# Estimation of Scour Propagation Rates around Pipelines While Considering Simultaneous Effects of Waves and Currents Conditions

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## Abstract

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## 1. Introduction

## 2. Overview of Databases

#### 2.1. Dimensional Analysis

_{H}, V

_{L}, and V

_{R}are, respectively, the non-dimensional scour propagation rate along the pipeline, at left-hand shoulder of the pipeline, and at right-hand shoulder of the pipeline, U

_{*}is the shear velocity of the seabed, e is the embedment depth, D is the pipeline diameter, U

_{W}is the maximum undisturbed orbital rate at the sea bottom just above the boundary of the waves, U

_{C}is the flow velocity due to current, d

_{50}is the median sediment size, T is the wave period, Φ is the repose angle for bed sediments, α is the flow incident angle to the pipeline, ρ is the mass density of water, ρ

_{s}is the mass density of bed material, μ is the dynamic viscosity of water, and g is the acceleration due to the gravity.

_{W}are selected as repeating variables, and therefore dimensionless parameters (π

_{1}, π

_{2}, π

_{3},…, π

_{12}, π

_{1}

_{3}, π

_{14}) were generated by the Buckingham theorem as follows:

_{5}, π

_{10}, and π

_{11}are the Keulegan–Carpenter (KC) number and the Reynolds number of the pipeline due to regular wave (Re

_{W}) and current (Re

_{C}), respectively. On the basis of Cheng et al.’s [1] research, many π-variables in Equation (2) need to be combined and categorized as follows:

_{H}*, V

_{R}*, and V

_{L}*, respectively. Additionally, ${\pi}_{11}^{\prime}$ is the Shields’ parameter for both wave and current conditions, which have a key role to play in the scour propagation below pipelines. Therefore, the functional relationship (2) is reduced to

_{H}*, V

_{L}*, and V

_{R}* are, respectively, the non-dimensional scour propagation rate along the pipeline, at the left-hand shoulder of the pipeline, and at the right-hand shoulder of the pipeline, KC is the Keulegan−Carpenter number, θ

_{C}is the Shields’ parameter due to steady current, θ

_{W}is the wave Shields’ parameter, Re

_{W}is the wave-induced Reynolds number, and Re

_{C}is the current-induced Reynolds number, which is calculated by the following function respectively; also, m is the velocity ratio [U

_{C}/(U

_{C}+ U

_{W})].

_{s}/ρ] and τ

_{w}is the wave-induced shear stress on the seabed, which is expressed as

_{w}is the wave-induced shear stress on the seabed and was proposed by Soulsby [18]. In particular,

#### 2.2. Experimental Case

_{s}/ρ] were 0.37 mm and 2.7, respectively. The tests were conducted under live-bed conditions. Additionally, the non-dimensional parameters are taken into account as constant parameters such as: Φ = 32°, α = 0°, ρ = 1000 Kg/m

^{3}, ρ

_{s}= 2650 Kg/m

^{3}, and μ = 0.001 Pa.s. Cheng et al. [1] reported that the maximum Shields’ parameter for the motion of the bed sediments was evaluated based on the Soulsby [18] approach. The only formula suggested by Cheng et al. [1] was employed to estimate the three-dimensional equilibrium scour propagation rate induced by combined waves/currents is expressed as

_{WC}is a constant and F, that was proposed by Sumer and Fredsøe (2002), is given by the following empirical equations:

## 3. Strategy on Selection of Effective Parameters

_{H}*

_{,}V

_{R}*

_{,}V

_{L}*). Hence, six non-dimensional parameters (KC, θ

_{C}, θ

_{W}, e/D, sin α, and m) were initially considered as input parameters to develop the AI models. As an alternative, according to Equation (15) given by Cheng et al. [1], regression terms of F and parameters of a and b are merged into the list of input variables. In addition to this, two typical parameters of Reynolds number (Re

_{W}and Re

_{C}) were excluded from the list of effective parameters because values of Reynolds number stood at the turbulent states, demonstrating insignificant influence on the scour rate propagation during the Cheng et al. [1] experiments. Hence, Equation (3) is re-written as,

_{W}, was thoroughly distributed, starting from θ

_{W}= 0.05 with four frequencies to ending θ

_{W}= 0.245 with seven frequencies. In Figure 2e, the Shields’ parameter was not symmetrically distributed due to the currents (θ

_{C}). For instance, three levels of θ

_{C}(i.e., 0.0233, 0.0656, 0.1079) have the same value of frequency (10). Figure 2 from (f) to (h) indicates that the patterns of dimensionless scour rate distributions are not symmetrical in a way that with an increase in VH* (or VR* and VL*) and the frequency generally had a downward trend. Consequently, it can be inferred from the fragmentation of histograms that five dimensionless parameters (as mentioned in Equation (17)) played a significant role in the scouring experiments.

## 4. Implementation of Soft Computing Models

#### 4.1. Gene-Expression Programming

_{H}*, V

_{R}*, and V

_{L}*, respectively. This study presented the results of the best GEP performances for three-dimensional scour rates. Table 4 gives characterizations of the GEP performance for the 3D scour rate propagation.

#### 4.2. Multivariate Adaptive Regression Splines

_{0}, χ

_{j}, BF, and NBF are the bias, the constant coefficients related to the basis functions, the basis functions, and the number of basis functions, respectively. The forward pass occasionally creates an overfitted model. After that, the backward pass is carried out to prune the primarily-calibrated MARS model. This methodology eliminates terms one by one, having insignificant effective terms at each stage until it yields the most promising sub-model. The performance of model subsets is assessed by the generalized cross-validation (GCV) criterion that has frequently been applied in the different fields of machine learning techniques [22,23].

_{H}* prediction was related to Set 2 with a GCV value of 1.498, whereas the input combination of Set 6 resulted in the most promising predictions for both V

_{L}* (GVC = 1.139) and V

_{R}* (GVC = 2.297). To further extend, Table 5 demonstrates the best performance of MARS models for seven combinations of dimensionless parameters in the calibration and validation for V

_{H}* (0.7705 and 0.7513), V

_{R}* (0.7325 and 0.5906), and V

_{L}* (0.8439 and 0.8949). The total effective number of parameters for all the best MARS models was 11. Consequently, the following relationships are the best MARS models for the prediction of the 3D scour propagation rates below pipelines:

_{W}+ mθ

_{C}) and four dimensionless parameters (1 − e/D, (1 − m)θ

_{W}, mθ

_{C}, KC) were incorporated into the prediction of V

_{H}* and V

_{L}*, respectively; whereas KC parameter has no contribution to the estimation of V

_{R}*. All the coefficients in Equations (24)–(26) were obtained using the Particle Swarm Optimization (PSO) algorithm, providing MSE = 0.636, 0.336, and 1.028 as the most promising results.

#### 4.3. M5 Model Tree

_{H}* for Set 1:

_{0}is the bias term and a

_{1}to a

_{5}are weighting coefficients that are computed by the least square technique.

_{H}*

_{,}V

_{R}*, and V

_{L}*, respectively. The most promising results of M5MT performances demonstrated that using Set 5 (e/D, F, m, θ

_{W}, θ

_{C}) provided V

_{H}* and V

_{R}* predictions with the most accurate results, whereas the most precise predictions of V

_{L}* values were simulated by Set 6. As seen in Table 8, three linear equations along with three rules were used to provide V

_{H}* predictions for Set 5. The performance of M5MT indicates that the F parameter has no role to play in the prediction of scouring rates in the longitudinal direction. Similarly, Table 9 indicated that the F parameter was not incorporated into the estimation of V

_{R}* and consequently, two rules were used to provide a linear regression equation along with the splitting parameter of θ

_{C}(0.087). In Table 10, equations extracted by M5MT demonstrated that when m is combined by two types of Shields’ parameters, the KC parameter has no role in the estimation of V

_{L}*.

#### 4.4. Evolutionary Polynomial Regression

_{j}is a set of coefficients, Q is a user-defined function, and ES function is a range of exponents explored by the EPR model. According to the most recent research works on scouring around pipelines, applying natural logarithmic as an inner function indicated the best performance in the evaluation of the scouring propagation rates due to waves and currents [15,16]. During the development of the EPR model, the number of generations was obtained in three levels: 6480 (Set 2 and Set 7), 8640 (Set 4, Set 3, and Set 6), and 10,800 (Set 2 and Set 7) for various combinations of dimensionless parameters.

_{H}*

_{,}V

_{R}*, and V

_{L}*, respectively. Moreover, Table 11 applying the fifth combination of inputs (e/D, F, m, θ

_{W}, θ

_{C}) yielded the best results of EPR models for V

_{R}* and V

_{L}* predictions whereas the most accurate predictions of V

_{H}* were obtained by the sixth combination (1 − e/D,KC,(1 − m)θ

_{W},mθ

_{C}). The optimum mathematical expressions given by EPR runs were given as:

_{H}* and V

_{L}*, respectively, whereas velocity ratio [m = U

_{C}/(U

_{C}+ U

_{W})]) has no role to play in the estimation of V

_{R}* parameter (as seen in Equation (30)).

## 5. Results and Discussion

#### 5.1. Definition of Statistical Indices

_{H,R,L}*

_{(Exp)}, V

_{H,R,L}*

_{(Est)}, $\overline{V}$

_{H,R,L}*, and N are the experimental values of non-dimensional scour propagation rate, the estimated values of non-dimensional scour propagation rate, the average values of non-dimensional scour propagation rate, and the number of scouring tests, respectively. In the case of correlation values, when CC values vary between +1 and −1, this means that the soft computing model indicates the best performance. In contrast, if CC is equal to 0, the worst performance is met. RMSE, MAPE, and SI values are known as error functions, varying from 0 to +∞. If the DR value is equal to 1, the soft computing technique indicates the most efficient performance. Where DR is greater than one, it shows over predictions, otherwise, if DR > 0, underprediction is met.

#### 5.2. Statistical Performance of Soft Computing Techniques

_{H}*. It can be inferred from statistical measures that the EPR model [Equation (29)] indicated the highest accuracy (CC = 0.879, RMSE = 0.595, and MAPE = 0.224) in the calibration phase when compared to other soft computing techniques. According to Table 12, GEP model (CC = 0.779, RMSE = 0.801, and MAPE = 0.348) stood at the second rank in terms of accuracy, followed by MARS (CC = 0.770, MAPE = 0.398, and DR = 1.167) and MTM5 (CC = 0.765, MAPE = 0.474, and DR = 1.286). Over/under predictions of the soft computing models were quantified by DR criterion. In this way, Equation (29) obtained by the EPR model has the lowest level of overprediction (DR = 1.046) whereas predictions of V

_{H}* were indicative of the highest level with DR = 1.286. Results of quantitative comparisons from both validation and calibration phases indicated that the EPR model had no sufficient potential of estimating V

_{H}*. In Table 12, values of three statistical criteria (RMSE, MAPE, and SI) showed that MARS could be chosen as the best soft computing model in the validation phase (RMSE = 0.929, MAPE = 0.498, and SI = 0.516). GEP model provided relatively accurate predictions (RMSE = 1.239, MAPE = 0.565, and DR = 1.403) than those obtained by M5MT (RMSE = 1.543, MAPE = 0.692, and DR = 1.661). In addition, the statistical criteria CC, RMSE, and DR indicated that EPR (CC = 0.884, RMSE = 1.509, and DR = 0.927) had a better potential for predicting V

_{H}* in comparison with M5MT. Overall, EPR (DR = 0.927) and MARS (DR = 0.897) had underprediction whereas M5MT (DR = 0.1661) and GEP (DR = 1.403) techniques indicated overprediction.

_{H}* in calibration and validation stages is depicted in Figure 3. In Figure 3a, all the soft computing techniques overpredicted the observed values of V

_{H}* between 0.5 and 1.5 for the calibration stage. For V

_{H}* = 5.89, the MARS model had significant underprediction in comparison with EPR and GEP models. In the validation stage, Figure 3b illustrated overprediction for observed values of V

_{H}* between 1 and 2.5. Additionally, for V

_{H}* = 5.89, the MARS model indicated remarkable underprediction when compared to other soft computing models.

_{R}*. More specifically, the accuracy metrics of the EPR model [Equation (30)] were CC = 0.746, RMSE = 1.003, and MAPE = 0.308, which were relatively close to those of the M5MT model (CC = 0.745, RMSE = 1.006, and MAPE = 0.312). According to Table 13, SI and DR proved marginal improvement of EPR (SI = 0.393 and DR = 1.120) over M5MT (SI = 0.394 and DR = 1.163). Additionally, Table 13 indicated that the MARS model stood at the third stage on the account of accuracy with CC, RMSE, and SI values of 0.732, 1.041, and 0.408, respectively. Overall, the GEP model resulted in the lowest value of correlation (0.664) and the highest values of error functions (RMSE = 1.142 and SI = 0.447). In the validation stage, GEP model demonstrated slightly better performance (CC = 0.742, MAPE = 0.315, and SI = 0.223) in the estimation of V

_{R}* than MARS (CC = 0.590, MAPE = 0.384, and SI = 0.446), and EPR (CC = 0.665, MAPE = 0.764, and SI = 0.878) did. With reference to all statistical measures except CC value, M5MT provided V

_{R}* predictions with comparatively lower accuracy in terms of SSE (0.1322) and MAE (0.0944) than those reported by the GEP model. As seen in Figure 4a, all the predicted values of V

_{R}* in the calibration stage were almost placed in ±25% error lines. Figure 4a depicted significant overprediction for the observed V

_{R}* of 1.321. In Figure 4b, the qualitative performance of the validation stage indicated relatively successful efficiency because the majority of data points were placed inside ±25% error lines. More specifically, Figure 4b indicated that EPR, MARS, and M5MT provided overprediction in the state of observed V

_{R}* varying from 1.334 to 3.277.

_{R}* for the validation stage. All the statistical measures obtained by the performance of MARS (CC = 0.895, RMSE = 0.440, and MAPE = 0.248) in the validation phases indicated its superiority over other soft computing techniques. In addition to this, SI and DR criteria proved the most efficient performance of the MARS model. In contrast, all the statistical measures except CC value showed that EPR had the lowest performance (RMSE = 2.203, MAPE = 0.991, and MAPE = 0.991). Moreover, the results of the validation stage demonstrated that the GEP model had a better assessment in the estimation of V

_{L}* compared with M5MT (RMSE = 1.197, MAPE = 0.679, and SI = 0.991). From DR values, it can be said that the MARS model had a lower overprediction of V

_{L}* (DR = 1.139) and followed by GEP (DR = 1.231), M5MT (DR = 1.666), and EPR (1.991). To make deeply qualitative comparisons in the calibration stage, Figure 5a indicated that all the soft computing techniques had remarkable underprediction for the observed value V

_{L}* = 0.427–1.321. As seen in Figure 5a, M5MT had no required capability of prediction in the V

_{L}* = 2.565 in comparison with other soft computing models. Additionally, for V

_{L}* = 0.984–1.515, the GEP model had overpredictions in the less level (see Figure 5b) than other soft computing models.

#### 5.3. Complexity of Soft Computing Techniques Performance

_{H}*, the EPR model was efficiently calibrated using Set 6 and consequently, this model provided complex expressions compared to M5MT and MARS [Equation (24)]. In fact, the sixth combination of dimensionless inputs (see Table 1) contains input parameters (KC, m, θ

_{W}), which are representative of wave impacts on the scour propagation rates. On the contrary, the best linear regression by M5MT (see Table 8) does not consist of the KC parameter, and additionally, linear equations by M5MT are simpler than expressions by EPR and GEP models. Equation (24) obtained by the MARS model had the most accurate assessment in comparison with using other performances by input combination. This means that the number of input parameters related to Set 4 (1 − e/D, F,(1 − m)θ

_{W}+ mθ

_{C}) is small and three parameters (θ

_{W}, m, θ

_{C}) were combined and converted into one dimensionless parameter.

_{R}* prediction, the EPR model provided a non-linear expression [Equation (30)] excluding velocity ratio (m), which generally had simpler mathematical structures than that developed by the GEP model [Equation (21)]. In the GEP models, all the sixth input parameters (see Table 1) were employed to obtain an expression with three genes; the GEP model demonstrated the lowest performance in the calibration stage than other soft computing models. In contrast, the MARS model [Equation (26)] with four inputs and more complex expressions (second polynomial as seen in Table 6) stood at a higher level of accuracy in the calibration stage than M5MT.

_{L}* prediction, MARS [Equation (25)] and EPR [Equation (31)] models have algebraic mathematics with lower complexity as well as superiority over the GEP performance in the validation stage. In the present research, the MARS model excluded the KC parameter even though experimental investigations carried out by Cheng et al. [1] proved that KC played a key role in the formation of the scour propagation around pipelines exposed to waves. As seen in Table 6, the MARS model created three linear splines (for instance, BF

_{1}= max(0, mθ

_{C}− 0.03044)) and one polynomial expression consisting of two splines.

## 6. Effects of Velocity Ratio on the Scour Propagation Rates

_{H}* values versus the velocity ratio m at four levels of e/D values varying from 0.2 to 0.5. In Figure 6a, with the exception of M5MT, all soft computing models provided the same pattern for e/D = 0.1. For instance, the EPR model indicated a downward trend, plummeting from V

_{H}* = 3.844 in m = 0.15 to V

_{H}* = 1.331 in m = 0.62, then, increasing to V

_{H}* = 4.531 in m = 0.79. Similar to what is inferred from Figure 6a, this trend has been seen in Figure 6b–d. Generally, for all levels of e/D, M5MT demonstrated that variation of V

_{H}* against velocity ratio remained slightly steady. From Figs.6a–d, it can be said that scour propagation rate along the pipeline increased for all values of e/D, thereby, m ratio increased over 0.5. Additionally, for m = 0.5–0.6, all the soft computing models except M5MT hit minimum values of V

_{H}* for all values of e/D.

_{R}* against m. As seen in Figure 7a, M5MT and EPR models significantly overpredicted for e/D = 0.2 and m = 0.45. In Figure 7b, the EPR model was under-predicted for two values of m (0.6 and 0.63) and e/D = 0.3, whereas the performance of soft computing models indicated well agreement with experimental observation for m = 0.15 and 0.20. Figure 7c illustrates all the soft computing models provided V

_{R}* values with significant overprediction for m = 0.6, whereas moderate overprediction can be seen for m = 0.79 and e/D = 0.4. Moreover, Figure 7c indicated a permissible harmony between soft computing models and experimental observation at m = 0.34 and 0.46. As depicted in Figure 7d, EPR and M5MT had remarkable underprediction in m = 0.63 and additionally, MARS indicated significant underprediction in m = 0.2. In m = 0.54, all the soft computing techniques had permissible performance in the prediction of V

_{R}*. Table 15 presented RMSE values of predictive models’ efficiency at various levels of e/D. EPR indicated the worst performance at three levels of e/D (RMSE = 1.366, 1.721, and 1.326 for e/D = 0.2, 0.3, and 0.4, respectively) than MARS, GEP, and M5MT models. Furthermore, M5MT resulted in the lowest accuracy level (RMSE = 0.577) in e/D = 0.5 than EPR (RMSE = 0.439), GEP (RMSE = 0.439), and MARS (RMSE = 0.495).

_{L}* values against m for all levels of e/D. For instance, in e/D = 0.2, Figure 8a indicated that V

_{L}* values first have a downward trend, decreasing from 2.863 in m = 0.15 to 0.810 in m = 0.6, then, increasing to 3.158 in m = 0.79. In Figure 8a, the EPR model significantly overpredicted for two values of m (0.63 and 0.79). M5MT had overprediction of V

_{L}* in m = 0.79, whereas the MARS and GEP models performed well in agreement with experimental observations. Additionally, Figure 8a illustrated all the soft computing models had a promising performance for m = 0.15. Figure 8b demonstrated that EPR had significant overpredictions of V

_{L}* in m = 0.66 whereas all the soft computing models had moderate overestimation in m = 0.6. Figure 8c depicted that M5MT had remarkable overprediction in m = 0.51 and 0.54 whereas, for m = 0.2, all the predictive models underpredicted values of V

_{L}*. More specifically, experimental observations simulated by M5MT have no certain patterns against m variations. In Figure 8c, MARS, GEP, and EPR models significantly overpredicted in m = 0.6 and 0.63. Figure 8d indicated that all the soft computing techniques had underprediction in m = 0.63. In m = 0.45, EPR and MARS models overpredicted V

_{L}* values, whereas M5MT and GEP models performed well and were in good agreement with experimental observations. According to the statistical results of Table 15, EPR model showed the best performance at e/D levels of 0.2 (RMSE = 0.437), 0.3 (RMSE = 0.671), 0.4 (RMSE = 0.623), and 0.5 (RMSE = 0.318).

## 7. Conclusions

_{C}, θ

_{W}, e/D, KC, and m. From these parameters, seven sets of non-dimensional parameters were provided to have a reasonable assessment of the scour propagation rates for the currents and waves at the same time. Overall, the following findings can be expressed as:

- -
- The developments of new equations by seven combinations of dimensionless parameters showed that the present predictive techniques raised three chief merits: (i) providing mathematical expressions by EPR and GEP with complicated terms (as naturally found in the scour propagation around offshore pipelines) when fed by a limited number of scouring tests, (ii) on the contrary, equations by M5MT and MARS models reduced the complexity of the evaluation of scouring process by providing simpler regression equations, and (iii) selecting optimal combination of the effective dimensionless parameters (from θ
_{C}, θ_{W}, e/D, KC, and m) played a key role in the prediction of the scour propagation rates; - -
- From the calibration and validation phases, the performance of AI models indicated a reasonable degree of efficiency for the estimation of the scour propagation rates. More specifically, statistical measures demonstrated that Equations (20) and (21) given by the GEP model had the best performance in the estimation of V
_{R}* and V_{H}*, respectively, whereas the MARS model [Equation (25)] indicated the most accurate efficiency for the evaluation of V_{L}*. On the other hand, the sixth combination of effective parameters (i.e., m θ_{C},(1 − m)θ_{W}, 1 − e/D, KC) provided the best results for the scour propagation rates in the right and left hands of pipelines (V_{R}* and V_{L}*) when the GEP and MARS models fed, respectively. In the case of V_{H}*, the MARS model fed by the fifth combination of dimensionless parameters (i.e., F, θ_{C}, θ_{W},m,e/D) demonstrated the best performance. Generally, it was inferred that the performance of AI models stood at the highest level of accuracy when θ_{C}, θ_{W}, and m were not converted to one dimensionless parameter [mθ_{C}+ (1 − m)θ_{W}]. - -
- The physical consistency of the predictive models’ results has been studied by analyzing scour propagation rates versus the ratio of pipeline embedment depth and pipeline diameter (e/D), the ratio of current velocity to orbital velocity (m), and Keulegan–Carpenter number (KC). From variations of the scour propagation patterns versus m, it was found that scour propagation rates followed a decreasing trend for m = 0.2–0.6 and all ranges of e/D and KC, then; increased up to m = 0.8.
- -
- The effects of complexity level on the performance of AI models for 3D scour propagation rates was investigated. The EPR model was developed by Set 6 for the V
_{H}* prediction with complex expressions in comparison with M5MT and Equation (24) (MARS). Multivariate linear equations by M5MT were simpler than those obtained expressions by EPR and GEP models. Additionally, the EPR model provided V_{R}* prediction with simpler mathematical structures than that developed by the GEP model [Equation (21)]. The sixth combination of input parameters has been applied to acquire a mathematical expression-based GEP model with three genes, indicating the lowest level of efficiency in the calibration stage in comparison with AI techniques. On the contrary, the MARS model with four dimensionless inputs (i.e., m θ_{C},(1 − m)θ_{W}, 1 − e/D, KC) and the second-order polynomial had better performance in the calibration phase than linear equations by M5MT. Furthermore, explicit equations given by MARS and EPR models have lower complexity for the V_{L}* estimation in comparison with the GEP model.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Illustrative sketch of 3D scour hole pattern with free span development at an underwater pipeline under orbital and current velocity profiles.

**Figure 2.**Distribution patterns for the effective parameters regarded in this research: (

**a**) ratio of the pipeline embedded depth to pipeline diameter, e/D; (

**b**) Keulegan−Carpenter number, KC; (

**c**) ratio of the maximum undisturbed orbital rate at the sea bottom just above the boundary of the waves to the flow velocity due to currents, m; (

**d**) Shields’ parameter due to waves, θ

_{W}; (

**e**) Shields’ parameter due to currents, θ

_{C}; (

**f**) dimensionless scouring propagation rate around offshore pipeline at the longitudinal direction, V

_{H}*; (

**g**) dimensionless scouring propagation rate around offshore pipeline at the right-hand shoulder of the pipeline, V

_{R}*; (

**h**) dimensionless scouring propagation rate around offshore pipeline at the left-hand shoulder of the pipeline, V

_{L}*.

**Figure 3.**Performance of soft computing models in the prediction of V

_{H}* for (

**a**) calibration and (

**b**) validation stages.

**Figure 4.**Performance of soft computing models in the prediction of V

_{R}* for (

**a**) calibration and (

**b**) validation stages.

**Figure 5.**Performance of soft computing models in the prediction of V

_{L}* for (

**a**) calibration and (

**b**) validation stages.

**Figure 6.**Comparisons of the soft computing models’ performance for variation of V

_{H}* versus m in for various levels of e/D: (

**a**) e/D = 0.2, (

**b**) e/D = 0.3, (

**c**) e/D = 0.4, and (

**d**) e/D = 0.5.

**Figure 7.**Comparisons of the soft computing models’ performance for variation of V

_{R}* versus m in for various levels of e/D: (

**a**) e/D = 0.2, (

**b**) e/D = 0.3, (

**c**) e/D = 0.4, and (

**d**) e/D = 0.5.

**Figure 8.**Comparisons of the soft computing models’ performance for variation of V

_{L}* versus m in for various levels of e/D: (

**a**) e/D = 0.2, (

**b**) e/D = 0.3, (

**c**) e/D = 0.4, and (

**d**) e/D = 0.5.

Set No. | List of Input Variables |
---|---|

Set 1 | e/D,KC,m,θ_{W},θ_{C} |

Set 2 | 1 − e/D,KC, (1 − m)θ_{W} + mθ_{C} |

Set 3 | 1 − e/D,F, θ_{W}, θ_{C} |

Set 4 | 1 − e/D, F, (1 − m)θ_{W} + mθ_{C} |

Set 5 | e/D, F,m,θ_{W},θ_{C} |

Set 6 | 1 − e/D,KC, (1 − m)θ_{W},mθ_{C} |

Set 7 | 1 − e/D,KC, [(1 − m)θ_{W} + mθ_{C}]^{5/3} |

Parameters | Data Range | Average | Standard Deviation | Coefficient of Variation |
---|---|---|---|---|

T (s) | 1–2 | 1.536 | 0.419 | 0.273 |

U_{W} (m/s) | 0.11–0.34 | 0.261 | 0.074 | 0.284 |

E (cm) | 0.5–2.5 | 1.57 | 0.586 | 0.3717 |

u* (m/s) | 0.0036–0.0282 | 0.0171 | 0.00769 | 0.449 |

U_{C} (m/s) | 0.06–0.47 | 0.291 | 0.130 | 0.445 |

V_{H} (mm/s) | 0.33–4.55 | 1.669 | 0.893 | 0.535 |

m | 0.15–0.79 | 0.505 | 0.187 | 0.371 |

θ_{W} | 0.0278–0.1315 | 0.08057 | 0.03094 | 0.361 |

θ_{C} | 0.00048–0.0297 | 0.01356 | 0.00908 | 0.669 |

KC | 2.2–13.6 | 8.314 | 3.723 | 0.448 |

e/D | 0.2–0.5 | 0.319 | 0.102 | 0.319 |

V_{H}* | 0.26–3.61 | 1.326 | 0.709 | 0.535 |

V_{R}* | 0.609–7.9023 | 2.411 | 1.371 | 1.634 |

V_{L}* | 0.233–4.59 | 1.952 | 1.036 | 0.6023 |

**Table 3.**Correlations of GEP performance in the calibration and validation stage for 3D scour rates.

Set No. | VH* | VR* | VL* | |||
---|---|---|---|---|---|---|

Calibration | Validation | Calibration | Validation | Calibration | Validation | |

1 | 0.5921 | 0.748 | 0.6482 | 0.8345 | 0.6115 | 0.4135 |

2 | 0.6739 | 0.5056 | 0.5175 | 0.5768 | 0.5692 | 0.6290 |

3 | 0.7481 | 0.6118 | 0.6160 | 0.7913 | 0.5090 | 0.6619 |

4 | 0.6151 | 0.8933 | 0.59359 | 0.8438 | 0.6587 | 0.7778 |

5 | 0.7796 | 0.7029 | 0.6755 | 0.6251 | 0.7823 | 0.7071 |

6 | 0.7185 | 0.8477 | 0.6645 | 0.7425 | 0.73617 | 0.6746 |

7 | 0.4607 | 0.4867 | 0.46055 | 0.6911 | 0.7552 | 0.7833 |

**Table 4.**Setting parameters of GEP performance in the calibration phases for 3D scour rates prediction.

Parameters | Description of Parameters | Setting of Parameters |
---|---|---|

P_{1} | Function set | +, −, ×, /, Power (x^{2}), (1 − x), 1/x, Average (x_{1},x_{2}), Atan (x), 3Rt, Ln, Min |

P_{2} | Linking function | Addition |

P_{3} | Mutation rate | 0.00138 |

P_{4} | Inversion rate | 0.00546 |

P_{5} | One-point and two-point recombination rates | 0.00277 |

P_{6} | Gene recombination rate | 0.00277 |

P_{7} | Permutation | 0.00546 |

P_{8} | Maximum tree depth | VH* (6), VR* (5), VL* (4) |

P_{9} | Number of gene | 3 |

P_{10} | Number of chromosomes | 30 |

P_{11} | Number of generation | VH* (1364), VR* (861), VL* (3916) |

P_{12} | Best fitness value | VH* (557.26), VR* (466.89), VL* (578.77) |

**Table 5.**Correlations of MARS performance in the calibration and validation stages for 3D scour rates.

Set No. | V_{H}* | V_{R}* | V_{L}* | |||
---|---|---|---|---|---|---|

Calibration | Validation | Calibration | Validation | Calibration | Validation | |

1 | 0.837 | 0.558 | 0.8113 | 0.4777 | 0.6361 | 0.44 |

2 | 0.7705 | 0.7513 | Nan | Nan | Nan | Nan |

3 | 0.8370 | 0.5586 | 0.3945 | 0.5965 | 0.6144 | 0.4405 |

4 | 0.6142 | 0.5622 | 0.3945 | 0.5965 | 0.7744 | 0.4408 |

5 | 0.7943 | 0.4148 | 0.7392 | 0.2263 | 0.6835 | 0.5542 |

6 | 0.7578 | 0.6955 | 0.7325 | 0.5903 | 0.8439 | 0.8949 |

7 | 0.4396 | 0.6438 | 0.8475 | 0.3638 | 0.6317 | 0.7374 |

Basis Functions Used in the Prediction of V_{H}* | |
---|---|

BF_{1} | max(0, 0.61751 − F) |

BF_{2} | max(0, 0.11276 − (1 − m)θ_{W} − mθ_{C}) |

BF_{3} | max(0, 0.73254 − (1 − e/D)) |

BF_{4} | max(0, 0.7 − (1 − e/D)) × max(0, F − 0.30369) |

Basis functions used in the prediction of V_{R}* | |

BF_{1} | max(0, mθ_{C} − 0.03862) |

BF_{2} | max(0, mθ_{C} − 0.064886) |

BF_{3} | max(0, 0.064886 − mθ_{C}) |

BF_{4} | max(0, 6.3 − KC) |

BF_{5} | max(0, 0.7 − (1 − e/D)) × max(0, 0.078754 − ((1 − m)θ_{W})) |

BF_{6} | max(0, (1 − e/D) − 0.6) × max(0, (1 − m)θ_{W} − 0.068215 ) |

Basis functions used in the prediction of V_{L}* | |

BF_{1} | max(0, mθ_{C} − 0.03044) |

BF_{2} | max(0, 0.7 − (1 − e/D)) |

BF_{3} | max(0, 0.0478720 − (1 − m)θ_{W}) |

BF_{4} | max(0, (1 − m)θ_{W} − 0.0478720)×max(0, mθ_{C} − 0.017761) |

**Table 7.**Correlations of M5MT performance in the calibration and validation stage for 3D scour rates.

Set No. | V_{H}* | V_{R}* | V_{L}* | |||
---|---|---|---|---|---|---|

Calibration | Validation | Calibration | Validation | Calibration | Validation | |

1 | 0.65 | 0.75 | 0.361 | 0.681 | 0.4866 | 0.6394 |

2 | 0.434 | 0.5962 | 0.361 | 0.680 | 0.4866 | 0.6394 |

3 | 0.756 | 0.775 | 0.361 | 0.680 | 0.486 | 0.639 |

4 | 0.4409 | 0.5960 | 0.7887 | 0.670 | 0.4866 | 0.6394 |

5 | 0.765 | 0.781 | 0.745 | 0.7956 | 0.6794 | 0.7740 |

6 | 0.753 | 0.774 | 0.723 | 0.820 | 0.6916 | 0.7760 |

7 | 0.4339 | 0.596 | 0.361 | 0.681 | 0.4944 | 0.6440 |

Rules of M5MT#1 with Focusing on Pruning and Smoothing Phases |
---|

Rule: 1 IF e/D < = 0.35 θ _{C} < = 0.087THEN V _{H}* = −5.6911 × θ_{W} + 27.9909 × θ_{C} − 6.8389 × m−2.8633 × e/D + 6.0401Rule: 2 IF e/D > 0.35 THEN V _{H}* = −6.015 × e/D + 4.3233Rule: 3 V _{H}* = +4.5309 |

Rules of M5MT#1 with Focusing on Pruning and Smoothing Phases |
---|

Rule: 1 IF θ _{C} < = 0.087THEN V _{R}*= −4.3517 × θ_{W} + 17.5692 × θ_{C} − 6.868 × m − 6.0857 × e/D + 7.0983THEN V _{R}*= −9.5347 × e/D + 6.8673 |

Rules of M5MT#1 with Focusing on Pruning and Smoothing Phases |
---|

Rule: 1 IF 1 − e/D > 0.65 mθ _{C} < = 0.052THEN V _{L}* = 2.4945 × (1 − e/D) + 8.6366 × (1 − m)θ_{W} + 8.5379 × mθ_{C} − 0.6196Rule: 2 V _{L}*= −10.8666 × (1 − e/D) − 4.5984 |

**Table 11.**Correlations of EPR performance in the calibration and validation stages for 3D scour rates.

Set No. | V_{H}* | V_{R}* | V_{L}* | |||
---|---|---|---|---|---|---|

Calibration | Validation | Calibration | Validation | Calibration | Validation | |

1 | 0.7780 | 0.4137 | 0.7413 | 0.6800 | 0.79173 | 0.2661 |

2 | 0.6512 | 0.8881 | 0.5898 | 0.8194 | 0.6771 | 0.7373 |

3 | 0.6887 | 0.6018 | 0.7219 | 0.6467 | 0.8788 | 0.3685 |

4 | 0.6382 | 0.7297 | 0.5323 | 0.8396 | 0.6921 | 0.7264 |

5 | 0.6887 | 0.6031 | 0.7458 | 0.6651 | 0.8725 | 0.7853 |

6 | 0.8794 | 0.8842 | 0.7432 | 0.6556 | 0.7378 | 0.6403 |

7 | 0.6258 | 0.5769 | 0.5800 | 0.8285 | 0.6956 | 0.6735 |

Soft Computing Models | Calibration Stage | ||||
---|---|---|---|---|---|

CC | RMSE | MAPE | SI | DR | |

MARS (Developed by Set 2) | 0.770 | 0.987 | 0.398 | 0.424 | 1.167 |

EPR (Developed by Set 6) | 0.879 | 0.595 | 0.224 | 0.256 | 1.046 |

M5MT (Developed by Set 5) | 0.765 | 0.817 | 0.474 | 0.348 | 1.286 |

GEP (Developed by Set 5) | 0.779 | 0.801 | 0.348 | 0.344 | 1.183 |

Soft Computing Models | Validation Stage | ||||

CC | RMSE | MAPE | SI | DR | |

MARS (Developed by Set 2) | 0.751 | 0.929 | 0.498 | 0.516 | 0.897 |

EPR (Developed by Set 6) | 0.884 | 1.509 | 0.796 | 0.832 | 0.927 |

M5MT (Developed by Set 5) | 0.781 | 1.543 | 0.692 | 0.529 | 1.661 |

GEP (Developed by Set 5) | 0.703 | 1.239 | 0.565 | 0.538 | 1.403 |

Soft Computing Models | Calibration Stage | ||||
---|---|---|---|---|---|

CC | RMSE | MAPE | SI | DR | |

MARS (Developed by Set 6) | 0.732 | 1.041 | 0.388 | 0.408 | 1.111 |

EPR (Developed by Set 5) | 0.746 | 1.003 | 0.308 | 0.393 | 1.120 |

M5MT (Developed by Set 5) | 0.745 | 1.006 | 0.312 | 0.394 | 1.163 |

GEP (Developed by Set 6) | 0.664 | 1.142 | 0.373 | 0.447 | 1.174 |

Soft Computing Models | Validation Stage | ||||

CC | RMSE | MAPE | SI | DR | |

MARS (Developed by Set 6) | 0.590 | 0.981 | 0.384 | 0.446 | 1.632 |

EPR (Developed by Set 5) | 0.665 | 2.236 | 0.764 | 0.878 | 1.234 |

M5MT (Developed by Set 5) | 0.795 | 1.388 | 0.581 | 0.527 | 1.415 |

GEP (Developed by Set 6) | 0.742 | 1.142 | 0.315 | 0.223 | 1.259 |

Soft Computing Models | Calibration Stage | ||||
---|---|---|---|---|---|

CC | RMSE | MAPE | SI | DR | |

MARS (Developed by Set 6) | 0.844 | 0.602 | 0.314 | 0.286 | 1.108 |

EPR (Developed by Set 5) | 0.872 | 0.544 | 0.290 | 0.259 | 1.129 |

M5MT (Developed by Set 6) | 0.691 | 0.807 | 0.598 | 0.383 | 1.374 |

GEP (Developed by Set 7) | 0.755 | 0.739 | 0.447 | 0.352 | 1.144 |

Soft Computing Models | Validation Stage | ||||

CC | RMSE | MAPE | SI | DR | |

MARS (Developed by Set 6) | 0.895 | 0.440 | 0.248 | 0.253 | 1.139 |

EPR (Developed by Set 5) | 0.785 | 2.203 | 0.991 | 0.991 | 1.991 |

M5MT (Developed by Set 6) | 0.776 | 1.197 | 0.679 | 0.439 | 1.666 |

GEP (Developed by Set 7) | 0.783 | 0.674 | 0.358 | 0.369 | 1.231 |

Soft Computing Models | Variation of V_{H}* vs. m for 2.32 < KC < 12.36 | |||
---|---|---|---|---|

e/D = 0.2 | e/D = 0.3 | e/D = 0.4 | e/D = 0.5 | |

MARS | 0.943 | 1.251 | 0.696 | 0.563 |

EPR | 0.947 | 1.038 | 0.812 | 0.390 |

M5MT | 1.105 | 1.229 | 0.741 | 0.835 |

GEP | 0.902 | 1.187 | 0.426 | 0.803 |

Soft Computing Models | Variation of V_{R}* vs. m for 2.32 < KC < 12.36 | |||

e/D = 0.2 | e/D = 0.3 | e/D = 0.4 | e/D = 0.5 | |

MARS | 1.029 | 1.141 | 1.061 | 0.495 |

EPR | 1.366 | 1.721 | 1.326 | 0.439 |

M5MT | 1.182 | 1.361 | 0.862 | 0.577 |

GEP | 0.842 | 1.323 | 1.008 | 0.439 |

Soft Computing Models | Variation of V_{L}* vs. m for 2.32 < KC < 12.36 | |||

e/D = 0.2 | e/D = 0.3 | e/D = 0.4 | e/D = 0.5 | |

MARS | 0.437 | 0.671 | 0.642 | 0.318 |

EPR | 1.685 | 1.157 | 0.623 | 0.626 |

M5MT | 1.035 | 0.910 | 0.792 | 0.891 |

GEP | 0.805 | 0.716 | 0.756 | 0.386 |

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**MDPI and ACS Style**

Najafzadeh, M.; Oliveto, G.; Saberi-Movahed, F.
Estimation of Scour Propagation Rates around Pipelines While Considering Simultaneous Effects of Waves and Currents Conditions. *Water* **2022**, *14*, 1589.
https://doi.org/10.3390/w14101589

**AMA Style**

Najafzadeh M, Oliveto G, Saberi-Movahed F.
Estimation of Scour Propagation Rates around Pipelines While Considering Simultaneous Effects of Waves and Currents Conditions. *Water*. 2022; 14(10):1589.
https://doi.org/10.3390/w14101589

**Chicago/Turabian Style**

Najafzadeh, Mohammad, Giuseppe Oliveto, and Farshad Saberi-Movahed.
2022. "Estimation of Scour Propagation Rates around Pipelines While Considering Simultaneous Effects of Waves and Currents Conditions" *Water* 14, no. 10: 1589.
https://doi.org/10.3390/w14101589