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 *F _{rm}* and

*d*in the limit of deposition state and

_{50}/y*F*in the deposition state had the more dominant role in prediction of bedload discharge in pipes than other parameters.

_{rm}## 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 < *y _{0}/D* < 0.80), median diameter of particles (0.46 <

*d*(mm) < 8.3), flow discharge (0.44 <

_{50}*Q*(L/s) < 115.04) and different bed roughness values (0.0 <

*k*(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

_{0}*d*= 0.72 mm, flow depths (0.5 <

_{50}*y*< 0.75) and flow discharge (28.75 <

_{0}/D*Q*(L/s) < 104.98).

*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 (

*d*= 0.72 mm) and flow velocity 0.082 <

_{50}*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

*S*and

_{0}, Q_{s}*Re*represent pipe slope, sediment discharge and flow Reynolds number, respectively.

. | . | Parameters . | No. of . | ||||||
---|---|---|---|---|---|---|---|---|---|

Condition . | Researcher . | D (mm) . | V (m/s) . | y_{0}/D
. | d_{50} (mm)
. | C_{v} (Q_{s}/Q)
. | Re (10^{5})
. | S_{0} (10^{−2})
. | data . |

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 (k_{0} = 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 (k_{0} = 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 | 5 |

. | . | Parameters . | No. of . | ||||||
---|---|---|---|---|---|---|---|---|---|

Condition . | Researcher . | D (mm) . | V (m/s) . | y_{0}/D
. | d_{50} (mm)
. | C_{v} (Q_{s}/Q)
. | Re (10^{5})
. | S_{0} (10^{−2})
. | data . |

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 (k_{0} = 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 (k_{0} = 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 | 5 |

### Support vector machine

^{T}. w in which w

^{T}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: subject to: t

_{i}−w

_{i}φ (x

_{i})−b≤ɛ + ξ

_{i}, w

_{i}φ (x

_{i}) + b−t

_{i}≤ ɛ + ξ

_{i}

^{*}, ξ

_{i}+ ξ

_{i}

^{*}≥ 0.

_{j=1}

^{n}(α

_{i}−α

_{i}

^{*}) = 0, 0 ≤ α

*, α*

_{i}_{i}

^{*}≤ 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.

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

_{gr}, m, C and n are empirically related to the dimensionless grain size D

_{gr}, C

_{v}: transport parameter (Q

_{S}/Q), d

_{50}: 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, W

_{0}: sediment bed width, specific density parameter, ρ

_{s}and ρ: sediment and flow density, : overall friction factor, R: hydraulic radius.

#### Formula of May

_{50}= 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: where : transport parameter, D: pipe diameter, W

_{b}: sediment bed width, R

_{e}: particle Reynolds number, θ: related transition factor, ν: kinematic viscosity.

#### Formula of Neilsen

_{b}: sediment transport rate, θ: Shields’ parameter, τ

_{0}: shear stress, G

_{s}: relative density of sediment, D

_{s}: diameter of particles and : specific weight of sediment and water.

#### Formula of Mayerle

#### Formula of Laursen

### 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.

_{n}, X, X

_{max}, X

_{min}, respectively, are: the normalized value of variable i, the original value, the maximum and minimum of variable i.

### Simulation and models development

#### Input variables

*u**

^{2}/(s − 1)

*gd*

_{50}; Particle Reynolds number Re

_{*}= u

_{*}d

_{50}/ν; depth particle size ratio R/d

_{50}; and the specific density parameter s = ρ

_{s}/ρ

_{w}.

*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: where Modified Froude number; Dimensionless particle number; = sediment width; = sediment bed depth.

_{b}and y

_{s}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.

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, F_{rm} | Db(I) | λs, F_{rm} |

(II) | R_{e} | LDb(II) | D_{gr}, F_{rm} | Db(II) | λs, F_{rm}, D_{gr} |

(III) | Fr, y_{0}/D | LDb(III) | λs, F_{rm}, D_{gr} | Db(III) | λs, F_{rm}, w_{b}/y_{0} |

(IV) | R_{e}, y_{0}/D | LDb(IV) | λs, F_{rm}, y/d_{50} | Db(IV) | y_{s}/D, F_{rm}, w/_{b}y_{0} |

LDb(V) | λs F_{rm}, D_{gr}, d_{50}/D | Db(V) | y/_{s}D, F_{rm}, w/y_{b}_{0}, λs | ||

LDb(VI) | λs, F_{rm}, D_{gr}, y/d_{50} | Db(VI) | λs, F_{rm}, D_{gr}, w/_{b}y_{0} | ||

LDb(VII) | λs, F_{rm}, D_{gr}, d_{50}/y | Db(VII) | λs, F_{rm}, D_{gr}, d_{50}/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, F_{rm} | Db(I) | λs, F_{rm} |

(II) | R_{e} | LDb(II) | D_{gr}, F_{rm} | Db(II) | λs, F_{rm}, D_{gr} |

(III) | Fr, y_{0}/D | LDb(III) | λs, F_{rm}, D_{gr} | Db(III) | λs, F_{rm}, w_{b}/y_{0} |

(IV) | R_{e}, y_{0}/D | LDb(IV) | λs, F_{rm}, y/d_{50} | Db(IV) | y_{s}/D, F_{rm}, w/_{b}y_{0} |

LDb(V) | λs F_{rm}, D_{gr}, d_{50}/D | Db(V) | y/_{s}D, F_{rm}, w/y_{b}_{0}, λs | ||

LDb(VI) | λs, F_{rm}, D_{gr}, y/d_{50} | Db(VI) | λs, F_{rm}, D_{gr}, w/_{b}y_{0} | ||

LDb(VII) | λs, F_{rm}, D_{gr}, d_{50}/y | Db(VII) | λs, F_{rm}, D_{gr}, d_{50}/y |

### SVM models development

*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.

. | Training . | Testing . | ||||
---|---|---|---|---|---|---|

Kernel types . | R . | DC . | RMSE . | R . | 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 . | R . | DC . | RMSE . | R . | 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.

## RESULTS AND DISCUSSION

### Developed models based on hydraulic parameters (Scenario 1)

*F*and

_{r}*y*

_{0}

*/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.

. | . | . | Performance criteria . | |||||||
---|---|---|---|---|---|---|---|---|---|---|

. | . | Optimal parameters . | Train . | Test . | ||||||

Condition . | SVM models . | c . | ɛ
. | γ . | R . | DC . | RMSE . | R . | DC . | RMSE . |

Limit of deposited bed | Smooth bed | |||||||||

(I) | 5 | 0.1 | 5 | 0.83 | 0.69 | 0.089 | 0.751 | 0.52 | 0.11 | |

(II) | 5 | 0.1 | 5 | 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) | 5 | 0.1 | 5 | 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) | 5 | 0.001 | 5 | 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) | 5 | 0.1 | 5 | 0.88 | 0.78 | 0.096 | 0.86 | 0.64 | 0.0964 | |

Deposited bed | Separate dunes bed | |||||||||

(I) | 10 | 0.1 | 2 | 0.828 | 0.721 | 0.117 | 0.81 | 0.57 | 0.136 | |

(II) | 10 | 0.1 | 2 | 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 | 8 | 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 | 4 | 0.81 | 0.719 | 0.1279 | 0.791 | 0.62 | 0.155 |

. | . | . | Performance criteria . | |||||||
---|---|---|---|---|---|---|---|---|---|---|

. | . | Optimal parameters . | Train . | Test . | ||||||

Condition . | SVM models . | c . | ɛ
. | γ . | R . | DC . | RMSE . | R . | DC . | RMSE . |

Limit of deposited bed | Smooth bed | |||||||||

(I) | 5 | 0.1 | 5 | 0.83 | 0.69 | 0.089 | 0.751 | 0.52 | 0.11 | |

(II) | 5 | 0.1 | 5 | 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) | 5 | 0.1 | 5 | 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) | 5 | 0.001 | 5 | 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) | 5 | 0.1 | 5 | 0.88 | 0.78 | 0.096 | 0.86 | 0.64 | 0.0964 | |

Deposited bed | Separate dunes bed | |||||||||

(I) | 10 | 0.1 | 2 | 0.828 | 0.721 | 0.117 | 0.81 | 0.57 | 0.136 | |

(II) | 10 | 0.1 | 2 | 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 | 8 | 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 | 4 | 0.81 | 0.719 | 0.1279 | 0.791 | 0.62 | 0.155 |

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

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

*λs*,

*F*

_{rm}

*, D*

_{gr}, and d

_{50}

*/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

*y*,

_{s}/D*F*

_{rm}, and

*W*

_{b}/y_{0}in separate dunes and continuous loose bedforms was more accurate than the other models. It could be inferred that using parameters

*y*and

_{s}/D*W*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.

_{b}/y_{0}. | . | . | Performance criteria . | |||||||
---|---|---|---|---|---|---|---|---|---|---|

. | . | Optimal parameters . | Train . | Test . | ||||||

Condition . | SVM model . | c . | ɛ
. | γ . | R . | DC . | RMSE . | R . | DC . | RMSE . |

Limit of deposited bed | Smooth bed | |||||||||

LDb(I) | 10 | 0.1 | 5 | 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 | 5 | 0.88 | 0.85 | 0.0796 | 0.873 | 0.813 | 0.083 | |

LDb(IV) | 8 | 0.02 | 10 | 0.89 | 0.887 | 0.0731 | 0.881 | 0.813 | 0.0812 | |

LDb(V) | 9.001 | 0.201 | 3 | 0.913 | 0.89 | 0.0668 | 0.883 | 0.816 | 0.0809 | |

LDb(VI) | 10 | 0.001 | 3 | 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) | 9 | 0.13 | 6 | 0.698 | 0.466 | 0.1542 | 0.659 | 0.438 | 0.157 | |

LDb(III) | 2 | 0.001 | 2 | 0.875 | 0.821 | 0.0838 | 0.871 | 0.8 | 0.0918 | |

LDb(IV) | 5 | 0.001 | 2 | 0.89 | 0.874 | 0.0761 | 0.879 | 0.805 | 0.082 | |

LDb(V) | 5 | 0.001 | 0.5 | 0.912 | 0.873 | 0.076 | 0.88 | 0.81 | 0.0813 | |

LDb(VI) | 2 | 0.001 | 2 | 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) | 8 | 0.2 | 6 | 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 | 3 | 0.791 | 0.548 | 0.108 | 0.78 | 0.532 | 0.144 | |

Db(II) | 10 | 0.001 | 5 | 0.918 | 0.89 | 0.059 | 0.9 | 0.849 | 0.062 | |

Db(III) | 8 | 0.001 | 3 | 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) | 8 | 0.1 | 6 | 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) | 8 | 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 | 3 | 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 | 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 . | c . | ɛ
. | γ . | R . | DC . | RMSE . | R . | DC . | RMSE . |

Limit of deposited bed | Smooth bed | |||||||||

LDb(I) | 10 | 0.1 | 5 | 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 | 5 | 0.88 | 0.85 | 0.0796 | 0.873 | 0.813 | 0.083 | |

LDb(IV) | 8 | 0.02 | 10 | 0.89 | 0.887 | 0.0731 | 0.881 | 0.813 | 0.0812 | |

LDb(V) | 9.001 | 0.201 | 3 | 0.913 | 0.89 | 0.0668 | 0.883 | 0.816 | 0.0809 | |

LDb(VI) | 10 | 0.001 | 3 | 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) | 9 | 0.13 | 6 | 0.698 | 0.466 | 0.1542 | 0.659 | 0.438 | 0.157 | |

LDb(III) | 2 | 0.001 | 2 | 0.875 | 0.821 | 0.0838 | 0.871 | 0.8 | 0.0918 | |

LDb(IV) | 5 | 0.001 | 2 | 0.89 | 0.874 | 0.0761 | 0.879 | 0.805 | 0.082 | |

LDb(V) | 5 | 0.001 | 0.5 | 0.912 | 0.873 | 0.076 | 0.88 | 0.81 | 0.0813 | |

LDb(VI) | 2 | 0.001 | 2 | 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) | 8 | 0.2 | 6 | 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 | 3 | 0.791 | 0.548 | 0.108 | 0.78 | 0.532 | 0.144 | |

Db(II) | 10 | 0.001 | 5 | 0.918 | 0.89 | 0.059 | 0.9 | 0.849 | 0.062 | |

Db(III) | 8 | 0.001 | 3 | 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) | 8 | 0.1 | 6 | 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) | 8 | 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 | 3 | 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 | 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.

### Combined data

*C*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

_{v}*λs*,

*F*

_{rm},

*D*

_{gr}, and d

_{50}

*/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

*C*, 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.

_{v}. | . | . | Performance criteria . | |||||||
---|---|---|---|---|---|---|---|---|---|---|

. | . | Optimal parameters . | Train . | Test . | ||||||

Condition . | SVM model . | c . | ɛ
. | γ . | R . | DC . | RMSE . | R . | DC . | RMSE . |

Limit of deposited and deposition state | Combined data | |||||||||

λs, F_{rm}, D_{gr} | 10 | 0.001 | 5 | 0.749 | 0.53 | 0.189 | 0.729 | 0.5025 | 0.191 | |

λs, F_{rm}, y/d_{50} | 10 | 0.12 | 10 | 0.82 | 0.674 | 0.154 | 0.81 | 0.672 | 0.159 | |

λs, F_{rm}, D_{gr},d_{50}/D | 5 | 0.1 | 4 | 0.803 | 0.635 | 0.158 | 0.782 | 0.54 | 0.162 | |

λs, F_{rm}, D_{gr}, y/d_{50} | 10 | 0.001 | 5.5 | 0.834 | 0.684 | 0.132 | 0.828 | 0.683 | 0.142 | |

λs, F _{rm}, D_{gr}, d_{50}/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 . | c . | ɛ
. | γ . | R . | DC . | RMSE . | R . | DC . | RMSE . |

Limit of deposited and deposition state | Combined data | |||||||||

λs, F_{rm}, D_{gr} | 10 | 0.001 | 5 | 0.749 | 0.53 | 0.189 | 0.729 | 0.5025 | 0.191 | |

λs, F_{rm}, y/d_{50} | 10 | 0.12 | 10 | 0.82 | 0.674 | 0.154 | 0.81 | 0.672 | 0.159 | |

λs, F_{rm}, D_{gr},d_{50}/D | 5 | 0.1 | 4 | 0.803 | 0.635 | 0.158 | 0.782 | 0.54 | 0.162 | |

λs, F_{rm}, D_{gr}, y/d_{50} | 10 | 0.001 | 5.5 | 0.834 | 0.684 | 0.132 | 0.828 | 0.683 | 0.142 | |

λs, F _{rm}, D_{gr}, d_{50}/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.

### 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 *F*_{rm} and in the rigid boundary condition (smooth and rough beds), *F*_{rm} and d_{50}*/y* had the most significant effect on bedload discharge respectively.

. | . | Performance criteria . | |||||
---|---|---|---|---|---|---|---|

. | . | Train . | Test . | ||||

SVM-best model . | Eliminated variable . | R . | DC . | RMSE . | R . | DC . | RMSE . |

Smooth bed | |||||||

LDb(IIV) | F_{rm} | 0.827 | 0.67 | 0.134 | 0.66 | 0.54 | 0.1735 |

D_{gr} | 0.939 | 0.857 | 0.08 | 0.845 | 0.7 | 0.129 | |

Λs | 0.94 | 087 | 0.077 | 0.852 | 0.73 | 0.12 | |

d_{50}/y | 0.918 | 0.83 | 0.095 | 0.83 | 0.69 | 0.137 | |

Rough bed | |||||||

LDb(IIV) | F_{rm} | 0.964 | 0.91 | 0.056 | 0.891 | 0.74 | 0.08 |

D_{gr} | 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 | |

d_{50}/y | 0.865 | 0.84 | 0.1056 | 0.812 | 0.728 | 0.097 | |

Separate dunes | |||||||

Db(IV) | F_{rm} | 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 | |

W/_{b}y_{0} | 0.945 | 0.925 | 0.046 | 0.943 | 0.89 | 0.06 | |

Continuous loose bed | |||||||

Db(IV) | F_{rm} | 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 | |

W/_{b}y_{0} | 0.972 | 0.952 | 0.0454 | 0.96 | 0.93 | 0.058 |

. | . | Performance criteria . | |||||
---|---|---|---|---|---|---|---|

. | . | Train . | Test . | ||||

SVM-best model . | Eliminated variable . | R . | DC . | RMSE . | R . | DC . | RMSE . |

Smooth bed | |||||||

LDb(IIV) | F_{rm} | 0.827 | 0.67 | 0.134 | 0.66 | 0.54 | 0.1735 |

D_{gr} | 0.939 | 0.857 | 0.08 | 0.845 | 0.7 | 0.129 | |

Λs | 0.94 | 087 | 0.077 | 0.852 | 0.73 | 0.12 | |

d_{50}/y | 0.918 | 0.83 | 0.095 | 0.83 | 0.69 | 0.137 | |

Rough bed | |||||||

LDb(IIV) | F_{rm} | 0.964 | 0.91 | 0.056 | 0.891 | 0.74 | 0.08 |

D_{gr} | 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 | |

d_{50}/y | 0.865 | 0.84 | 0.1056 | 0.812 | 0.728 | 0.097 | |

Separate dunes | |||||||

Db(IV) | F_{rm} | 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 | |

W/_{b}y_{0} | 0.945 | 0.925 | 0.046 | 0.943 | 0.89 | 0.06 | |

Continuous loose bed | |||||||

Db(IV) | F_{rm} | 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 | |

W/_{b}y_{0} | 0.972 | 0.952 | 0.0454 | 0.96 | 0.93 | 0.058 |

### Sediment rating curve

### Comparison of SVM and sediment transport classic equations

## 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 *F*_{rm} and d_{50}/*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*, *F*_{rm}*, D*_{gr}*,* and d_{50}*/y* presented better performance. Based on the obtained results from the sensitivity analysis, it was found that *F*_{rm} and d_{50}*/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 *y*_{s}*/D*, *F*_{rm}, and *W _{b}/y*

_{0}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,

*F*

_{rm}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.