Presented by: David T. Williams, Ph.D., P.E., P.H., CFM. D.WRE

1. Double Counting, Overconservative or Misapplication of Safety Factors for Stream Scour Scour AnalysesPresented to:U.S. Bureau of Reclamation Presented by: David T. Williams, Ph.D., P.E., P.H., CFM. D.WRE Senior Technical Advisor, Water Resources dtwilliams@pbsj.com

2. Scour Components We often apply safety factors to all of the above components (or to the sum of the components) to take into consideration: variability of equation coefficients (they are based upon regression analyses) field measurements variability of nature model capabilities numerous other reasons (warm and fuzzy)

3. Ways to Look at Safety Factors Determine total scour and multiply by a safety factor (SF) Determine individual scour components and apply a SF to each component weighted by the confidence in the accuracy of each scour component Reasonably vary an important parameter (or combination of parameters) that determines the scour component’s magnitude SF can depend on functional/catastrophic failure analyses SF can depend on uncertainty, acceptable risk and consequences

4. But if it was a perfectly uncertain world,it would be done like this … Determine the variability (uncertainty) of an important parameter (or combination of parameters) of a scour component (not easy to do!) and develop a probability density function of the parameter(s) – see next slide. Use a technique such as First Order Second Moment Analysis (FOSM) or Monte-Carlo Simulation to determine the probability density function (PDF) of the scour component. Using an acceptable probability of occurrence and the cumulative probability curve, determine the scour magnitude.

5. Parameter Value Probability of Occurrence

6. Scour Component Magnitude --- Probability of not occurring = 0.9 -------> --------- Design Magnitude of scour ------>

7. But it is not a perfect world – so we must work with what we have As pointed out earlier, several ways to apply safety factors – but the correct method is to vary the input variables of the scour components, not to apply a SF to the resulting scour magnitudes Variation of an input variable should be based upon: the method (does it already have a built-in SF?), experience in the method, the range of data used in its derivation, and confidence in the variable(s) (e.g., is numerical or normal depth used to get depth to which SF is applied?) that the SF is being applied to within that method.

8. Long-term Bed Elevation Change -Equilibrium (Stable) Slope Slope at which the sediment transport capacity is equal to the incoming sediment supply An undisturbed channel will tend to shift towards equilibrium slope over the long-term – does not mean it will eventually get there! Equilibrium slope equations provide a useful order-of-magnitude assessment of the likelihood of vertical channel adjustment and gives a lower bound slope Dominant discharge or “bankfull” discharge are often used for stable slope calculations

9. Long-term Degradation Degradation determined through Location of a downstream control point (e.g., bedrock outcrop, grade control structure, etc.) Pivoting the equilibrium slope around the control point Long-term degradation component = (Existing slope – Equilibrium Slope) x Distance

10. Common Equilibrium Slope Equations Schoklitsch Method Meyer-Peter, Muller Method Shield’s Diagram Method Lane’s Tractive Force Method Let’s look at Meyer-Peter and Muller

11. Meyer-Peter, Muller Method SL = Kmpm (Q / Qb) (ns / D901/6)3/2 D/d where: SL = stable slope, (ft/ft) Kmpm = 0.19 Q/Qb = ratio of total flow to flow over the channel Q = dominant discharge, (cfs) ns = Manning’s n for the stream bed D90 = bed sediment diameter for 90% finer, (mm) D = mean sediment diameter, (mm) d = mean depth, (ft)

12. Apply Variation to Inputs Variations to discharges can be based upon frequency analyses and confidence limits (see next slide) and since “n” and depth are related to discharge, use the resulting discharge (increase or decrease to see worst case) in a hydraulic model with same “n” and using resulting flow depth. Can vary the “n” value (increase/decrease to see worst case) and keep discharge constant. Can do multi-variant, but hard to apply unless using techniques like Monte Carlo. Equilibrium slope methods are inherently conservative (produce small slopes) since they assume either incipient motion or no sediment is being transported – no SF should be applied and used only to obtain a lower bound. Equilibrium slope results should not be used for designing a channel slope since it could result in excessive deposition.

13. t

14. General Scour – e.g., Zeller Equation ygs = ymax [(0.0685 Vm0.8) / (yh0.4 Se0.3) -1] where: ygs = general scour depth, (ft) ymax = maximum depth of flow, (ft) Vm = average velocity of flow, (ft/s) yh = hydraulic depth of flow, (ft) Se = energy slope (or bed slope for uniform flow), (ft) Applicable to sand bed streams

15. Apply Variation to Inputs Since the hydraulic parameters (velocity and flow depth) are determined by discharge, the variation could be applied to discharges as previously described. May want to vary the “n” value (increase/decrease to see worst case) and keep discharge constant – see previous presentation for recommended variations). Note that the Zeller General Scour usually results in smaller scour depths compared to other methods because it was based upon best fit of data, not “envelope” curves like others.

16. General Scour Methods by BOR Generally utilize four methods for estimating general scour Field measurements of scour Regime equations Mean velocity from field measurements Competent or limiting velocity

17. Field Measurements of Scour (Envelope Curve) Note that BOR Envelope Curve is based upon an envelope curve, not a regression curve, implying that there is already a SF inherent in the scour results. A small or no SF should be applied if using this method. Applicable for: Ephemeral, relatively steep, wide sand bed streams in southwestern U.S. D50 from 0.5 to 0.7 mm (coarse sand) Slopes from 0.004 to 0.008 ft If significantly outside these ranges, increase the SF or preferably, do not use this method.

18. Field Measurements of Scour (Envelope Curve) Navajo Indian Irrigation Project – Scour versus Unit Discharge

19. Recommendations for BOR Equations Should use variation on discharge as previously described. If using Blench for straight or moderate channels and not adjusting the input variables, the z multiplier is conservative - no SF. Blench method used incipient motion (zero bed factor), therefore it is already conservative and no SF is needed. Lacey and Neill methods are regression type methods and will generally be lower than Blench, so appropriate SFs should be used. Blench, Lacey and Neill methods include bend scour, so no bend scour method should be added if using these methods for combined general and bend scour.

20. General Scour:Competent or Limiting Velocity ys = ym (Vm / Vc -1) where: ys = scour depth below streambed, (ft) ym = mean depth, (ft) Vc = competent mean velocity of particle, (ft/s) Vm = mean velocity, (ft/s)

21. General Scour:Competent or Limiting Velocity Note that this method uses incipient motion (competent particle velocity). This assumes that once the particle moves, the general scour continues until the water velocity gets below the particle velocity threshold. Also assumes no sediment particles from upstream replaces the moving particle. Overtly conservative – no SF is required.

22. Recommendations for Zeller, Maynord and Thorne Bend Scour Methods Zeller and Thorne are regression type relationships. Therefore, need to vary input variables (ymax is suggested) as discussed earlier. Maynord is also a regression type relationship but he suggests using an SF of 1.08 to the results. For Maynord, may want to vary input variables to see if results are greater than 1.08. If so, use the larger. If less, use minimum of 1.08. Maynord and Thorne methods include general scour.

23. Corps of Engineers Bend Scour Scour depth in Bends for Sand Beds (separate chart for gravel beds)

24. Corps of Engineers Bend Scour Design curves for sour in bends are designated as safe design curves and includes general scour. Represents upper limit for channels with irregular alignment - use 10% reduction from bend scour design curve for relatively smooth alignment. This is an envelope type of method so it inherently has a SF - no SF is required. Could vary the discharge as previously discussed to obtain the variations in the water depths but this will result in only small changes since the ratio of the depths would stay about the same.

25. Bed Form Scour - Antidunes – Kennedy ha = 0.14 (2  V2) / g = 0.027 V2 ys = ½ ha where: ys = bed form scour depth below original bed, (ft) ha = antidune height form crest to trough bed, (ft) V = mean channel velocity (ft/s) g = acceleration of gravity, (32.2 ft/s2) y = actual depth of flow, (ft)

26. Bed Form Scour - Antidunes - Kennedy The antidune height can never be greater than the depth of flow – this limits the SF applied to flow depth Equation applies when ha < y Assume ha = ywhen calculated value of ha > y Vary discharge or “n” in hydraulic model as previously mentioned to obtain velocity

27. Bed Form Scour - Dunes - Yalin ys = 0.167 ymax where: ys = bed form depth below original bed, (ft) ymax = maximum depth of flow, (ft) ybf = depth of bed form scour, (ft) Note: Vary discharge or “n” to obtain ymax

28. First Example of Total Scour Bedform (Dune) Scour = 1.6 feet Long Term Scour = 2 feet Zeller Bend Scour = 2.5 feet Contraction (general) scour = 4.0 feet Pier (local) scour = 9.2 feet. Zeller general scour = 1.0 feet (disregard because less than contraction scour) Thalweg incisement = 0 (because detailed topography captures thalweg) Total Scour = (1.6 * 1.2) + (2 * 1.3) + (2.5 * 1.2) + (4.0 * 1.2) + (9.2 * 1.1) = 22.4 ft

29. Second Example of Total Scour Bedform (Antidune) Scour = 0.8 feet Long Term Scour = aggrading 2 feet (disregard) Local scour = 0 feet General scour (examination of methods and selecting appropriate method) = 1.5 feet Maynord Bend Scour = 0.9 feet (disregard because less than general scour – Maynord’s method includes general scour) Thalweg incisement = 1 Total Scour = (0.8 * 1.3) + (1.5 * 1.2) + (1 * 1.5) = 3.3 ft

30. Miscellaneous Recommendations Contraction scour contains general scour but not long term scour. For contraction scour, also calculate general scour; take the larger. For bend scour, also calculate general scour; take the larger result. The SF methods and values may be dictated by the local agency but generally establish minimum requirements to determine scour depths; still do the analyses as suggested and take the larger of the results. Vary inputs or apply SF to scour depth results, but not to both. To prevent double counting (e.g., adding bend scour to a general scour method that already includes it), use a accounting method as followings.

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