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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
2
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.
3
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
4
Common Dangers
SubsurfaceEmbankment
Voids
LocalizedThaw
Settlement
ThermalErosion
InducedKarsting
SubsurfaceMassiveIce-
IceWedge
TensionCrackingfromLateralEmbankment
Spreading
DrainageDitchThermalErosion
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Danger Identification
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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÷
9
Iqaluit Airport Location
IQA_13_08
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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
11
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
11
0,0
12
0,0
13
0,0
14
0,0
15
0,0
16
0,0
17
0,0
18
0,0
19
0,0
20
0,0
21
0,0
22
0,0
23
0,0
24
0,0
26
0,0
27
0,0
28
0,0
29
0,0
30
0,0
31
0,0
32
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)
17
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
Thanks to our partnersand collaborators!
www.arquluk.gci.ulaval.ca
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Potential Input Parameters
20
Lateral Embankment Spreading