Rock Mechanics & Experiment 암석역학 및 실험

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Rock Mechanics & Experiment 암석역학 및 실험

Lecture 7. Stress solutions for openings in rock Lecture 7. 암반 공동 ( 보어홀 ) 주변의 응력식

Ki-Bok Min, PhD

Associate Professor

Department of Energy Resources Engineering Seoul National University

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Stress distribution

• Elastic case

1) Internal pressure in the circular hole

2) Circular hole under uniaxial and biaxial boundary stress 3) Elliptical openings

4) Temperature change in the circular hole

5) Internal pressure + boundary stress + temperature change 6) Hollow cylinder with internal pressure + confining pressure 7) Spherical Cavity

• Elastoplastic case

1) Circular hole under hydrostatic boundary stress

(3)

Sliding wedge in sidewall Falling wedge in the roof

• Structurally-controlled instability (

불연속면에 따른 불안정

)

– 절리등에 의해 불연속면이

형성되어 암반블록이 이탈하여 생기는 불안정성

– 상대적으로 얕은 심도

• Stress-controlled instability (

높은 응력에 따른 불안정

)

– 높은 응력에 의해 암반의 파괴가 발생하여 생기는 불안정성

– 상대적으로 깊은 심도

V notched failure induced by high in situ stress, ~400 m deep, Winnipeg, Canada (Chandler, 2004)

Fracturing at the tip of V- notch (Chandler, 2004)

Introduction

Stability issues in underground opening

Chandler, N., 2004, Developing tools for excavation design at Canada's Underground Research Laboratory, IJRMMS;41(8): 1229-1249.

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Introduction

• Stress (and displacement) analysis important for underground openings (boreholes);

– Planning the location – Dimensions

– Shapes – Orientation

– Selecting supports

• Road/Railway tunnel, hydroelectric power station, pressure headrace tunnel, oil storage cavern, mining eng, petroleum eng (borehole for reservoir), geothermal eng, geological repository for nuclear waste, …

Lillehammer (Norway), Gjøvik Olympic Mountain Hall (62 m span, 25 m heigth, 91 m long, overburden 25-50 m, σ_H=3.5-4.5 MPa) (Scheldt et al. 2003)

Scheldt et al, 2003, Comparison of continuous and discontinuous modelling for computational rock mechanics, ISRM Congress, 1043-1048

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Introduction

Stress in Cartesian, Polar & Cylindrical Coordinates

• 2D & 3D Cartesian Coordinates

• Polar & Cylindrical coordinates

Stress transformation from Cartesian to Cylindrical coordinates?

cos sin cos sin

sin cos sin cos

T

x xy

r r

xy y

r θ

θ θ

σ τ

σ τ θ θ θ θ

τ σ

τ σ θ θ θ θ

 

  =   

  −  − 

   

( ) ( )

2 2

cos 2 sin cos sin

sin 2 sin cos cos

sin cos cos sin

r x xy y

x xy y

r y x xy

θ θ

σ σ θ τ θ θ σ θ

τ σ σ θ θ τ θ θ

= + +

= − +

= − + −

(6)

Side view

…

1 2

4 1

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

stress

strain

3

Nature of Underground Geomechanics

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Introduction

Saint Venant’s Principle

• Saint Venant’s Principle

– The stresses due to two statically equivalent loadings applied over a small area are significantly different only in the vicinity the area on which the loadings are applied, and at distances which are large in comparison with the linear dimensions of the area on which the loadings are applied, the effect due to these two loadings are the same

– In other words, local change doesn’t make a global change as long as the resultant forces are the same

– Fundamentally, local stress tends to diffuses

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Case 1

E=200 GPa Case 2

E=2 MPa

Introduction

Saint Venant’s Principle

Concentrated Load (1000 Pa, 2 m에 대해)

10 m

σ_y distribution

Unit: Pa

(9)

• Concept of “Stress Redistribution”

Stresses distribution around a borehole Stress equilibrium – Conservation of load

Hudson & Harrison, 1997, Engineering Rock Mechanics, Elsevier

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Stresses distribution around a borehole Requirements (2D)

• Stress equilibrium Eq.

• Compatability Eq.

• Boundary condition (inner hole, and/or far field (in situ) stress)

• Typically assume CHILE (Continuous, Homogeneous, Isotropic and Linearly-Elastic) materials

• DIANE (Discontinuous, Inhomogeneous, Anisotropic, and Non-Elastic) materials require more complex analysis

yx 0

xx

bx

x y

σ ^∂τ ρ

∂ + + =

∂ ∂

2 2

2

2 2

y xy

x

y x x y

ε γ

ε ^∂ ^∂

∂ + =

∂ ∂ ∂ ∂

(11)

Borehole stability problem

1) internal pressure in the circular hole

• Increase of internal mud/hydraulic pressure

– Water pressure – Mud pressure

– Injection pressure

2 r w 2

P a σ = r

2 w 2

P a

θ r σ = −

p_w Tangential

stress 접선응력

rθ 0 τ =

2

w r

u P a

= − G r 0 v_θ =

a:radius

r: distance from the center

(12)

• Stress concentration factor due to uniaxial stress is ‘3’.

• Influence of borehole is within 2~3 times of radius Stresses distribution around a borehole

2)Kirsch solution – circular hole under uniaxial/biaxial boundary stress

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Stresses distribution around a borehole

σ_y

σ_y -σ_y

3σ_y σ_x

3σ_x

-σ_x σ_x

+ =

^σ^x

-σ_y+3σ_x 3σ_y-σ_x

σ_x σ_y

σ_y

Under uniaxial stress condition Maximum Stress concentration 3 Minimum Stress concentration -1

Biaxial Stress condition:

By superposition

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– Stress distribution around a circular borehole Kirsch (1898)

– Homogeneous rock under plane strain condition within elastic range

a: radius of a hole

r: radial distance from the center of the hole θ: measured from P₁

P₁and P₂: boundary in situ stress

2 2 4

1 2 1 2

2 2 4

4 3

1 1 cos 2

2 2

r

P P a P P a a

r r r

σ = ⁺ ^ − ^+ ⁻ ^ − + ^ θ

2 4

1 2

2 4

2 3

1 sin 2

r 2

P P a a

r r

τ θ = ⁻ ^ + − ^ θ

σ_θ σ_r

τ_rθ

Stresses distribution around a borehole

2 4

1 2 1 2

2 4

1 1 3 cos 2

2 2

P P a P P a

r r

σθ = ⁺ ^ + ^ − ⁻ ^ + ^ θ

P₁

P₂

( )

2 2 2

1 2 1 2

4 1 2 cos 2

4 4

r

P P a P P a a

u = ⁺G r + ⁻G r ^ −ν − r ^ θ

( )

2 2

1 2

2 1 2 2 sin 2 4

P P a a

v^θ = − ⁻G r ^ − ν + r ^ θ

(15)

a

× r

a

×r

Error in the textbook

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• Some insights;

• Stresses distribution around a borehole is

– independent of size of radius

– independent of Elastic modulus of rocks

– Poisson’s ratio has some influence on vertical stress distribution (in vertical hole)

Stresses distribution around borehole

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• At boundaries (by putting a=r)

r 0 σ =

( )

max min 2 max min cos 2

H h H h

S S S S

σθ = + − − θ

rθ 0 τ =

2 4

max min max min

2 4

1 1 3 cos 2

2 2

H h H h

S S a S S a

r r

σθ = ⁺ ^ + ^− ⁻ ^ + ^ θ

max min

0, _θ S_H 3S_h θ = σ = − +

max min

90, _θ 3S_H S_h

θ = σ = −

σ_{θ, θ=0} σ_{θ, θ=90}

S_H,max

S_h,min

Stresses distribution around a borehole 2) Kirsch solution – at boundaries

( ) ( )

{

¹ ^{2 1} ^{cos 2}

}

P k k

σθ = + + − θ

max min H

h

k S

= S

( )

0, _θ P k 3

θ = σ = − +

( )

90, _θ P 3k 1

θ = σ = −

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Stresses distribution around a borehole 2) Kirsch solution – at boundaries

• Tangential stress changes along the boundaries under uniaxial stress

( )

1

1 2cos 2 P

P

σθ θ

σ^∞

= −

=

Jaeger, Cook & Zimmerman, 2007, Fundamentals of Rock Mechanics, Blackwell Publishing

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Stresses distribution around a borehole 2) Kirsch solution – effect of stress ratio, k

• Effect of stress ratio, k

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Stresses distribution around a borehole 2) Kirsch solution – hydrostatic case (k=1)

• Hydrostatic boundary stress (k=1)

– Stress concentration?

• Hydrostatic boundary stress (k=1) + internal pressure

– If P=Pw?

2

1 2 r

P a

σ = ^ − r ^

2

1 a2

P r

σθ = ^ + ^

2 2

r 1 w

a a

P P

r r

σ = ^ − ^+

2 2

1 a _w a

P P

r r

σθ = ^ + ^ −

 

(21)

borehole breakout and hydraulic fracturing

max min

3S_h −S_h >σ_c

max

SH min

Sh

max

SH

min

Sh Borehole breakout Borehole

breakout

Tensile stress/

hydraulic fracturing

min max 0

3 _h _H

Pw > S −S +T

Assumption: Impermeable reservoir & impermeable borehole wall

max

SH min

Sh

max

SH

min

Sh

Tensile stress/

hydraulic fracturing

Borehole breakout Hydraulic fracturing

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Stresses distribution around a borehole borehole breakout

– wellbore enlargements caused by stress-induced failure of a well occurring 180 degree apart – Induced by compressive (shear)

failure

– Occur in the direction of minimum horizontal stress

Zoback, 2007, Reservoir Geomechanics, Cambridge Univ Press

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Stresses distribution around a borehole borehole breakout (Pohang EGS site)

Borehole breakout (670 – 900 m)

by borehole acoustic scanner

Drilling induced fractures (DIFS) (770 – 810 m) Borehole

breakout

DIFS Hydraulic Fracturing

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Stresses distribution around a borehole borehole breakout – rock spalling

• Similar observation can be found in underground construction

V notched failure due to high in situ stress (400 m, Winnipeg, Canada, Chandler, 2004)

Winnipeg, Canada (Min, 2002)

Chandler, N., 2004, Developing tools for excavation design at Canada's Underground Research Laboratory, IJRMMS;41(8): 1229-1249.

(25)

Anisotropic boundary stress

(26)

Stresses distribution around a borehole

Isotropic boundary stress

(27)

0.5

Internal pressure from the opening/borehole

(28)

0.8

(29)

• Hydraulic fracturing occur when internal pressure is large.

• The location of the initiation of fracture can be decided depending on the boundary stress

2 or 4

Stresses distribution around a borehole

(30)

Stresses distribution around an opening 3) Elliptical opening

• Tangential stress concentration at peripheries

1 2

A

P k W

σ = ^ − + H ^ _A 1 2 H

P k k

σ = ^ − + W ^

(31)

• Thermally induced hoop stress at the wall of cylindrical borehole

r 0 σ =

( 0)

1 ^w

E T T

σθ α

= ν −

−

rθ 0 τ =

α: linear thermal expansion coefficient E: Elastic Modulus

T_w: well temperature T₀: reservoir temperature

T_w T₀

Stresses distribution around a borehole

(32)

Stresses distribution around a borehole 4) Temperature change in the circular hole

• 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) ( 0)

3 1 2

T

K T T E T T

σ α α

= − = ν −

−

( 0)

l T T

l α

∆ = −

( 0)

T E T T

σ =α −

(33)

(34)

• Thermal stress due to cold mud/water injection is important for geothermal project

– Ex) Injecting water T = 25°C, reservoir T = 75°C, Elastic modulus

= 50 GPa, ν=0.25, α = 1x10^-5/°C  hoop stress = 33 MPa  big influence!!!

(35)

• Stresses distribution around a borehole with;

– Principal in situ stress boundary

– Internal pore pressure (could be mud pressure) – Temperature change

2 2 4

max min max min

2 2 4

2 2

4 3

1 1 cos 2

2 2

H h H h

w r

S S R S S R R

r r

P R

r r

σ = ⁺ ^ − ^ + ⁻ ^ − + ^ θ +

( )

2 4

max min max min

2 4 0

2 2

1 1 3 co

s 2 1

2 2

H h

w

H h

w

S S R S S R

r r

P R T

r

E T

σθ θ α

ν

   

+ −

=  +  −  +  −

  +

 −

 −

2 4

max min

2 4

2 3

1 sin 2

2

H h

r

S S R R

r r

τ θ = ⁻ ^ + − ^ θ

Stresses distribution around borehole

5) General solution

(36)

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

– Without considering temperature change,

( )

,min min max 0

3 _h _H P_w 1E _w

T

S S T

θ α

σ = − − ⁺ −ν ⁻

( )

,max max min 0

3 _h _h P_w 1E _w

T

S S T

θ α

σ = − − ⁺ −ν ⁻

,min 3S_hmin S_Hmax P_w

σθ = − −

,max 3S_hmax S_hmin P_w

σθ = − −

max

SH min

Sh

max

SH

min

Sh

5) General solution

(37)

P_i P_o

a b

2 2 2 2

2 2 2 2 2

( ) ( )

o i i o

r

b P a P a b P P

b a b a r

σ = ⁻ + ⁻

− −

2 2 2 2

2 2 2 2 2

( ) ( )

o i i o

b P a P a b P P

b a b a r

σθ = ⁻ − ⁻

− −

6) Hollow cylinder

(38)

Stress distribution in a hollow cylinder with b = 2a

With externalpressure (P₀) alone With internal pressure (P_i) alone

Stresses distribution around borehole 6) Hollow cylinder

Jaeger, Cook & Zimmerman, 2007, Fundamentals of Rock Mechanics, Blackwell Publishing

(39)

Stresses distribution around borehole 7) Spherical cavity

• Spherical cavity with internal pressure

3 r 3

P a

σ _{= }^_ r ^__

 

3

2 3

P a

θ φ r

σ =σ = − ^ ^

rθ rφ θφ 0 σ =σ =σ =

0 u_θ = u_φ =

3

4 2 r

u P a

= − G r

(40)

• Spherical cavity in a 3D hydrostatic stress field

3

1 3 r

P a

σ = ^ −^ r ^^

3

1 3

2 P a

θ φ r

σ =σ = ^ +^ ^^

rθ rφ θφ 0 σ =σ =σ =

0 u_θ = u_φ =

3

4 2 r

u P a

= G r

P P

P

P P

P

(41)

Stresses distribution around borehole 7) Spherical cavity

• Comparison of 2D circular opening and 3D spherical opening (internal pressure P)

3

2 3

P a

θ φ r

σ =σ = − ^ ^

3

4 2 r

u P a

= − G r

2 w 2

P a

θ r σ = −

2

w r

u P a

= − G r

(42)

• Comparison of 2D circular opening and 3D spherical opening (hydrostatic case)

3

1 3

2 P a

θ φ r

σ =σ = ^ +^ ^^

3

4 2 r

u P a

= G r

2

1 a2

P r

σθ = ^ + ^

2 r 2

u P a

= G r

(43)

– A more general case around the borehole can be considered for inelastic case using failure criteria such as Mohr-Coulomb failure criteria.

– Analytical solution exists only for hydrostatic boundary stress condition.

Stresses distribution around borehole

Elasto-Plastic Solution with circular hole

(44)

( 0 / 2) ( / )² ( 0 / 2)

r Pi C r a C

σ = ^ + ^ −

( 0 ) ( )² ( 0 )

3 P_i C / 2 r a/ C / 2 σθ = ^ + ^ −

( ) ( )²

0 0 0

1 / 2 /

r P 2 P C r a

σ = − ^ + ^

( ) ( )²

0 0 0

1 / 2 /

P 2 P C r a

σθ = + ^ + ^ r b ρ < <

a < <r ρ

Borehole stability problem

Elasto-Plastic Solution with circular hole

( )

( ) ( )

( )

1/ 1

0 0

0

2 1

1 1

q

i

P q C

a P q C q

ρ = ^ ^ − +^{− +} +^ ^ ⁻

( )

1 0 3 0 3

1 sin Failure criterion: 2 tan

1 sin

C q S φ

σ σ β σ

φ

= + = + +

−

(45)

• Stress concentration differs depending on the anisotropy of

elastic modulus

3P -P

Isotropic rock (E/Eʹ= 1)

Transversely isotropic rock

(E/Eʹ = 2)

P P

Transversely isotropic rock

(E/Eʹ = 3)

3.3P P -0.7P

3.6P -0.6P

Hanna Kim, 2012, MSc Thesis, SNU

Stresses distribution around borehole

Effect of Anisotropy