new week 2(9 sept) fundamentals geomechanics (2)ocw.snu.ac.kr/sites/default/files/note/7658.pdf ·...

Topics in Energy and Environmental Geomechanics –Enhanced Geothermal Systems (EGS)

Lecture 2. Fundamentals of Geomechanics (2)

Ki-Bok Min, PhD, Associate ProfessorDepartment of Energy Resources Engineering

Seoul National University

IntroductionContents of the course

– Week 1 (2 Sept):Introduction to the course/Climate change & Emerging Subsurface Eng Application

– Week 2 (9 Sept): Fundamentals of Geomechanics– Week 3 (16 Sept):Borehole Stability– Week 4 (23 Sept): Borehole Stability– Week 5 (30 Sept): No lecture (business trip)– Week 6 (7 Oct) : Hydraulic Stimulation (focus on Hydraulic fracturing)– Week 7 (14 Oct): Hydraulic Stimulation– Week 8 (21 Oct): Induced seismicity– Week 9 (28 Oct): Induced seismicity

IntroductionContents of the course

– Week 10 (4 Nov): Drilling Engineering (invited lecture)– Week 11 (11 Nov): Well logging (invited lecture)– Week 12 (18 Nov): EGS Case studies– Week 13 (25 Nov): EGS Case studies– Week 14 (2 Dec): Student Conference– Week 15 (9 Dec): Final Exam (closed book or take-home exam)

Borehole stability problem

• Importance?– Economy, Safety, Environment– Optimal trajectory and mud weight determination

• Factors– In situ stress– Injection pressure or mud weight– Reservoir rock mechanics properties– Anisotropy– Thermal effect– …

Borehole stability problem

• Definitions – cylindrical coordinate

반경방향응력

접선응력

Dusseault, 2012

Side view

…

1

2

41

2

4

2

3

4

Civil structural problems: Mechanics of “Addition”

Underground Geomechanics problems: Mechanics of “Removal”

Before drilling/excavation

Start of drilling/excavation

Further advance of drilling/excavation

Monitoring points

stre

ss

strain

3

Nature of Underground Geomechanics

Borehole stability problemKirsch solution (1)

σθ

σr

τrθ

Sh,min

SH,maxR

– Stress distribution around circular borehole (Kirsch, 1898) – Homogeneous, isotropic rock under plane strain condition


• Stress redistribution(응력재분배)

Dusseault, 2012


R: 보어홀의반경r: 보어홀중심에서반경방향의거리

θ: SH,max으로부터반시계방향으로측정

SH,max , SH,max: 최대및최소수평응력

2 2 4max min max min

2 2 44 31 1 cos 2

2 2H h H h

rS S S SR R R

r r r

2 4max min max min

2 431 1 cos 2

2 2H h H hS S S SR R

r r

2 4max min

2 42 31 sin 2

2H h

rS S R R

r r

σθ

σr

τrθ

Sh,min

SH,maxR


Dusseault, 2012


• In the borehole surface (R=r)

0r

max min max min2 cos 2H h H hS S S S

0r

2 4max min max min

2 431 1 cos 2

2 2H h H hS S S SR R

r r

max min0, 3H hS S

max min90, 3 H hS S

σθ, θ=0

σθ, θ=90

SH,max

Sh,min


σy

σy

-σy

3σy σx

3σx

-σx σx+ =σx

-σy+3σx

3σy-σx

σx

σy

σy

Under uniaxial stress condition

Maximum Stress concentration 3

Minimum Stress concentration -1

Biaxial Stress condition:

By superposition


• The area of stress disturbance is within 2~3 times of borehole radius

Borehole stability problemInternal pressure (1)

• Increase of internal mud/hydraulic pressure

2

2r wRPr

2

2wRPr

pw

Tangential stress접선응력

Borehole stability problemInternal pressure (2)

• 이수밀도의증가 (or주입압력증가) 는공벽의응력을감소시킴.

Dusseault, 2012

Borehole stability problemExample (1)

• Anisotropic Boundary Stress


• Isotropic Boundary Stress


• Isotropic boundary stress + Injection pressure (0.5)

0.5


• Isotropic boundary stress + Injection pressure (0.8)

0.8

• Isotropic boundary stress + Injection pressure (2 -4)


• Hydraulic Fracturing can occur when internal pressure is big enough

2 or 4

Borehole stability problemBorehole Trajectory

• Drilling trajectory (최적시추궤적결정)

– Determine the trajectory that can induce the least stress concentration

Dusseault, 2012

Borehole stability problemThermal stress(열팽창및열응력 )

• Linear thermal expansion coefficient (unit: /K)

• Thermal stress thermal expansion + mechanical restraint– Thermal stress in 1D

– Thermal stress when a rock is completely (in all directions) restrained

0 031 2T

EK T T T T

0l T T

l

0T E T T

Borehole stability problemThermal stress(열팽창및열응력 )

• Downhole : Cooling stabilize• Uphole: Heating can induce failure

Dusseault, 2012

Stress around circular boreholeGeneralized Kirsch’s solution

• Boundary stress + internal pressure + temperature;– 응력경계 (Principal in situ stress boundary)– 내부주입압력 (Internal pore pressure (mud/water pressure))– 온도변화 (Temperature change)

2 2 4max min max min

2 2 4

2

24 31 1 cos 2

2 2H h H h

wrS S S SR R R

r rRP

r r

2 4

max min max min2 4 0

2

231 1 co

1s 2

2 2H h

wH h

wS S S SR R

r rRP Tr

E T

2 4max min

2 42 31 sin 2

2H h

rS S R R

r r

Stress around circular boreholeGeneralized Kirsch’s solution

– At the borehole wall (r = R), maximum and minimum hoop stresses are;

– Without considering temperature change,

,min min m 0ax 13 wwh H P E TS S T

,max max m 0in 13 wwh h P E TS S T

,min min max3 wh HS S P

,max max min3 wh hS S P maxHS

minhS

maxHS

minhS

Stress around circular boreholeborehole breakout vs. hydraulic fracturing

– Required internal pressure to induce tensile stress (neglect T effect);

– …. To induce hydraulic fracturing

– Required uniaxial compressive strength not to have borehole breakout

min max3w h HP S S

max min3 h h wc PS S

maxHS

minhS

maxHS

minhS

Tensile stress/hydraulic fracturing

Borehole breakout

Borehole breakout

Tensile stress/hydraulic fracturing

min ax 0m3 h HwP S TS

Borehole breakout

– wellbore enlargements caused by stress-induced failure of a well occurring 180 degree apart

– Induced by compressive failure– Occur in the direction of

minimum horizontal stress

Zoback, 2007

Borehole stability problemAnisotropy

• Some formulae considering anisotropy (internal pressure, uniaxial stress)

Lekhnitskii, 1963


• 탄성계수의이방성에따라응력집중의정도가달라짐.

3P-P

Isotropic rock(E/Eʹ= 1)

Transversely isotropic rock

(E/E ʹ = 2)

P P

Transversely isotropic rock

(E/E ʹ = 3)

P3.3P-0.7P

3.6P-0.6P

0 30 60 90-1

0

1

2

3

4y

σθθ (E/E'=1) σθθ (E/E'=2) σθθ (E/E'=3)

angle (o)

S

y

Kim H, 2012, MS thesis SNU


zy

x

P

P

1.8

2.0

2.20

30

60

90

120

150

180

210

240

270

300

330

1.8

2.0

2.2

o

o

o

o

o

o

o

o

o

o

o

o

Nor

mal

ized

tang

entia

l stre

ss,

P

E/E'=1 E/E'=2 E/E'=3



• 어떤파괴조건식을적용하느냐에따라파괴범위가다르게나타남

0

30

60

90

120

150

180

210

240

270

300

330

o

o

o

o

o

o

o

o

o

o

o

o

o

등방성파괴조건식

이방성파괴조건식


Borehole stability problemAnisotropy – North Sea Example (1)

• Extended Reach Drilling (ERD) hasbeen employed for increasing oilrecovery.

• Total Depth = 9,327 m

• Since 1979, total depth for wells hasincreased steadily.

Oseberg in North Sea (Norway)

Okland & Cook, SPE, 1998

Borehole stability problemAnisotropy – North Sea Example (2)

Loaded and unloadedat approximately 20 MPa

a) Draupne-similar Shale b) J2 (Outcrop shale)External Pressure= 13.6 MPa

External Pressure= 32.2 MPa

Severely Damaged!!

Undamaged

Okland & Cook, SPE, 1998

Borehole stability problemWork Flow chart

Fjaer et al., 2008, Petroleum related rock mechanics, 2nd ed., Elsevier

Reservoir subsidence (저류층침하)Mechanism

• Subsidence induced by the decrease of reservoir formation pressure

• P ↓ effective stress ↑ compression

Fjaer et al., 2008, Petroleum related rock mechanics, 2nd ed., Elsevier

Reservoir subsidence (저류층침하)Mechanism

• More serious when;– Bigger reservoir formation pressure decrease– Thick reservoir– Compressible reservoir– E.g.) Wilmington field in California: 9 m subsidence (Fjaer, 2008)

Reservoir subsidence (저류층침하)Poroelastic stress change

• Change of horizontal stress due to pore pressure change

(1 2 )(1 )

::

h p p

h

S p k p

S

수평방향응력(horizontal stress)

암석의 포아송비(Poisson's ratio)

k:응력경로계수(stress path factor, SPF)

Reservoir subsidence (저류층침하)Uniaxial compaction model

• Compaction coefficient or unaxial compressibility, Cm;

1 (1 )(1 2 )1m

h C p ph E

RESERVOIR

Reservoir subsidence (저류층침하)Example(2): In Salah (Gas/CO2 storage)

CO2

+P-P

UpliftSettlement

Annually, 5 mm (above injection holes)Heaving and Subsidence

Rutqvist et al., 2009, IJGG

XY

Z

0.012

0.011

0.01

0.009

0.008

0.007

0.006

0.005

0.004

0.003

0.002

0.001

0.XY

Z

0.012

0.011

0.01

0.009

0.008

0.007

0.006

0.005

0.004

0.003

0.002

0.001

0.

Prediction of Reservoir Heaving

3 years1 year 0.012

0.0

0.010

0.005

(m)0.012

0.0

0.010

0.005

(m)

100 m

Ground surface

Top of injection zone

TIME (days)

VE

RTI

CA

LD

ISP

LAC

EM

EN

T(m

)

0 250 500 750 10000

0.005

0.01

0.015

0.02

0.025

0.03

TIME (days)

VE

RTI

CA

LD

ISP

LAC

EM

EN

T(m

)

0 250 500 750 10000

0.005

0.01

0.015

0.02

0.025

0.03

Calculated

kcap = 1e-19 m2

kcap = 1e-21 m2

Measured KB503

Measured KB501

Rutqvist et al., 2009, IJGG

Pressure change with time near the injection point (Units: MPa)

Vertical displacement profile

After 10 years- the pore pressure : about 12 MPa- the vertical displacement : 0.87 m

Lee, Min, Rutqvist (2012), RMRE

Reservoir subsidence (저류층침하)Example(3): Heaving in CO2 storage reservoir

Example of Reservoir Geomechanics ApplicationNatural Gas storage in depleted Reservoir

Injection(*) Re-production(#)

Caprock

Reservoir

Reservoir Heaving (*) & Subsidence (#)

Shear slip of fracture in caprock(*, #) Tensile failure of caprock (*)

Sand production due to cyclic loading (*, #)

Borehole instability due to ΔT, ΔP (*, #)

• Natural Gas need to be stored due to unbalanced consumption during summer and winter.

Min, 2013, Geomechanics study for Underground Gas storage of depleted reservoir

SummaryContents – Applications

• Fundamentals of Geomechanics– Stress– Strain– Stress-strain relationship– Uniaxial, tensile & triaxial strength– In situ rock stress

• Stress around a cavities– Circular hole

Anisotropic case

Elastic-plastic rock

– Penny-shaped cracks

Things to remember

• Stress (Equilibrium, Principal stress, Transformation)• Hooke’s Law (compare with Darcy’s Law)• Stress concentration around circular hole

(3 & -1)• Concept of Effective Stress

new week 2(9 sept) fundamentals geomechanics (2)ocw.snu.ac.kr/sites/default/files/note/7658.pdf ·...

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