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Reservoir Geomechanics
In situ stress and rock mechanics applied to reservoir processes
Week 2 Lecture 4 Constitutive Laws Chapter 3
Mark D. Zoback Professor of Geophysics
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Section 1 Basic Definitions Poroelasticity and Effective Stress
Section 2 Viscoplasticity (Creep) in Weak
Sands
Section 3 Viscoplasticity (Creep) in Shales
Outline
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Laboratory Testing
Stre
ss (M
Pa)
Figure 3.2 pg.58
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Constitutive Laws
Figure 3.1 a,b pg.57
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Common Elastic Modulii
In all cases replace stress (S) with effective stress () for fluid saturated porous rock.
Youngs Modulus, E S11 only non-zero stress
11
11SE
=
Possions Ratio, S11 only non-zero stress
11
33
=
G = 12S1313
"
# $ $
%
& ' '
Shear Modulus, G Sij only non-zero stress
Bulk Modulus, K
(Compressibility, = K-1)
00
00SK
=
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Elastic Modulii and Seismic Waves
In an elastic, isotropic, homogeneous solid
+= 3
G4KVpP wave
=
GVsShear Wave
Liquid G = 0 , Vs = 0
3G4KVM 2p +==M Modulus
( )2s2p
2s
2p
VV2V2V
=
Liquid = 0.5
Poissons Ratio
*25.0= 73.13
1VV
s
p ==
Poisson Solid
= G
* common value for rocks
Equation 3.5 pg.63
Equation 3.6 pg.64
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Constitutive Laws
Figure 3.1 a,b pg.57
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Continuum Approach to Effective Stress
Stress = Force/AreaTotal S = F/AT
For an impermeable membrane:
Assumptions: Volume large compared to elements
Interconnected porosity
Statistically Averaged Volumes
a 0 lim ac = g Intergranular Stress:
Effective Stress: g = S - (1 - a) Pp = S - Pp
Force Balance at Grain Scale:
FT = Fg where a = Ac/AT S AT = Acc + (AT - Ac)Pp
S = ac + (1 - a)Pp where a = Ac/AT
Ac
g stress acting on grains Stanford|ONLINE gp202.class.stanford.edu
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Pp does not affect shear stress or shear strain, but does affect elastic moduli, rock strength, frictional strength
Simple (Terzaghi) form
pijijij PS =
Exact form
pijijij PS =
Biot Constant
g
b
KK
1= 10 Kb Drained bulk modulus of porous rock
Kg Bulk modulus of solid grains
Solid rock without pores. No pore pressure influence
Extremely compliant porous solid. Maximum pore pressure influence
Lim = 0
0
Lim = 1
Kb 0
Equations 3.8 & 3.10 pg.66 & 68
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Effective Stress
Figure 3.5 c pg.67
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Laboratory Measured Values of Alpha
ij =12G Sij ijS00( ) +
13K ijS00
3K ijPp
Shear strain not affected by Pp:
KP
KS p00
00
=
Elastic modulii (and strength) are dependent on effective stress
Complexity: Modulii are rate dependent because undrained rock is stiffer than drained rock (pore fluid supports external stress)
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Poroelasticity
Dispersion
2000
3000
4000
5000
4
5
Log Frequency (Hz)
1 cp
10 cp
100 cp
Vp
Vs
Veloc
ity (m
/s)
Log Lab
Figure 3.6 b pg.70
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Cycles of Hydrostatic Loading & Unloading Weak Sand
Figure 3.7 a,b pg.71
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Poro-Elastic Coupling Within a Reservoir How PpAffects SH
Using instantaneous application of force and pressure with no lateral strain:
( )pvpH PSPS
=1
Take the derivative of both sides and simplify
( )( ) pH
P121S
=
Pp32SH =1,25.0 == if
g
b
KK
=1
Sv
SH
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Section 1 Basic Definitions Poroelasticity and Effective Stress
Section 2 Viscoplasticity (Creep) in Weak
Sands
Section 3 Viscoplasticity (Creep) in Shales
Outline
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Constitutive Laws
Figure 3.1 c,d pg.57
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Viscoelastic/Viscoplastic Deformation of Unconsolidated Sands
The fact that the grains are not cemented allows these materials to creep (deform as a function of time at a constant stress or at constant strain, for stress to relax with time).
The presence of clay greatly exacerbates creep in uncemented sands.
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Loading History
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Ottawa Sand with Montmorillonite Clay
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Observations of Instantaneous and Viscous Deformation in Dry Wilmington Sand
510
1520
2530
00.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0 10 20 30 40
Drained Hydrostatic Load CyclingCleaned and Dried Wilmington Sand
Con
finin
g P
ress
ure
(MP
a)A
xial Strain (in/in)
Time (hr)
Confining Pressure
Instantaneous Strain
Creep Strain
Figure 3.8 a pg.73
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Creep and Clay Content
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Stress History Triaxial Conditions
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Attributes of Viscoelastic/Viscoplastic Materials
Figure 3.10 a-d pg.75
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Wilmington Sand Stress Relaxation
Figure 3.11 a pg.77
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Ideal Viscoelastic Materials (Time-Dependent Stress and Strain)
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Wilmington Creep and Standard Linear Solid
strain
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Figure 3.12 pg.78
Exploring Viscoelastic Models
Getting the Constitutive Law Right Matters
29 Figure 3.13a pg.79
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Experimental Procedure - Attenuation
510
1520
2530
35
-0.0
10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0 20 40 60 80 100
120
Constant Frequency Test ProcedureCleaned and Dried WIlmington Sand
Load Frequency = 1MPa/hr
Con
finin
g P
ress
ure
(MP
a)A
xial Strain (in/in)
Time (Hr)
Confining Pressure
Axial Strain
Stre
ss
Strain
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Attenuation Independent of Frequency
Figure 3.13b pg.79
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Experimental Procedure - Modulus Dispersion
510
1520
00.
005
0.01
0.01
50.
02
0 10 20 30 40 50 60
Frequency Cycling Test Procedure
Con
finin
g P
ress
ure
(MP
a)A
xial Strain (in/in)
Time(hr)
Axial Strain
Confining Pressure Pressure Amplitude
MeanPressure
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Best-fitting Model (Low Frequency)
Both the instantaneous (j) and time-dependent components of long term strain have power law functional forms. Written in terms of porosity (to simulate compaction), we have where the first term describes the instantaneous porosity change and the second term describes the normalized creep strain, where: Which leaves 4 unknowns:
2 constants (A, 0) and 2 exponents (b,d) Determinable with 2 experiments
bcjc tAPtP )/(),( =
dcj P0 =
Equation 3.16 pg.81
Equation 3.15 pg.80
i
i
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Best-Fitting Power Law Model
Fits very low frequency (reservoir compaction)
Intermediate frequency (laboratory testing)
High Frequency (seismic to sonic to ultrasonic modulus dispersion)
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Modeling Instantaneous Strain in Dry Wilmington Sand
i = 0Pcd
0.23
0.24
0.25
0.26
0.27
0.28
0.1 1 10 100
Wilmington Sand Dry/Drained/Hydrostatic
Constant Rate Test
Rate = 10 -6 /s
y = 0.27107 * x^(-0.046452) R= 0.99479 P
oros
ity
Effective Pressure (MPa)
0
d
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Modeling Creep Strain in Dry Field X (GOM) Sand
(Pc,t) = i - (Pc/A)tb
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Creep Parameters For Two Uncemented Sands
Reservoir sand
A
(creep)
b
(creep)
0
(instant)
d
(instant)
Notes
Wilmington
5410.3
0.1644
0.271
-0.046
Stiffer and more viscous
GOM Field X
6666.7
0.2318
0.246
-0.152
Softer and less viscous
Table 3.2 pg.82
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Best-Fitting Model: Wilmington
Best-Fitting Model: Field X, GOM
Maximum field compaction predicted: >10%
Maximum field compaction predicted: ~1.5% Observed field compaction ~ 2%
232.0152.0 )7.6666
(246.0),( tPPtP ccc =
164.0046.0 )3.5410
(271.0),( tPPtP ccc =
Equation 3.17 pg.81
Equation 3.20 pg.82
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Section 1 Basic Definitions Poroelasticity and Effective Stress
Section 2 Viscoplasticity (Creep) in Weak
Sands
Section 3 Viscoplasticity (Creep) in Shales
Outline
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Organic Rich Shales
Bedding plane and sample cylinder axis is either
parallel (horizontal samples) or
perpendicular (vertical samples)
3-10 % porosity
All room dry, room temperature experiments
Sample group Clay Carbonate QFP TOC (wt%)
Barnett-dark 29-43 0-6 48-59 4.1-5.8
Barnett-light 2-7 37-81 16-53 0.4-1.3
Haynesville-dark 36-39 20-23 31-35 3.7-4.1
Haynesville-light 20-22 49-53 23-24 1.7-1.8
Fort St. John 32-39 3-5 54-60 1.6-2.2
Eagle Ford-dark 12-21 46-54 22-29 4.4-5.7
Eagle Ford-light 6-14 63-78 11-18 1.9-2.5
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Recent Publications
Physical properties of shale reservoir rocks
Sone, H and Zoback, M.D. (2013), Mechanical properties of shale-gas reservoir rocksPart 1: Static and dynamic elastic properties and anisotropy, Geophysics, v. 78, no. 5, D381-D392, 10.1190/GEO2013-0050.1
Sone, H and Zoback, M.D. (2013), Mechanical properties of shale-gas reservoir rocksPart 2: Ductile creep, brittle strength, and their relation to the elastic modulus, Geophysics, v. 78, no. 5, D393-D402, 10.1190/GEO2013-0051.1
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Experimental Procedures
Hydrostatic, Triaxial Stage: Pressure applied in steps Held for 3 hrs 2 weeks
Failure & Friction: intact/frictional rock strength
Pc
Pax
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A Typical Experiment
Friction
Strength
Static Modulii
Dilatancy
Creep?
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Experimental Procedures
Hydrostatic, Triaxial Stage: Pressure applied in steps Held for 3 hrs 2 weeks
Failure & Friction: intact/frictional rock strength
From each pressure step,
The pressure ramp gives elastic modulus
The pressure hold gives the creep response
Pc
Pax
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39%clay
25%
22% clay
33%
5% clay
Creep Increases with Clay Content
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Eagleford Shale
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Creep Strain vs. Clay and E
Amount of creep (ductility) depends on clay content and orientation of loading with respect to bedding
Youngs modulus correlates with creep amount very well
Normal
To Bedding
Parallel
To Bedding
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Youngs Modulus
Youngs modulus falls within rough estimates of Voigt-Reuss bounds
Anisotropy exists between vertical and horizontal samples
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Analysis of Viscoplasticity
1. Describe the behavior quantitatively to
Creep Constitutive Relation
2. Relate the creep behavior to stress relaxation using Boltzmann Superposition
3. Investigate the implications of creep over
geologic time scales
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Long term creep experiments
)log(tAcreep =
ncreep Bt=
Most creep observed were only 3 hours long, and suggested logarithm function
Long experiments show that it is more closer to a power-law in the long term
Furthermore, the total response (elastic + creep) can be described by a power law
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Power-Law Parameters
nBt=
Parameters B and n are found for every creep step by fitting a line to the creep compliance, J(t), in log-log space
*J(t) determined by deconvolving creep data with stress ramp input
Compliant rocks have higher B and higher n
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Contours are % strain under 50 MPa differential load Reasonable axial strain magnitudes of 0.1~3%
Creep Strain over Geological Time
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q Stress Accumulation under constant strain rate q 150 Ma - Half of age
of Barnett shale q 10-19 s-1 - Stable
intraplate
q Significant stress relaxation observed for high n
ntnB
t
= 1)1(
1)(
Predicting Stress Anisotropy over Geological Time
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