Thesis Kumar Official

8/13/2019 Thesis Kumar Official

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Copyright

by

Raushan Kumar

2010


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The Thesis committee for Raushan Kumar

certifies that this is the approved version of the following thesis

Effect of Chemical Environments on Subcritical Crack Growth in

Geological Materials

Approved bySupervising Committee:

Jon E. Olson, Supervisor

Jon T. Holder


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Dedication

To my parents: for their unconditional love.

To my brother, Ravi and sister, Rashmi: for their presence and support.


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v

Acknowledgements

I would like to extend my sincerest gratitude to Dr. J.E. Olson for his invaluable

support, patience and guidance throughout my time at The University of Texas at Austin.

He had been a wonderful supervisor and provided intellectual stimuli for this work.

I would also like to offer sincere thanks to Dr. J. Holder for the assistance and

inputs during experiments. He was very instrumental in solving issues related to

hardware electronics and mechanical. I am also thankful to Glen Baum and Gary

Miscoe for their input on safety practices. And thanks to Autumn Kaylor for helping me

with point counts of my samples.

I would like to thank the sponsors of my work - the member companies of FRAC

(Fracture Research and Application Consortium) as well as a grant from ExxonMobil

Research and Engineering Company in New Jersey. Thanks to Gareth Block of

ExxonMobil for taking interest in my fracture mechanics research.

Special thanks to Libsen Castillo and Vinay Sahni for being wonderful friends

during my stay at Austin.

Lastly, I want to thank my father Prof. Suman Kumar, mother Smt. Bibha Sinha,

brothers Ravi and Subodh and sister Rashmi for their support.


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vi

Abstract

Effect of Chemical Environments on Subcritical Crack Growth in

Geological Materials

Raushan Kumar, MSE

The University of Texas at Austin, 2010

Supervisor: Jon E. Olson

Subcritical crack propagation experiments were performed using the double

torsion test methodology on sandstone and granite samples. The control environments

consisted of air, water, aqueous cationic surfactant solutions and aqueous caustic

solutions of high pH. The expected reduction in fracture toughness and subcritical index

going from air to water environments was observed, a result of water being a more

reactive environment. The reduction in subcritical index correlated strongly with the clay

content of the sandstones, but no other mineralogic parameters seemed to affect fracture

results significantly. Caustic solutions reduced strength and subcritical index more than

did water, but surfactant solutions performed approximately the same as water, even

though they were expected to be more reactive. Activation volume values were estimated

from velocity versus stress intensity factor curves using published values of crack-tip


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vii

radius of curvature, giving results on the order of 15.0 x 10-7

m3/mol for sandstone tested

in air, about 5 times higher than previously reported values for quartz. The values

dropped about 50% when measured in aqueous environments.

A systematic study of variability in subcritical index values in air showed that

subcritical index correlated inversely with the load drop observed during crack

propagation. Subcritical indices decreased as load drop increased, with stable values

usually resulting at load drops of 0.5 lbs or more for the 1.5 mm thick test samples. Tests

performed in more reactive environments (i.e., aqueous solutions) typically resulted in

higher load drops and less variable subcritical indices.

High rate data acquisition techniques were implemented to try to capture crack

propagation results beyond the stage I power-law velocity versus stress intensity factor

behavior. The stage II constant velocity regime was observed, but only in one sample

tested in air.

Fracture roughness of experimentally created crack paths was also analyzed using

fractal analysis. The degree of crack path tortuosity (actual rough path length divided by

straight-line length, a measure of roughness) was clearly dependent on the scale at which

it was measured. Using the modified divider technique, crack-path tortuosity exhibited a

clear fractal nature, with results for sandstone, shale and limestone exhibiting fractal

dimensions from D1.04 to D1.09. The fractal dimension indicated a negative

correlation with subcritical index, but the results showed a significant scatter.


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viii

Table of Contents

List of Tables ...........................................................................................................x

List of Figures ........................................................................................................ xi

Chapter 1: Introduction............................................................................................1

1.1 Objective................................................................................................1

Chapter 2: Mechanical and Thermodynamical Basis ..............................................3

2.1 Rock Fracture Mechanics ......................................................................3

2.1.1 Critical Crack Growth...................................................................3

2.1.2 Modes of Crack Surface Displacement.........................................42.1.3 Subcritical Crack growth ..............................................................5

2.2 Thermodynamical Basis.........................................................................7

2.3 Effect of Chemically-active Environment ...........................................13

2.3.1 Rebinder Effect ...........................................................................14

2.3.2 Westwood Approach...................................................................15

2.3.3 Stress Corrosion..........................................................................18

2.3.3.1 Chemical Reaction Rate Theory ..................................19

2.3.3.2 The Lawn-Cook Model................................................222.3.3.3 Power Law vs. Exponential Law .................................25

2.3.4 Other Mechanisms ......................................................................29

2.3.4.1 Diffusion ......................................................................29

2.3.4.2 Dissolution ...................................................................29

2.3.4.3 Ion Exchange ...............................................................30

2.3.4.4 Microplasticity .............................................................30

Chapter 3: Measurement of Subcritical Crack Index.............................................31

3.1 Test Description...................................................................................31

3.1.1 Mathematical Description of Stress Intensity Factor for DoubleTorsion Test ................................................................................33

3.1.1.1 Expression for Crack Velocity for Constant DisplacementMethod ...............................................................................36


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3.1.1.2 Data Analysis and Reduction Method .........................37

3.2 Sample Preparation ..............................................................................41

Chapter 4: Experimental Results and Analysis......................................................44

4.1 Petrographic Information.....................................................................44

4.2 Results Subcritical indices and Fracture Toughness.........................49

4.2.1 Dependence of Subcritical Indices on Loading Magnitude........52

4.2.2 Effect of Rock-types and Environmental Control ......................61

Chapter 5: Velocity vs. Stress-intensity Curves ....................................................68

5.1 Data Noise Reduction ..........................................................................68

5.1.1 Spline Average of Recorded Data-points ...................................68

5.1.2 Numerical Differentiation Technique .........................................695.2 Region-II Crack-growth.......................................................................70

5.3 Activation Volume Calculation ...........................................................73

Chapter 6: Fracture Roughness Analysis...............................................................76

6.1 Background..........................................................................................76

6.2 Fractal and Fractal-dimension..............................................................76

6.3 Fractal Applied to Fractures.................................................................78

6.4 Measuring Fracture Surface Roughness ..............................................79

6.4.1 Crack Imaging.............................................................................80

6.4.2 Crack-path Digitization...............................................................81

6.4.3 Data Analysis..............................................................................83

6.5 Fractal Analysis ...................................................................................85

6.5.1 Application of Modified Divider Technique ..............................85

6.5.2 Fractal Analysis Random Path.................................................88

6.5.3 Fractal Analysis Subcritical Test Samples...............................90

Chapter 7: Conclusions..........................................................................................96

Bibliography ..........................................................................................................99

Vita .....................................................................................................................108


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x

List of Tables

Table 2.1: Expressions for the crack velocity from literature................................27

Table 2.2: Expressions for time to failure from literature. ....................................28

Table 4.1(a): Point-count data. Numbers in each column denote raw count out of the

total number of counts in the right-most column..............................46

Table 4.1(b): Summary of point-count data reported percentages. .......................47

Table 4.2: A summary of the dual torsion tests performed for all samples. n is the

subcritical index, KIMAXis the stress-intensity factor corresponding to

peak load, and P is the load drop observed after 300 seconds. ......50Table 4.2: Contd.....................................................................................................51

Table 4.3(a): Mean and standard deviation of subcritical indices data for different

samples..............................................................................................64

Table 4.3(b): Youngs modulus and fracture toughness of different specimens in

different control environments..........................................................65

Table 5.1: Activation volume and subcritical index for three different sandstone

samples..............................................................................................75

Table 6.1: Comparisons of mean, standard deviation and skewness parameters of an

actual crack-path and a randomly generated crack-path with segment

lengths in a Gaussian distribution.....................................................88

Table 6.2: A summary of fractal analysis results and sub-critical indices for given

samples..............................................................................................94


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List of Figures

Figure 2.1: Three basic modes of loading for a crack: a) Mode I, b) Mode II, and c)

Mode III (adapted from Whittaker et al., 1992)..................................5

Figure 2.2: Schematic crack-velocity/stress-intensity diagram. K0is the stress

corrosion limit and KICis the fracture toughness; experiments yet to

confirm K0in rocks and minerals (Atkinson, 1984). ..........................6

Figure 2.3: a) Griffith crack system b) energetic of Griffith crack in uniform tension,

plane stress (adapted from Lawn, 1993). ............................................8

Figure 2.4: Irwin-Orowan extension of the Griffith concept: small-scale zone model

...........................................................................................................10

Figure 2.5: Thermodynamically admissible kinetic relations: a) thermally activated

crack growth in vacuum; lattice trapping, b) environmentally assisted

crack growth, c) three typical stages, d) environment is surface active

but unable to reach separating crack tip bonds, e) strongly surface active

environment makes negative but unable to reach tip (Rice, 1978)...12

Figure 2.6: Interaction between water molecules and strained crack-tip bond in glass:

(A) adsorption, (B) reaction and (C) separation (Michalske and Freiman,

1984). ................................................................................................19

Figure 2.7: Schematic of potential rate-limiting phenomena (Lawn, 1993)..........23

Figure 3.1: Loading configuration of the Double Torsion test (Nara and Kaneko,

2005). ................................................................................................32

Figure 3.2: Schematic diagram of a point loaded rectangular torsion bar (from

Williams and Evans, 1973)...............................................................34


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Figure 3.3: Load vs. relative time from Double Torsion test; blue triangles represent

actual data acquired and red diamond represents the power-law fit to

calculate the subcritical index...........................................................40

Figure 3.4: Load vs. relative time from Double Torsion test. Sharp drop in load

(dashed blue box) denotes critical failure. Maximum load is used to

calculate fracture toughness of the specimen....................................41

Figure 3.5: Schematic representation of the compliance relationship for DT test

method (adapted from Fuller, 1979). ................................................42

Figure 3.6: Evolution of the crack front with crack extension (Chevalier et al., 1996).

...........................................................................................................43

Figure 4.1: Q-F-L diagram for the point-count data for different specimens........48

Figure 4.2: Load-decay data of sample 5950_1A (Test a) tested in air shows bumps

during initial time..............................................................................56

Figure 4.3: Subcritical index vs. load-drop for Sandstone A1, A2 and A3 combined

(A) air, (B) water, and (C) caustic. ...................................................57

Figure 4.4: Subcritical index vs. load-drop for Canyon I, II and III combined (A) air,

(B) water, and (C) caustic. ................................................................57

Figure 4.5: Subcritical index vs. load-drop for MesaVerde (A) air, (B) water, and (C)

caustic. ..............................................................................................58

Figure 4.6: Subcritical index vs. load-drop for Granite (A) air, (B) water............58

Figure 4.7: Load-drop vs. maximum-KIfor Sandstone A1, A2 and A3 combined (A)

air, (B) water, and (C) caustic...........................................................59

Figure 4.8: Load-drop vs. maximum-KIfor Canyon I, II and III combined (A) air, (B)

water, and (C) caustic. ......................................................................59


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Figure 4.9: Load-drop vs. maximum-KIfor MesaVerde (A) air, (B) water, and (C)

caustic. ..............................................................................................60

Figure 4.10: Load-drop vs. maximum-KIfor Granite (A) air, and (B) water........60

Figure 4.11: Crack-velocity vs. load for sandstone sample. Leftward shift of the curve

indicates lower energy requirement for crack-growth in water and

surfactant...........................................................................................66

Figure 4.12: Percentage change in subcritical index from air to water, surfactant and

caustic vs. clay-content. ....................................................................67

Figure 5.1: Comparison between actual experimental data-points and spline data-fit.

Blue diamonds represents the spline-smoothed curve and the red squares

are the actual data..............................................................................69

Figure 5.2: (a) (a) Load-decay curve, and (b) velocity stress-intensity factor curve

for specimen MV3 (Test b). The rollover of velocity at higher KI

(earlier in the load decay) represents Stage II behavior. Start of Region-

III can also be observed (for first few load-decay data). ..................72

Figure 5.3: v-K curve for the specimen Sandstone-A1_3A (Test a) - test conducted in

caustic (a) power-law fit, and (b) exponential fit. ............................74

Figure 5.4: Activation volume vs. subcritical index for three sandstone types. ....75

Figure 6.1: Illustration of self-similar and self-affine fractals (Kulatilake, 1995).77

Figure 6.2: Patterns of a Microcrack coalescence process. (a) just before fracture, and

(b) a newly nucleated microcrack triggers the catastrophic failure

(adapted from Lu et al., 1994.) .........................................................78

Figure 6.3: Environmental Scanning Electron Microscope (ESEM) facility at JSG,

UT Austin..........................................................................................80


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xiv

Figure 6.4: A screenshot of the profile generated from the crack-path in a sample

from Canyon Sand tested in air, (a) BSE image, and (b) discrete

digitized image..................................................................................82

Figure 6.5: Effect of scale-length on estimation of length with the original divider

method...............................................................................................84

Figure 6.6: A screenshot of Excel VBA Tool developed for fractal analysis........84

Figure 6.7: Variation of log (Normalized length) with log (scale) for crack-roughness

data of Canyon samples from 6071 ft. depth for with the magnification

factor of (a) 100, and (b) 1000; Sub-critical test carried out in an

aqueous environment of pH 13. ........................................................86

Figure 6.8: Variation of fractal dimension with magnification factor for crack-path

roughness data (Canyon samples from 6071 ft. depth).....................87

Figure 6.9: Variation of fractal dimension with magnification factor for fracture

surface roughness data (adapted from Kulatilake, 2006)..................87

Figure 6.10: A comparison of fractal analysis from the actual crack-path and

randomly generated crack-path; red triangles are from actual crack-path

follow power law systematically; blue diamonds are from randomly

generated crack-path continuously curving. ..................................89

Figure 6.11: BSE images of crack-path in Canyon Sandstone and Barnett Shale, both

tested in air, at magnification 1000x.................................................91

Figure 6.12: Variation of log (Normalized length) with log (scale) for crack-

roughness data of Canyon samples from 6071 ft. depth with the

magnification factor of 100x in (a) air, and (b) caustic with pH 13. 92

Figure 6.13: A comparison of fractal analysis for Canyon sand (D = 1.092) and

Barnett shale (D = 1.045)..................................................................93


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Figure 6.14: Fractal analysis of a Yates sample; fractal dimension is 1.0849, which is

in the same range as Canyon sandstone; visual impression also gives the

same order of roughness for the samples..........................................93

Figure 6.15: (a) fractal dimension vs. subcritical index, (b) y-intercept vs. subcritical

index..................................................................................................95


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Chapter 1: Introduction

1.1 OBJECTIVE

Understanding the mechanics of rock fracture has been the continued interest of

engineers and scientists. In the oil and gas industry, fracture mechanics has

conventionally been applied to: 1) characterize naturally fractured reservoirs for better

modeling and prediction of flow-behavior, 2) stimulate oil and gas wells by hydraulic

fracturing, and 3) resolve issues related to wellbore integrity and stability. Applications

are being extended to fragmentation of oil shale beds for in-situ retorting, and stability of

thermal-stress induced fracturing of geothermal wells.

Understanding the mechanics of rock-fracture has helped in formulating laws to

explain how stresses remotely applied at the outer boundaries of a specimen transmit to

the crack tip (Tada, Paris and Irwin, 2000), and in defining criteria for extension of cracks

in terms of some parameter which characterizes the intensity of locally concentrated

stress fields at the crack-tip (Irwin and Wells, 1965).

Understanding the mechanism needs investigation on a micro-scale to understand

the processes involved in fracture at the crack-tip and the process-zone. Lawn (1983)

stated: For those who concern themselves primarily with the question of whenfracture

occurs, as engineers do, the methodology of fracture mechanics appears to be totally

adequate as a predictive tool. However, if we ask ourselves why fracture occurs, things

start to go wrong. Therefore, understanding the mechanism is vital to resolve the issues

related to engineering application of fracturing in a more effective way. In oil and gas

engineering, it can further be applied to explain the observations and to resolve the issues

related to increased rate of drilling in the presence of surface-active media, increased

injectivity during matrix-acidization with acid-surfactant mixture, and possible loss of


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ground stability after secondary oil recovery both during injection of water and

surfactant-water system.

During critical (irreversible catastrophic) failure, crack propagate unstably with a

velocity of the order of the shear wave velocity (103ms

-1). The most important parameter

based on the premise of linear elastic fracture mechanics to characterize this type of

rapid fracture growth is fracture toughness or critical stress intensity factor. Subcritical,

slow or stable crack growth at velocities less than 10-1

ms-1

can occur at stress intensities

far below the critical stress intensity factor. The presence of aggressive chemical

environment and elevated temperature, under sustained or residual loading condition,

further aggravates the kinetics of subcritical crack growth.

The purpose of this study is to document the mechanism of crack-growth (critical

and subcritical) in the presence of various chemical environments and to subsequently

characterize the observations in terms of fracture mechanics parameters. An attempt has

been made to couple the experimental observations with pertinent theoretical and

empirical models available in the area of material science of metals, glass, ceramics and

rocks to understand the processes involved at the crack-tip. An attempt has also been

made to relate sub-critical crack growth index to fractal dimension of the crack-path

roughness.


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Chapter 2: Mechanical and Thermodynamical Basis

2.1 ROCK FRACTURE MECHANICS

2.1.1 Critical Crack Growth

Linear Elastic Fracture Mechanics (LEFM) has been applied to provide insights

into the fracture of rocks and minerals (Atkinson, 1984; Meredith and Atkinson, 1985;

Swanson, 1984). Two alternative approaches to fracture analysis have been used: a) the

energy criterion, and b) the stress-intensity factor approach.

The energy criterion (Griffith criterion) states that crack-extension or fracture

occurs when the energy available for crack-growth is sufficient to overcome the

resistance of the material. At fracture of a linear elastic material, the crack driving force,

G equals a critical value Gc, which is a function of the fracture toughness.

( , , )c IC

G f E K = (2.1)

The stress intensity factor KIfor a sharp crack at in an infinite body is given by:

I IK c = (2.2)

where c is the fracture half-length and Iis the applied remote stress.

Near crack-tip stresses, that cause crack growth, are directly proportional to

stress-intensity factor and are given by a generalized equation:

( )2

Ii i

Kf

r

= (2.3)

where KI depend on the outer boundary conditions (i.e., on the applied loading and

specimen geometry). The remaining factors depend on the spatial coordinates about the


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tip, and determine the distribution of the field: the coordinates factors consist of a radial

component (r-1/2

) and an angular component (f()).

Thus, KIcharacterizes the crack-tip conditions, and therefore the fracture of the

material must occur at a critical stress intensity factor, KIC, also called the fracture

toughness of the material.

Irwin (1957) showed that the energy approach is equivalent to the strength

approach and the stress-intensity factor KICis related to Gcby:

2

'

ICc

KG

E= , (2.4)

where 'E =E in plane-stress, and

=E / (1-2) in plane strain,

E = Youngs modulus of the material,

= Poissons ratio

2.1.2 Modes of Crack Surface Displacement

Different loading configurations lead to different modes of crack surface

displacement. Generally, three different modes are defined (Figure 2.1): 1) Mode I, the

tensile opening mode corresponds to normal separation of the crack walls where

displacements of crack surfaces are perpendicular to the crack-plane 2) Mode II, the

sliding (in-plane shear) mode corresponds to longitudinal shearing of the crack walls

where displacements of the crack surfaces are in the crack plane and perpendicular to the

crack front and 3) Mode III, the tearing (out-of-plane) mode corresponds to lateral

shearing of the crack walls where displacements of the crack surfaces are in the crack

planes and perpendicular to the crack-front. Throughout this thesis, only Mode I fracture

has been taken into consideration while studying different fracture parameters.


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Figure 2.1: Three basic modes of loading for a crack: a) Mode I, b) Mode II, and c) ModeIII (adapted from Whittaker et al., 1992).

2.1.3 Subcritical Crack growth

Subcritical crack growth was first observed in glass by Grenet (1899). Gurney

(1947) and Gurney and Pearson (1949) studied the effect of environment on delayed

failure of glass and used thermodynamic concepts to explain moisture enhanced crack

growth. It has been subsequently investigated using glass (Charles, 1958a, b;

Wiederhorn, 1967; Kies and Clark, 1969; Wiederhorn and Bolz, 1970; Bhatnagar et al.,

2000), ceramics (Evans, 1974; Wu et al., 1978; Wiederhorn et al. 1980), metals

(Williams and Evans, 1973; Pabst and Weick, 1981), cement (Beaudoin, 1985a, 1985b,

1987), and rocks and minerals (Henry et al., 1977; Atkinson, 1979, 1980, 1981, 1984;

Atkinson and Meredith, 1981; Meredith and Atkinson, 1983; Holder et al., 2001; Rijken,

2005, Adefashe, 2006).

Stress-corrosion has been considered the main mechanism for subcritical crack-

growth in brittle rocks. Crack-velocity (v) stress intensity factor (KI) diagram

(Wiederhorn, 1967) has been used to study the subcritical crack growth and the effect of

(a) (b) (c)


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different control environments on it. Wiederhorn (1967) identified trimodal behavior of

log v - KIplot (Figure 2.2) and classified it into three distinct regions of crack-growth

behavior: 1) Region I, characterized by a strong dependence of crack-velocity on stress

intensity factor and considered to be controlled by the rate of stress-dependent chemical

reaction, 2) Region II, where the crack-velocity is almost independent of the stress

intensity factor and is limited by transport of reactive species to the crack-tip, and 3)

Region III, where crack-growth is controlled mainly by mechanical rupture and is

relatively insensitive to the chemical environment.

Figure 2.2: Schematic crack-velocity/stress-intensity diagram. K0is the stress corrosionlimit and KICis the fracture toughness; experiments yet to confirm K0inrocks and minerals (Atkinson, 1984).

Log10

crack-velocity

stress- intensity factor

Log10

crack-velocity


Corrosive + mechanical

cracking

Diffusion controlled

cracking

Reaction controlled

cracking

KO

KC

Log10

crack-velocity


Log10

crack-velocity



cracking


cracking


cracking


cracking

Reaction controlled

cracking

Reaction controlled

cracking

KO

KC


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These diagrams have been used to: a) predict time-dependent fracture properties

(Evans, 1972; Wiederhorn, 1974) and b) understand the micro-mechanism of subcritical

crack growth under different environments (Wiederhorn, 1974; Atkinson, 1979a).

2.2 THERMODYNAMICAL BASIS

Contemporary theories of the strength of brittle materials stem from the Griffith

energy balance description of fracture processes (1920). Griffiths idea to model a crack

as a reversible thermodynamic system (Griffith, 1920) coupled with his energy-balance

concept (pertaining to crack propagation) and flaw hypothesis (pertaining to crack

initiation) laid a strong foundation for a general theory of fracture.

Griffith considered a static crack as a thermodynamical system and defined a

criterion for crack propagation so as to minimize the total free energy of the system based

on the first law of thermodynamics (law of energy conservation). The system energy (U)

during crack propagation can be considered as a sum of mechanical-energy (UM) and

surface-energy (US) terms. Mechanical energy consists of strain potential energy (UE) in

the elastic media and the potential energy (UA) of the outer applied loading systems.

Surface energy (US) is the free energy consumed in creating new surfaces by overcoming

cohesive forces of molecular attraction. These can be mathematically expressed as

M SU = U + U , (2.5)

E A SU = (U + U ) + U , (2.6)

Differentiating Equation 2.5 with crack length, we get

SU= + 2MddUdU

Gdc dc dc

= + . (2.7)

where Gis the mechanical-energy release-rate,


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= free surface-energy per unit area.

Mechanical energy terms generally decrease during crack propagation

( / 0)MdU dc< and thus favor it. The surface energy term shall generally increase during

crack growth and thus oppose it. Griffiths concept assumes crack-growth as a reversible

process and proposes equilibrium condition as a criterion for predicting fracture behavior:

/ 0,dU dc= (2.8)

2cG = . (2.9)

The crack should propagate or not according to whether the (dU/dc) term is

negative or positive, respectively. Equation (2.4) provides an equivalence between

mechanical-energy release rate and critical stress-intensity factor, KIC. Equation (2.4) and

Equation (2.9) combined can be used to calculate free-surface energy or fracture-energy,

as a function of KIC

Figure 2.3: a) Griffith crack system b) energetic of Griffith crack in uniform tension,plane stress (adapted from Lawn, 1993).

Energy

c0 Crack length, c

SU = 2 c

22

MU = -

2c

E

M SU=U U+

Equilibrium

Energy

c0 Crack length, c

SU = 2 c

22

MU = -

2c

E

M SU=U U+

Equilibrium

Energy

c0 Crack length, c

SU = 2 c

22

MU = -

2c

E

M SU=U U+

Energy

c0 Crack length, c

SU = 2 c

22

MU = -

2c

E

M SU=U U+

Equilibrium


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9

2

'

ICK

E = (2.10)

Simpson (1973) proposed that in a porous material, the stress intensity in the solidwill be elevated because of the reduction in load-bearing area. KICshould be divided by

(1 ) to take porosity into account. Thus, the fracture energy normalized for porosity

should be

2(1 )N

=

(2.11)

where Nis normalized fracture energy.

By invoking linear elasticity theory and using Inglis solution of the stress and

strain fields, Griffith showed that the critical failure for an elastic plate with crack of

length 2c subjected at infinity to the action of uniformly distributed stress tension, 1,

occurs at

' 1/ 2(2 / )I E c = (2.12)

where Iis remotely applied stress

Irwin (1957) and Orowan (1949) independently proposed to add the work of

plastic deformation, pto the specific surface energy related to a unit area of the newly

formed surfaces, in the Equation (2.3). Plastic deformation zones are small zones in front

of the crack.

Poncelet (1965) challenged the basic Griffith Criteria (1920) assumption of

isothermal behavior of freshly cleaved surface. From the statistical thermodynamic

viewpoint, Poncelet (1965) argued that the surface energy of bodies is a free energy and

not a potential energy as stated by Griffith (1920). The breaking of bonds which are not

reformed, supplies the heat of sublimation of the solid to the surface and kinetic energy


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terms. But still it didnt take the thermodynamics of slow-crack growth into

consideration.

Zone boundary

crack

c dc

Zone boundary

crack

c dc

Figure 2.4: Irwin-Orowan extension of the Griffith concept: small-scale zone model

Two major works to advance the understanding on thermodynamical aspects of

crack propagation came from Pollet and Burns (1977) and Rice (1978). Pollet and Burns

(1977) expressed the crack extension force, neglecting thermodynamic surface energy of

the material, as sum of two components: an athermal and a thermal component. Pollet

and Burns (1977) proposed that the athermal component corresponds to the value of the

applied crack extension force below which crack propagation does not occur and is

required to overcome long-range, high-energy obstacles that cannot be activated by

thermal fluctuation, whereas the thermal component is used to assist thermal activation of

short-range barriers. The value of athermal component should correspond to the value of

crack extension force at which the crack velocity is zero.

Rice (1978) included the concept of irreversible thermodynamics in the formalism

of crack-propagation and proposed an extended Griffith criteria. Basing his arguments

on the second law of thermodynamics, i.e. non-negative entropy production, Rice

extended Griffith criterion to be


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11

.

( 2 ) 0G c (2.13)

where G = elastic energy release rate,

2 = reversible work to separate the fracture surfaces,

.

c = crack speed.

It was shown that the reversible work of separation (in vacuum) must lie in the range

between the critical levels of G for rapid growth (G+) and rapid healing (G

-), i.e.

2G G + . (2.14)

Thus, it paved the way for a thermodynamically unstable condition, which

described the crack propagation as stable within two extremes of applied stress-condition.

It was further shown that for crack growth in a reactive environment, which can adsorb

on the newly-created fracture surfaces, inequality remains valid when is interpreted as

a surface energy as altered appropriately to account for adsorption.

Thus, Rices extension of the Griffith criterion (Figure 2.5) helped to explain

lattice trapping of atomistic model (Thomson, Hsieh and Rana, 1971) and stress corrosive

effects (Charles and Hillig, 1962) within the theoretical formalism of thermodynamics.

Rice criterion provided a fundamental rationale for the construction of v-G or v-K

diagrams.


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Figure 2.5: Thermodynamically admissible kinetic relations: a) thermally activated crackgrowth in vacuum; lattice trapping, b) environmentally assisted crackgrowth, c) three typical stages, d) environment is surface active but unableto reach separating crack tip bonds, e) strongly surface active environmentmakes negative but unable to reach tip (Rice, 1978).


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2.3 EFFECT OF CHEMICALLY-ACTIVE ENVIRONMENT

Understanding the effect of chemically-active environment on mechanical

properties of rocks is paramount to application in oil and gas industry. Operations on

subterranean reservoir rocks include use of acidic media (matrix acidization), basic media

(Alkaline-Surfactant-Polymer Flooding) and surfactants (ASP Flooding, Hydraulic

fracturing and Drilling).

The effect of chemistry on polycrystalline-polyphase rocks is difficult to analyze

because of the complexity imposed due to the presence of microstructures, cements,

fabrics etc. There are various reversible and irreversible mechanisms at work during fast

and slow crack growth and it is imperative to understand the role of each of them, before

extrapolating the experimental results to macro-scale application. The interdisciplinary

nature of this field of study warrants integrating the complicated interplay of variables

arising out of chemistry of the rock constituents and environment, physics of the surface

of the rock and mechanics of rock-engineering. Different phenomena contributing to

critical and subcritical crack growth have been identified and various models have been

proposed to incorporate these mechanisms at work.

Rebinder (1928; cited in Rebinder and Shchukin, 1973) and Orowan (1944) were

among the earliest to identify the role of chemistry in brittle fracture. Rebinder (1928)

proposed reduced strength and plastic flow during deformation and failure of solids under

a definite stress-state in presence of surface-active media. The reversible physico-

chemical action of the media lowers the specific free surface energy of the solid and

therefore, the work of formation of new surfaces during deformation and failure

processes. Orowan (1944) observed a reduction by a factor of about three in the strength

of glass specimens in moist air relative to that in a vacuum under sustained loading. He

proposed that environmental molecules enter the crack and get adsorbed onto the walls in


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the adhesion zone, lowering the surface energy of the solid. Another approach to explain

reversible effect of the presence of surface-active agent on mechanical strength of

materials came from Westwood (1974). He proposed that the environments which tend to

bring zeta-potential to zero have the highest effect on lowering the mechanical strength of

materials. In particular, Mills and Westwood (1980) cited that the addition of 10-3

to 10-4

moles l-1

of cationic surfactant DTAB (dodecyl trimethyl ammonium bromide) to cutting

fluids increases drilling rate in quartz and westerly granite.

2.3.1 Rebinder Effect

The reversibility of physico-chemical effect and obligatory participation of

mechanical stress distinguishes Rebinder Effect from other chemical or electrochemical

processes such as stress-corrosion or dissolution of the solids in the surrounding media,

which are typically irreversible. It incorporates the idea of thermodynamically stable

interfaces between the given solid phase and the medium and therefore partial

cancellation of the intermolecular forces on the newly produced surfaces. This effect is

manifested both due to an adsorption monolayer and a liquid-phase layer, which can lead

to still stronger changes in the mechanical properties corresponding to very low values of

the inter-phase energy.

Rebinder Effect proposes reduction in materials strength due to combined action

of a reduction in surface energy of the material, a reduction in the bonding forces

(causing healing of the fracture) across the developing crack or fracture and a

chemomechanical wedging pressure exerted by the surfactant on the flanks of existing

cracks in the form of osmotic pressure. Parameters affecting these actions can be

expressed in terms of contact angle , wetting energy w, wedging pressure Pw and

capillary pressure Pcin order of importance. The contact angle is defined as,


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cos SV SL

LV

= , (2.15)

whereSV

= Interfacial tension between solid-vapor interfaces,

SL = Interfacial tension between solid-liquid interfaces,

LV = Interfacial tension between liquid-vapor interfaces.

Contact angle is a measure of wetting (ease of spread of liquid over solid surface).

Complete wetting occurs at a contact angle of zero; non-wetting means that the angle is

greater than zero and complete non-wetting occurs at a contact angle of 1800. Wetting

energy is defined as the difference between the solid-vapor and solid-liquid surface-

tensions. Wetting energy is also a measure of wetting and the ability of liquid to penetrate

irregularities. Wedging pressure is related to the liquid-vapor surface tension and the

crack dimension. The effect of wedging pressure is to increase the tensile stress

perpendicular to the tip of a crack or fracture. Capillary pressure is the pressure that

drives a wetting fluid into a crack or fracture.

2.3.2 Westwood Approach

Westwood (1974) proposed that the chemo-mechanical effects in the presence of

surface-active agents are result of electrostatic interaction between the adsorbed

monolayer of species from the chemical environment and the bulk fluid. Westwood

(1974) further proposed that, even if propagation of fast moving cracks in ceramic solids

is not affected by presence of surface active agents, slow-crack growth is affected. The

nucleation and mobility of dislocation near the crack tip are affected by a type of surface

electrostatic potential called the zeta-potential (). The zeta potential is the electrostatic

potential between the monolayer of adsorbed ions and molecules from the surfactant and

the bulk-fluid. The sign and magnitude of the zeta-potential are dependent on the

concentration of the surfactant in the fluid medium.


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If the zeta-potential is zero ( = 0), the mobility and nucleation of near-surface

dislocations are minimized. Westwood (1974) noted that the stresses required to

propagate a crack are reduced because all energy is directed toward breaking the bonds at

the tip of the crack. In a highly positive or negative zeta-potential environment, the

nucleation and mobilization of near-surface dislocations are enhanced. Energy is

consumed in the process of nucleation and mobilization of dislocations, which blunt the

tip of the crack or fracture and inhibit propagation of the crack

Any deformation that involves plastic deformation would be inhibited at = 0 and

enhanced at 0. Plastic deformation invariably involves the processes such as surface-

dislocations in the deformation-zone, which result in crack-tip blunting. The presence of

dislocations would result in stored elastic strain energy around the crack and therefore an

increase in the amount of work required propagating the crack.

Westwood (1974) noted that dislocation mobility in ionic solids such as MgO is

considerably influenced by dislocation extrinsic point defect interactions and the state

of ionization of these defects will be influenced by surface potential. For covalent solids

such as alumina and certain crystalline silicates, dislocation-lattice interactions dominate

dislocation mobility and therefore, they might be immune to any chemo-mechanical

effects. Wu et al. (1978) showed that the linking of microcracks is a significant

mechanism of crack growth in crystalline brittle materials and therefore, application of

Westwood approach alone to explain the effect of dislocation-mobility during crack

propagation might be insufficient.

The electrical potential (potential) is measured at the slipping plane, i.e. the

plane at which relative motion takes place. Dunning (1984) postulated possibility of

enhanced penetration of a liquid into cracks or flaws in a material at zero-zeta ()


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potential due to absence of any shearing resistance between the bulk fluid and the

adsorbed species on the solid material.

Westwood (1974) related an increasing rate of penetration during drilling through

silicate materials in presence of cationic surfactants to zero potential as compared to

silicates in water, which exhibit negative potential. Dunning, Lewis and Dunn (1980)

have linked changes in hydrofracture strength and microfracturing rate in orthoquartzite

to potential. Ishido and Mizutani (1980) showed that maximum reduction of strength

in quartz diorite is around zero potential.

However, Dunning et al. (1980) showed that Westwood (1974) and Rebinder et

al. (1944) do not accurately predict the crack-propagation behavior in quartz. Freiman

(1984) raised question whether zeta potentials measured on bulk solids or powders will

be similar in any way to those within the small confines of a crack tip.

During subcritical crack growth in rocks, various other mechanisms are also at

work, which are irreversible in nature, as opposed to the reversible effects of the

environment discussed in preceding paragraphs. Atkinson (1984) identified stress

corrosion, dissolution, diffusion, ion-exchange and microplasticity as the major

mechanisms, all of which are effected by chemical effects of the pore fluid. Various

phenomenological approaches using the theory of reaction rates in conjunction with

continuum mechanics description of tip geometry (Hillig and Charles, 1965; Wiederhorn

et al., 1980) have been taken to incorporate variables such as temperature, pressure and

concentration (activity). Yet, the fundamental understanding of chemical interactions at

the crack-tip is not complete. A complete description is yet to be formulated so that data

from one system could be used to predict the response of another.


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2.3.3 Stress Corrosion

Stress corrosion presumes that the chemical reaction between strained bonds and

an environmental agent produces a weakened state which can be broken at lower stresses

than the unweakened states. The general expression for weakening for silicate glasses and

quartz in water environments is proposed as (Scholz, 1968, 1972; Martin, 1972;

Atkinson, 1979; Martin and Durham, 1975):

[ ] [ ]2 .H O H Si O Si Si OH HO Si Si OH +

Charles (1958) proposed following expression for corrosion of silica glasses in basic

environments:

.Si O Si OH Si O Si OH + +

Michalske and Freiman (1982) approached the molecular interaction problem

from an electron orbital viewpoint. They postulated that for crystalline silicates and

silicate glasses, the strained Si-O bonds at crack tips can react more readily with the

environmental agents than unstrained bonds because of a strain induced reduction in the

overlap of atomic orbitals. It was envisaged that the incoming water molecule interacts

with the Si-O-Si crack-tip bond in three stages (Figure 2.6): 1) Step A involves

attachment and alignment of water molecule with the bridging bond, 2) Step B involves

reaction where water molecules donate an electron to the silicon and a proton to the

oxygen, in the stretched linkage unit and 3) Step C involves rupture of a weak hydrogen

bond and creation of a fracture surface saturated with hydroxyl groups.


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Figure 2.6: Interaction between water molecules and strained crack-tip bond in glass: (A)adsorption, (B) reaction and (C) separation (Michalske and Freiman, 1984).

Stress-corrosion of Si-O-Si bonds has been attributed to both ionized water

(Wiederhorn et al., 1980) and to molecular water (Michalske and Freiman, 1982).

Freiman (1984) correlated the rate of stress-corrosion to activity of the corrosive agent.

Stress-corrosion reactions in calcite rocks are even less understood. Possible chemical

reactions that accompany stress corrosion crack growth in the complex silicates biotite

and feldspar has been indicated by Barnett and Kerrich (1980), but these complex

reactions were also not well characterized.

2.3.3.1Chemical Reaction Rate Theory

The thermodynamic formulation of reaction rate theory is based on the

assumption that reactants are in a state of equilibrium with an activated complex formed

during the reaction and which in turn decomposes to form the reaction products:

*( )A B A B C+ (2.16)

The reaction rate, krfor such a single-step reaction is given by:


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( )*[ ] [ ] exp /rkT

k A B G RT h

= (2.17)

where k is Boltzmanns constant, T is the absolute temperature, h is Plancks constant, R

is the gas constant, [A] and [B] are the concentrations of the reactants, and G* is the

change in partial molar free energy between the initial and activated state of reaction.

G*can be expressed as:

* * * *G T S E P V = + + (2.18)

where S* is the activation entropy, E* is the activation energy, V* is the activation

volume and Pis the pressure at the crack-tip. Entropy (S) is a measure of the extent of

randomness or disorder in a system. The difference between the entropy of the transition

state and the sum of the entropies of the reactants is activation entropy S*. The

temperature dependence of the rate constant, kris characterized by the activation energy

of the experiment. The pressure dependence of the chemical reactions is characterized by

the activation volume V*(d(ln kr)/P = V*/RT).

The Charles-Hillig model (Charles and Hillig, 1962; Hillig and Charles, 1965)

proposes that catastrophic delayed failure is triggered by the interaction between

thermodynamics and chemical kinetics of a chemical corrosion process on a glass that is

subjected to a tensile stress. Wiederhorn et al. (1980), on the basis of the Charles-Hillig

model, treated the rupture process at crack tips as a chemical reaction. By assuming that

1) the crack tip can be modeled as an elastic continuum, 2) the crack tip has an elliptical

shape with a curvature equal to , 3) the pressure term in the Equation (2.17) can be

replaced by negative crack-tip stress (I= 2KI/()), and 4) the chemical potential of

reactants should be modified to include surface curvature, Wiederhorn et al. (1980)

obtained the following expression for the free energy of activation:


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( )

( )* ** * * *1/ 2

2 IV VK

G T S E V

= +

(2.19)

where KI is the applied stress intensity factor and V is the partial molar volume of the

material undergoing reaction.

Wiederhorn et al. (1980) further suggested that the experimental crack-growth

data in region I can be expressed by the following empirical relationship

*

exp IoE bK

v vRT

+=

(2.20)

where vo, E*, and b are empirical constants of the fit. Here, vo is approximately

proportional to the activity of the reactive species, E* is the related to the stress-free

activation-energy and b is related to the activation volume.

For chemical reaction rate theory to be consistent with the empirical relationship,

the term containing the stress intensity factor in Equation 2.19 must be equal to the terms

containing the stress intensity factor in Equation 2.20

*

2

bV = . (2.21)

Therefore by assuming a crack-tip radius and by measuring the slope of the v-K curve,

the activation volume of the reaction can be determined.

The limitations of the above model are the approximation of the the crack-tip

profile as a smooth, rounded ellipse. Further, only linear term for crack-tip stress has

been introduced into the energy barrier for activated crack-growth. Thus, the linearity

expressed between the logarithmic stress-intensity factor and the crack-velocity loses

sound physical basis. Further, there is no provision for crack-healing.


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To overcome these limitations, Lawn (1975) developed a model to explore crack-

motion kinetics at atomistic level, based on the lattice trapping theory of Thomson and

coworkers (Thomson 1973, Thomson et al. 1971). This model was further developed by

Cook et al. (1993) to derive atomistic variables from macroscopically measured

variables.

2.3.3.2The Lawn-Cook Model

Lawns atomistic model (Lawn, 1975) considers an ideally brittle fracture crack in

which sequential bond rupture occurs via the lateral motion of atomic kinks (considering

each atomic-scale jump as an energy-barrier), which are enhanced by thermal

fluctuations. Chemically enhanced subcritical cracking is a two stage process: transport

and reaction. Reactive species must be transported to the crack-tip before reactions can

occur that facilitate crack extension. The slower of these two steps will control the rate of

the overall process.

By considering as the sum of a linear reversible surface energy term and a non-

linear trapping term approximated by a harmonic function with atomic periodicity,

including statistical thermodynamics of a Maxwell-Boltzmann distribution to characterize

bond-rupture frequency and incorporating chemical potential term to modify surface

energy function, Lawn (1975) attempted to explain crack-growth on the atomic scale and

while incorporating the effect of some key control variables such as applied loading,

chemical concentration of reactive species and temperature on crack-velocity.


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Solute diffusion

(liquids)

Activated

diffusionMass F low

(viscous fluids)

Free

molecular flow

(dilute gases)

Solute diffusion

(liquids)

Activated

diffusionMass F low

(viscous fluids)

Free

molecular flow

(dilute gases)

Adsorptive

reaction

Solute diffusion

(liquids)

Activated

diffusionMass F low

(viscous fluids)

Free

molecular flow

(dilute gases)

Solute diffusion

(liquids)

Activated

diffusionMass F low

(viscous fluids)

Free

molecular flow

(dilute gases)

Adsorptive

reaction

Figure 2.7: Schematic of potential rate-limiting phenomena (Lawn, 1993).

The fundamental basis of this model is that the frequency of bond-rupture and

bond-healing in a reactive environment is modified by the magnitude of the energy

release rate. The net frequency of bond-rupture is represented by Maxwell-Boltzmann

statistics (Lawn, 1975) as:

* *

exp expoU U

f fkT kT

+

=

(2.22)

/of kT h= (2.23)

where fo= characteristic lattice vibration frequency,

k = Boltzmanns constant,

T = absolute temperature,

h = Plancks constant.

The activation energies for kink advance and retreat are represented by *U+ and*U terms

which are modulated by the mechanical energy release rate to promote macroscopic crack

motion.

For a simple chemical reaction of the form

*X B B + ,


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where X is the reactive environmental species, B and B* represent the unbroken and

activated complex state, the rate of change of surface potential or the fracture resistance

R, for an increment in crack area A (Cook and Liniger, 1993).

( )*( ). . . sin 2s B XBdU

R N N R NAdA

= + (2.24)

where X = chemical potential of the environmental specieX,

B= chemical potential of the reactant, B,

B*= chemical potential of the activated complex, B*,

N= number of bonds per unit area,

NA= total number of broken bonds.

The last term is included to account for trapping and gives a periodic fracture surface

energy term. On integrating the above equation, surface potential is obtained, which is

periodic in bond-separation:

( )1( ) cos 22

s o

uU u NA NA= , (2.25)

where uorepresents the energy required to break the bond in the reactive environment

*( ) .o B ABu = ,

and u1the intrinsic energy for bond rupture,

1

Ru

N

= .

Cook et al. (1993) showed that for small departures from the equilibrium Griffith

condition, the activation energy barriers can be expressed as a function of mechanical

energy release rate, G as

1

1

* /12

oG N uU uu

+=

(2.26)


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Further, the rate of increase of crack area, dA/dt can be expressed asf/Nwherefis given

by the Equation (2.22). An expression for crack velocity can be obtained by combining

Equation (2.21) and Equation (2.25):

2sinh

o

Gv v

=

(2.27)

Equation (2.26) can be used to fit the experimental v-G curves and the

macroscopic crack velocity parameters (vo, , ) obtained can be related to the atomistic

bond rupture parameters (fo,N, w, uo, u1) as

12 expoo f uvNw kT

= , (2.28)

2 ou N = , (2.29)

2NkT= . (2.30)

Here, it can be noted that Equation 2.27 contains a zero and sign reversal at the

equilibrium point G = 2. As discussed earlier, the energy per unit area 2is the surface

energy that is sensed experimentally. Cook et al. (1993) noted that Equation 2.27 has two

terms: a local, kinetic term vo, containing information about the mechanism of bond-

rupture in the intrinsic activation barrier u1; and a global, thermodynamic constraint, the

sinh function, containing information about the departure from equilibrium in (G-2). As

noted earlier, uo represents the energy required to break the bond in the reactive

environment (global term) and u1the intrinsic energy for bond rupture (local term). Cook

model is to be extended to the porous rock material.

2.3.3.3Power Law vs. Exponential Law

Charles (1958) fit experimental static fatigue data of glass to a power law and

proposed the following expression for subcritical crack growth velocity


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' *exp( / )n

ov v H RT K = (2.31)

where v is crack-velocity, H* is activation enthalpy, R is the gas constant, T is

temperature, vo and n are constants and n is called the subcritical index. Charles and

Hillig (Charles and Hillig, 1962; Hillig and Charles, 1965) favored reaction rate theory

for constitutive modeling of slow crack-growth. It was subsequently developed by

Wiederhorn et al (1980). A brief synopsis of their model is discussed in Section (2.3.3.1).

Equation (2.20) is a simple form of the exponential law. The Lawn-Cook model (Lawn,

1975; Cook and Liniger, 1993) proposed a hyperbolic sine functional relationship

(Equation 2.27) between velocity and fracture-mechanics parameter.

All these models fit quite well with the experimental data but diverge significantly

outside the range of observations (Atkinson, 1987). Despite the sophistications involved

in the reaction-rate theories and atomic theories, Charles power law is the most used

method to characterize subcritical growth, particularly for geological materials. Costin

(1987) noted that Equation 2.20 predicts a finite stress corrosion threshold, whereas

Equation 2.31 does not. No stress-corrosion threshold has been observed for rock

(Adefashe, 2006; Atkinson, 1984). The other advantage of using power-law model is that

time to failure can be calculated more easily, as integration operation on a power-law

model is easier.

Various authors have presented v-Krelationship in different mathematical forms.

The bases of all these models are extension of empirical power-law relationship (Charles,

1958), exponential relationship from chemical reaction-rate theory (Charles and Hillig,

1962) or, relationship from atomistic model (Thomson, 1973; Lawn, 1975). A

compendium of various models from literature is presented in Table 2.1.


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Table 2.1: Expressions for the crack velocity from literature.

Reference Expression

Charles, 1958a ( ) exp( / )n

mv k A RT =

Charles and Hillig, 1962 ( )exp * /o IE bK RT = +

Hillig and Charles, 1965exp * * /Mo

VE V RT

= +

Wiederhorn, 1967 exp( ) /IV H K RT = +

Kies and Clark, 1969 ( ){ }3

exp * / / ( )oo o s

ddxf kT N kT p N f

dt h = + + G G

Wiederhorn and Bolz, 1970 exp( * ) /o IE bK RT = +

Evans, 1972 ' exp( / )n

V K H RT =

Evans and Wiederhorn, 1974/

exp( / )onT T

o IV V K H RT =

Wiederhorn et al., 1974 exp( * ) /o IE bK RT = +

Atkins et. al, 1975 ( )exp ( ) /Ia A U R kT =

Pletka and Wiederhorn, 1982*

0

n

IKv vK

=

Cook and Liniger, 1993

2sinh

o

Gv v

=

Lockner, 1998 ( ) ( )*ln ,o c io

Ec f P gRT

= +


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Loading conditions present in the subsurface are generally favorable for

subcritical crack-growth. Rocks are loaded for long periods of time below their fracture-

toughness value (Anderson and Grew, 1977; Atkinson, 1976; Atkinson, 1984; Segall,

1984). Many authors have used subcritical fracture growth theory to explain observations

of natural fractures (Anderson and Grew. 1977; Olson, 1993, 2004). As noted earlier, v-K

diagram and associated relationship from the theories of stress-corrosion has been used in

the prediction of tensile failure. The most direct method for doing this is by a simple

integration of the v-K diagram (Evans, 1972). Various authors have used different v-K

relationship to deduce expression for time to failure. A brief summary of their final

results has been presented in Table 2.2.

Table 2.2: Expressions for time to failure from literature.

Reference Expression

Zhurkov, 1965 ( )exp /o oU kT =

Scholz, 1968 ( ).exp / ( * )f ot t E kT b S = +

Wiederhorn and Bolz, 1970

2

0.5 0.5(4 / ) ( / )IC Nt RTK bv = , for Griffith type crack

2

0.5 0.5( / )IC Nt RTK bv = , for penny-shaped crack

Kranz, 1980 ( ).m

f ot t

= ; ( ).exp 2.303f ot t b=


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2.3.4 Other Mechanisms

2.3.4.1Diffusion

Experimental evidence (Lewis and Karunaratne, 1981) shows that the dominant

mechanism of subcritical crack growth in ceramics at high homologous temperature can

be mass transport. Theoretical analysis of the observation has been done by Stevens and

Dutton (1971) and Dutton (1974). Potential diffusion paths identified are 1) lattice or

bulk diffusion, 2) surface diffusion, 3) vapor phase transport and 4) grain boundary

diffusion. For diffusion-controlled crack-growth, sub-critical index, n is often in the range

of 2-10, whereas for stress-corrosion crack-growth, n may be much higher (~40 or

higher).

2.3.4.2Dissolution

Dissolution of quartz in aqueous environment is given by reaction:

2 2 2 2 .. aqy SiO x H O y SiO x H O+

Quartz shows an increase in solubility with increase in temperature (Fyfe et al. 1978).

Solubility of quartz is largely unaffected by dissolved salts or changes in pH until a pH of

9, when there is a large increase in solubility.

Dissolution of calcite is described by

2

2 2 3 32CO H O CaCO Ca HCO+ + + +

Solubility is greater in NaCl solutions and sea water than in fresh water. Solubility

increases with either an increase in partial pressure of carbon dioxide or a decrease in

temperature (retrograde temperature dependence).


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2.3.4.3Ion Exchange

If the chemical environment contains species which can undergo ion exchange

with species in the solid phase and if there is a gross mismatch in the size of these

different species, then lattice strain can result from ion-exchange which can facilitate

crack extension, e.g. exchange of H+for Na+in silicate glasses (Atkinson, 1987a).

Another effect of ion-exchange is to modify the chemistry of the crack-tip

solution. For glass-water system, the exchange of hydrogen ions for alkali ions increases

pH near the crack-tip because of the restricted volume of fluid at the crack-tip. If crack-

tip pH exceeds 9, reaction becomes very fast.

2.3.4.4Microplasticity

At high homologous temperatures and low strain rates, in the presence or absence

of chemical environment, a damage zone may develop in the stress field ahead of

macrocrack tip. In the damage zone, microcracks first get nucleated by inhomogeneous

plasticity and subsequently link up to allow macrocrack extension. Available

experimental evidence from electron microscopy of quartz suggests that chemically

enhanced subcritical crack growth is not accompanied by any significant plastic flow at

crack tips up to a temperature of 250 C (Dunning et al. , 1980; Lawn 1983). Galena and

calcite show micro-plasticity at low stresses even at room temperature.


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Chapter 3: Measurement of Subcritical Crack Index

The double-torsion load-relaxation test method (Evans, 1972; Williams andEvans, 1973) has been the most widely technique for measuring subcritical crack growth,

particularly in opaque polycrystalline rocks, where crack-length measurements are

difficult to make. The wide applicability for this method lies in its low-cost setup with

simple test-specimen geometry and loading configuration, the simplicity in the

acquisition and analysis of data-sets, relative ease in providing specific environmental

controls and a complete K-v diagram from a single load decay measurement for a given

environmental condition.

In this study, this technique was used to study both critical (fracture-toughness

measurement) and subcritical (subcritical-index measurement) crack growth on rock

samples from reservoir and outcrops. This chapter provides an overview of the test

method, the theoretical basis of data analysis and the underlying assumptions, the

techniques used and the precautions taken during sample preparation.

3.1 TEST DESCRIPTION

Double-torsion testing techniques were initially proposed by Kies and Clark

(1969) to determine crack-velocity as a function of the driving force, and were

subsequently developed by Outwater et al. (1974). Various authors (Evans, 1972;

William and Evans, 1973; Evans, 1974; Evans and Johnson, 1975) subsequently

developed the complete description of the theoretical basis of this method. Excellent

critical studies were further done by Fuller (1979), Pletka el al. (1979), and Tait et al.

(1987) to focus on analytical, experimental and practical aspects of this test-technique

respectively.


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The test-specimen of DT test consists of a thin rectangular plate which is loaded

in bending via load-cell at one end across a notch or crack in double torsion(Figure 3.1).

Load is applied at one end of the specimen, which is supported at the outsides of the same

end of the specimen. The crack propagates along the center of the bottom of the specimen

under vertical loading from the end where the load is applied. The elegance of this simple

technique lies in the fact that the stress-intensity factor is independent of the crack length

over a substantial portion of the length of the specimen (Fuller, 1979).

Figure 3.1: Loading configuration of the Double Torsion test (Nara and Kaneko, 2005).

The fracture toughness of the test-specimen can be determined from the

maximum applied load without need for corrections for the applied load from plot of load

vs. crack mouth opening displacement (CMOD; Williams and Evans, 1973). Crack-

length does not enter into the equation for the derivation of fracture-toughness or, stress-

intensity.

Three loading methods for DT Tests constant load method (Kies and Clark,

1969), incremental displacement method (Evans, 1972), and load relaxation method


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(Evans, 1972; Williams and Evans, 1973) have been described in the literature, but the

load relaxation method is most commonly used. It has an advantage over other methods

because a range of stress intensity factor-velocity (K-v) data-points can be obtained in

one given test, whereas other methods provide for the determination of a single data-point

during a given experimental run.

3.1.1 Mathematical Description of Stress Intensity Factor for Double Torsion Test

Williams and Evans (1973) provided a complete description of the theoretical

development of the stress intensity factor for this test method. The double torsion

specimen is considered as two elastic torsion bars with a rectangular cross-section

subjected to load 2P as shown in Figure 3.2. For small deflections and for bars where

width is much greater than specimen thickness, it has been shown (Novozhilov, 1961)

that the torsional strain, , is given by:

3

6,

m

y Ta

w Wd G (3.1)

where T= torsional moment, (P/2) wm,

P/2= total load applied to one bar,

G = shear modulus of the material,

a= crack length,

d= bar thickness,

W/2= bar width,

wm= moment arm.

Equation 3.1 can be rearranged such that

2

3

3,m

w ayS

P Wd G (3.2)


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where, S is the elastic compliance. The strain-energy release rate for crack extension ,G

and the specimen compliance are related by (Irwin and Kies, 1954)

2

,2

P dSG

dA

=

(3.3)

where,Ais the area of the crack.

Figure 3.2: Schematic diagram of a point loaded rectangular torsion bar (from Williamsand Evans, 1973).

If the shape of the crack front is independent of crack length, then Equation 3.3 becomes,

2 2

3 2

3,

2 (1 )

m

n

EP wK

Wd d G

=

(3.4)

where dnis the plate thickness in the plane of the crack.


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For the plane-strain conditions, the stress-intensity factor, K, is related to the strain

energy release rate by (Paris and Sih, 1965)

12

2,

1

EGK

=

(3.5)

where E is the Youngs modulus and is the Poissons ratio. Thus substituting Equation

3.6 into 3.5 gives

2 2

3 2

3,

2 (1 )

m

n

EP wK

Wd d G

=

(3.6)

E and G are related by,

,2(1 )

EG

=

+ (3.7)

Equation 3.7 becomes

3

3

(1 )m

n

K PwWd d

=

(3.8)

Equation 3.9 is the expression for stress intensity factor under plane-strain condition for

double torsion load relaxation method (Pletka et al., 1979).

Fuller (1979) and Swanson (1981) analyzed the assumptions used in this

derivation: a) Mode-I failure, b) crack profile independent of the crack-length, c) no

frictional constraints along the sides of the torsion arms (i.e. the sides of the crack), d) the

elastic strain energy which provides the driving force for crack extension is derived

only from the strained torsion arms with a negligible amount of deformation occurring

ahead of the crack tip, e) plane-strain or plane stress, and f) the elastic constants of the


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sample are independent of the test environment. Swanson (1981) attributed the scattering

in the experimental observation to possible violations of some of these assumptions.

3.1.1.1Expression for Crack Velocity for Constant Displacement Method

An empirical compliance calibration for the double torsion specimen shows that

the specimen compliance, S, is linearly related to the crack length, a as (Evans, 1972;

Williams and Evans, 1973),

,o

yS S Ba

P= = + (3.9)

where y is the displacement of the loading point, P is the load, So is the elastic

compliance of the intact specimen and B is an experimental constant. Differentiating

Equation 3.10 as a function of time at constant displacement gives an expression for the

crack propagation velocity, vas (Williams and Evans, 1973)

( ),o

y y

S Baa Pv

t BP t

+ = =

(3.10)

From Equation 3.10, it can be seen that for constant displacement,

( ) ( ) ( ),o i o i f o f P S Ba P S Ba P S Ba+ = + = + (3.11)

where,i

P andi

a are the initial values of load and crack length and, fP and fa are the

corresponding values at the end of load relaxation. Combining Equations 3.11 and 3.12

gives,

, ,,

2 2

( )( ) ,

o

i f i f i f o

y y y

SP a

P S Baa P PBvt BP t P t

++ = = =

(3.12)


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In general, except for very low modulus materials (such as polymers), or for very

small crack lengths, oS

aB

< (Williams and Evans, 1973) so that

, ,

2

i f i f

y y

P aa Pv

t P t

= =

(3.13)

Equation 3.13 is the expression for the crack propagation velocity determined

from the load relaxation curve for a constant displacement double torsion test. Combining

Equation 3.9 and Equation 3.13 provides a unique description of the dynamics of

subcritical crack growth for a given specimen from the plot of crack velocity, V, versus

stress intensity factor, K.

From the equations discussed above, two major advantages of double-torsion load

relaxation techniques can be cited: a) the stress intensity factor is directly related to the

applied load over much of the specimen length (Equation 3.9), and b) crack velocity can

be determined without the need for multiple crack length measurements (Equation 3.14).

3.1.1.2Data Analysis and Reduction Method

A number of equations (Table 2.1) have been proposed to describe subcritical

crack-growth. Some of these are very complex because of their tendency to include a

comprehensive description of the competing mechanisms. The two most popular are

power law (Charles, 1958b) and exponential (Wiederhorn and Bolz, 1970) relationships

between crack-velocity and stress intensity factor. The Charles(1958b) power law is the

most popular equation (Atkinson, 1984) to describe subcritical crack growth in rocks and

minerals because of its ability to describe the whole range of K-v data with appropriate

changes in the empirical constants (Atkinson and Meredith, 1987a).

Test data for the load relaxation method is normally recorded in the form of

applied load versus time. Calculating the crack velocity (Equation 3.14) requires


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numerical differentiation of individual data points on the load-time curve. Because of the

inherent scatter in the acquired experimental data, direct numerical computation is not

suitable. Various schemes have been adopted to smooth the experimental data before

attempting any numerical computation. Swanson (1984) fitted load-time data to a sixth

order polynomial for glass. The drawback of this method is that before a good fit can be

obtained, the load-time curve may have to be segmented into separate regions with

polynomials of the same or different degrees fitted to each region. This method is not

only time consuming but also unreliable as it often computes data that deviates from the

well established power law and exponential relations.

Using a power law assumption, Holder (personal communication, 2001; described

by Rijken, 2005) developed an approach to obtain crack velocity from raw experimental

data by calculating a smooth load decay curve that permits numerical differentiation of

the load-time data. At the start of the test, the displacement is given as

,i

Sy

P= (3.14)

Combining the above equation with the compliance Equation 3.10 yields

,

1

i

o

PP

Ba

S

=

+

(3.15)

Differentiating Equation 3.16 with respect to time gives

2

2 2,

1 1

i i

o o o

i

o o

B B BP P P

S S SP a V Vt t PB B

a aS S

= = =

+ +

(3.16)


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Using power-law dependence of crack-velocity on the load, ( )nV A P= Equation 3.17 can

be written as

2

2( ) ( ) ,n no o

i i

B BP AS SP

A P Pt P P

= =

(3.17)

A ando

B

Sin Equation 3.18 can be related to the initial values of load Pi and its time

derivative 'iP as

'

( 1) ,

i

no i

PB

A S P += (3.18)

Integrating equation 3.18 gives

1

' ( 1)

1 ( 1)

i

ni

i

PP

Pn t

P

+

=

+

(3.19)

Equation 3.19, in principle, does provide a direct determination of subcritical index by a

least square fit of load and time to this power-law expression. An iterative procedure to

determine the parameters in Equation 3.19. Crack velocity is subsequently determined

from Equation 3.14. A typical load-decay data is shown in Figure 3.3.


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Figure 3.3: Load vs. relative time from Double Torsion test; blue triangles representactual data acquired and red diamond represents the power-law fit tocalculate the subcritical index.

The load relaxation becomes noisy at long times (low values of crack velocity)

(Beaumont and Young, 1975; Adefashe, 2006). Therefore all data-points, which give a

crack-velocity lower than ~10-7

ms-1

, were removed during data reduction. This reduces

the noise in the collected data-set and therefore, yields a better estimation of sub-critical

index from the given experimental run.


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Figure 3.4: Load vs. relative time from Double Torsion test. Sharp drop in load (dashedblue box) denotes critical failure. Maximum load is used to calculatefracture toughness of the specimen.

Figure 3.4 represents a typical load-decay for fracture-toughness measurements.

The load is ramped up to a point, where the stress intensity factor exceeds the critical

stress-intensity factor resulting in catastrophic failure of the rock sample. Maximum load

observed during such a catastrophic failure is used to calculate fracture toughness of the

given specimen.

3.2 SAMPLE PREPARATION

Test-specimens were cut into rectangular slab using an oil-cooled saw. It has been

shown that for rock sample fracture-toughness is independent of the specimen thickness

(Schmidt, 1980). However, Atkinson (1979) showed that to avoid erroneous estimation of

KI, width should be greater then twelve times thickness of the sample, and Pletka et


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al.(1979) suggested that the specimen length L should be greater than twice width, W,

i.e.,

12 / 2d W L (3.20)

A typical dimension for the test-specimens used in this study was 4x 1.1x 0.07

(L x W x d).

Crack Length, c

COMPLIANCE,

Crack Length, c

COMPLIANCE,

Crack length, a

Crack Length, c

COMPLIANCE,

Crack Length, c

COMPLIANCE,

Crack length, a

Figure 3.5: Schematic representation of the compliance relationship for DT test method(adapted from Fuller, 1979).

The double-torsion load relaxation method assumes that the stress intensity factor,

KI,is independent of crack length, a, for the double torsion test (Evans, 1972), Pletka et

al. (1979) showed that KI varies along the specimen. Pletka et al. (1979) proposed that in

order to ensure cracks are in the constant KI regime for a specimen of length L and

width W; the crack length should be between W and L-W. Therefore, pre-cracking a DT

specimen is necessary before performing any fracture mechanics studies (Pletka et al.,

1979).


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Trantina (1977), using finite element analysis, showed that the range for which KI

is independent of the crack length, a is

0.55 0.65W a L W (3.21)

All the samples used in this study were precracked in a controlled manner. The

precracking of the sample was allowed to continue to ensure that the crack has reached

the steady-state front (Chevalier et al., 1996). For fracture-toughness measurement, blunt

crack give higher value because for a given applied load the stress intensity at a blunt

crack tip is smaller than for a sharp tip.

Figure 3.6: Evolution of the crack front with crack extension (Chevalier et al., 1996).

Pabst and Weick (1981), performing tests on commercial alumina specimen,

showed that the level of reproducibility was highest for the specimen without guide

grooves. However, specimens without a guiding groove require a well-finished surface as

well as an extremely balanced loading device. Nara and Kaneko (2005) showed that the

shape of the guide groove affects experimental results. They conducted tests on

rectangular, semi-circular and rectangular grooves and observed the highest linearity and

least scattering for the specimens with a rectangular guiding groove. A rectangular

groove, approximately 0.02 inches deep, centered with respect to width, was cut along

the length of the test-specimens to provide a guide for the crack- propagation.

The dual torsion apparatus was assembled of stainless steel (SS 316) parts to

make the equipment amenable to hostile environments.


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Chapter 4: Experimental Results and Analysis

In the oil and gas industry, surface active agents at high pH are used instimulation operations (hydraulic fracturing) and EOR activities (chemical flooding). pH

of the fracturing fluid can go up to 12 (Economides and Nolte, 2000) depending upon the

nature of the crosslinker used in fracturing fluid. During Alkaline flooding and Alkaline-

Surfactant-Polymer flooding, pH of the injected fluid can go up to 13 (Green and Wilhite,

1998). Tests were carried out in a surfactant solution of 0.3% to 0.5% (w/w). Th

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