3 Crack Models

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2012.05.24

Crack models

ROSE

Rock Physics and Geomechanics

Course 2012

XÜÄ|Çz Y}—Ü



2012.05.24

Cracks have a strong impact on rock behavior

x

x

E

eff 1 x

x

E E Q

The presence of cracks

reduces the stiffness

of the rock



2012.05.24


x

x

E

eff 1 x

x

E E Q

Increasing stress

Crack closure

Increasing stiffness



2012.05.24


Increasing stress

Crack closure

Increasing stiffness

0

5

10

15

20

25

30

35

40

45

0 0.005 0.01 0.015 0.02

M e a n s t r e s s [ M P a ]

Volumetric strain



2012.05.24

Crack – or failed grain contact?



2012.05.24

Crack – or failed grain contacts?



2012.05.24

One large crack –



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– or several failed grain contacts?



2012.05.24

An assembly of failed grain contacts = a large crack

has a much stronger impact on rock stiffness

than the sum of the individual failed grain contacts



Crack density:2a

Drainage

parameter

Isotropic distribution of cracks:

Flat cracks: = c/a 0 D 0 for saturated rock

D = 1 for dry rock

2c

Open, flat cracks

n = number of cracks per unit volume

Not consistent with Biot, but

– we may use the model to predictthe properties of the dry material,

which gives us the frame moduli.



Non-isotropic distribution of cracks anisotropy

* 1o k

ij ij ij k

k

C C Q

k Q

z



13

Cracks control the velocities:

Large reduction

in velocityand amplitude

No effect (almost) on velocityand amplitude of P-wave

Some effect on vertically

polarized S-wave

Non-isotropic distribution of cracks anisotropy



14

Cracks can explain……….



* 1o k

ij ij ij k

k

C C Q

k Q

z

No drainage D = 0 (nearly) no P-wave anisotropy

"Saturation eliminates P-wave anisotropy"



Leon Thomsen (1995):

Pore pressure equalization between cracks and pores

Consequence: the drainage parameter will not vanish,regardless how thin the cracks are

"Saturation does noteliminate P-wave anisotropy

in porous and permeable rocks"



5/24/2012

General formalism for displacement discontinuities

(Sayers and Kachanov, 1995):

For open, penny-shaped cracks

This approximation is not valid in general,

hence the open crack model is too simple.

However, we may compensate for this by allowing the

drainage parameter D be an adjustable parameter.

0 N T ijkl B B



5/24/2012

Many cracks crack interactions

- the presence of one crack may

affect the influence of another

* 1s sK K Q

Self-consistent models:

Interactions are taken into account by giving the rock around

a crack the properties of the effective medium

* * *

s sK K K Q

*

*1

s

s

K K

Q



5/24/2012

Alternative, equally valid procedures

* 1s sK K Q

Self-consistency:

* * *

sK K K Q

*

*1

sK

K

Q

*

1 1 s

s s

Q

K K K

*

* *

1 1

s

Q

K K K

* *1s

K K Q

* 0 alwaysK * *0 for 1/K Q



There are many different self-consistent models,

giving different predictions – depending on the initial model.

The Differential Effective Medium (DEM) model resolves this discrepancy

by adding small numbers of cracks in many steps, and recalculating the

effective stiffness for each step (always working in the non-interacting limit)

This gives the same (unique!) solution for all initial models.

But – is it more correct because of that?

* * *dK K Q d



Models accounting for interactions

Many alternatives…..

- all are mathematically correct, but they give different predictions.

Which one is correct?

It depends….



If we do not knowhow the cracks are positioned

(and we usually don't),

the linear, non-interacting model

may be a good starting point.

The actual position of the cracks

relative to each other

determines which model is thecorrect one to use.

Unfortunately, no model comes

with a description of how the

cracks are positioned.



23

Stress effects on the rock framework

Cracks control the velocities:

Large reduction

in velocityand amplitude

No effect (almost) on velocityand amplitude of P-wave

Some effect on vertically

polarized S-wave



24

Rule of thumb:

Velocities are mostly affected by changes

in the normal stress in the direction of propagation

(and polarization)



25

Cracks are sensitive

to changes in stress

A compressive principal stress tends to

close a crack that is oriented normal to

the stress

0

500

1000

1500

2000

2500

3000

3500

-0.004 0 0.004 0.008 0.012

Axial strain

Velocity

[m/s]

0

50

100

150

200

250

Stress

[MPa] AxialP-wave

Radial

P-wave

Axial

S-wave

Axial

stress

Radial stress



Sliding cracks

- induce hysteresis,

permanent deformation,

and difference between

loading and unloading

modulus



27



Shear deformations

tend to open up cracks

0

500

1000

1500

2000

2500

3000

3500

-0.004 0 0.004 0.008 0.012

Axial strain

Velocity

[m/s]

0

50

100

150

200

250

Stress

[MPa] Axial

P-wave

Radial

P-wave

Axial

S-wave

Axial

stress

Radial stress



28



Shear deformations


0

500

1000

1500

2000

2500

3000

3500

-0.004 0 0.004 0.008 0.012

Axial strain

Velocity

[m/s]

0

50

100

150

200

250

Stress

[MPa] Axial

P-wave

Radial

P-wave

Axial

S-wave

Axial

stress

Radial stress



29

Solid, pores & cracks

Three sets of flat cracks

oriented normal to

the principal stresses

x

y

z

x

y

z

A model

Fjær (2006):



30



A compressive principal stress tends to

close a crack that is oriented normal to

the stress

We can not close a crack that is

already closed

i i id d

n

i i oT

Assumption:



31

Local failures induced by compressive stress:

shear

tensile

Simulation by PFC2D



Shear deformations




32

i i j i k d d d d d

2 i j k

i e



Shear deformations

tend to open up cracks Sensitivity to shear strain:

Assumption:



33

Very large shear strains

more turbulent crack development

Changes in crack density more sensitive to magnitude

than to orientation of shear strain

2

i e

Assumption:

= maximum shear strain



34

22 i j k

no

o i oi i

i o

T eT

1

o v

v

11 11 11 33 111o p

x y zC C Q Q Q

11 22 33

o o o

oC C C H 44 55 66

o o o

oC C C G

12 13 23 2o o o

o oC C C H G

H o = 80 GPa (fixed value)

Mathematics of the model:

etc.

Assumptions:

111

p

C V

etc.

Velocities:

= 0.2 (fixed value)

Velocity Stress



35

0

20

40

60

80

100

120

140

160

180

200

0 20 40 60 80

3

1 A

B

C

Test A

0

500

1000

1500

2000

2500

3000

3500

-0.004 0 0.004 0.008 0.012

Axial strain

Velocity

[m/s]

0

50

100

150

200

250

Stress

[MPa]

Axial

stress

Radial stress

Axial

P-wave

Radial

P-waves

Axial

S-wave

Scaling: , , ,o o o x y z oG

Velocity Stress



36

0

500

1000

1500

2000

2500

3000

3500

-0.004 0 0.004 0.008 0.012

Axial strain

[m/s]

0

50

100

150

200

250

[MPa]

Axial

stress

Radial stress

Axial

P-wave

Radial

P-waves

Axial

S-wave

0

20

40

60

80

100

120

140

160

180

200

0 20 40 60 80

3

1 A

B

C

Test A

0.15

800

n



2012.05.24

The model – based on flat cracks and sperical pores - matches

observations quite well

The match supports the claim that the stress dependency of

wave velocities may largely be explained in terms of opening and

closure of cracks



2012.05.24

References:

Fjær, E., Holt, R.M., Horsrud, P., Raaen, A.M. and Risnes, R. (2008) "Petroleum Related

Rock Mechanics. 2nd Edition". Elsevier, Amsterdam

Fjær, E. (2006) "Modeling the stress dependence of elastic wave velocities in soft rocks".

41st USRM Symposium, paper ARMA/USRM 06-1070

Sayers, C.M., Kachanov, M. (1995) "Microcrack-induced elastic wave anisotropy of brittle

rocks", J. Geophys. Res. B, 100, 4149-4156

Thomsen, L. (1995): "Elastic anisotropy due to aligned cracks in porous rock".

Geophysical Prospecting, 43, 805-829

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