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

5

Danger Identification

6

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

7

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

8

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

10

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

12

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

13

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

14

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

15

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)

16

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

19

Potential Input Parameters

20

Lateral Embankment Spreading