Mechanical properties of shale-gas reservoir rocks — Part 2: … · 2019-02-19 · brittle...

Mechanical properties of shale-gas reservoir rocks mdash Part 2 Ductilecreep brittle strength and their relation to the elastic modulus

Hiroki Sone1 and Mark D Zoback2

ABSTRACT

We studied the elastic moduli ductile creep behavior andbrittle strength of shale-gas reservoir rocks from Barnett Hay-nesville Eagle Ford and Fort St John shale in a series of triaxiallaboratory experiments We found a strong correlation betweenthe shale compositions in particular the volume of clay pluskerogen and intact rock strength frictional strength and visco-plastic creep Viscoplastic creep strain was approximately linearwith the applied differential stress The reduction in sample vol-ume during creep suggested that the creep was accommodatedby slight pore compaction In a manner similar to instantaneousstrain there was more viscoplastic creep in samples deformedperpendicular to the bedding than parallel to the bedding The

tendency to creep also correlated well with the static Youngrsquosmodulus We explained this apparent correlation between creepbehavior and elastic modulus by appealing to the stress parti-tioning that occurs between the soft components of the shales(clay and kerogen) and the stiff components (quartz feldsparpyrite and carbonates) Through a simple 1D analysis wefound that a unique relation between the creep complianceand elastic modulus independent of composition and orienta-tion can be established by considering the individual creepbehavior of the soft and stiff components that arises from thestress partitioning within the rock This appears to provide amechanical explanation for why long-term ductile deforma-tional properties can appear to correlate with short-term elasticproperties in shale-gas reservoir rocks

INTRODUCTION

We report here laboratory studies of the deformational propertiesof various shale-gas reservoir rocks through a suite of comprehensivelaboratory experiments Our objective was to investigate severalfundamental properties of shale that may be relevant to shale-gasproduction First we report observations of time-dependent creepdeformation of these rocks at differential stress conditions compa-rable with those in the reservoirs Intact and frictional strengthswere also reported to understand how brittle strength depends onshale composition Although the many published empirical rela-tions between rock strength and elastic properties imply that it isnot easy to accurately predict deformational properties from petro-physical parameters (see the review by Chang et al 2006) we alsoinvestigate the degree to which the deformational properties pre-sented here (for example rock strength and creep) compare withelastic properties (such as the Youngrsquos modulus) reported in our

companion paper (Sone and Zoback 2013) The relation betweenelastic stiffness and creep compliance is discussed quantitatively byappealing to the stress partitioning that occurs between the rockconstituents using a relatively simple 1D analysis A more accurate3D model was also performed which yielded similar results (Sone2012) Finally some discussion on the possible relation betweenelastic modulus and rock strength is provided

LABORATORY PROCEDURE

Laboratory experiments discussed here are the same experimentsas those discussed in Sone and Zoback (2013) The samples comefrom Barnett Haynesville Eagle Ford and Fort St John shalesMineralogy of the samples constrained by powder X-ray diffractionanalysis shows that clay quartzthorn feldspar thorn pyrite (QFP) and car-bonate contents vary between 5 volndash50 vol 5 volndash60 voland 0 volndash80 vol respectively representing a wide range of

Manuscript received by the Editor 12 February 2013 revised manuscript received 15 May 2013 published online 13 September 20131Formerly Stanford University Department of Geophysics Stanford California USA presently GFZ German Research Centre for Geosciences Potsdam

Germany E-mail sonegfz-potsdamde2Stanford University Department of Geophysics Stanford California USA E-mail zobackstanfordeducopy 2013 Society of Exploration Geophysicists All rights reserved

D393

GEOPHYSICS VOL 78 NO 5 (SEPTEMBER-OCTOBER 2013) P D393ndashD402 10 FIGS101190GEO2013-00511

Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg

mineralogy (Figure 1) Samples from Barnett Haynesville and Ea-gle Ford shale are further divided into two subgroups with distinctmineralogy in which subgroup-1 contains more clay and organiccontents than subgroup-2 Total organic contents range from04 wtndash58 wt and porosities estimated from the mineraland bulk density range between 15ndash9 As described in thecompanion paper (Sone and Zoback 2013) clay and kerogen con-tent roughly correlate with each other The porosity estimated basedon the bulk and average mineral densities increases with the in-crease of clay and kerogen content possibly because pore volumesin these shales mostly reside within the clay aggregates and solidorganics in the sample (Loucks et al 2009 Sondergeld et al2010) Microstructural observations from the companion paper(Sone and Zoback 2013) also showed that these shales exhibitvarious degrees of fabric anisotropy which is reflected on theanisotropy of their elastic propertiesCylindrical samples of 1-inch diameter and 12ndash21-inch length

from each sample groups were prepared with the axes either per-pendicular (vertical) or parallel (horizontal) to the bedding planesThese samples were pressurized in a servocontrolled triaxial defor-mation apparatus to observe its static and dynamic elastic propertiesand creep behavior Hydrostatic confining pressure Pc was first ap-plied in one to four steps (hydrostatic stage) and then axial differ-ential stress Pdiff was applied in two to five steps while Pc was heldconstant (triaxial stage) The duration of each stress step was either30 or 60 s after which the stress was held constant for 3 h to observethe creep response After the triaxial stage the samples were takento failure by loading the samples at a constant axial strain rate of10minus5 sminus1 to measure rock strengths (failure stage) After rock fail-ure we continued to slide the failure plane to measure the residualstrengths of the rock The constant confining pressure Pc during thetriaxial and failure stages was varied between 10 and 60 MPa sothat the Pc dependence of rock strengths and creep behavior couldbe evaluated The magnitude of the stress steps in the triaxial stageΔPdiff varied between 3 and 45 MPa to simulate differential stressstates above and below in situ levelsDuring the experiments the sample deformation in the direction

parallel to the cylindrical axis was measured by a pair of linear

variable differential transformer displacement transducers andthe deformation perpendicular to the sample cylindrical axis (lateraldeformation) was measured by a pair of spring-mounted strain-gauge transducers attached outside of the heat-shrink Viton jacketboth measurements had a displacement resolution of about 1 μmThe axial differential load was measured by an internal load cellyielding 03 MPa resolution for a 1-inch-diameter sample An ex-ample of the strain response to some stress steps during the triaxialstage is shown in Figure 2 We divide the total strain response to astress step into two parts elastic strain (εelastic) and creep strain(εcreep) We used εelastic to determine the static elastic constantsand we used εcreep to quantify the amount of creep strain that occursafter 3 h of constant stress Assuming that shales are transverselyisotropic with the symmetry axis (x3-axis) perpendicular to the bed-ding plane we are able to determine the vertical Youngrsquos modulusE3 and Poissonrsquos ratio v31 from the vertical samples and the hori-zontal Youngrsquos modulus E1 and Poissonrsquos ratios v13 and v12 fromthe horizontal samplesThe maximum axial differential stress during the triaxial stage

was kept below 50 of the ultimate rock strengths to assure thatcreep deformation did not enter its tertiary creep stage in whichstrain rate starts to accelerate and lead to unstable rock failure(Lockner 1993) We avoided the tertiary creep stage because ourfocus in the triaxial stage was to observe the long-term ductile prop-erty of the samples Also high differential stress magnitudes thatlead to tertiary creep are not pervasive in the crust because the crustis generally in equilibrium with the sliding frictional strength ofthe crustal materials (eg Townend and Zoback 2001) We alsonote that it is unlikely that time-dependent deformation is due toporoelastic effects because the fluid saturation of the cores wereat most 40 even including clay bound water

TRIAXIAL CREEP GENERAL CHARACTERISTICS

The axial and lateral creep strain responses during the triaxial stagefrom experiments using Haynesville-1 vertical and Barnett-1 hori-zontal samples are compared in Figure 3a After application of adifferential stress step the sample shrinks in the axial direction

Eagle ford-1Eagle ford-2Fort St JohnHaynesville-1

Haynesville-2

Barnett-1Barnett-2

00

0

02

02

02

04

04

04

06

06

06

08

08

08

1

1

1

Clay + K

erogen

Carbonates

Qua

rtz f

elds

par

pyrit

e (Q

FP)

Figure 1 Ternary plot representation of the sample material com-positions Barnett Haynesville and Eagle Ford samples are furtherdivided into two subgroups in which subgroup-1 samples havehigher claythorn kerogen content than subgroup-2 samples

0 5000 10000 15000 200000

20

40

60

80

4

6

8

x10ndash3

∆P2

Ultrasonic velocity measurements

εcreep1

εelastic1

∆P1 2

Axi

al s

trai

n (ndash

)

Diff

eren

tial s

tres

s (M

Pa)

Time (s)

εcreep2

εelastic2

Figure 2 An example of axial differential stress and axial straindata during the stress steps in the triaxial stage The elastic andcreep strain from each stress step are used to compute the Youngrsquosmodulus and the 3-h creep compliance

D394 Sone and Zoback

Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg

and dilates in the lateral direction over time However although theaxial strain continues to creep lateral strain appears to stabilize afterabout 10 min and stay constant The plots also show that the creepstrain response is much greater in the axial direction than in thelateral direction suggesting that the sample is losing volume Thusmost of the ductile response in these samples occurs as compactionin the direction of the applied stressFigure 3b displays several representative axial creep strain data

observed during the triaxial stage Even the sample that exhibitedthe least amount of creep (Barnett-2) shows some creep behavioralthough the magnitude is very small Note that the aluminum alloydata (Al7075 black data) tested under similar conditions showed nocreep behavior Figure 3b delineates how different the creep behav-ior can be between samples but we cannot directly compare thestrain responses to argue how one sample creeps more than anothersample because each set of triaxial creep data was collected underdifferent magnitudes of axial stress steps

In Figure 4 the cumulative amount of creep strain after each 3-hstep is plotted against cumulative differential stress during a testEach connected set of data points represents a single experimentwith one sample All samples show an approximately linear trenddespite the varying magnitudes of stress steps applied in each stepThis suggests that the magnitude of creep strain after 3 h roughlyscales linearly with the magnitude of differential stress The slope ofthe linear relation represents the tendency to creep for that sampleWe refer to this slope as the 3-h creep compliance Screep which isdetermined by linear regression and has units of MPaminus1 Note thatthe 3-h creep compliance does not describe the time-dependent con-stitutive relation but it merely represents the amount of creep after aspecific time of 3 h Nonetheless Screep becomes a useful proxy forus to infer the ductility of a sampleThe 3-h creep compliance determined from our data ranges be-

tween 1endash6 and 3endash5 MPaminus1 We note that these values are signifi-cantly less than are typically observed from creep of unconsolidatedreservoir rocks Laboratory results from Hagin and Zoback (2004)using Wilmington sands show that the 6-h volumetric creep strainper megapascal of hydrostatic pressure was about 6endash4 MPaminus1 andit was between 1endash3 and 3endash3 MPaminus1 for weak Gulf of Mexicoshales (Chang and Zoback 2009) Thus creep strains per unit stresschange for shale-gas reservoir rocks are an order of magnitudesmaller than those of uncemented sand reservoir rocks and weakGulf of Mexico shales We also find that the 3-h creep complianceis insensitive to the magnitude of the confining pressure during thetriaxial stage For instance the confining pressures during the tri-axial stage varied from 10 to 60 MPa in the Haynesville-1 verticalsamples but the slopes representing the creep compliances are sim-ilar to each other and do not exhibit any systematic variation with Pc

(Figure 4 red circles) The same is observed from all sample groupsexcept for the Eagle Ford vertical samples which we suspect hadgreater sample variability Therefore the 3-h creep compliances weobtain represent the intrinsic tendency to creep for these shalesthat is insensitive to confining pressure and differential stresswithin the time scale stress levels and temperature condition of ourexperiments

0 20 40 60 80 1000

05

1

15

2times10ndash3

Cumulative differential stress (MPa)

Cum

ulat

ive

axia

l cre

ep s

trai

n (ndash

)

Ver

Ef-1Ef-2

Hor

Hv-1Hv-2

Bn-1Bn-2

FSJ

Figure 4 Cumulative creep strain versus cumulative differentialstress Each set of connected data points represents a sampleand the slope of the data points represents the approximate 3-h creepcompliance Bn Barnett Hv Haynesville Ef Eagle Ford FSJ FortSt John

Barnett-1H Axial strain

Haynesville-1V Axial strain

Barnett-1H Lateral strain

Haynesville-1V Lateral strain

Com

pressionD

ilation

0 2000 4000 6000 8000 10000 12000Time [s]

ndash1

0

1

2

3

4

5

6

7

8

Cre

ep s

trai

n (ndash

)

x10ndash4a)

0 2000 4000 6000 8000 10000 12000Time [s]

ndash1

0

1

2

3

4

5

6

7

8

Cre

ep s

trai

n (ndash

)

x10ndash4

Barnett-1V Pdiff=46 MPa

Haynesville-1V Pdiff= 32 MPa

EagleFord-1V Pdiff=16 MPa

Barnett-2H Pdiff=45 MPa

Aluminum Alloy Pdiff=15 MPa

b)

ndash2

Figure 3 (a) Axial and lateral creep strain responses during the tri-axial stage from a Haynesville vertical sample (Pc frac14 30 MPaΔPdiff frac14 29 MPa) and a Barnett horizontal sample (Pc frac1420 MPa ΔPdiff frac14 48 MPa) Note that the lateral strain stops creep-ing after about 1000 s (b) Several representative axial creep straindata A test on an aluminum alloy standard is also shown in black

Mechanical properties of gas shale mdash Part 2 D395

Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg

TRIAXIAL CREEP RELATIONWITH OTHER PROPERTIES

The strain data suggest that the samples compact during creepThis is also confirmed from the slight increase in dynamic modulimeasured from ultrasonic velocities after 3 h of creep Figure 5ashows the positive correlation between creep compliance and thepercent change in P-wave modulus after 3 h of creep which sug-gests that the cause of creep compaction is related to the cause ofelastic stiffening Because the applied stress and the mineral proper-ties stay the same during creep the overall stiffening of the samplesshould be the result of porosity reduction and pore stiffening bothof these being caused by pore volume compaction Because nano-structural observations (Loucks et al 2009 Curtis et al 2010 Son-dergeld et al 2010) reveal that most of the pore spaces in theseshale-gas reservoir rocks reside in the clays and solid organics mostof the compaction responsible for the creep deformation should beoccurring within the clays and organics in the rock This is conceiv-able because an increase in clay content is known to enhance creepdeformation in unconsolidated shale sediments (Chang et al 1997)and clay minerals have low friction and velocity-strengthening fric-tional properties that favor stable time-dependent sliding behaviorunlike quartz and carbonate minerals (Moore and Lockner 2004)Also coal which is sometimes considered as a proxy for solid or-ganic materials is known to exhibit significant creep behavior fromlaboratory experiments (Hagin and Zoback 2010 Yang and Zo-back 2011)If creep deformation predominantly takes place within the clays

and organics it is naturally expected that the creep compliance ofthe rock increases with the amount of clays and organics in the sam-

ple When the 3-h creep compliance is compared with the amount oftotal clay and kerogen volume in the rock (Figure 5b) such a pos-itive correlation is observed when focusing on data from a singlereservoir (eg Barnett-1 versus Barnett-2 samples) However theredoes not seem to be a unique trend that explains all the data EagleFord vertical samples particularly have high creep compliances de-spite their relatively low clay and kerogen content This suggeststhat clay and kerogen content is not the only control on the creepcompliance In fact the creep compliance is fairly anisotropic justlike their elastic properties (see Sone and Zoback 2013) such thatthe rock creeps more in the direction perpendicular to the beddingthan in the direction parallel to the bedding Thus the bedding planeorientation with respect to the applied uniaxial differential stressalso has a significant control on the behaviorFigure 5c and 5d shows the 3-h creep-compliance data plotted

against some static elastic properties The static elastic constantsused here are those determined from the elastic strain (Figure 2)measured during the stress step Because multiple stress steps wereapplied during the triaxial stage this resulted in several measure-ments of static elastic constants from each sample We use theelastic constants derived from the stress step when the axial stress(confining pressurethorn differential stress) was closest to 50 MPa be-cause 50 MPa was when most of the pressure dependence of theelastic constants became insignificant due to the closure of cracks(see Sone and Zoback 2013) Figure 5c shows that there is a well-defined correlation between the creep compliance and the Youngrsquosmodulus regardless of the clay and kerogen content or the orienta-tion of the samples The elastic moduli of these rocks are also de-pendent on material composition and orientation thus evoking afundamental similarity in the cause of variation in elastic propertiesand creep compliances On the other hand all types of Poissonrsquosratios (v31 v13 v12) do not show any visible correlation with creepcompliance

INTACT AND FRICTIONAL ROCK STRENGTHS

As the samples were eventually taken to failure by loading in theaxial direction and all samples exhibited typical brittle behaviorcharacterized by the rapid breakdown of rock strength from theultimate- to residual-strength and the formation of a localized fail-ure plane cutting through the cylindrical sampleThe ultimate strength of the intact samples (maximum axial stress

upon failure) is plotted against confining pressure in Figure 6a As-suming a Mohr-Coulomb failure criterion a linear regression to thestrength data from each sample group gives an estimate of the uni-axial compressive strength (inferred UCS) from the y-intercept andthe internal coefficient of friction (μi) from the slope n The internalfriction is calculated from the slope n using the following equation

μi frac14n minus 1

2ffiffiffi

np (1)

Note that we do not distinguish between vertical and horizontalsamples here because it is generally regarded that the intact rockstrengths parallel to and perpendicular to the bedding plane arethe same when anisotropic rock strength are interpreted to be causedby a single plane of weakness in this case the bedding plane beingthe plane of weakness (Donath 1961 Paterson and Wong 2005)However the horizontal strength data tend to plot above the generaltrend determined by the linear regression (Figure 6a) Thus the

a)

d)c)

ndash1 0 1 2 30

05

1

15

2

25

3

35times10ndash5

Increase in P-wave modulus ()

3 h

cree

p-co

mpl

ianc

e (M

Pa

ndash1)

0 10 20 30 40 500

05

1

15

2

25

3

35times10ndash5

Clay + Kerogen volume ()

3 h

cree

p-co

mpl

ianc

e (M

Pa

ndash1)

0 20 40 60 800

05

1

15

2

25

3

35times10ndash5

Youngrsquos modulus (GPa)

3 h

cree

p-co

mpl

ianc

e (M

Pa

ndash1)

01 015 02 025 030

05

1

15

2

25

3

35times10ndash5

Poissonrsquos ratio ν31

3 h

cree

p-co

mpl

ianc

e (M

Pa

ndash1)

035

b)

Fort St JohnHor

Eagle ford-2Eagle ford-1

Ver HorHaynesville-1Haynesville-2

Barnett-1Barnett-2

Ver HorVer Hor

Figure 5 Three-hour creep compliance plotted against other prop-erties (a) Plotted versus increase in P-wave modulus after 3-h ofcreep (b) Plotted versus claythorn kerogen volume (c) Plotted versusYoungrsquos modulus (d) Plotted versus vertical Poissonrsquos ratio


Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg

intact rock strengths in our samples appear to be slightly stronger inthe direction parallel to the bedding planeFigure 6b shows the residual strength data collected in the failure

stage The angle between the failure plane and the sample cylinderaxis was measured after the experiment and this angle was used tocalculate the normal and shear tractions resolved on the failureplane during sliding We again assume a Mohr-Coulomb failure cri-terion and we obtain the sliding coefficient of friction μs and co-hesion from the slope and y respectively determined through linearregression Frictional sliding surfaces generally do not carry signifi-cant amounts of cohesion but we consistently see about 5ndash15 MPaof cohesion (Figure 6b) We interpret this apparent cohesion asthe sliding resistance due to the rough failure plane created by

the experiment Sliding along the failure plane reached at mostabout 25 mm in our experiments and this was not enough tosmooth out the failure plane For the residual strength data wedo not see any dependence on the orientation of the sampleFigure 7a and 7b compares the inferred UCS with the average

total clay-kerogen volume and the average static Youngrsquos modulusof each sample group We find that inferred UCS generally de-creases with increasing clay-kerogen volume and it has a positivecorrelation with the Youngrsquos modulus As the error bars indicatesome of these values were poorly constrained especially those fromEagle Ford-2 The uncertainty may be the result of sample variabil-ity within the sample group as the creep compliances for these sam-ples also varied The same trend is found when the internal

She

ar tr

actio

n (M

Pa)

0 20 40 60 80 1000

20

40

60

80

Normal traction (MPa)

Fort St John

Barnett-1Barnett-2

Haynesville-1Haynesville-2

Eagle ford-2

Ver Hor

Eagle ford-1

Ulti

mat

e st

reng

th (

MP

a)

0 10 20 30 40 50 60 70100

150

200

250

300

350

400

450

Confining pressure (MPa)

a)

b)

Figure 6 (a) Intact rock strength data Axial stress upon failure (ul-timate strength) plotted against confining pressure Data from eachsample group are fitted by a line to recover Mohr-Coulomb strengthparameters (b) Frictional strength data Shear and normal tractionresolved on the failure plane are plotted against each other Againdata from each sample group a fitted by a line to recover Mohr-Coulomb friction parameters Legends in (b) apply to (a)


Barnett-1Barnett-2

Fort St JohnEagle ford-2Eagle ford-1

0 10 20 30 40 500

50

100

150

200

250

Clay+Kerogen volume ()

Infe

rred

UC

S (

MP

a)

a)

0 20 40 60 800

50

100

150

200

250

Infe

rred

UC

S (

MP

a)

b)

0 10 20 30 40 500

02

04

06

08

1


Inte

rnal

fric

tion

0 20 40 60 800

02

04

06

08

1

Inte

rnal

fric

tion

0 10 20 30 40 500

02

04

06

08

1

Clay volume ()

Slid

ing

fric

tion

0 20 40 60 800

02

04

06

08

1




Slid

ing

fric

tion

d)c)

f)e)

Figure 7 Strength data against composition and vertical Youngrsquosmodulus All data are average values within the correspondinggroups (a) Inferred UCS versus claythorn kerogen volume (b) InferredUCS versus vertical Youngrsquos modulus (c) Internal friction versusclaythorn kerogen volume (d) Internal friction versus vertical Youngrsquosmodulus (e) Sliding friction versus claythorn kerogen volume (f) Slid-ing friction versus vertical Youngrsquos modulus Vertical Youngrsquosmodulus for Fort St John samples was estimated from the horizon-tal Youngrsquos modulus using the empirical relation defined in Soneand Zoback (2013) UCS values are obtained from the linear regres-sion shown in Figure 6a Vertical error bars correspond to one stan-dard deviation of the values determined from the regressions


Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg

coefficient of friction μi is compared with composition and modulus(Figure 7c and 7d) although the error is somewhat greater in thedetermination of μi Therefore in general the stiffer rocks havegreater intact strength in these rocks as expectedFigure 7e and 7f compares the sliding coefficient of friction μs

with the average clay volume and the average static Youngrsquos modu-lus We compared μs against the clay volume instead of the totalclay and kerogen volume because it is well known in the literaturethat sliding friction depends on the gouge composition (Tembe et al2010) and clay minerals exhibit anomalously low frictional coeffi-cients (Moore and Lockner 2004) However the correlation of μswith the clay volume is not as robust as expected The friction of theEagle Ford samples is specifically low despite their low clay con-tent Note that there is not a significant variation in clay mineralcomposition because all samples mainly contained illite Rather wesee that μs correlates with the elastic modulus fairly well

RELATION BETWEEN ELASTIC ANDDEFORMATIONAL PROPERTIES

Despite the fundamental differences in time scale between elasticand creep deformation or differences in strain magnitude betweenelastic deformation and rock failure the elastic modulus seems to be

a fairly good indicator of rock creep compliance and strength moreso than the material composition of the rock Although the simplenotion that (elastically) compliant rocks are weaker and creeps moremay be intuitive an explanation is not readily provided Here wedemonstrate that some correlation between the elastic modulus anddeformational properties holds because both reflect the effect stresspartitioning caused by the combined effect of rock composition andfabric anisotropy

CAUSE OF CREEP ANISOTROPY

We first focus on the laboratory observations that creep compli-ance is greater in the bedding-perpendicular direction than the bed-ding-parallel direction for a given type of sample To explain thisobservation we refer to a simple model in which an anisotropic gasshale rock is represented by a composite of soft and stiff layers withelastic stiffness of Csoft and Cstiff respectively (Figure 8a and 8b)As in our companion paper (Sone and Zoback 2013) soft layersrepresent clay and kerogen contents and stiff layers represent otherminerals such as quartz feldspars and carbonates We then examinethe stresses carried by each layer σsoft and σstiff when a far-fielduniaxial stress σ is applied to the rock Here we simplify the prob-lem to one-dimension treating stress strain and stiffness as scalarvalues Thus the layer properties are mechanically isotropic andshear tractions at the layer boundaries are ignoredWhen the far-field stress is loaded perpendicularly to the layers

(Figure 8a) representing a vertical sample in the lab this is an iso-stress condition in which the stresses carried by each layer are iden-tical to the far-field stress Thus

σ frac14 σsoft frac14 σstiff (2)

In this case the average stiffness of the whole rock is the Reuss(harmonic) average of the stiffness of each layer On the other handwhen the loading direction is parallel to the layers (Figure 8b) thisis an isostrain condition and the stresses carried by each layer willbe different from the far-field stress as follows

ε frac14 εsoft frac14 εstiff σ

Cfrac14 σsoft

Csoft

frac14 σstiffCstiff

there4 σsoft lt σ lt σstiff

(3)

where C is the average stiffness of the whole rock and equal to theVoigt (arithmetic) average of each layer stiffness If creep predomi-nantly occurs in the soft layer and the amount of creep per time isproportional to the magnitude of stress we would expect less creepin this setting representing a horizontal sample because σsoft issmaller in equation 3 than in equation 2 Because the Voigt averageis greater than the Reuss average we recover the laboratory obser-vation that horizontal samples are elastically stiffer and have lowercreep compliancesWe see from this simplified shale model that orientation affects

the overall creep compliance of an anisotropic rock because it de-termines how the far-field stress partitions within a sample Thisstress partitioning determines the stress acting on the individualcomponents and thus the magnitude of creep in each componentand ultimately the overall creep behavior of the whole rock Theaverage elastic moduli of the whole rock is also affected by thisstress partitioning (and strain partitioning) hence the negative cor-relation between elastic stiffness and creep compliance

Stiff

Soft

Stiff

Soft

σ

σσsoft = σ = σstiff

εsoft gt ε gt εstiff

a) Iso-stress

Stiff

Soft

σ

σσsoft lt σ lt σstiff


c) Intermediate

Stif

f

Sof

t

Stif

f

Sof

t

σ


εsoft = ε = εstiff

b) Iso-strain

0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90


You

ngrsquos

mod

ulus

(G

Pa)

Isostrain

Isostress


Barnett-1Barnett-2


Ver Hor

Fort St John

d)

Intermediate

Figure 8 (a) Schematic of a layered shale model loaded per-pendicular to the bedding representing the isostress condition(b) Schematic of a layered shale models loaded parallel to the bed-ding representing the isostrain condition (c) One example of shalemodels that could result in an intermediate state between isostressand isostrain conditions (d) Laboratory Youngrsquos modulus data plot-ted against the sum of clay and kerogen volume together with theVoigt and Reuss bounds calculated assuming Esoft frac14 54 GPaEstiff frac14 869 GPa


Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg

However these scenarios only represent the extreme cases inwhich the elastic moduli lie on the Voigt and Reuss theoreticalbounds (or more strictly the bounds calculated by the Backus aver-age (Backus 1962) if tractions between the boundaries are consid-ered) In reality the elastic moduli of the gas shale samples scatterwidely between the two theoretical bounds (Figure 8d) as found inSone and Zoback (2013) This means that the stress and strain con-ditions in the shale samples are neither isostress nor isostrain butsome intermediate-state in between Such an intermediate state canresult from almost any multidimensional heterogeneous rock fabricfor instance when one phase is suspended within the other(Figure 8c) or when one phase forms a 3D network within the otherFor further generalization and discussion we need to quantify thestress and strain partitioning in these intermediate states

QUANTIFYING STRESS PARTITIONING

We follow the approach by Hill (1963) and first introduce thequantity P which relates the local stress carried by one phase tothe average stress that acts on a representative volume of rockWe will refer to P as the stress-partitioning factor

σi frac14 Piσethi∶soft stiffTHORN (4)

Because stress is a tensor quantity Pi is generally a fourth-ranktensor that depends on the volumetric fractions spatial distributionand elastic properties of the individual constituents and the averageelastic property of the whole rock Frequently Pi (or the straincounterpart Qi) is used in effective medium theory (eg Bayuket al 2007 Bandyopadhyay 2009) and the forward determinationof the exact form is a fairly involved problem of elasticity Here wereduce our analysis to one dimension again treating stress as a sca-lar quantity and we also treat Pi as a scalar quantity Such simpli-fication of the problem ignores traction between phase boundariesand the stress heterogeneity within a given phase However wefound only small differences in the results when the followinganalyses were conducted numerically by 3D effective medium ap-proaches using tensor forms of Pi for ellipsoidal inclusions (Sone2012) Thus the following analyses provide a useful framework tounderstand our problem but only correct to the first orderMaintaining notations from the previous section and the idea of

viewing gas shales as binary mixtures of soft and stiff componentsthe definition of average stress and strain give the followingrelations

σ frac14 xsoftσsoft thorn xstiffσstiff (5)

ε frac14 xsoftεsoft thorn xstiffεstiff (6)

where xi is the volumetric fraction of the components (0 lt xi lt 1)By replacing individual stresses σi in equation 5 using equation 4we obtain

1 frac14 xsoftPsoft thorn xstiffPstiff (7)

By replacing individual strains εi in equation 6 with individualstress σi and individual stiffness Ci and also using equations 4and 7 we obtain an expression for the average stiffness Cethfrac14 σ∕εTHORN

1

Cfrac14 rsoft

Csoft

thorn 1 minus rsoftCstiff

ethri frac14 xiPiTHORN (8)

This equation takes a similar form to the Reuss average of the stiff-ness or the Voigt average of the compliance (frac14 1∕C) with the onlydifference being the ri that replaces the volume fraction xi Theaverage elastic moduli in the intermediate state are thus obtainedthrough modification of common averages (eg Voigt and Reuss)in which the weighting coefficients xi are further weighted by multi-plying Pi However because determining Pi from knowledge aboutthe rock fabric can be involved forward calculating the averagemoduli using equation 8 is not always easy Note that the reciprocitybetween stiffness and compliance leads to a similar set of equationsfor the strain-partitioning factors (ie Qi) and average complianceEquation 8 also shows that we can solve for the stress-partitioning

factor Pi if we know the average elastic property of the whole rockgiven volumetric fractions and elastic properties of the individualconstituents In other words by knowing the constituent mineralsand the outcome of mixing them we can infer how the constituentsare mixing mechanically (how stress and strain are partitioning)without detailed knowledge of the rock fabric that is causing itBy solving for Pi in equation 8 we obtain

Pi frac141

ximiddotCi

CmiddotΔC minus jC minus Cij

ΔCethΔC frac14 Cstiff minus CsoftTHORN (9)

We can now use equation 9 to quantify the stresses carried by eachcomponent (soft and stiff) from the material composition and labo-ratory-measured average modulus for any shale samples whosemoduli lie between isostrain and isostress conditions (Figure 8d)Figure 9 shows the range of values Pi taken in the intermediate

state between the two bounds for the Youngrsquos modulus shown inFigure 8d at 0ndash60 soft component volume We calculate Pi

using equation 9 based on composition xi and the average Youngrsquosmodulus as the average stiffness C As suggested from equation 2Psoft and Pstiff are equal to one along the Reuss lower bound of theYoungrsquos modulus confirming the isostress condition Then Psoft

and Pstiff change continuously to lower and higher values respec-tively as the average stiffness increases toward the Voigt upperbound When the average stiffness equals the Voigt upper boundPi is equal to Ci∕Cvoigt as determined by equation 3 Along theVoigt upper bound Pstiff∕Psoft frac14 Cstiff∕Csoft frac14 161 thus the stresscarried by the stiff components is 16 times higher than the stresscarried by the soft components

CREEP COMPLIANCE VERSUS YOUNGrsquoSMODULUS

Now we consider the forward determination of the bulk rockcreep compliance using the stress-partitioning behavior obtainedabove Honoring the definition of elastic strain in equation 6 weassume that the total 3-h creep strain of the rock can be describedsimilarly as

εcreep frac14 xsoftεcreepsoft thorn xstiffεcreepstiff (10)

Then if we define 3-h creep compliances for the individual com-ponents Ssoft and Sstiff similar to Screep defined for the whole rock


Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg

εcreep frac14 xsoftσsoftSsoft thorn xstiffσstiffSstiff (11)

Replacing individual stresses with stress-partitioning factorsusing equation 4 the overall 3-h creep compliance of the rock isdescribed as

Screep frac14 rsoftSsoft thorn eth1 minus rsoftTHORNSstiff (12)

Note that we have made an important assumption here that the elas-tic stress exerted on the individual components stays the samethroughout the creep deformation despite the plastic strain takingplace Because creep strain taking place in our experiments is onthe order of 1e minus 4 and the resulting change in elastic modulusis at most 2 (Figure 5a) this is small enough to preserve the in-ternal stress partitioning behavior within the rock after the creepFinally combining equations 9 and 12 by eliminating riethfrac14 xiPiTHORN

we obtain a unique relation between the 3-h creep compliance andthe average elastic modulus of the whole rock

ScreepethCTHORNfrac141

CmiddotCstiffCsoft

ΔCethSsoftminusSstiffTHORNthorn

CstiffSstiff minusCsoftSsoftΔC

(13)

Thus the 3-h creep compliance (Screep) is an inverse-proportionalfunction of the elastic modulus (C) independent of the compositionor the orientation of the rock Note that this relation appearsmuch similar to the inverse proportionality between the elastic-

compliance and elastic stiffness In fact if we set Si frac14 1∕Ciequation 13 becomes merely a redundant derivation of the inverseproportionality between average elastic compliance and averageelastic modulus (Selastic frac14 1∕C) The similarity in the functionalform between elastic and creep compliances arises because we de-fined the 3-h creep strain to be proportional to the applied differ-ential stress in equation 11 (based on the observations shown inFigure 4) similar to elastic strain But Screep has an additional offsetdefined by the second term in equation 13 which only vanishes inthe special case when the individual creep compliances are propor-tional to their elastic compliances (CiSi frac14 constant) Because elas-tic and creep deformation are caused by fundamentally differentphysical mechanisms such a coincidence will not be the case asshown in the next section and the offset created by the second termdistinguishes equation 13 from a simple inverse proportionality

COMPARISON WITH LABORATORY DATA

To validate equation 13 with our laboratory data we need to de-fine the individual elastic and creep properties of the soft and stiffcomponents (Ci and Si) We have thus far treated the rock stiffnessas a generic scalar property C in previous sections because the al-gebra was simplified to one dimension but from here on we use theYoungrsquos modulus E as the elastic stiffness representing C TheYoungrsquos modulus is appropriate because the 3-h creep-compliancedata reflect the axial creep strain caused by uniaxial loading similarto the axial elastic strain that defines Youngrsquos modulus We use thevalues E frac14 869 GPa and E frac14 54 GPa as the Youngrsquos modulus ofthe soft and stiffness components respectively because these valueswere found to best explain the scatter of static Youngrsquos modulusdata from Sone and Zoback (2013)There exists no direct measurement of Ssoft and Sstiff so we infer

these from several laboratory data First we refer to the stiffest sam-ples from Barnett-2 and we make a rather unrealistic assumptionthat the creep strains only take place in the stiff components(Ssoft frac14 0) to determine the upper limit value for Sstiff Using equa-tions 9 and 12 with E frac14 70 GPa for the average Youngrsquos modulusof Barnett-2 samples

maxethSstiffTHORNfrac14ScreepBarnettminus2rstiffBarnettminus2

asymp1510minus6

098asymp1510minus6 frac12MPaminus1

(14)

Now we refer to the Eagle Ford-1 vertical samples one of the mostcompliant samples whose Youngrsquos modulus lies on the Reussbound thus Pi frac14 1 Because the 3-h creep compliances of theserocks are on the order of Screep frac14 25e minus 5 MPaminus1 (Figure 5a)we easily recognize that the contribution to the whole rock creepcompliance from the stiff component is negligible even if Sstiff tookits maximum possible value estimated in equation 14 Thus we canignore the contribution by the stiff component and estimate thevalue of Ssoft as

Ssoft asympScreepEagleFordminus1rsoftEagleFordminus1

asymp25 10minus5

025frac14 1 10minus4frac12MPaminus1

(15)

Now coming back to the Barnett-2 samples with the above valueSsoft frac14 1e minus 4 MPaminus1 we recognize by using equations 9 and 12

02

04

06

08

1

0 02 04 06 080

20

40

60

80


Youn

grsquos

mod

ulus

(G

Pa)

a) Color-coded by value of Psoft

Reuss bound (Isostress)

Voigt bound (Isostrain)

1

12

14

16

18

2

22

0 02 04 06 080

20

40

60

80


Youn

grsquos

mod

ulus

(G

Pa)

b)


Color-coded by value of Pstiff


Figure 9 The Youngrsquos modulus versus soft component volumecolor coded by the value of the corresponding stress-partitioningfactor (a) Color coded by Psoft (b) Color coded by Pstiff The Voigtand Reuss bounds are the same as those shown in Figure 8d calcu-lated assuming Esoft frac14 54 Gpa Estiff frac14 869 GPa


Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg

that the contribution of creep from the soft component matches themagnitude of the whole rock creep behavior

rsoftBarnettminus2Ssoftasymp0016110minus4 frac12MPaminus1asymp1510minus6 frac12MPaminus1(16)

Therefore the stiff component is again not contributing to the wholerock creep deformation and Sstiff should be at least one order smallerthan determined by equation 14 or practically zero compared toSsoft Although these are coarse estimates of the individual creepcompliances these values for Ssoft and Sstiff confirm the idea thatthe majority of creep occurs in the soft componentsFigure 10 compares the laboratory data with the predicted rela-

tion between 3-h creep compliance and the Youngrsquos modulus Wefind that equation 13 captures the trend of the entire data set fairlywell Note that the two thin red curves in Figure 10 are inverse-proportional relations scaled vertically to match the data with thehighest elastic stiffness and the data with the highest creep compli-ance Both of these curves do not fit the overall trend of the labo-ratory data thus the full functional form of equation 13 with theoffset term is needed to explain the laboratory-observed trendTherefore the unique relation between 3-h creep compliance andthe Youngrsquos modulus found in the lab is successfully explainedby the derivations leading up to equation 13 which acknowledgesthe stress partitioning between the individual components within theshale and the contribution of creep strain by the individual com-ponentsFinally we note that a similar analysis in three dimensions was

performed with an effective medium approach in Sone (2012) Thenumerical differential effective medium model was used to charac-terize the stress partitioning between soft ellipsoidal inclusionsembedded in a stiff matrix The resulting relation between creepcompliance and the Youngrsquos modulus is not purely unique as inequation 13 due to the treatment of tractions between phases withinthe rock structure but the effective medium approach yielded thesame conclusions Thus our simplified 1D approach used in thispaper seems to successfully capture the essence of the problem

INSIGHTS ON INTACT ROCK STRENGTHANISOTROPY

Although appealing to the stress partitioning between the com-ponents constructing the rock is successful in predicting the creepbehavior it is quite difficult to apply the same concept to discuss theapparent correlation observed between rock strength and elasticproperties The discussion from the previous section was possiblebecause the creep deformation was small enough to preserve theelastic structure of the rock and thus the elastic calculation of stresspartitioning was valid throughout the creep process However rockstrengths are measured at the point of rock failure and processesleading up to rock failure do not preserve the elastic structure ofthe rock As observed from acoustic emissions studies rocks firststart to fail by hosting microscale failures distributed throughout therock specimen (Lockner et al 1992) Then as the number of micro-crack increases they coalesce into larger fractures that ultimatelydevelop into a macroscopic failure plane These microcracks arepermanent damage to the elastic structure of the rock and the proc-ess of fracture coalescence is a highly nonlinear process not cap-tured by simple elastic models Moreover the dependence ofstrength on the confining pressure can only be discussed by intro-

ducing some frictional process Therefore it does not seem straight-forward to argue why the apparent correlation between various rockstrength parameters and Youngrsquos modulus exist as in Figure 7 andobserved elsewhere in the literature (Fjaer et al 1992 Changet al 2006)However the stress-partitioning behavior examined in the pre-

vious sections may provide some insights into why rock strengthsappear to be stronger in the direction parallel to the bedding thanin the direction perpendicular to the bedding as observed here(Figure 6a) and in other data in the literature (see Paterson andWong [2005] and references therein) Because the stress-partition-ing behavior is different depending on the direction of the applieduniaxial load the timing of the onset of microfracturing could bedifferent between vertical and horizontal samples which may ulti-mately control when these microfractures coalesce into a macro-scopic fracture plane If we assume that the soft component failslocally before the stiff components because clay-rich shales aregenerally weaker than quartzcarbonate rich shale the onset ofmicrofracturing should occur at a lower stress level when a givenanisotropic shale rock is loaded perpendicular to the bedding planeThis may result in a weaker strength in the bedding-perpendiculardirection compared to the bedding-parallel direction for anisotropicrocks It is interesting to refer to the experiments using gneiss sam-ples by Rawling et al (2002) in which differences in microcrackdevelopment were observed between samples loaded perpendicularand parallel to the foliation Rawling et al (2002) together withdata from Shea and Kronenberg (1993) also suggest that the abun-dance of the weak phase (mica in their gneiss samples) correlatewith the initial damage state of the sample which stronglyinfluences the timing of crack coalescence and peak strength ofthe rock Observation of the microscale processes leading up to rockfailure in shales should help us understand the importance of fabricanisotropy on intact rock strengths of shales

10 20 30 40 50 60 70 800

05

1

15

2

25

3

35x 10ndash5

E (GPa)Youngrsquos modulus

3 h

cree

p co

mpl

ianc

e S

cree

p (M

Pandash1

)

(a)

(b)(c)

(a) Equation 13

(b) Screep = 50e-4E

(c) Screep = 15e-4E

Vertical samples

Horizontal samples

Figure 10 Comparison between predicted and observed relationbetween 3-h creep compliance and elastic Youngrsquos modulus Boldred curve is drawn by equation 13 using the following constant val-ues Esoft frac14 54 Gpa Estiff frac14 869 GPa Ssoft frac14 1e minus 4 MPaminus1 andSstiff frac14 0 MPaminus1 The two thin red curves are inverse-proportionalrelations scaled to match the data point with highest Youngrsquos modu-lus and highest creep compliance


Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg

CONCLUSIONS

We have shown through laboratory experiments that the ductilecreep property and brittle strengths of shale-gas reservoir rocks aredependent on material composition and sample anisotropy Creepdeformation is generally more pronounced in samples with higherclay and organic content but is also strongly affected by the direc-tion of loading with respect to the bedding plane Brittle rockstrengths were also affected by the clay and organic content anda slight anisotropy in intact rock strengths was also observed How-ever ductile creep and brittle strengths properties exhibited thestrongest correlation with the elastic modulus of the rocks althoughductile and brittle deformations are caused by fundamentally differ-ent mechanisms from elastic deformationWe showed through a shale model consisting of a binary mixture

of soft and stiff components that the apparent unique correlationbetween the creep compliance and elastic modulus can be explainedby appealing to the stress partitioning that occurs in the rock and byreconstructing the whole rock creep behavior according the stress-partitioning information Therefore although creep and elastic de-formation are fundamentally different processes their quantitativerelation can be established as long as the internal stresses causingthose deformations are the same and those stresses can be examinedquantitatively However the apparent relation between brittlestrengths and elastic properties of the shale rock still remains a dif-ficult observation to provide a quantitative explanation The internalstress heterogeneity caused by the stress-partitioning effect mayprovide some insights but the coalescence of microcracks into amacroscopic failure plane that takes place before rock failure cannotbe captured by elastic models Thus further characterization of theplastic processes leading up to rock failure is needed to understandthe possible relation between brittle rock strengths and elasticproperties

ACKNOWLEDGMENTS

We thank BP ConocoPhillips and Unconventional Gas Resour-ces for providing us with the core samples and data necessary tocarry out this research We also thank Gary Mavko and Jack Dvor-kin for the valuable discussions and the reviewers for their con-structive comments Financial support was provided by theStanford Rock Physics and Borehole Geophysics (SRB) IndustrialConsortium Chevron and ConocoPhillips

REFERENCES

Backus G E 1962 Long-wave elastic anisotropy produced by horizontallayering Journal of Geophysical Research 67 4427ndash4440 doi 101029JZ067i011p04427

Bandyopadhyay K 2009 Seismic anisotropy Geological causes and itsimplications to reservoir geophysics PhD thesis Stanford University

Bayuk I O M Ammerman and E M Chesnokov 2007 Elastic moduli ofanisotropic clay Geophysics 72 no 5 D107ndashD117 doi 10119012757624

Chang C D Moos and M D Zoback 1997 Anelasticity and dispersion indry unconsolidated sand International Journal of Rock Mechanics and

Mining Sciences and Geomechanics Abstracts 34 402 doi 101016S0148-9062(97)00094-6

Chang C and M D Zoback 2009 Viscous creep in room-dried uncon-solidated Gulf of Mexico shale (I) Experimental results Journal of Petro-leum Science and Engineering 69 239ndash246 doi 101016jpetrol200908018

Chang C M D Zoback and A Khaksar 2006 Empirical relationsbetween rock strength and physical properties in sedimentary rocksJournal of Petroleum Science and Engineering 51 223ndash237 doi 101016jpetrol200601003

Curtis M E R J Ambrose C H Sondergeld and C S Rai 2010Structural characterization of gas shales on the micro- and nano-scalesPresented at Canadian Unconventional Resources amp InternationalPetroleum Conference CUSGSPE 137693

Donath F A 1961 Experimental study of shear failure in anisotropic rockBulletin of the Geological Society of America 72 985ndash990 doi 1011300016-7606(1961)72[985ESOSFI]20CO2

Fjaer E R M Holt P Horsrud A M Raaen and R Risnes 1992Petroleum related rock mechanics Elsevier Science

Hagin P N and M D Zoback 2004 Viscous deformation of unconsoli-dated sands mdash Part 1 Time-dependent deformation frequencydispersion and attenuation Geophysics 69 731ndash741 doi 10119011759459

Hagin P N and M D Zoback 2010 Laboratory studies of the compress-ibility and permeability of low-rank coal samples from the Powder RiverBasin Wyoming USA Presented at 44th US Rock Mechanics Sympo-sium and 5th USndashCanada Rock Mechanics Symposium paper number10-170

Hill R 1963 Elastic properties of reinforced solids Some theoretical prin-ciples Journal of the Mechanics and Physics of Solids 11 357ndash372 doi1010160022-5096(63)90036-X

Lockner D 1993 Room temperature creep in saturated granite Journal ofGeophysical Research 98 475ndash487 doi 10102992JB01828

Lockner D A J D Byerlee V Kuksenko A Ponomarev and A Sidorin1992 Observations on quasistatic fault growth from acoustic emissionsinB Evans and TWong eds Fault mechanics and transport properties ofrocks Academic Press International Geophysics Series 3ndash32

Loucks R G R M Reed S C Ruppel and D M Jarvie 2009 Morphol-ogy genesis and distribution of nanometer-scale pores in siliceous mud-stones of the Mississippian Barnett shale Journal of SedimentaryResearch 79 848ndash861 doi 102110jsr2009092

Moore D E and D A Lockner 2004 Crystallographic controls on thefrictional behavior of dry and water-saturated sheet structure mineralsJournal of Geophysical Research 109 B03401 doi 1010292003JB002582

Paterson M S and T Wong 2005 Experimental rock deformation mdash Thebrittle field 2nd ed Springer-Verlag

Rawling G C P Baud and T F Wong 2002 Dilatancy brittle strengthand anisotropy of foliated rocks Experimental deformation and microme-chanical modeling Journal of Geophysical Research 107 2234 doi 1010292001JB000472

Shea W T and A K Kronenberg 1993 Strength and anisotropy offoliated rocks with varied mica contents Journal of Structural Geology15 1097ndash1121 doi 1010160191-8141(93)90158-7

Sondergeld C H R J Ambrose C S Rai and J Moncrieff 2010 Micro-structural studies of gas shales Presented at SPE Unconventional GasConference SPE 131771

Sone H 2012 Mechanical properties of shale gas reservoir rocks and itsrelation to the in-situ stress variation observed in shale gas reservoirsPhD thesis Stanford University

Sone H and M D Zoback 2013 Mechanical properties of shale gasreservoir rocks mdash Part 1 Static and dynamic elastic properties andanisotropy Geophysics 78 this issue doi 101190geo2013-00501

Tembe S D A Lockner and T Wong 2010 Effect of clay content andmineralogy on frictional sliding behavior of simulated gouges Binaryand ternary mixtures of quartz illite and montmorillonite Journal ofGeophysical Research 115 B03416 doi 1010292009JB006383

Townend J and M D Zoback 2001 Implications of earthquake focalmechanisms for the frictional strength of the San Andreas fault systemGeological Society London Special Publications 186 13ndash21 doi 101144GSLSP20011860102

Yang Y and M D Zoback 2011 The effects of gas adsorption onswelling visco-plastic creep and permeability of sub-bituminouscoal Presented at 45th US Rock Mechanics Symposium paper number11ndash433


Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg

mineralogy (Figure 1) Samples from Barnett Haynesville and Ea-gle Ford shale are further divided into two subgroups with distinctmineralogy in which subgroup-1 contains more clay and organiccontents than subgroup-2 Total organic contents range from04 wtndash58 wt and porosities estimated from the mineraland bulk density range between 15ndash9 As described in thecompanion paper (Sone and Zoback 2013) clay and kerogen con-tent roughly correlate with each other The porosity estimated basedon the bulk and average mineral densities increases with the in-crease of clay and kerogen content possibly because pore volumesin these shales mostly reside within the clay aggregates and solidorganics in the sample (Loucks et al 2009 Sondergeld et al2010) Microstructural observations from the companion paper(Sone and Zoback 2013) also showed that these shales exhibitvarious degrees of fabric anisotropy which is reflected on theanisotropy of their elastic propertiesCylindrical samples of 1-inch diameter and 12ndash21-inch length

from each sample groups were prepared with the axes either per-pendicular (vertical) or parallel (horizontal) to the bedding planesThese samples were pressurized in a servocontrolled triaxial defor-mation apparatus to observe its static and dynamic elastic propertiesand creep behavior Hydrostatic confining pressure Pc was first ap-plied in one to four steps (hydrostatic stage) and then axial differ-ential stress Pdiff was applied in two to five steps while Pc was heldconstant (triaxial stage) The duration of each stress step was either30 or 60 s after which the stress was held constant for 3 h to observethe creep response After the triaxial stage the samples were takento failure by loading the samples at a constant axial strain rate of10minus5 sminus1 to measure rock strengths (failure stage) After rock fail-ure we continued to slide the failure plane to measure the residualstrengths of the rock The constant confining pressure Pc during thetriaxial and failure stages was varied between 10 and 60 MPa sothat the Pc dependence of rock strengths and creep behavior couldbe evaluated The magnitude of the stress steps in the triaxial stageΔPdiff varied between 3 and 45 MPa to simulate differential stressstates above and below in situ levelsDuring the experiments the sample deformation in the direction

parallel to the cylindrical axis was measured by a pair of linear

variable differential transformer displacement transducers andthe deformation perpendicular to the sample cylindrical axis (lateraldeformation) was measured by a pair of spring-mounted strain-gauge transducers attached outside of the heat-shrink Viton jacketboth measurements had a displacement resolution of about 1 μmThe axial differential load was measured by an internal load cellyielding 03 MPa resolution for a 1-inch-diameter sample An ex-ample of the strain response to some stress steps during the triaxialstage is shown in Figure 2 We divide the total strain response to astress step into two parts elastic strain (εelastic) and creep strain(εcreep) We used εelastic to determine the static elastic constantsand we used εcreep to quantify the amount of creep strain that occursafter 3 h of constant stress Assuming that shales are transverselyisotropic with the symmetry axis (x3-axis) perpendicular to the bed-ding plane we are able to determine the vertical Youngrsquos modulusE3 and Poissonrsquos ratio v31 from the vertical samples and the hori-zontal Youngrsquos modulus E1 and Poissonrsquos ratios v13 and v12 fromthe horizontal samplesThe maximum axial differential stress during the triaxial stage

was kept below 50 of the ultimate rock strengths to assure thatcreep deformation did not enter its tertiary creep stage in whichstrain rate starts to accelerate and lead to unstable rock failure(Lockner 1993) We avoided the tertiary creep stage because ourfocus in the triaxial stage was to observe the long-term ductile prop-erty of the samples Also high differential stress magnitudes thatlead to tertiary creep are not pervasive in the crust because the crustis generally in equilibrium with the sliding frictional strength ofthe crustal materials (eg Townend and Zoback 2001) We alsonote that it is unlikely that time-dependent deformation is due toporoelastic effects because the fluid saturation of the cores wereat most 40 even including clay bound water

TRIAXIAL CREEP GENERAL CHARACTERISTICS

The axial and lateral creep strain responses during the triaxial stagefrom experiments using Haynesville-1 vertical and Barnett-1 hori-zontal samples are compared in Figure 3a After application of adifferential stress step the sample shrinks in the axial direction

Eagle ford-1Eagle ford-2Fort St JohnHaynesville-1

Haynesville-2

Barnett-1Barnett-2

00

0

02

02

02

04

04

04

06

06

06

08

08

08

1

1

1

Clay + K

erogen

Carbonates

Qua

rtz f

elds

par

pyrit

e (Q

FP)

Figure 1 Ternary plot representation of the sample material com-positions Barnett Haynesville and Eagle Ford samples are furtherdivided into two subgroups in which subgroup-1 samples havehigher claythorn kerogen content than subgroup-2 samples

0 5000 10000 15000 200000

20

40

60

80

4

6

8

x10ndash3

∆P2

Ultrasonic velocity measurements

εcreep1

εelastic1

∆P1 2

Axi

al s

trai

n (ndash

)

Diff

eren

tial s

tres

s (M

Pa)

Time (s)

εcreep2

εelastic2

Figure 2 An example of axial differential stress and axial straindata during the stress steps in the triaxial stage The elastic andcreep strain from each stress step are used to compute the Youngrsquosmodulus and the 3-h creep compliance


Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg






0 20 40 60 80 1000

05

1

15

2times10ndash3


Cum

ulat

ive

axia

l cre

ep s

trai

n (ndash

)

Ver

Ef-1Ef-2

Hor

Hv-1Hv-2

Bn-1Bn-2

FSJ






Com

pressionD

ilation

0 2000 4000 6000 8000 10000 12000Time [s]

ndash1

0

1

2

3

4

5

6

7

8

Cre

ep s

trai

n (ndash

)

x10ndash4a)

0 2000 4000 6000 8000 10000 12000Time [s]

ndash1

0

1

2

3

4

5

6

7

8

Cre

ep s

trai

n (ndash

)

x10ndash4






b)

ndash2



Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg









μi frac14n minus 1

2ffiffiffi

np (1)


a)

d)c)

ndash1 0 1 2 30

05

1

15

2

25

3

35times10ndash5


3 h

cree

p-co

mpl

ianc

e (M

Pa

ndash1)

0 10 20 30 40 500

05

1

15

2

25

3

35times10ndash5


3 h

cree

p-co

mpl

ianc

e (M

Pa

ndash1)

0 20 40 60 800

05

1

15

2

25

3

35times10ndash5


3 h

cree

p-co

mpl

ianc

e (M

Pa

ndash1)

01 015 02 025 030

05

1

15

2

25

3

35times10ndash5


3 h

cree

p-co

mpl

ianc

e (M

Pa

ndash1)

035

b)

Fort St JohnHor



Barnett-1Barnett-2

Ver HorVer Hor



Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg





She

ar tr

actio

n (M

Pa)

0 20 40 60 80 1000

20

40

60

80


Fort St John

Barnett-1Barnett-2


Eagle ford-2

Ver Hor

Eagle ford-1

Ulti

mat

e st

reng

th (

MP

a)

0 10 20 30 40 50 60 70100

150

200

250

300

350

400

450


a)

b)



Barnett-1Barnett-2


0 10 20 30 40 500

50

100

150

200

250


Infe

rred

UC

S (

MP

a)

a)

0 20 40 60 800

50

100

150

200

250

Infe

rred

UC

S (

MP

a)

b)

0 10 20 30 40 500

02

04

06

08

1


Inte

rnal

fric

tion

0 20 40 60 800

02

04

06

08

1

Inte

rnal

fric

tion

0 10 20 30 40 500

02

04

06

08

1

Clay volume ()

Slid

ing

fric

tion

0 20 40 60 800

02

04

06

08

1




Slid

ing

fric

tion

d)c)

f)e)



Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg












Cfrac14 σsoft

Csoft



(3)



Stiff

Soft

Stiff

Soft

σ



a) Iso-stress

Stiff

Soft

σ



c) Intermediate

Stif

f

Sof

t

Stif

f

Sof

t

σ



b) Iso-strain

0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90


You

ngrsquos

mod

ulus

(G

Pa)

Isostrain

Isostress


Barnett-1Barnett-2


Ver Hor

Fort St John

d)

Intermediate



Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg












1

Cfrac14 rsoft

Csoft





Pi frac141

ximiddotCi












Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg







CmiddotCstiffCsoft



(13)







asymp1510minus6


(14)



asymp25 10minus5


(15)


02

04

06

08

1

0 02 04 06 080

20

40

60

80


Youn

grsquos

mod

ulus

(G

Pa)




1

12

14

16

18

2

22

0 02 04 06 080

20

40

60

80


Youn

grsquos

mod

ulus

(G

Pa)

b)






Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg










10 20 30 40 50 60 70 800

05

1

15

2

25

3

35x 10ndash5


3 h

cree

p co

mpl

ianc

e S

cree

p (M

Pandash1

)

(a)

(b)(c)

(a) Equation 13

(b) Screep = 50e-4E

(c) Screep = 15e-4E

Vertical samples

Horizontal samples



Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg

CONCLUSIONS



ACKNOWLEDGMENTS


REFERENCES




























Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg






0 20 40 60 80 1000

05

1

15

2times10ndash3


Cum

ulat

ive

axia

l cre

ep s

trai

n (ndash

)

Ver

Ef-1Ef-2

Hor

Hv-1Hv-2

Bn-1Bn-2

FSJ






Com

pressionD

ilation

0 2000 4000 6000 8000 10000 12000Time [s]

ndash1

0

1

2

3

4

5

6

7

8

Cre

ep s

trai

n (ndash

)

x10ndash4a)

0 2000 4000 6000 8000 10000 12000Time [s]

ndash1

0

1

2

3

4

5

6

7

8

Cre

ep s

trai

n (ndash

)

x10ndash4






b)

ndash2



Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg









μi frac14n minus 1

2ffiffiffi

np (1)


a)

d)c)

ndash1 0 1 2 30

05

1

15

2

25

3

35times10ndash5


3 h

cree

p-co

mpl

ianc

e (M

Pa

ndash1)

0 10 20 30 40 500

05

1

15

2

25

3

35times10ndash5


3 h

cree

p-co

mpl

ianc

e (M

Pa

ndash1)

0 20 40 60 800

05

1

15

2

25

3

35times10ndash5


3 h

cree

p-co

mpl

ianc

e (M

Pa

ndash1)

01 015 02 025 030

05

1

15

2

25

3

35times10ndash5


3 h

cree

p-co

mpl

ianc

e (M

Pa

ndash1)

035

b)

Fort St JohnHor



Barnett-1Barnett-2

Ver HorVer Hor



Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg





She

ar tr

actio

n (M

Pa)

0 20 40 60 80 1000

20

40

60

80


Fort St John

Barnett-1Barnett-2


Eagle ford-2

Ver Hor

Eagle ford-1

Ulti

mat

e st

reng

th (

MP

a)

0 10 20 30 40 50 60 70100

150

200

250

300

350

400

450


a)

b)



Barnett-1Barnett-2


0 10 20 30 40 500

50

100

150

200

250


Infe

rred

UC

S (

MP

a)

a)

0 20 40 60 800

50

100

150

200

250

Infe

rred

UC

S (

MP

a)

b)

0 10 20 30 40 500

02

04

06

08

1


Inte

rnal

fric

tion

0 20 40 60 800

02

04

06

08

1

Inte

rnal

fric

tion

0 10 20 30 40 500

02

04

06

08

1

Clay volume ()

Slid

ing

fric

tion

0 20 40 60 800

02

04

06

08

1




Slid

ing

fric

tion

d)c)

f)e)



Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg












Cfrac14 σsoft

Csoft



(3)



Stiff

Soft

Stiff

Soft

σ



a) Iso-stress

Stiff

Soft

σ



c) Intermediate

Stif

f

Sof

t

Stif

f

Sof

t

σ



b) Iso-strain

0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90


You

ngrsquos

mod

ulus

(G

Pa)

Isostrain

Isostress


Barnett-1Barnett-2


Ver Hor

Fort St John

d)

Intermediate



Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg












1

Cfrac14 rsoft

Csoft





Pi frac141

ximiddotCi












Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg







CmiddotCstiffCsoft



(13)







asymp1510minus6


(14)



asymp25 10minus5


(15)


02

04

06

08

1

0 02 04 06 080

20

40

60

80


Youn

grsquos

mod

ulus

(G

Pa)




1

12

14

16

18

2

22

0 02 04 06 080

20

40

60

80


Youn

grsquos

mod

ulus

(G

Pa)

b)






Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg










10 20 30 40 50 60 70 800

05

1

15

2

25

3

35x 10ndash5


3 h

cree

p co

mpl

ianc

e S

cree

p (M

Pandash1

)

(a)

(b)(c)

(a) Equation 13

(b) Screep = 50e-4E

(c) Screep = 15e-4E

Vertical samples

Horizontal samples



Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg

CONCLUSIONS



ACKNOWLEDGMENTS


REFERENCES




























Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg









μi frac14n minus 1

2ffiffiffi

np (1)


a)

d)c)

ndash1 0 1 2 30

05

1

15

2

25

3

35times10ndash5


3 h

cree

p-co

mpl

ianc

e (M

Pa

ndash1)

0 10 20 30 40 500

05

1

15

2

25

3

35times10ndash5


3 h

cree

p-co

mpl

ianc

e (M

Pa

ndash1)

0 20 40 60 800

05

1

15

2

25

3

35times10ndash5


3 h

cree

p-co

mpl

ianc

e (M

Pa

ndash1)

01 015 02 025 030

05

1

15

2

25

3

35times10ndash5


3 h

cree

p-co

mpl

ianc

e (M

Pa

ndash1)

035

b)

Fort St JohnHor



Barnett-1Barnett-2

Ver HorVer Hor



Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg





She

ar tr

actio

n (M

Pa)

0 20 40 60 80 1000

20

40

60

80


Fort St John

Barnett-1Barnett-2


Eagle ford-2

Ver Hor

Eagle ford-1

Ulti

mat

e st

reng

th (

MP

a)

0 10 20 30 40 50 60 70100

150

200

250

300

350

400

450


a)

b)



Barnett-1Barnett-2


0 10 20 30 40 500

50

100

150

200

250


Infe

rred

UC

S (

MP

a)

a)

0 20 40 60 800

50

100

150

200

250

Infe

rred

UC

S (

MP

a)

b)

0 10 20 30 40 500

02

04

06

08

1


Inte

rnal

fric

tion

0 20 40 60 800

02

04

06

08

1

Inte

rnal

fric

tion

0 10 20 30 40 500

02

04

06

08

1

Clay volume ()

Slid

ing

fric

tion

0 20 40 60 800

02

04

06

08

1




Slid

ing

fric

tion

d)c)

f)e)



Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg












Cfrac14 σsoft

Csoft



(3)



Stiff

Soft

Stiff

Soft

σ



a) Iso-stress

Stiff

Soft

σ



c) Intermediate

Stif

f

Sof

t

Stif

f

Sof

t

σ



b) Iso-strain

0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90


You

ngrsquos

mod

ulus

(G

Pa)

Isostrain

Isostress


Barnett-1Barnett-2


Ver Hor

Fort St John

d)

Intermediate



Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg












1

Cfrac14 rsoft

Csoft





Pi frac141

ximiddotCi












Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg







CmiddotCstiffCsoft



(13)







asymp1510minus6


(14)



asymp25 10minus5


(15)


02

04

06

08

1

0 02 04 06 080

20

40

60

80


Youn

grsquos

mod

ulus

(G

Pa)




1

12

14

16

18

2

22

0 02 04 06 080

20

40

60

80


Youn

grsquos

mod

ulus

(G

Pa)

b)






Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg










10 20 30 40 50 60 70 800

05

1

15

2

25

3

35x 10ndash5


3 h

cree

p co

mpl

ianc

e S

cree

p (M

Pandash1

)

(a)

(b)(c)

(a) Equation 13

(b) Screep = 50e-4E

(c) Screep = 15e-4E

Vertical samples

Horizontal samples



Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg

CONCLUSIONS



ACKNOWLEDGMENTS


REFERENCES




























Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg





She

ar tr

actio

n (M

Pa)

0 20 40 60 80 1000

20

40

60

80


Fort St John

Barnett-1Barnett-2


Eagle ford-2

Ver Hor

Eagle ford-1

Ulti

mat

e st

reng

th (

MP

a)

0 10 20 30 40 50 60 70100

150

200

250

300

350

400

450


a)

b)



Barnett-1Barnett-2


0 10 20 30 40 500

50

100

150

200

250


Infe

rred

UC

S (

MP

a)

a)

0 20 40 60 800

50

100

150

200

250

Infe

rred

UC

S (

MP

a)

b)

0 10 20 30 40 500

02

04

06

08

1


Inte

rnal

fric

tion

0 20 40 60 800

02

04

06

08

1

Inte

rnal

fric

tion

0 10 20 30 40 500

02

04

06

08

1

Clay volume ()

Slid

ing

fric

tion

0 20 40 60 800

02

04

06

08

1




Slid

ing

fric

tion

d)c)

f)e)



Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg












Cfrac14 σsoft

Csoft



(3)



Stiff

Soft

Stiff

Soft

σ



a) Iso-stress

Stiff

Soft

σ



c) Intermediate

Stif

f

Sof

t

Stif

f

Sof

t

σ



b) Iso-strain

0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90


You

ngrsquos

mod

ulus

(G

Pa)

Isostrain

Isostress


Barnett-1Barnett-2


Ver Hor

Fort St John

d)

Intermediate



Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg












1

Cfrac14 rsoft

Csoft





Pi frac141

ximiddotCi












Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg







CmiddotCstiffCsoft



(13)







asymp1510minus6


(14)



asymp25 10minus5


(15)


02

04

06

08

1

0 02 04 06 080

20

40

60

80


Youn

grsquos

mod

ulus

(G

Pa)




1

12

14

16

18

2

22

0 02 04 06 080

20

40

60

80


Youn

grsquos

mod

ulus

(G

Pa)

b)






Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg










10 20 30 40 50 60 70 800

05

1

15

2

25

3

35x 10ndash5


3 h

cree

p co

mpl

ianc

e S

cree

p (M

Pandash1

)

(a)

(b)(c)

(a) Equation 13

(b) Screep = 50e-4E

(c) Screep = 15e-4E

Vertical samples

Horizontal samples



Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg

CONCLUSIONS



ACKNOWLEDGMENTS


REFERENCES




























Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg












Cfrac14 σsoft

Csoft



(3)



Stiff

Soft

Stiff

Soft

σ



a) Iso-stress

Stiff

Soft

σ



c) Intermediate

Stif

f

Sof

t

Stif

f

Sof

t

σ



b) Iso-strain

0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90


You

ngrsquos

mod

ulus

(G

Pa)

Isostrain

Isostress


Barnett-1Barnett-2


Ver Hor

Fort St John

d)

Intermediate



Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg












1

Cfrac14 rsoft

Csoft





Pi frac141

ximiddotCi












Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg







CmiddotCstiffCsoft



(13)







asymp1510minus6


(14)



asymp25 10minus5


(15)


02

04

06

08

1

0 02 04 06 080

20

40

60

80


Youn

grsquos

mod

ulus

(G

Pa)




1

12

14

16

18

2

22

0 02 04 06 080

20

40

60

80


Youn

grsquos

mod

ulus

(G

Pa)

b)






Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg










10 20 30 40 50 60 70 800

05

1

15

2

25

3

35x 10ndash5


3 h

cree

p co

mpl

ianc

e S

cree

p (M

Pandash1

)

(a)

(b)(c)

(a) Equation 13

(b) Screep = 50e-4E

(c) Screep = 15e-4E

Vertical samples

Horizontal samples



Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg

CONCLUSIONS



ACKNOWLEDGMENTS


REFERENCES




























Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg












1

Cfrac14 rsoft

Csoft





Pi frac141

ximiddotCi












Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg







CmiddotCstiffCsoft



(13)







asymp1510minus6


(14)



asymp25 10minus5


(15)


02

04

06

08

1

0 02 04 06 080

20

40

60

80


Youn

grsquos

mod

ulus

(G

Pa)




1

12

14

16

18

2

22

0 02 04 06 080

20

40

60

80


Youn

grsquos

mod

ulus

(G

Pa)

b)






Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg










10 20 30 40 50 60 70 800

05

1

15

2

25

3

35x 10ndash5


3 h

cree

p co

mpl

ianc

e S

cree

p (M

Pandash1

)

(a)

(b)(c)

(a) Equation 13

(b) Screep = 50e-4E

(c) Screep = 15e-4E

Vertical samples

Horizontal samples



Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg

CONCLUSIONS



ACKNOWLEDGMENTS


REFERENCES




























Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg







CmiddotCstiffCsoft



(13)







asymp1510minus6


(14)



asymp25 10minus5


(15)


02

04

06

08

1

0 02 04 06 080

20

40

60

80


Youn

grsquos

mod

ulus

(G

Pa)




1

12

14

16

18

2

22

0 02 04 06 080

20

40

60

80


Youn

grsquos

mod

ulus

(G

Pa)

b)






Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg










10 20 30 40 50 60 70 800

05

1

15

2

25

3

35x 10ndash5


3 h

cree

p co

mpl

ianc

e S

cree

p (M

Pandash1

)

(a)

(b)(c)

(a) Equation 13

(b) Screep = 50e-4E

(c) Screep = 15e-4E

Vertical samples

Horizontal samples



Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg

CONCLUSIONS



ACKNOWLEDGMENTS


REFERENCES




























Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg










10 20 30 40 50 60 70 800

05

1

15

2

25

3

35x 10ndash5


3 h

cree

p co

mpl

ianc

e S

cree

p (M

Pandash1

)

(a)

(b)(c)

(a) Equation 13

(b) Screep = 50e-4E

(c) Screep = 15e-4E

Vertical samples

Horizontal samples



Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg

CONCLUSIONS



ACKNOWLEDGMENTS


REFERENCES




























Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg

CONCLUSIONS



ACKNOWLEDGMENTS


REFERENCES




























Dow

nloa

ded

101

413

to 1

391

781

213

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht s

ee T

erm

s of

Use

at h

ttp

libra

rys

ego

rg

Date post:	20-Apr-2020
Category:	Documents
Upload:	others
View:	6 times
Download:	0 times

Mechanical properties of shale-gas reservoir rocks — Part 2: … · 2019-02-19 · brittle...

Documents