Rock Mechanics and Radioactive Waste Isolation.“One small step for geology, one giant leap for rock mechanics”

Fourth Müller LectureFourth Müller Lecture

presented by

Charles Fairhurst

Professor Emeritus, University of Minnesota, U.S.A.Senior Consultant Itasca Consulting Group IncSenior Consultant, Itasca Consulting Group, Inc.

Minneapolis, MN 55401, U.S.A. fairh001@umn.edu

Tenth Congress of the International Society for Rock MechanicsTenth Congress of the International Society for Rock MechanicsSandton, South Africa, September 8-12, 2003

International Society for Rock Mechanics

Early Members Journal

1 - Prof. Leopold Müller, Austria2 Mr F Pacher Austria 2 - Mr. F. Pacher, Austria

3 - Prof. L. V. Rabcewicz, Austria 4 - Mr. C. Lorber, Austria 5 - Prof. F. Kahler, Austria6 - Dr. W. Zanoskar, Austria 6 Dr. W. Zanoskar, Austria

7 - Mr. W. Finger, Austria 8 - Dr. A. Fuchs, Austria 9 - Prof. F. K. Müller, Austria10 - Mr. P. Reska, Austria11 - Dr. K. Waschek, Austria12 - Prof. G.B. Fettweis, Austria13 - Dr. Georg Beurle, Austria14 - Neue, Reformbaugesellschaft MbH, Austria

(Supporting Member) (Supporting Member)15 - Dr. Alois Kieser, Austria16 - Prof. H. Seelmeier, Austria17 - Dr. Adolf Bretterklieber, Austria18 - Mr. W. Wessiak, Austria,19 - Prof. A. Watznauer, East Germany20 - Prof. Charles Fairhurst, USA

Ph.D. degrees in Rock Mechanics, University of Minnesota (1960-2003) & Departmental Colleagues & Associates

1960-1969:

Paul GnirkBezalel Haimson

1970-1979:

Francois CornetSteve Crouch

1980-1989:

Gary CallahanChristine Detournay

Honorary Members:

Randall BarnesBarry BradyBezalel Haimson

William HustrulidHassan ImamHerbert KutterWilliam PariseauHil S h f ldt

Steve CrouchJaak DaemenMichael HardyJohn HudsonDarrell PorterJ Cl d R i

Christine DetournayEmmanuel DetournayRoger HartJose LemosErnest LindnerL L i

y yTed BrownNeville CookPeter CundallAndrew DrescherBojan Guzina

1990-present:

Hilmar von SchoenfeldtWolfgang Wawersik

Jean-Claude RoegiersRaymond SterlingEd Van EeckhoutMichael Voegele

Loren LorigPanos PapanastasiouSamuel SharpYongjia Wang

Bojan GuzinaJoe LabuzTom LangYoshi MizutaChuck Nelson1990 present:

Jose AdachiMargaret AsgianNathali BoukpetiIlya Berchenko

Robert SanturbanoAlexei SavitskiSakir SelcukEduard Siebrits

Anthony PearsonFritz RummelMiklos SalamonChris St. JohnTony Starfield

Ali FakimiDimitri GaragashMatt Handley Haiying HuangIlya Berchenko

Bjorn BirgissonMark BoardRoberto CarbonellCarlos Carranza-Torres

Eduard SiebritsKetan ShahKeh-Jian (Albert) ShouYou TianYiming Sun

ff h

Tony StarfieldOtto Strack

Haiying HuangMark LarsonChengho LeeMarc LokenMark Mack

Professorship

Fernanda CarvalhoLiangsheng ChengBranko Damjanac

Jeff WhyattSanchai MaitaimLee PetersenThomas Richard

The problem. Waste Decay

10 7

• Contains the byproducts offi i ti

s

10 6fission reactions.

• Is highly radioactive.

vity

Cur

ves

10 5• Must be isolated fromthe public for many thousands of years.

Rad

ioac

tiv10 4

y

Needed 1 million year

1010

0

3

10 1 10 2 10 3

radionuclide ‘container’

Geological Isolation?

Years

Dose-response relationships for solid cancer, Alternative hypothetical extrapolations of d th t f hi h t

23E-4

RR

)

ncer

p p ,all types combined, in atomic bomb survivors, 1958-1987 (from UNSCEAR, 1994)

cancer death rate curves from high tolow radiation dose rates

1

2E- 4

lativ

e R

isk (E

R

isk

of fa

tal c

an

LNT

LT

0 1 2 30

0

1E- 4

Exce

ss R

el

Ann

ual r

i

NLH? ?

B0

Weighted intestinal dose, Sv

25 mrem15 mrem4 mrem

0.001 S100 mrem

v 0.005 S500 mrem

v 0.01 S1 rem

vLinear, no thresholdLinear, threshold

LNTLT

Annual dose, Sv

100 mrem 500 mrem 1 remNon-linear, HormesisRange of natural backgroundradiation in U.S. (approximate)

NLHB

Radiation Risk and Waste Isolation Standard. LNTH and Natural Background

R f

Risk of death from radiation

Global Dose rates from Natural Background Radiation

Source of ExposureCosmic Rays

Terrestrial Sources

Radionuclides in Body

0.392.0

0.464.3

0.23

Typical ElevatedDose Rate (mSv/year)

Average Global EffectiveDose Rate (mSv/year)

Reference:ICRP, 1999. "Protection of the Public in Situations of Prolonged Radiation Exposure," Publication 82, Annals of the ICRP, Vol. 29, Nos. 1-2, International Radiation

(except Radon)Radon and its

Decay Products

0.61.3

10.0

Total Global Dose Rate From 2 4 ( d d)

Commission on Radiological Protection, Pergamon Press, New York, NY. Table A.1, page 73, and Table A.2, page 74.

Area

States of Rio de JaneiroEspirito Santo, Brazil

~30 (3.6 average)

Total Global Dose Rate From Natural Background Radiation 2.4 (rounded)

Monazite sand; coastal

Examples of Areas with High Natural

sp o S o,

Volcanic intrusions in 6 km scattered inland areas

~80 (13.3 average)

Kerala and Tamil naduIndia

Central Regionf F

Granitic schistous and sandstone

~6

Mineas Gerais andGoias, Brazil

Monazite sand; coastal area, 200 km long and 0.5 km wide ~30 (9 average)

2

gBackground Dose Rates

of France Granitic, schistous and sandstone

Niue Island(Cook Islands, NZ)

~5

MombasaKenya

Thorium bearing carbonalyte

~30

Ramsar 200

Volcanic soil

Maximum Total Dose Rate (mSv/year) Characteristics of Area

RamsarIran

Areas of radium-226 deposition from spring water

~200

MahallatIran

~20 Areas of radium-226 deposition from spring water

Dose as a function of tume, for SF, for a number of different realisations including those that gave the highest (samplerealisations, including those that gave the highest (sample 164) and the lowest (sample 151) dose maximum out of 1000 realisations.

The Base Case is included to allow an easy comparison. The other curves represent realisations that give the highest dose maximum or the lowest dose maximum in the period between 10 and 10 ears or 10 and 10 ears respecti el The bars4 5 6 710 and 10 years or 10 and 10 years, respectively. The bars beneath the graph indicate the radionuclides that make the highest contribution to dose at any particular time.

4 5 6 7

Predicted Performance of a Repository in Opalinus Clay Spent Fuel —from 02-05, p.229NAGRA Technical Report

“One small step for geology, one giant leap for rock mechanics”

Pre-closure Isolation

A f th E th

• Engineering Problem

Age of the Earth (years)10 10 10 10 100 2 4 6 8

Engineering Problem

Unprecedented Period of Time

Requires Improved Scientific BaseRequires Improved Scientific Base

Periods of Isolation. Geological Isolation Time Scales.

“We don’t know the rock mass strength. That is why we need an International Society” Muller, 24 May 1962

Deformation

Forc

e

Rock mass

De

Intact block

Single joint

Time [10 ~ 10 ye

2

6Forc

e Decreasingdeformationrat e

Continuummechanic s

Localization and disintegration

SIZE

years]

5

Ela sticity

Plasticity

Damagemecha-nics

Ela stic unloading

Energy excess(Seismic

Maximumdesign f

TIME

1 2 3

4

Plasticity (Seismic deformation)

Energy deficit(Aseismicd f ti )

force

Deformation

deforma tion)

Complete Load-Deformation Behavior

“In the field of geomechanics, granular media and block-jointed rockmasses are obvious examples where the concept of the ideal physicalcontinuum −one in which no gaps are formed− cannot be expected to

l h l d l ff l happly... The clastic model offers an alternative approach.

Indeed, it is this writer’s view that only with clastic models or some further development thereof can the problem of predicting the complete load-deformation behaviour of solids be tackled optimistically.”

D.H. Trollope (1968)p ( )

Choose a geologically simple, stable site!(low tectonic strain rate)

Geological Isolation

Overlying and underlying strata:Flat terrain

Borehole

Overlying and underlying strata:- Infrequent joints/faults.- Far-field stresses close to isotropic.- Groundwater discharge to large body of non-potable water (ocean, sea).

σ σ

σ

h v

v

= k (k 1)~

>200 m Repository stratum, simple host medium::- Low permeability.- Low hydraulic gradient.- High ion-exchange potential.

Establish and close the repository

Backfill EDZ (excavation disturbed zone)surrounds seal and backfillDetail of

seal and

SealSeal1 mm gap

seal andbackfill

0 5Exothermic

Seal1 mm gap

90

KK

1010

Hydraulic Conductivity

no gap

1mm gap

~~

-9

-3m/sm/s

o

0.5 m

Drift andcanister

Backfill 2.5 mg p

Compaction (6%) due to particlefriction reduction after t~10,000 years

Rock Charac eteris ticsRock Type

P bi lit I E h Therm al St th D tilit

Clay Low

Salt Low

Very Low

Im perme able

Permeabi lity

Very High

None

Ion Exchange

Low

High

Therm alConducti vity

Strength

High

High

Ductility

++ − − +

Crystalline High

Volc anic Tuff Low

p

Matrix LowFractures High

High

UsuallyNone

High

g

Variable

Low

g

Low

Moderate

+

ο

−

+

−

−

ο

−+ +

−ο +

+

Current Research (and Operational) Sites

Clay Indurated Bure, Franc eMont Terri, Switzerland

ο + − − ο

,Benken, Switzerl andNon-Indurated

Mol, BelgiumSalt DomalGorleben, Germany

Bedded WIPP, Carlsbad, NM, USA(intermediate level ope rational)

Be low water table:Re ducing environment.

( p )Crystalline Unfractured Granite Pinawa, Manitoba, Canada

Fractured Granit e Aspö, Sweden, F inland (cons truction approved May 2002)

Volcanic Tuff Yucca Mountain NV USA Above water table:Volcanic Tuff Yucca Mountain, NV, USA (license applicati on construct Dec 2004)

Preferred Respository Host Rocks and Current Research Sites

Oxidizing environment.

S ifi i

• Rock testing (and prediction)

Some specific issues

• Rock testing (and prediction)

- samples → coring, slotting- full scale → underground excavations

• Excavation technology

full scale → underground excavations

• Excavation damage• Time dependent deformation /rock mass strength• Transparent earth• Transparent earth

σ1 Limit surface

12(a)

Unloading path effects on coring,slotting and underground excavation

3(b)

σ

σ2

31Path 3 Coring (surface or underground)

Sl i (bl k )(a)(b)σ

σ

σ σ−

3

v

v H

031 2 3

Slotting (block center)(b)

σσ σ

σ σ−

h h

H vH

003

1Path 2 Underground excavation

D/R = 0.1 D/R = 0.2

σ σt o/ σ σt o/Maximum Maximum

Potential for core damage during coring operation

0 4

0.6

σ σσ σt

t t

o

o o

// /0.00

0.25

0.00

0.25

Maximum Maximum= 0.05 = 0.1

σσ

to

/or

e ,

coring operation

0.2

0.40.50 0.50

tens

ile st

ress

in c

oFa

r-fie

ld st

ress

DNumerical resultsBest fit curve

00

2 4

Core Depth / Core Radius, D/R

Max

. t F

R

D/R = 1D/R = 2

σ σσ σt to o/ /Maximum Maximum

= 0.5 = 1.0

Let n σ

σ

σ= =t

i

c

tensile strength

induced tension in core

compressive strength[n ~ 0.1]

σ σt o/0.0

0.5

σ σt o/0.0

0.5

Let m σσ= =t

o

induced tension in corein-situ horizontal stress

[m ~ 0.5]

Then core damage occurs if

1.0 1.0σ

σ

σ

σ

c

c

o

o

>

>

n/m

0.2i.e.,

Discing of a 1.3 m diameter core from highly stressed rock (Courtesy of URL/AECL, Canada)

N N

5545o

o

σ

σσ

σ

σ

σ 22

2112

11

2

1

= 45 MPa

= 60 MPa

σσ

σ σ

σσ2

22

21

2112

12

σσ 111

SE SENW NW

Directionof advance

Directionof advance

65

60 60

65 65

70 70

Tunnel parallel to Tunnel at ~10 toσ

σ σσ σ− −

σ2

1 12 2

2o

contours contours

Asymmetric breakout produced by shear stresses acting parallel to excavation (R.S., Read, 1995)

σ σσ

1 3o

−Contours of Maximum Shear Stress developed during slotting of a rock block

σ1

horizontalslot a/b=20

horizontalslot a/b=10

σ3

2.0

horizontalslot a/b=40

verticalslot a/b=10

3

1.0

0.0

verticalslot a/b=40

verticalslot a/b=20

σ σ σ1 3 oNote: Results are for = =

Wire saw slotting tool (Courtesy of URL/AECL, Canada)

W

W

WN

N

R< 1

= Stiffness of excavated slot (N)

W

N

R

( )

= Rock stiffness

[slot can be left open or backfilled]

WW WW

N

N R

R

< <

11

Slotting technique proposed by Kvapil (1962) to increase the stability of ‘critical’ underground excavations (e.g., intersections)in highly stressed rock.

Tunnel Near-field. Disintegration of EDZ

σc = 13 MPa σc = 30 MPawater circulated in inner borehole dry inner borehole

c 13 MPa

σ σ

c

σ σc c

Effect of water on the EDZ in the Opalinus Clay, Mont Terri, Switzerland (Courtesy of )

Bulkhead Open Drift8.0 1.5 5.0

1 81.8

7.01.53.0

5.79.0 1.0

Permeability [m/s]

10-21 10-20 10-19 10-18 10-17 10-16 10-15

ALOHA2: Permeability Distribution Around the Bulkhead Drift (Courtesy of )

FF

TBM Disc CMM Disc(undercutting)

Excavation face

Excavation face

(undercutting)

Plastic region Plastic region

Probablechip path

Excavationface

Note: Tensile crack can not propagate

Unstablecrack propagation

Stress distribution beneath a TBM Disc Indenterafter the stable vertical crack has developed

Crack path produced byCMM Disc Indenter

Disc tipradius

CMM Indenter Force (N/mm of contact length)

S = 25 mm S = 50 mm S = 100 mm S = ∞

until plastic region reaches critical size

6 mm

8 mm

12 mm

1,700 (74%) 2,310 (100%)

2,500 (100%)

3,150 (100%)

2,700 (117%)

3,150 (126%)

3,600 (114%)

3,000 (130%)

(3,500?) (140%?)

4,200 (133%)

N t I d t F I i ll f l i i tti d th SNote: Indenter Force Increase is small for large increase in cutting depth S.

Comparison of chip formation in classical TBM and in CMM (undercutting)

CMM Indentation F

L

λ

plasticrack

) len

gth,

λ

Stable crack propagation( i i )λ

L/λ increases

λ

plasticzone

flawcrack

Flaw

(c (F increases as increases)λ

Stable crack propagation(F decreases as increases)λ

2

TBMIndentation F

Force F

λ'πK

= i2

2θ

T

Infinite

= fracture toughness= pre-existing crack

= tensile strength of rock (q/K )= unconfined compressive strength Mohr Coulomb

Force, F

λ'K T

qi pθ

λ length in rock = (1+sin )/(1 sin )− φφ

φ = friction angle

Mohr-CoulombmaterialKp

Relationship between applied force on an indenter and the length of edge crack ( ) beneath the plastic crushed zone

λ

WIRTH-HDRK Continuous Mining Machine (CMM)

Rock cuttings produced by CMM

α

σ

αq q

F

F q

l

σ

2 a q qo

0.6

Wing crack growth under compression, Fairhurst and Cook (1966) model

KK FII oo = = = = 2π σ− π/ q ; a (1)0.64n

m F l lσ 0 ; crack closed

Crack growtharrested

0.4

K KI IC c

=

σ σCrack growth if

Rewriting (1)

> >[ ]

(after Germanovich and Dyskin, 2000)

n

mq

/ σ

00

0.2

5 10 15 20K

KK

I

I ICI

o o

oo=

= =

=

= σ − π

πa 2

2 0.64

nn

m

where n m

m

l / a q /

0

K

if ~

0.9

1σ

>KK

II IC o=

= = πn nn

ml / a

0K

if

Supression of crack growth by small confining pressure

( )>

aa

1

2σ

σσmax

minmax

3 0

= 0.6=‘harm onic’ hole:

a2a ξ2

σ i

2

= 2.0

1.0

3 .0

q u

σmean

a1

2 σ min

1

Inelastic region

aa

1

2= 1.5 ~ 1/0.6

a ξ 1

σ max

region

ξ 2

= 2.0

1 0

3.0 q u

σmean

a

a

1

2

a ξ 2

1

2 1 .0 σ min

Inelastic

a ξ 1region

“Harmonic Hole” Optimum shape of hole to minimize the extent of excavation damage?

aa

1

2

σσ

minmax

= 0.6=Mechanical propertiesThermal properties

φ = 30κ = 3 W/m C ooo q = 1 MPac = 844 83 J/kg Cσ

σminmax

Wall temperature T=100 C (fixed)

o

aa

1

2

= 1.2 m= 2.0 m

= 1.19 MPa= 0.71 MPa

o

o 3-6up

t

q = 1 MPac = 844.83 J/kg Cρ = 2650 kg/mα = 40 x10 1/ C

σσ

maxmax

100 o

a1 0o

σ σmin

min

Contours of t t

a2

1 0o

Extent of failure region (after 15 days)

temperature(after 15 days of heating)

“Harmonic Hole” Effect of thermal load on excavation damage

(a) (b)

(c) (d)

“Harmonic hole” - Kolar Gold Fied (Caw, 1956)

a)

b)

(a) Empirical stand-up time relations for tunnels as a function of span and(a) Empirical stand-up time relations for tunnels as a function of span and rock quality defined by the Rock Mass Rating System; (b) Theoretical stand-up time based on joint strength deterioration with time (after Fakhimi 1992).

Fracture mechanics models, a) single rock bridge under far fieldnormal and shear stresses; b) multiple rock bridge under far fieldnormal and shear stresses; b) multiple rock bridge under far fieldnormal and shear stresses.

Cohesion and safety factor as a function of time.From Kemeny, 2003

Constitutive behavior at contact between two particles

Linear contact lawContact law

Kn UnFn .=F

F

F

n

ncbond

breaks

(tension)Slip condition

nKs Us

F

F

F

F n

s

s

s

.

=

=Δ Δ

Fnμ

1

Kn

U

contact bondDeformation is assumed to occurat contact point only.

Un

Un

slip model(overlap)

contact

Contact bond Parallel bond2r

Fscontact bond

Fsmax

1

Ks n

Us

slip modelwhen U > 0

Models adhesion over vanishinglysmall area of contact point (does not resists bending moment); breaks ifnormal or shear force exceeds bond strength.

Models additional material deposited after particles are in contact (resists bending moment); breaks if normal or shear stress exceeds bond strength.

s

Particulate mechanics (PFC -Cundall and Potyondy)

Parallel-bonded stress corrosion modelParallel bonded stress corrosion model

2 r(t)Stress-dependent corrosion reaction occurs at micro-tension sites.

vCorrosion reaction occurs at the periphery of parallel bonds and removes bond material

vReaction rate is determined by local driving force and local energy barrier.

1) Driving force: bond stress (sigma)2) Energy barrier: micro-activation stress (sigma_a)

Express corrosive-front velocity as:

v =v =

β 1β 2 ( c)/σσe

0 if sigma < sigma_a

PFC simulation of time dependent weakening of rock

Cluster of particles

Compression testpon assembly ofparticles

PFC model of a particulate assembly (rock) and compression loading

y = -24.386x + 23.491R2 = 0.8779y = -20x + 2014

y = -31.056x + 31 0142

R2 = 1ds

]

12

14

y = -13.541x + 13.503R2 = 0.738

R2 = 0.7583

g( t

) f[s

econ

d

8

10

log

6

LdB (Pc=0)

2

4

Tuff (Pc=0) Assumed

Tuff (no fail)Tuff (Pc=5 MPa)LdB (Pc=5 MPa)

σ σ/

0.00

0.2 0.4 0.6 0.8 1.0

σ σ/ c

Static Fatigure Curves used as input to the UDEC analysis of collapse over time at Yucca Mountain

MagnitudeMagnitudeof displacements[meters]

0.0

After 1 year After 100 years

0.25

> 0.5

Collapse predicted around

After 1,000 years After 10,000 years

drifts over 10,000 years at Yucca Mountain assuming Category 2 Tuff Static Fatigue Curve.Note: thermal and seismic effects have not been considered.

MagnitudeMagnitudeof displacements[meters]

0.0

After 1 year After 100 years

0.25

> 0.5

Collapse predicted around

After 1,000 years After 10,000 years

drifts over 10,000 years at Yucca Mountain assuming Category 2 Lac du Bonnet Static Fatigue Curve.Note: thermal and seismic effects have not been considered.

v

Upper block

Bonds breaking in shearingBonds breaking

Upper block

Bonds breaking in tension

Rough joint

u

0.6

al

cem

ent,

u [m

m]

0.2

0.4

Dilation response of the joint

Horizontal displacement u [mm]

Verti

cadi

splac

0.0

0.0

0.5 1.0 1.5 2.0 2.5 3 .0

sample as measured duringthe PFC shearing experiment(no rel ative rotation of upper andlower blocks allowed)

Use of PFC to examine the shear behavior of a rough joint.

Horizontal displacement, u [mm]

220

230

Pink altered zone

230

240

250

-1 MPa

6 MPa

240 Level

250

260

270

Dep

th (m

) Major fracture zone

28 MPa

21 MPa24 MPa

RL

shaf

t

N l t ti F t

270

280

290

-1 MPa Cataclastic zonewith clay gouge

30 MPa19 MPa

? ? UR

Normal stress acting on FractureZone as determined from triaxialovercore measurements

290

300

30 MPa

Observed variability of normal stress across a thrust fault at the URL, Pinawa,Canada.

Subsidence?

Collapsed P

Monitoringsystem

prock

oad

capa

city

,

Pillar Pillar

Pillar degradationrate

Stable roomarch

Pilla

r l

Ti t ( )rate

P di ti f l t (50 100 ) b h i f Pill d L E ti i b d d i

Time, t (years)

Prediction of long-term (50~100 yr) behaviour of Pillars and Large Excavations in abandoned mines

Log10 seismicmoment

7.3

Time

55 days

Log10 seismicmomentTime

Mar 2 10:51 1992Feb 27 00:32Feb 22 14:14Feb 18 03:56

3 90 hours

4.80

4.353.90

3.453 00

Feb 18 03:56Feb 13 17:38Feb 9 07:20Feb 4 21:01Jan 31 10:43Jan 27 00:25Jan 22 14:07Jan 18 03:49

3.93.00Jan 13 17:31

Breakout PFC analysis

Keyed highly compactedclay-block bulkhead

3.5 m

12 MS7

3.5 m high by4.4 m wideelliptical tunnel

2.6 m

12 m Concretebulkhead key

MS4

MS13MS9

MS5

MS8

Mine-byTunnel

MS2

MS11MS7

Keyed concretebulkhead

MS3MS6

MS10

MS9 MS8

MS12

MS14MS1

AE13-16

AE1-4

AE5-8Claybulkhead key

σσ

13

Steel supportSand filler Highly compacted

backfill

Pressure supplyand withdrawalheaders

Room425

MS15

MS16

σ

σ

1

2

Transparent Earth. URL TSX AE (Courtesy of Professor Paul Young, University of Toronto)

C l iConclusions

• Remarkable and continuing computational and• Remarkable and continuing computational and observational advances.

• Predictive models should build from empirical rules; p ;rationalize and extend them.

• Advances will benefit all of rock mechanics.

• Waste isolation offers unprecedented opportunity to “know” the rock mass —Professor Müller’s prime concern.concern.

Slides left out

D/R = 0.1 D/R = 0.2

σ σt o/ σ σt o/Maximum Maximumσ σσ σ

σo

t to o/ /0.00

0.25

0.00

0.25

Maximum Maximum= 0.05 = 0.1

Dσt

σo0.50 0.50

R

/D/R = 0.4 D/R = 1

D/R = 2

σ σ σσ σ σ

t t to o o

/ / /Maximum Maximum Maximum

= 0.25 = 0.5 = 1.0

σ σt o/0.0

0.5

σ σt o/0.0

0.5

σ σt o/0.0

0.5

1.0 1.0 1.0