Quantifying Probability of Thaw Settlement Occurrence and ...
Transcript of Quantifying Probability of Thaw Settlement Occurrence and ...
Quantifying Probability of Thaw Settlement Occurrence and Vulnerability to Climate Warming
Heather Brooks, PE
Collaborators: Guy Doré PhD Eng; Michel Allard PhD; Chantal Lemieux
XI ICOP, Potsdam, Germany June 20-24, 2016
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PROJECT OBJECTIVETo create a quantitative risk analysis methodology and toolfor embankment-supported infrastructure on permafrost utilizing site conditions, physical and/or empirical engineering calculations and consequences.
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Definitions and Terminology
• Danger – an event or process causing damage
• Hazard – the probability of a danger’s occurrence within a time frame
• Random Variable - uncertain input parameter
• Probability Density Function – mathematical distribution of probable values
– Defined by the average and standard deviation
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Common Dangers
SubsurfaceEmbankment
Voids
LocalizedThaw
Settlement
ThermalErosion
InducedKarsting
SubsurfaceMassiveIce-
IceWedge
TensionCrackingfromLateralEmbankment
Spreading
DrainageDitchThermalErosion
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Risk Analysis
R = P x C• R = Risk• P = Probability of
Occurrence, Hazard• C = Consequence
Quantitative• P from past experience or
uncertainty calculation• C calculated from expected
damage
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Hazard - Calculation
First Order Second Moment (FOSM) Methodology
• Taylor series expansion of the failure equation – partial derivatives for each
input variable
• Result average– calculated from input
averages
• Result variation– summation of evaluated
partial derivatives
Monte Carlo Simulation Methodology
• Varies all random variables simultaneously – probability density functions
• Each calculation is a simulation
• Repeated simulations define the final– average
– standard deviation
• Accuracy dependent on simulation number
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Thaw Depth Calculation MethodsStefan Equation
• Partial derivatives
• Random variables– kt, w, γd, n, ATI, t
• Average properties for active layer depths
• FOSM Method Valid
Modified Berggren Equation
• Partial derivatives not possible
• Random variables– w, γd for each soil layer
– n, ATI, t, MASST
• Monte Carlo Method Required
X
t= 2i
kt
LT
t-T
0( )it
AIi=
Liid
i
24ili
2 in
dn
kn
+d
i
2ikim=1
i-1
åæ
èç
ö
ø÷
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Stefan FOSM Calculation Process
Variable Ave. STD
W (%) 11.2 5.3
γd (kg/m3) 2055.3 264.8
n 1.0 0.01
AI (°C-days) 704.8 99.2
Kt (W/m-K) 3.1 1.3
Xt (m) 2.6 0.9553
L = 344i
w
100ig
d
T
t=
niAI
t X
t= 2i
kt
LT
t-T
0( )it
Xt=
ktiniAI
1.72iwigd
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Excel Spreadsheet Tool
• Mod. Berg. Calculation
– Monte Carlo
– Climate & Soil Property Variation
• Normal Distribution
• VBA Program in Excel
Variation of 4 Climate Factors(AI, n, MASST, t)
Soil Parameters(w, γd) for each layer
Total Variables18
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Thaw Depth – Monte Carlo Results
0
200
400
600
800
1 000
1 200
1,5
4
1,6
8
1,8
1
1,9
5
2,0
9
2,2
2
2,3
6
2,4
9
2,6
3
2,7
7
2,9
0
3,0
4
3,1
8
3,3
1
3,4
5
3,5
9
3,7
2
3,8
6
4,0
0
4,1
3
Nu
mb
er
of
Cal
cula
tio
ns
wit
hin
Eac
h Z
on
e
Calculated Depth of Thaw (m)
Property Value
Ave. 3.14 m
MC Ave. (STD)
3.05 m (0.26) m
MC Min 1.47 m
MC Max 4.2 m
Iterations 10,000
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Thaw Depth - Analysis Method Comparison
0
0,2
0,4
0,6
0,8
1
0
0,4
0,8
1,2
1,6
2
2,4
2,8
3,2
3,6
4
0 0,5 1 1,5 2
Cu
mu
lati
ve P
rob
abili
ty F
un
ctio
n
No
rmal
ize
d N
orm
al D
istr
ibu
tio
n
Normalized Thaw Depth (m/m)FOSM Stefan MC Mod. Berg.
FOSM Stefan Cumulative MC Mod. Berg. Cumulative
Analysis STD - m
FOSM Stefan 0.95
MC Mod. Berg. 0.33
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Thaw Settlement – Analysis Method
• Layered Soil Analysis
• Luscher and Afifi 1973
– Empirical Relationships by soil type
– w and dry unit weight
• Monte Carlo Analysis
– Data from previous MC simulations
Figure 4 – Luscher and Afifi 1973
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Thaw Settlement – 1D Analysis Results
0
20
40
60
80
100
120
140
160
180
2000
,00
1
0,0
02
0,0
03
0,0
04
0,0
05
0,0
06
0,0
07
0,0
09
0,0
10
0,0
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0,0
12
0,0
13
0,0
14
0,0
15
0,0
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0,0
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0,0
18
0,0
19
0,0
20
0,0
21
0,0
22
0,0
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0,0
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0,0
26
0,0
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0,0
28
0,0
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0,0
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0,0
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0,0
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Nu
mb
er
of
occ
urr
en
ces
pe
r 2
00
0
Thaw Settlement (m)
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Vulnerability to Climate Warming
• Annual sinusoidal climate model
• Vary average air temp– New thawing Index
– Thaw season duration
• Recalculate– Thaw depth
– Thaw settlement
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
0 1 2 3 4 5
Air Temperature Increase (°C)
Thaw Settlement Average (cm)
Average Thaw Depth (m)
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Next Steps
• Analysis of differential thaw settlement– Spatial geostatistics
• Inclusion of additional dangers within the program
• Consequence analysis – Direct and indirect
• Documentation– Background calculations
– User guide