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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
Borehole stability problemKirsch solution (2)
• Stress redistribution(응력재분배)
Dusseault, 2012
Borehole stability problemKirsch solution (3)
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
Borehole stability problemKirsch solution (4)
Dusseault, 2012
Borehole stability problemKirsch solution (5)
• 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
Borehole stability problemKirsch solution (6)
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
Borehole stability problemKirsch solution (7)
• 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
Borehole stability problemExample (2)
• Isotropic Boundary Stress
Borehole stability problemExample (3)
• Isotropic boundary stress + Injection pressure (0.5)
0.5
Borehole stability problemExample (4)
• Isotropic boundary stress + Injection pressure (0.8)
0.8
• Isotropic boundary stress + Injection pressure (2 -4)
Borehole stability problemExample (5)
• 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
Borehole stability problemAnisotropy
• 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
Borehole stability problemAnisotropy
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
Kim H, 2012, MS thesis SNU
Borehole stability problemAnisotropy
• 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
Kim H, 2012, MS thesis SNU
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