Sensitivity Analysis of Effective Parameters in Empirical Equations for Estimating Local Scour Depth at Bridge Piers in Steady Flows

Scour around bridge piers is an important challenging issue in hydraulic engineering and one of the primary causes of bridge failures (Singh et al., 2022). To date, a large number of empirical equations have been proposed by researches for estimating local scour depth at bridge piers. However, their applicability and results are very different from each other, in such a way, calculated local scour depths using these equations, for a 1 m diameter pier in fine and coarse sand beds, could vary almost from 0.2 m to 3 m, and 0.25 m to 5 m, respectively (Sheppard et al. 2014). In these conditions, evaluation of the performance and accuracy of pier scour empirical equations is very important. In fact, it is necessary to assess the effect of errors in measurements of input properties on estimation of local scour at piers in order to specify the equations that are mostly affected by propagation of input data errors. Sensitivity analysis of these equations and determining the effect of data uncertainties on them could result in precise computation of pier scour that leads in safe designs and reduction in further damages.
Variables that are used in local scour equations at bridge piers, are flow and sediment properties, such as flow velocity and depth, and sediment grain sizes. Computation of local scour depth at bridge piers using empirical equations is based on the measurements of these properties. Errors in these measurements, that are common in engineering applications, affect the accuracy of estimations of pier scour. In the present research, propagation of input data errors and its effect on the pier scour estimation analyzed by Monte Carlo numerical method, that determines the sensitivity level, and relative strengths and weaknesses of the equations. In this research, sensitivity analysis carried out for four empirical equations of local scour depth around cylindrical piers in non-cohesive sediment beds and in both clear-water and live-bed scour conditions. Evaluated equations are: Froehlich (1991), Gao et al. (1993), S/M (2014), and HEC-18 (2016) equations.
Monte Carlo method is based on generation of random numbers. The goal of the Monte Carlo method is to simulate an existing model (in the present study, an empirical equation to estimate the local scour at bridge piers) by randomly sampling from the input physical properties distribution using a large number of samples and finally predicting the output response (Pinto et al., 2006). The physical properties selected for the error analysis include the depth-averaged flow velocity (v ̅_1), flow depth (y_1), and sediment median grain diameter (d_50). Bridge pier width (w) is a property in the equations that is supposed to be known precisely, therefore it is excluded from the analysis. In this research, pier’s reference width is supposed to be constant in the value of 1 m. The densities of water (ρ) and sediment (ρ_s) are variables that could be the error source in the computation of pier scour. However, they are both excluded from sensitivity analysis of pier scour equations, because their values are usually known and supposed to be constant. In order to analyze the results, four output quantities are defined: 1- local scour depth around bridge pier that is computed based on the mean values of physical properties (z_mp=z(v ̅_1,y ̅_1,d ̅_50)), 2- pier scour depths in each Monte Carlo simulation (z_i=z(v_1i,y_1i,d_50i );i=1,…,N), the number of computations (N) set to 10 000 in all simulations, 3- Mean pier scour depth (z_m=z ̅_i), and 4- standard deviation of pier scour depths (σ_z). The error analysis is based on the ratio between fourth and first quantities, that is defined as r ratio (r=σ_z/z_mp).
In evaluating velocity errors on local scour depth at piers, in all velocities, except in v ̅_1=0.5 m/s acceptable values of r (r≤0.3) obtained for all equations. In relative comparison of the equations, two equations, Gao et al. and HEC-18 show most and least sensitivity to the velocity variations, respectively. For flow depth (y_1) variations, small values of r obtained for all equations. In all cases, values of r were below 0.1. comparison of equations reveals that HEC-18 and S/M equations are the most and least sensitive to the flow depth errors, respectively. In analysis of sediment median grain size errors on pier scour depth, acceptable values of r observed for all the cases simulated. In all simulations, values of r were smaller than 0.07, except for Gao et al. equation, with d_50 = 0.2 mm and 0.5 mm and with α = 0.4, that values of r reached to 1.3 and 0.17, respectively.
Among four evaluated equations, Gao et al. and HEC-18 equations show most and least sensitivity to the sediment grain size variation, respectively. Therefore, Gao et al. equation is more sensitive to physical properties than the other equations analyzed. Results also show that among three input variables, errors in the measurements of flow depth and velocity have minimum and maximum effect on the estimation of pier scour depth, respectively.