201 0.913 0.89 0.0668 0.883 0.816 0.0809 
LDb(VI) 10 0.001 0.96 0.91 0.052 0.94 0.87 0.0662 
LDb(VII) 10 0.1 3 0.973 0.946 0.0415 0.962 0.94 0.062 
Rough bed          
LDb(I) 10 0.1 0.5 0.48 0.354 0.196 0.445 0.31 0.199 
LDb(II) 0.13 0.698 0.466 0.1542 0.659 0.438 0.157 
LDb(III) 0.001 0.875 0.821 0.0838 0.871 0.8 0.0918 
LDb(IV) 0.001 0.89 0.874 0.0761 0.879 0.805 0.082 
LDb(V) 0.001 0.5 0.912 0.873 0.076 0.88 0.81 0.0813 
LDb(VI) 0.001 0.953 0.913 0.0456 0.922 0.831 0.0761 
LDb(VII) 5 0.002 2 0.986 0.97 0.032 0.98 0.929 0.0472 
Mixed data          
LDb(III) 0.2 0.75 0.521 0.121 0.7 0.49 0.136 
LDb(IV) 10 0.1 10 0.84 0.78 0.0861 0.818 0.75 0.106 
LDb(V) 10 0.001 10 0.902 0.84 0.079 0.852 0.8 0.097 
LDb(VI) 10 0.1 10 0.92 0.85 0.073 0.876 0.821 0.089 
LDb(VII) 5 0.001 5 0.94 0.9 0.0678 0.93 0.89 0.0782 
Deposited bed Bed with separate dunes          
Db(I) 10 0.001 0.791 0.548 0.108 0.78 0.532 0.144 
Db(II) 10 0.001 0.918 0.89 0.059 0.9 0.849 0.062 
Db(III) 0.001 0.939 0.93 0.053 0.912 0.87 0.056 
Db(IV) 10 0.1 6 0.986 0.96 0.0318 0.981 0.951 0.033 
Db(V) 0.1 0.94 0.935 0.048 0.92 0.88 0.055 
Db(VI) 10 0.1 0.5 0.97 0.94 0.045 0.94 0.92 0.052 
Db(VII) 0.001 2.5 0.91 0.88 0.0599 0.9 0.83 0.64 
Continuous loose bed          
Db(I) 10 0.1 1.5 0.89 0.79 0.104 0.85 0.538 0.169 
Db(II) 10 0.1 0.914 0.88 0.09 0.89 0.759 0.122 
Db(III) 10 0.1 1.5 0.94 0.92 0.048 0.91 0.89 0.077 
Db(IV) 10 0.1 2.5 0.992 0.981 0.031 0.988 0.975 0.039 
Db(V) 10 0.1 0.945 0.93 0.046 0.931 0.92 0.065 
Db(VI) 10 0.1 0.5 0.97 0.94 0.044 0.96 0.93 0.052 
Db(VII) 10 0.1 0.5 0.92 0.9 0.05 0.89 0.88 0.079 

Note: The bold rows show the superior model in each state.

Figure 6

Comparison of observed and predicted sediment load for superior models of the limit of deposition state.

Figure 6

Comparison of observed and predicted sediment load for superior models of the limit of deposition state.

Figure 7

Comparison of observed and predicted sediment load for superior models of the deposition state.

Figure 7

Comparison of observed and predicted sediment load for superior models of the deposition state.

Combined data

An evaluation of the applicability of the applied methods for a wider range of data was tried. In other words, all data series were combined (simultaneous mode), then for predicting the Cv as the dependent variable, several models of Scenario 2 were reanalyzed for the combined data mode. The results for the SVM models are given in Table 7 and Figure 8. Based on the results of the combined data, the model with parameters λs, Frm, Dgr, and d50/y was more accurate. Comparison between Tables 6 and 7 indicated that SVM models for combined data set did not show the desired accuracy, especially for large values of Cv, and analyzing data sets separately led to more accurate results. However, it should be considered that the results of the combined data are capable of covering a wider range of data, and in this case without regarding the pipe boundary condition (i.e. the deposition and non-deposition states) the sediment transport process can be studied.
Table 7

The optimal parameters and statistical parameters of the SVM models; combined data

    Performance criteria
 
  Optimal parameters
 
Train
 
Test
 
Condition SVM model ɛ γ DC RMSE DC RMSE 
Limit of deposited and deposition state Combined data          
λs, Frm, Dgr 10 0.001 0.749 0.53 0.189 0.729 0.5025 0.191 
λs, Frm, y/d50 10 0.12 10 0.82 0.674 0.154 0.81 0.672 0.159 
λs, Frm, Dgr,d50/D 0.1 0.803 0.635 0.158 0.782 0.54 0.162 
λs, Frm, Dgr, y/d50 10 0.001 5.5 0.834 0.684 0.132 0.828 0.683 0.142 
λs, Frm, Dgr, d50/y 10 0.1 10 0.87 0.74 0.098 0.844 0.71 0.101 
    Performance criteria
 
  Optimal parameters
 
Train
 
Test
 
Condition SVM model ɛ γ DC RMSE DC RMSE 
Limit of deposited and deposition state Combined data          
λs, Frm, Dgr 10 0.001 0.749 0.53 0.189 0.729 0.5025 0.191 
λs, Frm, y/d50 10 0.12 10 0.82 0.674 0.154 0.81 0.672 0.159 
λs, Frm, Dgr,d50/D 0.1 0.803 0.635 0.158 0.782 0.54 0.162 
λs, Frm, Dgr, y/d50 10 0.001 5.5 0.834 0.684 0.132 0.828 0.683 0.142 
λs, Frm, Dgr, d50/y 10 0.1 10 0.87 0.74 0.098 0.844 0.71 0.101 

Note: The bold row shows the superior model.

Figure 8

Comparison of observed and predicted sediment load for superior model of combined data.

Figure 8

Comparison of observed and predicted sediment load for superior model of combined data.

Sensitivity analysis

To investigate the impacts of the different employed parameters from the best proposed models on bedload discharge prediction via SVMs, a sensitivity analysis was performed. The significance of each parameter was evaluated by eliminating them. Based on the results from Table 8, it could be deduced that in a loose boundary condition (i.e. separate dunes and continuous loose bedforms) parameter Frm and in the rigid boundary condition (smooth and rough beds), Frm and d50/y had the most significant effect on bedload discharge respectively.

Table 8

Relative significance of each of the input parameters of the best models

  Performance criteria
 
  Train
 
Test
 
SVM-best model Eliminated variable DC RMSE DC RMSE 
Smooth bed 
 LDb(IIV) Frm 0.827 0.67 0.134 0.66 0.54 0.1735 
Dgr 0.939 0.857 0.08 0.845 0.7 0.129 
Λ0.94 087 0.077 0.852 0.73 0.12 
d50/y 0.918 0.83 0.095 0.83 0.69 0.137 
Rough bed 
 LDb(IIV) Frm 0.964 0.91 0.056 0.891 0.74 0.08 
Dgr 0.96 0.921 0.0485 0.951 0.903 0.0512 
λs 0.968 0.913 0.051 0.905 0.768 0.077 
d50/y 0.865 0.84 0.1056 0.812 0.728 0.097 
Separate dunes 
 Db(IV) Frm 0.78 0.67 0.119 0.61 0.46 0.146 
ys/D 0.94 0.91 0.048 0.93 0.78 0.079 
Wb/y0 0.945 0.925 0.046 0.943 0.89 0.06 
Continuous loose bed 
 Db(IV) Frm 0.89 0.81 0.077 0.88 0.78 0.096 
ys/D 0.951 0.922 0.057 0.94 0.876 0.095 
Wb/y0 0.972 0.952 0.0454 0.96 0.93 0.058 
  Performance criteria
 
  Train
 
Test
 
SVM-best model Eliminated variable DC RMSE DC RMSE 
Smooth bed 
 LDb(IIV) Frm 0.827 0.67 0.134 0.66 0.54 0.1735 
Dgr 0.939 0.857 0.08 0.845 0.7 0.129 
Λ0.94 087 0.077 0.852 0.73 0.12 
d50/y 0.918 0.83 0.095 0.83 0.69 0.137 
Rough bed 
 LDb(IIV) Frm 0.964 0.91 0.056 0.891 0.74 0.08 
Dgr 0.96 0.921 0.0485 0.951 0.903 0.0512 
λs 0.968 0.913 0.051 0.905 0.768 0.077 
d50/y 0.865 0.84 0.1056 0.812 0.728 0.097 
Separate dunes 
 Db(IV) Frm 0.78 0.67 0.119 0.61 0.46 0.146 
ys/D 0.94 0.91 0.048 0.93 0.78 0.079 
Wb/y0 0.945 0.925 0.046 0.943 0.89 0.06 
Continuous loose bed 
 Db(IV) Frm 0.89 0.81 0.077 0.88 0.78 0.096 
ys/D 0.951 0.922 0.057 0.94 0.876 0.095 
Wb/y0 0.972 0.952 0.0454 0.96 0.93 0.058 

Sediment rating curve

SRC is a common approach for predicting sediment discharge in river engineering. In this method, there is a regression relationship between water and sediment discharge, which is mostly a power function. In the present study, a rating curve was used for all data sets in order to investigate its capability in estimating sediment discharge in circular pipes. Figure 9 shows SRC for rough and continuous loose beds. According to Figure 9, no reasonable relationship between water and sediment discharge in pipes could be found.
Figure 9

Sediment rate curve; (a) rough bed, (b) continuous loose bed.

Figure 9

Sediment rate curve; (a) rough bed, (b) continuous loose bed.

Comparison of SVM and sediment transport classic equations

The accuracy of the best proposed models of Scenario 2 developed in this study, and some bedload semi-theoretical formulae available in literature, have been compared to evaluate the performance of the applied approach. The results of the comparison for any data set and testing procedure are represented in Figure 10. According to three evaluated criteria (R, DC, RMSE) which are shown in Figure 10, it can be seen that the estimated values via SVM models had more accurate results than equations. It should be noted that the existing equations are developed based on special flow conditions and sediment particle features, therefore the application of the equations is limited to special cases of their development and did not show uniform results under different conditions. The mentioned issue can be seen in Figure 10, which shows that the results obtained from the equations differ from each other and from the measured data, and have less preciseness than the models proposed in this study. In the limit of deposition state, the Mayerle and Laursen approaches showed better results than others, respectively, and in the deposition state the May formula had a more accurate outcome. However, the obtained results confirmed the capability of the SVM as an efficient machine learning approach in the modeling of bedload discharge in pipes.
Figure 10

Comparison of equations and SVM model; (a) smooth bed, (b) rough bed, (c) separate dunes bedform, (d) continuous loose bedform.

Figure 10

Comparison of equations and SVM model; (a) smooth bed, (b) rough bed, (c) separate dunes bedform, (d) continuous loose bedform.

CONCLUSION

In the present study, the capability of the SVM approach was verified for predicting non-cohesive sediment transport in circular channels. The SVM was applied for different data sets in the limit of deposition and deposition states. To develop the models, two different scenarios were proposed and evaluated. Furthermore, the results of the SVM models were compared with several semi-theoretical approaches. The obtained results revealed that Scenario 2, which used both hydraulic and sediment parameters as inputs in modeling sediment transport in pipes, was more accurate than Scenario 1, which applied only hydraulic parameters as inputs. In the limit of deposition state, it was observed that adding Frm and d50/y as an input parameters improved the efficiency of the models. In this state, for both smooth and rough beds and also mixed data, the model including parameters λs, Frm, Dgr, and d50/y presented better performance. Based on the obtained results from the sensitivity analysis, it was found that Frm and d50/y had a more dominant role in predicting bedload discharge than the other parameters (see Table 8). For the deposition state in separate dunes and continuous loose bedforms, the model with parameters ys/D, Frm, and Wb/y0 led to a better outcome. This model demonstrated the influence of flow depth and width and the depth of the sediment bed on bedload transport in deposited beds. For the deposited bed, Frm had a dominant role (see Table 8). From the results, it was found that analyzing data sets separately led to a more accurate outcome (see Tables 6 and 7). The results also revealed that for evaluating the sediment load in pipes, SRC did not capture a reasonable relationship between flow and sediment discharge. However, the proposed approach was found to be able to predict bedload discharge in both limits deposited and deposited beds more accurately and successfully than classic equations. Since SVM is a data-driven model, it is suggested to investigate the sufficiency of the models proposed via SVM for data ranges outside this study to find the merits of the models in estimating bedload sediment.

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