Reliability Analysis for Dams and Levees

1. Reliability Analysis for Dams and Levees Thomas F. Wolff, Ph.D., P.E. Michigan State University Grand Rapids Branch ASCE September 2002

2. Hodges Village Dam

3. Walter F. George Dam

4. Herbert Hoover Dike

5. Some Background • Corps of Engineersmoving to probabilistic benefit-cost analysis for water resource investment decisions (pushed from above) • Geotechnical engineersmust quantify relative reliability of embankments and other geotechnical features • Initial implementationmust build on existing programs and methodology and be practical within resource constraints

6. Some Practical Problems • Given possibility of an earthquake and a high pool, what is the chance of a catastrophic breach ? (Wappapello Dam, St. Louis District, 1985) • Given navigation structures of differing condition, how can they be ranked for investment purposes ? (OCE, 1991+ ) • What is the annualized probability of unsatisfactory performance for components of Corps’ structures ? (1992 - 1997)

7. Some More Practical Problems • For a levee or dam, how does Pr(f) change with water height ? (Levee guidance and Hodges Village Dam) • How to characterize the annual probability of failure for segments of very long embankments ? (Herbert Hoover Dike) • How to characterize the annual risk of adverse seepage in jointed limestone ? (Walter F. George Dam)

8. General Approaches: Event Tree Close to levee p = 0.6 given some water level : 0.09 Carries material p=0.3 Sand Boil p = 0.5 Not close p = 0.4 0.06 Doesn’t p = 0.7 0.35 Most problems of interest involve or could be represented by an event tree..

9. Probabilities for the Event Tree • f (Uncertainty in parameter values) • Monte Carlo method • FOSM methods • point estimate • Taylor’s Series • Mean Value • Hasofer-Lind • Frequency Basis • Exponential, Weibull, or other lifetime distribution • Judgmental Values • Expert elicitation

10. Pr(f) = Function of Parameter Uncertainty • Identify performance functionand limit state, typically ln(FS) = 0 • Identify random variables, X i • Characterize random variables, • E[X], sx, r • Determine E[FS], sFS • Determine Reliability Index, b • Assume Distribution and calculate Pr(f) = f(b)

11. The Probability of Failure Answers the question, how accurately can FS be calculated?, given measure of confidence in input values

12. The Reliability Index, b E [ln FS ] b sln FS b = s ln FS Pr (U)

13. Taylor’s series, mean-value FOSM approach

14. Slope Stability Results, Lock & Dam No. 2

15. Lognormal distribution on FS, L&D 2 E[FS] = 2.41 s FS = 0.51 b = 4.11

16. Change in FS and Pr(f) ( Duncan’s Mine Problem from Uncertainty ‘96 Conference) • Evaluate shape change of probability density function due to drainage. • Provide enough drainage to obtain b > 4 FS = 1.3, VFS - 10% FS = 1.5, VFS = 10%

17. Advantages “Plug and Chug” fairly easy to understand with some training provides some insight about the problem Disadvantages Still need better practical tools for complex problems Non-unique, can be seriously in error No inherent time component only accounts for uncertainties related to parameter values and models Pros and Cons of b, Pr(U)

18. Physical Meaning of b, Pr(f) • Reliability Index, b • By how many standard deviations of the performance functions does the expected condition exceed the limit state? • Pr(f) or Pr(U) • If a large number of statistically similar structures (were designed) (were constructed) (existed) in these same conditions (in parallel universes?), what fraction would fail or perform unsatisfactorily? • Has No Time or Frequency Basis !

19. Frequency-based Probabilities • Represent probability of event per time period • Poisson / exponential model well-recognized in floods and earthquakes • Weibull model permits increasing or decreasing event rates as f(t), well developed in mechanical & electrical appliactions • Some application in material deterioration • Requires historical data to fit

20. Advantages Can be checked against reality and history Can obtain confidence limits on the number of events Is compatible with economic analysis Disadvantages Need historical data Uncertainty in extending into future Need “homogeneous” or replicate data sets Ignores site-specific variations in structural condition Pros and Cons of Frequency Models

21. Judgmental Probabilities • Mathematically equivalent to previous two, can be handled in same way • Can be obtained by Expert Elicitation • a systematic method of quantifying individual judgments and developing some consensus, in the absence of means to quantify frequency data or parameter uncertainty

22. Advantages Gives you a number when nothing else will May be better reality check than parameter uncertainty approach permits consideration of site-specific information Some experience in application to dams Disadvantages Distrusted by some (including some within Federal Agencies) Some consider values “less accurate” than calculated ones Non-unique values Who is an expert? Pros and Cons of Judgmental Probabilities

23. An Application:Levee Reliability = f (Water Level) • Previous Corps’ policy treated substandard levees as not present for benefit calculations • New policy assumes levee present with some probability, a function of water level • First approach by Corps took relationship linear, R = 1 at base, R = 0 at crown • New research to develop functional shape

24. Levee Failure Modes • Underseepage • Slope Stability • Internal erosion from through-seepage • External erosion • through-seepage • current velocity • wave attack • animal burrows, cracking, etc., may require judgmental models • Combine using system reliability methods

25. Pervious Sand Levee Example

26. FOSM Underseepage Analysis

27. Pr (underseepage failure) vs H

28. Probabilistic Case HistoryHodges Village Dam • A dry reservoir • Notable seepage at high water events • Very pervious soils with no cutoff

29. Probabilistic Case HistoryHodges Village Dam • Required probabilistic analysis to demonstrate economic justification • Random variables • horizontal conductivity • conductivity ratio • critical gradient • FASTSEEP analyses using Taylor’s series to obtain probabilistic moments of FS

30. Probabilistic Case HistoryHodges Village Dam

31. Probabilistic Case HistoryHodges Village Dam • Pr (failure) = Pr (FS < 1) • This is a conditional probability, given the modeled pool, which has an annual probability of occurrence

32. Probabilistic Case HistoryHodges Village Dam • Annual Pr (failure) = Pr [(FS < 1)|pool level] * Pr (pool level)Integrated over all possible pool levels

33. Probabilistic Case HistoryHodges Village Dam

34. Probabilistic Case HistoryWalter F. George Lock and Dam

35. Probabilistic Case HistoryWalter F. George Lock and Dam • Has had several known seepage events in 40 year history • From Weibull or Poisson frequency analysis, can determine the probability distribution on the number of future events

36. Probabilistic Case HistoryWalter F. George Lock and Dam

37. Probabilistic Case HistoryWalter F. George Lock and Dam

38. Probabilistic Case HistoryHerbert Hoover Dike • 128 mile long dike surrounds Lake Okeechobee, FL • Built without cutoffs or filtered seepage control system • Boils and sloughing occur at high pool levels • Failure expected in 100 yr event (El 21)

39. Probabilistic Case HistoryHerbert Hoover Dike • Random variables • hydraulic conductivities and ratio • piping criteria • Seepage analysis • FASTSEEP • Probabilistic model • Taylor’s series

40. Probabilistic Case HistoryHerbert Hoover Dike • Pr (failure) = Pr (FS < 1) • Similar to Hodges Village, this is a conditional probability, given the occurrence of the modeled pool, which is has an annual probability • Consideration of length effects • long levee is analogous to system of discrete links in a chain; a link is hundreds of feet or meters

41. Yes Comparative reliability problems Water vs. Sand vs. Clay pressures on walls, different b for same FS Event tree for identifying relative risks No Tools for complex geometries Absolute reliability Spatial correlation where data are sparse Time-dependent change in geotechnical parameters Accurate annual risk costs Questions Has the theory developed sufficiently for use in practical applications?

42. FOSM Reliability Index Reliability Comparisons structure to structure component to component before and after a repair relative to desired target value Insight to Uncertainty Contributions Questions When and where are the theories used most appropriately?

43. Frequency - Based Probability Earthquake and Flood recurrence, with conditional geotechnical probability values attached thereto Recurring random events where good models are not available: scour, through-seepage, impact loads, etc. Wearing-in, wearing-out, corrosion, fatigue Questions When and where are the theories used most appropriately?

44. Expert Elicitation “Hard” problems without good frequency data or analytical models seepage in rock likelihood of finding seepage entrance likelihood of effecting a repair before distress is catastrophic Questions When and where are the theories used most appropriately?

45. Define purpose of analysis Select simplest reasonable approach consistent with purpose Build an event tree Fill in probability values using whichever of three approaches is appropriate to that node Understand and admit relative vs absolute probability values Questions What Methods are Recommended for Reliability Assessments of Foundations and Structures ?

46. YES Conditional probability values tied to time-dependent events such as earthquake acceleration or water level NO variation of strength, permeability, geometry (scour), etc; especially within resource constraints of planning studies Questions Are time-dependent reliability analysis possible for geotechnical problems? How?

47. Needs • A Lot of Training • Develop familarity and feeling for techniques by practicing engineers • Research • Computer tools for practical probabilistic seepage and slope stability analysis for complex problems • Characterizing and using real mixed data sets, of mixed type and quality, on practical problems, including spatial correlation issues • Approaches and tools for Monte Carlo analysis

48. How accurately can Pr(f) be calculated? • Not very accurately (my opinion) --Many ill-defined links in process: • variations in deterministic and probabilistic models • different methods of characterizing soil parameters • f - c strength envelopes are difficult • slope is a system of slip surfaces • distributions of permeability and permeability ratio • difficult to quantify spatial correlation in practice • difficult to account for length of embankments • difficult to account for independence vs correlation of multiple monoliths, multiple footings, etc.