Introduction

Local scour around single pier is a phenomenon that was studied by many researcher and hydraulic engineers in different hydraulic laboratories and research centers. But few studies have been done about local scour around group piers.

Many parameters can affect on local scour around group piers, which include the approaching flow characteristics (steady or unsteady flow conditions, flow velocity, and flow depth), the geometry of the pier (width, shape, and orientation to the flow direction), bed sediment particles (size, density, sediment load), and fluid properties (density and viscosity).

A number of performed researches about local scour around group piers include Ataie-Ashtiani and Beheshti (2006), Zounemat-Kermani et al. (2009), Amini et al. (2012, 2014), Baghbadorani et al. (2018), Hamidi and Siadatmousavi (2018), Hosseini and Amini (2015), Liao et al. (2018), Moussa (2020), Wang et al. (2016), and Zhang et al. (2017).

Also, researchers applied different artificial intelligence methods for determination of local scour depth around piers or piles. Kaya (2010) used MLP-ANN for determination of local scour depth around bridge pier, and sensitivity analyses showed that pier shape and skew, flow depth, and velocity were the most effective factors on local scour. Ebtehaj et al. (2018) developed an artificial intelligence method based on extreme learning machines (ELM) and utilized it for calculation of local scour depth around pile groups. Results stated that accuracy of this method is better than those of ANN and SVM. Azimi et al. (2017) prepared a hybrid multi-objective method for determination of local scour depth around pile groups in clear water condition. They introduced the singular value decomposition (SVD) and the differential evolution (DE) algorithms to ANFIS and considered two contradictory objective functions, training error (TE) and prediction error (PE), in Pareto curve. Results of ANFIS-DE/SVD are more accurate than those of GA and empirical equations. Sensitivity analysis showed that the most effective parameters are the pile diameter and number of piles normal to the flow. Najafzadeh (2015) applied two methods for determination of local scour depth around pile groups in clear water condition. They were neuro-fuzzy-based group method of data handling (NF-GMDH) that used particle swarm optimization (PSO) and gravitational search algorithm (GSA). NF-GMDH-PSO was more accurate than NF-GMDH-GSA, and sensitivity analysis showed that the most effective parameter was pier diameter. Chou and Pham (2017) applied a novel hybrid smart artificial firefly colony algorithm (SAFCA)–based support vector regression (SAFCAS) model for determination of local scour around bridge piers. They observed that results of this model are more accurate than those of cross-fold validation algorithms and hypothesis testing. Najafzadeh et al. (2013) developed the group method of data handling (GMDH) for calculation of local scour around piles. This method used LM training method. They showed that results of this method are more accurate than those of ANFIS and RBF-ANN. Ghodsi et al. (2018) used harmony search (HS) algorithm for calculation of local scour around complex bridge piers at clear water conditions. They stated that results of this method were better than those of empirical equations. Bateni et al. (2019) used genetic expression programming (GEP) and multivariate adaptive regression splines (MARS) for calculation of local scour around pile groups at clear water conditions. Using of dimensional data improved results of two models and results of MARS method are better than those of GEP method. Sensitivity analysis showed that the most effective parameter is pile diameter. Ebtehaj et al. (2016) used self-adaptive evolutionary extreme learning machine (SAELM) for determination of local scour around bridge cylindrical and square piers. Sensitivity analysis showed that the most effective parameters are pier dimensions and bed-sediment size. Results of this method were better than those of ANN and SVM. Vaghefi et al. (2019a) used the box plot, histograms, linear regression, k-nearest neighbors, local outlier factor, k-medoid clustering, multilayer perceptron, and self-organizing map for detection of outliers in flow pattern experiments. For comparison of the results of these data mining method, they considered a spur dike in a 90° bend. They showed that the box plot and local outlier factor methods are the best methods for this purpose. Vaghefi et al. (2019b) utilized feed-forward ANN for determination of velocity in a 180° sharp bend. They illustrated that spur dike increases turbulence of flow and decreases accuracy of ANN method. Ahmadianfar et al. (2019) applied the locally weighted linear regression (LWLR), supported vector regression (SVR), and multivariate linear regression (MLR) for estimation of the local scour depth around piles due to waves. They observed that LWLR is more accurate than other methods.

This research uses four methods for calculation of local scour around group piers. The MLP-ANN, RBF-ANN, ANFIS, and SVM method will be applied for this purpose, and these methods will be optimized by GA. These networks utilize momentum and LM training methods. These four artificial intelligence methods have different natures. ANFIS is a combination of ANN and fuzzy logic principles and uses linear and nonlinear functions, while SVM method is a non-probabilistic binary linear method that is used for classification of data and regression analysis. MLP is a feed-forward ANN and uses back propagation technique for training, and it is a nonlinear method, while RBF uses radial basis functions and its output is a linear combination of these functions. Therefore, four different methods were considered in this research (linear (RBF), nonlinear (MLP), binary linear (SVM), and probabilistic (ANFIS) methods).

The necessary dataset for this study was prepared using of a constructed physical model in the Water Research Institute of Iran (WRI). The shapes of group piers were cylindrical and square piers. The piers were perpendicular to the flow direction. The piers are behind each other. This study will distinguish the best method for determination of local scour depth around group piers using of performance criteria such as R, MSE, and MAE; also, sensitivity analysis will show the most effective parameter for calculation of local scour depth around group piers.

Advantages and differences of this study relation to other studies are as follows:

  1. 1

    At the most of studies, piers are parallel to each other and are at a line perpendicular to the flow direction.

  2. 2

    Evaluation of using of different transfer functions and fuzzy models and selection the best of them for calculation of local scour around group piers.

  3. 3

    Preparation of observed data in a large flume. This matter eliminates effects of walls, expansion, and contraction of cross section of flume on flow characteristics.

  4. 4

    Dividing flume to four sections (in front of the first pier, between the first and second pier, between the second and third piers, and behind the third pier). The local scour depth in each section is different. For example, the maximum value of local scour depth is observed in front of the first pier, while sedimentation can occur between piers and behind the third pier. This matter can affect on selection the best artificial intelligence method.

Materials and methods

Experimental tests

Experimental tests were performed at the Water Research Institute (WRI) of Iran. Water was supplied by four pumps from underground reservoir. A basket filled with rocks is located at the upstream section of the flume to create more uniform flow within the flume. The cross section of the main flume is trapezoidal with 1.45 m width, 1 m deep and bank slope of 1.5H: 1.0V. Flow discharge is control by an inlet butter fly valve and measured by a V-notch weir. A system of rail was installed to carry the required apparatus such as point gage for water surface and bed level measurements and flow velocity meter (Valeport 801 made in England). The plan, section, and photo of constructed experimental flume are illustrated in Fig. 1.

Fig. 1
figure 1

The plan, section, and photo of constructed experimental flume

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The median size of uniform sands (\( {\sigma}_g=\sqrt{\frac{d_{84}}{d_{16}}} \) < 1.3) is 1.31 mm. Diameters of cylindrical piers are 6, 8, and 10 cm, and widths of square piers are 6 and 8 cm, and flow discharges are 50, 65, and 80 lit/s with flow depths 9.5, 11, and 12.5 cm. The values of U and Uc are 0.351 and 0.448 m/s (for Q = 50 lit/s), 0.394 and 0.459 m/s (for Q = 65 lit/s), and 0.426 and 0.469 m/s (for Q = 80 lit/s). Uc is dependent to critical shear velocity and grain size of sand particles (it is not a function of flow velocity). Also, Uc is dependent to logarithm of flow depth and depth flow changes have not important effect on changes of Uc (logarithm function mitigates effects of flow depth changes).

$$ {U}_c\ \left(\mathrm{m}/\mathrm{s}\right)=5.75\ u\ast {}_c\ \log \left(5.53h/0.1{d}_{50}\right) $$
(1)
$$ u\ast {}_c=0.0115+0.0125\ {\left({d}_{50}\right)}^{1.4}\ \mathrm{if}\ {d}_{50}<1\ \mathrm{mm} $$
(2)
$$ u\ast {}_c=0.0305\ {\left({d}_{50}\right)}^{0.5}-0.0065/{d}_{50}\ \mathrm{if}\ {d}_{50}>1\ \mathrm{mm} $$
(3)

where, d50 is median size of sand particles (mm), h is flow depth (cm), and u*c is critical shear velocity (m/s).

The relation between U and Uc is shown in Fig. 2.

Fig. 2
figure 2

The relation between U and Uc

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Group piers (three piers) in series are illustrated in Fig. 3 schematically.

Fig. 3
figure 3

Group piers

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Before the start of each test, the channel bed was filled with sediment (25 cm thick) and its bed surface was leveled and piers were located at the desired location in the centerline of the flume. For finding of equilibrium time, measurement of scour depths was performed in different times (10, 15, 20, 30, 60, 120, and 360 min after starting of test). Observations illustrated that equilibrium time is equal to 360 min, and measured scour depths in this time were used for training and testing of different artificial intelligence methods.

Artificial neural networks

Artificial neural networks (ANNs) are composed of simple elements, which operate parallel and are a simplified model of the biological nervous systems. A neural network is known by the pattern of connection between nodes (architecture), the transfer function, and connection weights determining method, training method (Fausett (1994)). In this study, two ANN types, MLP and RBF, were applied. The LM and momentum training methods were used. Furthermore, the different transfer functions were selected for hidden layers and an output layer. Trial-and-error process was utilized to determine the optimal number of neurons and hidden layers. Also, GA method was applied for optimization of parameters of MLP and RBF.

Adaptive neuro-fuzzy inference system

Zadeh (1965) developed fuzzy theory in University of California, Berkeley. Fuzzy set is a theory for operating in unreliability conditions. This theory is able to give a mathematical form to many of the concepts, variables, and systems, which are inexact and indeterminate, and makes possible to reason, conclude, control and decide in unreliability conditions.

The ANFIS was developed by Jang (1993). The neuro-adaptive learning is similar to neural networks, is a feed-forward multi-layer network, and provides a method to learn information about a data set in the fuzzy modeling procedure. The basic structure of ANFIS is based on the Sugeno fuzzy inference system (Takagi and Sugeno 1985). The Sugeno fuzzy inference system (FIS) method is a more computationally efficient representation in comparison to Mamdani system (Drake 2000). The parameters associated with the membership functions are determined in the training phase of a neural network (Cobaner et al. 2009). ANFIS employs two learning algorithms for updating membership function (MF) parameters: (1) back propagation algorithm and (2) a hybrid algorithm, which is the combination of gradient descent method and least-squares.

Support vector machines

Support vector machine (SVM) has been used in information categorization and lately in regression problems. Mathematically, support vector machine is placed in classification and regression algorithms range, which is formulated using the principals of statistical learning theory by Vapnik (1995). The main relationship for statistical learning process is as follows:

$$ y=f(x)=\left\{\sum \limits_{i=1}^N{w}_iK\left({X}_i,X\right)\right\}-b $$
(4)

where K is the Kernel function, wi and b are the parameters of the model, N is the total number of learning patterns, and Xi is the data vector for network learning and X is an independent vector. Support vector machines use some of the specific Kernel functions that convert the input vector as the input data from nonlinear function in this model. Selection of an appropriate Kernel function is a complex stage and often is used from standard kernel function.

There are four standard conversion of Kernel function mostly used in regression and modeling including the following:

  1. (a)

    Linear kernel. The simplest Kernel function (Han et al. 2007):

$$ K\left(x,z\right)=\left(x,z\right) $$
(5)
  1. b

    Polynomial kernel. Polynomial mapping is a common method for nonlinear modeling:

$$ \mathrm{K}\left(x,z\right)={\left(x,z\right)}^{\mathrm{d}},\mathrm{K}\left(x,z\right)={\left(\left(x,z\right)+1\right)}^{\mathrm{d}} $$
(6)
  1. c

    Radial basis function (RBF) kernel. A function based on RBF more similar to Gaussian (bell shape) (Han et al. 2007):

$$ K\left(x,z\right)=\exp \left(-\alpha {\left|x-z\right|}^2\right) $$
(7)
  1. d

    Sigmoid kernel. The Sigmoid kernel function:

$$ K\left(x,z\right)=\tan \mathrm{h}\left(-\alpha {x}^Tz+c\right) $$
(8)

Performance criteria

For evaluation of performance of models, below criteria are applied:

  1. (a)

    Correlation coefficient (R):

$$ R=\frac{\sum \limits_{i=1}^N\left({x}_i-\overline{x}\right)\left({y}_i-\overline{y}\right)}{\sqrt{\sum \limits_{i=1}^N{\left({x}_i-\overline{x}\right)}^2{\left({y}_i-\overline{y}\right)}^2}} $$
(9)
  1. b

    Mean square error (MSE):

$$ \mathrm{MSE}=\frac{\sum \limits_{i=1}^N{\left({x}_i-{y}_i\right)}^2}{N} $$
(10)
  1. c

    Mean absolute error (MAE):

$$ \mathrm{MAE}=\frac{\sum \limits_{i=1}^N\left|{x}_i-{y}_i\right|}{N} $$
(11)

where xi and yi are the observed and estimated values, respectively; \( \overline{x} \) and \( \overline{y} \) \( \overline{\mathrm{Y}} \)are the averages of xi and yi; and N is the number of data. For good fitness between observed and estimated values, MSE and MAE should be close to zero and R should be close to one.

Genetic algorithm method

The characteristics of applied GA in this research are as follows:

Objective function = Minimize MSE, Rate of crossover = 0.8, Type of mutation = Uniform, Type of crossover = Heuristic, Selection method = Stochastic universal sampling, Number of generations = 3000, Population of each generation = 120

Applied GA in this research used of variable mutation rate for different generations. Mutation rates for different generations are

  • Mutation rate = 0.3 if (no of generation < 700)

  • Mutation rate = (− 0.295/1300) × (no of generation-700) + 0.3 if (700 < no of generation < 2000)

  • Mutation rate = 0.005 if (no of generation > 2000)

Results and discussion

In this research flume divided to four sections around the group piers (Fig. 4). Also, this figure illustrates the vertical profile of velocity in sections 1 to 4.

Fig. 4
figure 4

Considered sections around group piers and the vertical profile of velocity in different sections

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The inputs of ANNs were coordinates of considered points in the flume (X, Y), size of pier (d), flow depth (h), flow discharge (Q), flow velocity (U), and critical velocity for inception of sediment transport (Uc). Output of ANNs is local scour depth (Z); 70% of data is applied for training stage and 30% for testing stage.

  1. (a)

    Characteristics of the best MLP-GA for calculation of local scour depth around the group piers:

The characteristics of the best MLP-GA network for different sections of flume are shown in Table 1.

Table 1 The characteristics of the best MLP-GA network
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For training of MLP-GA network, momentum and LM training methods were used. The performance criteria showed that accuracy of trained network by LM is more than those of trained network by momentum method. For section S1, sigmoid transfer function and one hidden layer with 10 nodes, R, MSE, and MAE are 0.991, 9.22 mm2, and 2.23 mm, respectively, in training stage for LM training method and 0.975, 26.45 mm2, and 3.94 mm, respectively, in training stage for momentum training method.

Also, using of GA method improved the performance criteria. For section S1, sigmoid transfer function, one hidden layer with 10 nodes, and LM training method, R, MSE, and MAE are 0.991, 9.22 mm2, and 2.23 mm, respectively, in training stage for optimized ANN by GA and 0.973, 27.84 mm2, and 4.03 mm, respectively, in training stage for ordinary ANN.

  1. b

    Characteristics of the best RBF-GA for calculation of local scour depth around the group piers:

The characteristics of the best RBF-GA network for different sections of flume are shown in Table 2.

Table 2 The characteristics of the best RBF-GA network
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For selection of the best RBF-GA network, different competitive rules, metrics, and values of cluster center were evaluated. Evaluation of different competitive rules, metrics and values of cluster center for different sections illustrates that conscience Full, Euclidean and 35 are the best competitive rules, metrics, and values of cluster center in this study, respectively.

  1. c

    Characteristics of the best ANFIS for calculation of local scour depth around the group piers:

The characteristics of the best ANFIS network for different sections of flume are shown in Table 3.

Table 3 The characteristics of the best ANFIS network
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For selection of the best ANFIS, different membership functions, fuzzy models, transfer functions, training methods, and number of MFs are tested. For example, the optimum membership function and fuzzy model for section S1 with sigmoid transfer function, momentum training method, and two MFs are Bell membership function and Takagi–Sugeno–Kang (TSK) fuzzy model.

  1. d

    Characteristics of the best SVM for calculation of local scour depth around the group piers:

The characteristics of the best SVM network for different sections of flume are shown in Table 4.

Table 4 The characteristics of the best SVM network
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Step size of applied SVM is 0.00002 in this research. This value was selected using of trial and error method, and it has the least value of MSE and MAE and the most value of R. For section S1 and step size 0.00001, R, MSE, and MAE are 0.94, 420.12 mm2, and 12.85 mm, respectively, in training stage and 0.81, 454.33 mm2, and 14.25 mm, respectively, for testing stage. Also, the used kernel is RBF kernel in this study.

Based on results of application of four methods, the best method for S1 is MLP-GA method and for other sections is RBF-GA method. These artificial intelligence methods have two hidden layer, and their training method is LM training method.

  1. e

    Comparison between the obtained results from four artificial intelligence methods

Tables 1, 2, 3, and 4 illustrate that the ANFIS and SVM methods are not appropriate methods for calculation of local scour depth around group piers. R of these methods is low, and MSE and MAE of these methods are high. Based on these tables, the MLP-GA and RBF-GA are appropriate methods for calculation of local scour depth around group piers. These methods are optimized by GA method, and optimization increases R and reduces MSE and MAE. Also, these methods normalized inputs and reduce effects of outlier data. The outlier data can enlarge MSE very much. MSE is the mean of square error. The SVM and ANFIS cannot mitigate effects of the outlier data, and MSE of these methods is very high. Also, in MLP-GA and RBF-GA methods, the values of MSE are relative high for section S2. In this section, complexity of flow is more than other sections. Extreme erosion and sedimentation can occur in this section and the maximum difference between values of erosion depth and sedimentation depth can be observed in this section.

Among eight sections, the maximum value of local scour depth is related to S1 section and MLP-GA method is suitable method for this section, while RBF-GA method is appropriate method for other sections. The best architecture of RBF-GA has one hidden layer for calculation of scour depth around group piers in section S1, while for other sections RBF-GA has two hidden layers. Therefore, the results of MLP-GA method are better than those of RBF-GA method in this section.

Table 5 illustrates the range and average of observed measurements and calculated values of local scour depth around group piers by four methods.

Table 5 The range and average of observed measurements and calculated values of local scour depth around group piers in mm (d = 6 cm and Q = 50 l/s)
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  1. f

    Sensitivity analysis: Inputs of networks are five parameters (d, h, Q, U, and Uc). Sensitivity analysis showed that the most effective parameter on local scour depth around group piers is Uc. Results of sensitivity analysis are shown in Table 6.

    Table 6 Results of sensitivity analysis based on the best selected artificial intelligence method in each section
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For sensitivity analysis in this study, each parameter doubled and other parameters remained constant. The effect of parameter doubling on the local scour depth around group piers distinguishes the importance of this parameter. Sensitivity analysis shows the ratio of variations of the local scour depth around group piers. For example, if ratio of variations is equal to 3, the local scour depth around group piers will triple. Positive sign showed increasing of scour depth, and negative sign showed reduction of scour depth. Increasing of Uc reduces scour depth; also, it is observed that Uc will decrease by increasing of flow discharge and velocity in flume.

Fig. 5 illustrates that calculated local scour depth around group piers using of the best artificial intelligence method has good fitness to the observed local scour depth around group piers in flume.

Fig. 5
figure 5

Comparison between the calculated and observed local scour depth around group piers (d = 8 cm and Q = 80 lit/s): a circular group piers and b square group piers

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Conclusion

This study showed that MLP-GA and RBF-GA can simulate local scour depth around group piers with acceptable accuracy, while accuracy of ANFIS and SVM is not appropriate for this purpose. Also, these methods can show sedimentation and erosion around group piers very exact. Optimization of MLP-ANN and RBF-ANN methods by GA increased R and reduced MSE and MAE. Therefore, it is observed that the R of these methods is more than 0.93 and MSE and MAE of these methods are less than 61.68 mm2 and 4.63 mm. However, the minimum values of R are 0.511 and 0.348 for ANFIS and SVM methods. The maximum values of MSE and MAE of ANFIS method are 433.31 mm2 and 13.9 mm. The maximum values of MSE and MAE of SVM method are 444.75 mm2 and 34.02 mm.

Sensitivity analysis illustrates which dependent parameters to sediment characteristics such as Uc are most effective parameters on local scour depth around group piers. Based on the observed measurements in experimental model and results of artificial intelligence methods, increasing flow discharge and velocity and decreasing of Uc increase local scour depth around group piers. The sensitivity analysis confirms this matter. The second effective parameter is flow velocity, and influence of flow discharge on local scour depth is negligible. In circular group piers, effect of pier size on local scour depth is more than those of square group piers. Uc is dependent to critical shear velocity, and increasing of Uc is resulting from increasing of critical shear velocity and grain size of sand particles. When shear velocity become more than critical shear velocity, erosion will occur. The shear velocity and grain size of sand particles are the most effect factors on local scour depth and sensitivity analysis shows this subject. Among parameters of flow, flow velocity is the most effect parameter on local scour depth. The oscillation of flow velocity can cause sedimentation or erosion. While flow discharge variations are dependent to flow depth changes (its dependency to flow velocity changes is less), flow depth is not an important effective factor on local scour depth. Therefore, flow discharge is not an effective parameter on local scour depth.

Fig. 5 illustrates that maximum sedimentation occurs between middle and gear piers and maximum erosion occurs in front of the first pier.