Week 2(9 Sept) Fundamentals Geomechanics (1) · 2018. 1. 30. · – 1D example – Stress transformation is conducted by multiplying the direction cosine twice (응력의변환식은방향코사인이두번

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

Lecture 2. Fundamentals of Geomechanics (1)

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


– 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)


• 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

Physical variables for various physical problems

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

Physical problem

ConservationPrinciple

State Variable Fluxσ

Material properties

Source Constitutive equation

Elasticity 힘의평형(Equilibrium)

변위(Displacement), u

응력(Stress)σ

탄성계수및포아송비(Young’s modulus & Poisson’s ratio)

체적력(Body forces)

후크의법칙(Hooke’s law)

Heat conduction

Conservation of energy

TemperatureT

Heat fluxq’

Thermal conductivityk

Heat sources

Fourier’s law

Porous media flow

Conservation of mass

Hydraulic headh

Fluid fluxq

Permeabilityk

Fluid source

Darcy’s law

Mass transport

Conservation of mass

ConcentrationC

Diffusive fluxq

Diffusion coefficientD

Chemical source

Fick’s law

Reservoir Geomechanicscomparison with fluid flow - a convenient truth

Porous media fluid flow (다공성매질의유체유동) Geomechanics(암반역학)

Darcy’s Law Hooke’s Law

Fluid Flux Stress (응력)Pressure gradient strain (변형율)Permeability Elastic modulus (Young’s Modulus)

& Poisson’s ratioTime dependent Not time dependent (elastic)

time dependent creep

Conservation of mass Equilibrium Equation

Pdl

k dq

E E dudx

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=GPa145 psi = 1 MPa = 10 bar = 10 kg중/cm2

nFA

sFA

– Stress in 2D

– Normal stress: acting perpendicular to the plane– Shear stress: acting tangent to the plane– Stress is a 2nd order tensor

Force is 1st 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: xyDirection of the stress component

Direction of surface normal upon which the stress acts

– Stress component in 2D


x xy

yx y

xy yx 3 component is independent4성분


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 components9 components

- Stress in 3D

Stress (응력) & Force(힘)stress transformation

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

N, Normal Force

V, Shear Force

Stress (응력) & Force(힘)Stress Transformation

– 1D example

– Stress transformation is conducted by multiplying the direction cosine twice (응력의변환식은방향코사인이두번곱해져서구해짐).

2

1

cosN PA A

1

sin cosV PA A

2 1cos 1 cos 22x x sin cos sin 2

2x

x

1 1 1

1 1 1

cos sin 0 cos sinsin cos 0 0 sin cos

Tx x y x

x y y

Stress (응력) & Force(힘)Stresses on inclined sections (Transformation)

• A different way of obtaining transformed stresses– For vector

– For tensor (stress)

1 1 1

1 1 1

cos sin cos sinsin cos sin cos

Tx x y x xy

x y y xy y

1

1

cos sinsin cos

x x

y y

F FF F

1cos 2 sin 2

2 2x y x y

x xy

1 1sin 2 cos 2

2x y

x y xy

=

x

y

x'y'θ

Stress (응력) & Force(힘)Stresses on inclined sections (Transformation)

• Stresses acting on inclined sections assuming that σx, σy, τxyare known.

– x1y1 axes are rotated counterclockwise through an angle θ

1cos 2 sin 2

2 2x y x y

x xy

1 1sin 2 cos 2

2x y

x y xy


– 1D example

– Stress transformation is conducted by multiplying the direction cosine twice (응력의변환식은방향코사인이두번곱해져서구해짐).

2

1

cosN PA A

1

sin cosV PA A

2 1cos 1 cos 22x x sin cos sin 2

2x

x

1 1 1

1 1 1

cos sin 0 cos sinsin cos 0 0 sin cos

Tx x y x

x y y


• 예)


주각 (Principal angles)

최대주응력(maximumPrincipal stresses)

최소주응력(minimumPrincipal stresses)


• Principal stress (주응력)– The largest (or smallest) normal stress and shear stress at that plane is 0 (가장큰수직응력이며전단응력이 0 임)

– Vertical stress in the earth is usually principal stress and the other two principal stress is in horizontal direction (통상지각의수직방향이주응력이며, 수평방향으로도두개의주응력정의가능)

– Usually denoted as σ1, σ2, σ3 (크기순으로 σ1, σ2, σ3으로표시)

Principal stress주응력상태

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

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 22 2x

1 2 sin 22xy

sin 2xy cos 2xy 0

0

1 2

2

1 2

2

1 2

2

Mohr’s Circle for Plane StressAlternative 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

1cos 2 sin 2

2 2x y x y

x xy

1 1sin 2 cos 2

2x y

x y xy

1 1 1

1 2 1 2 1 2co s 2 sin 22 2 2x x y

1 1 1

2 2 21 2( ) ( )2 2

x yx x y

2220)( Ryxx 1x

11yx

ABO C

R

1 1 1( , )x x y

1( ,0)2( ,0)2


A

D


• Mohr’s Circles for 3D – 밑금친부분이해당면에서의응력의값

S

N1.5 3 3.5

N

S

Stress (응력) & Force(힘)Summary

• Stress = Force/Area, two types: normal stress & shear stress (응력은힘/면적 -수직응력과전단응력)

• Stress can be expressed differently depending on reference axis (기준좌표에정의된응력과회전각에따라응력이달리표시됨)

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

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

• Geometric expression of deformation caused by stress (dimensionless)

L duL dx

L ∆L

1D

, ,yxxx yy

yxxy

uux y

uuy x

Strain (변형율) & Displacement (변위) 3D

xx xy xz

yx yy yz

zx zy zz

xx

yy

zz

yz

xz

xy

Tensor form

, , ,

12

12

12

yx zxx yy zz

yxxy

y zyz

x zxz

uu ux y z

uuy x

u uz y

u uz x

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

Matrix form

Mechanical Properties elastic properties

• Elastic– Loaded deformation recovers when

unloaded

• Plastic– Not recoverable

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 σc = 20-50 MPa E ~ 25 GPa

– Granite σc = 100-200 MPa E ~ 60 GPa

– Alloy steel σc = >500 MPa E ~ 200 GPa

y

yE

x

y

lateral strainaxial strain

duE Edx

LL

xdTq kdx

x xhq Kx


• Hooke’s Law

• Shear modulus G

• Generalized Hooke’s law (isotropy)

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

E

1 0 0 0

1 0 0 0

1 0 0 0

10 0 0 0 0

10 0 0 0 0

10 0 0 0 0

x x

y y

z z

yz yz

xz xz

xy xy

E E E

E E E

E E E

G

G

G

1D

2(1 )EG

xy xyG


• Normal strain in plane stress (평명응력하에서의수직변형율)

– Similarly

• Normal strain in plane stress(평면응력하에서의전단변형율)

– Shear strain is a change of angle(전단변형율은각도의변화)

– No influence from σx & σy (수직응력 σx과 σy은영향이없다.)

1x x yE

Normal strain, εx1

xE yE

+=

1y y xE

z x yE

xyxy G

G: 전단계수

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 – 3rd dimensional strain goes zero– Stresses around drill hole or 2D tunnel

00

0 0 0

xx xy

yx yy

00

0 0

xx xy

yx yy

zz

2

2

(1 ) (1 ) 0

(1 ) (1 ) 0

2(1 )0 0

xx xx

yy yy

xy xy

E E

E E

E

1 0 0 0

1 0 0 0

1 0 0 0

10 0 0 0 0

10 0 0 0 0

10 0 0 0 0

x x

y y

z z

yz yz

xz xz

xy xy

E E E

E E E

E E E

G

G

G

1 0 0 0

1 0 0 0

1 0 0 0

10 0 0 0 0

10 0 0 0 0

10 0 0 0 0

x x

y y

z z

yz yz

xz xz

xy xy

E E E

E E E

E E E

G

G

G


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

00

0 0 0

xx xy

yx yy

00

0 0

xx xy

yx yy

zz

1 0

1 0

2(1 )0 0

xx xx

yy yy

xy xy

E E

E E

E

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'

10 0 0 0 0'

2(1 )0 0 0 0 0

10 0 0 0 0'

x x

y y

z z

yz yz

zx zx

xy xy

E E E

E E E

E E E

G

E

G

(H. Gercek, 2006)

x z

y

xz zx

yx yz

xy yz

E E EE E

G G G

Conservation EquationEquilibrium equation

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

– bx, by, bz are components of acceleration due to gravity.

0

0

0

yxxx zxx

xy yy zyy

yzxz zzz

bx y z

bx y z

bx y z

현재 이이미지를 표시할 수 없습니다 .

0iM xy yx

2i

iuF mt

Very slow loading

Governing Equation3D – Navier’s Equation

• In 3D - Navier’s equation

(1 )(1 2 )E

22 2 2 2 2

2 2 2 2

2 2 2 22 2

2 2 2 2

2 2 2 2

2 2 2

( ) 0

( ) 0

( )

yx x x x zx

y y y yx zy

x x x x

uu u u u uG G bx y x zx y z x

u u u uu uG G bx y y zx y z y

u u u uG Gx zx y z

2 2

2 0y zz

u u by z z

Governing Equation1D

• Strain-displacement relationship• Stress-strain relationship• Static Equillibrium Equation

• Final equation for elasticity

dudx

1E

0xxxbx

2

2 0xx

uE bx

Governing Equation1D - example

– With no body force, and static case

2 2

2 2x x

xu uE bx t

2

2 0xuEx

x = 0Fixed u =0

1xuE Cx

2xEu x C

From σ = 1, C1=1

From u = 0 at x=0, C2=0x

xuE

x = 1σ = 1

σ = 1

x

1/E

1

Stress-Strain Relationship (Hooke’s Law)Relationship with Vp and Vs

• In really, Dynamic E ≠ Static E. Typically Dynamic E > Static E.• Difference decrease with stronger, larger confinement

2 22

2 2

2 2

2 2

3 4

22( )

: Dynamic Elastic Modulus

: Dynamic Poisson's Ratio

: P wave velocity

:S wave velocity

p sdyn p

p s

p sdyn

p s

dyn

dyn

p

s

V VE V

V V

V VV V

E

V

V

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

강도 (strength)Uniaxial compressive Strength

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

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

maxcFA

Fmax

강도 (strength)Uniaxial compressive Strength

– Computer simulation of rock failure using discrete element method (개별요소법에의한파괴모사 (컴퓨터로만든돌?))

2D 3D

강도 (strength)취성(brittle)과연성(ductile)

Brittle rock Ductile rock

강도 (strength)취성(brittle)과연성(ductile)

• 취성-연성전이현상 (Brittle-ductile transition)– 봉압증가로보다연성에가깝게거동

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

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 loadingmax

tTA


• 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

maxtTA


• The reason why Brazilian Test Works…

– Stress distribution along the x-axis

Jaeger et al., 2007

2 20

02 4 20

(1 )sin 22 (1 )arctan tan(1 2 cos 2 ) (1 )rr

P

2 20

02 4 20

(1 )sin 22 (1 )arctan tan(1 2 cos 2 ) (1 )

P

When 2θ0=15°

/r a


Failure CriteriaFriction coefficient

sF N

Failure CriteriaMohr-Coulomb Failure Criterion

– τ: Shear stress– S0=cohesion (or cohesive

strength)– σn: normal stress– μi: coefficient of internal friction =

tanΦ

0 n iS

강도 (strength)파괴조건식 (Mohr-Coulomb Failure Criterion)

0 n iS

Relationship between β and φ?

강도 (strength)파괴조건식 (Mohr-Coulomb Failure Criterion)

Increase of pore fluid pressure

σNormal stress

τmax = C0 + σn,θ tan(Φ)Φ

Shea

r stre

ss

σNormal stress


Shea

r stre

ss

σNormal stress


Shea

r stre

ss

Increase of major principal stress

σ1

σ3

Failure CriteriaMohr-Coulomb Failure criteria (Example)

– Examples of measured cohesive strength (cohesion) and coefficient of internal friction

Zoback, 2007

강도 (strength)Anisotropy (이방성)

• 각도에 따른 강도 및 탄성계수의 변화 (보령셰일)

보령셰일의 역학적, 탄성파적, 열전도도적 이방성비

압축강도 이방성비(UCS/UCS’)

인장강도 이방성비(BTS/BTS’)

탄성 계수 이방성비(E/E’)

P파 속도 이방성비(VP(90°) /VP(0°) ’)

열전도도 이방성비(K(90°) /K(0°) ’)

2.6 2.2 2.1 1.5 2.1

(a) 일축압축강도 (b) 인장강도 (c) 탄성계수

(Cho et al., 2012)

(Cho, Kim, Min and Jeon, 2012)

Strength (강도)Relationship with other parameters

• Porosity and UCS

Chang, Chandong, Mark D. Zoback, and Abbas Khaksar. "Empirical Relations between Rock Strength and Physical Properties in Sedimentary Rocks." Journal of Petroleum Science and Engineering 51, no. 3–4 (2006): 223-37.

2254(1 2.7 )

277 exp( 10 )

Effective Stress (유효응력)Definition

• Behavior of fluid-saturated reservoir will be controlled by the effective stress (Terzaghi, 1923): total stress minus pore pressure.

p

Pp

σcAc

AT

PpFF

T

T

FA

T

T

FA

Effective Stress (유효응력)Definition

• Biot Coefficient, α– measured during drained test– Varies from 0 to 1

p

V:volume change of sampleV :volume change of fluid

f

f

VV

State of Stress

Uniaxial stress Triaxial stress

1

1

1

1

2

3

2

Close to the reality in deep underground

In Situ Stress

• Water pressure with respect to depth (z)

전석원, 2008

( )p gz density acceleration due to gravity depth

0.01 ( )p gz z MPa

In Situ Stress

p gz 밀도 중력가속도 심도

0.01 ( )p gz z MPa 0.027 ( )p gz z MPa 전석원, 2008

In Situ StressMagnitude of Vertical stress

Vertical stress vs depth

– Vertical component of in situ stress

– More or less similar to predicted stress

( ) 0.027 ( )v MPa gh h m

In Situ StressMagnitude of Horizontal stress

– Horizontal components of insitu stress

– Average horizontal stress is usually 0.3 ~ 4.0 times of vertical stress

– High horizontal stress: tectonic stress, erosion, topography

In Situ StressState of Stress

• 세가지종류의응력상태

Itasca Consulting Group, 2011)

In Situ StressWorld stress map

• http://dc-app3-14.gfz-potsdam.de/

Heidbach, O., Tingay, M., Barth, A., Reinecker, J., Kurfeß, D. and Müller, B., The World Stress Map database release 2008 doi:10.1594/GFZ.WSM.Rel2008, 2008.

Thermal expansion and thermal stress

• 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 2TEK T T T T

0l T Tl

0T E T T

Summary

• Fundamentals of rock mechanics– Concept of Stress, Strain, & their relationships (Hooke’s Law)– Mohr’s Circle– Mechanical properties (Elastic modulus, Poisson’s ratio, Strength)– Governing Equations– In situ stress

OutlineApplication of reservoir geomechanics

• Applications of reservoir geomechanics: – Stress distribution around a circular hole– Borehole stability – stability of geothermal wellbore– Hydraulic fracturing & Hydroshearing– Subsidence

Documents

Week 2(9 Sept) Fundamentals Geomechanics (1) · 2018. 1. 30. · – 1D example – Stress transformation is conducted by multiplying the direction cosine twice (응력의변환식은방향코사인이두번