STRENGTHS OF GRANITE UNDER VARIED TEMPERATURESsutir.sut.ac.th:8080/sutir/bitstream/123456789/5314/2/Fulltext.pdf · กฤติกร รอดกลาง :...

STRENGTHS OF GRANITE UNDER VARIED

TEMPERATURES

Kittikron Rodklang

A Thesis Submitted in Partial Fulfillment of the Requirements for the

Degree of Master of Engineering in Geotechnology

Suranaree University of Technology

Academic Year 2013

กาํลงักดของหินแกรนิตภายใต้อุณหภูมิแปรผนั

นายกฤตกิร รอดกลาง

วทิยานิพนธ์นีเ้ป็นส่วนหน่ึงของการศึกษาตามหลกัสูตรปริญญาวศิวกรรมศาสตรมหาบัณฑิต สาขาวชิาเทคโนโลยธีรณี

มหาวทิยาลัยเทคโนโลยสุีรนารี ปีการศึกษา 2556

_______________________________

(Dr. Prachya Tepnarong)

Chairperson

_______________________________

(Prof. Dr. Kittitep Fuenkajorn)

Member (Thesis Advisor)

_______________________________

(Dr. Decho Phueakphum)

Member

STRENGTHS OF GRANITE UNDER VARIED TEMPERATURES

Suranaree University of Technology has approved this thesis submitted in

partial fulfillment of the requirements for a Master’s Degree.

Thesis Examining Committee

(Prof. Dr. Sukit Limpijumnong) (Assoc. Prof. Flt. Lt. Dr. Kontorn Chamniprasart)

Vice Rector for Academic Affairs Dean of Institute of Engineering

and Innovation

กฤติกร รอดกลาง : กาํลงักดของหินแกรนิตภายใตอุ้ณหภูมิแปรผนั

(STRENGTHS OF GRANITE UNDER VARIED TEMPERATURES)

อาจารยท่ี์ปรึกษา : ศาสตราจารย ์ดร.กิตติเทพ เฟ่ืองขจร, 77 หนา้.

กาํลงักดของหินแกรนิตเป็นปัจจยัสําคญัในการกาํหนดเสถียรภาพในระยะยาวของหินรอบ

หลุมเจาะท่ีใชก้กัเก็บกากของเสีย วตัถุประสงคข์องการศึกษาน้ีคือ ศึกษาผลกระทบของอุณหภูมิต่อ

กาํลงักดของหินแกรนิตชุดตาก การทดสอบความแข็งดาํเนินการภายใตอุ้ณหภูมิท่ีผนัแปรและความ

เคน้ลอ้มรอบ ความเคน้ลอ้มรอบในขณะทดสอบมีค่าคงท่ีเท่ากบั 0, 3, 7 และ 12 เมกะปาสคาล โดย

ใชโ้ครงกดทดสอบในสามแกนจริง ตวัอยา่งนาํมาจดัเตรียมเป็นรูปแท่งส่ีเหล่ียมผืนผา้ท่ีมีขนาดเฉล่ีย

เท่ากบั 5×5×10 ลูกบาศก์เซนติเมตร อุณหภูมิท่ีใช้ทดสอบผนัแปรจาก 273 ถึง 773 เคลวิน การ

ทดสอบตวัอย่างหินท่ีอุณหภูมิแตกต่างกัน จะแสดงผลกระทบของอุณภูมิท่ีมีต่อพฤติกรรมทาง

กลศาสตร์ของหิน พลงังานความเครียดเบ่ียงเบนท่ีจุดวิบติั ไดน้าํมาใช้เพื่อการศึกษาความแข็งของ

หินในฟังก์ชนัของพลงังานความเครียดเฉล่ีย เกณฑ์การแตกท่ีนาํเสนอจะเป็นประโยชน์ ในการ

คาดคะเนความแขง็และการเปล่ียนแปลงรูปร่างของหินกกัเก็บภายใตอุ้ณหภูมิสูง

สาขาวชิา เทคโนโลยธีรณี ลายมือช่ือนกัศึกษา

ปีการศึกษา 2556 ลายมือช่ืออาจารยท่ี์ปรึกษา

KITTIKRON RODKLANG : STRENGTHS OF GRANITE UNDER

VARIED TEMPERATURES. THESIS ADVISOR : PROF. KITTITEP

FUENKAJORN, Ph.D., P.E., 77 PP

TEMPERATURE/STRENGTH/CONFINIGN PRESSURE/GRANITE

Strength of granite is an important parameter dictating the long-term stability

of rock around disposal boreholes. The objective of this study is to experimentally

determine the effect of elevated temperatures on the compressive strengths of Tak

granite. Failure strengths are determined for various temperatures and confining

pressures. The confining stresses are maintained at 0, 3, 7, to 12 MPa using a

polyaxial load frame. The specimens are prepared to obtain rectangular block

specimens with nominal dimensions of 5×5×10 cm3. The testing temperatures will be

varied from 303 to 773 Kelvin. Loading the specimens at different temperatures will

reveal the effects of temperature on the mechanical behavior of rocks. The

distortional strain energy density at failure will be determined to describe the rock

strength as a function of mean strain energy. The proposed strength criterion will be

useful to predict the strength and deformation of rock around boreholes under

elevated temperatures.

School of Geotechnology Student’s Signature

Academic Year 2013 Advisor’s Signature

ACKNOWLEDGEMENTS

The author wishes to acknowledge the support from the Suranaree University

of Technology (SUT) who has provided funding for this research.

Grateful thanks and appreciation are given to Prof. Dr. Kittitep Fuenkajorn,

thesis advisor, who lets the author work independently, but gave a critical review of

this research. Many thanks are also extended to Dr. Prachya Tepnarong and Dr.

Decho phueakphum, who served on the thesis committee and commented on the

manuscript. Grateful thanks are given to all staffs of Geomechanics Research Unit,

Institute of Engineering who supported my work.

Finally, I most gratefully acknowledge my parents and friends for all their

supported throughout the period of this research.

Kittikron Rodklang

TABLE OF CONTENTS

Page

ABSTRACT (THAI) ..................................................................................................... I

ABSTRACT (ENGLISH) ............................................................................................. II

ACKNOWLEDGEMENTS .........................................................................................III

TABLE OF CONTENTS ............................................................................................ IV

LIST OF TABLES .................................................................................................... VIII

LIST OF FIGURES .................................................................................................... IX

SYMBOLS AND ABBREVIATIONS ..................................................................... XIII

CHAPTER

I INTRODUCTION ............................................................................1

1.1 Rationale and background..........................................................1

1.2 Research objectives ....................................................................2

1.3 Research methodology ...............................................................2

1.3.1 Literature review .............................................................2

1.3.2 Sample preparation .........................................................3

1.3.3 Laboratory testing ...........................................................3

1.3.4 Development of the strength criteria ..............................3

1.3.5 Computer simulation ......................................................3

1.3.6 Discussion, conclusions and thesis writing ....................3

1.4 Scopes and limitations ...............................................................5

V

TABLE OF CONTENTS (Continued)

Page

1.5 Thesis contents ...........................................................................5

II LITERATURE REVIEW ................................................................6

2.1 Introduction ................................................................................6

2.2 Literature review ........................................................................6

2.2.1 Tak batholith ...................................................................6

2.2.2 Effects of temperature on rock strength..........................7

2.2.3 Strength calculation ......................................................14

2.2.4 Stress distribution around circular hole ........................17

2.2.5 Deep hole injection technology ....................................17

III SAMPLE PREPARATION ...........................................................21

3.1 Introduction ..............................................................................21

3.2 Sample preparation ..................................................................21

IV LABORATORY TESTING ...........................................................23

4.1 Introduction ..............................................................................23

4.2 Test apparatus ..........................................................................23

4.3 Test method ..............................................................................25

4.3.1 Heating method .............................................................25

4.3.2 Uniaxial and triaxial compression tests ........................26

4.3.3 Brazilian tension test.....................................................26

VI


Page

4.4 Test results ...............................................................................28

4.5 Calculation ...............................................................................38

V MATHEMATICAL EQUATIONS ...............................................42

5.1 Introduction ..............................................................................42

5.2 Empirical equations .................................................................42

5.3 Hoek and Brown criterion ........................................................45

5.4 Coulomb criterion ....................................................................48

5.5 Elastic parameters ....................................................................49

5.6 Strain energy density criterion .................................................51

VI COMPUTER SIMULATIONS .....................................................55

6.1 Introduction ..............................................................................55

6.2 Numerical simulations .............................................................55

6.3 Results ......................................................................................61

6.3.1 Coulomb criterion .........................................................61

VII DISCUSSIONS AND CONCLUSIONS .......................................75

7.1 Discussions and conclusions ....................................................75

7.2 Recommendations for future studies .......................................77

VII


Page

REFERENCES ............................................................................................................78

BIOGRAPHY ..............................................................................................................82

LIST OF TABLES

Table Page

2.1 Average of compressive strength and elastic modulus at different

temperature 14

3.1 Properties of granite 22

4.1 Compressive strengths of granite 37

4.2 Brazilian tensile strengths of granite 37

4.3 Strengths of granite 40

4.4 Elastic parameters 41

5.1 Shear stress and normal stress at failure 54

6.1 Material properties used in FLAC simulation 59

6.2 The series of computer simulation 60

6.3 The results in FLAC simulation 73

6.3 Tangential stress at point A from FLAC simulation in long-term 74

LIST OF FIGURES

Figure Page

1.1 Research methodology 4

2.1 Failure limits from multiple failure state tests on sandstones from Acu

formation 7

2.2 Compressive strength with temperature 9

2.3 Tensile strength with temperature 9

2.4 (a) Uniaxial compressive strength of salt as a function of temperature

(b) Brazilian tensile strength of salt as a function of temperature

(c) Major principal stress at failure as a function of confining pressure 11

2.5 Octahedral shear strength of salt as a function of mean stress 12

2.6 Small triangular slab of rock used to derive the stress transformation 15

2.7 Plane of weakness with outward normal vector oriented at angle β to the

direction of maximum principal stress 16

3.1 The specimens prepared for uniaxial and triaxial compression testing (a)

and Brazilian tension testing (b) 22

4.1 Steel platen dimensions 24

4.2 Heater coil entwine around steel platen 24

4.3 Thermostat with digital controller 25

X

LIST OF FIGURES (Continued)

Figure Page

4.4 Temperatures measured and regulated by thermocouples and Thermostats

while the specimen installed in polyaxial load frame 27

4.5 Polyaxial load frame 28

4.6 Some post-test granite specimens from compressive strength testing under

difference confining pressure and temperature 29

4.7 Some post-test granite specimens from Brazilian tensile strength testing as

a function of temperature 30

4.8 Stress-strain curves obtained from some granite specimens at 273 K.

Numbers in brackets indicate [σ1, σ2, σ3] at failure 31









4.13 Major principal stresses at failure as a function of confining pressure 36

4.14 Elastic parameters as a function of temperature 39

XI


Figure Page

5.1 Major principal stresses of granite as a function of temperature 44

5.2 Octahedral shear strengths at failure of granite as a function of mean stress 44

5.3 Parameter m and uniaxial compressive strength (σc,HB) as a function of

temperatures 46

5.4 Shear strengths of granite as a function of normal stress with different

temperatures 48

5.5 Comparisons cohesion and internal friction angle between Hoek and Brown

criterion and Coulomb criterion of granite as a function of temperatures 49

5.6 Elastic parameters of granite as a function of temperature 51

5.7 Distortional strain energy as a function of mean strain energy 53

6.1 Finite difference mesh constructed to simulate a horizontal plane normal to

disposal borehole at depth of 1,000 m 57

6.2 Tangential stress at point A and maximum displacement at point B 58

6.3 Factor of safety contour under isotropic condition at 273 K 63

6.4 Factor of safety contour under anisotropic condition at 273 K 63





XII


Figure Page





6.13 Maximum stress under isotropic condition at 273 K 68

6.14 Maximum stress under anisotropic condition at 273 K 68









SYMBOLS AND ABBREVIATIONS

a = Inside radius

ATh = Empirical constant

BTh = Empirical constant

c, So = Cohesion

E = Elastic modulus

G = Shear modulus

K = Bulk modulus

P1 = Vertical stress

P2 = lateral stress

r = Variable radius

Wd = Distortional strain energy

Wm = Mean strain energy

α = Empirical constant

β = Angle between direction of σ1 and σ3

φ = Friction angle

λ = Empirical constant

σ1 = Maximum principal stress

σ2 = Intermediate principal stress

σ3 = Minimum pressures

σc = Uniaxial compressive strength

XIV

SYMBOLS AND ABBREVIATIONS (Continued)

σm = Mean stress

σn = Normal stress

σt = Tensile stress

σr = Radial stress

σθ = Tangential stress

τ = Shear stress

τoct = Octahedral shear stresses

τoct,f = Octahedral shear stresses at failure

τrφ = shear stress around tunnel

τxx = Stress in direction x

τxy = Shear in direction x-y

τyy = Stress in direction y

γoct = Octahedral shear strains

γoct,f = Octahedral shear strains at failure

η = Empirical constant

µ = Coefficient friction

ν = Poisson’s ratio

ω = Empirical constant

χ = Empirical constant

XV

SYMBOLS AND ABBREVIATIONS (Continued)

δ = Empirical constant

ξ = Empirical constant

CHAPTER I

INTRODUCTION

1.1 Rationale and background

The effects of temperature on deformability and strength of rocks have long

been recognized (Vosteen and Schellschmidt, 2003; Shimada and Liu, 2000). A

number of new topics have been raised, related to rock mechanics, given an

increasing demand for underground storage of nuclear waste, natural gas and

petroleum, as the rate of exploration for energy resources on a worldwide scale

accelerates (Xu et al., 2008). The rock temperature around the nuclear waste in such

conventional storage, may not rise beyond 523 Kelvin (Bergman, 1980; US

Department of Energy, 1980). But, in the case of non-conventional or direct burial of

nuclear waste, the rock temperature may be very high and sometimes exceeds the

melting point of the rock (Logan, 1973; Heuze, 1981). The mechanical behavior of

rocks essentially depends upon mineralogy, structure, temperature, stress and time

(Etienne, 1989). The knowledge of thermo-mechanical behavior of rock is imperative

because high temperature leads to development of new micro-cracks or

extension/widening of pre-existing micro-cracks within the rocks. This phenomenon

affects the strength of rocks (Dwivedi et al., 2008). Strength and deformability of

rock is an important parameter used in the design analysis of engineering structures.

The rock strength and deformation dictate the stability of openings.

The advantages of the deep hole injection in granite technology are that the

2

rock at depth usually has less fracture, low permeability and resistant high temperature

that may come from the decay of radioactive waste. It is desirable to study the

stability in granite mass under elevated temperatures for the assessing safety of the

repository for heat generating nuclear waste in a deep geological rock formation.

1.2 Research objectives

The objectives of this study are to determine the compressive strength of

granite subjected to confining pressure and to assess the predictive capability of some

failure criteria that can be readily applied in the design and stability analysis of granite

under temperatures ranging from 273 to 773 Kelvin (0-500 Celsius). The efforts

involve determination of the maximum principal stress at failure of the granite

specimens under various minimum principal stresses, and determination of the most

suitable multi-axial strength criterion. The test results are used to calibrate the

deformability and compressive strength of granite specimens, stress-strain with

confining pressure under various constant temperatures.

1.3 Research methodology

The research methodology shown in Figure 1.1 comprises 7 steps; including

literature reviews, sample preparation, laboratory testing, assessment of strength

criteria, discussions and conclusions and thesis writing.

1.3.1 Literature review

Literature review is carried out to study the previous researches on

compressive strength as affected by mechanical and thermal loading. The sources of

3

information are from text books, journals, technical reports and conference papers. A

summary of the literature review is given in the thesis.

1.3.2 Sample preparation

Rock samples used here have been obtained from the intrusive rock of

the Tak batholiths of western Thailand. Sample preparation is carried out in the

laboratory at the Suranaree University of Technology. Samples prepared for

compressive strength test are 5×5×10 cm3.

1.3.3 Laboratory testing

The laboratory testing involves triaxial compressive strength tests in

polyaxial load frame. The specimen is first confined under hydrostatic stress

equivalent to the pre-defined confining pressure while the axial stress is increased

until failure occurred. The specimens are tested under constant temperatures ranging

from 273, 373 (ambient temperature), 573 to 773 Kelvin. Each temperature is tested

at confining stresses at 0, 3, 7 and 12 MPa. Failure stresses and strains are recorded.

The stress-strain curves are developed for each specimen.

1.3.4 Development of the strength criteria

Results from laboratory measurements in terms of the principal

stresses at failure are used to formulate mathematical relations. The temperature

effect is incorporated to the strength criteria. The mean misfit between the criterion

and test data is determined.

1.3.5 Computer simulation

A computer simulation is performed to determine the effects of

temperature on the granite strength and deformation around waste disposal borehole.

1.3.6 Discussions, conclusion and thesis writing

4

Discussions are made to analyze the impacts of the temperature and

determine a new failure criterion for strength under elevated temperatures. All

research activities, methods and results are documented and complied in the thesis.

The research finding is published in the conference proceedings or journals.

Figure 1.1 Research methodology

5

1.4 Scope and limitations

The scope and limitations of the research include as follows:

1. All testing will be conducted on granite specimens obtained from the Tak

batholiths of western Thailand.

2. The specimens are prepared with nominal dimensions of 5×5×10 cm3.

3. The applied confining pressures are varied from 0, 3, 7, to 12 MPa.

4. The nominal temperatures vary from 273, 303, 373, 573 and 773 Kelvin.

5. A new failure strength criterion is derived from the test results.

1.5 Thesis contents

Chapter I introduces the thesis by briefly describing the background of

problems and significance of the study. The research objectives, methodology, scope

and limitations are identified. Chapter II summarizes the results of the literature

review. Chapter III describes the sample preparation. Chapter IV proposes the

laboratory experiment and presents the results obtained from the laboratory testing.

Chapter V describes mathematical equations to determine the temperatures effect on

the strength of rock. Chapter VI shows the method and results of computer

simulation performed to determine the effects of temperature on the granite strength

and deformation around waste disposal borehole. Chapter VII discusses and

concludes the research results, and provides recommendations for future research

studies.

CHAPTER II

LITERATURE REVIEW

2.1 Introduction

Relevant topics and previous research results are reviewed to improve an

understanding of rock compressive strength as affected by temperature. The topics

reviewed here include geology of Tak batholiths, effects of temperature on rock

strength, strength criteria, and deep hole injection technology.

2.2 Literature review

2.2.1 Tak batholith

Mahawat et al. (1988) study the origin of Tak Batholith of western

Thailand, showing that it is made up of four zoned plutons, the youngest of which is

210 Ma old. The composition of the plutons changes from granodioritic in the oldest,

through monzonitic and monzogranitic to syenogranitic in the youngest. The earliest

pluton (Eastern) shows an extended compositional range and is calc-alkaline-

granodioritic, medium K in the sense of Lameyre and Bowden. It is similar to classic

Andean Cordilleran Batholith plutons. The three later plutons (Western, Mac Salit

and Tak), the first of which also shows an extended compositional range, are all calc-

alkaline- monzonitic, high K plutons characterised by the early precipitation of U, Th,

REE rich accessory minerals and K-feldspar. The changes in major and trace element

composition of the plutons with time are similar to that seen in an increasing K

volcanic series at an active continental margin. The final member is chemically

7 similar to shoshonitic volcanic rocks and by analogy the plutonic sequence is

considered to be due to a change from subduction to strike-slip and then uplift, which

brought mantle of different composition into the source region beneath the Tak

Batholith. In such situations simple classifications relating a batholith to end-member

source and geotectonic setting may be misleading.

2.2.2 Effects of temperature on rock strength

Araújo et al. (1997) investigate the mechanical properties of reservoir

rocks obtained in laboratory at room temperature. The investigation was based on

triaxial tests performed by a servo-controlled system on samples of friable sandstone

cored from a reservoir in the Potigure Basin, Northwastern Brazil. The samples were

tested at 24°C, 80°C and 150°C with confining pressures varying from 2.5 MPa to 20

MPa. The resulting values for bulk compressibility indicate a significant decrease

with the temperature increasing from 24°C to 150°C. Figure 2.1 presents the

regression lines in the plane σ1-σ3, for the temperature 24°C and 150°C.

0 5 10 15 200

1020

30

40

50

6070

σ3 MPa

σ 1 M

Pa

24 °C150 °C

Figure 2.1 Failure limits from multiple failure state tests on sandstones from Acu

formation (Araujo et al., 1997).

8 Results show an average reduction of about 18% in the compressive strength for an

increase of temperature from 24°C to 150°C.

Dwivedi et al. (2008) study various thermo-mechanical properties of

Indian granite at high temperatures in the range of 30-160°C, keeping in view the

highest temperatures expected in underground nuclear waste repositories. The rock

temperature around the nuclear waste in such conventional storage, may not rise

beyond 250°C. But, in the case of non-conventional or direct burial of nuclear waste

the rock temperature may be very high and sometimes exceeds the melting point of

the rock. Cylindrical specimens having visible cracks have been discarded. Samples

from the same depth of borehole were selected for one set of uniaxial compressive

strength and Brazilian tensile strength test. These properties are Young’s modulus,

uniaxial compressive strength, tensile strength, Poisson’s ratio, coefficient of linear

thermal expansion, creep behavior and the development of micro-crack on heating

using scanning electron microscope. Figure 2.2 shows variation of compressive

strength of Indian granite with temperature, whereas Figure 2.3 shows variation of

tensile strength decreases with increase in temperature. Normalized compressive

strength (σc/σc0) values decrease at faster rate from 400°C on wards up to 600°C. The

normalized tensile strength (σt/σt0) decreases at the fastest rate between 400 and

650°C. The information collected may be useful to the researchers engaged in the

modeling of important geological processes in granites. There is a mixed trend in

normalized compressive strength (σc/σc0) up to 400°C but it decreases from 500°C.

Ultimate compressive strength increases with increase in confining pressure. On the

other hand, tensile strength of all granites decrease with increase in temperature for

the reported temperature range 30–1050°C. In addition, information gathered would

9 be useful in thermo-mechanical characterisation of granites for modelling of several

geological phenomena.

Figure 2.2 Compressive strength with temperature (Dwivedi et al., 2008)

Figure 2.3 Tensile strength with temperature (Dwivedi et al., 2008)

Inada et al. (1997) used specimens in the form of cylinder specimens for

simulating stress condition around the opening excavated to study the strength and

deformation characteristics of granite and tuff after undergoing thermal hysteresis of

high and low temperatures. The cylinder specimens are used in uniaxial compression

test and Brazilian tensile strength test apparatus. The conditions of thermal hysteresis

10 are wet and dry specimens. Compressive and tensile strength of rocks decrease with

the increasing number of thermal hysteresis. The values of tangential Young's

modulus and Poisson's ratio of rocks after undergoing thermal hysteresis have the

same tendency as those of compressive and tensile strength. The results of measuring

strain, the residual strains at room temperature can be seen for all specimens after

undergoing thermal hysteresis. It is considered that the residual strain tends to

converge to a constant value as thermal hysteresis is repeated. Elastic wave

propagation velocity of rocks decreases with the increasing number of thermal

hysteresis, but the ratio of decreasing decreases. It is considered that the value will

converge to a constant value.

Sriapai et al. (2013) study of the temperature effects on salt strength by

incorporating empirical relations between the elastic parameters and temperatures of

the tested specimens to describe the distortional strain energy density of rock salt

under different temperatures and deviation stresses. The results are obtained under

temperatures ranging from 273 to 467 Kelvin. The results indicate that the uniaxial

compressive strengths (σc) of salt decrease linearly with increasing temperature. The

tensile strength (σB) decreases linearly with increasing specimen temperature. The

triaxial compressive strength results, under the same confining pressure (σ3), the

maximum principal stress at failure (σ1) decreases with increasing specimen

temperature as shown in Figure 2.4. The diagram in Figure 2.5 clearly indicates that

the effect of temperature on the salt strength was larger when the salt was under

higher confining pressures. When σm is below 20 MPa, the octahedral shear strengths

for salt were less sensitive to the temperature. The effect of temperature on the salt

strength may be enhanced for the salt cavern with high frequency of injection-

11 retrieval cycles. To be conservative the maximum temperature (induced during

injection) and the maximum shear stresses (induced during withdrawal) in salt around

the cavern should be determined (normally by numerical simulation). The salt

stability can be determined by comparing the computed temperature distribution and

mechanical and thermal stresses against the criterion proposed above. The results

should lead to a conservative design of the safe maximum and minimum storage

pressures.

Figure 2.4 (a) Uniaxial compressive strength of salt as a function of temperature.

(b) Brazilian tensile strength of salt as a function of temperature.

(c) Major principal stress at failure as a function of confining pressure.

(Sriapai et al., 2012)

(c) (b)

(a)

12

Figure 2.5 Octahedral shear strength of salt as a function of mean stress

(Sriapai et al., 2012)

Wai and Lo (1982) study the effects of temperature up to 350°C on the

strength and deformation properties of rock. Particular attention was paid to the

experimental procedure to avoid premature thermal cracking of the specimens. It was

shown that the thermal-mechanical behavior varies with the rock type. For granitic

gneiss, the deformation modulus increases slightly with temperature up to 120°C, then

decreases at a rate of about 25% per 100°C. Poisson’s ratio generally decreases with

increasing temperature up to 250°C. The uniaxial compressive strength of granitic

gneiss decreases with increasing temperature at a rate of the order of 30 MPa per

100°C. The deformation properties of the granitic gneiss are also dependent on the

temperature history of the specimen. In contrast, both the deformation and strength

behavior of the limestone appear to be insensitive to temperature change.

Xu et al. (2008) propose the relationships between mechanical

characteristics of rock and microcosmic mechanism at high temperatures which were

13 investigated by MTS815, Based on a micropore structure analyzer and SEM, the

changes in rock porosity and micro structural morphology of sample fractures and

brittle-plastic characteristics under high temperatures were analyzed. The results are

as follows: 1) Mechanical characteristics do not show obvious variations before

800°C; strength decreases suddenly after 800°C and bearing capacity is almost lost at

1200°C, 2) Rock porosity increases with rising temperatures; the threshold

temperature is about 800°C; at this temperature its effect is basically uniform with

strength decreasing rapidly, and 3) The failure type of granite is a brittle tensile

fracture at temperatures below 800°C which transforms into plasticity at temperatures

higher than 800°C and crystal formation takes place at this time. Chemical reactions

take place at 1200°C. Failure of granite under high temperature is a common result of

thermal stress as indicated by an increase in the thermal expansion coefficient,

transformation to crystal formation of minerals and structural chemical reactions.

Xu et al. (2009) state that the effect of temperature on mechanical

characteristics and behaviors of granite was analyzed by using experiments as

scanning electron microscope (SEM), X-ray diffraction and acoustic emission (AE)

and the micromechanism of brittle-plastic transition of granite under high temperature

was discussed as well. The rock specimens are 50 mm long, with diameters of 25

mm. The specimens of each were heated to the temperature ranging from 25 to

1300°C. The failure from of granite changes from abruptly brittle fracture to semi-

brittle shear fracture gradually with the rise of temperature. The average of

compressive strength and elastic modulus at different temperatures is seen as Table

2.1, which decreases with the rise of temperature.

14 Table 2.1 Average of compressive strength and elastic modulus at different

temperatures (Xu et al., 2009).

Rock properties Temperature(°C)

25 200 500 800 900 1000 1100 1200 Compressive strength

σc (MPa) 191.9 135.96 151.9 185.22 89.94 71.61 77.98 36.09

Elastic modulus E (GPa) 38.57 28.68 31.25 25.11 11.02 8.39 6.61 2.87

Zhao et al. (2012) perform triaxial compression system for rock testing

under high temperature and high pressure. The performance and technological

innovations of a self-developed 20 MN rock triaxial test machine XPS-20MN for high

temperature and high pressure are presented. The tests revealed the stress-strain

characteristics of the coal specimens at high temperature, particularly the enhanced

plastic features. Young’s modulus decreases in a negative exponential function in

regarding to the temperature. Test results revealing the thermal deformation and

failure mode of the large-size granite specimens at high temperature and high

pressure, and the changing of the thermodynamic parameters of the specimen, such as

Young’s modulus, with temperature, are also present as examples of test carried out

using the testing machine.

2.2.3 Strength calculation

Jaeger et al. (2007) state that in order to derive the laws that govern the

transformation of stress components under a rotation of the coordinate system, we again

consider a small triangular element of rock, as in Figure 2.6. We arrive at the following

15

ey

tn

ex

τ σ

x

y

τxx

τxy

τyx

τyy(a) (b)

φ φ

Figure 2.6 Small triangular slab of rock used to derive the stress transformation

(Jaeger et al., 2007)

equations for the normal and shear stresses acting on a plane whose outward unit

normal vector is rotated counterclockwise from the x direction by an angle θ:

σ = ½ (τxx + τyy)+ ½ (τxx − τyy) cos 2θ + τxy sin 2θ (2.1)

τ = ½ (τyy − τxx) sin 2θ + τxy cos 2θ (2.2)

An interesting question to pose is whether or not there are planes on which the shear

stress vanishes, and where the stress therefore has purely a normal component. The

answer follows directly from setting τ = 0, and solving for

yyxx

xy

ττ2

tan2−

τ=θ (2.3)

16 A simple graphical construction popularized by Mohr (1914) can be used to represent

the state of stress at a point. Recall that (1) and (2) give expressions for the normal

stress and shear stress acting on a plane whose unit normal direction is rotated from

the x direction by a counterclockwise angle θ. We replace τxx with σ1, replace τyy

with σ2, replace τxy with 0 in the principal coordinate system, and interpret θ as the

angle of counterclockwise rotation from the direction of the maximum principal

stress. We thereby arrive at the following equations that give the normal and shear

stresses on a plane whose outward unit normal vector is rotated by θ from the first

principal direction:

( ) ( )cos2θ2

σσ2

σσσ 2121 −+

+= (2.4)

( )sin2θ2

σστ 21 −= (2.5)

The rock has a preexisting plane of weakness whose outward unit normal vector

makes an angle β with the direction of the maximum principal stress, σ1 (Figure 2.7).

τ σ

σ1

σ2

β

Figure 2.7 Plane of weakness with outward normal vector oriented at angle β to

the direction of maximum principal stress (Jaeger et al., 2007).

17 The criterion for slippage to occur along this plane is assumed to be

|τ| = So + μσ (2.6)

where σ is the normal traction component acting along this plane and τ is the shear

component. By (2.4) and (2.5), σ and τ are given by

σ = ½ (σ1 + σ2) + ½ (σ1 − σ2) cos 2β (2.7)

τ = − ½ (σ1 − σ2) sin 2β (2.8)

2.2.4 Stress distribution around circular hole

Hoek and Brown (1990) state that the stress distributions around a

borehole (σr, σθ, τrθ) can be calculated by Kirsch solution:

σr = [(P1 + P2) / 2)][1 - (a2/r2)] + [(P1 - P2) / 2)][(1 - (4a2/r2)) +( 3a4/r4)] cos2θ (2.9)

σθ = [(P1 + P2) / 2)][1 + (a2/r2)] - [(P1 - P2) / 2)][1 + (3a4/r4)] cos2θ (2.10)

τrθ = - [(P1 + P2) / 2)][(1 + (2a2/r2) - ( 3a4/r4)] cos2θ (2.11)

where σr is the stress in the direction of changing r, σθ is tangential stress, τrθ is

shear stress, P1 is vertical stress, P2 is lateral stress, a is inside radius of opening, r is

variable radius and θ is the angle between vertical axis and radius.

2.2.5 Deep hole injection technology

Gibb et al. (2008) study waste actinides, including plutonium, present a

long-term management problem and a serious security issue. Immobilisation in

18 mineral or ceramic waste forms for interim storage is a widely proposed first step.

The safest, most secure geological disposal for Pu is in very deep boreholes and they

propose that the key step to combination of these immobilisation and disposal

concepts is encapsulation of the waste form in cylinders of recrystallized granite.

They discuss the underpinning science, focusing on experimental work, and consider

implementation. Finally, they present and discuss analyses of zircon, UO2 and Ce-

doped cubic zirconia from high pressure and temperature experiments in granitic

melts that demonstrate the viability of this solution and that actinides can be isolated

from the environment for millions, maybe hundreds of millions, of years.

Gibb et al. (2008) states that deep (4–5 km) boreholes are emerging as

a safe, secure, environmentally sound and potentially cost-effective option for

disposal of high-level radioactive wastes, including plutonium. One reason this

option has not been widely accepted for spent fuel is because stacking the containers

in a borehole could create load stresses threatening their integrity with potential for

releasing highly mobile radionuclides like 129I before the borehole is filled and

sealed. This problem can be overcome by using novel high-density support matrices

deployed as fine metal shot along with the containers. Temperature distributions in

and around the disposal are modelled to show how decay heat from the fuel can melt

the shot within weeks of disposal to give a dense liquid in which the containers are

almost weightless. Finally, within a few decades, this liquid will cool and solidify,

entombing the waste containers in a base metal sarcophagus sealed into the host rock.

Gibb (2003) proposes an alternative strategy for the disposal of spent

nuclear fuel (SNF) and other forms of high-level waste (HLW) whereby the integrity

of a mined and engineered repository for the bulk of the waste need be preserved for

19 only a few thousand years. This is achieved by separating the particularly

problematic components, notably heat generating radionuclides (HGRs) and very long

lived radionuclides (VLLRs) from the waste prior to disposal. Such a solution

requires a satisfactory means of disposing of the relatively minor amounts of HGRs

and VLLRs removed from the waste. This could be by high-temperature very deep

disposal (HTVDD) in boreholes in the continental crust. However, the viability of

HTVDD, and hence the key to the entire strategy, depends on whether sufficient

melting of granite host rock can occur at suitable temperatures and whether the melt

can be completely recrystallized. The high-temperature, high-pressure experiments

reported here demonstrate that granite can be partially melted and completely

recrystallized on a time scale of years, as opposed to millennia as widely believed.

Furthermore, both can be achieved at temperatures and on a time scale appropriate to

the disposal of packages of heat generating HLW. It is therefore concluded that the

proposed strategy, which offers, environmental, safety and economic benefits, could

be a viable option for a substantial proportion of HLWs.

Gibb (1999) states that safe disposal of radioactive waste, especially

spent fuel, ex-military fissile materials and other forms of high-level waste (HLW), is

one of the major challenges facing contemporary science. Currently, the

internationally preferred solution is for geological disposal by interment in a mined

and engineered, multi-barrier repository. Although this is often referred to as ``deep''

disposal, the depths involved are usually quite shallow in geological terms. A new

scheme, currently under development, for the high-temperature disposal of

concentrated HLW in very deep boreholes is outlined. Recent advances in the

knowledge of continental crustal rocks and fluids at depths of several kilometres

20 suggest that much deeper disposal might offer a safer and environmentally more

acceptable solution to the HLW problem. The new scheme seeks to capitalise on this

potential while turning the problematical heat output of the waste to advantage. It

appears to offer significant benefits over both mined repositories and earlier borehole

scenarios, particularly in terms of safety.

Sundberg et al. (2009) state that Swedish Nuclear Fuel and Waste

Management Company (SKB) are organized with the objective of sitting a deep

geological repository for spent nuclear fuel. The spent fuel is encapsulated in 5 m

long cylindrical copper/steel canisters with an outer diameter of 1.05 m. The canisters

are deposited in vertical 8 m-deep, 1.75 m-diameter deposition holes excavated in the

floor of horizontal deposition tunnels at 400–700 m depth in crystalline rock below

ground surface. The insulation and protection, the canisters are surrounded by barrier

of bentonite clay. The temperature peak occurs some 10 years after deposition and

amounts to 87°C.

Witherspoon et al. (1979) investigate granite at Stripa, Sweden, for

nuclear waste storage and state that the designing waste repositories must understand

how such stresses are affected by the behavior of the fracture systems. The fractures

affect the thermo-mechanical of the rock mass, but they provide the main pathways

for radionuclides to migrate away from the repository.

CHAPTER III

SAMPLE PREPARATION

3.1 Introduction

Tak granite from Tak province, Thailand, has been selected for use as rock

sample here primarily because it has low permeability, less fracture and resistant

heating. Mahawat et al. (1990) give detailed description and origin of the Tak granite

that Tak pluton cuts the Eastern and Western plutons, with faulted and strongly

sheared contacts with Lower Palaeozoic sediments. It is surrounded by extensive

microgranite, porphyry and pegmatite dykes. The mineral compositions of the sample

tested are plagioclase 16.2%, quartz 5.4%, K-fieldspar 5%, biotite 2.7%, hornblende

0.5%, Ore/Rest tr, groundmass 70% (Atherton et al.,1992).

3.2 Sample preparation

The specimens have been prepared to obtain rectangular blocks with nominal

dimensions of 5×10×10 cm3 for the uniaxial and triaxial compression tests. The

cutting and grinding comply with ASTM (D 4543-85) as shown in Figure 3.1a. Table

3.1 shows the properties of granite. The Brazilian tension test uses disk specimens

with a nominal diameter of 5.4 cm with a thickness-to-diameter ratio of 0.5 to comply

with ASTM (D 3967-95a) as shown in Figure 3.1b. All specimens are prepared for

each temperature level. The thermal properties are determined by National Metal and

Materials Technology Center (MTEC).

22

Table 3.1 Properties of granite.

Properties Value

Density (g/cc) 2.61±0.02

Thermal diffusivity (mm2/s) 1.71±0.04

Thermal conductivity (W/mK) 2.22±0.01

Specific heat (MJ/m3K) 1.30±0.03

Thermal expansion (K-1) 1.8×10-7

Figure 3.1 The specimens prepared for uniaxial and triaxial compression testing (a)

and Brazilian tension testing (b).

10 cm

5 cm 5 cm

5.4 cm

2.7 cm

(a) (b)

CHAPTER IV

LABORATORY TESTING

4.1 Introduction

The objective of the laboratory testing is to determine the effect of

temperatures on strengths of granite. This chapter describes the method, test results

and calculation. The strength results are obtained under temperatures of 273, 303,

373, 573 and 773 Kelvin, and confining pressures of 0, 3, 7 and 12 MPa. The

temperatures effect on strength results is analyzed to evaluate deep disposal borehole.

4.2 Test apparatus

Steel platens with heater coil are the key component for this experiment. Its

dimensions are shown in Figure 4.1. The heater coil is wrapped around the steel

platen (Figure 4.2). Electric heating is through a resistor converts electrical energy

into heat energy. Electric heating devices use Nichrome (Nickel-Chromium Alloy)

wire supported by heat resistant. A thermostat (Figure 4.3) is a component of

a control system which senses the temperature of a system so that the system's

temperature is maintained near a desired setpoint. A heating element converts

electricity into heat through the process of resistive. Electric current passing through

the element encounters resistance, resulting in heating of the element. The thermostat

is SHIMAX MAC5D-MCF-EN Series DIGITAL CONTROLLER. The digital

controller is 48×48 mm with panel depth of 62-65 mm. Power supply is a 100-240V

http://en.wikipedia.org/wiki/Control_system

http://en.wikipedia.org/wiki/Temperature

http://en.wikipedia.org/wiki/System

24

± 10%AC on security surveillance system. The accuracy is ± 0.3%FS + 1digit. The

thermocouple is type E1 that can measure the temperatures ranging of 0-700°C.

Figure 4.1 Steel platen dimensions.

Figure 4.2 Heater coil entwine around steel platen.

25

Figure 4.3 Thermostat with digital controller.

4.3 Test method

4.3.1 Heating method

To test the granite specimens under elevated temperatures, the

prepared specimens and loading platens are heated by heater coil, thermocouple and

thermostat for 24 hours before testing at 373, 573 and 773 Kelvin (Figure 4.4). The

temperature is measured and regulated by using thermocouple and thermostat. A

digital temperature regulator is used to maintain constant temperature to the specimen.

The low temperature specimens are prepared by cooling them in a freezer for 24

hours. The changes of specimen temperatures between before and after testing are

less than 5 kelvin. As a result the specimen temperatures are assumed to be uniform

and constant with time during the mechanical testing (i.e., isothermal condition).

26

4.3.2 Uniaxial and triaxial compression tests

The uniaxial and triaxial compression tests are performed to determine

the compressive strength and deformation of granite specimen under various

confining pressure and temperature. A polyaxial load frame (Figure 4.5) (Fuenkajorn

and Kenkhunthod, 2010) has been used to apply axial stress and lateral stresses to the

granite specimens. The test frame utilizes two pairs of cantilever beams to apply

lateral stresses to the specimen while the axial stress is applied by a hydraulic cylinder

connected to an electric pump. The uniform lateral stresses on the specimens range

from 0, 3, 7 to 12 MPa, and the constant axial stress rate of 1 MPa/s until failure

occurs. The specimen deformation monitored in the three principal directions is used

to calculate the principal strains during loading. The failure stresses are recorded and

mode of failure examined. The frame has an advantage over the conventional triaxial

(Hoek) cell because it allows a relatively quick installation of the test specimen under

triaxial condition, and hence the change of the specimen temperature during testing is

minimal.

4.3.3 Brazilian tension test

The Brazilian tensile strength of the granite has been determined from

disk specimens with temperatures ranging from 273, 303, 373, 573 to 773 kelvin, and

the constant axial stress rate of 1 MPa/s. Except the pre-heating and cooling process,

the test procedure, sample preparation and strength calculation follow the (ASTM

D3967-08). The tests are performed by applying continuous increasing compressive

load to the rock specimen until failure occurs and measuring the failure load.

27

Figure 4.4 Temperatures measured and regulated by thermocouples and

Thermostats while the specimen installed in polyaxial load frame.

28

Figure 4.5 Polyaxial load frame (Fuenkajorn and Kenkhunthod, 2010).

4.4 Test results

Figure 4.6 shows some post-test granite specimens from the triaxial

compression test under confining pressures from 0, 3, 7 to 12 MPa with temperatures

from 273 to 773 kelvin. Table 4.1 shows the compressive strength results. Post-test

observations indicate that under low confining pressure, the specimens fail by the

extension failure mode. Under the high confining pressure shear failure mode is

observed. The Brazilian tensile strength of the granite has been determined from disk

specimens with temperatures ranging from 273 to 773 kelvin (Figure 4.7 and Table

4.2). Figure 4.8 – 4.12 show stress-strain curves monitored from the granite

specimens under various temperatures and confining pressures. For all specimens the

two measured lateral strains induced by the same magnitude of the applied lateral

29

stresses are similar. It is assumed that the tested granite is linearly elastic and

isotopic. Some discrepancies may be due to the intrinsic variability of the granite.

The maximum principal stresses at failure from the compressive and tensile testing

can be presented as a function of the minimum principal stress in Figure 4.13.

Figure 4.6 Some post-test granite specimens from compressive strength testing under

difference confining pressure and temperature.

30

Figure 4.7 Some post-test granite specimens from Brazilian tensile strength testing as

a function of temperature.

31

T = 273 K [169.3, 3.0, 3.0]

ε1εvε2ε3

σ1 (

MPa

)

150

300

250

200

100

50

0

milli-strains0 5 10 15-5-10 20

T = 273 K [222.2, 7.0, 7.0]

σ1 (

MPa

)

150

300

250

200

100

50

0


ε1εvε2ε3

T = 273 K [277.8, 12.0, 12.0] σ

1 (M

Pa)

150

300

250

200

100

50

0


ε1εvε2ε3

T = 273 K [131.1, 0.0, 0.0] σ

1 (M

Pa)

150

300

250

200

100

50

0


ε1εvε3

Figure 4.8 Stress-strain curves obtained from some granite specimens at 273 K.

Numbers in brackets indicate [σ1, σ2, σ3] at failure.

32

T = 303 K [215.2, 7.0, 7.0]

ε1εvε2ε3

σ1 (

MPa

)

150

300

250

200

100

50

0


T = 303 K [269.2, 12.0, 12.0]ε1εvε2ε3

σ1 (

MPa

)

150

300

250

200

100

50

0


T = 303 K [161.1, 3.0, 3.0]

σ1 (

MPa

)

150

300

250

200

100

50

0


ε1εvε2ε3

T = 303 K [118.8, 0.0, 0.0] σ

1 (M

Pa)

150

300

250

200

100

50

0


ε1εvε3

Figure 4.9 Stress-strain curves obtained from some granite specimens 303 K.


33

T = 373 K [143.4, 3.0, 3.0]

ε1εvε2ε3

σ1 (

MPa

)

150

300

250

200

100

50

0


σ1 (

MPa

)

0 5 10 15-5-10 20

T = 373 K [198.5, 7.0, 7.0]

ε1εvε2ε3

150

300

250

200

100

50

0

milli-strains

T = 373 K [250.6, 12.0, 12.0] σ

1 (M

Pa)

150

300

250

200

100

50

0


ε1εvε2ε3

T = 373 K [104.1, 0.0, 0.0] σ

1 (M

Pa)

150

300

250

200

100

50

0


ε1εvε3



34

σ1 (

MPa

)

150

300

250

200

100

50

0

milli-strains

T = 573 K [231.8, 12.0, 12.0]

ε1εvε2ε3

0 5 10 15-5-10 20

T = 573 K [180.1, 7.0, 7.0]

ε1εvε2ε3

σ1 (

MPa

)

150

300

250

200

100

50

0


T = 573 K [127.7, 3.0, 3.0]

σ1 (

MPa

)

150

300

250

200

100

50

0


ε1εvε2ε3

T = 573 K [90.4, 0.0, 0.0] σ

1 (M

Pa)

150

300

250

200

100

50

0


ε1

εvε3



35

T = 773 K [104.2, 3.0, 3.0]

ε1εvε2ε3

σ1 (

MPa

)

150

300

250

200

100

50

0


σ1 (

MPa

)

150

300

250

200

100

50

0

milli-strains

T = 773 K [211.7, 12.0, 12.0]

ε1εvε2ε3

0 5 10 15-5-10 20

T = 773 K [73.1, 0.0, 0.0] σ

1 (M

Pa)

150

300

250

200

100

50

0


ε1εvε3

T = 773 K [157.9, 7.0, 7.0]

σ1 (

MPa

)

150

300

250

200

100

50

0


ε1εvε3ε2



36

σ1 (MPa)

σ3 (MPa)

50

150

200

250

100

300

155-5-10 0 10-15

T = 273 K

773 K573 K373 K303 K

Figure 4.13 Major principal stresses at failure as a function of confining pressure.

37

Table 4.1 Compressive strengths of granite.

σ3 (MPa) σ1 (MPa)

273 K 303 K 373 K 573 K 773 K 0 131.09 118.81 104.07 90.43 73.08 3 169.26 161.13 143.41 127.68 104.16 7 222.21 215.15 198.53 180.14 157.92 12 277.77 269.20 250.64 231.83 211.68

Table 4.2 Brazilian tensile strengths of granite.

Specimen Number Density (g/cm3) Temperature (K) σB (MPa) BZ-GR-1 2.54 273 8.64 BZ-GR-2 2.53 273 8.67 BZ-GR-3 2.50 273 8.64 BZ-GR-4 2.57 303 7.64 BZ-GR-5 2.58 303 7.86 BZ-GR-6 2.57 303 7.78 BZ-GR-7 2.64 373 6.85 BZ-GR-8 2.63 373 6.58 BZ-GR-9 2.51 373 6.32 BZ-GR-10 2.48 573 5.31 BZ-GR-11 2.45 573 5.46 BZ-GR-12 2.50 573 5.54 BZ-GR-13 2.53 773 4.29 BZ-GR-14 2.60 773 4.43 BZ-GR-15 2.52 773 4.25

38

4.5 Calculation

The test results can be presented in terms of the octahedral shear stress at

failure (τoct,f) as a function of mean stress (σm), as shown in Table 4.3, where (Jaeger

et. al, 2007):

( ) ( ) ( )[ ]0.5

213

232

221foct, 3

1

σ−σ+σ−σ+σ−σ=τ (4.1)

( )321m 31

σ+σ+σ=σ (4.2)

The elastic parameters are calculated for the three-dimensional principal stress-

strain relations. It is assumed that the tested granite is linearly elastic and isotopic.

The shear modulus (G) and bulk modulus (K) can therefore be determined for each

specimen using the following relations:

G = (1/2) (τoct/γoct) (4.3)

λ = (1/3) [(3σm /∆) − 2G] (4.4)

E = 2G (1+ν) (4.5)

ν = λ/2(λ+G) (4.6)

K = E/3(1-(2ν)) (4.7)

where τoct, γoct, σm and ∆ are octahedral shear stress, octahedral shear strain,

mean stress, and volumetric strain.

39

Figure 4.14 shows elastic modulus (E), Poisson’s ratio (ν), shear modulus and

bulk modulus as a function of temperature. Table 4.4 summarizes the measurement

results for each specimen in terms of the elastic modulus, shear modulus, bulk

modulus, and Poisson’s ratio.

773 K

E (G

Pa)

T (K)200 300 400 500 600 700 800

0

5

10

15

20

T = 273 K 373 K573 K

303 K ν

T (K)200 300 400 500 600 700 800

0

0.1

0.2

0.3

0.4

T = 273 K

303 K373 K

573 K 773 K

G (G

Pa)

T (K)200 300 400 500 600 700 800

0

1

7

2

3

4

5

6

303 K

373 K773 K

T = 273 K 573 K

K (G

Pa)

T (K)200 300 400 500 600 700 800

0

2

14

4

6

8

10

12T = 273 K

303 K

573 K 773 K373 K

Figure 4.14 Elastic parameters as a function of temperature.

40

Table 4.3 Strengths of granite.

T (K) σ3 (MPa) σ1 (MPa) σm (MPa) τoct (MPa)

273

0 131.1 43.67 61.79 3 169.3 58.67 78.37 7 222.2 78.67 101.45 12 277.8 100.33 125.28

303

0 118.8 39.67 56.01 3 161.1 55.67 74.54 7 215.2 76.33 98.12 12 269.2 98.00 120.00

373

0 104.1 34.67 49.06 3 143.4 50.00 66.18 7 198.5 71.33 90.28 12 250.6 91.00 112.49

573

0 90.4 30.00 42.63 3 127.7 44.33 58.77 7 180.1 64.67 81.62 12 231.8 84.33 102.04

773

0 73.1 24.33 39.00 3 104.2 37.00 47.68 7 157.9 57.67 71.14 12 211.7 78.00 94.13

41

Table 4.4 Elastic parameters.

T (K) σ3 (MPa) E (GPa) ν G (GPa) K (GPa)

273

0 13.35 0.28 5.17 10.41 3 14.05 0.28 5.47 10.81 7 14.87 0.29 5.72 12.34 12 14.53 0.28 5.68 10.94

303

0 13.63 0.28 5.33 10.23 3 13.93 0.28 5.44 10.55 7 13.92 0.29 5.39 11.15 12 13.74 0.29 5.31 11.12

373

0 12.41 0.28 4.85 9.36 3 12.24 0.28 4.77 9.36 7 12.33 0.30 4.76 10.02 12 12.38 0.30 4.78 10.11

573

0 11.00 0.28 4.29 8.46 3 11.40 0.27 4.48 8.37 7 11.35 0.29 4.42 8.80 12 11.00 0.30 4.23 9.17

773

0 10.67 0.28 4.17 8.08 3 10.62 0.26 4.20 7.47 7 10.60 0.30 4.08 8.79 12 10.61 0.29 4.10 8.54

CHAPTER V

MATHEMATICAL EQUATIONS

5.1 Introduction

The mathematical equations derived in this study aim at predicting the effects

of temperatures on strength and elasticity of granite specimens. The regression

analysis is performed by on the Hoek and Brown criterion (Hoek and Brown, 1980)

and the Coulomb criterion (Jaeger et al., 2007) to assess their predictive capability.

5.2 Empirical equations

The results indicate that under the same confining pressure the maximum

principal stresses at failure (σ1f) of granite decrease non-linearly with increasing

temperature (T) and can be best represented by (Figure 5.1):

3f1, /T)(exp σ⋅α+β⋅κ=σ (MPa) (5.1)

The parameters κ, β, and α are empirical constants. A good correlation is

obtained between the test results and the proposed equation (R2 = 0.996).

The maximum principal stresses at failure from the compressive and tensile

testing can be presented as a function of the minimum principal stress in Figure 5.1.

Linear relations can be observed at all temperature levels. The higher temperature

imposed on the granite specimen, the lower failure envelope is obtained.

43

To incorporate the intermediate principal stress (σ2) the test results can be

presented in terms of the octahedral shear stress at failure (τoct,f) as a function of mean

stress (σm), as shown in Figure 5.2, where (Jaeger et. al, 2007):

( ) ( ) ( )[ ]0.5

213

232

221foct, 3

1

σ−σ+σ−σ+σ−σ=τ (5.2)

( )321m 31

σ+σ+σ=σ (5.3)

The diagram in Figure 5.2 clearly indicates that the effect of temperature on

the granite strength is larger when the granite is under higher confining pressures.

Nonlinear relations can be observed at all temperature levels, and can be represented

by

mfoct, T)/ ( exp σ⋅λ+η⋅δ=τ (MPa) (5.4)

The parameters δ, η, and λ are empirical constants. A good correlation is

obtained between the test results and the proposed equation (R2 = 0.990).

44

σ1,

f (M

Pa)

0

50

150

200

250

100

300

T (K)200 300 400 500 600 700 800

σ3 = 12 MPa

7

30

σ1,f = 53.109 × exp (251.533 / T)+(12.4 × σ3 )(R2 = 0.996)

Figure 5.1 Major principal stresses of granite as a function of temperature.

τoc

t,f (M

Pa)

σm (MPa)

80

40

0

20

60

100

120

140

0 40 12080 1006020

303 K

373 K

773 K

573 K

T = 273 K

τoct,f = 5.19 x (exp(252.059 / T))+(σm x 1.12) (R2 = 0.999)

Figure 5.2 Octahedral shear strengths at failure of granite as a function of mean

stress.

45

5.3 Hoek and Brown criterion

The empirical relationship between the principal stresses associated with the

failure of rock can be calculated from Hoek and Brown criterion given by equation

(Hoek and Brown, 1980):

2c3HBc,1 sm σ+σσ+σ=σ 3 (5.5)

where σ1 is the major principal stress at failure,

σ3 is the minor principal stress applied to the specimen,

σc,HB is the uniaxial compressive strength of the intact rock material in the

specimen,

m and s are constants which depend upon the properties of the rock and upon

the extent to which it has been broken before being subjected to the stresses σ1

and σ3.

For intact rock, s = 1 and the uniaxial compressive strength σc,HB and the material

constant m are given by (Hoek and Brown, 1980):

)( ( )( )( ) ( )nΣxnΣxΣxnyΣxyΣxnΣy i2

i2

iiiiii2

HBc, ⋅−−−σ = (5.6)

)( ( )( )( )nΣxΣxnyΣxyΣxm 2i

2iiiii

2HBc,1 −−⋅σ= (5.7)

where 3x σ=

( )231y σ−σ=

n is the total number of test specimens

46

It is found that the m constants increase with increasing temperature and

uniaxial compressive strength (σc,HB) decrease with increasing temperature (Figure.

5.3)

The shear stress (τ) and normal stress (σn) can be calculated from Mohr

envelope given by equation (Hoek and Brown, 1980):

( )8HBc,m2m3n mσ+ττ+σ=σ (5.8)

( ) mHBc,3n 41 m τσ+σ−σ=τ (5.9)

m

T (K)

0

10

30

20

40

200 300 400 500 600 700 800

m = 5.90×T 0.24

(R2 = 0.955) σc,

HB (M

Pa)

T (K)

0

50

150

100

200

σc,HB = 3299.10×T -0.55

(R2 = 0.987)

200 300 400 500 600 700 800

Figure 5.3 Parameter m and uniaxial compressive strength (σc,HB) as a function of

temperatures.

47

where ( )31m 21 σ−σ=τ

The shear stress of granite as a function of normal stress with different

temperatures is shown in Figure 5.4. Results for the σn, τ, m, s, and σc,HB are

summarized in Table 5.1.

The instantaneous cohesion (ci) and instantaneous friction angle (φi) can be

determined from the strength results for each temperature level using the following

relations (Hoek and Brown, 1980):

tan φi = A⋅B ((σn/ σc) − (σ t/ σc,HB)) B−1 (5.10)

ci = τ − σn tan φi (5.11)

The values of the constants A and B are found by linear regression analysis

They are determined from the τ − σn curves at σn = 20 MPa. It is found that

the cohesion decrease non-linearly and friction angle decrease linearly with increasing

temperature, and can be represented by (Figure 5.5):

c = 450.29⋅T −0.57 (MPa) (5.12)

φ = −0.001⋅T + 58.70 (Degree) (5.13)

48

τ (M

Pa)

σn (MPa)

303 K

373 K

773 K573 K

T = 273 K

0 10 20 30 40 50 60 70 80 90-10-20

10

20

30

40

50

60

70

80

90

T (K) φ (°) c (MPa) 273 58.0 19.0 303 58.5 17.2 373 58.0 15.4 573 57.4 13.5 773 56.4 10.7

Figure 5.4 Shear strengths of granite as a function of normal stress with different

temperatures.

5.4 Coulomb criterion

Based on Coulomb criterion the cohesion (c) and internal friction angle (φ)

can be determined from the strength results for each temperature using the following

relation (Jaeger et. al, 2007):

σ1 = σc + σ3 tan2 [(π/4) + (φ/2)] (5.14)

σc = 2c tan [(π/4) + (φ/2)] (5.15)

They are determined from the tangent of the σ1 − σ3 curves at σ3 = 5 MPa. It

is found that the cohesion decrease non-linearly and friction angle decrease linearly

with increasing temperature, and can be represented by (Figure 5.5).

49

c = 455.69⋅T −0.57 (MPa) (5.16)

φ = −0.0011⋅T + 58.17 (Degree) (5.17)

c (M

Pa)

T (K)200 300 400 500 600 700 800

0

5

15

20

10c = 455.69×T -0.57

(R2 = 0.985)

Coulomb criterion

Hoek & Brown criterion

Hoek & Brown criterionc = 450.29×T -0.57

(R2 = 0.984)φ

(°)

T (K)200 300 400 500 600 700 800

0

30

40

20

70

φ = -0.0011×T + 58.17(R2 = 0.709)

10

50

60

Coulomb criterion


φ = -0.001×T + 58.70(R2 = 0.633)


Figure 5.5 Comparisons cohesion and internal friction angle between Hoek and

Brown criterion and Coulomb criterion of granite as a function of

temperature.

5.5 Elastic parameters

The elastic modulus (E) and Poisson’s ratio (ν) are determined by assuming

that the tested granite is linearly elastic and isotopic. The shear modulus (G) and bulk

modulus (K) can therefore be determined for each specimen using the following

relations.

G = (1/2) (τoct/γoct) (5.18)

λ = (1/3) [(3σm /∆) − 2G] (5.19)

50

E = 2G (1+ν) (5.20)

ν = λ/2(λ+G) (5.21)

K = E/3(1-(2ν)) (5.22)

Therefore, E, ν and K can be determine as (Jaeger et. al, 2007). Nonlinear

variations of the three parameters and Poisson’s ratio with respect to temperature are

observed from the results, as shown in Figure 5.6. Good correlations are obtained

when they are fitted with the following linear equations (R2 = 0.962, 0.959, 0.975,

0.699, respectively):

0.28TE 68.11 −⋅= (GPa) (5.23)

0.28TG 26.09 −⋅= (GPa) (5.24)

0.29TK 57.07 −⋅= (GPa) (5.25)

( ) 0.29106 T 6 +⋅−=ν −⋅ (5.26)

The results indicate that the elastic, shear, and bulk moduli decrease with

increasing temperature, and tend to be independent of the confining pressure. The

Poisson’s ratios tend to be independent of the temperature and confining pressure.

51

E (G

Pa)

T (K)200 300 400 500 600 700 800

0

5

10

15

20

303 K

T = 273 K 373 K573 K 773 K

E = 68.11×T -0.28

(R2 = 0.962)

G (G

Pa)

T (K)200 300 400 500 600 700 800

0

1

7

2

3

4

5

6

(R2 = 0.959)

G = 26.09×T -0.28

303 K

T = 273 K 573 K

373 K773 K

K (G

Pa)

T (K)200 300 400 500 600 700 800

0

2

14

4

6

8

10

12

(R2 = 0.975)

K = 57.07×T -0.29

T = 273 K

303 K

573 K 773 K373 K

ν

T (K)200 300 400 500 600 700 800

0

0.1

0.2

0.3

0.4

ν = (-6x10-6)×T + 0.29

(R2 = 0.699)

T = 273 K

303 K373 K

573 K 773 K

Figure 5.6 Elastic parameters of granite as a function of temperature.

5.6 Strain energy density criterion

The strain energy density principle is applied here to describe the rock strength

and deformation under different temperatures. Assuming that for each temperature

level the granite is linearly elastic prior to failure, Wd and Wm can be determined for

each specimen using the following relations (Sriapai et al., 2013):

=

Gτ

W2oct

d43 (5.27)

52

σ=

KW

2

2m

m (5.28)

The elastic parameters G and K can be defined as a function of the testing

temperature, and hence the granite strengths from different temperatures can be

correlated. By substituting equations (5.24) into (5.27) and (5.25) into (5.28) the Wd

at failure can therefore incorporate the effect of specimen temperature into the

strength calculation.

The distortional strain energy for each granite specimen at failure, that

implicitly takes the temperature effect into consideration, is plotted as a function of

the mean strain energy in Figure 5.7. The data can be described best by a linear

equation:

ThmThd BWAW +⋅= (5.29)

The parameters ATh and BTh are empirical constants depending on the strength

and thermal response of the rock. For the Tak granite ATh = 4.49 and BTh = 0.09 MPa.

A good correlation is obtained between the test results and the proposed criterion (R2

= 0.992).

53

Wd (

MPa

)

0 0.2 0.60.4 Wm (MPa)

T = 273 K

0

1

2

3

(R2 = 0.992) Wd = 4.49×Wm + 0.09

773 K

303 K

573 K373 K

Figure 5.7 Distortional strain energy as a function of mean strain energy.

54

Table 5.1 Shear stress and normal stress at failure.

T (K) σ3 (MPa) σc,HB

(MPa) m R2 τ (MPa)

σn (MPa)

273

−8.57

153.40 21.86 0.945

60.27 25.73 0 229.04 131.00 3 291.56 170.00 7 374.92 222.00

12 463.09 277.00

303

−7.79

143.15 23.34 0.943

55.36 23.37 0 211.61 119.00 3 280.24 161.00 7 368.47 215.00

12 458.33 270.00

373

−6.58

124.61 24.55 0.954

46.99 19.74 0 181.94 104.00 3 246.01 144.00 7 335.70 200.00

12 414.18 249.00

573

−5.43

105.46 25.96 0.962

38.95 16.29 0 154.12 90.00 3 211.93 127.00 7 294.73 180.00

12 371.29 229.00

773

−4.33

84.06 28.95 0.957

30.28 12.99 0 120.69 73.00 3 168.90 105.00 7 250.26 159.00

12 327.09 210.00

CHAPTER VI

COMPUTER SIMULATIONS

6.1 Introduction

The finite difference analyses are performed using FLAC 4.0 (Itasca, 1992) to

assess the stability of waste disposal boreholes in Tak granite. The Coulomb strength

criterion calibrated from the compressive strength test results and the uniaxial and

triaxial strength test data under elevated temperatures are applied to determine the

stability conditions of the rock around the boreholes. The stress distributions are

compared against the Coulomb criterion under isotropic and anisotopic stress states.

The main emphasis is placed at the borehole boundary when the deviation and shear

stress are greatest.

6.2 Numerical simulations

A finite difference analysis is performed to demonstrate the impact of the

temperatures on the granite strength around borehole. The boreholes are simulated

under temperatures ranging from 273 to 773 K. The simulated disposal depth is at

1,000 m with a borehole radius of 1 m. The analysis of the horizontal plane is made

in plane strain under isothermal condition, assuming that no nearby underground

structure. The mesh consists of 1400 elements covering an area of 100 m2. The finite

difference mesh and boundary conditions are designed as shown in Figure 6.1. The

material properties used in the simulations are shown in Table 6.1. Elastic modulus

(E) and Poisson’ s ratio (ν) are determined by assuming that the testing in granite is

56 linear elastic equation and all testing is under isotropic condition. Shear modulus (G)

and bulk modulus (K) are in Chapter V. The thermal properties of the granite are

given in Table 6.1.

The estimated in-situ stresses proposed by Brady and Brown (2006) are applie

here;

σv = 0.027⋅z (MPa) (6.1)

σh = k⋅σv (MPa) (6.2)

k = (1500/z)+0.5 (6.3)

where σv is vertical stress (MPa), σh is horizontal stress (MPa), and z is depth (m).

The value σh,x is the horizontal stress in x-direction and σh,y is the horizontal

stress in y-direction. The horizontal stress can be calculated by equation (6.2). In this

study, the in-situ stress in the simulation is defined as isotropic condition (σh,x = σh,y)

and anisotropic condition (σh,x = 3σh,y), as shown in Table 6.1.

The computer simulation is divided in 5 series, each series is under both

isotropic and anisotropic condition as shown in Table 6.2.

This study indicates the tangential stress at point A is maximum and

displacement at point B, shows the most movement. Tangential stress is maximum at

the borehole boundary. The radial stress is minimum at the borehole boundary. It is

reduced to zero, as shown in Figure 6.2.

57

0 1 2 4 53 (m)

0

1

2

4

5

3

(m) σh,y

σh,x

Figure 6.1 Finite difference mesh constructed to simulate a horizontal plane normal

to disposal borehole at depth of 1,000 m

58

σh,y

σh,x

(m)0 1 2 5

(m)

0

1

2

5

A

B

Figure 6.2 Tangential stress at point A and displacement at point B are of interest

here

59 Table 6.1 Material properties used in FLAC simulation.

σh,x (MPa)

σh,y (MPa)

Density (g/cc)

Thermal conductivity

(W/mK)

Specific heat

(MJ/m3K)

Thermal expansion

(K-1)

T (K)

K (GPa)

G (GPa)

E (GPa) ν c

(MPa) φ

(Degrees)

13.5 40.5

2.61 2.22 1.30 1.8×10-7

273 11.13 5.51 14.19 0.29 18.62 57.87 27 27 13.5 40.5 303 10.76 5.37 13.81 0.29 17.55 57.84 27 27 13.5 40.5 373 9.71 4.79 12.34 0.29 15.59 57.76 27 27 13.5 40.5 573 8.70 4.35 11.19 0.29 12.20 57.54 27 27 13.5 40.5 773 8.22 4.14 10.63 0.28 10.29 57.32 27 27

60 Table 6.2 The series of computer simulation.

Series T (K) σv (MPa σh,x (MPa) σh,y (MPa)

I 273

27

13.5 40.5 27 27

II 303 13.5 40.5 27 27

III 373 13.5 40.5 27 27

IV 573 13.5 40.5 27 27

V 773 13.5 40.5 27 27

61 6.3 Results

The FLAC simulations determine the stress distribution around the disposal

boreholes in granite. The factor of safety (FS) is determined for criterion by using

Coulomb criterion.

6.3.1 Coulomb criterion

The basic criterion for material failure in FLAC is the Mohr-Coulomb

relation, which is a linear failure surface corresponding to shear failure (fs or FS) as

(Hoek and Brown, 1980):

φφ− +σσ= N2cNf s

31 (6.4)

where Nφ = (1 + sinφ)/(1 - sinφ)

σ1 = major principal stress

σ3 = minor principal stress

φ = friction angle

c = cohesion

Figures 6.3 through 6.12 show the factor of safety (FS) under isotropic

(k = 1) and anisotropic (k = 3) condition under various temperature levels. The

results indicate that the factor of safety decreasing with increasing temperature. The

factor of safety at isotropic condition is higher than anisotropic condition, as show in

Table 6.3.

62 The factor of safety contour (fsc or FS,c)is determined by using

Coulomb criterion in the form of shear stress which can be expressed as (Hoek and

Brown, 1980):

canf 22sc +φσ−τ−= t' (6.5)

where θσ+θθσ−θσ=σ 22212

21122 coscossin2sin'

τ is shear stress

Figure 6.13 through 6.22 show the maximum stress under isotropic (k = 1) and

anisotropic (k = 3) condition various temperature level. The results indicate that the

distribution of the tangential stress is lowest at point A and at point B found the most

displacement of the storage around borehole boundary. When horizontal stress in y-

direction is three time the horizontal stress in x-direction (k = 3); the tangential stress

at point A is slightly decreasing with increasing temperature. However, when the

horizontal stress in y-direction equal to horizontal stress in x-direction (k = 1);

tangential stress at point A with respect to temperature. The factor of safety at σv =

σh condition is greater than when displacement at point B reduce, which is about 50

% of the σv = 3σh condition, as shown in Table 6.3.

Tables 6.4 summarizes the simulation results for two conditions. The

simulation model using thermal condition at 773 kelvin and granite properties at room

temperature. For all case the magnitudes of the tangential stress at point A decrease

with time increase.

63

5.63

9.01

7.51 11.26

15.01

22.52

45.04

6.43

σh,y

σh,x

FLAC 4.00 (K = 1; T = 273 K)

Figure 6.3 Factor of safety contour under σh,y = σh,x at 273 K, FS,c = 5.63

FLAC 4.00 (K = 3; T = 273 K)σh,y

σh,x

2.90

3.31

3.87

4.64

11.6

23.2

5.80 7.73

Figure 6.4 Factor of safety contour under σh,y = 3σh,x at 273 K, FS,c = 2.90

64

FLAC 4.00 (K = 1; T = 303 K)σh,y

σh,x

5.50

6.88

9.17

13.75

27.50


FLAC 4.00 (K = 3; T = 303 K)σh,y

σh,x

2.833.30

3.96

4.95

6.60

9.91

19.81


65

FLAC 4.00 (K = 1; T = 373 K)σh,y

σh,x

5.19 6.23

7.7910.38

15.57

31.14


FLAC 4.00 (K = 3; T = 373 K)σh,y

σh,x

2.693.23

4.045.38

8.07

16.14


66

FLAC 4.00 (K = 1; T = 573 K)σh,y

σh,x

4.67

5.84

7.78

11.68

23.35


FLAC 4.00 (K = 3; T = 573 K)σh,y

σh,x

2.442.75

3.14

3.66

7.3210.98

21.96

5.494.39


67

FLAC 4.00 (K = 1; T = 773 K)σh,y

σh,x

4.36

4.985.81

6.98

8.72

11.63


FLAC 4.00 (K = 3; T = 773 K)σh,y

σh,x

2.272.593.033.63

4.54

6.059.08


68

FLAC 4.00 (K = 1; T = 273 K)σh,y

σh,x

55 MPa

5045 40

35

30

25

Figure 6.13 Maximum stress under isotropic condition at 273 K

FLAC 4.00 (K = 3; T = 273 K)σh,y

σh,x

100 MPa

8060

40

20

Figure 6.14 Maximum stress under anisotropic condition at 273 K

69

FLAC 4.00 (K = 1; T = 303 K)σh,y

σh,x

55 MPa

5045 40

35

30

25


FLAC 4.00 (K = 3; T = 303 K)σh,y

σh,x

100 MPa

8060

40

20


70

FLAC 4.00 (K = 1; T = 373 K)σh,y

σh,x

55 MPa

5045 40

35

30

25


FLAC 4.00 (K = 3; T = 373 K)σh,y

σh,x

100 MPa

8060

40

20


71

FLAC 4.00 (K = 1; T = 573 K)σh,y

σh,x

55 MPa

5045 40

35

30

25


FLAC 4.00 (K = 3; T = 573 K)σh,y

σh,x

100 MPa

8060

40

20


72

FLAC 4.00 (K = 1; T = 773 K)σh,y

σh,x

55 MPa

5045 40

35

30

25


FLAC 4.00 (K = 3; T = 773 K)σh,y

σh,x

100 MPa

8060

40

20


73 Table 6.3 The results in FLAC simulation.

K (σh/σv) σh,x (MPa) σh,y (MPa) T (K) σ1,A (MPa) σ1,UCS (MPa) Displacement

s,B (mm) FS,c

3 13.5 40.5

273 107.2 131 5.84 2.90 303 107.2 119 6.01 2.83 373 107.2 104 6.38 2.69 573 106.5 90 7.19 2.44 773 105.4 73 7.84 2.27

1 27 27

273 53.5 131 2.69 5.63 303 53.5 119 2.77 5.50 373 53.5 104 2.93 5.19 573 53.5 90 3.31 4.67 773 53.5 73 3.59 4.36

74 Table 6.4 Tangential stress at point A from FLAC simulation in long-term.

K (σh/σv) σh,x (MPa) σh,y (MPa) T (K) (10

cycle/day) σ1,A (MPa)

3 13.5 40.5 773

3 108.2 10 107.5 30 107.3 60 107.3 120 107.3 215 107.3 365 107.3

1 27 27 773

3 54.5 10 53.7 30 53.6 60 53.6 120 53.6 215 53.6 365 53.6

CHAPTER VII

DISCUSSIONS AND CONCLUSIONS

7.1 Discussions and conclusions

This study experimentally determines the granite strengths under different

constant temperatures from 273 K to 773 K. For each temperature level the testing is

assumed to be under isothermal condition (constant temperature with time during

loading). For this simplified approach the granite specimens subject to different

temperatures have been taken as different materials. As a result the induced thermal

stress or thermal energy imposed on the granite specimens has not been explicitly

incorporated into the initial strength calculation. This approach is different from the

conventional thermo-mechanical analysis, but it allows a simple assessment of

temperature effect on the rock strength.

The decrease of the granite strength as the temperature increases suggest that

the applied thermal energy before the mechanical testing makes the granite weaker.

The result agrees with those obtained by Araújo et al. (1997), Dwivedi et al. (2008),

Inada et al. (1997) and Xu et al. (2008). More plastic – failing at lower stress and

higher strain with lower elastic and shear moduli an observed which agree with those

obtained by Wai and Lo (1982), Xu et al. (2009) and Zhao et al. (2012). The

temperature effect is larger when granite is under higher mean stresses. In order to

consider the temperature dependency of the failure stress and strain and elastic

properties, the strain energy density concept is applied. Assuming that the granite is

76 non-linearly elastic before failure, the distortional strain energy (Wd) at failure can be

calculated as a function of mean strain energy (Wm). In Figure 5.7 at a given Wm the

Wd decreases with increasing temperature. The differences of Wd from one

temperature to the other therefore correspond to the difference of thermal energy

imposed on the specimens.

The single multi-axial strength criterion (5.29) for granite under various

confining pressures and temperatures implicitly considers the effect of the thermal

energy by incorporating the empirical equations between the elastic parameters and

temperature into the Wd – Wm relation (Figure 5.7). The strain energy criterion

agrees well with the strength results from different temperature levels (Sriapai et al.,

2012). Since the analysis is intended to determine the short-term strength, the long-

term deformations induced by the mechanical and thermal loadings are not considered

here.

A finite difference analysis is performed to demonstrate the impact of the

temperature on the granite around the disposal borehole subject to uniform and

deviatoric field stresses. The analysis is made in plane strain under isothermal

condition, and assuming that no nearby underground structure. Coulomb criterion is

developed from triaxial test results, and hence, they can predict the stability condition.

The computer simulation results show that the factor of safety decrease and

displacement at point B increase with increasing temperature when isotropic and

anisotropic condition. The factor of safety however is less than 1 when the

temperature is more than 373 kelvin under anisotropic condition. The factor of safety

under isotropic condition is higher and displacement is less than when compared with

anisotropic condition. Under field condition, deep borehole is under hydrostatic

77 condition. In long-term, the magnitudes of the tangential stress at point A decrease

with the withdrawal rate. However, tangential stress at point A is constant or steady-

state in long-term.

7.2 Recommendations for future studies

The uncertainties and adequacies of the research investigation and results

discussed above lead to the recommendations for further studies. More laboratory

testing should be performed using the higher temperatures with larger specimens.

The effect of temperature should be considered on the true triaxial compressive test

and observe rock fracture characteristics. The effect of loading rate and stress parth

should be investigated experimentally in order to assess the disposal borehole stability

under long-term period. The effect of friction at the interface between the loading

platen and rock surfaces should be investigated.

REFERENCES

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on the influence of temperature on the mechanical properties of reservoir

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34(3–4): 298.e1-298.e16.

ASTM D3967-95a. Standard test method for splitting tensile strength of intact rock

core specimens. In Annual Book of ASTM Standards (Vol. 04.08).

Philadelphia: American Society for Testing and Materials.

ASTM D4543-85. Standard practice for preparing rock core specimens and

determining dimensional and shape tolerances. Annual Book of ASTM

Standards (Vol. 04.08). Philadelphia: American Society for Testing and

Materials.

Bergman M.S. (1980). Nuclear waste disposal. Subsurf Space 2: 791–1005.

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mechanical properties of Indian and other granites. International Journal of

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Etienne, F. H. and Houpert, R. (1989). Thermally induced microcracking in granites:

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alternative for high-level radioactive waste? Waste Management 19(3): 207-

211.

79

Gibb, F. G. F. (2003). Granite Recrystallization – The Key to an Alternative Strategy

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BIOGRAPHY

Mr. Kittikron Rodklang was born on July 27, 1989 in Ubon Ratchathani

province, Thailand. He received his Bachelor’s Degree in Science (Geosciences) from

Mahidol University, Kanchanaburi Campus in 2011. For his post-graduate, he study

with a Master’s degree in the Geological Engineering Program, Institute of

Engineering, Suranaree university of Technology. During graduation, 2012-2014, he

was a part time worker in position of research assistant at the Geomechanics Research

Unit, Institute of Engineering, Suranaree University of Technology.

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