angle between maximum principal stress and vertical. [°]. βFB ...... fractures is optical microscopy, which has typically a maximal magnification of 500x. ...... Thus an element is eligible for activation when the shear stress in a collocation point.
INFLUENCE OF PLANAR HETEROGENEITIES ON THE FRACTURE BEHAVIOUR OF ROCK
Bjorn DEBECKER
Promotor: Prof. A. Vervoort
Proefschrift voorgedragen tot het behalen van de graad van Doctor in de ingenieurswetenschappen
Leden van de examencommissie: Prof. C. Vandecasteele, voorzitter Prof. D. Van Gemert Prof. M. Wevers Prof. C. Schroeder (ULB) Prof. R. Holt (NTNU, Trondheim)
Oktober 2009
© 2009 Katholieke Universiteit Leuven, Groep Wetenschap & Technologie, Arenberg Doctoraatsschool, W. de Croylaan 6, 3001 Heverlee, België Alle rechten voorbehouden. Niets uit deze uitgave mag worden vermenigvuldigd en/of openbaar gemaakt worden door middel van druk, fotokopie, microfilm, elektronisch of op welke andere wijze ook zonder voorafgaandelijke schriftelijke toestemming van de uitgever. All rights reserved. No part of the publication may be reproduced in any form by print, photoprint, microfilm, electronic or any other means without written permission from the publisher.
D/2009/7515/115 ISBN 978-94-6018-132-0
Acknowledgments First I would like to thank my girlfriend Maaike, who was and is with me from the beginning to the end of this thesis. Without her endless support, this would not have been possible. I am of course grateful to my promoter, prof. Vervoort. The many discussions we had on the research were indispensible for the outcome of this work. Also to some other members of my jury, I am grateful for their valuable input during the years, which contributed to the quality of this work: prof. Wevers, prof. Van Gemert and prof. Holt. I would like to thank my entire jury for reading the draft version of this thesis and for helping me with their recommendations. I would also like to thank prof. John Napier for the enlightening discussions we had on his code and its applications. Another warm thanks goes to Alexandre Lavrov at SINTEF, Norway for providing me with some very valuable input at key moments in my research. Spasieba. For the experimental work, I got help from several people throughout the years, and I also wish to thank them: Stephan, Frank, Luc, Jean, and Roger. No research on an island, the research years would not have been the same without my colleagues: thanks to Annelies, Abbas, Patrick and Colette. I would also like to thank my parents, family, friends and other colleagues at Bouwkunde, especially for the last six months when more or less all I could talk about was the fracturing of rock. My last thoughts here go to two friends somewhere on the road who placed this thesis, among other things in the larger framework of life.
Bjorn Debecker Leuven, October 2009.
Abstract (in English) Fracture patterns in rock are often a complex combination of fractures along preexisting planar heterogeneities and fractures in other directions. However for layered anisotropic rock, most research is limited to the determination of the anisotropic strength properties on sample scale. This study relates the strength anisotropy on μ-scale in layered rock to the strength anisotropy on sample scale. It also relates the strength anisotropy to the deformation behaviour and to the fracture pattern. The work presented in this thesis is based on two complementary approaches. The first part is the experimental work, where different loading tests are combined with observation techniques to gain insight in the fracture mechanisms in layered rock. In the second part, numerical simulations of these tests are discussed and related to the experimental results, providing a more detailed knowledge. The aim of this study is twofold. The main aim is to contribute further to the understanding of the fundamental processes in layered rock fracturing. The different features and properties in the fracture process in layered rock are identified and described. With this knowledge, a framework can be constructed that relates the key properties in layered rock. The second aim of this study is to develop and to validate a methodology for the numerical simulation of fracturing around planar heterogeneities. For this purpose, a numerical model is constructed that can simulate fracture growth in random directions, as well as parallel to the layers. Conclusions can be drawn based on both approaches (experimental and numerical), and with their combined knowledge, a general framework can be constructed. First, on the experimental side, the analysis of digital video recorded images is successful to map the evolution of the fracture pattern and in most cases to identify the growth direction of the different fractures. These images are insufficient to determine the failure mode of the fractures. To do this, the fracture orientation and the orientation of the load direction relative to the layer direction have to be taken into account. In addition, a clear correlation is made between the occurrence of new fractures and variations in the external load. An algorithm is also developed to perform Acoustic Emission (AE)-localisation in a layered medium, where the wave velocities are transverse isotropic. The results of AElocalisation are in agreement with the visual observations and provide information on the interior of the sample. However, there is room for several improvements to enhance the quality of this localisation.
iv Second, in the part on the numerical simulations, the fracture process is successfully simulated. A first study on a sample with one planar heterogeneity identifies the importance of layer stiffness to the stress distribution and consequently to the fracture pattern. For the simulations of layered slate rock, the input parameters are based on experimentally determined properties. The results between experiments and simulations correspond well. Numerical configurations also confirm the failure modes. Detailed illustrations from simulated fracture patterns are generated. These illustrations show different fracture features, which are also experimentally observed (e.g. transition of failure mode, sample crushing,…). Finally, with the combined knowledge from simulations and experiments, a framework is formulated that relates the key properties in layered rock. Depending on the degree of μ-scale anisotropy, the sample strength and fracture pattern are determined by the load configuration on the one hand, and by the µ-scale strength in layer direction and/or in non-layer direction, on the other hand. It is also found that there are two transitions of dominant fracture mechanism in function of the load configuration. Rock fracturing occurs typically around excavations or boreholes, creating hazardous situations. Therefore, knowledge on the fracture processes in rock is essential for engineering applications in this area. The results from this study can be used in future applications where layered rock or planar heterogeneities are involved.
Abstract (in Dutch) Invloed van vlakke heterogeniteiten op het breukgedrag van gesteente Breukpatronen in gesteente zijn meestal een complexe combinatie van breuken langs bestaande vlakke heterogeniteiten en breuken in andere richtingen. Voor gelaagd anisotroop gesteente is het meeste onderzoek in de literatuur beperkt tot het bepalen van de anisotrope sterkteparameters op laboschaal. Deze studie relateert de anisotrope sterkte op microschaal aan de sterkte op laboschaal. Daarenboven wordt de anistropie van de sterkte ook in verband gebracht met het vervormingsgedrag en met het breukpatroon. Het werk in dit doctoraat is gebaseerd op twee complementaire benaderingen. Ten eerste is er het experimentele gedeelte waarbij verschillende belastingstesten gecombineerd worden met observatietechnieken. In het tweede gedeelte worden numerieke simulaties van deze testen besproken en gerelateerd aan de experimentele resultaten. Zo krijgt men een gedetailleerde kennis van de achterliggende breukmechanismen. Het doel van deze studie is tweeledig. De voornaamste betrachting is een verdere bijdrage te leveren aan de kennis over de fundamentele breukprocessen in gelaagd gesteente. De verschillende aspecten van het breukgedrag in gelaagd gesteente worden geïdentificeerd en beschreven. Met deze kennis kan een schema worden opgesteld dat de verschillende parameters in gelaagd gesteente met elkaar in verband brengt. Het tweede doel van dit doctoraat is om een methodologie te ontwerpen en te valideren voor de simulatie van breukgedrag rond vlakke heterogeniteiten. Hiervoor is een numeriek model opgesteld dat breukgroei in willekeurige richtingen alsook langsheen gelaagdheden kan simuleren. De conclusies van dit werk komen uit de twee onderdelen (experimenten en simulaties), en met de gecombineerde kennis kan een algemeen schema worden opgesteld. Ten eerste, in het experimenteel gedeelte is de analyse van digitale video-opnames van breukpatronen geschikt voor het bepalen van de groeirichting van de verschillende breuken. Deze beelden volstaan niet om rechtstreeks de faalmodes van de verschillende breuken te bepalen. Om dit te doen, neemt men ook de oriëntatie van de breuken en de oriëntatie van de belastingsrichting tegenover de gelaagdheid in rekening. Daarnaast kan er een duidelijke correlatie gemaakt worden tussen het ontstaan van afzonderlijke breukjes en variaties in de externe belasting. Er is ook een algoritme ontwikkeld voor Akoestische Emissie (AE) – lokalisatie in een gelaagd medium, waar de snelheiden transversaal isotroop zijn. De resultaten van de
vi lokalisatie stemmen overeen met de visuele observaties en leveren ook informatie over de binnenkant van het monster. Er is echter nog ruimte om de kwaliteit van deze lokalisatie te verbeteren. Ten tweede, in het gedeelte van de numerieke simulaties, is het breukgedrag met succes gesimuleerd. Een eerste studie van een monster met een enkele vlakke heterogeniteit toont het belang van de stijfheid van de gelaagdheid aan. Daarbij wordt ook de invloed hiervan op de spanningsverdeling en het breukgedrag duidelijk. De input parameters bij de simulaties van gelaagd gesteente zijn gebaseerd op experimentele resultaten. De resultaten van experimenten en simulaties komen goed overeen. De numerieke simulaties bevestigen ook de faalmodes. Er worden eveneens gedetailleerde illustraties gegenereerd van breukpatronen. Deze afbeeldingen tonen duidelijk de verschillende kenmerken van het breukgedrag, die ook in de experimenten worden waargenomen (bijv. verandering van faalmode, vergruizen van het monster,…) Tenslotte, met de gecombineerde kennis van simulaties en experimenten wordt er een schema opgesteld dat de belangrijkste eigenschappen in gelaagd gesteente met elkaar in verband brengt. Afhankelijk van de graad van anisotropie op microschaal, zijn sterkte en breukpatroon bepaald door enerzijds de belastingsconfiguratie en anderzijds de sterkte op microschaal in de richting van de gelaagdheid en/of in andere richtingen. Verder zijn er ook nog twee zones waar het dominante breukmechanisme verandert in functie van de belastingsconfiguratie. Het breken van gesteente komt veelvuldig voor rond uitgravingen en rond boorgaten, en creëert daar gevaarlijke situaties. Daarom is de kennis van breukprocessen in gesteente essentieel voor ingenieurstoepassingen in dit gebied. De resultaten van deze studie kunnen gebruikt worden in toekomstige toepassingen in gelaagd gesteente of rond vlakke heterogeniteiten.
List of Symbols and Abbreviations Symbols a b1 b2 coh coh// d kmax kn ks m nj rx’y’ s ti ű (t) ú (t) u(t) un us ux, uy v v v// vi vx’y’ x’ xi xref xα y’ yC yi yQ yref yα z’ zi
half crack length first layer plane quality parameter second layer plane quality parameter cohesion cohesion in layer direction distance to shear criterion maximum contact stiffness in the model normal contact stiffness shear contact stiffness mass normal vector component (j = y, z) direction of wave propagation shear strength arrival time at sensor i acceleration at time step t velocity at time step t displacement at time step t normal displacement at contact shear displacement at contact x- and y-displacement component wave velocity wave velocity in the direction normal to the layering wave velocity in the layer direction wave velocity in the direction from hit origin to sensor i wave velocity in the plane normal to layering Cartesian coordinate Cartesian coordinate of sensor i Cartesian coordinate in reference grid Cartesian coordinate of hit origin Cartesian coordinate local coordinate of collocation point Cartesian coordinate of sensor i local coordinate at displacement discontinuity Cartesian coordinate in reference grid Cartesian coordinate of hit origin Cartesian coordinate Cartesian coordinate of sensor i
[mm] [-] [MPa] [MPa] [MPa] [MPa] [Pa/m] [Pa/m] [Pa/m] [kg] [-] [-] [MPa] [μs] [m/s2] [m/s] [m] [m] [m] [mm] [mm/μs] [mm/μs] [mm/μs] [mm/μs] [mm/μs] [mm] [mm] [mm] [mm] [mm] [mm] [mm] [mm] [mm] [mm] [mm] [mm]
viii zref zα
Cartesian coordinate in reference grid Cartesian coordinate of hit origin
[mm] [mm]
ATDi,j C D Dx, Dy E F Ficontact Fiext Fn Fr Fs H I K0 K1 Ki Kic Kyz, Kzz L L’min Lmin M Mn N Q S T UCS UCS
arrival time difference between sensors i and j stiffness matrix diameter of a circular sample x- and y-component of displacement discontinuity Young’s modulus vertical load contact force at gridpoint i external applied force at gridpoint i normal contact force Felicity ratio shear contact force average arc size/thickness moment of inertia self-effect influence coefficient influence coefficient of second collocation point stress intensity factor for mode i (i = 1, 2, 3) critical stress intensity factor for mode i (i = 1, 2, 3) influence coefficient of far field stresses thickness of circular sample minimal distance between field point and structural point minimal distance between two field points torque number of sectors in annulus n number of annuli triangle quality parameter element length between two structural points surface energy uniaxial compressive strength uniaxial compressive strength normal to the layer direction
[μs] [GPa] [mm] [mm] [GPa] [N] [N] [N] [N] [-] [N] [mm] [kg m2] [MPa/mm] [MPa/mm] [MPa mm0.5] [MPa mm0.5] [MPa/mm] [mm] [mm] [mm] [N m] [-] [-] [-] [mm] [mJ/mm2] [MPa] [MPa]
α β βFB δ γ εH εij εV θ ν ξ ρ σ1 σ2 σ3 σI1,max σII1,KE
minimum angle in mesh triangle angle between maximum principal stress and vertical angle between failure bands and normal to loading direction Thomsen parameter for elastic anisotropy angle between schistosity and direction of wave propagation horizontal strain components of the strain tensor (i,j = x, y, z) vertical strain angle between normal to the schistosity and loading direction Poisson’s ratio geometry factor for stress intensity factor density minimum principal stress intermediate principal stress maximum principal stress peak value for σ1 in the first loading cycle σ1 in the second loading cycle at the start of acoustic activity
[°] [°] [°] [-] [°] [-] [-] [-] [°] [-] [-] [kg/m3] [MPa] [MPa] [MPa] [MPa] [MPa]
List of Symbols and Abbreviations
ix
σij σmin σn σT σv σv,max τ τhor φ φ// ψ
components of the stress tensor (i,j = x, y, z) minimal tensile stress for crack propagation in a plate normal stress tensile strength vertical stress vertical peak stress during uniaxially loading shear stress shear stress in the horizontal direction friction angle friction angle in layer direction angle between wave direction and plane normal to layering
[MPa] [MPa] [MPa] [MPa] [MPa] [MPa] [MPa] [MPa] [°] [°] [°]
Δt Θ Ω
distinct time increment rotation allocation efficiency
[s] [°] [μs]
Abbreviations AE AR BEM CT DD-BEM DEM DIC ESPI FDM FEM SEM UW μCT
Acoustic Emission Anisotropic Ratio Boundary Element Method Computed Tomography Displacement Discontinuity Boundary Element Method Distinct Element Method Digital Image Correlation Electronic Speckle Pattern Interferometry Finite Difference Method Finite Element Method Scanning Electron Microscopy Ultrasonic Wave Microfocus Computed Tomography
Table of Contents Acknowledgments Abstract (in English) Abstract (in Dutch)
i iii v
List of Symbols and Abbreviations
vii
Table of Contents
xi
1
Introduction 1.1 General context 1.1.1 Fracturing around planar heterogeneities 1.1.2 Research Unit Mining 1.2 Aim of the study 1.3 Outline of the research
1 1 1 2 3 4
2
Literature Survey – State of the Art 2.1 Anisotropy and heterogeneity in rock 2.1.1 Heterogeneity 2.1.2 Anisotropy 2.1.3 Areas of relevance 2.2 Fracture theory 2.2.1 Fracture process in brittle rock 2.2.1.1 Microcracks: origin 2.2.1.2 Griffith’s theory, fracture modes and stress intensity factors 2.2.1.3 Coalescence of fractures into fracture patterns 2.2.2 Strength and fracture process in transverse isotropic rock 2.2.2.1 Strength models 2.2.2.2 Experimental observations 2.3 Fracture observational techniques 2.3.1 Sample surface observation techniques 2.3.1.1 Direct visual observation 2.3.1.2 Digital image correlation 2.3.1.3 Interferometry 2.3.2 Inside observation techniques 2.3.2.1 Microscopy 2.3.2.2 Microfocus computed tomography 2.3.2.3 Acoustic emission 2.3.2.4 Ultrasonic wave testing 2.3.3 Conclusion
7 7 8 9 9 11 12 12 13 15 17 17 20 22 22 22 23 24 25 25 26 27 29 30
xii 2.4 Numerical modelling of fracture process in rock 2.4.1 Finite element method (FEM) 2.4.2 Finite difference method (FDM) 2.4.3 Distinct element method (DEM) 2.4.4 Displacement discontinuity boundary element method (DD-BEM) 2.4.5 Others 2.4.5.1 Lattice Model 2.4.5.2 Fuzzy sets
31 32 34 35 37 38 38 39
3
Experimental Work 3.1 Preceding study on fracture patterns in siltstone 3.1.1 Description of the rock material and experimental set-up 3.1.2 Discussion of the anisotropy of the fracture patterns 3.1.3 Analysis of the fracture lengths 3.1.4 Influence of alternating mineralogical composition 3.1.5 Conclusions 3.2 Determination of the rock mechanical parameters of slate 3.2.1 Description of the rock material 3.2.2 UCS tests 3.2.3 Shear loading tests 3.2.4 Brazilian tensile tests 3.3 Fracture process in diametrically loaded slate discs by image analysis 3.3.1 Experimental set-up 3.3.2 Discussion on strength anisotropy and fracture patterns 3.3.3 Conclusions 3.4 Fracture process in uniaxial load tests by image analysis and AE analysis 3.4.1 Experimental set-up: image analysis and AE 3.4.2 Fracture process observation by image analysis 3.4.3 Results of the AE localisation 3.4.4 Discussion and concluding remarks 3.5 Conclusions on the experimental work
41 41 41 44 48 50 51 52 52 53 56 58 59 59 60 69 70 70 72 78 80 82
4
Numerical Simulations 83 4.1 Description of the numerical codes 83 4.1.1 DIGS 83 4.1.2 UDEC 87 4.1.3 Tessellation 91 4.2 Influence of stiffness on fracture initiation 93 4.2.1 Implementation of the model with one central discontinuity 93 4.2.2 Model of discontinuity activation only (Model 1) 95 4.2.2.1 UDEC (stiffness is modelled) 95 4.2.2.2 DIGS (stiffness is not modelled) 97 4.2.3 Model of discontinuity activation and new crack growth (Model 2) 99 4.2.3.1 UDEC (stiffness is modelled) 99 4.2.3.2 DIGS (stiffness is not modelled) 101 4.2.4 Comparison of the results from both codes 103 4.2.5 Conclusions 104
Table of Contents
5
xiii
4.3 The fracture process in layered slate during uniaxial compression 4.3.1 Description of the model 4.3.2 Simulation of a uniaxial load test 4.3.2.1 Strength and fracture process 4.3.2.2 Deformation of the discrete elements 4.3.3 Influence of natural variability on parameters 4.3.3.1 Influence of spatial variability 4.3.3.2 Influence of strength anisotropy 4.3.4 Conclusions 4.4 Fracture process in layered rock during diametrical load tests 4.4.1 Description of the model 4.4.2 Influence of inclination angle on the fracture pattern in slate 4.4.3 Investigation of tensile failure in the schistosity direction 4.4.4 Influence of strength anisotropy 4.4.5 Summary of results 4.5 Conclusions on the numerical work
105 105 107 107 110 111 111 112 114 116 116 119 126 131 134 135
Conclusions and Recommendations for Future Research 5.1 Conclusions 5.2 Recommendations for future research
137 137 139
Appendix A: Algorithm Construction for AE-localisation in Anisotropic Media 141 A.1 Determination of the direction dependent wave velocities in 3D 141 A.2 A model for position estimation 143 A.3 Validation of the model in 2D 145 A.4 Sensitivity of the localisation to the velocity 148 Appendix B: Mesh Quality B.1 Relevance of variation of element size B.2 Model A (initial algorithm) B.3 Model B and Model C (modified algorithm)
149 149 150 153
References
155
List of Publications
167
1 Introduction The fracture process in brittle rock has been studied extensively in the past decades, both by experimental work and by numerical simulations. However for layered anisotropic rock, most research is limited to the determination of the anisotropic strength properties on sample scale. This study relates the strength anisotropy on μ-scale in layered rock to the strength anisotropy on sample scale, the deformation behaviour and the fracture pattern. In the first paragraph the general context of this research is outlined, and related to earlier and ongoing work in the Research Unit Mining. In the second paragraph the aim of the study is formulated. Finally the outline of the work is presented in the third paragraph.
1.1 General context 1.1.1 Fracturing around planar heterogeneities The start of this research is a series of tensile loading tests performed on coal samples (Vervoort et al., 1992). Coal is not homogeneous, but consists of a network of planar heterogeneities (e.g. calcite veins) and pre-existing lines of weakness, called cleats. These cleats can be caused among others by contraction during dewatering or by tectonic uplift (Laubach et al., 1998). From pictures and Computerized X-Ray Tomography (CT) scans of the samples, it is apparent that fracture patterns in coal are often a complex combination of fractures along these cleats and new fractures in the intact material (Figure 1.1). To understand the complex fracture mechanism in such a rock, a detailed analysis of the different strength properties and a good knowledge of the stress distribution under a given load configuration is required.
Figure 1.1 CT-scan of a coal sample with pre-existing cleat system (black lines) and cemented fractures (white lines), before testing (a) and after a Brazilian tensile test (b) (induced fractures are also black lines).
2
Chapter 1
The cleat network in coal is also very often complex and to a certain extent irregular, making it difficult for a systematic study of the fracture mechanisms involved. Therefore, in this study it is chosen to study fracturing in rock with a specific type of pre-existing structure, namely (parallel) layering. In rock, as in any other brittle material, when stresses become sufficiently large, fractures start to appear. Depending on the properties of the rock, the resulting fracture pattern can evolve at constant or increasing deformation level. Apart from the strength properties of rock, the actual deformation behaviour is also important for engineering applications. The surface under the strain-deformation curve is a measure for the energy a rock can absorb before and after failure. Pure brittle behaviour results in a complete loss of strength once the peak strength is reached (Andreev, 1995), while other rock types can have a considerable amount of post-peak strength, allowing additional deformation with very little or no loss of strength. Fracturing is also important for the hydraulic properties of a rock. A previously impermeable rock can become permeable when fractured. The main focus in this study is on the fracture behaviour of slate. Slate is a layered rock that is at its weakest along its layer direction, also called schistosity. Thus, the fracture pattern in slate depends on the anisotropic rock mechanical parameters. Fractures can grow parallel to the weaker layers, or in other directions, or as a combination of both. Experiments are applied to determine the strength parameters on sample scale, to describe the fracture process and to identify the different types of fractures. Simulations use the experimental elastic and strength properties as a starting point for the calibration of element scale properties. Detailed information on stress distributions and failure modes of fractures is then provided by simulations. It is also important that there is a good agreement between fracture patterns from simulations and experimental fracture patterns. This combined approach from experiments and simulations results in a detailed description of the fracture behaviour and of the importance of the different properties involved. The mechanical behaviour and permeability of slate is, among others, important in oil drilling context. Slate or its less metamorphic variant, shale can be found as cap rock on top of hydrocarbon reservoirs (Al Bazali et al., 2005) or in fractured state as reservoir rock (Sircar, 2004). Borehole instability in oil and gas drillings results every year in considerable financial losses during exploration as well as production. Due to its layered structure, slate is a typical transverse isotropic rock. Transverse isotropy is a specific type of anisotropy. Within a plane parallel to the layering, properties are isotropic, while the normal to the layers serves as an axis of symmetry for the 3D-anisotropy. This study focuses on transverse isotropy.
1.1.2 Research Unit Mining The research presented in this work is accomplished in the Research Unit Mining of K.U.Leuven. In this research group, fracturing in brittle rock has been a primary focus of research and over the years an expertise is developed in the fields of experimental work and numerical simulations of individual fracture growth. In order to better understand the process of fracture initiation and growth, the problem is split up in different research topics, focusing on different aspects (Vervoort et al., 2008).
Introduction
3
Van de Steen (2001) studied the influence of heterogeneous stress distributions on the fracture pattern of crinoidal limestone. He examined fracturing in configurations involving stress gradients, as in diametrical load tests on discs with a hole. The experiments were simulated successfully by introducing a set of randomly positioned weaker elements, so-called flaws. The set of flaw elements with weaker strength properties represent the natural heterogeneity in the rock material. The micromechanical flaw concept is used for simulations of laboratory loading tests on samples, as well as for fracturing near large-scale excavations. He concluded that the observed fracture pattern depends on the magnitude of the stresses, as well as on the stress gradients. Ganne (2007) quantified the effect of different stress paths on brittle fracturing. He contributed to the understanding of the influence of previously induced heterogeneities (cracks) on brittle fracturing, both in a micro-mechanical and in a phenomenological perspective. He observed that compressive stresses cause the occurrence of intragrain cracks, i.e. within one single grain (e.g. activated cleavage planes in limestone). On the other hand, tensile stresses cause the occurrence of both intragrain cracks and intergrain cracks, i.e. cracks extending from one grain into another. Tensile loading after compressive loading of a sample shows that the previously induced intragrain cracks are used as a path to grow along for intergrain cracks during tensile loading, resulting in a zig-zag pattern. As a consequence, the amount of additional damage induced during this second loading in tension is limited. Alternatively, when a sample is loaded in compression after in tension, more damage is induced by the occurrence of more and longer intragrain cracks. These conclusions are based on attenuation of ultrasonic waves, acoustic emission, statistical analysis of cracks observed on thin slices and numerical simulations. Tavallali (Tavallali et al., 2007) also works with layered rock, but mainly layered sandstone. He studies the effect of layer orientation on macro scale strength and on the fracture pattern, in Brazilian tensile tests. A detailed petrographical study of thin slices from loaded samples identifies the properties that cause different fracture behaviour in similar sandstones (Tavallali et al., 2008). The effect of micro scale parameters on the macro scale behaviour of layered sandstones is quantified.
1.2 Aim of the study The aim of this study is twofold. The principal interest is to identify and describe the different features and properties that are directly involved in the fracture process in layered rock. This work contributes to the understanding of the fundamental processes in layered rock fracturing and to serve as solid reference framework for future applications. Most fracture theories traditionally focus on a single fracture or a regular fracture pattern in a homogeneous, isotropic material (see Paragraph 2.2). The existing fracture theories for layered rock mainly focus on macro strength. This work studies how μ-scale anisotropy not only determines sample scale anisotropy, but also deformation behaviour and fracture patterns. The second goal of this thesis is to develop and to validate a methodology for the numerical simulation of fracturing around planar heterogeneities. Most rock mechanical numerical codes focus on the activation of pre-existing discontinuities or on the continuous behaviour of rock. In this work, a numerical model is required that can simulate fracture growth in random directions, as well as parallel to the layers. A first study on fracturing around one planar heterogeneity is simulated by two different
4
Chapter 1
numerical codes. Based on the results from this study, one numerical code is retained for further research and new models are stepwise constructed. The continuous interaction with experimental results is important to ensure realistic input parameters and to validate the results from the simulations. Also on the experimental part, a systematic approach is developed, more specifically on the continuous observation of the fracture pattern evolution by both optical techniques and acoustic emission technique.
1.3 Outline of the research The work presented in this thesis is constructed around two different, complementary approaches. The first part is the experimental work, where different loading tests are combined with observation techniques to gain insight in the fracture mechanisms in layered rock. In the second part, numerical simulations of these tests are discussed and related to the experimental results, providing a more detailed knowledge. To begin with, a comprehensive literature study is given in the next chapter. This starts with a discussion on heterogeneity and anisotropy in rock, and their areas of relevance. Next, a brief overview is presented on existing fracture theories in isotropic and anisotropic rock. This serves as a frame of reference to which the results of this study can be related. Subsequently, different experimental techniques that are used for the observation of rock fracturing are reviewed. The literature study is concluded by a discussion on the numerical methods that have been applied so far for the simulation of the fracture process in rock. The third chapter describes the experimental work. The chapter is introduced by a preliminary study of the fracture patterns in siltstone from earlier diametrical load tests. This serves to identify the important features in the fracturing of layered rock and to develop a methodology for further tests. The focus of the experimental work is however on the fracturing of slate. Therefore, in a second paragraph the general properties of slate are discussed and its rock mechanical properties are determined by a series of tests. Further on, an extensive study of the fracture patterns in slate during diametrical load tests, respectively uniaxial load tests is performed. All tests are monitored by means of digital video recording, providing a continuous stream of image for further analysis. In addition to this, 3D localisation by the acoustic emission technique is performed during the uniaxial load test. The relation is established between the dominant fracture type (parallel or non-parallel to layering) and the angle between direction of loading and direction of transverse isotropy (i.e. layer direction). This relation is part of a larger framework that relates the different strength properties on μ-scale to the deformation behaviour, the fracture pattern and the strength on sample scale. The fourth chapter treats the numerical work in this study. A first paragraph discusses the theoretical background of the two codes applied in this study, namely UDEC, a 2D distinct element code and DIGS, a 2D displacement discontinuity boundary element code. In the second paragraph, a comparative study between DIGS and UDEC on the simulation of rock fracturing around one central discontinuity is performed. Here the importance of layer stiffness on the fracture behaviour is revealed. Since this parameter is not modelled in DIGS, it is chosen to continue with UDEC for further simulations. In
Introduction
5
the third and fourth paragraph uniaxial load tests, respectively diametrical load tests on slate are simulated. The strength and elastic properties that are determined by the experiments serve as input values for the model. There is a good agreement between experiments and simulations. Thus, the simulations also provide valid detailed knowledge of the failure modes of the generated fractures, as well as of the stress distributions within the samples. Finally, in the fifth chapter, the conclusions of this work are presented and recommendations are formulated for further research in this area.
2 Literature Survey – State of the Art The study on the occurrence of fractures and fracture growth in literature is extensive in the field of rock mechanics as well as in material sciences. In general, there are two ways of treating the subject, namely by physical observations in laboratory experiments or on site and through numerical simulations of the fracture process. Often results from both approaches are discussed independently. However, these two approaches should be complementary since observations provide data for input and verification for numerical analysis. With the combined knowledge from both approaches, different theoretical frameworks are developed that describe and explain the mechanisms that govern fracturing. The literature survey starts with a brief description of two important concepts in this study, namely anisotropy and heterogeneity in rock. Next, the theoretical framework for fracture growth in brittle rock is discussed, together with existing fracture theories in anisotropic rock. Subsequently an overview is given on the observational techniques relevant for this study. Finally, the different numerical codes that are applied within this area are reviewed.
2.1 Anisotropy and heterogeneity in rock Anisotropy and heterogeneity are two terms that are sometimes confused with one another. However, they each refer to a different specific quality, essential in this research and therefore it is important that they are clearly defined at the beginning of this chapter. Isotropy means that a given property is identical in every direction. Anisotropy consequently implies that the concerned property is direction dependent. Transverse isotropy is a form of 3D-anisotropy, typical for a layered medium and implies that parallel to the layering plane, the concerning property is isotropic, while every line orthogonal to this plane serves as an axis of symmetry (Brady and Brown, 1993). For example, elastic properties as Young’s modulus and Poisson’s ratio are isotropic in common steel, or in a limestone rock. However, several sedimentary rocks (e.g. mudstone, sandstone,…) or metamorphic rocks (e.g. schists, gneiss) can consist of a layered structure, and have transverse isotropic elastic properties. Homogeneity refers to the uniformity in a material’s composition, throughout its volume. A glass of pure water or a piece of pure chalk rock are both homogeneous and isotropic. However, with the presence of dirt particles within the glass of water or fossils within the chalk rock, the media become heterogeneous. Thus a constituent that differs from the surrounding material is called heterogeneity.
8
Chapter 2
A homogeneous material is not necessarily isotropic. For example, layered schist (with no inclusions) is fairly homogeneous, i.e. it has the same layered structure in each point of its volume. At the same time, its elastic properties are anisotropic, namely transverse isotropic due to this layered structure. Both anisotropy and heterogeneity are scale dependent. Consider once more a layered rock, e.g. sandstone that consists of several layers formed by depositions of different sandy materials. Within one layer the rock can considered to be isotropic and homogeneous, while on a scale larger than interlayer distance, the rock is anisotropic and heterogeneous. This study is a rock mechanical approach to layered rock with an interlayer distance ranging from μm- to cm-scale, and thus not on a wider regional scale as in the field of structural geology. Since anisotropy and heterogeneity in rock have an important influence on the fracture process, they are each discussed more in detail in the next part. 2.1.1 Heterogeneity Rock is no man-made material and therefore very few rocks are completely homogeneous. Heterogeneities are numerous under many forms. Examples are particles of minerals different from the surrounding rock matrix (e.g. pyrite in slate), fossils (e.g. in limestone, chalk,…) or cavities due to enclosure of gasses (e.g. in basalt) or dissolution of the rock (typical for limestone). Heterogeneities in rock can have a profound influence on the fracture process. While weaker heterogeneities can make a fracture pattern deviate towards them, stronger heterogeneities can make a growing fracture turn away or even be stopped. Van de Steen (2001) conducted diametrical load tests on crinoidal limestone disks with a hole, i.e. an artificial heterogeneity. He observed that the fracture pattern deviated towards the hole, and was able to simulate this by numerical models. Kudo et al. (1987) studied fracture patterns in granite and the interaction between fracture paths and mineral grains in a double-torsion test. Here it was seen that quartz grains are often an obstacle while the cleavage planes of feldspar could change the direction of the fracture. Contrary, biotite grains had no significant effect. Chen (2008) performed uniaxial compressive loading tests on granite and made similar findings. Several authors have modeled the fracture process in heterogeneous rock with different numerical codes (Van de Steen, 2001; Liu et al., 2007; Fjaer and Ruistuen, 2002). They compared their models with experimental data in order to better understand the role of heterogeneities within the fracture process as described above. It should be noted that heterogeneities can also initiate fracture growth due to the local stress concentrations that they induce. However, other mechanisms for the nucleation of fractures exist as e.g. twinning crystals, or planes of weakness (cf. anisotropy). All these mechanisms are in literature often grouped into one concept, called defects or flaws and modeled as such (Sellers and Napier, 1997). Pre-existing fractures, on micro scale or on a larger scale, can also be seen as heterogeneities. In this study, the focus is on 2D planar heterogeneities such as a preexisting fracture or alternating parallel layers of different material (siltstone, Paragraph 3.1). In Paragraph 4.2 it is shown by numerical simulations that not only the strength properties but also the elastic properties of planar heterogeneities can have a considerable effect on (new) fracture initiation.
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2.1.2 Anisotropy An important and frequently found type of anisotropy in rock is transverse isotropy, which is the main interest of this work. The discussion is from here on limited to transverse isotropy. It is possible to divide layered rocks in two main classes, based on their formation history. First, there are the rocks that have their layered structure due to sequential sedimentation. When sedimentation takes places in distinct sequences, the layers can differ from each other in grain size, type of material, type of cementation, degree of compaction,… . Cementation results in lithification, the diagenetic process by which soft sediment is hardened into rock. Common examples are layered sandstone, mudstone and siltstone (Reading, 1996). Layers in this type of rock are often referred to as ‘bedding planes’. A second class consists of metamorphic rocks that obtained a layered structure due to high stress often caused by the weight of overburden. A typical example is slate or phyllite, that is formed by metamorphose of clay or shale. In this rock, the layered structure is formed by the parallel rearrangement of the platy clay minerals, orthogonally to the maximal stress direction (Press and Siever, 1998). The layer direction in such rock is called foliation, cleavage direction or schistosity. Typical for this class is that the material is rather homogeneous, contrary to the heterogeneous nature of the first class. Often, bedding planes are still visible as well in such a metamorphic layered rock. Bedding planes and schistosity can vary in direction, when the bedding planes are folded due to tectonic activity and consequently the orientation of the stress vector relative to the bedding plane changes. In this thesis, two rock types, one from each class, are studied experimentally. The first one is sedimentary siltstone, the lithified equivalent of silt and the second one is slate, a low-metamorphic rock originated from shale. In literature, most research is done on anisotropic elasticity and anisotropic strength of layered rock. However, this research examines the mechanisms (i.e. fracture process) that govern the mechanical anisotropic behaviour, rather than just quantify it, as is mostly done in literature. Generally, it is found for most types of layered rocks that they are the most compressible in the direction orthogonal to the layering and the least compressible parallel to the layering. This relation is observed experimentally, among others, for gneiss (Saroglou et al., 2004), coal (Pomeroy et al., 1971) and greywacke schist (Pinto, 1970). Concerning strength, the relation is a bit more complicated, but this will be discussed in Paragraph 2.2.3, after a brief introduction on fracture and strength theories in general. 2.1.3 Areas of relevance The strength and deformation behaviour of anisotropic rock is an important issue in oil drilling context. Slate or its less metamorphic variant, shale can be found as cap rock on top of hydrocarbon reservoirs (Al Bazali et al., 2005) or in fractured state as reservoir rock (Sircar, 2004). Anisotropic sandstone is also a typical reservoir rock. Borehole instability in oil and gas drillings results every year in considerable financial losses during exploration and production. Neglecting the anisotropic nature of the rock strength can result in an overestimation of the strength under some conditions, with all detrimental results. In addition, there is a mutual interaction between elastic anisotropy and heterogeneity on the one hand and rock stress on the other hand. Rock fabric
10
Chapter 2
controls the build-up of in situ stresses in the Earth crust, their magnitude and orientation. On the other hand, compressive stresses can close microcracks and planar discontinuities, thus making rock anisotropy pressure dependent (Amadei, 1996). In mining operations, one is often confronted with layered rock or elongated planar heterogeneities. For example, the roof material around coal excavations, which is derived from the deposit and the subsequent burial of organic material, exhibits mostly a layered structure. This bedding controls the way in which stresses are distributed around coal mine openings. In addition, the orientation and strength anisotropy of bedded coal defines which beams may span or not span openings (Seedsman, 2001). In drilling operations, usually the highest drilling velocities are obtained normal to the layer direction. Schormair et al. (2006) studied the fracture patterns of borehole cores and performed numerical simulations of the drilling process. They observed that drilling parallel to the layer direction produces smaller fragments, because the cracks follow the layer direction. Normal to the layer direction, fracturing is more omnidirectional, resulting in a more effective fragmentation. Another example comes from the limestone quarry in Clypot (B). In this limestone, stylolites are present, which are distinct thin seams of residual clay material, often mixed with organic material (Van der Pluijm and Marshak, 1997). These thin layers are in fact elongated planar heterogeneities that are much weaker than the surrounding material. In this quarry, the stylolites are situated subhorizontally (i.e. almost horizontally), and the rock can be broken along these stylolites with relatively little effort in order to excavate them. Furthermore, an area where layered rock is frequently encountered is tunneling. The issues on layered rock around a circular excavation are to a certain extend similar as for borehole applications. However, obviously safety requirements are much more demanding especially when the tunnel serves for transportation of vehicles or people. Research is done, among others, in the implementation of rock layers in numerical modeling of tunnel support (Yin and Yang, 2000). The importance of heterogeneity and anisotropy in rock is of course not limited to mechanical features. In the field of hydrogeology, often the layered structure of a rock defines anisotropy in permeability and conductivity, which has to be taken into account in e.g. groundwater flow modeling, hydraulic properties of hydrocarbon reservoirs (Louis et al., 2003), sequestration of C0 gasses (Li et al, 2009) or storage of nuclear waste (Takeda et al, 2003). For this last application, also the anisotropy in thermal conductivity is an important parameter (Choi, 2003). In geophysics, a layered medium composed of isotropic layers behaves like a transverse isotropic medium when the seismic wavelengths are large compared to the bed thickness (Telford et al., 1990). It is no surprise that the wave velocity in layered rock is anisotropic, since the velocity depends directly on the elastic properties. In addition, planar heterogeneities can cause anomalies due to reflections or delays of the source waves.
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2.2 Fracture theory Fracture processes have been widely studied in the past decades, on different scales. This rock mechanical study focuses on fractures at μm to cm-scale, contrary to the field of structural geology where one studies fractures with a size ranging from centimeters to kilometers. First, the terminology as applied in this work is given. The term microfracture, microcrack or crack is used for fractures on sub-mm-scale (Simmons & Richter, 1976). These fractures cannot be seen by naked eye, but nevertheless play an important role since they characterize the initiation of the fracture process. Backers (2005) defines a mesocrack as a discontinuity spanning a larger number of grains than a microcrack, formed by a complicated rupture event and eventually connecting several microcracks. The extension is several hundreds of microns to few millimeters. In this work a mesocrack is a fracture with a length within the range of 1 mm to 1-2 cm. If these fractures in turn connect into an even larger fracture that extends e.g. from one end of a test sample towards the other end (cm-scale), it is called a macrofracture. Note that the term fracture is used in this text for microcrack, mesocrack and macrofracture while crack refers to both micro- and mesocracks. Figure 2.1 shows several macrofractures and mesocracks observed on a slate sample, on three different magnifications. The fracture evolution of one single microcrack consists of two stages, namely crack initiation (or nucleation) and crack propagation. The term fracture process includes the fracture initiation and propagation of all microcracks involved, together with the coalescence of microcracks into mesocracks and/or macrofractures.
Figure 2.1 Two different scales of fractures in a slate sample. The arrows indicate respectively a macrofracture (a), and two mesocracks on the second magnification (b). The third magnification (c) shows the detailed structure of a mesocrack, where the arrows show sequential parts in different directions.
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Chapter 2
2.2.1 Fracture process in brittle rock Rock is typically a brittle material under common stress conditions. Brittle failure takes place when a material shows (quasi) linearity until the failure moment, i.e. there is little or no irreversible deformation (Bieniawski et al., 1969). At failure, a sudden loss of strength occurs. Contrary to this, in ductile deformation, a material can sustain further permanent deformation without losing all load-carrying capacity (Brady and Brown, 1993). A fundamental attribute of brittle failure is the initiation and propagation of microcracks that disturb the continuity of the material and transform it to a discontinuum. Fracture processes have been extensively studied for different brittle materials e.g. ceramics (Danzer et al., 2008), polymers (Beaumont and Young, 1975), concrete (Hsu, 1984) and many types of brittle rock (e.g. Andreev, 1995; Kranz, 1983; Bobet and Einstein, 1998a; Wawersik and Fairhurst, 1970; Friedman et al., 1970; Bombolakis 1973). 2.2.1.1 Microcracks: origin Griffith (1924) developed the concept that failure in brittle material occurs through crack propagation. For this, the presence of microcracks as pre-existing crack nucleation centers is essential. Microcracks can be classified in 3 main categories (Simmons and Richter, 1976), namely intragrain cracks (within one grain), intergrain cracks (from one grain into another) and grain boundary cracks (following the grain boundary). An important subset of intragrain cracks are cleavage cracks, parallel with the cleavage planes in minerals (Kranz, 1983). An intragrain crack, spanning several grains is often called a transgranular crack. Alternatively, Kranz (1983) divided the mechanisms that form these microcracks into six categories: [1] interaction of twin crystals with grain boundaries [2] energy release due to deformation bands [3] cleavage separation [4] stress concentrations around grain boundaries and cavities [5] elastic mismatch between grains [6] grain translations and rotation The most important mechanisms related to our research, are [3], [4] and [5] and are discussed here further. Cleavage planes as e.g. in slate are generally the atomic planes with the lowest bond strength and the lowest surface energy. A weak bond is usually accompanied by a large interplanar spacing because the attractive force cannot hold the planes closely together (Klein and Hurlbut, 1993). Thus when external stress is applied, the strain energy stored in the rock grains are relieved on those planes first (Brace and Walsh, 1962). As will be shown later in this work, cracks parallel to the cleavage direction are not limited to microcracks, but also observed on meso- and macro-scale. Cleavage cracks have been observed by scanning electron microscopy (SEM), among others, in gneiss (Rasolofosaon et al., 2000), breunnerite (a magnesium-rich ore rock) (Bogoch et al., 1982) and slate (Abad et al., 2003). Figure 2.2 shows a SEM-image from a chlorite slate where microcracks (cleavage cracks), with a length smaller than 1μm are visible.
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Figure 2.2 SEM image from chlorite in slate. The cleavage direction is clearly visible (parallel lines), as well as microcracks (white lines, indicated by arrows) (Abad et al., 2003).
Stress concentrations are found where grain boundaries form point and line contacts. The associated tensile stresses often exceed local tensile strength, resulting in extensional microcracks along the boundary. Hoek (1964) assumed grain boundary cracking and cleavage cracking the only important mechanisms of microcracking in anisotropic rock. Schild et al. (2001) confirmed this thesis by observations through microscopy of undisturbed ansitropic granite. Elastic mismatch can occur when two neighboring but different minerals are subjected to the same externally applied stress. This induces additional tensile stresses in the stiffer mineral, resulting in extensional cracks in this mineral (Dey and Wang, 1981). An example of microscopic observation of microcracks in feldspar due to elastic mismatch can be found in Lee et al. (2006). In Paragraph 3.1, fracture patterns in siltstone are studied. Siltstone consists of alternating layers that are rich either in mica or in organic material. It is assumed that elastic mismatch might be an important mechanism in the fracture process of layered siltstone. Although apparently little research on elastic mismatch - microcracks is performed in siltstone, Pollard and Aydin (1988) observed elastic mismatch between siltstone and underlying shale on a larger macro scale (i.e. meters). 2.2.1.2 Griffith’s theory, fracture modes and stress intensity factors In brittle fracture mechanics, it is accepted that pre-existing microcracks act as stress concentrators that initiate local failure and crack growth (Dyskin, 1998). Griffith (1924) developed this theory, based on the energy conservation principle. If a new crack is formed or extended due to the action of stress, the surface energy of the material is increased. In order to preserve the total potential energy of the system, this surface energy increase equals the decrease in potential of the strain energy and applied forces. For a ‘perfectly’ homogeneous material under stress, the theoretical amount of energy to form a crack in the material can be calculated. However, it is observed that this theoretical energy is 10 to 100 times larger than the decrease in potential energy measured in experiments. Subsequently, Griffith derived that at the moment of fracture, energy is not uniformly distributed but concentrated at the tips of pre-existing microcracks. The amount of energy for a microcrack to propagate is much smaller than to induce a fracture in a ‘perfectly’ homogeneous material, and hence the energy balance is respected.
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Chapter 2
Based on the above theory, Griffith derived σmin, the minimal tensile stress at a plate, required for the propagation of an elliptic microcrack (perpendicular to the loading direction): 2 ET (1 2 ) (2.1) a where E is Young’s modulus, ν Poisson’s ratio, T the surface energy of the crack (= surface energy divided by crack area), and a half the crack length (Figure 2.3). This formula is only valid for perfectly brittle behaviour. However, it is observed that for most brittle materials Formula (2.1) provides an underestimation of σmin. This is because secondary inelastic deformation mechanisms consume an additional amount of energy (Lajtai, 1971).
min
Figure 2.3 Griffith crack with length 2a in a 2D-plate under tensile stress σ3 (Andreev, 1995).
Griffith’s theory is later also adapted for a compressive stress state (more common in rock). Wang and Shrive (1995) give a comprehensive overview on this adaptation and later modifications by several researchers. Until now, only fracture growth through tensile opening of microcracks is considered. However, a crack can close, and still grow by means of shearing or tearing. Generally, it is accepted that 3 modes of micro-failure exist, as illustrated in Figure 2.4 (Rao et al., 2003): opening, sliding and tearing, respectively called mode I, mode II and mode III failure.
Figure 2.4 Three modes of micro-failure: mode I, opening (a), mode II, sliding (b) and mode III, tearing (c) (Rao et al., 2003).
Irwin (1957) introduced the concept of ‘stress intensity factor’. The stress intensity factor expresses for a given fracture mode the relevant stress at the crack tip in function of the external stresses , the half-crack length a and a dimensionless geometry factor ξ. K i f i , a, (2.2)
Thus, for a given geometry and loading condition, there is a stress intensity factor for each failure mode, called KI, KII and KIII.
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In the example of Figure 2.3 the stress intensity factor for mode I becomes: KI 3 a (2.3) Each of the three factors attains a critical value KiC when the corresponding type of failure occurs in the crack tip (Ingraffea, 1987). The critical stress intensity fracture is also called fracture toughness. Fracture toughness is a material parameter of brittle materials, next to macroscopic strength parameters. For simple loading conditions (e.g. uniaxial tension as in Figure 2.3), one critical factor KiC prevails over the two others. However, in more complicated loading configurations, all three values KIc, KIIc and KIIIc have to be taken into account. The stress intensity factor Ki that attains first its critical value KiC subsequently determines the mode of failure. Most research is done on mode I failure (opening) and KIc, though mode II failure (shearing) is very common in rock. Recently, Rao et al. (2003) proposed a new mixed mode fracture criterion, based on KIc and KIIc. 2.2.1.3 Coalescence of fractures into fracture patterns When a rock is loaded, and stresses become sufficiently large, multiple microcracks start to grow. As their growth continues, such a crack can branch or coalescence with other cracks. At a certain moment, one or more fractures of macro dimensions develop in this way, resulting in loss of structural integrity of the rock material and consequently loss of strength. Extensive research has been done on the interaction of microcracks, often by loading samples with artificial microcracks in order to control geometrical parameters as mutual orientation, distance,… in 2D (Bobet and Einstein, 1998a; Baud and Reuschlé, 1997; Sagong and Bobet, 2002) and 3D (Dyskin et al., 1994; Wong et al., 2004). A term often used is wing crack, i.e. a tensile microcrack that propagates from the tip of pre-existing microcracks in a stable manner and in a curvilinear path to realign with the maximum stress axis (Bobet and Einstein, 1998a).
Figure 2.5 Five types of fracture coalescence observed under uniaxial load tests (after Bobet and Einstein, 1998a).
Figure 2.5 summarizes the types of fracture coalescence observed by Bobet and Einstein (1998a) during uniaxial compression for two pre-existing parallel cracks. Two microcracks can connect when they grow in pure shear mode (type I) or in pure tensile mode as wing cracks (type IV), or as a combination of both (type II and type III). The mechanism behind type V is believed to be shear growth, though the authors are not sure about this. Adams and Sines (1978) brought the phenomenon of interlocking under the attention. Due to the stress concentration around a given crack, a smaller neighbouring crack that lies within this stress concentration zone of the larger crack can be inhibited to grow.
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Chapter 2
In general, two main types of macrofractures can develop during uniaxially loading. A first type is tensile fractures, parallel to the loading direction. They develop as a result of the tensile stresses that are induced orthogonal to the compressive load. This type of fracture is also induced during a Brazilian tensile test, where one loads a cylindrical sample diametrically. In such a test, the tensile strength can be calculated from the compressive load, assuming an elastic deformation state (ISRM, 1978b). A second type is shear fractures that form an acute angle with the (vertical) loading direction. Coulomb developed a (compressive) strength criterion that predicts these shear fractures to develop at an angle of [45°-φ/2] with the loading direction, where φ is the friction angle of the intact rock material (Brady and Brown, 1993). This is also commonly observed in laboratory testing. Figure 2.6 shows a schematic overview of the fracture pattern evolution observed by naked eye in granite by Wawersik and Fairhurst (1970) where one can see several tensile mesocracks and the development of two distinct shear macrofractures.
Figure 2.6 Evolution of the fracture pattern in granite during a uniaxial load test (after Wawersik and Fairhurst, 1970).
Wawersik and Fairhust (1970) determined two classes of post peak strength behaviour for brittle rock under compressive stress. Their distinction is directly related to the fracture pattern. On the one hand they define class I, where fracture propagation is stable, and thus incremental displacement must be imposed for incremental decrease in strength (Figure 2.7). The fracture pattern of granite as shown in Figure 2.6 belongs to class I behaviour. On the other hand, there is class II for which the fracture evolution is unstable or self-sustaining, i.e. the elastic strain energy stored in the sample at peak strength is sufficient to maintain fracture propagation until all strength is lost. For class I behaviour, they indentified several stages during the fracture propagation, as indicated on Figure 2.7. Stages A, B and C were already described earlier by Bieniawski (1967a and 1967b) and are considered similar for both classes. At the onset of loading, first the existing microcracks close (A), resulting in an increase of elasticity, and hence the nonlinearity in this stage. Next, virtually no crack activity occurs and the stress increases linearly with the deformation in this stage (B). Before peak strength is reached, microcracks start to grow in a stable way and when they become mesocracks they can be observed by naked eye (C). In stages C and D however, these mesocracks are predominantly tensile and local, thus no macrofractures are yet formed. However, Bombolakis (1973) did observe the beginning of microcrack growth in shear when the axial stress approaches peak strength. Due to the opening of these tensile cracks, the rock extends orthogonal to the load direction, this is called dilatancy. Stages F, G and H are characterized by appearance of shear fractures and their development into
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macrofractures until finally all strength is lost. The fracture evolution during these last 3 stages corresponds with the images in Figure 2.6. Class II behaviour is related to more homogeneous rock (e.g. basalt, limestone). They state that for very homogeneous materials tensile fracturing does not occur since the absence of heterogeneities prevents local stress concentrations large enough for local tensile fractures. Subsequently the rock cannot dissipate energy through local tensile cracks (contrary to class I behaviour in stages C and D) and rapid strength release occurs with the formation of shear macrofractures.
Figure 2.7 Stress-strain curve of uniaxial compression of brittle rock with two classes of post-peak behaviour and indication of different crack growth stages (Wawersik and Fairhust, 1970).
Concerning triaxial stress, several authors have reported that the brittleness of a rock decreases with increasing confining pressure (e.g. Chen et al., 2006; Wawersik and Fairhurst, 1970; Ghazvinian et al., 2008; El Bied et al, 2002). Due to lateral stresses, tensile microcracks stabilize by tip blunting and do not grow to meso-dimensions. This results in increasing peak strength with increasing confinement. At sufficiently high confining pressure, several ductile deformation mechanisms arise, e.g. grain readjustment or shear banding of deformation (El Bied et al, 2002) and brittleness disappears altogether (Andreev, 1995). An important exception to this is reported by Chang and Haimson (2005). They observed little influence of confining pressure on peak strength and deformation behaviour for brittle hornfels and metapelite. In these types of rock they observed no tensile microcracking, and thus no dilatancy. The authors assume that macrofailure occurs through non-dilatant shear microracks that coalesce. 2.2.2 Strength and fracture process in transverse isotropic rock 2.2.2.1 Strength models In transverse isotropic rock, the fracture mechanism can be considerably altered due to the anisotropy, creating strength anisotropy. In order to understand and model the resulting strength anisotropy, it is essential to understand the fracture process. The research further in this thesis contributes to this understanding. Here, experimental approaches as well as proposed theoretical frameworks are reviewed. In Paragraph 2.4.3, numerical models on fracture processes are discussed. As described earlier, the governing factors in the fracture process in isotropic rock are: - density of pre-existing microcracks - stress situation and failure mode - strength (i.e. fracture toughness)
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Chapter 2
Now, in transverse isotropic rock, most authors consider two types of microcracks: the cleavage cracks parallel to the layers of isotropy and cracks in any other directions. Often the former and the latter are called respectively primary and secondary cracks (e.g. Hoek, 1964), though this does not indicate any chronological order of formation. Thus, in anisotropic rock, the fracture pattern can be formed by growth of primary cracks, secondary cracks or a combination of both. Two key factors govern the fracture process and consequently the macro strength, namely the stress orientation relative to the layer direction and the strength difference between primary cracks and secondary cracks. This last factor is quite complicated, since both types of fractures can fail in mode I (tension) or in mode II (shear), and thus a complex interaction of 4 types of crack failures occur. A shortcoming of most theoretical strength models is that within given ranges, they assume only one failure type to occur, as is illustrated hereafter. Jaeger (1960) developed a first theoretical approach of strength anisotropy, based on Coulomb’s strength criterion. In his analysis, a rock with well-defined parallel planes of weakness is considered. According to Coulomb’s criterion, each plane has a limiting shear strength s: s c/ / n tan / / (2.4) where σn is the normal stress, and c// and φ// respectively cohesion and friction angle of the layer planes. Shear failure thus occurs when shear strength τ equals s. Consider a stress state (σ1, σ2 = σ3) as shown in Figure 2.8a with θ the angle between the maximal stress direction and the normal to the layer direction. Normal stress σn and shear stress τ can be formulated in function of principal stresses (σ1, σ3) as: n 0.5 ( 1 3 ) ( 1 3 ) cos 2 (2.5)
0.5( 1 3 ) sin 2
(2.6)
Taking (2.5) and (2.6) into account, (2.4) can be rewritten as: Jaeger ) ( 1 3 )(max
2(cl 3 tan l ) (1 tan l cotan )sin2
(2.7)
Jaeger ) where ( 1 3 )(max is the deviatoric stress at which shear failure occurs, according to Jaeger’s theory. A boundary condition for slip to occur, is that the plane inclination θ is smaller than the Jaeger ) becomes friction angle φl. In addition, for large values of θ, the value for ( 1 3 )(max sufficiently large so that shear failure along a different angle or tensile fracturing is more likely to occur. The former conditions thus limit the range for which (2.7) is valid as the peak strength for the rock. Figure 2.8b shows the peak strength in function of θ for a triaxial stress state (σ2 = σ3 = 5 MPa). Note also that generally peak strength for small values of θ (layers quasi perpendicular to σ1) are found to be larger than for values of θ close to 90° (layers quasi parallel to σ1).
Among others, Duveau and Shao (1998) found that experimental data for triaxial compression of Ordivican schist did not completely agree with the theoretical predictions from Jaeger (Figure 2.8b).
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Figure 2.8 (a) Schematic drawing of a transverse isotropic rock under triaxial compression (σ1, σ3) with angle θ between the maximal stress direction and the normal to the layer direction. (b) Peak devatoric stress (σ1-σ3) versus angle θ for experimental values, Jaeger’s theory and the modified Jaeger theory (after Duveau and Shao, 1998).
They proposed a modification where the shear strength criterion for the planes is no longer linearly dependent on the normal strength, according to the theory of Barton (1976): UCS (2.8) ) s n tan(b1 log n b2 where UCS is the uniaxial compressive strength perpendicular to the layer direction and b1 and b2 two quality parameters for the layer planes. Experimental values and theoretical values according to Jaeger, respectively Duveau and Shao can be seen on Figure 2.9b. It remains debatable which model approximates the experimental data the best. Except for θ
