Fractured Rock Masses As Equivalent Continua – a Numerical Study

Fractured Rock Masses as Equivalent Continua – A Numerical Study

Ki-Bok Min

April 2004

TRITA-LWR PHD 1011 ISSN 1650-8602 ISRN KTH/LWR/PHD 1011-SE ISBN 91-7283-757-8

To my father and mother in Korea

and

my lovely wife, Mi-Suk Lee

Fractured Rock Masses as Equivalent Continua – A Numerical Study

ACKNOWLEDGEMENT

It is my great pleasure to be able to thank the people who have helped and supported me in various ways.

First of all, I would like to express my deepest gratitude to my main supervisor, Dr. Lanru Jing of the Royal Institute of Technology (KTH) for his academic guidance throughout the duration of the study. I am greatly indebted to his advice, suggestion and all the support for my research work. I also would like to express my sincere gratitude to my co- supervisor, Prof. Ove Stephansson of GeoForschungsZentrum (GFZ), Germany for the trust on my work and endless encouragement, which led me to come to this stage with confidence. I also thank Dr. Jonny Rutqvist of Lawrence Berkeley National Laboratory (LBNL), USA as my third supervisor for his enlightening advice and discussion, through my two visits to Berkeley and numerous email and telephone correspondences.

Main financial support for the study was given by European Commission and additional support was provided by the Swedish Nuclear Inspective (SKI) and the French National Radioactive Waste Management Agency (ANDRA). I would like to thank Dr. Henning von Maravic of EC, Dr. Fritz Kautsky of SKI and Dr. Kun Su of ANDRA for their en- gagements.

I was very fortunate to be involved in the prestigious international projects, DECO- VALEX and BENCHPAR. I would like to thank the colleagues in the projects for their comments, fruitful discussion and cooperation during the course of workshops. In particular, I would like thank Prof. John Hudson of Imperial College, UK for his insightful comments and especially encouragement. I would like to extend my sincere gratitude to Dr. Chin-Fu Tsang of LBNL for suggesting many ideas on my research, collaboration on the two papers and hospitality during my visits to Berkeley. I also would like to thank Philipp Blum of then Univ. of Birmingham, Johan Öhman of Uppsala Univ., Dr. Johan Anderson of JA Streamflow, Dr. Leslie Knight of Nirex, Dr. Auli Niemi of Uppsala Univ. for very fruitful interactions.

I am grateful to the former and current colleagues at Engineering Geology and Geophysics (EGG) group at KTH for all the comments, discussion and cooperations. Ms. Ulla Eng- berg is specially thanked for having been supportive of many aspect of my life in Stock- holm, especially my early days. Diego Mas of Itasca Geomekanik is acknowledged for our collaborations and fruitful discussions. I greatly appreciate the expert review of the summary of the thesis by Tomofumi Koyama and I wish him the best of luck in both profes- sional and private life in Stockholm. I would like to thank Britt Chow for effective and kind help with all administrative matters.

I would like to thank my ex-supervisor, Prof. Chung-In Lee of Seoul National University, for giving me enormous moral support and advice for my PhD study through many email correspondences. Prof. Jae-Dong Kim of Kangwon National University is acknowledged for the help in finding this wonderful place and encouragement. I would like to convey my special thanks to a number of seniors in rock mechanics community in Korea, including Dr. Hee-Suk Lee of SK E & C, for their encouragement for the initiation and continuation of my PhD study. i

Ki-Bok Min TRITA-LWR PHD 1011

Contribution by my wife, Mi-Suk Lee, is immeasurable if invisible in this thesis. Her exceptional support and understanding during my thesis work have been essential and it made me to complete the thesis successfully. Also I would like to share much of this pleasure with Mi-Suk’s family in Korea, especially new-born nephew and nieces.

I am deeply grateful to my parents for their exceptional moral support on my study and I would like to thank my sisters and brother for encouraging my study. I am excited and so happy to give this thesis to my parents who have been extremely enthusiastic for the higher education of their four children.

Tack så mycket!!

Ki-Bok Min, Stockholm, April 2004

Fractured Rock Masses as Equivalent Continua – A Numerical Study

ABSTRACT

In this thesis, fractured rock masses are treated as equivalent continua for large-scale analyses of rock engineering projects. Systematic developments are made for the determination of equivalent mechanical and hydraulic properties of fractured rock masses using a hybrid discrete fracture network - distinct element method (DFN-DEM) approach. The determined equivalent properties are then used for a far-field finite element analysis of the thermo-mechanical impacts on the stress, deformation and permeability of fractured rocks surrounding a hypothetical geological repository of nuclear waste. The geological data were extracted from the results of an extensive site investigation programme at Sellafield, UK, conducted by Nirex UK Ltd.

The scale dependencies of the hydraulic and mechanical properties were investigated by using multiple realizations of the fracture system geometry with increasing model sizes until properly defined hydraulic and mechanical representative elementary volumes (REVs) were reached. The validity of the second order permeability tensor and the fourth- order mechanical compliance tensor were tested for continuum analyses at larger scales. The REV was determined to be around 5 m for mechanical and hydraulic data in this study.

Analysis of the stress-dependent mechanical and hydraulic properties shows that the effect of rock stresses is crucial. The elastic moduli increase significantly with the increase of stress and an empirical equation of stress-dependent elastic modulus is suggested based on results of numerical experiments. Calculations of the Poisson’s ratios suggest greater values than are normally assumed in practice. Depending on the state of stress, permeability decreases or increases with increasing compressive stress. Stress-induced flow channeling effect is captured by numerical modeling for the first time and detailed mechanisms of shear dilation of fractures are provided. Based on the numerical experiments, a set of empirical equations was suggested for the stress-dependent permeability, considering both normal deformation and shear dilation of fractures.

Thermo-mechanical impact on the performance of a hypothetical repository at a far-field scale (5 km by 1 km) was investigated with the stress-dependent equivalent properties determined at the REV scale. This analysis shows that mechanical responses vary significantly depending on how the mechanical properties were determined. The change of permeability due to the thermal loading is, however, not significant in this particular case.

The thesis provides a framework for systematic analysis of large-scale engineering applications in fractured rock masses, such as geological repositories of nuclear wastes.

Keyword: Fractured rock masses, Equivalent Continuum, Discrete Fracture Network (DFN), Distinct Element Method (DEM), Finite Element Method (FEM), Nuclear Waste Disposal, Coupled Thermo-Hydro-Mechanical Processes

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Ki-Bok Min TRITA-LWR PHD 1011

Fractured Rock Masses as Equivalent Continua – A Numerical Study

SAMMANFATTNING

I denna avhandling behandlas metoder för analys av spruckna bergmassor som kontinuer- liga material. En systematisk utveckling av metoder för ekvivalenta mekaniska och hydrauliska egenskaper hos spruckna bergmassor har skett med användninga av diskreta spricknätsmodeller och distinkt elementmetod (DFN-DEM). De ekvivalenta materialmo- dellerna har sedan tillämpats på storskaliga analyser av ett hypotetiskt slutförvar för kärn- avfall med hjälp av finit elementmetod. Spänningar, deformationer, flöden och termome- kanisk respons hos bergmassan har analyserats. Geologiska data för studien baseras på resultaten av platsundersökningarna vid Sellafield i Storbritannien, som genomfört av Ni- rex UK Ltd.

Skalberoendet hos de hydrauliska och mekaniska egenskaperna har undersökts genom an- vändande av upprepade analyser av spricksystem med samma statistiska egenskaper men med ökande modellstorlek tills representativa hydrauliska och mekaniska elementarvoly- mer (REV) uppnåddes. Skalan för den fastställda elementarvolymen REV för den testade bergmassan är ungefär 5m för konstanta mekaniska och hydrauliska egenskaper.

Studien av spänningsberoende mekaniska och hydrauliska egenskaper visar att effekten av bergspänningarna är kritisk. Elasticitetsmodulen ökar signifikant med ökad spänning och ett empiriskt samband för elasticitetsmodulens spänningsberoende presenteras. Det beräk- nade Poissonsförhållandet är mycket högre för bergmassorna än de typiska värden som används i praktiken. Permeabiliteten hos bergmassan minskar eller ökar med ökad spän- ning och är beroende av spänningstillståndet. Effekter av spänningsinducerad kanalström- ning i sprickorna fångas upp genom numerisk modellering och redovisas för första gången och mekanismerna bakom dilatation av sprickorna i samband med skjuvning redovisas. Baserat på numeriska experiment föreslås en uppsättning av empiriska ekvationer för spänningsberoende permeabilitet i samband med normalbelastning och skjuvning av sprickorna och bergmassan.

Den termomekaniska responsen hos bergmassan från ett hypotetisk slutförvar för radioak- tivt avfall i skalan 5km x 1 km har studerats med den utvecklade DFN-DEM tekniken för ekvivalenta materialegenskaper hos bergmassan. Den mekaniska responsen är starkt beroende av sprickgeometrin och valda sprickegenskaper. Förändringarna i permeabiliteten är mindre känslig för temperaturbelastningen från slutförvaret.

Avhandlingen presenterar en metod för systematisk analys av bergtekniska och bergme- kaniska problem för spruckna bergmassor.

Nyckelord: Spruckna bergmassor, ekvivalenta kontinuummodeller, diskreta spricknät, distinkt elementmetod, finit elementmetod, slutförvaring av kärnavfall, kopplade termo- hydro-mekaniska processer

Ki-Bok Min TRITA-LWR PHD 1011

Fractured Rock Masses as Equivalent Continua – A Numerical Study

초록

본 논문에서는 균열암반을 등가연속체로 모사하는 방법에 관한 총괄적 연구가 수 행었다. 불연속 균열망(Discrete Fracture Network)과 개별 요소법(Distinct Element Method)을 혼합한 수치실험을 통하여 균열암반의 역학적, 수리적 등가물성치를 결정하는 방법이 제시되었다. 결정된 등가물성치는 균열암반에 건설된 가상 지하 핵폐기물 처분장의 열역학적 영향을 살펴보기 위한 유한요소해석을 위해 이용되 어, 균열암반의 응력, 변형 그리고 투수율의 변화에 대한 해석이 실시되었다. 본 연구의 데이터로는 영국 셀라필드 지역의 부지조사 자료가 이용되었다.

균열암반의 수리적, 역학적 물성치의 크기의존성을 규명하기 위하여 복수로 생성 된 불연속 균열망에 대하여 대표요소체적에 이를때까지 모델의 크기를 점차 증가 시켜가면서 수치실험이 실시되었다. 수치설험을 통해 얻어진 균열암반의 투수율 텐서와 역학적 컴플라이언스 텐서가 연속체해석에 이용될 수 있는 지를 검증하기 위한 해석이 또한 실시되었다. 본 연구에서 이용된 역학적, 수리적 데이터를 이용할 경우 균열암반이 등가연속체 해석에 이용될 수 있는 최소 대표요소체적의 한변은 5 m 정도였다.

본 연구에서의 수치실험 결과 균열암반의 탄성계수와 투수율은 응력의존성이 매 우 큰 것으로 나타났다. 균열암반의 탄성계수는 응력의 증가에 따라 크게 증가하 였으며 수치적 실험결과에 기초하여 응력의존성 탄성계수의 경험적 수식이 제시 되었다. 수치실험을 통해 얻어진 균열암반의 포아송비는 통상 가정되어지는 값보 다 매우 크게 나타났다. 투수율의 값은 응력증가에 따라서 증가하기도, 감소하기 도 하였는데 이는 수직응력과 수평응력의 비율에 따라 결정되었다. 본 연구에서 처음으로 응력의 변화에 의해 유발되는 균열암반내 유체유동의 채널링 현상이 수 치실험을 통해 재현되었고, 이에 수반된 균열의 팽창 현상에 대한 자세한 메커니 즘이 설명되었다. 수치실험에 기초하여 균열의 수직거동과 급격한 전단팽창을 모 두 고려할 수 있는 응력의존성 투수율의 경험식이 또한 제시되었다.

대표요소체적 상에서 결정된 물성치와 응력의존 관계를 가지고 가로 5 킬로, 세로 1 킬로의 규모로 가상 핵폐기물 처분장의 성능에 대한 열역학적 해석이 실시되었 다. 해석결과 암반의 역학적 거동은 역학적 물성이 어떻게 결정되는가에 따라서 매우 다른 결과를 보여준다. 본 연구에서는 열응력에 의한 투수율의 변화는 작은 것으로 나타났다.

본 연구는 균열암반에 건설되는 핵폐기물 지하처분장과 같이 대단위 해석을 위해 연속체 해석을 실시하는 경우의 해석방법과 결과를 보여준다.

핵심어: 균열암반, 등가연속체, 불연속 균열망, 개별요소법, 유한요소법, 방사성 폐기물 지층처분, 열-수리-역학적 연동작용

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Ki-Bok Min TRITA-LWR PHD 1011

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Fractured Rock Masses as Equivalent Continua – A Numerical Study

TABLE OF CONTENTS

LIST OF APPENDED PAPERS...... XI 1 INTRODUCTION...... 1 1.1 Motivation and objectives of the study ...... 1 1.2 Mechanical properties of fractured rock masses...... 3 1.3 Hydraulic properties of fractured rock masses ...... 5 1.4 Coupled analysis and equivalent continuum approach ...... 7 2 METHODOLOGY AND SCOPE OF THE STUDY ...... 11 2.1 Numerical experiments on fractured rock masses (DFN-DEM approach)...... 11 2.1.1 Mechanical compliance tensor of fractured rock masses...... 11 2.1.2 Hydraulic permeability of fractured rock masses ...... 12 2.2 Appropriateness of equivalent continuum approach...... 13 2.2.1 Unified two criteria for the equivalent continuum approach ...... 13 2.2.2 Methodology of investigation...... 15 2.3 Overview of appended papers...... 16 3 GEOLOGICAL DATA AND GEOMETRY OF THE FRACTURE SYSTEM ... 21 3.1 Geological data...... 21 3.2 Discrete Fracture Network (DFN) generation...... 22 4 THEORY AND NUMERICAL CODE DESCRIPTION ...... 27 4.1 Constitutive equation of fractured rock masses...... 27 4.2 UDEC code (Universal Distinct Element Code)...... 28 4.2.1 Basic features ...... 29 4.2.2 Mechanical behavior of a fracture ...... 30 4.2.3 Fluid Flow in fractures ...... 32 4.3 ROCMAS code (ROCk Mass Analysis Scheme) ...... 33 5 MECHANICAL PROPERTIES OF FRACTURED ROCK MASSES (PAPER I) 35 5.1 Methodology ...... 35 5.2 Verification ...... 35 5.3 Results of calculated elastic moduli and Poisson’s ratio ...... 35 5.4 Results of the tensor quantity investigations of the mechanical properties ...... 37 6 PERMEABILITY OF FRACTURED ROCK MASSES (PAPER II)...... 41 6.1 Methodology ...... 41 6.2 Results of calculated permeability tensors...... 42 6.3 Results of an investigation on tensor expression of permeability ... 42 6.4 Determination of REV and obtained properties from Paper I and Paper II ...... 43 7 STRESS DEPENDENT MECHANICAL PROPERTIES OF FRACTURED ROCK MASSES (PAPER III)...... 45 7.1 Methodology ...... 45 ix

Ki-Bok Min TRITA-LWR PHD 1011

7.2 Results of stress dependent elastic modulus and empirical equation 45 7.3 Results of stress-dependent Poisson’s ratio and the bounds...... 45 8 STRESS DEPENDENT PERMEABILITY OF FRACTURED ROCK MASSES (PAPER IV)...... 47 8.1 Methodology ...... 47 8.2 Results of stress dependent permeability ...... 47 8.3 Results of stress-induced channeling and anisotropy...... 48 8.4 Proposed empirical equation of stress-dependent permeability ...... 49 9 THERMOMECHANICAL IMPACT ON PERFORMANCE OF A NUCLEAR WASTE REPOSITORY (PAPER V) ...... 51 9.1 Methodology ...... 51 9.2 Result of Mechanical response caused by thermal loading...... 52 9.3 Result from permeability change from thermal loading ...... 53 10 CONCLUSION ...... 55 11 DISCUSSION AND FURTHER RESEARCH...... 59 12 REFERENCES ...... 61

Fractured Rock Masses as Equivalent Continua – A Numerical Study

LIST OF APPENDED PAPERS

I. Min KB, Jing L, Numerical determination of the equivalent elastic compliance tensor for fractured rock masses using the distinct element method, Int J Rock Mech Min Sci; 2003;40(6):795-816.

II. Min KB, Jing L, Stephansson O, Determining the Equivalent Permeability Tensor for Fractured Rock Masses Using a Stochastic REV Approach: Method and Appli- cation to the Field Data from Sellafield, UK, Hydrogeology Journal (in press).

III. Min KB, Jing L, Stress dependent mechanical properties and bounds of Poisson’s ratio for fractured rock masses investigated by a DFN-DEM technique, Int J Rock Mech Min Sci; 2004;41(3):431-432, special issue of SINOROCK2004, Int Symp on Rock Mechanics, Rock Characterization, Modelling and Engineering Design Methods, Three Gorges Project Site, China (Paper No. 2A13).

IV. Min KB, Rutqvist J, Tsang C-F, Jing L, Stress-dependent permeability of fractured rock masses: a numerical study, Int J Rock Mech Min Sci (submitted).

V. Min KB, Rutqvist J, Tsang CF, Jing L, Thermally induced mechanical and permeability changes around a nuclear waste repository – a far-field study based on equivalent properties determined by a discrete approach, Int J Rock Mech Min Sci (to be submitted for the special issue of DECOVALEXIII/BENCHPAR projects).

Ki-Bok Min TRITA-LWR PHD 1011

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Fractured Rock Masses as Equivalent Continua – A Numerical Study

often too numerous to be represented explicitly in computer models. Proper ap- 1 INTRODUCTION proaches of property homogenization and upscaling are then needed to derive such 1.1 Motivation and objectives of the equivalent properties with both numeri- study cal reliability and capacity to simulate Fractures are common in rock (Figure 1). the processes of stress, deformation and Any engineering application in and on fluid flow in fractured rock masses. rock masses must take account of the presence of fractures for design, con- Existence of fractures alters the hydraulic struction and operation. Environmental and mechanical properties of rock masses impacts on the biosphere induced by the significantly. Fractures are more compli- rock engineering have to be investigated ant than intact rock and this makes the considering the discontinuous nature of mechanical properties of fractured rock the fractured rock masses. Especially, the masses substantially different from that inclusion of environmental factor has be- of intact rock. Fractures also act as main come increasingly important for the dis- pathways of fluid flow, especially for posal of toxic materials in underground hard crystalline rocks. With the inclusion such as nuclear waste repositories. Frac- of fractures, the permeability of fractured tured rock masses are often treated as rock masses can increase greatly. Aper- equivalent continua for large-scale analy- tures of fractures can change due to nor- ses because of the fact that the fractures mal stress-induced closures or openings in rock masses in practical problems are and due to shear stress-induced dilations; hence permeability of fractured rock

Figure 1. An Example of fractured rock masses in the dimensions about 5 m × 5 m (picture taken at a rock exposure at Forsmark, Sweden). In this thesis, ‘fractures’ are defined as any form of discontinuities such as joint, fault, fissure and veins that exist in rocks. The compos- ite entity of intact rock and fractures are described as ‘fractured rock masses’.

Ki-Bok Min TRITA-LWR PHD 1011

masses is stress-dependent. The deform- efforts to answer above questions with ability of rock fractures is highly non- the purpose of providing fundamental linear with much stiffer deformation at contributions rather than site-specific higher stress, and therefore, the deform- analysis. ability of fractured rock masses is stress- dependent. Given this importance of The objective of the thesis is to provide a fractures in behavior of the fractured systematic framework for conducting the rock masses, application of continuum equivalent continuum analysis of frac- modeling without consideration of the tured rock masses for large-scale analy- effects of fractures cannot warranty its ses. Systematic investigations were con- success and proper modeling of fractures ducted to develop the methodologies of in terms of its geometry and hydraulic determinations of equivalent mechanical and mechanical behavior is needed. and hydraulic properties of fractured rock masses, the conditions for their applica- In considering the effect of fractures, the tions to equivalent continuum analyses, discrete approach emerged as a powerful and the stress-dependencies of the hy- tool and has been enjoying its popularity draulic and mechanical properties, using since 1970s (Cundall, 1971). However, a hybrid discrete fracture network - one of its drawbacks is the limitation of distinct element method (DFN-DEM) computing power for modeling large- approach. The determined equivalent scale problems, usually in the order of properties are then used for a far-field kilometer scales, with extremely large analysis for the thermo-mechanical im- number of fractures. One alternative to pacts on the stress, deformation and this limitation is the equivalent contin- change of permeabilities of fractured uum approach. The equivalent continuum rocks surrounding a hypothetical geo- approach models the fractured rock logical repository of nuclear waste, using masses as a continuum with the equiva- a finite element method (FEM). The geo- lent properties that represent implicitly logical data of this study was extracted the effects of the fractures. However, from the results of an extensive site in- there are unresolved or difficult questions vestigation programme at Sellafield, UK, that have to be considered when applying conducted by Nirex UK Ltd. the equivalent continuum approach. This thesis begins with brief surveys on • How to determine the mechanical and the previous studies on the mechanical hydraulic properties of fractured rock and hydraulic properties of fractured rock masses considering the fracture sys- masses, coupled analysis and equivalent tem geometry and scales? continuum approach (Chapter 1.2 – • Is it appropriate to use equivalent Chapter 1.4). In chapter 2, the basic ‘continuum’ approach for the model- methodology adopted for this study is ing of ‘discontinuous’ fractured rock presented with the overview of the ap- masses? pended papers. Detailed description of • How significant is the influence of geological data is provided in Chapter 3 stress on hydraulic and mechanical and background theory and explanation properties of fractured rock masses? of numerical code is given in Chapter 4. The contents from Chapter 5 to Chapter 9 Above questions are especially challeng- are summaries of each paper appended in ing partly because it is very difficult to the thesis. obtain relevant information from in situ experiment. This thesis is the results of 2

Fractured Rock Masses as Equivalent Continua – A Numerical Study

1.2 Mechanical properties of frac- long history and several analytical solu- tured rock masses tions were proposed for cases of simple Considerable efforts have been made in fracture system geometry such as strati- the past to establish methodologies for fied rock (Salamon, 1968), staggered determining the equivalent mechanical fracture sets (Singh, 1973), orthogonally properties (essentially the elastic fractured rock masses (Amadei and modulus 1 and Poisson’s ratio) of frac- Goodman, 1981), stratified orthorhombic tured rock masses. Direct measurements layers (Gerrard, 1982) and randomly by in situ experiments with samples of fractured rock mass (Fossum, 1985). The large sizes are technically possible, but closed-form solutions have the advantage are costly and often involve some uncer- of being compact and straightforward, tainties related to the control of boundary but they work only for regular and often conditions and interpretation of results persistent and orthogonal fracture system (Bieniawski, 1978). Therefore, it is not geometries. It is difficult, often even im- surprising that significant research efforts possible, to derive closed-form solutions have been devoted to indirect ways such with general irregular fracture systems. as empirical, analytical and numerical The exception in this class of approach is methods. the crack tensor theory that has been applied to find anisotropic elastic properties The empirical methods ‘infer’ the rock with irregular fracture systems of differ- mass properties from the rock mass clas- ent sizes, orientations and mechanical sification or characterization results such properties (Oda, 1986). However, like all as RMR (Bieniawski, 1978), Q (Barton, analytical methods for fractured rock 2002), GSI (Hoek and Brown, 1997), masses, it does not consider the interac- RMi (Palmström, 1996) and Joint Factor tions between the fractures and the (Ramamurthy, 1993). Although they blocks divided by the fractures, which have gained wide popularity in practical may have significant impacts on the applications for design, especially in tun- overall behavior of rock masses because neling, it often gives too conservative the intersections of the fractures are often estimates for property characterizations, the locations with the largest stress and largely because it makes use of catego- deformation gradients, damages and fail- rized parameters based on case histories. ures. The main shortcoming of this approach is that it lacks a proper mathematical plat- With the rapid growth of our computing form to establish constitutive models and capacity, numerical methods are expand- the associated properties. Furthermore, ing their application from conventional the anisotropy and effect of stress on the stability analysis to various applications mechanical properties cannot be gener- such as understanding in situ stress (Hart, ally represented. 2003), shear behavior of rough rock fracture (Cundall, 2001), rock mass strength The efforts to find analytical solutions (Pouya and Ghoreychi, 2001) and for estimating the equivalent properties equivalent mechanical and hydraulic of fractured rock masses have a rather properties (Stietel et al., 1996). For the simulation of discontinuous rock, the ex-

1 The fact that fractured rock masses do not behave plicit approach of finite difference type elastically prompted the usage of the term ‘modulus method is very effective because it mod- of deformation’ (Bieniawski, 1978). In this thesis, els the physical instability without nu- however, the large-scale analysis is conducted as- merical instability. Especially, the capa- suming fracture rock masses behaves elastically bility of DEM (Distinct or Discrete Ele- and, therefore, the term ‘elastic modulus’ is used. 3

Ki-Bok Min TRITA-LWR PHD 1011

ment Method) allows the explicit repre- tive scales from large-scale experiments. sentation of fracture systems and the Therefore, a research on the stress- constitutive behaviors of fractures can be dependent mechanical behavior of frac- effectively incorporated for the determi- tured rock masses in a representative nation of mechanical properties of frac- scale, with explicit consideration of fractured rock masses, by performing ‘nu- tures, is needed. An instructive example merical experiments’ to emulate the in is introduced by Martin et al. (2003) who situ field testing conditions (Hart et al., re-evaluated the extensometer results at 1985; Bhasin and Hoeg, 1998; Ivars et the Olympic Ice Hockey Cavern in al., 2001). In comparison with the em- Gjøvik, Norway using a continuum pirical and analytical approaches, the model with equivalent rock mass numerical approach has a certain advan- modulus. They showed that predictions tage that the influence of irregular frac- using rock mass modulus from rock mass ture system geometry and complex con- classification were far smaller than the stitutive models of intact rocks and frac- measured results and accurate prediction tures can be directly included in the deri- was possible only when adjustment con- vation of the equivalent properties of sidering the nonlinear deformation of rock masses. However, in determining fractures was made. Therefore, consid- the mechanical properties, most of the eration of stress-dependent mechanical previous studies did not consider the ani- properties is needed for the design and sotropy of fractured rock masses, which analysis. By including appropriate consti- can be significant due to the role of frac- tutive models of fractures, numerical ex- tures. Stietel et al. (1996) presented a periments can be effectively employed to numerical experiment approach to de- investigate the stress-dependency, which termine all components of the elastic cannot be considered in current empirical compliance tensor of a fractured rock or analytical methods with adequate effi- mass by three independent boundary ciency and flexibility. conditions using a 2D DEM code, UDEC (Universal Distinct Element Code, Itasca, The Poisson’s ratio is one of the two 2000). However, the shear stress bound- most important elastic parameters of ary conditions prescribed by displace- rocks. However, it has not been given ments in only two sides in their model due attention because of its smaller influ- did not ensure the moment equilibrium ence compared with elastic modulus on and this will lead to inaccurate estimation the stress analysis and final displacement of compliance tensors. Therefore, estab- fields. However, it can be shown from lishing a more systematic methodology the Kirsch solution that Poisson’s ratio for the numerical experiments is required can have an important role in evaluating for characterization of the fractured rock the final displacement fields (e.g., Jaeger masses. and Cook, 1969) and a recent study suggests its larger role in radial displacement Behaviors and properties of fractured around the face of a tunnel (Unlu and rock masses are stress dependent due to Gercek, 2003). The problem, as was the highly nonlinear mechanical proper- pointed out by Boyle (1992), is that there ties of the fractures. While there exist is no ISRM (International Society for studies on the stress dependency of the Rock Mechanics) suggested method of intact rocks (Brown et al., 1989) and measuring the in situ Poisson’s ratio. In fractured rock (Ramamurthy, 1993), it is the ISRM suggested method for ‘deter- difficult to obtain stress-dependent rela- mining in situ deformability of rock’ tionships at large enough, i.e., representa- (Brown, 1981), Poisson’s ratio is simply 4

Fractured Rock Masses as Equivalent Continua – A Numerical Study

assumed without actual measurement. rock exposures of limited areas or bore- Therefore, it is a common practice to use holes, the reliability of DFN analysis is the values of intact rocks or assume the largely constrained by the quality of value of the Poisson’s ratio in the range characterization techniques and much of of 0.2-0.3. Further, a study suggesting research efforts has been spent to obtain the method of determining Poisson’s ra- more representative statistics of fractures tio recommended that the Poisson’s ratio from the limited data sets (e.g., La be discarded when it is higher than upper Pointe, 2002). limit of isotropic case (0.5) (Ünal, 1997). However, some literature indicated larger The DFN approach is, so far, an irre- Poisson’s ratios of fractured rocks (e.g., placeable tool for modeling fluid flow Bhasin and Hoeg, 1998; Wu and Wang, and transport phenomena at the ‘near- 2001) and, in fact, theoretical bounds for field’ scale because the dominance of the Poisson’s ratio for anisotropic rock are fracture geometry at small and interme- not limited to 0.5 (Pickering, 1970; diate scales can be approximated explic- Amadei et al., 1987). More systematic itly and in detail. This advantage dimin- investigations are required on the possi- ishes for ‘far-field’ problems at large ble range of Poisson’s ratio of fractured scales when explicit representation of rock masses from actual measurement. large numbers of fractures makes the computational model less efficient, and 1.3 Hydraulic properties of fractured the continuum model with equivalent rock masses properties become more attractive (Jing, In fractured hard rocks, the permeability 2003). In such large-scale problem, com- of rock matrix is generally very low and bination of DFN analysis and continuum fractures act as the dominating fluid con- porous analysis can be effective with the ducting pathways. In this case, under- equivalent permeability calculated by standing on the geometry of the fracture numerical experiments if the condition system and fluid flow along the fractures for the application of fractured rock is essential for hydraulic analyses. The masses to continuum approach is satis- key issue is the proper representation of fied (Long et al., 1982). Detailed discus- fracture geometry and apertures. Since sion about the applicability of fractured fracture exists in various forms of orien- rock masses to continuum analysis is tation, size and locations, systematic dealt with in next sections. analysis, including such geometrical factors, is needed using discrete approaches On the other hand, aperture values in a such as discrete fracture network (DFN) single fracture are not uniform (Hakami, methods. 1995) and field observations from Sella- field area indicate that the flowing fea- The DFN technique has been developed tures observed in the boreholes show in both two and three-dimensional forms marked spatial clustering (Andersson and and has been applied for many applica- Knight, 2000; Öhman and Niemi, 2003). tions in civil, environmental and other The apertures of fractures may also be engineering geosciences (Herbert, 1996). related to the sizes of the fractures The key process is to create probability (Renshaw and Park, 1997) and it is likely density functions (PDFs) of fracture pa- that the longer fractures are more con- rameters related to the geometry and ap- ductive even though its number is fewer ertures (Priest, 1993; Herbert, 1996; Der- than shorter fractures. Therefore the spa- showitz et al., 1998). Because mapping tial distributions of aperture values, non- of fractures can only be conducted at conducting fractures and role of long 5

Ki-Bok Min TRITA-LWR PHD 1011

fractures may change the flow character- much higher than that of other fractures istics of the rock masses concerned. not critically (optimally) oriented for These factors are basically site-specific shear failure (Barton et al., 1995). These information and are additional difficul- observations are supported by applica- ties in performing DFN analyses. tions to geothermal reservoirs to detect flow pathways (Ito and Hayashi, 2003). The permeability of fractured rock masses change according to the deforma- Despite much research works, an ade- tion of fractures caused by stress. This quate understanding of stress-induced change of permeability due to stress can permeability change has yet to be be viewed as an “indirect” hydro- achieved, especially regarding clear ex- mechanical coupling that occurs when planation of field observations. A study the applied stresses produce a change in of the excavation-induced disturbed zone the hydraulic properties, whereas a “di- at the Äspö Hard Rock Laboratory in rect” coupling occurs when the applied Sweden showed that transmissivities of stresses produce a change in fluid pres- the fractures around a tunnel sometimes sure and vice versa (Rutqvist and increased and sometimes decreased as a Stephansson, 2003). Laboratory investi- result of excavation, and it was not pos- gations on single rock fractures show that sible to make a firm statement about un- normal closure and shear dilation can derstanding of this change in fracture significantly change fracture transmissiv- transmissivity (Bäckblom and Martin, ity (Makurat et al., 1990; Olsson and 1999). The difficulty is mainly how to Barton, 2001). Beyond the observations represent the complex fracture system in single fractures, a number of studies geometry, with various orientations and have been conducted about the perme- finite sizes of fractures, and how to rep- ability alterations caused by fracture resent the complex mechanical deforma- normal deformation around the exca- tion mechanisms that are much influ- vated openings regarding the effect of enced by the interactions between indi- redistributed stresses and blast damage vidual fractures. Therefore, numerical (Kelsall et al., 1984; Pusch, 1989). A modeling has to be applied to deepen the zone where the failure of fractures can understanding of the critical mechanisms occur can be generated around the exca- and the contributions to the overall per- vated openings (Daemen, 1983), and the meability from both normal and shear corresponding fracture dilations in this stresses (Monsen et al., 1992; Damjanac zone can also be a source of permeability et al., 1999). change. A study based on orthogonal fracture system geometry shows that the Analytical models of stress-dependent extent of the disturbed zone that can permeability of fractured rock masses cause permeability changes can be sig- based on orthogonal fracture (Bai and nificant, depending on the fracture sys- Elsworth, 1994), arbitrarily oriented frac- tem geometry and the in situ stress condi- ture (Chen and Bai, 1998) and non- tions (Shen and Barton, 1997). orthogonal fracture (Bai et al., 1999) exist that consider the normal closures of Regarding the concentrated flow affected fractures and constant shear dilations in by stress, an illuminating investigation both fractured media and fractured- shows that there is very high correlation porous media. However, models based between the stress state and transmissivi- on persistent fractures have certain limi- ties of fractures and the transmissivities tations in simulating the shear dilations of critically oriented fractures can be and highly clustered flow paths resulting 6

Fractured Rock Masses as Equivalent Continua – A Numerical Study

from stresses. Oda’s permeability tensor of another and the behavior of the reposi- approach considers stress-dependency in tory system cannot be predicted with complex fracture networks (Oda, 1986). confidence by considering each process However, fracture connectivity and com- independently (Jing et al., 1995). Con- plex fracture interactions, which are im- siderable progress has been made in the portant factors affecting the overall hy- study of coupled THM processes during dro-mechanical behaviors of the frac- the past two decades (Stephansson et al., tured rock masses, cannot be considered 1996). A number of numerical codes in this approach. Especially, when the have been developed to model the fully shear failures and dilations of fractures coupled THM processes of fractured rock are to be considered, analytical solutions masses and engineered barrier systems do not generally exist. using finite element method (Noorishad et al., 1984; Ohnishi and Kobayashi, A series of works by Zhang and Sander- 1993; Nguyen and Selvadurai, 1995) and son show that the method of ‘numerical explicit finite difference method (Hart experiment’ using the distinct element and St John, 1986). code, UDEC, is effective in modeling fluid flow and deformation of fractured However, one of the remaining challeng- rock masses (Zhang and Sandersson, ing tasks is to characterize the physical 2002). Such numerical experiments will properties of the geological materials, be increasingly important due to the especially the fractured hard crystalline strength of DEM (Distinct or Discrete rocks, with consideration of the effects of Element Method) or modeling that can the fractures. Since the discrete approach incorporate both hydraulic and mechani- are not practical for large-scale problems cal analysis with explicit representation with explicit representation of extremely of fractures. The influence of stresses on large number of fractures due to the limi- permeability of fractured rock masses tation of computing power, equivalent was extensively investigated considering continuum approach must be used. In various geometries of fracture systems particular, the stress-permeability rela- (Zhang et al., 1996; Zhang and Sanders- tion is a key component for coupled hy- son, 1996; Zhang and Sandersson, 1997). dro-mechanical analysis when fractured However, the realistic fracture normal rock masses are modeled as continua. stress-displacement relationship and frac- Given that current numerical THM code ture shear dilations were not systemati- normally incorporate the relationship cally considered with properly selected based on intact rock experiments or regu- representative scales and therefore, fur- larly fractured rock masses, separate, ther research is needed. elaborate analysis to determine the realistic stress-dependent permeability of rock 1.4 Coupled analysis and equivalent masses with irregular fracture system ge- continuum approach ometry is needed. Studies exist that im- For the geological disposal of nuclear plicitly consider fracture effects for hy- wastes, the interactions between the rock dro-mechanical analysis (Cho et al., masses and repositories (especially with 1991) and thermo-mechanical analysis placement of high-level nuclear wastes) (Sasaki and Morikawa, 1996). However, are coupled phenomena involving ther- this class of approach is limited in prop- mal (T), hydrological (H), mechanical erly modeling the explicit geometry of (M) and chemical (C) processes (Tsang, fractures, the interactions among frac- 1987). Coupled processes imply that one tures and the behavior changes of frac- process affects the initiation and progress tures such as shear dilation. 7

Ki-Bok Min TRITA-LWR PHD 1011

tions of the local effects of the fractures, Equivalent continuum approach assumes especially near the excavations or other that macroscopic behavior of fractured sources of man-made or natural distur- rock masses can be described by princi- bances, are sacrificed by representing the ples of continuum mechanics, as long as averaged overall behavior. its constitutive relations and associated properties/parameters can be properly Long et al (1982) suggested that the fol- established according to the basic laws of lowing two conditions must be satisfied continuum mechanics. This is a common to justify the equivalent continuum ap- and often implicitly assumed modeling proach for fractured rock hydraulics. approach used in the fields of rock mechanics and hydrogeology, especially for • Firstly, there is an insignificant large-scale problems (Sitharam et al., change in the value of the equivalent 2001; Long et al., 1982). The equivalent permeability with a small addition or continuum approach has the advantage of subtraction to the test volume. being more suitable for representing the • Secondly, an equivalent symmetric overall behavior of fractured rock masses permeability tensor exists which pre- for problems of large scales, where the dicts the correct flux when the direc- effects of the fractures are implicitly con- tion of gradient in a representative tained in the equivalent constitutive elementary volume (REV) is models and associated properties and pa- changed. rameters. The more accurate representa-

Figure 2. Illustration of REV. (a) general concept. (b) Example data scatter (from Hudson and Harrison, 1997).

Fractured Rock Masses as Equivalent Continua – A Numerical Study

First criterion is related to the scale dependency and a ‘REV’ is defined as the minimum volume (or a range) beyond which the characteristics of the domain remain basically constant (Bear, 1972), as illustrated in Figure 2. In fractured rock masses, scale dependence of the properties is much affected by the existence of fracture. Second criterion was investigated by comparing the predicted flux and calculated flux (Long et al., 1982).

However, some studies indicate that great care should be taken regarding the application of equivalent continuum approach (Aydan, 1995; La Pointe, 1996). Regarding the existence of REV, contro- versy also exists for the justification of the REV concept for fractured rock mass (Pariseau, 1995). Generally, there is no guarantee that a REV always exists for a given fractured rock mass at a given scale (Neuman, 1987). Whether or not a REV can be established depend on the fracture system geometry, scale of models and properties of the individual fractures (La Pointe et al., 1996).

Therefore, careful examination on the appropriateness of equivalent continuum representation of fractured rock masses is needed, and the problem is basically site- specific. Moreover, even though two criteria suggested by Long et al. (1982) has been widely used in hydraulic analysis, systematic investigations for the mechanical analysis are rare except some efforts to determine the equivalent mechanical properties. Given that many continuum analyses are being conducted for mechanical problems of fractured rock masses, more systematic investigation regarding the applicability of equivalent continuum approach is required.

Ki-Bok Min TRITA-LWR PHD 1011

Fractured Rock Masses as Equivalent Continua – A Numerical Study

By adopting a contracted matrix form of Sijkl, Eq.(1) can be expressed as 2 METHODOLOGY AND SCOPE OF ε x S11 S12 S13 SSS14 15 16 σ x  THE STUDY   ε σ yySS21 22 S23 S24 S25 S26  First two sections of this chapter present  ε z SSSSSS σ z  the basic methodology of the thesis; 1) = 31 32 33 34 35 36  γ SSSSSS τ how to determine the equivalent me- yz 41 42 43 44 45 46 yz  γ SSSSSS τ  chanical and hydraulic properties (Chap- xzx51 52 53 54 55 56 z γ SSSSSS τ  ter 2.1) and 2) how to investigate the ap- xyx61 62 63 64 65 66 y propriateness of continuum approach for (2) fractured rock masses (Chapter 2.2). Fi- nally, overview of the appended papers is where matrix Sij is called the compliance given in the last section. matrix and above equation is called generalized Hooke’s law (Lekhnitskii, 2.1 Numerical experiments on frac- 1963). The symbols of εi and γij (i, j = x, tured rock masses (DFN-DEM y, z) denote the normal and shear strains, approach) and symbols of σi and τij (i, j = x, y, z) As a methodology to determine the me- denote the normal and shear stresses, re- chanical and hydraulic properties of frac- spectively. The compliance matrix can be tured rock masses, discrete fracture net- described explicitly by giving the physi- work-distinct element method (DFN- cal meaning of each element as combina- DEM) approach is used for the numerical tions of elastic moduli, Poisson ratios, experiments. DFN-DEM approach uses shear moduli and other technical con- the fracture system realizations as the stants of the solids (see Paper I). geometrical models of the fractured rock Since the shear strain and shear stress masses and conduct numerical experi- components associated with z-direction ments using DEM for the calculation of can be removed for two-dimensional mechanical and hydraulic properties. plane strain analyses, generalized Hooke’s law can be reduced to the fol- 2.1.1 Mechanical compliance tensor of lowing form: fractured rock masses Mechanical compliance tensor is calcu- ε xxSS11 12 S13 S16 σ   lated through numerical experiments us- ε SSSSσ ing a DEM code, UDEC (Universal Dis- yy= 21 22 23 26  (3) ε zzSSSSσ  tinct Element Code, 2000). Due to the 31 32 33 36  γ τ  induced anisotropy caused by fractures, xy SS61 62 S63 S66 xy  general anisotropic elasticity has to be considered for the numerical experiment. Considering the fact that fractures are modeled to have strikes only normal to The constitutive relation for general lin- the model plane in UDEC, the elastic ear elasticity can be expressed as modulus in z-direction (Ez) and Poisson’s ratio under z-directional stress can be

ε ij = Sijklσ kl (1) predetermined as the values of intact rock. Further, shear stress (τxy) does not induce any deformation in z-direction where εij and σkl are stress and strain ten- (S = 0). Therefore, considering the sors of a second order rank and Sijkl is the 36 compliance tensor of a fourth-order rank, symmetry conditions, S13 = S31, S23 = S32 involving 21 independent coefficients. and S36 = S63, which are associated with 11

Ki-Bok Min TRITA-LWR PHD 1011

y y+ ∆ y y+ ∆ y xy

x x x x x x

xy y y+ ∆ y y+ ∆ y B.C.(1) B.C.(2) B.C.(3) Figure 3. Three linearly independent boundary conditions for numerical experiments (Pa- per I, Paper III). z-directional stress and strains, eventually Consideration of stress-dependent me- only three components are independent chanical properties can be achieved using in each row of the matrix. Therefore, the relevant fracture model (BB model in three linearly independent stress bound- the thesis) through the repeated numeri- ary conditions are sufficient to determine cal experiments at different stress levels all components in the matrix shown in (used in Paper III). Eq. (3). 2.1.2 Hydraulic permeability of frac- Figure 3 illustrates the three linearly in- tured rock masses dependent boundary conditions (BC1, The basic assumptions for this analysis BC2 and BC3) used to produce the two- are that the rock matrix is impermeable dimensional compliance matrix. BC1 and the fluid flow occurs only through consists of biaxial normal stresses and the fractures and obeys the cubic law. A BC2 is created by sequentially increasing generalized Darcy’s law for anisotropic the normal stress in the y-direction. Simi- and homogeneous porous media (Bear, larly, BC3 is created by sequentially su- 1972) is used for the calculation of perimposing shear stress increments over equivalent permeability tensors. When the stress conditions of the final BC2. elevation head is neglected, the equation For calculation of the average strain val- can be described as followings ues in the domain, eleven parallel sampling lines within each model are set in both x- and y-directions, with the same kij ∂P distance in between. Strain components QA= (5) i µ ∂x are then calculated from the displacement j values according to the following strain- displacement relationship: where Qi = the flow rate, A = the cross- 1 section area of the DFN model, kij = the ε = (u + u ) (4) ij 2 i, j j,i permeability tensor, µ = the dynamic viscosity and P = the hydraulic pressure applied. where ui (i = x, y) is the displacement components along the sampling lines. Finally, the full components of compli- Figure 4 shows the boundary conditions ance matrix in two-dimensions can be for the calculation of permeability ten- obtained using final stresses and strains sors. The flow rates in the x- and y- of the model for each boundary condi- directions were calculated with a con- tion. stant hydraulic pressure gradient in the x- direction and the same calculation was conducted with the same constant hy- 12

Fractured Rock Masses as Equivalent Continua – A Numerical Study

P2 P1

Figure 4. Generic boundary conditions for calculation of the permeability tensor. P1 and P2 indicate the hydraulic pressure (Paper II). draulic pressure gradient in the y- continuum approach direction. Complete components of the 2D permeability tensors of the DFN 2.2.1 Unified two criteria for the models, kxx, kyy, kxy and kyx can be ob- equivalent continuum approach tained using these two sets of linearly Two criteria suggested by Long et al. independent generic boundary conditions (1982) for the application of equivalent (Long et al., 1982). porous medium are extended to mechanical problems that include the fourth-order Consideration of stress-dependent mechanical compliance tensor. Subse- permeability can be achieved by quently, the two criteria were investi- alternating the mechanical experiments gated to justify the application equivalent and permeability experiments as shown continuum approach to fractured rock in Figure 5 (used in Paper IV). Note that masses. the lateral boundaries are set impermeable unlike boundary conditions Two criteria for the continuum analysis in Figure 4 in order to avoid the effect of application can be generalized as follows possible overestimation of dilation in the to include general tensor property (Min, corners of the blocks (Figure 20 of Paper 2002). IV). 2.2 Appropriateness of equivalent • Firstly, there is an insignificant change in the value of the property (both equivalent permeability and P1 mechanical compliance tensor) with a small addition or subtraction to the Y impermeable test volume (criteria for REV). • Secondly, the derived equivalent

X X property can be represented in tensor P2 form to be used for the constitutive equation for continuum analysis (cri- Y Y P2 P1 teria for tensor quantity). X The extension of both criteria to me- Figure 5. Applications of stress boundary chanical property is natural in view of the conditions and calculation of equivalent definition of REV and tensor. The differ- permeability in the x- and y- directions ence between hydraulic and mechanical (Paper IV). properties is that compliance tensor in elasticity has a fourth-order rank whereas 13

Ki-Bok Min TRITA-LWR PHD 1011

permeability tensor in hydraulics has a ping operation with a 6 by 6 matrix is second-order rank. The comparison of introduced in a simplified form (Lekhnit- transformation of permeability and com- skii, 1963) pliance tensor is presented in Figure 6.

Since the rank of the tensor is defined SSij′ = mnqmiqnj (7) according to the number of direction cosines associated with the transformation where S′ is the compliance matrix in the (Fung, 1994), mechanical compliance ij transformed axes and Smn is the one in the tensor needs four direction cosine to de- original axes, respectively. The compo- fine the compliance tensor in transformed nent of the qij matrix is described in Ap- axis while permeability needs two direc- pended Paper I. Above equation is the tion cosine to define the permeability in basis for the investigation of tensor quan- transformed axis. tity investigation for the mechanical compliance tensor. The transformation of compliance tensor is defined by the following mapping op- The transformation of a permeability ten- erations, sor can be similarly defined as follows,

Sijk′ l = βim ββjn kp βlp Smnpq (6) kk′ = β β (8) ij mn im jn where S′ and S are the compliance ijkl mnpq tensors in the transformed and the origi- where k and k ′ are the permeability nal axes, respectively, and β is direc- mn ij im tensors in the original and rotated axes, tion cosines representing the transformation. When the compliance tensor is ex- respectively, and βim and βjn are the direc- pressed in matrix form, following map- tion cosines.

When the permeability has a tensor qual- k ∂p ity, an ellipse equation of the directional QA= ij kk′ = ββ i µ ∂x ij im jn mn permeability, in the direction of pressure j gradient, is given by (Bear 1972) Permeability tensor in 2nd order tensor generalized darcy’s law transformation xy22 + =1 (9) εσ= S SSijk′ l = βββim jn kp βlq mnp ij ijkl kl 1/kkxy1/ Compliance tensor in 4th order tensor generalized Hooke’s law transformation where kx and ky are principal permeabili- y ties in the direction of pressure gradient x y′ and y, respectively. Above equation is x′ Two coordinate often used for the validity of tensor quan- x systems for tensor tity of permeability. transformations z In this thesis, two measures of trunca- z′ tions have been suggested in order to evaluate the two criteria for the appropri- Figure 6. Comparison of transformations of permeability and compliance tensor ateness of equivalent continuum ap- and two coordinate systems (Paper I). proach: ‘coefficient of variation’ to be used to evaluate the variation from the 14

Fractured Rock Masses as Equivalent Continua – A Numerical Study

multiple realization of stochastic discrete Eq.(12) was used as the collective meas- fracture networks and ‘mean prediction ure of prediction error for comparisons at error’ to be used for the evaluation of er- different scales. ror involved in the prediction of compliance tensor in rotated axes. For hydraulic permeability tensor, two measures can be similarly defined fro A ‘coefficient of variation’ is defined as coefficient of variation and prediction the ratio of standard deviation over the error. The prediction error for permeabil- mean value of a certain property (say, ity tensor is given as elastic modulus or Poisson’s ratio) ob- NN tained from all random DFN models at a rr ∑∑kk11 −−11 k22 k22 given scale. 11rr==11 EPp12==, EPp Nk11 Nk22 For mechanical compliance tensor, pre- (13) diction errors in two-dimensional analysis are defined as where EPpi is the prediction error of permeability tensor in the i-direction (i = N 2 r x, y), k and k are the diagonal com- ∑∑s ij − sij 11 22 1 rj==11 EPci = (10) ponents in the average permeability ten- N 2 r r ∑ sij sor, and k 11 and k 22 are the diagonal j=1 components in the permeability tensors from numerical experiments at the r-th where EPci is the prediction error of rotated state. The average tensor at refer- compliance matrix in i direction (i = x, r ence axis are similarly calculated as y), s ij is the compliance matrix from numerical experiments at rotated DFN N 1 r models and N denotes the number of kkij = ∑ pq βip β jq (14) N r=1 cases of DFN model rotations (N = 6 for this study). sij is the average compliance r where k pq is the calculated permeability matrix defined at reference axis (original at rotated p-q axis. x-y axes in this study) as Note that, in evaluating the prediction errors, the influence of non-diagonal 1 N components of permeability tensor was S = S r q q (11) ij ∑ pq pi qj omitted for simplicity. The mean predic- N r=1 tion error ( µEPp ) was then defined as r where S pq are the calculated compliance matrices at rotated p-q axes, and qpi and 1 2 µEPp = ∑ EPpi (15) qqj are the matrices of direction cosines 2 i=1 of the rotations as defined in Paper I. Note that comparison is conducted at the These measures give a tool for the quan- same reference axes. titative evaluation of the uncertainties A mean prediction error (µEPc) can then related to the REV sizes and tensor rep- be defined as the mean values of the two resentations for the equivalent continuum EPci, given by application.

1 2 2.2.2 Methodology of investigation µEPc = ∑ EPci (12) 2 i=1 To investigate the two criteria for the application of the equivalent continuum 15

Ki-Bok Min TRITA-LWR PHD 1011

Figure 7. Schematic view of generations of fracture networks. The size of the square DFN model in the figure is 1 m × 1 m, 3 m × 3m and 5 m × 5 m, respectively (Paper I, II). approach, a series of numerical experi- clockwise direction with a 30-degree in- ments are conducted. terval (0˚, 30˚, 60˚, 90˚, 120˚, 150˚) and mechanical and hydraulic calculations For the investigation of the first criteria are conducted. The results of rotated for REV, multiple fracture system ge- model are then compared with the pre- ometry models were generated by a DFN diction based on the tensor transforma- generator based on the geological data tions (Eqs. (7) and (8)). provided, through Monte Carlo simulation processes. To properly represent the 2.3 Overview of appended papers stochastic nature of fracture system, ten Following the principles and methodolo- series of DFN models are generated to gies introduced above, this thesis is com- ensure that the calculated results are not posed of a series of works conducted as a dependent on one single realization of part of a BenchMark Test (BMT2) of the fracture geometry and to produce more international co-operative research pro- representative stochastic behavior of the jects DECOVALEX III (Acronym of fractured rock mass. From each gener- DEvelopment of Coupled models and ated parent fracture network, increasingly their VAlidation against EXperiments in large models were cut out from the center nuclear waste isolation)/BENCHPAR of the parent model, in sizes from 0.25 m (Acronym of BENCHmark tests and × 0.25 m to 10 m × 10 m scale (Figure guidance on coupled processes for Per- 7). The small size of 0.25 m and 0.5 m formance Assessment of nuclear waste are chosen to see the effect of the mini- Repositories). The objective of the mum cutoff length of 0.5 m. BMT2 was to determine how the upscaling process impacts the performance as- For the investigation of the second crite- sessment of geological repository at a ria for the tensor quantity, one of the se- far-field scale (Andersson and Knight, ries of the DFN models were rotated in 2000). 16

Fractured Rock Masses as Equivalent Continua – A Numerical Study

DFN-DEM FEM

Fractured Rock Masses Equivalent Continua

Mechanical properties (Paper I) Coupled THM analysis in far field for nuclear waste repository ε = S σ ij ijkl kl T (Paper V) Hydraulic properties (Paper II)

kij ∂P QAi = µ ∂x j H M

Established Determined REV methodology

Stress-dependent mechanical properties 11 1 =+ Stress-dependent properties EEmiSm⋅σ (Paper III)

Stress-dependent hydraulic properties

ffxd33x kbxx=+() ()dx 12 12 (Paper IV) ffyd33y kbyy=+() ()dy 12 12

Figure 8. Layout of the thesis.

The layout of the guiding principle, the Paper I Min KB, Jing L, Numeri- contents of each paper and its inter- cal determination of the equivalent elas- relations are presented in Figure 8. As tic compliance tensor for fractured rock noted, the combination of discrete and masses using the distinct element continuum approach is the key conceptu- method, Int J Rock Mech Min Sci; alization factor for this study and exten- 2003;40(6):795-816. sive investigations are carried out on the mechanical (Paper I) and hydraulic (Pa- Paper II Min KB, Jing L, per II) properties of fractured rock Stephansson O, Determining the Equiva- masses and their stress dependencies lent Permeability Tensor for Fractured (Paper III and Paper IV). In this thesis, Rock Masses Using a Stochastic REV large-scale THM analysis is confined to Approach: Method and Application to thermo-mechanical and hydro- the Field Data from Sellafield, UK, mechanical processes induced by thermal Hydrogeology Journal (in press). loading (Paper V, considered processes are indicated as solid line in T-H-M diagram in the Figure 8).

The followings five papers are appended in the thesis.

Ki-Bok Min TRITA-LWR PHD 1011

Table 1. Summary of main methodology and geometry of each paper.

Paper I Paper II Paper III Paper IV Paper V Geometry DFN DFN DFN DFN Hypothetical repository Fracture Constant - Barton- Step-wise - normal stiffness Bandis (BB) nonlinear behavior model model Fracture Constant - Barton- Elasto- - shear stiffness Bandis (BB) perfectly- behavior model plastic Aperture - Constant - Deformable - for (nonlinear) hydraulics Process Stress & Fluid flow Stress & Coupled Thermally considered deformation deformation stress-flow induced mechanical & permeability change Numerical UDEC UDEC UDEC UDEC ROCMAS Code

Paper III Min KB, Jing L, Stress the special issue of DECO- dependent mechanical properties and VALEXIII/BENCHPAR projects). bounds of Poisson’s ratio for fractured rock masses investigated by a DFN-DEM Paper I presents a methodology to deter- technique, Int J Rock Mech Min Sci; mine the equivalent elastic properties of 2004;41(3):431-432, special issue of SI- fractured rock masses and to investigate NOROCK2004, Int Symp on Rock its appropriateness for the equivalent Mechanics, Rock Characterization, Mod- continuum approach for representing the elling and Engineering Design Methods, mechanical behavior of fractured rock Three Gorges Project Site, China (Paper masses. Equivalent mechanical compli- No. 2A13). ance tensor is calculated using the proposed methodology and the investigation Paper IV Min KB, Rutqvist J, shows that the representative elementary Tsang C-F, Jing L, Stress-dependent volume (REV) can be defined at around permeability of fractured rock masses: a 5 m x 5 m scale for the equivalent con- numerical study, Int J Rock Mech Min tinuum approach. Sci (submitted). Paper II presents an evaluation of the Paper V Min KB, Rutqvist J, equivalent permeability tensor of frac- Tsang C-F, Jing L, Thermally induced tured rock masses. The methodology pre- mechanical and permeability changes sented by Long et al. (1982) is used to around a nuclear waste repository – a far- determine the equivalent 2D permeability field study based on equivalent properties tensor at the multiple realizations of determined by a discrete approach, Int J models. The investigation shows that the Rock Mech Min Sci (to be submitted for representative elementary volume (REV)

Fractured Rock Masses as Equivalent Continua – A Numerical Study

Table 2. List of generated DFN models and their employment for each paper. Capital Roman character indicates the paper number. DFN1_30 ~ DFN1_50 are rotated models from DFN1 in 30 degrees intervals. DFN11~DFN50 are additional forty models used for Hydraulic analysis.

Side length of square model (m) Geometry 0.25 0.5 1 2 3 4 5 6 7 8 9 10 DFN1 I, II I, II I, II I, II I, II I, II I, II, III, IV I, II I, II I, II II II DFN2 I, II I, II I, II I, II I, II I, II I, II, III I, II I, II I, II II II DFN3 I, II I, II I, II I, II I, II I, II I, II, III I, II I, II I, II II II DFN4 I, II I, II I, II I, II I, II I, II I, II, III I, II I, II I, II II II DFN5 I, II I, II I, II I, II I, II I, II I, II, III I, II I, II I, II II II DFN6 I, II I, II I, II I, II I, II I, II I, II, III I, II I, II I, II II II DFN7 I, II I, II I, II I, II I, II I, II I, II, III I, II I, II I, II II II DFN8 I, II I, II I, II I, II I, II I, II I, II, III I, II I, II I, II II II DFN9 I, II I, II I, II I, II I, II I, II I, II, III I, II I, II I, II II II DFN10 I, II I, II I, II I, II I, II I, II I, II, III I, II I, II I, II II II DFN11~ II II II - - - II - - - - II DFN50 DFN1_30 I, II I, II I, II I, II I, II I, II I, II I, II I, II II II II DFN1_60 I, II I, II I, II I, II I, II I, II I, II I, II I, II II II II DFN1_90 I, II I, II I, II I, II I, II I, II I, II I, II I, II II II II DFN1_120 I, II I, II I, II I, II I, II I, II I, II I, II I, II II II II DFN1_150 I, II I, II I, II I, II I, II I, II I, II I, II I, II II II II can be defined at around 5 m x 5 m scale pressive stress. Fluid flow under various for the equivalent continuum approach. stress conditions suggests a phenomenon of stress-induced flow channeling and In Paper III, the stress dependent me- permeability anisotropy. chanical properties and bounds of Pois- son’s ratio are investigated using the Bar- Paper V presents a numerical investiga- ton-Bandis (BB) fracture model based on tion on the impacts of the thermal load- the methodology established in Paper I. ing history on the evolution of mechani- The results show that mechanical proper- cal response and permeability field of a ties are highly stress-dependent and the fractured rock mass containing a hypo- bounds of Poisson’s ratio can be well thetical nuclear waste repository. The above the upper limit of the isotropic results obtained from Paper I – Paper IV case. An empirical equation of stress- are passed on to the far-field study in the dependent elastic modulus is proposed. scale of 5 km (width) by 1 km (height) and the importance of proper determina- Paper IV presents the stress-dependent tion of properties are demonstrated. permeability and proposes a set of empirical equations for a more general de- For Paper I, II, III and IV, UDEC is used scription of stress-dependent permeabil- as a numerical tool of and a FEM code, ity. Results show that, depending on ROCMAS is used for Paper V for the stress state, permeability can both in- large-scale application. The summary of crease or decrease with increasing com- main methodology and geometry for 19

Ki-Bok Min TRITA-LWR PHD 1011

50 m 150 m T=11°C Vertical Fault 0.5 km Formation 2 No heat flow No heat flow No fluid flow No fluid flow Repository 0.5 km Formation 1

Basal heat flow=54mW/m2 No fluid flow 5 km

Vertical Fault Formation 2 5 m

10 m 20 m 10 m 2.5 km to the 100 m left boundary Repository Formation 1

Figure 9. Reference model and boundary conditions for the Coupled THM analysis in far field used for Paper V (adapted from Andersson and Knight, 2000). each paper are listed in Table 1. Different excavation effect in the repository. In this fracture constitutive models are selected study, the properties of the model is se- for the given purpose. Since current Bar- lected at corresponding depth and the ton-Bandis model is incomplete in prop- properties of fault zone are actually not erly considering the hydromechanical distinguished. coupling during the shear deformation (Olsson and Barton, 2001), it was used only for mechanical calculation

Table 2 presents the complete lists of generated models used for Paper I, II, III and IV. In Paper I and Paper II, numerical experiments are conducted on multiple realizations in varying sizes and rotated models for investigation of continuum approach. In Paper III and Paper IV, stress-dependencies are investigated on selected REV sizes (5 m scale in this study). For Paper IV, only one realization (DFN1) is selected due to the excessive computing time required to conduct both mechanical and hydraulic analysis, while Paper III conducted numerical experiments on ten realizations at REV scale.

The reference model used for Paper V is shown in Figure 9 as defined in the BMT2. The disturbance is only by the thermal loading without consideration of 20

Fractured Rock Masses as Equivalent Continua – A Numerical Study

CH37 in Figure 10) from one- dimensional scan-line data and outcrop 3 GEOLOGICAL DATA AND GE- trace mappings. For this study, the minimum and maximum cut-offs of trace- OMETRY OF THE FRACTURE lengths are set to be 0.5m and 250m, re- SYSTEM spectively, which correspond to the observed fracture distributions. 3.1 Geological data

The fracture system for this study is Table 3 shows the basic information based on the results of a site characteriza- about the fracture systems and properties tion programme at the Sellafield area, of intact rocks and fractures. Fracture Cumbria, England undertaken by the data are based on the Coupled Shear United Kingdom Nirex Limited (Nirex, Flow Tests (CSFT) of rock fractures, 1997), concerning the formations in the which measures the permeability change Borrowdale Volcanic Group, a thick se- during the closure and shearing of frac- quence of Ordovician volcanoclastic tures. The normal stiffness of fractures rocks. Only a limited part of the charac- was taken as the mean value of the nor- terization results were taken for defining mal stiffness of four samples, determined the BenchMark Test (BMT2) for the from the fourth-cycle of displacement- DECOVALEX III/BENCHPAR projects, load curves to consider the effects of the and therefore the data and conclusions in this thesis and appended papers do not necessarily represent the complete results and conclusions from the entire site characterization at Sellafield area.

From the site investigation, four sets of fractures were identified and the orientations of fracture sets follow Fisher distributions. The data set shows highly fractured rock condition with high fracture densities and the highly dispersed patterns with low Fisher constants. Fracture trace lengths are characterized by a power law between the number (N) of fractures of trace-length larger than a given value L (m) per unit area (m2), as given by

N=×4 L−D (16) where D is the fitted fractal dimension Figure 10. Plot of length against number equal to 2.2 ± 0.2; the value 2.2 was used per m2 for fractures from surface map- for this study. Figure 10 shows the fractal ping sites CH22 and CH37, and for long nature of fracture trace-length distribu- fracture samples from the photolinea- tion as fitted by Eq.(16). The relation was ment. Data fall on a line establishing a obtained from combined analysis of ae- power-law scaling relationship for trace- rial photography lineaments and mapping length in the range of 0.5 to 250 m (Ni- rex, 1997). results at two sites (named CH22 and 21

Ki-Bok Min TRITA-LWR PHD 1011

Table 3. Geological data and model parameters used in this study.

Parameter Elastic modulus (GPa) 84.6 Intact rock Poisson’s ratio 0.24 Dip/dip direction (4 sets) 8/145, 88/148, 76/21, 69/87 Fisher constants (4 sets) 5.9, 9.0, 10.0, 10.0 Fracture density per set (m-2) 4.6* Normal stiffness (GPa/m) 434 Shear stiffness (GPa/m) 434, 86.8 Cohesion (MPa) 0 Friction angle (˚) 24.9 Dilation angle (º) 5 Critical shear displacement for dilation, U 3 Fractures cs (mm) Joint Roughness Coefficient (JRC, scale 3.85 0.3 m) Joint wall compressive strength (MPa) 112.21 Initial mechanical aperture at first cycle 77 (µm) Initial hydraulic aperture at first cycle (µm) 65 Initial aperture at fourth cycle (µm) 30** Maximum aperture at fourth cycle (µm) 50** Residual aperture at fourth cycle (µm) 5**

*Fracture density is calculated from Eq. (16).

** No distinction was made between hydraulic and mechanical aperture. disturbance during the specimen acquisi- for handling of large amount of data. In tion. The shear stiffness values were as- order to construct a DFN model, Monte sumed to be 20% and 100% of normal Carlo Simulation is performed on each stiffness for sensitivity studies. These Cumulative Density Function (CDF) of mechanical properties of fractures were location, orientation and trace length of assigned to all four fracture sets. fractures.

3.2 Discrete Fracture Network A Poisson process was applied to gener- (DFN) generation ate the locations of the fracture centers DFN (Discrete Fracture Network) mod- and therefore, the locations of fractures els are generated to represent the frac- are random in space. Figure 11 presents tured rock masses and generated geome- the locations of centers of fractures in a tries are passed on to UDEC model for DFN model of 10 m × 10 m in size, as an the numerical experiments. An independ- example. The number of generated frac- ent DFN Generator was developed for tures are based on the density of fractures the thesis based on the original program calculated by Eq.(16). by Jang et al. (1996). Further development was made to include the boundary Fracture trace lengths are generated with effect, Fisher distribution and improve- the cumulative probability density func- ment of efficiency of the code structure tion of the trace length, which can be 22

Fractured Rock Masses as Equivalent Continua – A Numerical Study

where cutmin and cutmax denote the mini- 5 mum and maximum cut-offs of the trace lengths, F denotes the random probability of a uniform distribution in the range 0 ≤ F ≤ 1 and L is the trace length of the fractures. 0 Figure. 12 shows the curves of cumulative probability of fracture trace lengths Y coordinate (m) with the given fractal dimension 2.2. Due to its fractal nature, more fractures of smaller sizes are concentrated in the -5 modeling region. For a fractal dimension -5 0 5 X coordinate (m) of 2.2, more than 95% of fractures have the trace-lengths less than 2 m and the Figure 11. Locations of centers of fractures calculated mean trace length is 0.92 m. determined by a Poisson process (example For the purpose of comparison, the curve at 10 m × 10 m scale, Paper II). for a smaller fractal dimension of 1.2 is also plotted in Figure. 12. In this case, determined from the given cumulative more than 95 % of fractures are less than density of fractures with minimum and 6 m in length, with a mean trace length maximum cutoff lengths. The following of 2.1 m. cumulative probability density function of trace length (L) is derived using the Orientations of the fractures are assumed fractal dimension (D) from Eq.(16) to follow the Fisher distributions and the deviation angle (θ) from the mean orien- 1 − tation angle is generated by the following L =−cut −−DDF(cut −cut −D)D (min min max ) cumulative probability density function (17) for the Fisher constant K, given by (Priest, 1993)

1.0

0.8 ty i

0.6 babil ve pro i 0.4 Fractal dimension,D=2.2 Fractal dimension,D=1.2

Cumulat 0.2

0.0 0 5 10 240 250 0.5 m Fracture trace length (m)

Figure 12. Fracture length distribution with fractal dimension, D. D=2.2 is for this study and D=1.2 is plotted for the purpose of comparison.

Ki-Bok Min TRITA-LWR PHD 1011

fact that the centers of some large frac- KK−K −1 ln(eF−−(ee) tures may not necessarily lie inside the θ = cos  (18) K domain of computational models and this can result in the underestimation of ef- The Fisher distribution is one of the most fects of larger fractures intersecting the popular ways of describing a spherical computational domain. To avoid this data such as fracture orientations. When boundary effect, the parent DFN models K is large, the distribution will have very should be at least larger than the maxi- concentrated form around the true angle mum cutoff length of fractures. In this and, with small K, the distribution will study, test models were cut from the cen- have larger variance (Mardia, 1972). ter of sufficiently large parent DFN models of 300 m × 300 m in size to account As the deviation angle is an one- for effects of the maximum trace length dimensional expression measured from of 250 m. the mean normal direction of a fracture set, this must be converted to a three di- Ten series of DFN models are generated mensional form by rotating the generated to ensure that the calculated results are normals about the mean normal of the not dependent on one single realization fracture set through a random angle taken and to produce more representative sto- from a uniform distribution in the range chastic behavior of the fractured rock of 0 to 2π (Priest 1993). This process is mass. Figure 13 shows examples of gen- usually facilitated by the use of spherical erated fractures of the ten realizations coordinates and transformation of axis (DFN1 to DFN10) at the 5 m × 5 m (Dershowitz et al., 1998). The generated scale. It should be noted that all models orientations are then converted to a two- have slightly different fracture patterns dimensional form for this study. depending on the individual Monte Carlo Simulations even though they have the A boundary effect can be caused by the same fracture statistics. In UDEC code, the generated fractures are ‘regularized’

Figure 13. Discrete fracture network models with 10 realizations (5 m × 5 m scales are shown here as examples). DFN1, DFN2…DFN10 denotes the different random realization.

Fractured Rock Masses as Equivalent Continua – A Numerical Study

so that the ‘dead-ends’ and ‘singly connected’ fractures are deleted. This makes no difference for hydraulic analysis since they would not contribute to fluid flow for permeability evaluation, and it is assumed that their effects is not significant for mechanical analysis.

Ki-Bok Min TRITA-LWR PHD 1011

Fractured Rock Masses as Equivalent Continua – A Numerical Study

When there is a plane of transverse isotropy parallel to the xy plane, and this ma- 4 THEORY AND NUMERICAL terial is called a transversely isotropic material, with the stress-strain relation CODE DESCRIPTION given by (e.g. Amadei et al., 1987) 4.1 Constitutive equation of fractured 1 νν′ rock masses −− 000 EEE′  In most practical cases, anisotropic rocks νν1 ′ −−000 ε x σ x  are modeled as orthotropic or trans-  EE E′  ε σ versely isotropic materials in a co- yyνν′′1  −− 000 ε z EE′′E′ σ z coordinate system attached to their direc- =  γ 1 τ tions of symmetry. The material, which yz 000 00yz  γ G′ τ  has three orthogonal planes of elastic xzxz γ 1 τ  symmetry at each point, is called xyx0000 0y G′ orthotropic (Lekhnitskii, 1963). For the 1 00000 orthotropic material, generalized G Hooke’s law, Eq.(2) has the following (20) form,

where, E=Ex=Ey, E′=Ez, ν=νxy=νyx, 1 ν yx ν  −−zx 000 E EExyEz ν′=νzx=νzy, G = 2(1+ν ) ννxy 1 zy −−000 ε x EExyEz σ x   ε σ The strain energy (W) can be defined, yν xz ν yz 1 y −− 000which must be positive, as shown in the ε z EEE σ z  = xyz  following. γ yz 1 τ yz  γ 000 00τ  xzxGyz z γ τ  1 xy1 xy W = σ ε (21) 0000 02 ij ij Gxz 1 00000   Gxy σ ij Sijklσ kl > 0 (22) (19) This condition implies that the 6 × 6 ma- where, Ex, Ey and Ez are the Young’s trices of elastic constants must be posi- moduli with respect to direction x, y, z, tive definite (Ting, 1996). For isotropic respectively. Gyz, Gzx and Gxy are the case, following constrains can be ob- shear moduli for elastic symmetry tained. planes, which are parallel to the yz, zx, xy planes, respectively. The Poisson’s ratios E > 0 ν determine the ratio of strain in the j 1 ij −1<<ν (23) direction to the strain in the i direction 2 due to a stress acting in the i direction. For example, νxy used in Paper III is the Undoubtedly, negative Poisson’s ratio is ratio of increase of strain in y-direction to theoretically possible and experimental the decrease of strain in x-direction with evidences exist (Baughman et al., 1998). the compression in x-direction. However, this may not be applicable for engineering rock materials in relatively larger scales. Above constraints become 27

Ki-Bok Min TRITA-LWR PHD 1011

invalid when the assumption of isotropy The superscript, i, denote the properties is not adopted. For example for the of intact rock and definitions of elastic transversely isotropic rock defined in moduli, shear moduli and Poisson’s ratio (20), the bounds of elastic constants are is the same as Eq.(19). Knx, Kny, Knz are as follows (e.g., Amadei et al., 1987), the normal stiffness of fractures in x, y and z directions, Ksx, Ksy, Ksz are the EE,,′′G> 0 shear stiffness of fractures in x, y and z directions, and Sx, Sy, Sz are the spacing −<1ν <1 (24) of fracture sets measured in x, y and z directions, respectively (cf. Figure 5 of EE′′(1 −−ν ) (1 ν ) Paper I). −<ν ′ < EE22 4.2 UDEC code (Universal Distinct Element Code) When orthotropic material is considered above constraints needs to be more re- In the distinct element method, rock mass laxed and it is obvious that Poisson’s ra- is represented as an assembly of discrete tio of anisotropic material is not confined bodies and fractures are considered as to below the upper limit of isotropic ma- interfaces between distinct bodies. The terial, 0.5. formulation and development of the distinct element method has progressed The compliance tensor of a rock mass since the initial presentation in Cundall with three sets of orthogonal persistent (1971, 1980) and the method has also fractures in three-dimension is con- been developed for applications in parti- structed by superimposing the fracture cle flow processes for granular and constitutive relations on the compliance bonded materials (Cundall and Strack, matrix of intact rock, by treating the rock 1979). Details of the principles of the mass as an orthotropic elastic material method can be seen in Itasca (2000). (Amadei and Goodman, 1981), given by

i  ν i  11 yx νzx  +− − 000 iii  EEKSnx x E   xyz   ii  ννyx 11 zy  −+− 000 ii i  EEKSny y E   yy z  i  i ν  νzx zy 11  −− + 000 iii  EEEKSnz z  (25)  zzz   11 1  000++ 0 0  i KS KS   Gyz sy y sz z   11 1   000 0 ++ 0 i KS KS  Gxz sx x sz z    11 1  000 0 0 ++  i KS KS  Gxy sxx syy

Fractured Rock Masses as Equivalent Continua – A Numerical Study

Figure 14. Components of the explicit, dynamic solution scheme: (a) explicit finite difference method; and (b) distinct element method. (Hart, 2003). F ()t 4.2.1 Basic features uu()tt+∆ 22= ()t−∆t ++∑ i g∆t iii UDEC uses an explicit time-marching m scheme to solve the equations of motion (26) directly. Every derivative in the set of governing equations is replaced directly where u is the velocity components of by an algebraic expression written in i block centroid, F is contact force vector, terms of field variables (i.e., stress or i g is the components of gravitational ac- force and displacement) at discrete points i celeration (body forces). The new veloci- in space. Using the complete dynamic ties in Eq. (26) are used to determine the equations ensures that the numerical new block location according to the cen- scheme is stable when the physical sys- tral difference scheme. The motion of tem being modeled is unstable. rotation can be similarly formulated.

The explicit calculation cycle solves two For the deformable blocks, the law of sets of equations: motion (Newton’s sec- balance of angular momentum is auto- ond law of motion) and constitutive. The matically satisfied due to the symmetry equation components are shown in Fig- of the stress tensor and the translational ure 14. equations of motion at a grid point are

given by For rigid block, the equations of motions for translation is given by σ ij ndj s+ Fi ug =+∫s (27) iim

Ki-Bok Min TRITA-LWR PHD 1011

where s is the surface enclosing the mass, form and invert a stiffness matrix and m is the mass at the gridpoint, nj is the non-linear problems involving large dis- unit normal to s, Fi is the resultant of all placements can be solved with little in- external forces applied to the gridpoint crease in computational cost relative to and gi is the gravitational acceleration. linear problems. During each time step, strains and rotations are related to nodal displacements 4.2.2 Mechanical behavior of a frac- in the usual fashion. ture The basic constitutive model for fractures In both sets in Figure 14, variables on the used in UDEC captures several of the right-hand side of expressions are known features, which are representative of the and can be regarded as fixed for the dura- physical behaviors of rough rock frac- tion of a calculation step, i.e. forces and tures. In the normal direction, the stress- strain rates or relative velocities are fixed displacement relation is assumed to be alternately. Consequently, each element linear and governed by the stiffness (kn) in this type of model appears to be physi- such that cally isolated from its neighbors during one calculation step. Thus, nonlinear ∆σn = − kn ∆un (28) constitutive relations can be implemented without difficulty, because only local where ∆σn is the effective normal stress conditions are relevant during the calcu- increment, and ∆un is the normal dis- lation step. No iterations are necessary to placement increment. Tensile strength is follow nonlinear laws, and no matrices also defined, above which the stress be- are formed. The solution scheme applies comes zero. Similarly, in shear direction, equally well for both continuum and dis- the response is controlled by constant continuum numerical models. In contin- shear stiffness, ks. Step-wise stiffness uum models, the scheme is implemented model can also be used by changing the as an explicit finite-difference (EFD) stiffness at designated stages and this formulation. In discontinuum models, the model is used for Paper IV to consider distinct element method (DEM) embod- the stress-dependent fracture behavior ies the explicit, dynamic scheme. UDEC (see Figure 4 of Paper IV). uses the DEM scheme to simulate motion at fracture contacts between rock blocks Shear behavior is modeled as an elasto- and the EFD formulation to simulate de- perfectly-plastic model where the shear formation within the blocks (Hart, 2003). stress,τ , is limited by a combination of s cohesion (c) and frictional angle (φ) fol- The equations of motion are solved using lowing the Coulomb failure criteria as the procedure of ‘dynamic relaxation’ follows. with numerical damping, and then inte- grated in time to reach steady state condi- ∆τ =−k∆ue for tions using the central difference integra- s ss tion scheme. τ sn≤+c σφtan =τmax (29)

e The DEM has been found to be an effi- Where ∆us is the elastic component of cient method for numerical analysis of the incremental shear displacement. geomechanical problems due to relatively modest computer memory re- A series of empirical relations of fracture quirements to analyze large discrete behavior developed by Barton and models – because it is not necessary to Bandis (Barton, 1982) is implemented in 30

Fractured Rock Masses as Equivalent Continua – A Numerical Study

the UDEC (UDEC-BB model, Itasca, 2000) and it is used for the analysis of The shear stress approaches the shear Paper III. Barton-Bandis (BB) model is strength incrementally by multiplying the capable of modeling the hyperbolic shear displacement increment by the stress-displacement paths of fracture shear stiffness. The shear stiffness is de- normal behavior and dilation of fracture fined by two initial linear segments of the due to shear stress, as a function of nor- load path, depending on shear displace- mal stress and shear displacement, ment. among other characteristics. Figure 15 shows an example of fracture The equation that controls the normal normal and shear behavior calculated by stress-displacement path for the Barton- Barton-Bandis equation with the geo- Bandis (BB) model is: logical data used in this study (Paper III).

Normal stiffness of fracture (GPa/m) 0 20000 40000 60000 80000 100000 − u K 50 σ = nc ni (30) n u nc 1st cycle 4th cycle 1 − 40 vmi ) a P M where unc is the current normal dis- ( 30 s s re placement (mm), Kni is the initial normal t s l stiffness (MPa/mm), and vmi is the a 20

rm Mechanical aperture

o Normal stiffness maximum allowable closure (mm) for N

10 load cycle i. The initial normal stiffness 1st cycle and maximum allowable closure are 4th cycle 0 determined by the fracture wall 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 compressive strength (JCS), initial Mechanical aperture (mm) (a) aperture of fracture and Joint Roughness 25 Coefficient (JRC). Normal stress = 40 MPa The shear resistance of a fracture is cal- 20

) 30 MPa culated using the concept of mobilized a P

M 15 roughness in the BB model. The mobi- ( ss

e 20 MPa lized roughness coefficient, JRC , is a r

mob st

r 10 a function of the joint properties: length, e h normal load, current shear displacement, S 10 MPa and shear displacement history. The rela- 5 tion between normalized shear displace- 0 0 0.001 0.002 0.003 ment (δ/δpeak) and the normalized rough- Shear displacement (mm) ness coefficient (JRCmob/JRCpeak) is set in (b) a form of table and this determines the l shear behavior. Shear strength, σ s, is cal- Figure 15. Fracture deformation charac- culated by: teristics studied in this paper, calculated by Barton Bandis (BB) model. (a) Normal, and (b) shear deformation (Paper III). l σ s = σ n tan()JRCmob log10 ()JCS n σ n + φr (31)

where φr is residual friction angle.

Ki-Bok Min TRITA-LWR PHD 1011

Figure 16. Fluid/solid interaction in UDEC (ITASCA, 2000).

where kj is a fracture permeability factor 4.2.3 Fluid Flow in fractures (whose theoretical value is 1/12µ), µ is UDEC has the capability to perform the the dynamic viscosity of the fluid, a is analysis of fluid flow through the frac- the contact hydraulic aperture and l is the tures of a system of impermeable blocks. length assigned to the contact between A fully coupled mechanical-hydraulic domains. analysis can be performed, in which frac- The hydraulic aperture is then given by ture permeability is dependent on mechanical deformation and, conversely, a = a + u (33) water pressures in fractures affect the 0 n mechanical computations. The effects modeled in UDEC are summarized in where a0 is the fracture aperture at zero Figure 16. Note that pressure effect for normal stress and un is the normal dis- effective stress of fracture was not placement of the fracture (positive denot- needed in this study since only the effect ing opening). A minimum value ares and of stress to aperture change (one-way maximum value amax is assumed for aper- coupling) is considered (Paper IV). tures, below and beyond which mechani- In the case of an edge-edge contact, the cal closure does not affect the contact flow rate is given as follows using the permeability. cubic law in a planar fracture The pressure (p) of a domain in the frac- ∆p ture is defined as q = − k a 3 (32) j l

Fractured Rock Masses as Equivalent Continua – A Numerical Study

∆tV∆ of different applications such as heat pp=+KQ −K (34) 0 wVVw tests, tunnel excavations, a flow injection m test and modeling the performance of a

nuclear waste repository (Noorishad et where p is the domain pressure in the 0 al., 1984; Noorishad et al., 1992; preceding time-step, Q is the sum of flow Rutqvist et al., 1992; Rutqvist et al., rates into the domain from all surround- 2001b). In this study, thermo-mechanical ing contacts, K is the bulk modulus of w analysis is conducted without consider- the fluid, ∆V is the change of volume due ing thermo-hydraulic coupling, with the to the deformation, V is the average m focus placed on the impact of thermal volume of new and old domain areas. loading on mechanical response and The mass continuity equation was then permeability change. described at the fracture intersections and regions (named ‘domains’ in UDEC) between intersections. This equation is then solved through an iterative scheme in combination with the prescribed boundary conditions. Since the purpose of the current study (Paper II, IV) is to evaluate permeability of the network models, steady state flow with generic hydraulic pressure boundary conditions is used. In steady state flow calculation, the change of volume is neglected and the fluid bulk modulus is automatically defined in the code.

4.3 ROCMAS code (ROCk Mass Analysis Scheme) For the large-scale analysis, the ROCMAS code (ROCk Mass Analysis Scheme) is used. ROCMAS is a finite- element code developed for analysis of coupled THM processes in saturated- unsaturated fractured porous media. The original formulation of coupled thermo- hydroelasticity in terms of Biot's theory of consolidation (Biot, 1941) is extended to include partially saturated media for heat and moisture flow. In the formulation of ROCMAS, three balance equations—water mass balance, energy con- servation and linear momentum balance—and a number of constitutive relations are required for a full description of the THM states, leading to a set of governing equations. Detailed descriptions can be found in Rutqvist et al. (2001a). ROCMAS has been applied in a number

Ki-Bok Min TRITA-LWR PHD 1011

Fractured Rock Masses as Equivalent Continua – A Numerical Study

s 1.0 u l 60 K0.1 u K0.2 od 0.8 K0.5

ECHANICAL PROPERTIES OF m 5 M K1.0 c i FRACTURED ROCK MASSES 0.6 K2.0 ast K5.0 l 30 e

APER d (P I) e 0.4 s i al

m Y r 0.2

5.1 Methodology o X N Numerical experiments are conducted on 0.0 0 the DFN models taken from the ten realizations of square models with varying Figure 17. Verification of UDEC model- side lengths from 0.25 m to 8 m (Table 2, ing results against analytical solution for Figure 7). The numerical experiments a fractured rock mass with two orthogo- were repeated for the rotated DFN mod- nal sets of fractures. Points of symbols els for the investigation of tensor quan- correspond to numerical results with dif- tity of compliance tensor. Since constant ferent K ratio values and the lines are the analytical solutions. K is defined as the fracture stiffness was used in this study ratio of shear stiffness to normal stiffness (Paper I), the calculated elastic compli- of fracture. ance tensor is independent of stress.

5.2 Verification 5.3 Results of calculated elastic Verification of the methodology is made moduli and Poisson’s ratio against the closed-form solutions using Figure 18 present the normalized elastic Eqs. (7) and (25). A model of orthogonal moduli in the x- directions with ten DFN fractures with equal space (Figure 11 of realizations of increasing side lengths Paper I) is rotated in intervals of 10 de- from 0.25 m to 8 m, with the K ratio of grees to evaluate the variation of elastic 1.0 and 0.2, respectively. At the side moduli in rotated directions. Parameters length less or equal to 1 m, the ranges of used for this verification is listed in Table the value changes are notably larger. This 4 of Paper I. Three boundary conditions scattered data implies that the model at described in Figure 3 are applied to ob- this size cannot be described as statisti- tain the compliance matrix. Since the cally homogeneous because one cannot model is symmetric about x- and y-axes assume a possible stable range of the when properties of two fracture sets are properties. However, the scattering of the the same, only values from 0 to 90 de- results clearly narrows down with in- grees are compared. As shown in Figure crease of the side lengths of the DFN 17, the two sets of results show a perfect models, and points to the possible exis- agreement with different K2 ratio values. tence of a REV. The mean value of the This proves that UDEC can simulate the normalized elastic moduli in the x- deformation of fracture-rock interaction direction is reduced to about 43% and properly and methodology to calculate 31% of that of the intact rock with K=1.0 the compliance tensor is applicable. and K=0.2, respectively. Anisotropy of elastic modulus was not significant and the elastic modulus is reduced considera-

bly with the decrease of K ratio (Figure 2 A capital letter ‘K’ is defined as the ratio of shear stiffness to normal stiffness of fracture in Paper I. 16 of paper I). Note that, in Paper IV, a small letter ‘k’ is defined as the ratio of horizontal stress to vertical stress The results of the Poisson’s ratio also and ‘kx’ and ‘ky’ are defined as the permeability in x- show a similar pattern and the values and y- directions, respectively. 35

Ki-Bok Min TRITA-LWR PHD 1011

0.90

DFN1 0.80 DFN2

0.70 DFN3 DFN4

0.60 DFN5

DFN6 0.50 DFN7

0.40 DFN8

DFN9 0.30

Normalized elastic modulus DFN10

0.20 012345678 Side length of square model (m)

(a) 0.60

DFN1

0.50 DFN2 DFN3

DFN4 0.40 DFN5

DFN6 0.30 DFN7

DFN8

0.20 DFN9

Normalized elastic modulus DFN10

0.10 012345678 Side length of square model (m)

(b)

Figure 18. Variation of elastic modulus in the x-direction with the increase of side lengths of square models (a) K=1.0, (b) K=0.2.

converge to about 0.1 and 0.35 for K ra- Figure 18 and Figure 19 demonstrate the tio of 1.0 and 0.2, respectively (Figure scale dependency of mechanical proper- 19). With the K ratio of 1.0, the derived ties of fractured rock masses. The proper- Poisson’s ratios are less than half of that ties to be determined have very wide of the intact rock (= 0.24) whereas the ranges until a certain scale of model size derived Poisson’s ratio with K ratio of is reached. In both cases, beyond side 0.2 is about 50% larger than that of intact length of 2 - 3 m, the scattering range of rock. Compared to the trend of elastic the properties becomes notably smaller modulus, Poisson’s ratio varies more and more constant. sensitively with the change of K ratio.

Fractured Rock Masses as Equivalent Continua – A Numerical Study

0.20 DFN1

DFN2

DFN3

0.15 DFN4 DFN5

DFN6

DFN7

Poisson's ratio 0.10 DFN8

DFN9

DFN10

0.05 012345678 Side length of square model (m)

(a) 0.60

DFN1

0.50 DFN2 DFN3

DFN4 0.40 DFN5

DFN6

0.30 DFN7

Poisson's ratio DFN8

0.20 DFN9

DFN10

0.10 012345678 Side length of square model (m)

(b)

Figure 19. Variation of Poisson's ratio (νxy) with the increase of side lengths of square models (a) K=1.0, (b) K=0.2. calculated elastic properties can be represented by an elastic compliance tensor, 5.4 Results of the tensor quantity an average compliance matrix (Eq.(11) is investigations of the mechanical calculated by averaging six compliance properties matrixes from the six rotated DFN mod- The DFN models are rotated in six els, at all different scales and these val- clockwise directions with a 30º interval ues are rotated to compare with the actual (0º, 30º, 60º, 90º, 120º and 150º, respec- values obtained from numerical experi- tively) to perform similar UDEC simula- ments. tions for the calculation of the compliance matrixes of the six rotated DFN models. In order to examine whether the

Ki-Bok Min TRITA-LWR PHD 1011

45 45 Pa) Pa) G G

us ( 40 us ( 40 odul odul

c m 35 c m 35 asti asti El El

30 30 0 30 60 90 120 150 180 0 30 60 90 120 150 180 Angle (degrees) Angle (degrees) (a) (b)

45 45 Pa) Pa) G G

us ( 40 us ( 40 odul odul

c m 35 c m 35 asti asti El El

30 30 0 30 60 90 120 150 180 0 30 60 90 120 150 180 Angle (degrees) Angle (degrees) (c) (d)

45 45 Pa) Pa) G G

us ( 40 us ( 40 odul odul

c m 35 c m 35 asti asti El El

30 30 0 30 60 90 120 150 180 0 30 60 90 120 150 180 Angle (degrees) Angle (degrees) (e) (f)

PExrepdecictteedd Ex b yfrom avera averageg etens tensoorr MeasurCalculateded Ex by b numy numeriericcalal eexxpperierimement

Figure 20. Comparison between expected and calculated elastic moduli (Ex) in rotated axes. The expected values are calculated from averaged compliance tensor. DFN1 models from side lengths of 0.25 m to 7 m were used for the analysis. (a) 0.25 m, (b) 0.5 m, (c) 1 m, (d) 3 m, (e) 5 m, (f) 7 m.

responding rotation directions are plotted Figure 20 presents the averaged and cal- as separated symbols. culated trace curves of the elastic moduli in the x-direction from all the rotated The figure shows that the numerical re- models at different scales for the DFN1 sults from the small scales models do not series. The solid curve in the figure for match well with the averaged values. the averaged values is the elastic moduli This, in other words, means that the calculated properties do not have a tensor in the x-direction from Sij with rotations, quantity at smaller scales. However, as and the numerical results obtained from the size of model increases, the numeri- the six rotated DFN models at their cor- cal results match increasingly well with the averaged values. At the side length of 38

Fractured Rock Masses as Equivalent Continua – A Numerical Study

5-7 meters, the numerical and the averaged trace curves agrees very well, indicating that the property at these sizes can be approximated by a fourth-order tensor. This means that the elastic compliance matrix obtained at the REV scale can be approximated by a fourth-order elastic compliance tensor at a certain degree of accuracy through a proper homogenization (averaging) process.

Ki-Bok Min TRITA-LWR PHD 1011

Fractured Rock Masses as Equivalent Continua – A Numerical Study

with varying side lengths from 0.25 m to 10 m (Table 2, Figure 7). Boundary con- 6 PERMEABILITY OF FRACTURED ditions for the calculation of permeability ROCK MASSES (PAPER II) tensors were shown in Figure 4. The numerical experiments were conducted for 6.1 Methodology the rotated DFN models for calculation Equivalent permeability tensor is calcu- of directional permeability in order to lated by the numerical experiments based check whether the calculated directional on the same DFN models of ten realiza- permeability in the pertinent regions can tions of square models used in Paper I be represented approximately as a tensor.

3.0E-13

DFN1

2.5E-13 DFN2 )

2 DFN3 2.0E-13 (m DFN4 xx DFN5 1.5E-13 DFN6

DFN7 1.0E-13 DFN8 Permeability, k DFN9 5.0E-14 DFN10

0.0E+00 0246810 Side length of square model (m) (a) 2.5E-13

DFN1

2.0E-13 DFN2 )

2 DFN3

(m DFN4 yy 1.5E-13 DFN5

DFN6 1.0E-13 DFN7

DFN8 Permeability, k 5.0E-14 DFN9

DFN10

0.0E+00 0246810 Side length of square model (m) (b) 1.5E-13

1.0E-13 ) 2 (m

yx 5.0E-14 & k

xy kxy 0.0E+00 0246810kyx -5.0E-14 Permeability, k -1.0E-13

-1.5E-13 Side length of square model (m) (c)

Figure 21. Results of calculated permeability tensor elements. (a) kxx, (b) kyy, (c) kxy & kyx

Ki-Bok Min TRITA-LWR PHD 1011

6.2 Results of calculated permeabil- 5 m and 10 m, the number of multiple ity tensors realizations was extended to fifty in order Figure 21 shows the results of calculated to derive the statistical range of perme- values of permeability elements kxx, kyy, ability values (Figure 10 of Paper II). kxy and kyx from the ten random realiza- The results show that the calculated val- tions (from DFN1 to DFN10), at differ- ues are more concentrated around the ent model sizes. The variances of calcu- mean values when the side lengths be- lated permeability components become come larger. smaller as the model size increases, and the permeability values maintain constant 6.3 Results of an investigation on ranges after a certain size. At the side tensor expression of permeability length of 2 m drastic reduction of vari- In order to justify the existence of an ance can be observed in the figure. The equivalent permeability tensor, the direc- mean values of both kxx and kyy with fur- tional permeability values were calcu- ther increase of the model sizes do not lated with the DFN models rotated at a show significant changes, indicating the 30-degree interval, and their reciprocals existence of a REV. The values of kxy and of square roots were plotted in a polar kyx are more symmetrical with the in- diagram to see whether they would form crease of side lengths as shown in Figure an ellipse. An average permeability ten-

21c. sor, kij , was first calculated at a given side length scale by averaging over the At the side lengths of 0.25 m, 0.5 m, 1 m, permeability values from all rotated

0 0 0 6 6 6 5x10 330 30 5x10 330 30 5x10 330 30 4x106 4x106 4x106 6 6 6 3x10 300 60 3x10 300 60 3x10 300 60 2x106 2x106 2x106 1x106 1x106 1x106 0 270 90 0 270 90 0 270 90

1x106 1x106 1x106

2x106 2x106 2x106 6 240 120 6 240 120 6 240 120 3x10 3x10 3x10 4x106 4x106 4x106 6 210 150 6 6 210 150 5x10 5x10 210 150 5x10 180 180 180 Side length 0.25 m Side length 0.5 m Side length 1 m

0 0 0 6 6 6 5x10 330 30 5x10 330 30 5x10 330 30 4x106 4x106 4x106 6 6 6 3x10 300 60 3x10 300 60 3x10 300 60 2x106 2x106 2x106 1x106 1x106 1x106 0 270 90 0 270 90 0 270 90

1x106 1x106 1x106 2x106 2x106 2x106 6 240 120 6 240 120 6 240 120 3x10 3x10 3x10 6 4x106 4x10 4x106 6 210 150 6 210 150 6 210 150 5x10 5x10 5x10 180 180 180 Side length 2 m Side length 5 m Side length 8 m Average 1/K1/2 Calculated 1/K1/2

Figure 22. Approximation of equivalent permeability tensor with increasing model size (expressed in 1/K1/2(θ)).

Fractured Rock Masses as Equivalent Continua – A Numerical Study

models (Eq.(14) and transforming the As practical improvement of DFN analy- average tensor to the pertinent rotation sis, the effect of size of parent DFN angles. model (boundary effect) is investigated as shown in Figure 14 of Paper II. The directional permeability from a permeability tensor will appear as a perfect 6.4 Determination of REV and ob- ellipse on a polar diagram when it is ex- tained properties from Paper I pressed in the reciprocal of square root. and Paper II The calculated reciprocals of square roots Table 4 shows the mechanical and hy- of the directional permeability values by draulic properties obtained at suggested numerical experiments using the rotated REV based on two measures from both DFN models were then plotted on the mechanical and hydraulic analysis. The same polar diagram to see how they will suggested REV size is used for the fur- match against each other. Figure 22 ther analysis of stress-dependency as a shows the comparison of the calculated representative scale so that the investiga- directional permeability values (the solid tion made will have more general impli- line with symbols) and the average per- cation without the effect of scale. Deter- meability tensor (dashed lines) at given mined REV is also used for choosing the side lengths of the DFN models. The re- minimum mesh size for the large-scale sults show that the directional permeabil- finite element method. The acceptable ity does not conform to an ellipse for variance shown in the Table 4 is rather models of side length < 5 m, indicating subjective, however, can be useful for the that a permeability tensor cannot be justi- quantitative comparison with other appli- fied at such scales. As the side length of cation concerning equivalent continuum the model increases, the tendency of ap- analysis. proaching an ellipse by the numerical data becomes stronger.

Table 4. Mechanical and hydraulic properties obtained at suggested REV.

Acceptable Variation (%) Coefficient Mean predic- Determined value at REV of 5 m × 5 m scale of Variation tion error (Scale de- (Tensor quan- pendency) tity evaluation) Mechaical compliance tensor*

SS11 12 S13 S16 2.7772 −−0.2946 0.2801 −0.0446  SSSS −−0.2927 2.3914 0.2816 −0.0451 6.2 2.6 21 22 23 26 =×10−2 (1/ GPa) SS31 32 S33 S36 −−0.2801 0.2816 1.1820 −0.0048  SS61 62 S63 S66 −−−0.0584 0.0639 0.0048 5.5654 Hydraulic permeability tensor

kkxx xy 1.0384 0.0393 −13 2 18.1 3.8 =×10 (m ) kkyx yy 0.0393 1.3258

*: taken from the mechanical analysis (Min and Jing, 2004b).

Ki-Bok Min TRITA-LWR PHD 1011

Fractured Rock Masses as Equivalent Continua – A Numerical Study

were observed with the stress increase. Normalized elastic modulus (Em/Ei) was 7 STRESS DEPENDENT MECHANI- about 3 % at 1 MPa and it increases to CAL PROPERTIES OF FRAC- near 50 % at 40 MPa. Elastic moduli in TURED ROCK MASSES (PAPER the y-direction show similar results due III) to the near-random fracture system geometry and isotropic stresses conditions 7.1 Methodology used in the numerical experiments. Con- sidering the strong stress-dependency, In order to consider the stress depend- anisotropic elastic modulus is anticipated ency, the boundary conditions of BC(1), if anisotropic stress conditions are con- BC(2) and BC(3) (Figure 3) are repeated sidered. Even at very high stress levels, it at different stress levels. Intact rock is is generally below 50% of intact rock. assumed to be a linearly isotropic elastic Considering that the normal stiffness of material and the Barton-Bandis (BB) fracture is very high at this stress level, model is used as the constitutive model the reduction of moduli at high stress of the fractures (Barton, 1982) in order to comes mainly from the shear deforma- properly consider the effect of stress on tion. the resulting mechanical behavior.

A simple empirical equation is proposed Geometry of this paper is taken from the in order to relate the rock mass elastic models used in Paper I with the side modulus (E ) to the magnitude of length of 5 m, which is determined to be m REV. stresses (σ), as

11 1 7.2 Results of stress dependent elas- =+ (35) tic modulus and empirical equa- EEmiSm⋅σ tion Figure 23 presents the elastic moduli in where Ei is the elastic modulus of the in- the x-direction with the increase of the tact rock and Sm is a sensitivity parame- boundary stress from 1 MPa to 40 MPa. ter. Above empirical equation is pro- Significant increases of elastic moduli posed based on the observation that both shear and normal stiffness are nearly linear functions of stress and therefore 0.6 adopting a new sensitivity parameter Sm would be applicable in equivalent media 0.5 as an approximation. Determined S

us m from a least square regression was odul 0.4 m

c 1895.9 for this study and the fitted curve i t s a

l 0.3 shown in Figure 23 captures the trend of e d

e rock mass modulus well in spite of the z i l

a 0.2 simple form. m r o

N DFN models 0.1 fitted curve 7.3 Results of stress-dependent Pois- son’s ratio and the bounds 0 0 10 20 30 The Poisson’s ratios decrease from over Stress (MPa) 0.8 at 1 MPa to 0.65 at 40 MPa (Figure 24). However, its stress dependency is Figure 23. Elastic moduli with stress in- not as strong as in the case for elastic crease. 45

Ki-Bok Min TRITA-LWR PHD 1011

modulus. It is emphasized that the calcu- 0.9 lated results are far bigger than the con- ventionally assumed values of 0.2 to 0.3 0.8 and what is startling is that all of Pois-

son’s ratio values are higher than 0.5 re- io t

a 0.7 r

gardless of the magnitude of stress. It is s ' n

noted that the DFN models used in this so

is 0.6 study have high fracture densities and o P connectivities, which allow larger degrees of freedom for motion and defor- 0.5 Upper limit for isotropic rock (0.5) mation of the block-fracture system.

0.4 0 10 20 30 40 Since two-dimensional approach may Stress (MPa) have influence to obtain this high Pois- son’s ratio, a limitation of the two- Figure 24. Poisson's ratios with stress in- dimensional approach used in this study crease. is investigated by analytical solution. Figure 25 shows the Poisson’s ratio rotated around both the z-axis and the y- axis. In Figure 25a, the radii in terms of the spherical coordinate correspond to the Poisson’s ratio (νxy, defined as the ratio of increase of strain in y-direction to the decrease of strain in x-direction with the compression in x-direction.) and this plot is constructed using Eq.(7) and Eq.(25). The plan view (Figure 25b) seen from the z-axis direction has the same form as the 2D case (Figure 4b of Paper III). It can be seen that the 2D plot fol- (a) lows the maximum trace of the surface in Figure 25a. This can be physically under- stood by the geometry in a 2D set-up where the orientation of fractures is expressed as an apparent dip in the cutting planes and the strike of the fracture is assumed to be parallel to the z-axis. The lateral deformation in inclined fractures in 2D case become maximum compared to the case where strike is not parallel to the z-axis. Therefore, the 2D approach gives a more conservative (larger Pois- (b) son’s ratio) evaluation compared to 3D Figure 25. The Poisson’s ratio of the trans- approach. versely isotropic model rotated around both z- and y-axis. Kn=43.4GPa/m, Ks=4.34GP/m. (a) the 3D view, (b) the 2D view (plan view from z-axis).

Fractured Rock Masses as Equivalent Continua – A Numerical Study

1010-14 kx (MC model) 8 STRESS DEPENDENT PERME- 10-14 9 kx (elastic) ky (MC model) ABILITY OF FRACTURED ROCK 8 ky (elastic) ) y kx 7 /k MASSES APER ) ky

(P IV) 2 x Contribution k

m -15

( from dilation ( 10 6 kx o y t ti i l i a -15 8.1 Methodology 10r 5 eab py o

m 4 A series of numerical experiments were r e otr s P 10i -136

conducted using various stress conditions An Contribution 2 from dilation for calculating the corresponding flow ky Development of fields and changes in the permeability of 1 anisotropic permeability 10-1067 the region (Figure 5). Numerical experi- 0 1 10 2 20 3 304 540 ments were conducted in two ways: (1) Ratio of hMoreizaonnsttarletsso v(eMrticaPa)l stress, k increasing the overall stresses with a fixed ratio (stress ratio, k) of horizontal Figure 26. Permeability (kx and ky) to vertical stresses components; and (2) change versus stress change with the fixed ratio of horizontal to vertical increasing the differential stresses (i.e., stresses = 1.3. Stress is expressed as mean the difference between the horizontal and of horizontal and vertical stresses. vertical stresses) while keeping the magnitude of vertical stress constant (Figure nificant, mainly because the stress ratio k 2 of Paper IV). A step-wise normal stiff- is close to the isotropic boundary stress ness model (Figure 4 of Paper IV) of condition. The application of the non- fracture is implemented to represent the linear normal stiffness of fractures in the normal stress-normal closure deforma- fracture model in this study led to more tion response and special emphasis is sensitive responses of permeability given to the role of fracture shear dilation change at lower normal stress magni- and associated channeling of flow for the tudes. This implies that permeability in investigation of stress-dependent perme- shallow depth is more sensitive to stress ability, adopting an elasto-perfectly- changes than in greater depth, as reported plastic shear behavior of the fracture. in Rutqvist and Stephansson (2003).

The size of 5 m × 5 m is selected for the Figure 27 presents the calculated equiva- model based on the previous investiga- lent-permeability changes with the in- tions for the calculations of the equiva- creasing k ratio. To evaluate the effect of lent mechanical and hydraulic REVs shear dilation, results are compared with (Paper I, II). the results from a pure elastic fracture model that excludes failure and dilation 8.2 Results of stress dependent per- (the dashed lines in Figure 27). The pure meability elastic and the elastoplastic models show Figure 26 shows the calculated equiva- a similar response until the stress ratio k lent permeability change with increases reaches approximately 2.5. At this point in mean stresses with stress ratio k = 1.3. and afterwards, some fractures in the Because of the dominating fracture clo- fractured rock masses start to fail in shear sure with the increasing normal stresses, and, with continued shear dilation, nota- the permeability of the model decreases ble differences between the models are accordingly. The reduction of permeabil- observed. The increase of permeability ity is more than two orders of magnitude, stabilizes after a certain k ratio, because and anisotropy in permeability is not sig- the shear dilation of a fracture does not

Ki-Bok Min TRITA-LWR PHD 1011

10-14 zontal hydraulic pressure gradient (Fig- kx (MC model) kx (elastic) ure 28a) more manifest than the vertical ky (MC model) ky (elastic) pressure gradient (Figure 28 b). ) 2 Contribution

m from dilation ( kx y lit

i -15

b 10

a Direction e m

r of Flow e P

Contribution from dilation ky Development of anisotropic permeability

-16 σx = 0 MPa 10 0 1 2 3 4 5 σ = 0 MPa Ratio of horizontal to vertical stress, k y

Figure 27. Equivalent permeability (kx and ky) change due to the change in σ = 5 MPa stress ratio. Differential stress is in- x σy = 5 MPa creased while keeping the magnitude of vertical stress constant. Mohr Coulomb (MC) model (solid lines) is compared with the pure elastic model with no shear σx = 10 MPa failure (dashed lines). σy = 5 MPa continue after their critical shear displacement is reached.

8.3 Results of stress-induced chan- σx = 15 MPa σ = 5 MPa neling and anisotropy y Figure 28 shows the change in flow patterns with the increasing stress ratio. A notable channeling flow effect caused by σx = 20 MPa stress-induced fracture dilation is ob- σy = 5 MPa served, analogous to the channeling effect observed in field for fractured rocks. As large shear dilations are concentrated in a smaller part of the fracture popula- σx = 25 MPa tion with near-critical orientations, good σy = 5 MPa connectivity, and long trace lengths, the rest of the fracture population, especially the subvertical ones, still undergo the (a) (b) normal closures without any shear dila- Figure 28. Fluid pathways during stress tion. This situation causes a high contrast applications with the direction of hydraulic in aperture values between the fractures. pressure gradient (a) from right to left,(b) Therefore, a few fractures with much in- from top to bottom. Thickness of the line creased apertures become the major represents the magnitude of flow rates. pathways of localized fluid flow. The One line indicates the flow rate of 10-9 dilated fractures tend to be those inclined m3/sec and the flow rates smaller than this more horizontally with the increasing value are not drawn. horizontal stress, and this tendency makes the channeling effect with hori-

Fractured Rock Masses as Equivalent Continua – A Numerical Study

1010-14 kx (MC model) 10-14 9 kx (elastic) where kx and ky are the final permeabili- ky (MC model) 8 ky (elastic) ties in the x- and y-direction, respec- ) y kx 7 tively. knx and kny are the permeabilities /k ) ky 2 x Contribution k

m -15

( from dilation ( 10 of normal stress-induced deformation in 6 kx o y i t lit i a -15 the x- and y-direction, respectively. k r Startofdilation dx b 10 5 a e

opy and kdy are the permeabilities from shear m r 4 r e ot s P -16 stress-induced dilation. bx and by are the 10 3 Ani Contribution equivalent apertures in the x- and y- 2 from dilation ky Development of directions, respectively. dx and dy are the anisotropic permeability 1 equivalent apertures of dilating fractures 10-16-170 0 1 10 2 20 3 304 5 40 in the x- and y-directions, respectively. Ratio of hMeorizaonnstatrletossve(MrtiPcaa)l stress, k More explicit forms containing the stress components are described in Eqs. (13a) Figure 29. Anisotropy ratio (kx/ky) of the and (13b) of Paper IV. The dilation terms equivalent permeability with the increase of horizontal to vertical stress ratio. in Eqs (36a) and (36b) become zero when the stress ratio is below the critical With the increase in horizontal stress, the stress ratio. The strength of above equa- anisotropy for the equivalent permeabil- tions is that they can consider the abrupt ity of the model also becomes significant change of shear-induced permeability (Figure 29). This is because the subverti- change more effectively. cal fractures are more vulnerable to closure by the horizontal stresses, and its The associated parameters are obtained effect on the x-directional permeability of from the laboratory test and numerical the model is more pronounced. The ratio experiments by multiple regression of anisotropic permeability, defined as analysis. Constructed models are com- kx/ky, increased from about 0.5 to 2 with pared with the numerical experiments in the increase of stress ratio k from 0.5 to Figure 30. The constructed models match 2.5. When dilation starts, anisotropy be- reasonably well with the results of nu- comes much more significant, and the merical experiments and capture the ob- maximum anisotropy ratio reaches ap- served dilation behavior with an accept- proximately 8. The fracture network used able degree of agreement. in this study shows a near-random fracture pattern with slightly more vertical fractures.

8.4 Proposed empirical equation of stress-dependent permeability To consider the stress effect, the stress- dependent equivalent permeability in x- and y-directions (kx, ky) are proposed by superimposing the contributions from normal closures and shear dilations of fractures in the following forms:

ff kk=+k=xb3+dxd3 (36a) x nx dx 12 x 12 x ff kk=+k=yb3+dyd3 (36b) ynydy12 y12 y 49

Ki-Bok Min TRITA-LWR PHD 1011

10-14 10-14

kx numerical experiments ky numerical experiments kx suggested equation ky suggested equation ) ) 2 2 (m -15 (m -15 y x 10 10 k k , , y y t i l lit i i b b a a e e m r rm

e 10-16 10-16 Pe P

10-17 10-17 0 10 20 30 40 0 10 20 30 40 Mean Stress (MPa) Mean Stress (MPa)

-14 10-14 10

ky numerical experiments ky numerical experiments ) ) 2 2 (m (m y x k k , , y y t -15 10-15 ilit

ili 10 b b a a e e m rm r e e P P

kx numerical experiments kx suggested equation

10-16 10-16 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Ratio of horizontal to vertical stress, k Ratio of horizontal to vertical stress, k

Figure 30. Comparison of equivalent permeabilities from numerical experiments and empirical equations.

Fractured Rock Masses as Equivalent Continua – A Numerical Study

draulic properties corresponding to different stress levels with depth are then 9 THERMOMECHANICAL IMPACT passed on to the large-scale model to examine the impacts of thermo-mechanical ON PERFORMANCE OF A NU- processes on performance assessment of CLEAR WASTE REPOSITORY a hypothetical nuclear waste geological (PAPER V) repository.

9.1 Methodology The geometry of reference model and Equivalent continuum analysis is con- boundary conditions are shown in Figure ducted with the stress-dependent me- 9. A hypothetical reference problem is chanical and hydraulic properties ob- defined with simplifications in two di- tained from DFN-DEM approach de- mensions. The performance of the re- scribed in Papers I to IV. The conceptual pository system is investigated in terms approach adopted for this study is dis- of mechanical responses and permeabil- played in Figure 8. Mechanical and hy- ity changes induced by the thermal load-

Thermal stress, σ (MPa) Thermal stress, σ (MPa) VeY,rticaXV3,cloordVisp4,dilaV5,nacetemV(y6,emn)V7t (m) VeY,rticaV3,l dVisp4,laV5,cemVy6,enV7t (m) 0 0 2 0.014 0.026 0.8 03 10 0 0 2 0.014 0.026 0.8 03 10 200 200

Case 2 Case 3 Case 1 Case 2 Case 3 Case 1 20100 20100

) 15 Case 2 Case 1 ) 15 Case 2 Case 1 a a

P Case 3 P Case 3 1,000 years 1,000 years (M (M x x

) 200 ) 200

σ 400 σ 400

m 100 years m 100 years ( 1,000 years ( 1,000 years ss, ss, h h X X t 10 t 10 e 100 years e 100 years p p tr tr s s

l 300 l 300 De 600 De 600 a a m m r r e e

h Case 3 h Case 3

T 5 T 5 400 100 years 400 100 years 800 Case 3 1000 years 800 Case 3 1000 years Case 3 Case 3 Case 3 Case 3 Case 2 Case 2 CaCassee22 CaCassee22 Case 2 Case 2 CasCaeCa1sCaese1s1e 1 CasCaeCa1sCaes1ese11 500 500 10000 10000 2600 2700 2800 2900 3000 2600 2700 2800 2900 3000 X coordinate (m) X coordinate (m)

(a) (b)

Thermal stress, (MPa) Thermal stress, σx (MPa) σx -5 0 5 10 15 20 25 -5 0 5 10 15 20 25 0 0

200 200 101,0000 yeayerasrs 1001,000yeyaerasrs ) ) 400 400 m m ( ( h h t t Case 1 Case 1 p p Case 1 Case 2 Case 3 Case 3 Case 1 Case 2 Case 3 Case 3 De De 600 600 Case 2 Case 2

800 800

1000 1000

Figure 31. Horizontal thermal stresses in different cases. (a) Along horizontal reference line in 100 years, (b) Along horizontal reference line in 1,000 years, (c) Along vertical reference line in 100 years, (d) Along vertical reference line in 1,000 years.

Ki-Bok Min TRITA-LWR PHD 1011

ing caused by the heat releasing from nu- A maximum temperature of about 50°C clear waste decay as a function of time. occurs at about 100 years after the waste To investigate the effect of mechanical placement (Figure 8 of Paper V). Gener- properties, three cases (Case 1, Case 2 ated temperature induces thermal stresses and Case 3) are considered, whose values and Figure 31 shows the horizontal ther- differ only in their mechanical properties mal stresses in horizontal and vertical (Table 3 of Paper V). reference lines across the repository. As can be seen, induced thermal stresses 9.2 Result of Mechanical response vary greatly with the mechanical proper- caused by thermal loading ties, which demonstrates the importance 100 yrs 0.01

0.0

400 yrs 0.01 0.02

0.0

0.02 0.03 1,000 yrs 0.01

0.0

0.04 10,000 yrs 0.05 0.02 0.03 0.01

100,000 yrs

0.02 0.01

0.05 0.04 0.03 0.02 0.01 0.0 -0.005 (m

Figure 32. Vertical displacement distribution at different time scales (Case 1).

Fractured Rock Masses as Equivalent Continua – A Numerical Study

of proper method of determination of the to the permeability at the initial effective mechanical properties. When only the stress state. The scale of changed perme- intact rock properties were used (Case 3), ability increased from 100 years to 1,000 the maximum thermal stress in the hori- years and then becomes smaller again at zontal direction was nearly 20 MPa, 10,000 years. The permeability generally whereas it is only 5 MPa when the effect decreases owing to the induced compres- of fractures was considered through sive stresses around the repository; the equivalent mechanical properties. Such a magnitude of normalized permeability difference can also be observed for the was less than 0.5. Empirical equations of vertical reference lines and what is inter- stress-dependent permeability (Table 2 of esting is the generated tensile stress in Paper V) indicate that the shear dilation the surface (Figure 31d). The induced of fractures plays a significant role in the tensile stress is small in Case 1, however, increase of permeability. However, no much bigger tensile stress is induced in dilation was induced from the thermal Case 3 (about 3 MPa at 1,000 years). loading in this study. This difference is due to the large difference of elastic modulus near the surface. The addition of thermal stress was not Figure 32 shows the vertical displace- substantial enough to reach the critical ment distributions over time. The general stress ratio (about 2.45 for conditions in trend shows a heaving of rock above the this study). Furthermore, this relative repository of a few centimeters. This smaller change even around the reposi- magnitude of displacement may or may tory largely results from the mechanical not cause safety concerns at the ground excavation effect not being considered in surface. However, analysis may be this study. needed for confidence-building measures such as for post-closure monitoring. On account of the slow diffusion of thermal processes, maximum displacement is observed at about 10,000 years; after this period, the heaved ground settles with the decreased temperature. The location of maximum displacement shifts from directly above the repository (e.g., at 10 years) upward as time passes (Figure 15 of Paper V). The displacement field also depends on mechanical properties and the heaving becomes large when large Poisson’s ratio is used due to the confined lateral sides (Figure 16 of Paper V).

9.3 Result from permeability change from thermal loading Using the proposed stress-dependent permeability equations (Table 2 of Paper V), the permeability changes induced by the thermal stress are investigated. Figure 33 shows the permeability change distribution at different time. The changed permeability is normalized with respect 53

Ki-Bok Min TRITA-LWR PHD 1011

kk/kk/k10/k1,,400100000100yyeeyaarrsrss yxkxy/ki xyi 1,i 000 years

1,000100yyrrss k10,x/k1,i 000400000yyerasrs

(a)

1,000100yeayrrss 10k10,,x000/k1,i 00040000y0eyyaerasrs

1,000100yeyayrrsrss k10y/kki,x10,000/k1,i 000400yeyyaerasrs

1,000100yeyayrrsrss 10,k10,x000/k1,i 000400yeyyaerasrs

0.5 0.7 0.9 1.1 (b)

Figure 33. Permeability change due to the induced thermal stress (Case 1). Permeability is normalized with respect to the initial permeability (a) horizontal permeability change (b) vertical permeability change. 54

Fractured Rock Masses as Equivalent Continua – A Numerical Study

Applicability of the derived equivalent 10 CONCLUSION properties for continuum analysis of the fractured rock masses is systematically The results in the thesis provide a investigated through the two criteria sug- framework for systematic analysis of gested in this study. It is demonstrated large-scale engineering applications in that the permeability and elastic compli- fractured rock masses such as geological ance tensors in rotated axes can be pre- repository of nuclear waste. All five pa- dicted from the one in original axis as pers are closely inter-related, but they long as it is estimated at supporting size can also serve as independent references of REV. This implies that fractured rock on the behavior of fractured rock masses. masses of equivalent statistical homoge- neity of fracture systems can be repre- Main conclusions are summarized as fol- sented as a homogeneous continuum lows. above the REV scale.

The methodology for the determination The study suggests that mechanical prop- of mechanical properties of fractured erties of fractured rock masses are highly rock masses is developed using the DFN- stress-dependent. The elastic modulus DEM approach. The method has strength increases substantially with the increase of considering the explicit geometry of of stress magnitude. Therefore, engineer- fractures with their constitutive behavior. ing applications in various in situ stress This method can substitute or comple- conditions should consider this aspect for ment the existing analytical and empiri- design and analysis. A simple empirical cal methods for determination of rock equation of stress-dependent elastic mass properties, with much greater flexi- modulus is proposed (Eq.(35)). The bility in considering fracture system ge- equation is promising in that it is simple ometry and fracture constitutive behav- but contains the stress dependency of iors. normal and shear behavior of fracture in a lumped form, which led to good Scale dependency of hydraulic and me- agreement with the results from chanical properties is thoroughly investi- numerical experiments. gated by the DFN-DEM approach using multiple realizations of fracture systems The Poisson’s ratio in fractured rock of increasing model sizes. The results masses can be larger than upper bounds demonstrate the existence of REV for the of isotropic case (0.5) and much larger fractured rock masses with the given than the typical values used in practice. geological data at Sellafield. This This suggests that the common practice achieved from the converging ranges of of assuming the values as 0.2 – 0.3 needs mechanical properties with the increasing to be carefully re-evaluated with respect side lengths of square DFN models. to the effect of fracture. The results dem- However, the existence of REV should onstrate that consideration of anisotropy be considered site-specific. The obtained should be duly given not only to elastic mechanical and hydraulic REV is deter- modulus but also to Poisson’s ratio of mined to be 5 m x 5 m scale above which fractured rock masses. the uses of continuum approach is war- ranted. An improvement for the DFN Permeability can increase or decrease analysis is made to avoid the boundary with stress increase depending on the effect. state of stress rather than a single com- 55

Ki-Bok Min TRITA-LWR PHD 1011

ponent or an averaged value of stresses. important factors in determining these Hence, changes in permeability should major fluid-carrying features. It is the be investigated in the context of changes first time that such a clear demonstration in stress tensor. of stress-induced flow channeling by numerical modeling in fractured rocks • Equivalent permeability decreases was reported. with increase in stresses, when the differential stress is not large enough Increase in permeability anisotropy can to cause shear dilations of fractures. become important with continued in- For the rock mass concerned in this crease of differential stresses. This ani- thesis, the reduction of permeability sotropy can be more prominent with the was more than two orders of magni- onset of stress-induced shear dilation of tude, under the stress increase up to fractures. For the problem studied in this 40 MPa (from zero stress). The more thesis, the permeability in the horizontal sensitive permeability change at direction was eight times larger than that lower stress levels was captured due in the vertical direction when the stress to the hyperbolic behavior of the ratio was five, due to the highly mobi- normal stress-normal closure of rock lized shear dilations of the subhorizontal fracture model. fractures, even though the permeability • The equivalent permeability increases was isotropic before the stress change with the increase in differential was imposed. stresses, when the stress ratio was large enough to cause continued A set of empirical equations that account shear dilation of fractures. In this for both normal closure and abrupt shear case, shear dilation is the dominating dilation are suggested for modeling the mechanism in characterizing the stress-dependent permeability. In addi- stress-dependent permeability. The tion to the parameters determined from maximum contribution of dilation is the laboratory, a few other parameters more than one order of magnitude in need to be obtained from numerical ex- permeability in this study. The in- periments, as conducted in this study. crease of permeability stabilizes Predicted permeability from the sug- when shear dilation reaches its sta- gested equations provides reasonable tionary value. agreement with the results obtained from numerical experiments. Stress-induced channeling effects of fluid flow were found from the numerical A combined discrete and continuum modeling. The results show that fluid modeling approach is presented. The flow becomes uneven and clustered as a procedures for determining the equiva- result of localized shear dilations of frac- lent mechanical and hydraulic properties tures, and a major portion of the flow of fractured rock masses are introduced, may be carried by only a few connected with their stress-dependencies applied as fractures. The results from this paper input data for large-scale analysis. Stress- confirm that high differential stresses, dependent permeability, as is an impor- causing fracture shear failure at certain tant hydromechanical component in frac- orientations, is one of the major reasons tured rock masses, is determined by for highly channeled flow in fractured DFN-DEM approach and incorporated in rock masses. The numerical analysis in large-scale analysis. This methodology this paper suggests that the lengths of provides a more systematic platform for fractures and their connectivity are also large-scale engineering applications in 56

Fractured Rock Masses as Equivalent Continua – A Numerical Study

fractured rock masses, such as a geological repository of nuclear waste.

The responses of fractured rock masses vary significantly according to how mechanical properties are determined, which may have significant conse- quences for the performance of the repository. One special issue raised by the results of this paper is the differences in thermal stresses when properly derived equivalent mechanical properties and intact rock properties are used. For repositories in intact or extremely sparsely fractured hard rocks, the thermal stress increment will be very significant in terms of repository safety. In fractured rocks, this safety concern has much less significance.

The calculated results also show the development of vertical heaving and horizontal tensile displacement, which may be important issues for confidence- building and the performance assessment of repository systems. Observed displacement fields also depend on how the mechanical properties (i.e., elastic modulus and Poisson’s ratio) are characterized considering the effects of fractured systems.

Coupled effects were investigated considering the permeability change induced by thermo-mechanical processes. Perme- ability changes induced by thermal loading indicate that they generally decreased by a factor of two close to the repository. Thermally induced fracture dilation was not observed. Permeability change on the large scale is small and the effects dimin- ish over time. However, this result has to be interpreted carefully in relation to the role of in situ stress conditions, and to the assumptions that did not consider excavation effects.

Ki-Bok Min TRITA-LWR PHD 1011

Fractured Rock Masses as Equivalent Continua – A Numerical Study

data in this study. For the cases where fracture systems does not have high de- 11 DISCUSSION AND FURTHER RE- gree of connectivity, the REV may not exist or the size of it may be so large that SEARCH practical application of equivalent con- While previous applications of DEM tinuum approach may not applicable. (e.g., UDEC) approach have been applied mainly in simple geometry models Constant aperture values were used for to identify important mechanisms of frac- the evaluation of hydraulic REV and in- tures (Hart, 1993), the current DFN- vestigation of the applicability of contin- DEM approach introduces a more realis- uum approach (Paper II). However, as tic, however, complex fracture geometry shown in the Paper IV, the distribution of system. It should be pointed out that aperture values vary to a great extent due much can be obtained through more real- to the effect of stress. Other factors such istic representation of fracture geometry. as roughness, fracture size and chemical For example, realistic mechanical proper- effects also make the aperture distribu- ties can be achieved considering fracture tion heterogeneous. Further, often only interactions. The abrupt shear dilation parts of the fracture population actually observed in this thesis cannot be properly carry fluid, as demonstrated by the dila- modeled with persistent fracture geome- tion-induced channeling shown in Paper try At the same time, increased complex- IV. Therefore, the decided hydraulic ity of fracture geometry make it difficult REV and applicability of continuum to interpret the observed behavior. Mod- analysis need further investigation with eling of more complex geometry such as more realistic aperture statistics. In this the ones presented in this study should regards, the conclusion provided in Paper proceed from simple geometry for identi- II should be viewed in relation with the fication of main mechanisms to increased effect of fracture geometry and be con- complexity of fracture geometry as pre- sidered to be a baseline study for further sented in Figure 12 of Paper I. investigation.

The result of the thesis needs to be exam- Two-dimensional DEM modeling is lim- ined in relation to the geological data ited because fractures are modeled to used in this study, especially the fracture have strikes only normal to the model statistics. The fracture data in this study plane. Hence, three-dimensional model- is characterized as relatively high density ing is necessary for the true representa- and random orientations, which results in tion of fracture geometry. The effect can high connectivity and no dominating be partially estimated with persistent fracture orientation. However, when a fracture lengths. For mechanical analysis, few dominance fracture sets exist with lateral deformation under axial stress is moderate fracture connectivity, the maximized in two-dimensional analysis methodology presented in this thesis is and generally this makes the elastic still applicable since it can consider ani- modulus smaller and Poisson’s ratio (as sotropy caused by the dominance of a shown in Paper III) higher than three- certain fracture sets. Hence the method- dimensional analysis. Further, shear- ology presented in this study should be induced channeling can be reproduced considered general. The fact that REV more realistically in three dimensional existed in this study and the size of de- analysis as fluid flow perpendicular to termined REV is basically site-specific the direction of shearing can be much with much influence from the geological higher than fluid flow in parallel with 59

Ki-Bok Min TRITA-LWR PHD 1011

shearing which is measured in the current oped for the equivalent continuum two-dimensional analysis. Nonetheless, analysis of fractured rock masses further experimental study is needed as • Consideration of heterogeneous frac- to the extent of such anisotropy (Koyama ture apertures distribution for the va- et al., 2003; Kim and Inoue, 2003) for lidity of existence of REV and tensor more proper investigation of the limita- quantity investigations for permeabil- tion of this two-dimensional study. ity and mechanical compliance tensor. In the hypothetical repository model (Pa- • Fully coupled THM analysis with per V), the current study focused on the transport of radionuclides for the far- thermo-mechanical impacts. Inclusion of field model of the geological nuclear mechanical excavation effect requires waste repository considering the un- true three-dimensional representation of certainties related to the determina- repository with drifts and deposition tion of properties and coupling holes, which will also need much more mechanisms in fractured rock masses. computing time. Main performance measure was mechanical response and permeability change due to thermal loading. Hydraulic analysis is not conducted with the changed permeability. In fact, the maximum temperature was only about 50 degrees and thermo-hydraulic effect is not generally expected to be high. However, for the purpose of confidence building, full thermo-hydromechanical and nuclide transport analysis needs to be conducted for more complete assessment of performance for the geological repository.

Based on the above discussions and achievement in the thesis, further research is recommended in the following subjects.

• Validation of proposed DFN-DEM approach against in situ field experiments of mechanical properties. • Developing three-dimensional DFN- DEM approaches for the investigation of the coupled hydro-mechanical properties of fractured rock masses, and the limitation of two-dimensional analysis. • Extension of current DFN-DEM approach to the study of strength of the fractured rocks so that constitutive models of plasticity may be devel-

Fractured Rock Masses as Equivalent Continua – A Numerical Study

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