More reliable predictions of clear-water scour depth at pile groups by robust artificial intelligence techniques while preserving physical consistency

Abstract

This paper presents the results of an investigation on scour around pile groups under steady currents using artificial intelligence (AI) models. Namely, EPR (Evolutionary Polynomial Regression), GEP (Gene-Expression Programming), MARS (Multivariate Adaptive Regression Spline), and M5MT (M5 Model Tree) approaches were used to develop nonlinear regression equations for estimating the maximum equilibrium clear-water scour depth. In total, 321 datasets were collected from various literature sources for different pile group configurations also including the gap between piles and pile groups non-aligned with the flow direction. Results through training and testing phases showed that the MARS technique with Index of Agreement (IOA) of 0.984, Root Mean Square Error (RMSE) of 0.483, and Mean Absolute Error (MAE) of 0.250 provides more accurate estimates of the scour depth (normalized by the pile diameter) than EPR (IOA = 0.976, RMSE = 0.579, and MAE = 0.195), GEP (IOA = 0.972, RMSE = 0.628, and MAE = 0.295), and M5MT (IOA = 0.965, RMSE = 0.704, and MAE = 0.259) models. Conversely, the most frequently used literature formulas demonstrate unconvincing efficiency when wide range experimental data are considered. The sensitivity analysis, in terms of Sobol’s index, revealed that the ratio U/U_c, between the approach flow velocity, U, and the flow velocity, U_c, at the inception of sediment motion, is the most influential parameter with Total Sobol Index (TSI) of 0.514 and an opposite trend of scour with the ratio m/n (TSI = 0.023), between the number, m, of piles inline with the flow and that, n, of piles normal to the flow, was found.

Appendix

The empirical equations used in this study are listed. The HEC-18 scour equation is as follows (Arneson et al. 2012)

$$ \frac{{d_{{\text{s}}} }}{y} = 2KK^{\prime } \left( {\frac{{D_{{\text{e}}} }}{y}} \right)^{0.65} \left( {Fr_{{\text{h}}} } \right)^{0.43} $$

(14)

in which K and K^’ denote correction factors for pile nose shape and bed condition (e.g., clear-water scour, small dunes, large dunes), respectively. D_e is indicative of an equivalent diameter as scour process occurs at pile group, and Fr_h is the Froude number due to the approaching flow. In the present study K is equal to 1, because all the piles have a circular cross section. Additionally, all the experimental investigations were performed in the clear-water condition and, as a result, K^’ = 1.1. According to Arneson et al. (2012) investigations, D_e can be computed as

$$ D_{{\text{e}}} = K_{{{\text{Smn}}}} W $$

(15)

where K_Smn and W are the correction factors for the arrangement and projected width of the pile group, respectively. Equation (14) has the merit of being straightforward and of impact from a physical point of view. It highlights that the approach Froude number and the equivalent diameter D_e (normalized by the flow depth) are the main parameters governing the scouring process, as can be expected. However, the effects of the bed sediment characteristics (e.g., d₅₀ and sediment gradation σ_g) are neglected, which leaves some doubts.

Several experimental investigations were conducted for estimating K_Smn for various pile group configurations. Arneson et al. (2012) have proposed the following equation

$$ \begin{aligned} K_{{{\text{Smn}}}} = & \left( {1 - \frac{4}{3} \cdot \left( {1 - \frac{1}{n}} \right) \cdot \left( {1 - \left( {\frac{G + D}{D}} \right)^{ - 0.6} } \right)} \right) \\ & \cdot \left( {0.9 + 0.1m - 0.0714\left( {m - 1} \right) \cdot \left( {2.4 - 1.1 \cdot \left( {\frac{G + D}{D}} \right) + 0.1 \cdot \left( {\frac{G + D}{D}} \right)^{2} } \right)} \right) \\ \end{aligned} $$

(16)

Alternatively, Ataie-Ashtiani and Beheshti (2006) proposed the following expression for the correction factor K_Smn in the case of pile groups aligned to the flow

$$ K_{{{\text{Smn}}}} = 1.11(m)^{0.0396} \times (n)^{ - 0.5225} \times (G/D)^{ - 0.1153} . $$

(17)

Furthermore, Ataie-Ashtiani and Beheshti (2006) extended the Melville and Coleman’s (2000) equation (related to bridge piers) to pile groups by using the following K_Smn factor

$$ K_{{{\text{Smn}}}} = 1.118(m)^{0.0895} \times (n)^{ - 0.8949} \times (G/D)^{ - 0.1195} . $$

(18)

Specifically, the Melville and Coleman’s (2000) equation is as follows

$$ d_{{\text{s}}} = K_{{\text{S}}} \times K_{{{\text{y,}}D_{{\text{e}}} }} \times K_{{d_{50} }} \times K_{{\text{I}}} $$

(19)

where K_S, K_y,De, K_d50, and K_I are multiplication factors accounting for: pier/pile shape, flow depth-pier/pile size, bed sediment size, and flow intensity, respectively. For cylindrical pile, K_S is equal to 1. Furthermore, K_y,De is categorized into three classes as: 2.4D_e for D_e/y < 0.7, 2(y·D_e)^0.5 for 0.7 ≤ D_e/y < 5, and 4.5y for D_e/y ≥ 5. Moreover, K_d50 is computed as: 0.57Log (2.24D_e/d₅₀) for D_e/d₅₀ ≤ 25 and K_d50 = 1 for D_e/d₅₀ > 25. Finally, for clear-water conditions, K_I is equal to U/U_C. The second modification on the HEC-18 equation is due to Amini et al. (2012) according to

$$ K_{{{\text{Smn}}}} = 1.31\left( m \right)^{0.05} (n)^{ - 0.44} \left( {\frac{G + D}{D}} \right)^{ - 0.38} . $$

(20)

Later, Ghaemi et al. (2013) applied Multivariable Linear Regression (MLR) to predict the scour depth at pile groups. They propose the following scour equation

$$ \frac{{d_{s} }}{D} = 2.09\left( m \right)^{0.03} \times (n)^{0.14} \times \left( \frac{G}{D} \right)^{ - 0.14} \times \left( \frac{y}{D} \right)^{0.38} \times \left( {Fr_{h} } \right)^{0.34} . $$

(21)

Also Eq. (21) has the merit of being straightforward and of impact from a physical point of view. The approach Froude number and the pile diameter D normalized by the flow depth are the main parameters governing the scouring process, as can expected. Moreover, Eq. (21) shows the effects of the pile group characteristics. In particular, the scour depth decreases with increasing the spacing G between piles normal to the flow and increases with increasing of the number of piles normal to the flow (i.e., the overall width of the pile group). Conversely, the effect of the number of piles inline with the flow (i.e., the overall length of the pile group) is negligible, though this cannot be true when the pile group is not aligned with the approach flow (i.e., effect of the skew angle α). All this is physically understandable and in harmony with the well-known behavior of bridge piers and abutments.

Howard and Etemad-Shahidi (2014) introduced a regression-based equation for prediction of the local scour depth at pile groups as

$$ \frac{{d_{s} }}{D} = 2.368\left( n \right)^{0.07} \times \left( \frac{G}{D} \right)^{ - 0.42} \times \left( \frac{y}{D} \right)^{0.25} \times \left( {Fr_{h} } \right)^{0.37} . $$

(22)

From a physical point of view, Eq. (22) confirms the above comments on Eq. (21).

Moreover, Sheppard and Renna (2005) proposed a regression-based equation for prediction of K_Smn as follows

$$ K_{Smn} = \left( {1 - \frac{4}{3} \times \left( {1 - \frac{1}{n}} \right) \times \left( {1 - \left( {\frac{G + D}{D}} \right)^{ - 0.6} } \right)} \right) \times \left( {0.045m + 0.96} \right). $$

(23)

Finally, the recent equation used by the Florida Department of Transportation (FDOT) for pile groups is structured as follows (Sheppard and Renna 2005)

$$ \frac{{d_{s} }}{{D_{e} }} = 2.5\tanh \left[ {\left( {\frac{y}{{D_{e} }}} \right)^{0.4} } \right] \times \left( {1 - 1.75\left[ {Ln\left( {\frac{U}{{U_{C} }}} \right)} \right]^{2} } \right) \times \left( {\frac{{D_{e} /d_{50} }}{{0.4(D_{e} /d_{50} )^{1.2} + 10.6(D_{e} /d_{50} )^{ - 0.13} }}} \right) $$

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