# Analysis of Variables Influencing Scour on Large Sand-Bed Rivers Conducted Using Field Data

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## Abstract

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## 1. Introduction

_{s}), pier width (b), pier length (l), angle of flow attack (θ), effective pier width (b

_{ef}), pier type and nose shape, approach flow depth (y), approach flow velocity (v), bed material type, median sediment size (d

_{50}), geometric standard deviation of sediment particles (σ

_{g}), approximate recurrence interval for measured flow rate (RI), stream slope (S), date, type, time, and location of measurements, etc.

## 2. Methodology

#### 2.1. Influencing Variables

_{s}, b, l, θ, b

_{ef}, pier type, nose shape, y, v, d

_{50}, σ

_{g}, RI, and S. Since this dataset contains only directly measured variables, it is of keen interest to include several other variables commonly used in scour prediction equations. Thus, this study introduces 8 additional variables calculated using the measured data from the PSDB-2014 database: channel width (B), Froude number (Fr), local bed shear stress (τ), critical bed shear stress (τ

_{c}), critical velocity (v

_{c}), critical Froude number for incipient motion (Fr

_{c}), densimetric Froude number (Fr

_{d}), and particle Reynolds number (Re

_{p}) (Table 1). By including the aforementioned additional variables, information about the flow regime, clear-water or live-bed scour, and particle entrainment threshold conditions are taken into account. These additional variables are not given in the original PSDB-2014 database, but they are often used in scour research (e.g., [32,33]). With this addition, a comprehensive dataset of 21 independent variables was created with essential information on flow, sediment, and morphology at each location.

#### 2.2. Data Filtering

_{50}) ranging between 0.0625 mm and 2 mm. The geometrical standard deviation of the sediment (σ

_{g}) presents a measure of the non-uniformity of the sediment [28]. Uniform sediment gradation associated with uniform sand is considered to be less than 1.3 [27] because larger σ

_{g}values would indicate the presence of coarser gravel particles. After applied filtration, the average value of σ

_{g}is 3.0, which shows slightly non-uniform sediments, which indicates potential bed armoring and consequently reduces scour depth [39]. To restrain the dataset to large rivers, measurements of flow depth (y) lower than 1.5 m were excluded from further analysis. The original PSDB-2014 database [26] contains no information regarding the river widths. Benedict and Knight [40] have applied basin data to correlate known channel widths against drainage area, flow depth, and stream slopes. To validate filtering criteria for large rivers, correlation by Benedict and Knight [40] was used herein to calculate channel widths, of which 69% are greater than 30 m.

_{c}> 0.2 by Jain and Fischer is used for validation [38]. Interestingly, after filtering by using the aforesaid expression, 98% of the 348 remaining measurements fall within the clear-water scour conditions, even though the clear-water criterion was not applied in the filter, which validates the reliability of the applied data filters. The range of original and additional variables before and after filtration is shown in Table 2.

#### 2.3. Variable Reduction

_{s}). PCA interprets data with principal components—linear combinations of the input variables that are orthogonal to each other. In this study, there are 21 principal components as well as 21 input variables. The first principal component explains the largest amount of variability in the dataset; the second principal component accounts for the next largest variance; etc. The principal components are lines in the coordinate system with a corresponding eigenvalue—the sum of the squared distances between the orthogonally projected observed data onto the line and the origin of the coordinate system.

_{ef}, y, d

_{50}, v, v

_{cr}, τ, τ

_{cr}, Fr, Fr

_{c}, Fr

_{d}, Re

_{p}, B, b, l, and S), while other variables (σ

_{g}, θ, RI, pier type, and nose shape) can be excluded from further analysis. The results of the PCA analysis are presented in the form of a loading plot, in which the variable vectors are positioned with respect to the first two principal components (Figure 2a). Variables located near the center of the coordinate system have the smallest impact on the variance of the dataset (within the inner dashed circle) and are therefore eliminated from further analysis. When two variable vectors are perpendicular to each other, they are not correlated at all, while variables whose vectors are close to each other are highly correlated. The contribution of each variable to the first three principal components can be evaluated by the squared cosine values (Figure 2b). Since the principal components are axes of the rotated coordinate system, the squared cosine represents the quality of the variables after rotation. The values in bold are the maximum squared cosine values for each variable. Therefore, only the variables that have the maximum squared cosine for one of the first three principal components are retained for further analysis.

_{ef}), hydraulic (y, v, τ, and Fr), compounded hydraulic-sediment variables (v

_{c}, Fr

_{c}, and Fr

_{d}), sediment properties (d

_{50}, τ

_{c}, and Re

_{p}), and channel properties (B and S). Previous studies have concluded that d

_{50}can be eliminated from the development of the scour equation since its range of values is small when compared to the other variables, which consequently leads to biased results [19]. Since only field data are used in this study, d

_{50}is expected to be negligible due to reaching large values of the ratio b/d

_{50}[45,46]. In order to obtain information on sediment properties, only τc and Rep were retained for developing the equation, while d

_{50}was discarded.

_{ef}) are also close together. The angle of flow attack (θ) is not a reliable variable for scour prediction because flow direction depends on the water level and changes over time. This makes the measurement of θ arbitrary and should be used only when it can be reliably estimated [24]. The former debate was supported by the PCA results, which suggested that θ and l should not be used in scour prediction, leaving b and b

_{ef}as the only descriptors of pier geometry. Taking into consideration the difficult approximation of complex pier geometry (in the case of a group of piers, pile caps, etc.), this study uses b

_{ef}as the pier geometry variable, which is reduced to b when no other data is known but can be adjusted for pier alignment when necessary.

_{c}, Re

_{p}). Although τ has been previously recognized as an influential variable for scour prediction [47], it is usually omitted from conventional scour equations due to difficulties in obtaining direct measurements in a complex flow environment [19]. However, τ and v are both variables that reflect drag forces in front of the pier. Since it is challenging to measure v in situ during the flood, perhaps it can be replaced by τ. Locally measured variables (y and S) in the original PSDB-2014 database allow τ to be estimated for each pier. Since it is not sufficiently investigated how much local value of τ contributes to d

_{s}compared to v, its importance for the local scour process will be evaluated in this study. The incipient motion of sediment starts when τ exceeds the critical shear stress (τ

_{c}). Instead, in equations for estimating scour depth, v

_{c}is mainly used because it is easier to measure and is proportional to τ

_{c}. In this study, it is decided to retain both critical variables τ

_{c}and v

_{c}, in order to evaluate their contribution to d

_{s}estimation. Particle Reynolds number (Re

_{p}), which is a measure of eddy currents around particles, is an important variable for estimating the transport of sediment mixtures [48]. Previous research by Vonkeman and Basson [49] demonstrates that replacing v

_{c}with Re

_{p}in the HEC-18 scour equation can improve scour depth prediction.

## 3. Results

#### 3.1. MNLR—Multiple Nonlinear Regression

_{s}and the explanatory influential variables, the following dimensional and non-dimensional models were proposed:

^{2}). R

^{2}shows how closely the scour depth, predicted by the MNLR dimensional and nondimensional equations, resembles the observed scour depth with values of 0.51 and 0.45, respectively. The same observation was made by Ali and Günal [30], who noticed that dimensional scour data were more accurate than those based on dimensionless data. Based on R

^{2}, the dimensional MNLR model was adopted for further analysis. Too many influencing variables reduce the number of degrees of freedom and also increase R

^{2}. Hereby, the adjusted R

^{2}is useful because it corrects the original R

^{2}in such a way that if the number of variables increases, the number of degrees of freedom decreases, as does the adjusted R

^{2}. For the developed model, the adjusted R

^{2}= 0.50 does not change significantly in comparison with the original R

^{2}, which confirms that the number of selected variables is sufficient and not overused.

_{s}) and those predicted by the MNLR dimensional model (d

_{s,pred}) is presented in Figure 3. The gray continuous line shows ideal agreement between measured and predicted values, i.e., d

_{s,pred}= d

_{s}. Data points that lie below the line of agreement indicate overprediction (54% of predicted values are greater than actual measured values), while data points that lie above the abscissa represent underprediction (46% of predicted values are lower than measured values).

_{95}is the two-tailed Student’s T-Distribution for 95% fit, SE is the standard error, which considers the squared ratio of the residual sum of squares by degrees of freedom, n is the number of entries, ($\overline{{d}_{s}}$) is the average of all measured scour depths, and finally SS

_{xx}is the sum of squares of the deviations of measured scour depths from their mean value.

_{c}= 0.06). Therefore, of the initial 15 potential outliers, two were detected as true outliers and removed from further analyses. When outliers were removed from the model, the performance of the dimensional model increased from R

^{2}= 0.50 to R

^{2}= 0.54.

#### 3.2. Comparison with Different Scour Models

_{s,pred}= d

_{s}. The circle points, obtained by using our dimensional model and the filtered dataset, are the same as in Figure 3. However, Figure 5 also displays the lines that were constructed by using predicted d

_{s,pred}data obtained by utilizing different models from Table 6. The following discussions will explain the performance of all models from Table 6.

_{2}), and live-bed vs. clear water correction factor (K

_{1}). Considering that their approach was similar to ours (the same database and the same modeling technique), their equation almost matches the dimensional model developed in this study (Equation (17)). According to Figure 5, there are similarities between the performance of our model and that of the AL model that can be assigned to the fact that in the AL model there was no restrictive data filtering, i.e., almost the whole span of measurements was considered.

_{s}, followed by the ratio of velocities v/v

_{c}, the ratio of pier width B/b, the pier Froude number Fr

_{pier}, and finally the pier shape factor K

_{s}with the weakest influence. According to Figure 5, the HJ model tends to underpredict scour depth. Unfortunately, only ranges of nondimensional variables are given in their study, so the real span of variable values remains unknown. However, it can be assumed that Hassan and Jalal [50] collected numerical data in terms of lower water depths since flow depth turned out to be the most significant variable. This assumption is supported by the work of Melville and Sutherland [52], as they claim that flow depth does not play an important role in scour when the y/b

_{ef}ratio is above 2.6. The final difference is that they have taken into account the ratio of pier width B/b, so it can be assumed that contraction scour has a more important role in their dataset.

_{50}values ranging from 0.12 to 108 mm and a median of 54 mm. Although information regarding the angle of flow attack is not available in their study, the authors included pier length in their equation. However, it can be assumed that piers were skewed, as otherwise, it would have no effect on scour depth [53]. Azamathulla et al. also provided an explicit equation for the GEP model, but when the PSDB-2014 measurements were input into the equation, the computed results were unreasonable, such as negative values of scour depth. In addition, parts of an equation of the developed MNLR model are vague, such as the exponent of the variable σ

_{g}.

#### 3.3. Variable Sensitivity Analysis

_{s}). The Pearson correlation coefficient varies in range from −1 to 1. A value of −1 indicates a perfect negative correlation, a value of 1 a perfect positive correlation, and a value of 0 indicates no correlation at all. If variables are classified into four categories: pier geometry (b

_{ef}), hydraulic (y, v, τ, and Fr), compounded hydraulic-sediment variables (v

_{c}, Fr

_{c}, and Fr

_{d}), and sediment (τ

_{c}, and Re

_{p}), then the pier geometric variables are the most influential, followed by compounded hydraulic-sediment, hydraulic, and finally the sediment variables. Based on Pearson correlation coefficients, b

_{ef}proved to be the most influential variable for estimating d

_{s}(r = 0.625), which is the same observation that was already stated in previous investigations [28,30]. The second most influential variable is y (r = 0.492), followed closely by v

_{c}and v, r = 0.474 and r = 0.436, respectively. In the compounded hydraulic-sediment category, v

_{c}has the greatest impact on scour depth in comparison to Fr

_{d}and Fr

_{c}, which are defined by r = 0.427 and r = 0.323, respectively. The remaining four variables (τ, τ

_{c}, Fr, and Re

_{p}) show a very weak correlation with scour depth, with r < 0.3.

## 4. Discussion

_{ef}), hydraulic (y, v, τ, and Fr), sediment (τ

_{c}, and Re

_{p}), and compounded hydraulic-sediment variables (v

_{c}, Frc, and Fr

_{d}), it was elucidated that pier geometry variables are the most influential ones. Rating pier geometry variables above hydraulic variables was expected because more than 70% of the data used in this study exceeds the threshold value of y/b

_{ef}= 2.6, over which the flow depth is no longer significant for the pier scour process [52].

_{50}was excluded from analysis, suspecting that it has a negligible effect, so two new sediment variables (τ

_{c}, Re

_{p}) were included in the scour depth equation. Even though it was expected that Re

_{p}would play a more significant role owing to a larger standard deviation than d

_{50}, both sediment variables τ

_{c}and Re

_{p}yielded the lowest Pearson correlation coefficients, which shed light on sediment properties in general (d

_{50}, τ

_{c}, and Re

_{p}) as being insignificant on scour depth variance. In order to consider the effect of sediment properties, it was recommended to use compound hydraulic-sediment variables such as v

_{c}, Fr

_{c}, and Fr

_{d}that consist of both the acting forces of the flow and sediment properties. In this study, the compounded hydraulic-sediment variables and hydraulic variables had a similar impact on the scour prediction because the average values of their Pearson correlation coefficients were 0.408 and 0.340, respectively. A comparable observation was made by Török et al. [54], who deem that shear Reynolds number, a function of grain size and shear velocity, is a more adequate variable for evaluating sediment transport than Re

_{p}, d

_{50}, or other variables whose evaluation is based only on sediment properties. Furthermore, at the beginning of this paper, there was a doubt concerning which variable should be used to assess the incipient motion of sediment, v

_{c}or τ

_{c}. The doubt originates from the fact that v

_{c}is a function of y and d

_{50}, while τ

_{c}is a function of d

_{50}. The assumption that the compound hydraulic-sediment variable would be more significant was verified by Pearson correlation coefficients of v

_{c}and τ

_{c}with values of 0.474 and −0.006, respectively. After v

_{c}, which proved to be the most influential variable in the category of compounded hydraulic-sediment variables, Fr

_{d}and Fr

_{c}are next in the sequence. It was expected that Fr

_{d}would show a stronger association with scour depth since it has been previously determined as the most influencing variable in the group of variables that describe sediment properties [55,56,57].

^{2}value (R

^{2}= 0.5) due to the presence of scatter in the dataset. Previous equations that used the same PSDB-2014 dataset (AL and RM) performed better, as indicated by higher R

^{2}or lower RMSE values. The reason could be that they used almost all field measurements without excluding multiple measurements for the same bridge. Such a collection of data with similar measurements could lead to overfitting. However, in this study, the independence of the measurements was maintained in the final set of filtration criteria by retaining only one measurement for each bridge, which eventually led to the exclusion of 46% from the original dataset. Another reason for the scatter is that the filtered dataset contains only field data where the scour regime is unknown; it is difficult to determine whether some measurements have reached an equilibrium or maximum state.

## 5. Conclusions

_{50}, as they showed low impact on the scour depth. Since the measurement of θ is arbitrary and changes with flow severity, it should be used only when it can be reliably estimated, which is not often the case. Since the pier length was also eliminated by the PCA and taking into consideration the difficult approximation of complex pier geometry with a single variable, this study selected b

_{ef}as the only pier geometry variable, combining the information of pier geometry and alignment with the flow, if available. PCA has eliminated the characteristic sediment size as well, which can be explained by focusing on sand-bed rivers where sand is uniform and therefore has a significantly smaller range than other variables. To take into account riverbed composition, compounded hydraulic-sediment variables (v

_{c}, Fr

_{c}, and Fr

_{d}) were retained, and consequently, 10 influential variables were classified into four categories: pier geometry (b

_{ef}), hydraulic (y, v, τ, and Fr), sediment (τ

_{c}, and Re

_{p}), hydraulic-sediment variables (v

_{c}, Fr

_{c}, and Fr

_{d}), and compounded. Afterward, the variables from these categories were used to determine the scour model variables.

^{2}value of 0.5 due to the scatter of the dataset, which could be a consequence of excluding multiple measurements for the same bridge locations. Since, at the time of the field measurement, it is not known whether the equilibrium or maximum state has been reached, dispersion in the results is expected.

_{c}, v, Fr

_{d}, and Fr

_{c}. The remaining four variables (τ, τ

_{c}, Fr, and Re

_{p}) exhibit a very weak correlation with scour depth, probably resulting from errors in the measurement of variables used for their calculation. The fact that compounded hydraulic-sediment variables were highly influential for our dimensional model indicates that it is necessary to use sediment size when filtering data to reduce the uncertainty associated with the acquisition of a representative bed sample.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Glossary

Symbol | Unit | Description |

d_{s} | [m] | scour depth |

v | [m/s] | local approach flow velocity (upstream of the pier) |

v_{c} | [m/s] | critical velocity |

y | [m] | approach water depth (upstream of the pier) |

b | [m] | nominal pier width |

l | [m] | pier length |

b_{ef} | [m] | effective pier width normal to the flow |

θ | [°] | angle of attack |

B | [m] | the channel width |

S | [1] | stream slope |

τ | [Pa] | local bed shear stress |

τ_{c} | [Pa] | critical bed shear stress |

Fr | [1] | Froude number |

Fr_{c} | [1] | critical Froude number for incipient motion |

Fr_{d} | [1] | densimetric Froude number |

Re_{p} | [1] | particle Reynolds number |

RI | [years] | recurrence interval for measured flow rate |

g | [m/s^{2}] | gravitational acceleration |

d_{50} | [mm] | sediment median grain size |

d_{95} | [mm] | the size at which 95% of the sediment particles are smaller |

σ_{g} | [1] | geometrical standard deviation of sediment (measure of non-uniformity) |

ρ_{rel} | [1] | submerged relative mass density of sediment particles (ρ _{rel} = [(ρ_{s} − ρ)/ρ] − 1 = 1.65) |

ρ_{s} | [kg/m^{3}] | mass density of sediment particles (equal to 2650 kg/m^{3}) |

ρ_{w} | [kg/m^{3}] | mass density of water (equal to 1000 kg/m^{3}) |

γ_{s} | [N/m^{3}] | specific gravity of sediment (equal to 25,996.5 N/m^{3}) |

γ_{w} | [N/m^{3}] | specific gravity of water (equal to 9810 N/m^{3}) |

K_{1} | [1] | the live-bed vs. clear-water correction factor |

K_{2} | [1] | the pier shape correction factor |

ν | [m^{2}/s] | kinematic viscosity of fluid (ν = 1.6 × 10^{−6} m^{2}/s) |

Φ | [°] | angle of repose for sediments |

r | [1] | Pearson correlation coefficient |

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**Figure 1.**Flowchart of the proposed methodology to identify the most influential variables before developing a new model.

**Figure 2.**Results of PCA analysis: (

**a**) Loading plot that presents a total of 21 variables in the coordinate system with the first two principal components on the axes; (

**b**) Contribution of each variable to the retained first three principal components throughout the squared cosines of the variables. Values in bold correspond for each variable to the principal component for which the squared cosine is the largest.

**Figure 3.**Scattered plot of measured (d

_{s}) and scour depth predicted (d

_{s,pred}) by MNLR dimensional model. Isolated points are outliers where red dots present underprediction and green dots present overprediction.

**Figure 4.**Dependence of influential variables on the scour depth residual. Upper and lower limits illustrated by the red dashed lines represent the 95% prediction intervals.

**Figure 5.**Comparison of dimensional model developed in this study with various previously developed scour equations (Table 6). The displayed data (i.e., circles) are obtained by using our non-dimensional model and the filtered data. The lines present data computed by different models when using the filtered data.

**Figure 6.**Pearson correlation coefficients (r) for selected influential variables that are classified into four variable categories. The selected variables are taken from the filtered dataset.

Additional Variable | Equation | |
---|---|---|

Froude number | $Fr=\frac{v}{\sqrt{g\xb7y}}$ | (1) |

local shear stress | $\tau ={\gamma}_{w}\xb7y\xb7S$ | (2) |

critical shear stress [34] | ${\tau}_{c}=0.25\xb7{\left({d}_{50}\xb7{\left[\frac{\left(\frac{{\gamma}_{s}}{{\gamma}_{w}}-1\right)\xb7g}{{\nu}^{{2}^{*}}}\right]}^{\frac{1}{3}}\right)}^{-0.6}\xb7g\xb7\left({\rho}_{s}-{\rho}_{w}\right)\xb7{d}_{50}\xb7tan\left({\Phi}^{*}\right)$ | (3) |

critical velocity [35] | ${v}_{c}=\sqrt{{\rho}_{rel}{}^{*}\xb7g\xb7{d}_{50}}\xb7\left(0.0024\xb7\left(\frac{y}{{d}_{50}}\right)+2.34\right)$ | (4) |

critical Froude number | $F{r}_{c}=\frac{{v}_{c}}{\sqrt{g\xb7y}}$ | (5) |

densimetric Froude number | $F{r}_{d}=\frac{v}{\sqrt{\left({\rho}_{rel}{}^{*}-1\right)\xb7g\xb7{d}_{50}}}$ | (6) |

particle Reynolds number | ${\mathrm{Re}}_{\mathrm{p}}=\frac{{\mathrm{d}}_{50}\xb7\sqrt{{\mathsf{\rho}}_{\mathrm{rel}}{}^{*}\xb7\mathrm{g}\xb7{\mathrm{d}}_{50}}}{{\mathsf{\nu}}^{*}}$ | (7) |

**Table 2.**Range of original and additional variables with their symbols, measurement units, ranges, average values, and standard deviations before and after filtering.

Variable | Before Filtering | After Filtering | |||||
---|---|---|---|---|---|---|---|

Range | Average | Standard Deviation | Range | Average | Standard Deviation | ||

Original | d_{s} | 0–10.4 | 1.1 | 1.3 | 0.21–7.8 | 1.5 | 1.3 |

b_{ef} | 0.21–28.7 | 2.3 | 2.5 | 0.24–11.6 | 2.0 | 1.9 | |

b | 0.21–19.5 | 1.6 | 1.6 | 0.24–11.6 | 1.3 | 1.4 | |

l | 0.21–39.6 | 6.3 | 5.3 | 0.37–25.3 | 6.3 | 4.7 | |

θ | 0–85.0 | 6.1 | 10.3 | 0–600 | 6.6 | 9.6 | |

d_{50} | 0.001–228.6 | 14.7 | 25.0 | 0.06–1.82 | 0.59 | 0.41 | |

y | 0–22.5 | 3.9 | 3.2 | 1.5–22.5 | 5.6 | 3.4 | |

v | 0–5.4 | 1.4 | 0.8 | 0.20–3.9 | 1.4 | 0.71 | |

RI | 1–500 | 53.6 | 50.9 | 1–500 | 63.1 | 69.4 | |

σ_{g} | 1.2–20.3 | 3.3 | 2.8 | 1.4–20.3 | 3.0 | 1.2 | |

S | 0.00007–0.02 | 0.00086 | 0.00152 | 0.00007–0.0036 | 0.00052 | 0.00044 | |

Additional | B | 5.4–692.5 | 71.5 | 67.1 | 10.5–692.5 | 75.3 | 80.7 |

v_{c} | 0.15–55.7 | 2.4 | 3.2 | 0.76–11.6 | 2.8 | 1.7 | |

τ | 0–180 | 21.3 | 21.8 | 0.015–1.7 | 0.25 | 0.21 | |

τ_{c} | 0.025–3.9 | 0.87 | 0.77 | 0.13–0.56 | 0.32 | 0.10 | |

Fr | 0–1.98 | 0.28 | 0.21 | 0.039–0.55 | 0.20 | 0.10 | |

Fr_{c} | 0.19–5.77 | 0.40 | 0.34 | 0.19–1.0 | 0.38 | 0.13 | |

Fr_{d} | 0–629 | 11.1 | 23.4 | 1.6–88.9 | 16.0 | 9.9 | |

Re_{p} | 0.0025–274,834 | 8222 | 19,074 | 1.2–195.2 | 42.1 | 43.7 | |

Original | Pier type | Single and group | Not affected | ||||

Pier nose shape (drag coefficient) [42] | Cylindrical (1.2), Round (1.33), Square (2.0), Sharp (1.0), Triangular (1.72) |

Influential variables | b_{ef} | y | v | v_{c} | τ | τ_{c} | Fr | Fr_{c} | Fr_{d} | Re_{p} |

**Table 4.**Parameters obtained applying MNLR technique on both dimensional and non-dimensional models.

a | b | c | d | e | f | g | h | i | j | k | |
---|---|---|---|---|---|---|---|---|---|---|---|

dimensional | 3.29 | 0.49 | 1.19 | −0.91 | −0.99 | −0.011 | 0.019 | 0.38 | 0.26 | 0.97 | −0.44 |

non-dimensional | 0.002 | 0.48 | −0.90 | −0.047 | −0.33 | −2.50 | 1.16 | −0.094 |

**Table 5.**List of potential outliers with corresponding measured and calculated influential variable values.

ID | b_{ef} | y | v | v_{c} | τ | τ_{c} | Fr | Fr_{c} | Fr_{d} | Re_{p} | d_{s} | d_{s,pred} | Residuals |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Obs96 | 4.88 | 9.81 | 2.13 | 5.11 | 38.51 | 0.27 | 0.22 | 0.52 | 27.57 | 17.90 | 0.67 | 3.50 | −2.83 |

Obs293 | 11.58 | 4.85 | 2.68 | 2.30 | 95.08 | 0.31 | 0.39 | 0.33 | 29.82 | 28.11 | 2.50 | 5.15 | −2.65 |

Obs252 | 5.54 | 13.47 | 2.35 | 7.44 | 72.45 | 0.26 | 0.20 | 0.65 | 32.61 | 14.39 | 1.80 | 4.08 | −2.28 |

Obs52 | 9.63 | 3.60 | 0.79 | 1.30 | 7.76 | 0.49 | 0.13 | 0.22 | 5.46 | 117.86 | 0.43 | 2.36 | −1.93 |

Obs57 | 2.50 | 9.33 | 1.55 | 4.09 | 17.38 | 0.33 | 0.16 | 0.43 | 16.63 | 31.55 | 0.43 | 2.22 | −1.79 |

Obs340 | 5.97 | 8.35 | 1.22 | 3.28 | 11.47 | 0.37 | 0.13 | 0.36 | 11.37 | 47.57 | 5.15 | 2.95 | 2.20 |

Obs307 | 0.61 | 2.77 | 1.13 | 1.14 | 5.17 | 0.42 | 0.22 | 0.22 | 8.86 | 79.52 | 2.87 | 0.65 | 2.22 |

Obs346 | 6.10 | 22.52 | 2.43 | 9.11 | 66.29 | 0.34 | 0.16 | 0.61 | 24.65 | 36.96 | 7.10 | 4.73 | 2.37 |

Obs324 | 1.07 | 4.54 | 1.77 | 2.17 | 53.46 | 0.31 | 0.26 | 0.33 | 19.65 | 28.11 | 3.66 | 1.27 | 2.38 |

Obs343 | 4.33 | 5.67 | 2.93 | 3.32 | 32.81 | 0.25 | 0.39 | 0.45 | 41.99 | 13.07 | 6.22 | 3.57 | 2.65 |

Obs329 | 1.07 | 4.15 | 1.71 | 2.00 | 48.80 | 0.31 | 0.27 | 0.31 | 18.97 | 28.11 | 4.05 | 1.23 | 2.82 |

Obs316 | 0.61 | 2.19 | 0.67 | 0.97 | 4.52 | 0.42 | 0.14 | 0.21 | 5.27 | 79.52 | 3.29 | 0.47 | 2.82 |

Obs344 | 4.37 | 5.21 | 2.44 | 2.98 | 16.62 | 0.26 | 0.34 | 0.42 | 33.88 | 14.39 | 6.43 | 3.26 | 3.17 |

Obs347 | 4.27 | 9.78 | 2.90 | 5.62 | 9.60 | 0.25 | 0.30 | 0.57 | 41.55 | 13.07 | 7.65 | 3.86 | 3.79 |

Obs348 | 10.09 | 9.00 | 0.65 | 3.60 | 48.40 | 0.36 | 0.07 | 0.38 | 6.24 | 43.61 | 7.80 | 2.90 | 4.90 |

**Table 6.**Summary of selected previously developed scour models and two models developed in this paper.

Author | Dataset | Method | Equation | |
---|---|---|---|---|

Annad and Lefkir, 2022 [19] | field (PSDB-2014) | MNLR | ${d}_{s}=0.318\xb7{K}_{1}{}^{*}\xb7{K}_{2}{}^{*}\xb7{b}^{0.76}\xb7{y}^{0.515}$ | (11) |

Rathod and Manekar, 2022 [28] | field and laboratory (PSDB-2014) | GEP | ${d}_{s}=y\xb7\left[\left(\frac{b}{b+1.6\xb7y}\right)+\left(\frac{b\xb7v}{y\xb7\left(b+v\right)}\right)+\left(\frac{\raisebox{1ex}{$b$}\!\left/ \!\raisebox{-1ex}{$y$}\right.}{\raisebox{1ex}{$34.12\xb7y$}\!\left/ \!\raisebox{-1ex}{$F{r}_{d}$}\right.+2{\sigma}_{g}}\right)\right]$ | (12) |

Hassan and Jalal, 2021 [50] | numerical on a large scale | GEP | ${d}_{s}=b\xb7\left[7.24\xb7\frac{v}{{v}_{c}}\xb7\frac{b}{B}-{\left(\frac{v}{{v}_{c}}\right)}^{2}\xb7\frac{b}{B}-\frac{v}{{v}_{c}}\xb7{\left(\frac{b}{B}\right)}^{2}+\frac{\frac{b}{B}}{\frac{b}{B}\xb7\frac{v}{{v}_{c}}-\frac{y}{b}-\frac{b}{B}}\phantom{\rule{0ex}{0ex}}+\left\{\frac{y}{b}\xb7\frac{b}{B}\xb7F{r}_{pier}{}^{*}+{\left(\frac{b}{B}\right)}^{2}\xb7F{r}_{pier}{}^{*}\right\}\xb7{K}_{2}{{}^{2}}^{*}\xb7\frac{v}{{v}_{c}}\right]$ | (13) |

Jain and Fischer, 1979 [38] | field and laboratory | conventional MNLR | ${d}_{s}=1.84\xb7b\xb7{\left(\frac{y}{b}\right)}^{0.3}\xb7{\left(F{r}_{c}\right)}^{0.25}$ | (14) |

Azamathulla et al., 2010 [37] | field | MNLR | ${d}_{s}=1.82\xb7y\xb7{\left(\frac{{d}_{50}}{y}\right)}^{0.042}\xb7{\left(\frac{b}{y}\right)}^{-0.28}\xb7{\left(\frac{L}{y}\right)}^{-0.37}\xb7F{r}^{0.42}\xb7{\sigma}_{g}{}^{-0.031}$ | (15) |

Our non-dimensional model | field (PSDB-2014) | MNLR | $\frac{{d}_{s}}{y}=0.002\xb7{\left(\frac{{b}_{ef}}{y}\right)}^{0.48}\xb7{\left(\frac{v}{{v}_{c}}\right)}^{-0.90}\xb7{\left(\frac{\tau}{{\tau}_{c}}\right)}^{-0.047}\xb7{\left(Fr\right)}^{0.33}\xb7{\left(F{r}_{c}\right)}^{-2.5}\xb7{\left(F{r}_{d}\right)}^{1.16}\phantom{\rule{0ex}{0ex}}\xb7{\left(R{e}_{p}\right)}^{-0.094}.$ | (16) |

Our dimensional model | field (PSDB-2014) | MNLR | ${d}_{s}=3.29\xb7{\left({b}_{ef}\right)}^{0.49}\xb7{\left(y\right)}^{1.19}\xb7{\left(v\right)}^{-0.91}\xb7{\left({v}_{c}\right)}^{-0.99}\xb7{\left(\tau \right)}^{-0.011}\xb7{\left({\tau}_{c}\right)}^{0.019}\phantom{\rule{0ex}{0ex}}\xb7{\left(Fr\right)}^{0.38}\xb7{\left(F{r}_{c}\right)}^{0.26}\xb7{\left(F{r}_{d}\right)}^{0.97}\xb7{\left(R{e}_{p}\right)}^{-0.44}$ | (17) |

_{1}is 1.2 for live-bed scour and 1 for clear-water scour. K

_{2}is 1.1 for round, 1 for square, and 0.9 for a sharply pierced nose. Fr

_{pier}is the pier Froude number, calculated as follows: $F{r}_{pier}=\frac{v}{\sqrt{g\xb7{b}_{ef}}}$.

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**MDPI and ACS Style**

Harasti, A.; Gilja, G.; Adžaga, N.; Žic, M.
Analysis of Variables Influencing Scour on Large Sand-Bed Rivers Conducted Using Field Data. *Appl. Sci.* **2023**, *13*, 5365.
https://doi.org/10.3390/app13095365

**AMA Style**

Harasti A, Gilja G, Adžaga N, Žic M.
Analysis of Variables Influencing Scour on Large Sand-Bed Rivers Conducted Using Field Data. *Applied Sciences*. 2023; 13(9):5365.
https://doi.org/10.3390/app13095365

**Chicago/Turabian Style**

Harasti, Antonija, Gordon Gilja, Nikola Adžaga, and Mark Žic.
2023. "Analysis of Variables Influencing Scour on Large Sand-Bed Rivers Conducted Using Field Data" *Applied Sciences* 13, no. 9: 5365.
https://doi.org/10.3390/app13095365