Time-Dependent Reliability Modelingfor Use in Major Rehabilitation of

Embankment Dams and Foundation

Robert C. PatevNAD Regional Technical Specialist

Navigation/Risk and Reliability

Outline

� Time-Dependent Reliability� Hazard Functions� Wolf Creek Major Rehabilitation Report

(MRR)� Time-Dependent Reliability Modeling� Expert-Opinion Elicitation

Reliability� Probability of unsatisfactory performance

(PUP)� Limit state defined before failure occurs� Problem – snapshot in time� Not cumulative (does not account for previous

loadings)� Must account for degradation of structures

� Mechanisms� Corrosion� Fatigue� Freeze-Thaw� Wear� Abrasion/Erosion

Time-Dependent Reliability� Geotechnical Aspects

� Difficult task-at-hand� Foundations

� Karst� Used Expert-Opinion Elicitation to define PUPs� Snapshot

� Alluvial� Used Taylor Series� Snapshot

� Degradation� Data/Rates� Models

Time-Dependent Reliability

f(R)

f(D)

Time, t

R , D L(t) = f(R)*g(t) f(D)

h(t) = - ln d/dt (L(t))

Time-Dependent Reliability

� Hazard Function� Developed by actuaries in 1880’s� Used by aerospace industry in early 1950’s� Conditional probability

� h(t) = PUP [t + dt, t]� PUP in time, t+ dt, given you have survived up to

time, t� Based on efforts on Ohio River Mainstem Study

ttime,toupsurvivedofNumber1ttime,inesperformanctoryunsatisfacofNumberh(t) +=

Hazard Function

Time, t

Burn-in

h(t) dec.

h(t)

Burn-out

h(t) inc.

Life phase

h(t) constant to inc.

Wolf Creek Dam Project

� Concrete 240’ ht 1836’ length� Earth Emb 200’ ht 3900’ length

Wolf Creek Dam - History� Designed in late 1930’s� Construction began 1941� Completion delayed until 1951 due to WWII� 1967-68 Sinkholes near Switchyard + right

d/s abutment wet areas D/S of embankment,muddy flows

� 1968-72 Emergency exploration / grouting� 1975-79 Diaphragm walls construction

(ICOS)

Wolf Creek Dam – Typical Section

0 1+00A 2+00A 3+00A 4+00A 5+00A1+00B2+00B3+00B4+00B5+00B5+00B

EL. 660

EL. 690

EL. 730

EL. 763EL. 773

EL. 730

EL. 690

EL. 665

EL. 640

Flow

Axis of Dam

C.L. Roadway & Embankment Sta. 0+16B

Foundation of Toe Drain

End of Drainage Blanket

Core trench

Drainage

Blanket

Bedrock

Emb

Qal

Wolf Creek Dam – 1967/1968Events

Wet Area

No. 1

Wet Area

No. 2

Wet Area

No. 5

Wet Area

No. 3

Wet Area

No. 6

Sinkhole 3’ Dia.

15 August 1967

Sta. 56+75L,4+36B

Wet Area No. 4

Sinkhole No. 1

13 March 1968

Sta. 32+16L, 4+13B

13’ Dia. & 10’ Deep

Sinkhole No. 2

22 April 1968

Sta. 32+58L, 4+13B

6’ Dia. & 6’ Deep

Muddy

Water

Downward Slope of

Cut Off Trench

Cut Off Trench

Toe of Random Fill

Wolf Creek Dam – Piezometers

0 1+00A 2+00A 3+00A 4+00A 5+00A1+00B2+00B3+00B4+00B5+00B5+00B

EL. 660

EL. 690

EL. 730

EL. 763EL. 773

EL. 730

EL. 690

EL. 665

EL. 640

Flow

Axis of Dam

C.L. Roadway & Embankment Sta. 0+16B

Foundation of Toe Drain

End of Drainage Blanket

Diaphragm Wall

PZ No. DC-248R

Sta. 35+49L 0+06A

PZ No. D-323

Sta. 35+55L 0+35B

March 03March 03

Wolf Creek Dam – CutoffTrench

Wolf Creek Dam – CutoffTrench

Wolf Creek Dam – DiaphragmWall, 1975-1979

Diaphragm Wall in Emb Concrete

Limestone

Wolf Creek Dam – DistressIndicators

C o m p aris o n of D is tres s Ind ic ato rs 1 96 8-20 04 at W o lf C re ek P ro jec t

3

8

0 .2 0 0 0 0

6

1

37

4 .2

2

64

3

7

0

5

10

15

20

25

30

35

40

N o. o f S ink s N o . o f W etA reas

% of W et A reasD /S

S eepageIns tab ility o fR ive r B ank

Z ones , N o . o fLoca tions

N o . o f A rtes ianP Z 's

C racks per 100F ee t a t R oad

S urface (100 '=1 )S ta . 35+11-

39+00

E m bankm entS ettlem en t

(0 .1 '=1 ) M axS e ttlem en t atS ta tion 37+00

P Z H ead(10% =1)

H ighest H eadsBe tw een 35+11-

40+00Q u alita tiv e D is tress In d icato r

* T em pera ture m easurem ents from P Z s in 1968 and 2004 show anom a lis tic co ld a reas dow ns tream of dam axis .

Dis

tres

s

19682004

Wolf Creek Dam – CrestSettlement

W olf C re ek S ettlem en t R ate s

0 .00

0 .05

0 .10

0 .15

0 .20

0 .25

0 .30

0 .35

0 .40

0 .45

0 .50

J-81 J-82 J -83 J -84 J-85 J -86 J -87 J-88 J-89 J -90 J -91 J-92 J -93 J -94 J-95 J-96 J -97 J -98 J-99 J-00 J -01 J -02 J-03 J -04 J -05

D ate

Settl

emen

t (Fe

et)

E M -1E M -2E M -3E M -4

E M -5E M -6E M -7E M -8E M -9E M -1 0E M -1 1E M -1 2E M -1 3E M -1 4E M -1 5E M -1 6E M -1 7E M -1 8E M -1 9E M -2 0E M -2 1E M -2 2E M -2 3E M -2 4E M -2 5E M -2 6

Wolf Creek Dam – PiezometricData - Section 1

WOLF CREEK DAMSelected PZ's (20 Year trend)

540

550

560

570

580

590

600

610

620

630

640

650

660

670

680

690

700

710

720

730

740

750

760

J-84 J-85 J-86 J-87 J-88 J-89 J-90 J-91 J-92 J-93 J-94 J-95 J-96 J-97 J-98 J-99 J-00 J-01 J-02 J-03 J-04

DATE

PZ

Ele

vatio

ns in

Fee

t (M

SL)

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

Rai

nfal

l in

Inch

es

D-321D-322D-323HeadwaterTailwaterRainfallLinear (D-322)

Wolf Creek Dam – PiezometricData -Section 2

WOLF CREEK DAMSelected PZ's (20 Year trend)

540

550

560

570

580

590

600

610

620

630

640

650

660

670

680

690

700

710

720

730

740

750

760

J-84 J-85 J-86 J-87 J-88 J-89 J-90 J-91 J-92 J-93 J-94 J-95 J-96 J-97 J-98 J-99 J-00 J-01 J-02 J-03 J-04

DATE

PZ

Ele

vatio

ns in

Fee

t (M

SL)

DC-258DC-254RWA-25HeadwaterTailwaterWA-49Linear (WA-49)

Wolf Creek Dam – PiezometricData - Section 3

WOLF CREEK DAMSelected PZ's (20 Year trend)

540

550

560

570

580

590

600

610

620

630

640

650

660

670

680

690

700

710

720

730

740

750

760

J-84 J-85 J-86 J-87 J-88 J-89 J-90 J-91 J-92 J-93 J-94 J-95 J-96 J-97 J-98 J-99 J-00 J-01 J-02 J-03 J-04

DATE

PZ E

leva

tions

in F

eet (

MSL

)

D-275AWA-99FWO-35HeadwaterTailwaterLinear (WA-99F)

Wolf Creek Dam – Time-Dependent Reliability Concepts

PiezometricHead,Flow, Surface

Water

DefinedLimitState

Time1951 1968 1979 2004

Groutwall

Diaphragmwall

1984 2060

0

?

PiezometricHead,Flow, Surface

Water

DefinedLimitState

Time1951 1968 1979 2004

Groutwall

Diaphragmwall

1984 2060

0

?

Time1951 1968 1979 2004

Groutwall

Diaphragmwall

1984 2060

0

?

Wolf Creek Dam MRR

� Time-Dependent Reliability Model� Utilize permanent rise in piezometric

pressures in the foundations� Reliable and most consistent data� Model three different embankment sections� Assume that selected limit state occurs before

any decrease or fall in piezometric pressurewhich would indicate a more critical situation

Wolf Creek Dam MRR

� Time-Dependent Reliability Model� Damage Accumulation Model (DAM)

� Used in fatigue and wear rate analyses� Model cyclic variations of headwater above EL

710

El 710

AnnualDuration

Annual Intensity

Cyclic Headwater

El 710

AnnualDuration

Annual Intensity

Cyclic Headwater

Wolf Creek Dam MRR

� Time-Dependent Reliability Model� DAM Calibration

Annual and Accumulated Damages Versus Time

0.000

2.000

4.000

6.000

8.000

10.000

12.000

1950 1960 1970 1980 1990 2000 2010

Time

Piez

omet

ric R

ise

(in fe

et)

Annual DamagesAccumulated DamagesActual Piezometric Rise

Wolf Creek Dam MRR� Time-Dependent Reliability Model

� Developed Monte Carlo Simulation Model� Accounted for entire life cycle including past remedial

repairs� Performed 50,000 iterations for reliability calculations

� Random Variables� Annual Intensity –

� Based on historical records from 1950 to 2004� Truncated lognormal distribution

� Annual Duration –� Based on historical records from 1950 to 2004� Truncated lognormal distribution

Wolf Creek Dam MRR� Time-Dependent Reliability

� Random Variables (cont’)� Spatial Variability Factor-

� Accounts for variation in piezometric pressures in thefoundation

� Based on a wide range of piezometer data for each section� Incorporated as quadratic modifier to annual damage

accumulation� Uniform distribution

� More uncertainty in Section 3 than Sections 1 or 2 dueto continued wet spots downstream and lack ofdiaphragm wall

Wolf Creek Dam MRR

� Time-Dependent Reliability Model� Limit state

� Defined limit state for unsatisfactoryperformance on piezometric rise in 3 sections

� Used Expert-Opinion Elicitation to quantifythose values

Section Average PZValue in 2004

(in feet)

Rise of PZfrom EOE(in feet)

UnsatisfactoryPerformance Limit State

(in feet)

1 9.3 5 14.32 4.1 5 9.13 1.2 3 4.2

Wolf Creek Dam MRRWolf Creek Expert Elicitation

Event Full Description Expert-opinion elicitation SummaryName of Issue Table

First SecondResponse Response

Unsatisfactoryperformance occurs

in Section 1

At what future change inpiezometric pressure in thefoundation would you expectunsatisifactory performance inSection 1?

Median = 5 Median = 5

ConfidenceMinimum = 1

Expert #1 0.5 1 med 25 Percentile = 3Expert #2 5 5 highExpert #3 10 7 med Median = 5Expert #4 3 2 medExpert #5 5 5 med 75 Percentile = 5Expert #6 5 4 high

90 Percentile = 6

High = 7

Minimum = 0.5 1Median = 5 5

Maximum = 10 7

Wolf Creek Dam MRRTime Dependent Reliability for Sections 1, 2 and 3 - Wolf Creek

Dam

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1940 1960 1980 2000 2020 2040 2060 2080

Time (years)

Rel

iabi

lity Section 1

Section 2Section 3

Wolf Creek Dam MRR

Summary of Hazard Rates for Wolf Creek Dam

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1940 1960 1980 2000 2020 2040 2060 2080

Time (years)

Haz

ard

Rat

e Section 1Section 2Section 3

Wolf Creek Dam MRRWolf Creek Dam Event Tree

Baseline Condition - Section 1

Limit State - PUP Annual Hazard Rate Event Event Probability Repair Scenarios/Costs Effect on Hazard Rate(from Reliabilty Model) (from EOE)

Full UP 0.15 Rebuild New Dam AHR changed to 0 for remainder of lif(Dam breach) 5 Years Design

5 Years Construction

AHR Drain Reservior to El 680Time, T 5 ft unacceptable rise in X Partial UP 0.6 Grout of Dam AHR adjusted back to 0.1

piezometric pressure (Settlement, sinkholes, piping, wet spots) 1 Year Construction Degrades using AHR

Increased Surveillance and Monitoring 0.25 O&M Costs, Instrumentation Time T + 1(Reach limit state but no observable damages) and increased monitoring (AHR changes to next year)

1-X O&M Costs Time T + 11-AHR (AHR changes to next year)

Time-Dependent Reliability� Conclusions

� Time-dependent geotechnical models are difficult� Need to think well outside the box (deductive versus

inductive thinking)� Incorporate and define distress indicators

� My require some data processing

� Understand - models are for major rehabilitationpurpose to gain funding to rehab structure

� Don’t get lost in the fine details� Not the true probability of failure

� Within an order of magnitude� Not for dam safety