12 static and dynamic moduli
DESCRIPTION
Static and Dynamic ModuliTRANSCRIPT
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