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26 3 Data 2.5 Sheppard & Miller 2006 S/M 2 ys/a 1.5 1 0.5 0 0 0.2 0.4 0.6 0.8 1 V1/Vc Figure 22. Measured field data at low velocities compared to the upper limit of Sheppard and Miller (2006) and S/M equations. a discontinuity in predicted scour depth with increasing pier known. Several of the scour depths in the field data set are width as shown in Figure 25. extremely small for the pier size, water depth, and flow veloc- ity. This observation suggests that the duration of the flow at the reported velocity was short; that the bed material was Evaluation of Equilibrium Scour cohesive although reported as cohesionless; or perhaps both. Predictive Methods With the uncertainty associated with even the most reliable Evaluating predictive equations using laboratory data is rel- field data, it is not appropriate to use it directly to evaluate the atively straightforward because, in most cases in the database, predictive equations. However, with only a few exceptions, either the scour has reached equilibrium or the measured the predicted values should not be less than the measured val- depth has been extrapolated to an equilibrium value. In addi- ues, assuming the measured values are accurate and do not tion, the input values (water depth, flow velocity, sediment include other types of scour (contraction, degradation, etc.). properties, etc.) are all accurately known. Obtaining accurate One exception is the case where the measured scour depth measurements of input parameters in the field is much more is due to a previous, more severe, flow event. Figure 26 difficult and the maturity of the scour hole is almost never gives some insight into the flow duration problem for field 1.8 With Wide-Pier Corr. 1.6 With Wide-Pier Corr. W/O Wide-Pier Corr W/O Wide-Pier Corr 1.6 1.4 1.4 1.2 * Predicted ys/a Predicted ys/a 1 1.2 0.8 1 0.6 0.8 0.4 0.6 0.2 0.4 0 0 0.5 1 1.5 2 0 0.5 1 1.5 * Measured ys/a Measured ys/a Figure 23. Effect of wide-pier correction on Figure 24. Effect of wide-pier correction on HEC-18 normalized scour depth predictions HEC-18 normalized scour depth predictions for for laboratory data. field data.

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27 35 W/O Wide-Pier Corr 30 With Wide-Pier Corr. Equilibrium Scour (ft) 25 20 15 10 5 0 0 5 10 15 20 25 30 Pier Width (ft) Figure 25. An example of HEC-18 scour predictions (y1 = 15 ft, D50 = 0.3 mm, V1 = 8 ft/s). Jain 1981 Froehlich 1988 2 2 measured/computed ys 1.5 measured/computed ys 1.5 1 1 0.5 0.5 0 0 -2 0 2 -2 0 2 Melville 1997-mod. HEC-18 2 2 measured/computed ys measured/computed ys 1.5 1.5 1 1 0.5 0.5 0 0 -2 0 2 -2 0 2 HEC-18-no wp corr S/M 2 2 measured/computed ys measured/computed ys 1.5 1.5 1 1 0.5 0.5 0 0 -2 0 2 -2 0 2 log( t90 in days) log( t90 in days) Note: The lines are best linear fits to the data. Figure 26. Measured/predicted versus predicted time to reach 90% of equilibrium scour depth for the field data.

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28 conditions. In this figure, the ratio of measured divided by pre- The desired method/equation is the one that has the least dicted scour is plotted versus t90 (time required to reach 90% overall error and least underprediction. Because these of equilibrium) for six equilibrium scour equations. Best-fit equations are being recommended for design, underpre- lines to the data are also shown. The time to reach 90% of equi- diction errors must be weighted heavier. That is, greater librium scour depth is computed using the M/S equation dis- emphasis must be given to reducing underprediction than cussed later in this report. Note that t90 is more than 100 days to total error. for some situations. Underpredictions increase for all equa- Table 6 gives percentage errors (SSE and SSEn) for the labo- tions with increasing t90. This result suggests that a large por- ratory and field data sets. Underpredictions are much smaller tion of the field cases have not reached equilibrium. compared to the total error for all predictive equations because The errors associated with predicting the measured equilib- they are all conservative by design. For the reasons discussed rium scour depths were computed using the following equation previously, total errors for the field data are not presented for dimensional scour depths, ys: because the maturity of these data are not known. The fact that all the equilibrium scour equations overpredict many SSE% = ( y measured s - y computed s ) 2 100 (26) of the measured depths, especially for the larger structures, indicates the need to account for the design flow event dura- (y s ) measured 2 tion in the prediction. However, even the best scour evolu- tion equations are not sufficiently accurate for this task at The corresponding equation for normalized scour depths, this time. Large-scale, live-bed scour evolution tests and the ys/a, is development of improved predictive scour evolution equa- tions are definitely needed. 2 Six different measures of error were used for each method/ y measured y model a s - s a equation. The overall fit of all equations can be improved by SSEn% = measured 2 100 (27) modifying their coefficients, but, in general, such modifica- ys tion increases their underpredictions as can be seen between a Sheppard and Miller (2006) and its modified form, S/M. The Table 6. Absolute and normalized errors for predictive equilibrium scour equations. Laboratory (441 points) Field (760 points) Reference SSE% SSEn% SSE% SSEn% Total Under Total Under Under Under Inglis (1949) 77.0 30.1 297.0 57.3 32.7 106.1 Ahmad (1953) 167.7 56.2 957.1 19.8 70.3 127.8 Laursen (1958, 1963) 24.0 21.4 24.3 12.6 4.1 7.2 Chitale (1962) 123.3 31.1 921.1 19.2 25.8 67.1 Larras (1963) 21.1 1.8 27.2 0.5 8.0 4.2 Breusers (1965) 31.0 1.8 10.9 5.7 0.2 0.2 Blench (1969) 18.6 18.3 60.6 60.0 5.7 7.8 Shen et al. (1969) 19.1 6.4 55.5 0.9 3.0 1.9 Coleman (1971) 87.2 87.2 97.7 97.7 25.2 49.8 Hancu (1971) 130.2 20.8 250.9 7.7 3.1 6.6 Neill (1973) 38.7 1.2 11.1 4.0 0.1 0.1 Breusers et al. (1977) 18.9 13.0 12.5 6.2 4.9 7.2 Jain (1981) 9.1 0.2 24.5 0.6 2.2 1.1 Froehlich (1988) 21.0 0.6 10.3 1.7 0.0 0.0 May & Willoughby (1990) 17.9 6.5 18.4 2.6 4.1 6.2 Gao et al. (1993) 75.7 16.2 158.5 8.1 7.4 12.9 Ansari & Qadar (1994) 52.9 52.5 97.8 97.8 0.6 12.0 Wilson (1995) 9.1 6.9 15.6 5.3 2.8 1.9 Melville (1997) 34.9 0.3 24.8 0.1 0.2 0.1 Melville (1997)mod. 34.9 0.2 24.8 0.1 0.2 0.1 HEC-18 7.9 1.8 21.0 0.3 1.1 1.1 HEC-18no wp corr 9.5 1.2 21.1 0.3 0.4 0.5 Sheppard & Miller (2006) 5.9 0.1 11.7 0.3 1.9 2.6 S/M 6.8 0.1 13.0 0.3 0.3 0.2

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29 Table 7. Ranking of all predictive equilibrium scour equations. Laboratory (441 points) Field (760 points) Reference SSE SSEn SSE SSEn Total Under Total Under Overall Under Under Overall Inglis (1949) 20 20 22 21 20 23 23 23 Ahmad (1953) 24 23 24 20 24 24 24 24 Laursen (1958, 1963) 13 19 11 18 16 15 17 16 Chitale (1962) 22 21 23 19 22 22 22 22 Larras (1963) 12 9 15 7 13 20 13 17 Breusers (1965) 14 11 2 14 12 5 5 5 Blench (1969) 8 17 17 22 17 18 18 19 Shen et al. (1969) 10 12 16 9 15 13 11 12 Coleman (1971) 21 24 18 23 23 21 21 21 Hancu (1971) 23 18 21 16 19 14 15 14 Neill (1973) 17 7 3 12 10 2 2 2 Breusers et al. (1977) 9 15 5 15 14 17 16 18 Jain (1981) 5 4 12 8 6 11 8 9 Froehlich (1988) 11 6 1 10 5 1 1 1 May & Willoughby (1990) 7 13 8 11 11 16 14 15 Gao et al. (1993) 19 16 20 17 18 19 20 20 Ansari & Qadar (1994) 18 22 19 24 21 8 19 13 Wilson (1995) 4 14 7 13 9 12 10 10 Melville (1997) 15 5 13 2 8 4 4 4 Melville (1997)mod. 16 3 14 1 7 3 3 3 HEC-18 3 10 9 4 3 9 9 8 HEC-18no wp corr 6 8 10 3 4 7 7 7 Sheppard & Miller (2006) 1 2 4 6 1 10 12 11 S/M 2 1 6 5 2 6 6 6 prediction of the absolute scour depth is important because scour depths are not equilibrium values. Note that an equa- the design scour depth will be computed using the dimen- tion that grossly overpredicts can have small underpredic- sional form of the equation. tion errors. The results of the error statistics are presented in Table 6 Six equations were chosen for final evaluation based on their for all equations. The error statistics confirm the screening performances listed in Tables 6 and 7: Jain (1981), Froehlich process in that the equations eliminated are the ones with the (1988), modified Melville (1997), HEC-18, HEC-18 without greatest errors. Table 7 gives the ordering of the equations wide-pier correction, and S/M. Only the modified/final according to the values given in Table 6. There are also two versions of the equations were considered. However, both columns for overall error order for field and laboratory forms of HEC-18 are presented because they are currently data. Each overall error order was calculated by averaging widely used in the United States. The error statistics and the columns to its left and ordering the results. Field data rankings for these six equations, based on performance errors are not very informative by themselves because they with all data, are given in Tables 8 and 9, respectively (sta- do not include total errors. Total errors for the field data tistics are the same as those given in Table 6, but they are are also not meaningful because so many of the reported repeated for easier access). Tables 10 and 11 give the error Table 8. Errors for selected predictive equations. Laboratory (441 points) Field (760 points) Reference SSE% SSEn% SSE% SSEn% Total Under Total Under Under Under Jain (1981) 9.1 0.2 24.5 0.6 2.2 1.1 Froehlich (1988) 21.0 0.6 10.3 1.7 0.1 0.0 Melville (1997)mod. 34.9 0.2 24.8 0.1 0.2 0.1 HEC-18 7.9 1.8 21.0 0.3 1.1 1.1 HEC-18no wp corr 9.5 1.2 21.1 0.3 0.4 0.5 S/M 6.8 0.1 13.0 0.3 0.3 0.2

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30 Table 9. Ranking of selected predictive equations based on performance with all data. Laboratory (441 points) Field (760 points) Reference SSE SSEn SSE SSEn Total Under Total Under Overall Under Under Overall Jain (1981) 3 3 5 5 5 6 5 5 Froehlich (1988) 5 4 1 6 6 1 1 1 Melville (1997)mod. 6 2 6 1 3 2 2 2 HEC-18 2 6 3 3 2 5 6 6 HEC-18no wp corr 4 5 4 2 4 4 4 4 S/M 1 1 2 4 1 3 3 3 Table 10. Errors for selected predictive equations for wide piers (a/D50 > 100, a/y1 > 2). Laboratory (17 points) Field (142 points) Reference SSE% SSEn% SSE% SSEn% Total Under Total Under Under Under Jain (1981) 15.4 0.0 10.9 0.0 5.6 5.4 Froehlich (1988) 94.8 0.0 55.2 0.0 0.0 0.0 Melville (1997)mod. 3.3 1.3 7.8 0.4 0.7 1.0 HEC-18 4.2 3.9 2.4 1.9 3.6 6.2 HEC-18no wp corr 8.7 0.6 4.9 0.2 0.3 0.9 S/M 5.1 0.0 8.5 0.0 0.5 1.5 Table 11. Ranking of selected predictive equations based on performance for wide piers (a/D50 > 100, a/y1 > 2). Laboratory (17 points) Field (142 points) Reference SSE SSEn SSE SSEn Total Under Total Under Overall Under Under Overall Jain (1981) 5 1 5 1 1 6 5 5 Froehlich (1988) 6 2 6 2 6 1 1 1 Melville (1997)mod. 1 5 3 5 3 4 3 3 HEC-18 2 6 1 6 5 5 6 6 HEC-18no wp corr 4 4 2 4 4 2 2 2 S/M 3 3 4 3 2 3 4 4 statistics and ordering among these six equations based on for the dimensional scour (Figure 27), the normalized scour performance for wide piers. performance is significantly reduced (Figure 28). Figures 27 through 30 show underprediction error ver- Figures 31 through 34 are plots of predicted versus mea- sus total error for the six selected methods. These plots sured scour depths for six of the predictive equations for the show the performance of the different methods and how various data sets. their errors can be modified by a multiplicative constant. Based on these analyses, the best-performing equation The symbols indicate the errors produced by the method and the one that attempts to account for the most important with a multiplier of one. As the multiplier changes from local scour mechanisms is the S/M equation. To illustrate one, the symbol simply moves along the curve on which the this equation's dependence on the various dimensionless symbol lies. Note that shifting the position of the symbol groups, three example plots are presented in Figures 35 along its line for one data set does, however, shift its position through 37. The HEC-18 equation is also presented in these on the other plots. For example, if the HEC-18 equation is equations. multiplied by 1.2, its position moves from that shown with Figures 38 through 40 are additional comparisons between the left-pointing open triangle to that with the closed tri- the S/M and HEC-18 equations for three different prototype angle. Note that even though there is improved performance design conditions ( = 2.7, a/D50 = 10,000).

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31 2 Jain 1981 HEC-18 Froehlich 1988 Underprediction Error (%) Melville 1997-mod. 1.5 HEC-18 HEC-18-no wp corr HEC-18-no wp corr S/M 1 Froehlich 1988 0.5 Jain 1981 Melville 1997-mod. S/M 0 0 5 10 15 20 25 30 35 40 45 Total Error (% ) Figure 27. Underprediction scour error versus total error for laboratory data. 2 Jain 1981 Froehlich 1988 Froehlich 1988 Underprediction Error (%) Melville 1997-mod. 1.5 HEC-18 HEC-18-no wp corr S/M 1 Jain 1981 0.5 HEC-18 S/M HEC-18-no wp corr 0 Melville 1997-mod. 0 10 20 30 40 50 Total Error (%) Figure 28. Underprediction normalized scour error versus total error for laboratory data. 2 Jain 1981 Field Underprediction Error (%) Froehlich 1988 Melville 1997-mod. 1.5 Jain 1981 HEC-18 HEC-18-no wp corr S/M HEC-18 1 0.5 HEC-18-no wp corr S/M Melville 1997-mod. Froehlich 1988 0 0 5 10 15 20 25 30 35 40 45 Total Lab Error (% ) Figure 29. Underprediction scour error for field data versus total error for laboratory data.

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32 2 Jain 1981 Froehlich 1988 Field Underprediction Error (%) Melville 1997-mod. 1.5 HEC-18 HEC-18-no wp corr S/M HEC-18 1 Jain 1981 HEC-18-no wp corr 0.5 S/M Melville 1997-mod. Froehlich 1988 0 0 5 10 15 20 25 30 35 Total Lab Error (% ) Figure 30. Underprediction normalized scour error for field data versus total error for laboratory data. Jain 1981 Froehlich 1988 Melville 1997-mod. 5 5 5 4 4 4 Predicted ys (ft) Predicted ys (ft) 3 3 Predicted ys (ft) 3 2 2 2 1 1 1 %cases U.P.=17.2 %cases U.P.=30.3 %cases U.P.=7.25 0 0 0 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 Measured ys (ft) Measured ys (ft) Measured ys (ft) HEC-18 HEC-18-no wp corr S/M 5 5 5 4 4 4 Predicted ys (ft) Predicted ys (ft) Predicted ys (ft) 3 3 3 2 2 2 1 1 1 %cases U.P.=14.7 %cases U.P.=13.8 %cases U.P.=8.16 0 0 0 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 Measured ys (ft) Measured ys (ft) Measured ys (ft) U.P. = underpredicted Figure 31. Predicted versus measured equilibrium scour depths for six predictive equations for laboratory data.

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Jain 1981 Froehlich 1988 Melville 1997-mod. 3 3 3 2.5 2.5 2.5 Predicted ys/a Predicted ys/a Predicted ys/a 2 2 2 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5 %cases U.P.=17.2 %cases U.P.=30.3 %cases U.P.=7.25 0 0 0 0 1 2 3 0 1 2 3 0 1 2 3 Measured ys/a Measured ys/a Measured ys/a HEC-18 HEC-18-no wp corr S/M 3 3 3 2.5 2.5 2.5 Predicted ys/a Predicted ys/a Predicted ys/a 2 2 2 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5 %cases U.P.=14.7 %cases U.P.=13.8 %cases U.P.=8.16 0 0 0 0 1 2 3 0 1 2 3 0 1 2 3 Measured ys/a Measured ys/a Measured ys/a U.P. = underpredicted Figure 32. Predicted versus measured normalized equilibrium scour depths for six predictive equations for laboratory data. Jain 1981 Froehlich 1988 Melville 1997-mod. 40 40 40 30 30 30 Predicted ys (ft) 20 20 20 10 10 10 %cases U.P.=3.68 %cases U.P.=0.78 %cases U.P.=1.44 0 0 0 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 HEC-18 HEC-18-no wp corr S/M 40 40 40 30 30 30 Predicted ys (ft) 20 20 20 10 10 10 %cases U.P.=9.34 %cases U.P.=6.31 %cases U.P.=2.89 0 0 0 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 Measured ys (ft) Measured ys (ft) Measured ys (ft) U.P. = underpredicted Figure 33. Predicted versus measured equilibrium scour depths for six predictive equations for field data.

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34 Jain 1981 Froehlich 1988 Melville 1997-mod. 3 3 3 2.5 2.5 2.5 * Predicted ys/a 2 2 2 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5 %cases U.P.=6.82 %cases U.P.=1.13 %cases U.P.=1.39 0 0 0 0 1 2 3 0 1 2 3 0 1 2 3 HEC-18 HEC-18-no wp corr S/M 3 3 3 2.5 2.5 2.5 * Predicted ys/a 2 2 2 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5 %cases U.P.=11.1 %cases U.P.=6.95 %cases U.P.=5.05 0 0 0 0 1 2 3 0 1 2 3 0 1 2 3 * * * Measured ys/a Measured ys/a Measured ys/a U.P. = underpredicted Figure 34. Predicted versus measured normalized equilibrium scour depths for six predictive equations for field data. 2.5 S/M HEC-18 2 ys/a 1.5 1 D50=0.4 mm, y1/a= 3, V1/Vc=1 0.5 1 2 3 4 10 10 10 10 a/D50 Figure 35. Comparison of S/M and HEC-18 predictions of normalized scour depth versus a/D50.

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35 1 S/M HEC-18 0.8 0.6 ys/a 0.4 D50=0.4 mm, a/D50=10000, V1/Vc=1 0.2 0 0 1 2 3 4 5 y1/a Figure 36. Comparison of S/M and HEC-18 predictions of S/M normalized scour depth versus y1/a. 2 S/M HEC-18 1.5 ys/a 1 0.5 D50=0.4 mm, y1/a= 3, a/D50=10000 0 0 1 2 3 4 5 6 V1/Vc Figure 37. Comparison of S/M and HEC-18 predictions of normalized scour depth versus V1 /Vc. 35 S/M 30 HEC-18 25 20 ys (ft) 15 10 y1=30 ft, V1=3Vc, D50=0.2 mm 5 0 0 10 20 30 40 50 60 70 a (ft) Figure 38. Comparison of S/M and HEC-18 equation predictions of equilibrium scour depth as a function of pier diameter (for D50 = 0.2 mm).

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36 50 S/M HEC-18 40 30 ys (ft) 20 10 y1=30 ft, V1=3Vc, D50=0.8 mm 0 0 10 20 30 40 50 60 70 a (ft) Figure 39. Comparison of S/M and HEC-18 equation predictions of equilibrium scour depth as a function of pier diameter (for D50 = 0.8 mm). 70 S/M 60 HEC-18 50 40 ys (ft) 30 20 10 y1=30 ft, V1=3Vc, D50=5 mm 0 0 10 20 30 40 50 60 70 a (ft) Figure 40. Comparison of S/M and HEC-18 equation predictions of equilibrium scour depth as a function of pier diameter (for D50 = 5 mm).