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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 Stanford|ONLINE gp202.class.stanford.edu
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Constitutive Laws
Figure 3.1 a,b – pg.57 Stanford|ONLINE gp202.class.stanford.edu
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Common Elastic Modulii
In all cases replace stress (S) with effective stress (σ) for fluid saturated porous rock.
Young’s Modulus, E S11 only non-zero stress
11
11SEε
=
Possion’s Ratio, ν S11 only non-zero stress
11
33
ε
ε−=ν
G =12
S13ε13
"
# $ $
%
& ' '
Shear Modulus, G Sij only non-zero stress
Bulk Modulus, K (Compressibility, β = K-1)
00
00SKε
=
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Elastic Modulii and Seismic Waves
In an elastic, isotropic, homogeneous solid
ρ
+= 3
G4KVpP wave
ρ=
GVsShear Wave
Liquid G = 0 , Vs = 0
3G4KVM 2
p +=ρ=“M” Modulus
( )2s2p
2s
2p
VV2V2V
−
−=ν
Liquid ν = 0.5
Poisson’s Ratio *25.0=ν 73.131
VV
s
p ==
Poisson Solid λ = 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 Stanford|ONLINE gp202.class.stanford.edu
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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 aσc = σg
Intergranular Stress:
Effective Stress: σg = S - (1 - a) Pp = S - Pp
Force Balance at Grain Scale:
FT = Fg where a = Ac/AT
S AT = Acσc + (AT - Ac)Pp
S = aσc + (1 - a)Pp
where a = Ac/AT
Ac
σg 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 δ−=σ
“Exact form”
pijijij PS αδ−=σ
Biot Constant
g
b
KK
1−=α 10 ≤α≤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 α = 0 φ → 0 Lim α = 1
Kb → 0
Equations 3.8 & 3.10 – pg.66 & 68
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Effective Stress
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Figure 3.5 c – pg.67 Stanford|ONLINE gp202.class.stanford.edu
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Laboratory Measured Values of Alpha
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ε ij =12G
Sij −δ ijS00( ) + 13K
δ ijS00 −α3K
δ ijPp
Shear strain not affected by Pp: KP
KS p00
00
α−=ε
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
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Dispersion
2000
3000
4000
5000
4 5 Log Frequency (Hz)
1 cp
10 cp
100 cp
Vp
Vs
Velo
city
(m/s)
Log Lab
Figure 3.6 b – pg.70 Stanford|ONLINE gp202.class.stanford.edu
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Cycles of Hydrostatic Loading & Unloading – Weak Sand
Figure 3.7 a,b – pg.71 Stanford|ONLINE gp202.class.stanford.edu
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Poro-Elastic Coupling Within a Reservoir How ΔPpAffects ΔSH
Using instantaneous application of force and pressure with no lateral strain:
( )pvpH PSPS ανν
α −⎟⎠
⎞⎜⎝
⎛−
=−1
Take the derivative of both sides and simplify
( )( ) pH P121S Δν−
ν−α=Δ
Pp32SH Δ=Δ1,25.0 == ανif
g
b
KK
−=1α
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 Stanford|ONLINE gp202.class.stanford.edu
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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 Stanford|ONLINE gp202.class.stanford.edu
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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 Stanford|ONLINE gp202.class.stanford.edu
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Wilmington Sand Stress Relaxation
Figure 3.11 a – pg.77 Stanford|ONLINE gp202.class.stanford.edu
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Ideal Viscoelastic Materials (Time-Dependent Stress and Strain)
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Wilmington Creep and Standard Linear Solid
strai
n
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Figure 3.12 – pg.78
Exploring Viscoelastic Models
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Getting the Constitutive Law Right Matters
29 Figure 3.13a – pg.79 Stanford|ONLINE gp202.class.stanford.edu
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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 Stanford|ONLINE gp202.class.stanford.edu
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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 (φj) 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, φ0) and 2 exponents (b,d) Determinable with 2 experiments
bcjc tAPtP )/(),( −=φφ
dcj P0φφ =
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
φi = φ0Pcd�
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)�
φ0 d�
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Modeling Creep Strain in Dry Field X (GOM) Sand
φ(Pc,t) = φi - (Pc/A)tb
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Creep Parameters For Two Uncemented Sands
Reservoir sand
A (creep)
b (creep)
Φ0
(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 −= −φ
164.0046.0 )3.5410
(271.0),( tPPtP ccc −= −φ
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 rocks—Part 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 rocks—Part 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
• Young’s modulus correlates with creep amount very well
Normal To Bedding
Parallel To Bedding
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Young’s Modulus
• Young’s modulus falls within rough estimates of Voigt-Reuss bounds
• Anisotropy exists between vertical and horizontal samples Stanford|ONLINE gp202.class.stanford.edu
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Analysis of Viscoplasticity
1. Describe the behavior quantitatively to
à Creep Constitutive Relation
2. Relate the creep behavior to stress relaxation using à Boltzmann Superposition
3. Investigate the implications of creep over
geologic time scales
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Long term creep experiments
)log(tAcreep =ε
ncreep Bt=ε
• 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=ε
• Parameter’s 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 Stanford|ONLINE gp202.class.stanford.edu
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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)( εσ
Predicting Stress Anisotropy over Geological Time
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