12 static and dynamic moduli

2012.05.24

Static and Dynamic Moduli

ROSERock Physics and Geomechanics

Course  2012

2012.05.24

What do we mean with "static" and "dynamic" moduli?

Elastic wave velocities

2e eP

eS

GV

GV

Dynamic moduli

2

2 22e s

e p s

G V

V V

Stress and strain measured in a rock mechanical test

z

z

E

Static moduli

3

Estat Edyn

Estat < Edyn

In general:

Note: is not a constant ratio – it changes with stress! :stat dynE E

2012.05.24

In saturated rocks,

- Static deformation is often drained

- Dynamic moduli are always undrained

Occasionally used definition:

"Static modulus = drained modulus"

"Dynamic modulus" = undrained modulus"

5

Strain

Stre

ss

Strain

Stre

ss

Static modulus

Alternative definition, also used:

"Static modulus" = slope of stress-strain curve measured during unloading

6

Strain

Stre

ss

Strain

Stre

ss

Static modulus

Static modulus

Our definition:

"Static modulus" = slope of stress-strain curve

7

Standard triaxial set-up + acoustics

StressStrainAcoustic wave velocities

Laboratory tests:

Enables simultaneous measurements of- static moduli (slope of stress-strain curve)- dynamic moduli (density x velocity2)

Measurements:

Static and dynamic moduli of soft rocks are different.

The difference changesalong the stress path.

Potential causes for the differencebetween static and dynamic moduli:

- Strain rate dispersion- Length of stress path- Stress history- Rock volume involved- Drainage conditions- Anisotropy

9

Strain

Stre

ss

Strain

Stre

ss

Static modulus

First: consider the static modulus measured during initial loading

10

We introduce a parameter P, defined as:

P is a measure of the inelastic part of the deformation caused by a compressive hydrostatic stress increment.

Dry rock

,

3v v eP

,v eeK

v - total volumetric strain

Hydrostatic test

- elastic strain

K = Static bulk modulusKe = Dynamic bulk modulusK K

PKe

e

1 3

Fjær (1999):

11

Observations

0

5

10

15

20

25

0 5 10 15 20 25

Hydrastatic stress [MPa]

1/P

[GPa

-1]

Hydrostatic test

gPT

PFC3D-simulationLi & Fjær, 2008

13

We introduce a parameter F, defined as:

F is a measure of the inelastic part of the deformation caused by a shear stress increment.

Dry rock

, ,z z e z p

z

F

,z eeE

,z p z zP

z - total axial strain

Uniaxial loading test

- elastic strain

E = Static Young’s modulusEe = Dynamic Young’s modulus 1

1e

z e

EE FP E

Fjær (1999):

14

Observations

0.00

0.01

0.02

0.03

0.04

0.000 0.005 0.010 0.

Shear strain

F** z rF F S

z r o

z r

F AS

Uniaxial loading test

15

Discussion: the F - parameter

Since E (1 - F)

when F =1 then E = 0

peak stress

11

e

z e

EE FP E

0

20

40

60

80

100

120

140

160

0.005 0.010 0.015 0.020 0.025St

ress

(MPa

)

Peak stress

Axial

Radial

Note:

F = 1 rock strength

16

We have a set of equations……

These represent a constitutive model for the rock

We may use it to predict rock behavior, and thereby derive mechanical properties for the rock gP

T

11

e

z e

EE FP E

K KPKe

e

1 3

z r o

z r

F AS

17

Porosity, Density, Sonic, . . . .

Constitutive model

Application for logging purposes

Simulates rock mechanical test on fictitious core

0

2

4

6

8

10

12

-5 0 5 10 15

Strain (mStrain)

Stre

ss (M

Pa)

Strength

Providescalibrationfor the model

Providesstrengthand stiffness

18

0

50

100

150

200

0 25 50 75 100 125

Strength (MPa) @ 2MPa

Dep

th (m

- fr

om a

refe

renc

e po

int)

Courtesy of Statoil

Prediction from logs

Core measurements

… an example:

Static and dynamic moduli of soft rocks are different.

The difference changesalong the stress path.

Potential causes for the differencebetween static and dynamic moduli:

- Strain rate dispersion- Length of stress path- Stress history- Rock volume involved- Drainage conditions- Anisotropy

20

Potential causes for the differencebetween static and dynamic moduli:

- Strain rate- Length of stress path- Stress history- Rock volume involved- Drainage conditions- Anisotropy

1. Stress path:

Static modulus = slope ofstress-strain curve:

Dynamic modulus given by axial P-wave velocity:

z

z

H

2e PH V

Uniaxial strain (K0)

21

Potential causes for the differencebetween static and dynamic moduli:

- Strain rate- Length of stress path- Stress history- Rock volume involved- Drainage conditions- Anisotropy

2. Saturation:

Static modulus: usually drained

Dynamic modulus: undrained

Irrelevant for dry rocks

22

Potential causes for the differencebetween static and dynamic moduli:

- Strain rate- Length of stress path- Stress history- Rock volume involved- Drainage conditions- Anisotropy

3. Loading direction:

Friction controlled shear slidingof closed cracks- the only non-elastic processactive during unloading

23

0

0.005

0.01

0.015

0.02

-25 -20 -15 -10 -5 0

Axial stress change [MPa]1/

H -

1/H

e [1

/GPa

]

SH [1/GPa]

1 1H

e

SH H

Observation:

Non-elastic compliance

increases linearly with decreasing stress

1 1H

e

SH H

Static modulus

Strain

Stre

ss

Strain

Stre

ss

Static modulus

Consider the slope of thestress-strain curve during unloading

Fjær et al. (2011):

Observation:

24

Non-elastic compliance

increases linearly with decreasing stress

0

0.005

0.01

0.015

0.02

-25 -20 -15 -10 -5 0

Axial stress change [MPa]1/

H -

1/H

e [1

/GPa

]

SH [1/GPa]

1 1H

e

SH H

1 1H

e

SH H

Assumption:Linear extrapolation towards thebeginning of the unloading pathprovides estimate of behavior for vanishing length of stress path

25

Potential causes for the differencebetween static and dynamic moduli:

- Strain rate- Length of stress path- Stress history- Rock volume involved- Drainage conditions- Anisotropy

4. Loading direction:

Linear extrapolation towards thebeginning of the unloading pathprovides estimate of behavior for vanishing length of stress path

26

Strain rate

4 f

Average strain rate for dynamic measurements:

710 = strain amplitude

= frequency55 10 Hzf

1 110 s For "static" deformations6 110 s

1 Hzf Corresponds to an acoustic wave with

If strain rate is the only cause for the differencebetween the static and dynamic moduli, then

static modulus dynamic modulus at 1 Hz

27

0

0.002

0.004

0.006

0.008

0.01

0.012

20 30 40 50 60

Non

‐elastic co

mpliance [GPa

‐1]

Axial stress [MPa]

Castlegate sandstoneDry, clay free – presumably no significant dispersion

Uncertainties: , 1%, 1%Phb V No measurable dispersion

0

20

40

60

80

100

120

140

160

0 5000 10000 15000 20000 25000

Stress [M

Pa]

Time [s]

• K0 stress path

• Unloading sequences

01 1H z z

e

S a bH H

28

Berea sandstoneDry, 8% clay – possibly some dispersion

0

0.002

0.004

0.006

0.008

0.01

0.012

10 30 50 70

Non

‐elastic co

mpliance [GPa

‐1]

Axial stress [MPa]

01 1H z z

e

S a bH H

0

20

40

60

80

100

120

140

0 20 40 60 80

V ph-

V pl[m

/s]

Axial stress [MPa]

Uncertainties: , 1%, 1%Phb V

Significant, measurable dispersion, decreasing with increasing stress

Fjær et al. (2012):

29

Mancos shaleSaturated, with 13% illite/smectite, 5% kaolinite, etc. – probably significant dispersion

0

0.005

0.01

0.015

0.02

0.025

0.03

20 30 40 50 60

Non

‐elastic co

mpliance [GPa

‐1]

Axial stress [MPa]

Uncertainties: , 1%, 1%Phb V

Significant dispersion, far beyond the resolution limit for the method

Other applicationsWe measure,

- the axial P-wave velocity

- the axial S-wave velocity

- the axial stress

- the radial stress

We want

Thomsen’s

We have , we need

33e

PzCV

44e

SzrCV

z

r13

1333

r

z

CrC

2

,4444

33 ,

eS zre

eP z

VCrC V

2 2

13 44 33 44

33 33 442

e e e e

e e

C C C C

C C C

2 2

13 44 44

44

1

2 1

e e e

e

r r r

r

44 13& er r 13er

1313

33

r

z

CrC

0

0.002

0.004

0.006

0.008

0.01

0.012

-25 -20 -15 -10 -5 0

Non

-ela

stic

com

plia

nce

[1/G

Pa]

Stress change [MPa]

We have seen that the non-elastic compliancevanish at the turning point on the stress path, (the approach is linear during unloading)

- which means: static modulus dynamic modulusat the turning point

0

0.05

0.1

0.15

0.2

0.25

-25 -20 -15 -10 -5 0

r13

Stress change [MPa]

Therefore –We assume thatstatic r13 dynamic r13at the turning point

13er

SH [1/GPa]

13r

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

00 10 20 30 40 50

Thom

sen'

s

Shear stress [MPa]

Example (dry Castlegate sandstone):

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. (1999) "Static and dynamic moduli of weak sandstones". In Proceedings of the 37th U.S. Rock Mechanics Symposium, eds Amadei et al., 675-681

Li, L. and Fjær, E. (2008) "Investigation of the stress-dependence of static and dynamic moduli of sandstones using a discrete element method". 42nd US Rock Mechanics Symposium and 2nd U.S.-Canada Rock Mechanics Symposium, ARMA 08-191

Fjær, E. (2009) "Static and dynamic moduli of a weak sandstone". Geophysics, 74,WA103-WA112

Fjær, E., Holt, R.M., Nes, O.M. and Stenebråten, J.F. (2011) "The transition from elastic to non-elastic behavior". 45rd US Rock Mechanics Symposium, ARMA 11-389

Fjær, E., Stroisz, A.M. and Holt, R.M. (2012) "Combining static and dynamic measurements for evaluation of elastic dispersion". 46rd US Rock Mechanics Symposium, ARMA 11-537