Data-driven-based determination of influential parameters on local energy loss of slope-tapered culvert

Culverts provide an efficient solution for flowing water through man-made cross barriers (e.g., roads). There is a wide variety of solutions of the inlet part of the culverts, among which, slope-tapered inlet is the focus of this paper. Hydraulic characteristic and sizing of culverts are complex problems for several reasons (Khalil & Zein 1995; Graziano et al. 2001a, 2001b; Jones et al. 2006; Mohammed 2016; Aly 2017; Wilhere et al. 2017; Jaeger et al. 2019a, 2019b, 2019c; Rao & Raju 2019; Zeng et al. 2019; Nakhostin et al. 2020). The optimal culvert could be considered the one in which the head loss is minimal and has the ability to transfer the design flow rate. Culverts are among the drainage structures in which proper and desirable performance is essential. Therefore, the failure of culvert structures is a catastrophic phenomenon. As a result, a detailed study of the causes of failure is recommended (Rigby et al. 2002; French & Jones 2018; Günal et al. 2019). One of the most recent comprehensive guidelines is Hydraulic Design Series Number 5 (HDS-5) (Norman et al. 2001). Jaeger et al. (2019a, 2019b, 2019c) presented findings regarding how different inlet designs increase the discharge rate. The flume flow experiments showed that a slope-tapered inlet can have a greater capacity than a conventional culvert with a square edge. This type of inlet is designed to increase the culvert performance by providing a depression and a more efficient control section at the throat (Graziano 2001a, 2001b; Khaled 2004). In order to prevent flooding, estimation of water levels with reasonable accuracy to determine free-board and top elevation of culverts is essential. Therefore, several studies have tested a variety of culvert configurations in laboratory models and other research has investigated loss coefficient, aquatic organism passage, and culvert blockage (Apelt 1983; Graziano et al. 2001a, 2001b). It was found that culvert local losses accrue for a variety of reasons, such as outlet loss, inlet loss, or bend loss (Tullis et al. 2005; Malone & Parr 2008; Tullis 2012). Previous studies on the determination of the energy dissipation at culvert outlets has focused on simplified methods to estimate the head loss coefficient. As a general guideline, the head loss is considered to be comparable to the velocity head within the conduit (Simmons 1964; Liu & Zhu 2000; Tullis & Robinson 2008; Schall 2012; Habibzadeh & Rajaratnam 2016). Also, the evolution of the effective parameters on energy losses of buried-invert culverts via kernel-based approaches was carried out by Roushangar et al. (2019). Examining the bend loss of culverts showed that the parameters affecting the energy loss were the Froude number and bend angle whereas the parameters affecting the inlet loss were Froude number and depth ratio.

The structural choice of a culvert and corresponding inlet is based on environmental considerations, risk to property, cost of construction and maintenance and also esthetic considerations. Many U.S. agencies and researchers require that culverts be designed and engineered to meet specific federal, state, or local regulations and guidelines to ensure proper function and to protect against culvert failures (Smith 1957; Chen 1970; Simons & Stevens 1972; Rigby et al. 2002; Environmental Protection Agency EPA Management 2003; Alberta Transportation 2004; Federal Highway Administration Department of Interior Bureau of Land Management 2006; Günal et al. 2019). Defects in the conveying of water in culverts can lead to overflows and structural damage, therefore, to mitigate these risks, the capacity of many culverts needs to be increased. It is clear that there are many tools for enhancing the discharge efficiency of culverts (Harrison et al. 1972; Normann 1975). However, many researchers believe that the best way to increase culvert capacity is to improve inlet function. Relying on this belief, the use of inclined headwalls or misaligned culverts has been recommended (Aly 2017; Jaeger 2019a, 2019b, 2019c). Others researchers believed that utilizing improved inlets, such as slope-tapered ones, enhances the flow transient (McGrath & Heger 1983; Graziano et al. 2001a, 2001b). It is worth noting that energy loss is among one of the reasons affecting culvert capacity (Kotowski et al. 2011). Furthermore, head loss falls into two categories, general and local losses, which in culverts due to their short length the local loss is outstanding. Experimental study of bend loss in rectangular culverts has been developed by Malone & Parr (2008). From another viewpoint, there are other studies that have focused on outlet loss of culverts and have examined various types of culvert systems, such as elliptical, circular, and rectangular (Liu & Zhu 2000; Larson 2004; Robinson 2005; Habibzadeh & Rajaratnam 2016). On the other hand, several experimental studies on inlet loss have stated that the inlet loss coefficient is significantly affected by the inlet configuration and usually has a constant value (Graziano et al. 2001a, 2001b; Jones et al. 2006; Tullis 2012). However, in contrast, taking advantage of intelligence methods has shown that the local loss coefficient is a dependent parameter, which in addition to geometric parameters, is influenced by hydraulic parameters (Roushangar et al. 2019).

Many studies have considered the entrance local energy loss coefficient of the slope-tapered culvert to be constant at 0.2 (Norman et al. 2001). However, some researchers have considered this coefficient as a function of geometric parameters and flow characteristics (Graziano et al. 2001a, 2001b; Roushangar et al. 2019). Energy losses are a complex process in which the existing regression models do not show favorable accuracy and the results are associated with large error. Therefore, the present study proposed artificial intelligence approaches to predict the entrance loss coefficient and evaluate the best input variables with the most impact on inlet loss coefficient of slope-tapered culverts. In the past decades, the application of artificial intelligence approaches such as artificial neural networks (ANNs), neuro-fuzzy models (NF), genetic programming (GP), gene expression programming (GEP), support vector machine (SVM), and Gaussian process regression (GPR) have become popular in water resources engineering, leading to numerous publications in this field (Liriano & Day 2001; Hazi & Ghani 2011; Hazi & Haque 2012; Najafzadeh 2016; Huang et al. 2017; Carvalho et al. 2018; Amaranto et al. 2018; Tayyebi et al. 2018; Owen & Liuzzo 2019; Poursorkhabi & Ghasempour 2019; Roushangar et al. 2019; Tinelli & Juran 2019; Zhu et al. 2019; Roushangar & Shahnazi 2020). Detailed review of previous studies (Graziano et al. 2001a, 2001b; Tullis 2012) revealed that the local loss coefficient of slope-tapered culverts was monitored experimentally and, to the best of the authors’ knowledge, there is a lack of comprehensive research on estimating the entrance loss coefficient using artificial intelligence. As a result, this study aims to evaluate the efficiency of GPR and GEP for predicting the entrance loss coefficient of slope-tapered culverts. Also, a new equation to predict the local loss coefficient was provided utilizing the GEP method and the most important input variables were identified using OAT sensitivity analysis.

In order to investigate the parameters affecting the local loss coefficient in the culverts with a slope-tapered inlet, the laboratory dataset of Graziano et al. (2001a, 2001b) was used. All of the laboratory testing was done at the Federal Highway Administration's Turner-Fairbank Highway Research Center located in McLean, Virginia, USA and inlet control design constants and entrance loss coefficients were calculated for the circular slope-tapered culverts with various reducer lengths. The experimental setup was constructed primarily of plywood and consisted of a 2.43 m long by 2.43 m wide headbox, and a 1.21 m wide by 2.43 m long tailbox, which was located 4.5 m downstream of the headbox. The slope-tapered inlet and culvert barrel spanned the 4.5 m between the headbox and the tailbox. A sketch of the slope-tapered culvert is shown in Figure 1. Fourteen pressure ports were inserted along the bottom of the culvert setup to measure hydraulic depth. The total entrance loss (H_etot) was measured for outlet control tests by projecting the energy grade line (EGL) in the headbox and the EGL for the culvert barrel to a common plane at the upstream end of the culvert barrel.

Providing an appropriate dataset is a critical step in the prediction of local loss coefficient via artificial intelligence methods. Examination of models showed that considering 75% of the dataset for training goals and the remaining 25% for testing goals leads to more accurate results. Owing to more accurate estimate of out-of-sample accuracy and more efficient use of data, as every observation is used for both training and testing, and in order to avoid model bias, the train and test dataset was divided using v-fold cross validation which was developed in STATISTICA software (v. 8). The separated dataset is then utilized in further model developments. V-fold cross validation is a standard approach for model selection and the main idea of cross validation is data splitting (Arlot & Lerasle 2016).

Kernel-based approaches (such as GPR) are one of the common methods for solving the nonlinear problems which are based on statistical learning theory, and the appropriate selection of kernel type in these approaches is the most important step due to its direct impact on the training and classification precision. Availability of sufficient input data enables these models to predict any variable. They are also fairly robust against overfitting, especially in high-dimensional space (Roushangar et al. 2019).

To assess the hyper-parameters, the partial derivation of Equation (4) can be obtained with respect to σ² and k. For more detailed discussion of GP regression, see the study of Kuss (2006). The optimal value of capacity constant (c) and the size of error-intensive zone (ɛ) in GPR are required due to their high impact on the accuracy of the mentioned regression approaches. The optimum values of these parameters were obtained after the trial-and-error process. In order to develop the GPR approach, a code written in MATLAB software was executed.

Gene expression programming was developed by Ferreira (2001) using fundamental principles of the genetic algorithms (GA) and GP. The strength of the proposed approach includes the simplicity of creating genetic diversity, and a unique and multi-genic nature which allows the evaluation of more complex programs composed of several subprograms. GEP as GA mimics the biological evolution to create a computer program for simulating a specified phenomenon. A GEP algorithm begins by selecting five elements, including the function set, terminal set, fitness function, control parameters, and stopping condition. There is a comparison between predicted values and actual values in each subsequent step. When the desired results are obtained in accordance with previously selected error criteria, the GEP process is terminated. After the desired fitness score is achieved, the process terminates and then the chromosomes are decoded for the best solution of the problem. The most important advantages of GEP are as follows (Ferreira 2001): (1) the chromosomes are simple entities and (2) the expression trees are exclusively the expression of their respective chromosomes. It should be noted that the software used to develop this approach is GeneXproTools 4.0 Release 2.

The design of GP-based regression approach involves use of the concept of kernel functions. In order to select the best kernel function, model M (I) was predicted using various kernels. According to the statistical parameters, using the kernel function of squared-exponential led to more accurate prediction. Figure 2 indicates the results of the statistical parameters of different kernels for model M (I).

Furthermore, GEP was trained for local loss coefficient prediction with basic arithmetic operators of (+, − ,*, /) and several mathematical functions (exp, x², x3,⁠) as the function set. Different combinations of chromosomes’ structure presented in Table 2 were tested. Then the model was run for a number of generations and stopped when there was no significant change in the fitness function value and coefficient of correlation. Accordingly, the models with 30 chromosomes head size of 8 and 3 numbers of genes led to the better results. In addition, one of the important steps in preparing the GEP model is choosing the appropriate set of genetic operators, therefore, a combination of all genetic operators was also tested. Characteristics of optimized GEP models are shown in Table 2.

In order to evaluate the impact of various parameters on inlet loss coefficient of slope-tapered culverts, several models were developed based on flow characteristics (Froude number) and geometric parameters (reducer's length and depth ratio). For predicting the inlet loss coefficient, all models were GPR and GEP models which were trained and tested. The obtained results are listed in Table 3 as well as Figures 2 and 3. From the R, DC, and RMSE viewpoints (highest R and DC and lowest RMSE), it can be concluded that among the seven models, the model with input parameters of Fr, H_w/D, and L_r/L leads to more accurate results (0.847, 0.777, 0.2) compared to others, and therefore, it was identified as the best model. However, it can still be seen that models M (II) and M (III) produce an acceptable result with R, DC, and RMSE equal to 0.827, 0.747, 0.224 and 0.815, 0.69, 0.24, respectively.

The results obtained from GPR model developments are illustrated in Table 3 and Figures 5 and 6. The results were similar to the GEP models, in which the superior model was M (I) with performance criteria (R, DC, RMSE) equal to 0.76, 0.718, and 0.25. By comparing R, DC, and RMSE in GEP and GPR models of this case, it seems that the accuracy of GEP models was slightly higher.

The impact of the different employed parameters on inlet loss coefficient of a slope-tapered culvert is evaluated using sensitivity analysis. There are different approaches of sensitivity analysis, including local or global, quantitative or qualitative, or one-at-a-time (OAT). One of the simplest and most common approaches is that of OAT, to determine what effects this produces on the output. OAT sensitivity analysis essentially consists of selecting an initial parameter setting (nominal set) and varying one parameter at a time while keeping all the other parameters fixed. Consequently, OAT reveals a form of the relationship between the varied parameter and the output, given that all other parameters have their nominal values (Holvoet et al. 2005).

The current study utilized the OAT approach to evaluate the impact of each parameter. To do so, the models were run with all input parameters, then, one of the parameters was eliminated and the process was repeated. Based on the results from Figure 7, it could be deduced that in predicting the inlet loss, with eliminating L_r/L, H_w/D, and Froude, the statistical parameters of R and DC decreased to 0.82, 0.74, 0.815, 0.69, and 0.43, 0.45, and the amount of RMSE error increased to 0.227, 0.24, and 0.36, respectively. Consequently, variable Fr, would be the most significant impact in the current study and model M (I) with input parameters of Fr, L_r/L, H_w/D would be the superior model. Also, investigating the developed models revealed that variables H_w/D and L_r/L had scant impact on local loss coefficient and the impact of H_w/D was slightly more than L_r/L.

Inlet edge configuration is one of the prime factors influencing the performance of a culvert. As the inlet configuration is improved, the flow contraction is reduced, thus, a slope-tapered inlet improves culvert performance by providing a more efficient control section. The defections due to the flow of water in culvert systems can cause problems such as overflow of water on the road. The optimum design of culverts depends on many factors, including but not limited to, the entrance loss coefficient, which is the focus in this paper. Since many previous studies have pointed out the dependency of local loss coefficient on geometric and hydraulic parameters, it seems that having sufficient knowledge of parameters affecting the entrance loss coefficient could be useful in the accurate calculation of inlet energy loss of culvert systems and the upcoming optimum design of culverts. In the present study, the capability of artificial intelligence methods including GEP and GPR in the estimation of entrance local energy loss coefficient of a slope-tapered culvert was assessed. For this purpose, laboratory data of Graziano et al. (2001a, 2001b) of a slope-tapered culvert were used. In order to perform a detailed investigation on entrance loss coefficient, seven models with various input parameters were selected.

According to the results of the GEP approach, it was found that model M (I) with input variables of Fr, H_w/D, and L_r/L leads to better results, and it was identified as the superior model (R = 0.847, DC = 0.777, and RMSE = 0.2). Unacceptable results of model M (IV) showed that adding Froude numbers increases the efficiency, significantly. Also, by comparing model M (V) with model M (I), it was observed that parameters H_w/D and L_r/L had a slight contribution in increasing the accuracy of the model. Moreover, an explicit expression of the relationship of variables was developed using the GEP method.

The results obtained using the GPR method also provided similar results, in which model M (I) with performance criteria of 0.76, 0.718, and 0.254 was the best model in estimating the entrance loss coefficient of the slope-tapered culvert. Also, comparison of evaluation criteria (R, DC, RMSE) of the GEP and GPR methods illustrated a relatively high accuracy of the GEP method in predicting the entrance loss.

From the obtained results of OAT sensitivity analysis, it was found that the correlation coefficient between K and Fr (in the state of entrance loss of a slope-tapered culvert) was significantly higher than other parameters. Therefore, Froude number had the most impact on entrance loss coefficient. By investigating M (II) to M (IV) models, it can be deduced that eliminating input parameters of L_r/L, H_w/D, and Fr increases the amount of RMSE error up to 0.227, 0.24, and 0.36, respectively. Also, scrutinizing models M (VI) and M (VII) revealed that, in the absence of Froude number, input variables of H_w/D and L_r/L are the most affecting parameters, respectively.

It was concluded that the proposed approaches were found to be able to predict the entrance loss coefficient of the slope-tapered culvert successfully. This issue confirms that in the presence of insufficient information, the inlet loss coefficient of the slope-tapered culvert could be estimated via these approaches. However, it is worth noting that the selected models are data-driven ones, hence, in order to discover the merits of the models to estimate the entrance loss coefficient in real conditions of flow, further studies using data ranges beyond the scope of this study are essential.

All relevant data are included in the paper or its Supplementary Information.