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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

Stanford|ONLINE gp202.class.stanford.edu

2

2

Section 1 • Basic Definitions • Poroelasticity and Effective Stress

Section 2 • Viscoplasticity (Creep) in Weak

Sands

Section 3 • Viscoplasticity (Creep) in Shales

Outline


3

Laboratory Testing

Stre

ss (M

Pa)

Figure 3.2 – pg.58 Stanford|ONLINE gp202.class.stanford.edu

4

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

=


6

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



7

Constitutive Laws


8

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


10

Effective Stress

Figure 3.5 c – pg.67 Stanford|ONLINE gp202.class.stanford.edu

11

Laboratory Measured Values of Alpha

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)


12

Poroelasticity

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

13

Cycles of Hydrostatic Loading & Unloading – Weak Sand


14

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


15



Sands


Outline


16

Constitutive Laws

Figure 3.1 c,d – pg.57 Stanford|ONLINE gp202.class.stanford.edu

17

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.


18

Loading History


19

Ottawa Sand with Montmorillonite Clay


20

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

21

Creep and Clay Content


22

Stress History – Triaxial Conditions


23

Attributes of Viscoelastic/Viscoplastic Materials

Figure 3.10 a-d – pg.75 Stanford|ONLINE gp202.class.stanford.edu

24

Wilmington Sand Stress Relaxation

Figure 3.11 a – pg.77 Stanford|ONLINE gp202.class.stanford.edu

25

Ideal Viscoelastic Materials (Time-Dependent Stress and Strain)


26

Wilmington Creep and Standard Linear Solid

strai

n


27

Figure 3.12 – pg.78

Exploring Viscoelastic Models

Getting the Constitutive Law Right Matters

29 Figure 3.13a – pg.79 Stanford|ONLINE gp202.class.stanford.edu

29

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


30

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


32

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 3.15 – pg.80 i

i


33

Best-Fitting Power Law Model

Fits very low frequency (reservoir compaction)

Intermediate frequency (laboratory testing)

High Frequency (seismic to sonic to ultrasonic modulus dispersion)

34

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�


35

Modeling Creep Strain in Dry Field X (GOM) Sand

φ(Pc,t) = φi - (Pc/A)tb


36

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


37

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




38



Sands


Outline


39

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


40

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


41

Experimental Procedures

Hydrostatic, Triaxial Stage: Pressure applied in steps Held for 3 hrs – 2 weeks

Failure & Friction: intact/frictional rock strength

Pc

Pax


42

A Typical Experiment

Friction

Strength

Static Modulii

Dilatancy

Creep?


43

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


44

39%clay

25% 22% clay 33%

5% clay

Creep Increases with Clay Content


45

Eagleford Shale


46

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


47

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

48

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


49

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


50

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

51

Contours are % strain under 50 MPa differential load Reasonable axial strain magnitudes of 0.1~3%

Creep Strain over Geological Time


52

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


53

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