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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

Section 1 • Basic Definitions • Poroelasticity and Effective Stress

Section 2 • Viscoplasticity (Creep) in Weak

Section 3 • Viscoplasticity (Creep) in Shales

Outline

Laboratory Testing

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

Constitutive Laws

Figure 3.1 a,b – pg.57 Stanford|ONLINE gp202.class.stanford.edu

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

11SEε

Possion’s Ratio, ν S11 only non-zero stress

ε−=ν

S13ε13

Shear Modulus, G Sij only non-zero stress

Bulk Modulus, K (Compressibility, β = K-1)

00SKε

Elastic Modulii and Seismic Waves

In an elastic, isotropic, homogeneous solid

G4KVpP wave

GVsShear Wave

Liquid G = 0 , Vs = 0

3G4KVM 2

p +=ρ=“M” Modulus

( )2s2p

VV2V2V

−=ν

Liquid ν = 0.5

Poisson’s Ratio *25.0=ν 73.131

Poisson Solid λ = G

* common value for rocks

Equation 3.5 – pg.63

Constitutive Laws

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

σg stress acting on grains Stanford|ONLINE gp202.class.stanford.edu

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

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

Effective Stress

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

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

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)

Poroelasticity

Dispersion

4 5 Log Frequency (Hz)

100 cp

Log Lab

Figure 3.6 b – pg.70 Stanford|ONLINE gp202.class.stanford.edu

Cycles of Hydrostatic Loading & Unloading – Weak Sand

Poro-Elastic Coupling Within a Reservoir How ΔPpAffects ΔSH

Using instantaneous application of force and pressure with no lateral strain:

( )pvpH PSPS ανν

α −⎟⎠

⎞⎜⎝

⎛−

Take the derivative of both sides and simplify

( )( ) pH P121S Δν−

ν−α=Δ

Pp32SH Δ=Δ1,25.0 == ανif

−=1α

Outline

Constitutive Laws

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

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.

Loading History

Ottawa Sand with Montmorillonite Clay

Observations of Instantaneous and Viscous Deformation in Dry Wilmington Sand

00.005

0 10 20 30 40

Drained Hydrostatic Load CyclingCleaned and Dried Wilmington Sand

xial Strain (in/in)

Time (hr)

Confining Pressure

Instantaneous Strain

Creep Strain

Figure 3.8 a – pg.73 Stanford|ONLINE gp202.class.stanford.edu

Creep and Clay Content

Stress History – Triaxial Conditions

Attributes of Viscoelastic/Viscoplastic Materials

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

Wilmington Sand Stress Relaxation

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

Ideal Viscoelastic Materials (Time-Dependent Stress and Strain)

Wilmington Creep and Standard Linear Solid

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

Experimental Procedure - Attenuation

0 20 40 60 80 100

Constant Frequency Test ProcedureCleaned and Dried WIlmington Sand

Load Frequency = 1MPa/hr

xial Strain (in/in)

Time (Hr)

Confining Pressure

Axial Strain

Strain

Attenuation Independent of Frequency

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

Experimental Procedure - Modulus Dispersion

0 10 20 30 40 50 60

Frequency Cycling Test Procedure

xial Strain (in/in)

Time(hr)

Axial Strain

Confining Pressure Pressure Amplitude

MeanPressure

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

Best-Fitting Power Law Model

Fits very low frequency (reservoir compaction)

Intermediate frequency (laboratory testing)

High Frequency (seismic to sonic to ultrasonic modulus dispersion)

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

ity�

Effective Pressure (MPa)�

φ0 d�

Modeling Creep Strain in Dry Field X (GOM) Sand

φ(Pc,t) = φi - (Pc/A)tb

Creep Parameters For Two Uncemented Sands

Reservoir sand

A (creep)

b (creep)

(instant) d (instant)

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

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

Outline

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

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

Experimental Procedures

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

Failure & Friction: intact/frictional rock strength

A Typical Experiment

Friction

Strength

Static Modulii

Dilatancy

Creep?

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

39%clay

25% 22% clay 33%

5% clay

Creep Increases with Clay Content

Eagleford Shale

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

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

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

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

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

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

Creep Strain over Geological Time

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

−= 1

)1(1)( εσ

Predicting Stress Anisotropy over Geological Time

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