Upload
others
View
1
Download
0
Embed Size (px)
Citation preview
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
TABLE OF CONTENTS (Continued)
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
TABLE OF CONTENTS (Continued)
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
4.9 Stress-strain curves obtained from some granite specimens at 303 K.
Numbers in brackets indicate [σ1, σ2, σ3] at failure 32
4.10 Stress-strain curves obtained from some granite specimens at 373 K.
Numbers in brackets indicate [σ1, σ2, σ3] at failure 33
4.11 Stress-strain curves obtained from some granite specimens at 573 K.
Numbers in brackets indicate [σ1, σ2, σ3] at failure 34
4.12 Stress-strain curves obtained from some granite specimens at 773 K.
Numbers in brackets indicate [σ1, σ2, σ3] at failure 35
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
LIST OF FIGURES (Continued)
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
6.5 Factor of safety contour under isotropic condition at 303 K 64
6.6 Factor of safety contour under anisotropic condition at 303 K 64
6.7 Factor of safety contour under isotropic condition at 373 K 65
6.8 Factor of safety contour under anisotropic condition at 373 K 65
XII
LIST OF FIGURES (Continued)
Figure Page
6.9 Factor of safety contour under isotropic condition at 573 K 66
6.10 Factor of safety contour under anisotropic condition at 573 K 66
6.11 Factor of safety contour under isotropic condition at 773 K 67
6.12 Factor of safety contour under anisotropic condition at 773 K 67
6.13 Maximum stress under isotropic condition at 273 K 68
6.14 Maximum stress under anisotropic condition at 273 K 68
6.15 Maximum stress under isotropic condition at 303 K 69
6.16 Maximum stress under anisotropic condition at 303 K 69
6.17 Maximum stress under isotropic condition at 373 K 70
6.18 Maximum stress under anisotropic condition at 373 K 70
6.19 Maximum stress under isotropic condition at 573 K 71
6.20 Maximum stress under anisotropic condition at 573 K 71
6.21 Maximum stress under isotropic condition at 773 K 72
6.22 Maximum stress under anisotropic condition at 773 K 72
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
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
milli-strains0 5 10 15-5-10 20
ε1εvε2ε3
T = 273 K [277.8, 12.0, 12.0] σ
1 (M
Pa)
150
300
250
200
100
50
0
milli-strains0 5 10 15-5-10 20
ε1εvε2ε3
T = 273 K [131.1, 0.0, 0.0] σ
1 (M
Pa)
150
300
250
200
100
50
0
milli-strains0 5 10 15-5-10 20
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
milli-strains0 5 10 15-5-10 20
T = 303 K [269.2, 12.0, 12.0]ε1εvε2ε3
σ1 (
MPa
)
150
300
250
200
100
50
0
milli-strains0 5 10 15-5-10 20
T = 303 K [161.1, 3.0, 3.0]
σ1 (
MPa
)
150
300
250
200
100
50
0
milli-strains0 5 10 15-5-10 20
ε1εvε2ε3
T = 303 K [118.8, 0.0, 0.0] σ
1 (M
Pa)
150
300
250
200
100
50
0
milli-strains0 5 10 15-5-10 20
ε1εvε3
Figure 4.9 Stress-strain curves obtained from some granite specimens 303 K.
Numbers in brackets indicate [σ1, σ2, σ3] at failure.
33
T = 373 K [143.4, 3.0, 3.0]
ε1εvε2ε3
σ1 (
MPa
)
150
300
250
200
100
50
0
milli-strains0 5 10 15-5-10 20
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
milli-strains0 5 10 15-5-10 20
ε1εvε2ε3
T = 373 K [104.1, 0.0, 0.0] σ
1 (M
Pa)
150
300
250
200
100
50
0
milli-strains0 5 10 15-5-10 20
ε1εvε3
Figure 4.10 Stress-strain curves obtained from some granite specimens 373 K.
Numbers in brackets indicate [σ1, σ2, σ3] at failure.
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
milli-strains0 5 10 15-5-10 20
T = 573 K [127.7, 3.0, 3.0]
σ1 (
MPa
)
150
300
250
200
100
50
0
milli-strains0 5 10 15-5-10 20
ε1εvε2ε3
T = 573 K [90.4, 0.0, 0.0] σ
1 (M
Pa)
150
300
250
200
100
50
0
milli-strains0 5 10 15-5-10 20
ε1
εvε3
Figure 4.11 Stress-strain curves obtained from some granite specimens 573 K.
Numbers in brackets indicate [σ1, σ2, σ3] at failure.
35
T = 773 K [104.2, 3.0, 3.0]
ε1εvε2ε3
σ1 (
MPa
)
150
300
250
200
100
50
0
milli-strains0 5 10 15-5-10 20
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
milli-strains0 5 10 15-5-10 20
ε1εvε3
T = 773 K [157.9, 7.0, 7.0]
σ1 (
MPa
)
150
300
250
200
100
50
0
milli-strains0 5 10 15-5-10 20
ε1εvε3ε2
Figure 4.12 Stress-strain curves obtained from some granite specimens 773 K.
Numbers in brackets indicate [σ1, σ2, σ3] at failure.
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
Hoek & Brown criterion
φ = -0.001×T + 58.70(R2 = 0.633)
Hoek & Brown criterion
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
Figure 6.5 Factor of safety contour under σh,y = σh,x at 303 K, FS,c = 5.50
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
Figure 6.6 Factor of safety contour under σh,y = 3σh,x at 303 K, FS,c = 2.83
65
FLAC 4.00 (K = 1; T = 373 K)σh,y
σh,x
5.19 6.23
7.7910.38
15.57
31.14
Figure 6.7 Factor of safety contour under σh,y = σh,x at 373 K, FS,c = 5.19
FLAC 4.00 (K = 3; T = 373 K)σh,y
σh,x
2.693.23
4.045.38
8.07
16.14
Figure 6.8 Factor of safety contour under σh,y = 3σh,x at 373 K, FS,c = 2.69
66
FLAC 4.00 (K = 1; T = 573 K)σh,y
σh,x
4.67
5.84
7.78
11.68
23.35
Figure 6.9 Factor of safety contour under σh,y = σh,x at 573 K, FS,c = 4.67
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
Figure 6.10 Factor of safety contour under σh,y = 3σh,x at 573 K, FS,c = 2.44
67
FLAC 4.00 (K = 1; T = 773 K)σh,y
σh,x
4.36
4.985.81
6.98
8.72
11.63
Figure 6.11 Factor of safety contour under σh,y = σh,x at 773 K, FS,c = 4.36
FLAC 4.00 (K = 3; T = 773 K)σh,y
σh,x
2.272.593.033.63
4.54
6.059.08
Figure 6.12 Factor of safety contour under σh,y = 3σh,x at 773 K, FS,c = 2.27
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
Figure 6.15 Maximum stress under isotropic condition at 303 K
FLAC 4.00 (K = 3; T = 303 K)σh,y
σh,x
100 MPa
8060
40
20
Figure 6.16 Maximum stress under anisotropic condition at 303 K
70
FLAC 4.00 (K = 1; T = 373 K)σh,y
σh,x
55 MPa
5045 40
35
30
25
Figure 6.17 Maximum stress under isotropic condition at 373 K
FLAC 4.00 (K = 3; T = 373 K)σh,y
σh,x
100 MPa
8060
40
20
Figure 6.18 Maximum stress under anisotropic condition at 373 K
71
FLAC 4.00 (K = 1; T = 573 K)σh,y
σh,x
55 MPa
5045 40
35
30
25
Figure 6.19 Maximum stress under isotropic condition at 573 K
FLAC 4.00 (K = 3; T = 573 K)σh,y
σh,x
100 MPa
8060
40
20
Figure 6.20 Maximum stress under anisotropic condition at 573 K
72
FLAC 4.00 (K = 1; T = 773 K)σh,y
σh,x
55 MPa
5045 40
35
30
25
Figure 6.21 Maximum stress under isotropic condition at 773 K
FLAC 4.00 (K = 3; T = 773 K)σh,y
σh,x
100 MPa
8060
40
20
Figure 6.22 Maximum stress under anisotropic condition at 773 K
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
Araújo, R. G. S., Sousa, J. L. A. O. and Bloch, M. (1997). Experimental investigation
on the influence of temperature on the mechanical properties of reservoir
rocks. International Journal of Rock Mechanics and Mining Sciences
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.
Dwivedi, R. D., Goel, R. K., Prasad, V. V. R. and Sinha, A. (2008). Thermo-
mechanical properties of Indian and other granites. International Journal of
Rock Mechanics and Mining Sciences 45(3): 303-315.
Etienne, F. H. and Houpert, R. (1989). Thermally induced microcracking in granites:
characterization and analysis. International Journal of Rock Mechanics
and Mining Sciences 26(2): 125-134.
Gibb, F. G. F. (1999). High-temperature, very deep, geological disposal: a safer
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
for HLW Disposal? In Proceedings of the Twenty-seventh Material
Research Society Symposium, Scientific Basis for Nuclear Waste
Management (vol. 807, pp. 937-942). Warrendale, Pa. MRS Online
Proceedings Library.
Gibb, F. G. F., McTaggart, N. A., Travis, K. P., Burley, D. and Hesketh, K. W.
(2008). High-density support maes: Key to the deep borehole disposal of
spent nuclear fuel. Journal of Nuclear Materials 374(3): 370-377.
Gibb, F. G. F., Taylor, K. J. and Burakov, B. E. (2008). The ‘granite encapsulation’
route to the safe disposal of Pu and other actinides. Journal of Nuclear
Materials 374(3): 364-369.
Heuze, F. E. (1981). On the geotechnical modelling of high-level nuclear waste
disposal by rock melting. Lawrence Livermore National Laboratory
UCRL-53183.
Hoek, E. and Brown, E. T. (1990). Underground excavations in rock (pp. 102-
106). London. UK. Institution of Mining and Metallurgy.
Inada, Y., Kinoshita, N., Ebisawa, A. and Gomi, S. (1997). Strength and deformation
characteristics of rocks after undergoing thermal hysteresis of high and low
temperatures. International Journal of Rock Mechanics and Mining
Sciences 34(3-4): 140.e1-140.e14
Itasca. (1992). User Manual for FLAC-Fast Langrangian Analysis of Continua,
Version 4.0. Itasca Consulting Group Inc., Minneapolis, Minnesota.
Jaeger, J. C., Cook, N. G. W., Zimmerman, R. W. (2007). Fundamentals of Rock
Mechanics Fourth Edition (pp 17-73). Blackwell Publishing, Oxford.
80
Logan, S. E. (1973). Deep self-burial of radioactive wastes by rock melting capsules.
Nuclear Technology 21:111-124.
Mahawat, C., Atherton, M. P. A. and Brotherton, M. S. (1990). The Tak Batholith,
Thailand: the evolution of contrasting granite types and implications for
tectonic setting. Journal of Southeast Asian Earth Sciences 4(1): (11-27).
Shimada, M. and Liu, J. (2000). Temperature dependence of strength of rock under
high confining pressure. Disaster Prevention Research Institute Annuals
43B-1:75-84.
Sriapai, T., Walsri, C. and Fuenkajorn, K. (2013). True-triaxial compressive strength
of Maha Sarakham salt. International Journal of Rock Mechanics and
Mining Sciences 61:256-265.
Sundberg, J., Back, P. E., Christiansson, R., Hökmark, H., Ländell, M. and Wrafter, J.
(2009). Modelling of thermal rock mass properties at the potential sites of a
Swedish nuclear waste repository. International Journal of Rock
Mechanics and Mining Sciences. 46(6): 1042-1054.
US Department of Energy. (1980). Statement of the position of the United States
Department of Energy in the matter of rulemaking on the storage and disposal
of nuclear wastes, Report DOE/NE-0007.
Vosteen, H. and Schellschmidt, R. (2003). Influence of temperature on thermal
conductivity, thermal capacity and thermal diffusivity for different types of
rock. Physics and Chemistry of the Earth 28(9-11): 499-509.
Wai, R. S. and Lo, K. Y. (1982). Temperature effects on strength and deformation
behaviour of rocks in Southern Ontario. Canadian Geotechnical Journal
19(3): 307-319
81
Witherspoon, P. A., Gale, J. E. and Cook, N. G. W. (1979). Investigations in granite
at stripa, Sweden for nuclear waste storage. Annual Meeting of the
American Nuclear Society. June 3-8, Atlanta, Georgia.
Xu, X., Ga, F., Shen, X. and Xie, H. (2008). Mechanical characteristics and
microcosmic mechanisms of granite under temperature loads. Journal of
China University of Mining and Technology 18(3): 413-417.
Xu, X., Kang, Z., Ming, j., Ge, W. and Jing, C. (2009). Research of microcosmic
mechanism of brittle-plastic transition for granite under high temperature.
Procedia Earth and Planetary Science 1(1): 432-437.
Zhao, Y., Wan, Z., Feng, Z., Yang, D., Zhang, Y. and Qu, F. (2012). Triaxial
compression system for rock testing under high temperature and high pressure.
International Journal of Rock Mechanics and Mining Sciences 52: 132-
138.
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.