New formulations for prediction of velocity at limit of deposition in storm sewers based on a stochastic technique

Dry weather seasons may lead to a decrease of flow rate and water level, causing deposition of solid materials in many storm sewer networks. If the deposits remain for a relatively long duration within the sewer system, sediment characteristics change, and the deposits may permanently become consolidated. This phenomenon not only influences the sewer hydraulic resistance, flow properties, and the capacity of sediment motion, it may also significantly reduce the flood conveyance capacity of the sewer system, and even in severe conditions it might lead to network deterioration (Ab. Ghani & Md. Azamathulla 2011; Ebtehaj et al. 2014).

Hydraulic designers usually take into account two main criteria in the design of sewer systems. These criteria are defined based on low discharge and high flow conditions: the sewer should convey the design floods, while the sediment deposition in the network should be minimized during the low flow seasons. Combined sewer systems, which carry rainfall and wastewater, are mostly designed based on frequency and intensity of rainfall and runoff to flush out the deposited sediment of the dry weather periods (Vongvisessomjai et al. 2010).

Further studies showed that the mechanism of sediment transport in sewers is a complex three-dimensional phenomenon and additional parameters than those assumed by Camp (1942) should be considered to obtain acceptable results. It is worth noting that the overall mechanism of sediment transport in sewer pipes is almost similar to what occurs in natural channels. Two main differences in the mechanism of sediment transport in alluvial channels and sewers are generally considered. First, the supply amount of sediments in a natural channel is practically unlimited and originates from the channel itself. In contrast, in sewers, the sediment supply rate is derived from the catchment of the sewer network. Second, the effective roughness of sewers may vary depending on whether the internal surface of sewer pipes is clean, or it is made up of deposits of sediment particles (Ab. Ghani 1993).

The available criteria were modified, and more reliable design methodologies were suggested based on non-deposition of sediment materials rather than scouring of the deposited sediment. To this end, several regression-based equations including more parameters have been derived during the past 40 years. Those parameters, as well as the proposed equations, will be described in the following section.

Owing to the complex nature of some hydraulic phenomena, researchers in recent decades utilized several sophisticated artificial intelligence (AI) models to achieve more accurate predictions than the conventional regression-based formulas (Sharafati et al. 2019a). For instance, an AI technique, named group method of data handling (GMDH), has been optimized to estimate longitudinal dispersion coefficient in rivers (Najafzadeh & Tafarojnoruz 2016). Application of other AI methodologies, like evolutionary polynomial regression (EPR), gene-expression programming (GEP), and model tree, have been widely evaluated by researchers in the field of hydraulic engineering to predict sediment transport, scouring around hydraulic structures, design of riprap stone size, etc. (Azamathulla & Zakaria 2011; Azamathulla 2012; Guven & Azamathulla 2012; Najafzadeh et al. 2013, 2014, 2017, 2018; Najafzadeh 2016; Kargar et al. 2019; Sharafati et al. 2019b).

According to Azamathulla & Ghani (2011), Azamathulla et al. (2012), Ebtehaj & Bonakdari (2013, 2017), and Najafzadeh & Bonakdari (2017), artificial neural networks (ANNs), adaptive neuro-fuzzy inference systems (ANFIS), GEP, and neuro-fuzzy GMDH were successfully utilized to predict velocity at limit of deposition. More specifically, some investigations devoted to applying EPR, support vector regression, and the firefly algorithm resulted in accurate sediment transport prediction in pipes (Ebtehaj & Bonakdari 2016; Montes et al. 2020). Authors of these studies reported significant improvement in the prediction of the selected parameter comparing with the conventional regression-based formulas. However, some of the selected AI techniques are not capable of introducing a precise predictive method for practical purposes, or in some cases the proposed equation has an extremely complicated mathematical formulation.

Apart from use of a regression-based equation or an AI technique, some countries defined a single minimum non-deposition velocity magnitude for design purposes. For example, in the USA, the minimum velocity within the foul sewers must be 0.60 m/s; however, a larger value of 0.90 m/s should be considered in the design of storm sewers. It is interesting to note that such values vary in other countries. Take the UK as another example; different values of 0.75 and 1 m/s were, respectively, adopted for storm and combined sewers. A survey of available studies clarifies that a dozen of such velocity limits have already defined by different authorities, yet it is obvious that taking into account a single minimum velocity looks inadequate and unsafe to design storm sewer pipelines of various roughness, diameter, gradient and sediment properties. These criteria may be only acceptable for the design of small sewer networks (Vongvisessomjai et al. 2010; Ebtehaj et al. 2014).

The mentioned methods and the available formulas in the design of non-deposition sediments have been developed according to the flow and sediment characteristics, while engineers usually assess a combination of operation, maintenance and construction in the design of a sewer system. For example, to reduce the construction cost, a limited sediment disposition may be allowed within the sewer pipelines. This less conservative design significantly diminishes the slope of sewer lines and, as a result, reduces the construction cost, even though it requires specific maintenance and operation.

All the methodologies, as mentioned earlier (i.e., regression-based equations, AI techniques, pre-defined non-deposition velocity values) have been developed assuming that the ‘sediment transport’ is a deterministic phenomenon. This hypothesis stands in contrast to the physical nature of the sediment transport process. Most available environmental models are developed to simulate the mechanism of a system response to some influencing factors which are not necessarily unique. In this case, a stochastic methodology may offer a more reliable relationship between the system inputs and the model output (Sadegh et al. 2018; Shojaeezadeh et al. 2018). The sediment transport process, whether in nature (e.g., rivers, coastal zones) or in manmade structures (e.g., pipelines, sewer networks) is inherently stochastic. Nevertheless, a little research has been already conducted to take into account the intrinsic property of bed material motion, which may considerably improve sediment transport prediction (Kleinhans & Van Rijn 2002; Dodaro et al. 2014, 2016; Sharafati et al. 2018).

To quantify the uncertainty of random variables, several techniques such as generalized likelihood uncertainty estimation (GLUE) and sequential uncertainty fitting (SUFI) have already been proposed (Beven & Binley 1992; Abbaspour et al. 2004). In the field of water resource engineering and sediment transport, GLUE has been proven as a useful tool to include the uncertainty in a wide range of predictions. In particular, in hydrological models, it may recognize the fundamental limitations in the rainfall-runoff process (Beven 1993; Freer et al. 1996). For example, the GLUE methodology was employed to analyze the uncertainty of a fully distributed physically based code in terms of streamflow prediction limits in France (Vázquez et al. 2009). It has also been applied to flood estimation by fitting the model and identifying the model parameters in river flow prediction in Vietnam (Cu et al. 2018). The GLUE technique does not assume only one optimal parameter set in deriving a model; this is one key issue and merit of this methodology (Duan et al. 1992; Beven 1993). With this advantage, Sharafati et al. (2018) improved the prediction accuracy of wave-induced scour depth around a pipeline, assuming the scouring process and the attributable sediment transport are stochastic phenomena.

The main objective of the present investigation is to revise the most common sediment transport mathematical formulation in sewer pipelines utilizing the stochastic technique GLUE. Two mathematical structures have been considered in the present study, which include the most effective dimensionless groups. The performance of the new stochastic formulas in the prediction of non-deposition velocity limit is evaluated by means of several statistical indices.

To develop and evaluate the new model, and compare its performance with those available in the literature, a large number of datasets, 257, were extracted from two experimental studies (Ab. Ghani 1993; Vongvisessomjai et al. 2010). To train the proposed stochastic model, 193 datasets (approximately 75% of the entire data) were selected randomly. In comparison, the 64 datasets (25% of the available data) were allocated for the testing phase and verifying the proposed model. Table 1 shows the range of each dimensionless parameter employed for training and testing stages while their frequency is presented in Figure 1.

is in range of 0 to 1: the zero value indicates high accuracy in output prediction.

To obtain a simple GLUE-based model, the values of from different two variate combinations are represented in Table 2.

The prior probability density functions (PDFs) of the random variables are obtained using the cross-validation method. Parameters of prior functions are presented in Table 3. To compute the random variables using the GLUE algorithm, the posterior probability density functions of the random variables are obtained by Equation (8) and an approach which is demonstrated in Figure 2.

This study aims to propose a novel formulation to provide a more accurate estimation of the densimetric Froude number at the limit of deposition in comparison with the available equations (Ab. Ghani 1993; Vongvisessomjai et al. 2010; Ebtehaj et al. 2014). Owing to high randomness of the densimetric Froude number, a stochastic approach (herein, the GLUE methodology) is utilized to tune the defined random variables. Ultimately, the accuracy of the proposed stochastic models are compared with the existing well-known formulas.

To obtain the GLUE-based models (i.e., GBFP and GBTP models) for simulating the densimetric Froude number, a number of parameters (i.e., ⁠, and ⁠) in Equations (9) and (15) are defined as random variables. These variables are estimated by means of the GLUE approach, where the uniform distributions are assigned as their prior distributions (Table 3). The posterior distributions of the random variables are obtained using the 20,000 simulations (Figure 4) through the GLUE approach (Figure 3).

Figure 3 indicates that the coefficient and exponent of in GBFP model (i.e., and ⁠) have negative skewness implying that the accuracy of Froude number estimation would be increased through the larger values of the parameter, while coefficients of the other variables ⁠, ⁠, and (i.e., ⁠, ⁠, ⁠, and ⁠, ⁠, ⁠) have relatively symmetric distributions. This implies that the prediction accuracy of the Froude number increases, taking into account the mean value of these parameters.

To assess the reliability of the proposed GLUE-based models, the corresponding 95 percent prediction uncertainty (95 PPU) band (i.e., the interval between the 97.5% and the 2.5% percentiles) are generated using the obtained 20,000 simulations in the training phase (Figure 4). The prediction reliability is measured using the index: the percentage of the bracketed experimental data by the 95 PPU band (Sharafati & Azamathulla 2018).

The value is an indicator to assess the stochastic models. The minimum acceptable range for the bracketed data by the 95 PPU band is 50% (Abbaspour et al. 2007). In accordance with the GBFP and GBTP model prediction results in the training phase (Figure 4), it is clear that significant reliability is achieved in the predictability of both models (⁠⁠). In fact, the GBFP and GBTP models are able to offer accurate densimetric Froude number estimations.

Another uncertainty index, ⁠, is related to the average thickness of the 95 PPU band and standard deviation of the observed data (Sharafati & Azamathulla 2018). Figure 4 clarifies that the uncertainty in predicted densimetric Froude number associated with the GBFP model (⁠⁠) is relatively more than the GBTP model (⁠⁠).

The scatter plots (Figure 5) illustrate the consistency between the observed densimetric Froude number and the predicted values by the proposed GLUE-based models as well as the previous investigations (Vongvisessomjai et al. 2010; Ebtehaj et al. 2014)). Figure 5 indicates that a significant agreement is observed between the measured and predicted Froude number values by means of all the predictive models in both the training (⁠⁠) and testing (⁠⁠) phases. This agreement is stronger in small Froude numbers (⁠⁠), whereas the points corresponding to the larger Froude numbers (⁠⁠) are noticeably more spread. Also, the GBTP points are located at the closest regions to the best-fit line, indicating the best agreement with the experimental data in training (⁠⁠) and testing (⁠⁠) phases. Furthermore, the least agreement with the measured data is observed using the Ab. Ghani (Ab. Ghani 1993) equation.

The predictive models were also comparatively evaluated by the Taylor diagram (Figure 6). In this diagram, several metrics, e.g., correlation coefficient, standard division and the root mean, are considered simultaneously (Taylor 2001). From Figure 6, it is clear that in both phases, GBTP (yellow point) has a slightly shorter distance to the observed point (cyan point) while the point corresponding to the Ab. Ghani (Ab. Ghani 1993) results (blue diamond) has a relatively longer distance from the observed point in comparison with the other models. Indeed, the Taylor diagram analysis confirms the highlighted results from the scatter plots.

Following the computed normalized ⁠, and indices, the heat maps over the training and testing phases (Figure 7) are obtained to provide a visual comparison between the performance of the predictive models.

From Figure 7, it is clear that all the cells relating to the GBTP model have dark blue color in both the training and testing stages, noting that the GBTP model offers the best performance indices. Furthermore, the Vongvisessomjai et al. (2010) model is also ranked as the second accurate model in terms of ⁠, and indices in training phase (Figure 7(a)) while in testing stage, the GBFP model performs better than the remaining models (i.e., Ab. Ghani 1993; Vongvisessomjai et al. 2010; Ebtehaj et al. 2014) in terms of all the indices, except the index (Figure 7(b)). In fact, in agreement with the scatter plots and the Taylor diagrams, heat maps also indicate that a significant prediction superiority is expected in the use of the GBTP model in comparison with the other predictive models.

The boxplots, depicting the percentiles (i.e., ⁠, and ⁠) and the interquartile ranges (⁠⁠) of the observed and predicted Froude number values, are exhibited in Figure 8 to quantify their variability in training and testing phases.

Figure 8 indicates that the closest values between the predicted and observed median (⁠⁠) of the Froude number in the training phase are obtained from the Ebtehaj et al. (2014) and GBTP models, while a better performance to capture the observed IQR = 2.81 is evidenced in the GBTP model with ⁠. Relatively similar results are also obtained in the testing phase, noting that the GLUE-based models offered the best overall prediction performance. Indeed, that is a normal result where the input and output parameters have stochastic nature.

To quantify the performance improvement by the proposed models, the index (Equations (21) and (22)) is assessed in training and testing phases (Table 4). To this end, the Ab. Ghani (1993) model is considered as the benchmark model. Hence, a positive value of means the model has better performance than the Ab. Ghani (1993) model, and vice versa. Table 4 demonstrates that the highest improvement in all the performance indices (1.60–19.19%) as well as the index (9.2%) are obtained by GBTP over the training phase. Similar superiority is also achieved in the testing phase, where the performance indices (2.16–24.65%) and the index (11.3%) are improved through the GBTP.

From Figure 9, it is clear that the highest values of the are found respectively, in the R_ℎ/d₅₀ (RCV = 0.55), and C_vL (RCV = 0.29). It means that the non-deposition velocity limit is most affected by those variables. This finding is consistent with those obtained by Mayerle et al. (1991).

Although the formulas provided by the GLUE approach for predicting the non-deposition velocity limit are simple as the empirical equations, those formulas preserve the stochastic nature of the deposition phenomenon. Hence, the primary advantage of the GLUE-based approach is simple modeling with acceptable accuracy. At the same time, the AI models comprise a complicated structure with many unknown parameters, and, thus, accurate tuning of those parameters is a crucial issue. Furthermore, it is easy to identify the attributes of GLUE modeling while it is a difficult task in AI modeling due to its close-box nature.

Clean sewer design is a key issue for engineers. Deposition and consolidation of solid materials in sewer pipes may reduce the system efficiency or even block a part of the network. The velocity at limit of sediment deposition is embedded in the densimetric Froude number, a function of dimensionless grain size, friction factor in the presence of sediment motion, sediment concentration, and the ratio of hydraulic radius over the median sediment size. Fundamental analysis is made through the GT to identify the most effective parameters. Results reveal that among the four mentioned dimensionless parameters, sediment concentration at limit of deposition and the ratio of hydraulic radius over median sediment size are the most effective parameters on the prediction of velocity at limit of deposition. The internal part of many sewer pipes is not rough enough to affect sediment transport. Moreover, the other less effective parameter, i.e., dimensionless grain size, particularly expresses the effect of sediment size on the densimetric Froude number. Noting that median sediment size is also included in the ratio of hydraulic radius over median sediment size, one may conclude that the dimensionless grain size is a redundant parameter in the prediction of velocity at limit of deposition.

To evaluate the results of the GT, two separate equations are derived based on the GLUE methodology: (1) GBFP (Equation (28)) and (2) GBTP (Equation (29)). These two formulas consist of all the four influencing parameters and the selected two variables, respectively. Further analysis by means of technique assured that both models are able to offer reliable predictions. In contrast, the analysis reveals that the prediction uncertainty by the GBTP model (Equation (29)) is considerably less than the GBFP model. Other analyses by several error indices, e.g., SI, LMI, NSE, support the superiority of the GBTP (Equation (29)) with respect to the four-parameter GLUE-based formula (i.e., GBFP), as well as the conventional regression-based equations in this context.

From the methodology used in this study, a new equation was derived for the estimation of velocity at the limit of deposition for sewer design purposes. The suggested equation is more accurate than the available regression-based formulas in the literature. Thus, one may also employ this technique in some other predictions if the phenomena are not fully deterministic. For example, local scouring is an interaction result of two stochastic processes (i.e., transient vortices and sediment motion) and use of a stochastic method may enhance the prediction accuracy and reliability.

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