Shear stressShear stressShear stress

(1)

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

Lecture 2. Fundamentals of Geomechanics (1)

Ki-Bok Min, PhD, Associate Professor

Department of Energy Resources Engineering Seoul National University

(2)

Introduction

Contents 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

(3)

Introduction

Contents 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)

(4)

Introduction

Contents of the course

• 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

(5)

Physical variables for various physical problems

Structure of state variables and fluxes are mathematically similar – a convenient truth!

Physical problem

Conservation Principle

State Variable Flux σ

Material properties

Source Constitutive equation

Elasticity ^{힘의 평형}

(Equilibrium)

변위

(Displacement), u

응력 (Stress) σ

탄성계수 및 포아송비

(Young’s modulus

& Poisson’s ratio)

체적력 (Body forces)

후크의 법칙 (Hooke’s law)

Heat

conduction

Conservation of energy

Temperature T

Heat flux q’

Thermal conductivity k

Heat sources

Fourier’s law

Porous media flow

Conservation of mass

Hydraulic head h

Fluid flux q

Permeability k

Fluid source

Darcy’s law

Mass transport

Conservation of mass

Concentration C

Diffusiv e flux q

Diffusion coefficient D

Chemica l source

Fick’s law

(6)

Reservoir Geomechanics

comparison with fluid flow - a convenient truth

Porous media fluid flow (다공성매질의 유체유동) Geomechanics^{(암반역학)}

Darcy’s Law Hooke’s Law

Fluid Flux Stress (응력)

Pressure gradient strain (변형율)

Permeability Elastic modulus (Young’s Modulus)

& Poisson’s ratio

Time dependent Not time dependent (elastic) time dependent creep Conservation of mass Equilibrium Equation

P dl k d

q ^{ }  ^ ^ ^E^ ^ ^E ^du_dx

(7)

Stress (응력) Definition

– Stress

: a force acting over a given area, F/A  simple definition

the internal distribution of force per unit area that balances and reacts to external loads applied to a body exact definition

– Normal stress: Normal force/Area – Shear stress: Shear force/Area

– Unit: N/m²=Pa, 10⁶Pa=MPa, 10⁹Pa=GPa 145 psi = 1 MPa = 10 bar = 10 kg중/cm²

Fn

  A Fs

  A

(8)

– Stress in 2D

– Normal stress: acting perpendicular to the plane – Shear stress: acting tangent to the plane

– Stress is a 2^nd order tensor

Force is 1^st order tensor (=vector)

Can be defined according to the reference axis

Principal stresses are defined

Stress (응력) & Force(힘) stress in 2D

x xy

yx y

 

 

 

 

 

2D:



xy

Direction of the stress component Direction of surface normal

upon which the stress acts

(9)

– Stress component in 2D

Stress (응력) & Force(힘) stress in 2D

x xy

yx y

 

 

 

 

 

xy yx

  3 component is independent 4성분

(10)

Stress (응력) & Force(힘) stress in 3D

x xy xz

yx y yz

zx zy z

  

  

  

 

 

 

x y z yz xz xy





 

 

 

텐서형식

(Tensor form) 행렬형식

(matrix form)

xz zx

 

xy yx

 

yz zy

 

6 independent components 9 components

- Stress in 3D

(11)

Stress (응력) & Force(힘) stress transformation

• Stress magnitude varies with respect of the direction of reference plane = stress is ‘2

^nd

order tensor’

N, Normal Force

V, Shear Force

(12)

Stress (응력) & Force(힘) Stress Transformation

– 1D example

– Stress transformation is conducted by multiplying the direction cosine twice (응력의 변환식은 방향 코사인이 두 번

곱해져서 구해짐).

2 1

N Pcos A A

   

1

sin cos

V P

A A

      

 

2 1

cos 1 cos 2

x 2 x

       sin cos sin 2

2

x

 x

       

1 1 1

cos sin 0 cos sin

sin cos 0 0 sin cos

T

x x y x

x y y

      

     

     

     

 

(13)

Stress (응력) & Force(힘)

Stresses on inclined sections (Transformation)

• A different way of obtaining transformed stresses

– For vector

– For tensor (stress)

1 1 1

cos sin cos sin

sin cos sin cos

T

x x y x xy

x y y xy y

       

       

     

     

   

1

cos sin sin cos

x x

y y

F F

 

    

     

   

1 cos 2 sin 2

2 2

x y x y

x xy

   

  ^  ^    _{1 1} sin 2 cos 2

2

x y

x y xy

 

   ^   

=

x y

y' x'

θ

(14)

Stress (응력) & Force(힘)

Stresses on inclined sections (Transformation)

• Stresses acting on inclined sections assuming that σ

_x

, σ

_y

, τ

_xy

are known.

– x₁y₁ axes are rotated counterclockwise through an angle θ

1 cos 2 sin 2

2 2

x y x y

x xy

   

  ^  ^   

1 1 sin 2 cos 2

2

x y

x y xy

 

   ^   

(15)

Stress (응력) & Force(힘) Stress Transformation

– 1D example

– Stress transformation is conducted by multiplying the direction cosine twice (응력의 변환식은 방향 코사인이 두 번

곱해져서 구해짐).

2 1

N Pcos A A

   

1

sin cos

V P

A A

      

 

2 1

cos 1 cos 2

x 2 x

       sin cos sin 2

2

x

 x

       

1 1 1

cos sin 0 cos sin

sin cos 0 0 sin cos

T

x x y x

x y y

      

     

     

     

 

(16)

Stress (응력) & Force(힘) Stress Transformation

• 예)

(17)

Stress (응력) & Force(힘) Stress Transformation

주각 (Principal angles)

최대 주응력 (maximum

Principal stresses)

최소주응력 (minimum

Principal stresses)

(18)

Stress (응력) & Force(힘) Stress Transformation

• Principal stress (주응력)

– The largest (or smallest) normal stress and shear stress at that plane is 0 (가장

큰 수직응력이며 전단응력이 0 임)

– Vertical stress in the earth is usually principal stress and the other two principal stress is in horizontal direction (통상 지각의 수직방향이 주응력이며, 수평방향으로도 두 개의 주응력 정의 가능)

– Usually denoted as σ₁, σ₂, σ₃ (크기 순으로 σ₁, σ₂, σ₃ 으로 표시)

Principal stress 주응력 상태

(19)

Stress (응력) & Force(힘)

Principal Stresses and Maximum Shear Stresses

• Stress element is three dimensional

– Three principal stresses (σ₁ ,σ₂ and σ₃) on three mutually perpendicular planes

(20)

Stress (응력) & Force(힘) Mohr’s Circle

– One can always find an axis where shear stress goes zero  principal axis & principal stress

– Using principal stresses (σ₁ and σ₂),

– Radius:

– Center:

– Maximum shear stress:

1 2 1 2

cos 2

2 2

x

   

   ^  ^  ¹ ² sin 2

xy 2

    ^  sin 2

xy 

 _xy cos 2

0

1 2

2

 

1 2

2

 

1 2

2

 

(21)

Mohr’s Circle for Plane Stress

Alternative way of understanding

• The transformation Equations for plane stress

– In terms of principal stresses (shear stress becomes zero)

– Square both sides of each equation and sum the two equations – Equation of a circle in standard algebraic form

1 cos 2 sin 2

2 2

x y x y

x xy

   

  ^  ^   

1 1 sin 2 cos 2

2

x y

x y xy

 

   ^   

1 1 1

1 2 1 2 c o s 2 1 2 s in 2

2 2 2

x x y

     

  ^  ^     ^ 

1 1 1

2 2 1 2 2

( ) ( )

2 2

x y

x x y

   

  ^    ^

2 2

0)2

(x  x  y  R  ^x¹

1 x1y



B A

O C

R

1 1 1

( _x , _{x y} )

(1, 0) (2, 0)

2

(22)

Stress (응력) & Force(힘) Mohr’s Circle

A

D

(23)

Stress (응력) & Force(힘) Mohr’s Circle

• Mohr’s Circles for 3D

– 밑금친 부분이 해당면에서의 응력의 값

S

1.5 3 3.5 N

 

N S

(24)

Stress (응력) & Force(힘) Summary

• Stress = Force/Area, two types: normal stress & shear stress (응력은 힘/면적 - 수직응력과 전단응력)

• Stress can be expressed differently depending on reference axis (기준좌표에 정의된 응력과 회전각에 따라

응력이 달리 표시됨 )

• Mohr’s Circle is a graphical expression of the state of stress

(모어 서클은 응력상태를 표시하는 유용한 도구)

(25)

Strain (변형율) & Displacement (변위) 1D & 2D

• Geometric expression of deformation caused by stress (dimensionless)

L du L dx

  ^ 

L ∆L

1D

, ^y ,

xx x yy

x y xy

u u

x y

u u

y x

 



 

 

 

 

 

(26)

Strain (변형율) & Displacement (변위) 3D

xx xy xz

yx yy yz

zx zy zz

  

 

 

 

xx yy zz yz xz xy





 

 

 

Tensor form

, , ,

1 2 1 2 1 2

x y z

xx yy zz

x y xy

y z

yz

x z

xz

u u u

x y z

u u

y x

u u

z y

u u

z x

  



  

  

  

   

     

   

     

 

 

     

Strain is also a 2^nd order tensor and symmetric by definition.

Matrix form

(27)

Mechanical Properties elastic properties

• Elastic

– Loaded deformation recovers when unloaded

• Plastic

– Not recoverable

(28)

Mechanical Properties

Constitute Equation (Hooke’s Law)

• Hooke’s law

• Elastic (Young’s) modulus, E (N/m²=Pa)

• Poisson’s ratio, v (dimensionless)

• Typical range of properties

– Concrete σ_c = 20-50 MPa E ~ 25 GPa – Granite σ_c = 100-200 MPa E ~ 60 GPa – Alloy steel σ_c = >500 MPa E ~ 200 GPa

y y

E 

 

x y

lateral strain axial strain

 

   

E E du

    dx

L

  ^L

x

q k dT

   dx

x x

q K h x

  



(29)

Mechanical Properties

Constitute Equation (Hooke’s Law)

• Hooke’s Law

• Shear modulus G

• Generalized Hooke’s law (isotropy)

• 2 independent parameters (E, ν) for isotropic material

  E

1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

x x

y y

z z

yz yz

xz xz

xy xy

E E E

G G

G

 

 

   

 

 

   

 

  

     

     

      

     



     

   

     

 

 

1D

2(1 ) G E

 



xy G xy

  

(30)

Mechanical Properties

Constitute Equation (Hooke’s Law)

• Normal strain in plane stress

(평명응력하에서의 수직변형율)

– Similarly

• Normal strain in plane stress

(평면응력하에서의 전단변형율)

– Shear strain is a change of angle(전단변형율은 각도의 변화)

– No influence from σ_x & σ_y^{(수직응력 σ}x과σ_y은 영향이 없다.)

 

1

x x y

  E  

Normal strain, ε_x ¹ x

E _y

 E

+



=

 

1

y y x

  E   ^^z ^{ }^_E



^^x ^^^y



xy

xy G

  G: 전단계수

(31)

Plane Strain (평면변형율)

Plane strain versus plane stress

(32)

Plane Strain (평면변형율)

Plane strain versus plane stress

• Stress and strain in different dimensions are coupled. Therefore, we need a special consideration –plane strain and plane stress

• Plane strain

– 3^rd dimensional strain goes zero

– Stresses around drill hole or 2D tunnel

0 0

0 0 0

xx xy

yx yy

 

 

 

 

0 0

xx xy

yx yy

zz

 



 

 

 

2

(1 ) (1 )

0

(1 ) (1 )

0 2(1 )

0 0

xx xx

yy yy

xy xy

E E

E

  

 

  

 

  

   

  

 

   

 

       

   

 

   

     

 

 

1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

x x

y y

z z

yz yz

xz xz

xy xy

E E E

G G

G

 

 

   

 

 

   

 

  

     

     

      

     



     

   

     

 

 

1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

x x

y y

z z

yz yz

xz xz

xy xy

E E E

G G

G

 

 

   

 

 

   

 

  

     

     

      

     



     

   

     

 

 

(33)

Plane Strain (평면변형율)

Plane strain versus plane stress

• Plane stress

– 3rd dimensional stress goes zero – Thin plate stressed in its own plane

0 0

0 0 0

xx xy

yx yy

 

 

 

 

0 0

xx xy

yx yy

zz

 



 

 

 

1 0

2(1 )

0 0

xx xx

yy yy

xy xy

E E

E



 

  

  

  

 

    

     

    

     

 

(34)

Mechanical Properties

Constitute Equation (Hooke’s Law): Anisotropic case

• There are five independent elastic parameters for transversely isotropic rock

1 '

0 0 0

'

' 1 '

0 0 0

' ' '

' 1

0 0 0

'

0 0 0 1 0 0

'

2(1 )

0 0 0 0 0

0 0 0 0 0 1

'

x x

y y

z z

yz yz

zx zx

xy xy

E E E

G

E

G

   

 

 

 

  

 

    

      

    

     

    

     

    

     

    

   

    

   

   

  

   





(H. Gercek, 2006)

x z

y

xz zx

yx yz

xy yz

E E E

E E

G G G

  

 

 

 

  

(35)

Conservation Equation Equilibrium equation

– Sum of traction, body forces (and moment) are zero (static case)

– b_x, b_y, b_z are components of acceleration due to gravity.

0 0 0

xx yx zx

x

xy yy zy

y

xz yz zz

z

x y z b

   

  



   

  

   

  

   

  

   

  

현재 이 이미지를 표시할 수 없습니다.

i 0

M  _xy  _yx

2 i i

F m u t

  



Very slow loading

(36)

Governing Equation 3D – Navier’s Equation

• In 3D - Navier’s equation

(1 )(1 2 ) E

 ^   

2 2 2 2 2 2

2 2 2 2

2 2 2 2 2 2

2 2 2 2

2 2 2

( ) 0

( )

x x x x y z

x

y y y x y z

y

x x x x

u u u u u u

G G b

x y x z

x y z x

u u u u u u

G G b

x y y z

x y z y

u u u u

G G

x y z x z

 



  

      

        

 

           

   

                

   

           

   

     

   

 

      

 

2 2

2 0

y z

z

u u

y z z b

   

   

 

    

 

(37)

Governing Equation 1D

• Strain-displacement relationship

• Stress-strain relationship

• Static Equillibrium Equation

• Final equation for elasticity

du

  dx 1

  E 

xx 0

bx

x

 

  



2

2^x _x 0

E u b

x 

  



(38)

Governing Equation 1D - example

– With no body force, and static case

2 2

x x

x

u u

E b

x   t

 

 

 

2

2^x 0 E u

x

 



x = 0

Fixed u =0

1

ux

E C

x

 



x 2

Eu  x C

 From σ = 1, C₁=1

 From u = 0 at x=0, C₂=0

x

u x

 E

x = 1 σ = 1

σ = 1

x

1/E

1

(39)

Stress-Strain Relationship (Hooke’s Law) Relationship with V

_p

and V

_s

• In really, Dynamic E ≠ Static E. Typically Dynamic E > Static E.

• Difference decrease with stronger, larger confinement

2 2

2

2 2

3 4

2

2( )

: Dynamic Elastic Modulus : Dynamic Poisson's Ratio : P wave velocity

: S wave velocity

p s

dyn p

p s

dyn

p s

dyn dyn

p s

V V

E V

V V

V V E

V V





 



 



Dry red Wildmoor sandstone (Fjaer et al., 2008)

(40)

강도 (strength)

Uniaxial compressive Strength

– Uniaxial Compressive Strength (UCS): Maximum sustainable compressive stress

– Unit: same as stress (단위: 응력과 같음) (Pa or MPa).

max c

F

  A _F

max

(41)

강도 (strength)

Uniaxial compressive Strength

– Computer simulation of rock failure using discrete element method (개별요소법에 의한 파괴 모사 (컴퓨터로 만든 돌?))

2D 3D

(42)

강도 (strength)

취성 (brittle)과 연성(ductile)

Brittle rock Ductile rock

(43)

강도 (strength)

취성 (brittle)과 연성(ductile)

• 취성-연성 전이현상 (Brittle-ductile transition)

– 봉압 증가로 보다 연성에 가깝게 거동

Fjaer et al., 2008 Jaeger et al., 2007

(44)

강도 (strength) Tensile Strength

• Tensile strength: Maximum tensile stress that the specimen can sustain

• Tensile strength of rock is usually 1/10 ~ 1/20 of UCS

Tensile loading

max t

T

  A

(45)

강도 (strength) Tensile Strength

• Tensile strength (인장강도) : Maximum sustainable tensile stress

• Measured by ‘Brazilian Test’ in case of rock (indirect tension,간접인장)

• Tensile strength is 1/10 ~ 1/20 of UCS

Tensile loading

max t

T

  A

(46)

강도 (strength) Tensile Strength

• The reason why Brazilian Test Works…

– Stress distribution along the x-axis

Jaeger et al., 2007

2 2

0

2 4 2 0

0

(1 ) sin 2

2 (1 )

arctan tan

(1 2 cos 2 ) (1 )

rr

P   

 

    

    

 

       

2 2

0

2 4 2 0

0

(1 ) sin 2

2 (1 )

arctan tan

(1 2 cos 2 ) (1 )

P



  

 

    

    

 

        

When 2θ₀=15°

r a/

 

(47)

강도 (strength)

Tensile Strength

(48)

Failure Criteria

Friction coefficient

Fs  N

(49)

Failure Criteria

Mohr-Coulomb Failure Criterion

– τ: Shear stress

– S₀=cohesion (or cohesive strength)

– σ_n: normal stress

– μ_i: coefficient of internal friction = tanΦ

0 n i



 S 

 

(50)

강도 (strength)

파괴조건식 (Mohr-Coulomb Failure Criterion)

0 n i



 S 

 

Relationship

between β and φ?