Department of Land and Water Resources Engineering DETERMINATION OF EQUIVALENT HYDRAULIC AND. Ki-Bok Min

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1 Department of Land and Water Resources Engineering DETERMINATION OF EQUIVALENT HYDRAULIC AND MECHANICAL PROPERTIES OF FRACTURED ROCK MASSES USING THE DISTINCT ELEMENT METHOD Ki-Bok Min Stockholm 2002 TRITA-LWR LIC: 2007 ISSN ISRN KTH/LWR/LIC 2007-SE ISBN

2 Acknowledgement The research work presented in the thesis was carried out from September 2000 to October 2002 at Royal Institute of Technology. I am delighted to take this opportunity to thank many of contributors. First of all, I d like to express my sincere gratitude to my main supervisor Dr Lanru Jing for his academic guidance, stimulating discussions throughout my research work. Professor Ove Stephansson is a co-supervisor and examiner of my study and I d like to thank him for accepting me as PhD student, introducing me to wonderful international projects and encouraging comments during my research work. My third supervisor, Dr Jonny Rutqvist of LBNL, UC Berkeley, USA is acknowledged for our fruitful discussion and his hospitality during my six-week visit to LBNL in Financial support provided by European Commission through BENCHPAR (FIKW-CT ) project and Swedish Nuclear Power Inspectorate (SKI) through DECOVALEX III project are gratefully acknowledged and I wish to thank Dr. Fritz Kautsky of SKI for his engagement in the project. The thesis work has been conducted as a part of international projects (DECOVALEX III/BENCHPAR) and I would like to thank my colleagues in the project: Drs Johan Andersson of JA Streamflow and Leslie Knight of nirex, UK, are acknowledged for constantly reviewing my research approach. Student colleagues, Philipp Blum of University of Birmingham, UK, and Johan Öhman of Uppsala University are especially acknowledged for constructive comments, which I am sure to reciprocate for their work. Also I would like to thank Professor John Hudson of Imperial College, UK for his interest and encouragement on my research work. More than warm atmosphere at our EGG (Engineering Geology and Geophysics) Group in KTH has been a firm foundation for my life in Sweden. I would like to thank my former and current American, Chinese, Finnish, Iranian, Italian, Korean, Spanish, Swedish colleagues for being cooperative, fruitful comments during seminar, and sometimes pulling me out of my computer, which I feel more thankful. Diego Mas Ivars is particularly thanked for our cooperation at the early stage of my work. Ulla Engberg, former colleague of mine who is now in Stockholm University, merits special thanks for unforgettable help especially when I was totally stranger in this new world. I would like to thank Professor Chung-In Lee of Seoul National University, my ex-supervisor, and Professor Jae-Dong Kim of Kangwon National University in Korea for making it possible to connect two far countries and their respect for my potentials. If it is honour to receive this degree, my lovely wife, Mi-Suk Lee, deserves at least half of it and I would like to thank her for her exceptional patience, love and support. Lastly, I hope this thesis would be a small gift to my and Mi-Suk s families in Korea. Stockholm, October 2002 Ki-Bok Min i

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4 Abstract The equivalent continuum approach uses equivalent properties of rock mass as the input data for a continuum analysis. This is a common modeling method used in the field of rock mechanics and hydrogeology. However, there are still unresolved questions; how can the equivalent properties be determined and is the equivalent continuum approach suitable for modeling the discontinuous fractured rock mass. The purpose of this paper is to establish a methodology to determine the equivalent hydraulic and mechanical properties of fractured rock masses by explicit representations of stochastic fracture systems, to investigate the scale-dependency of the properties, and to investigate the conditions for the application of the equivalent continuum approach for the fractured rock masses. Geological data used for this study are from the site characterization of Sellafield, Cumbria, UK. A program for the generation of stochastic Discrete Fracture Network (DFN) is developed for the realization of fracture information and ten parent DFN models are constructed based on the location, trace length, orientation and density of fractures. Square models with the sizes varying from 0.25 m 0.25 m to 10 m 10 m are cut from the center of the each parent network to be used for the scale dependency investigation. A series of the models in a parent network are rotated in degrees interval to be used for investigation of tensor characteristic. The twodimensional distinct element program, UDEC, was used to calculate the equivalent permeability and compliance tensors based on generalized Darcy s law and general theory of anisotropic elasticity. Two criteria for the applicability of equivalent continuum approach were established from the investigation: i) the existence of properly defined REV (Representative Elementary Volume) and ii) existence of the tensor in describing the constitutive equation of fractured rock The equivalent continuum assumption cannot be accepted if any one of the above two criteria is not met. Coefficient of variation and mean prediction error is suggested for the measures to quantitatively evaluate the errors involved in scale dependency and tensor characteristic evaluation. Equivalent permeability and mechanical properties (including elastic modulus and Poisson s ratios) determined on realistic fracture network show that the presence of fracture has a significant effect on the equivalent properties. The results of permeability, elastic moduli and Poisson's ratio show that they narrow down with the increase of scale and maintain constant range after a certain scales with some acceptable variation. Furthermore, Investigations of the permeability tensor and compliance tensor in the rotated model show that their tensor characteristics are satisfied at a certain scale; this would indicate that the uses of the equivalent continuum approach is justified for the site considered in this study. The unique feature of the thesis is that it gives a systematic treatment of the homogenization and upscaling issues for the hydraulic and mechanical properties of fractured rocks with a unified approach. These developments established a firm foundation for future application to large-scale performance assessment of underground nuclear waste repository by equivalent continuum analysis. Keywords : Equivalent continuum approach, Equivalent property, Representative Elementary Volume (REV), Distinct Element Method, Discrete Fracture Network (DFN) iii

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6 Table of contents ACKNOWLEDGEMENT...I ABSTRACT...III TABLE OF CONTENTS...V 1 INTRODUCTION DISCRETE FRACTURE NETWORK (DFN) ANALYSIS DATA USED FOR ANALYSIS DISCRETE FRACTURE NETWORK (DFN) METHODOLOGY HYDRAULIC ANALYSIS (PAPER A, B) MECHANICAL ANALYSIS (PAPER C) RESULTS HYDRAULIC PROPERTIES - PERMEABILITY TENSORS (PAPER A, B) MECHANICAL PROPERTIES - COMPLIANCE MATRIX (PAPER C) COMPARISON BETWEEN AVERAGE AND CALCULATED PROPERTIES IN ROTATED MODEL (PAPER B, C) E VALUATION OF ACCEPTABLE ERROR IN DETERMINING REV (PAPER B, PAPER C) CONCLUDING REMARKS REV, EQUIVALENT PERMEABILITY AND COMPLIANCE MATRIX (PAPER B, C) APPROPRIATENESS OF EQUIVALENT CONTINUUM APPROACH COMPARATIVE STUDY ABOUT THE CHARACTER OF HYDRAULIC AND MECHANICAL PROPERTIES OF FRACTURED ROCK RECOMMENDATION FOR FURTHER RESEARCH THE EFFECT OF FRACTURE CONSTITUTIVE MODEL, ROCK MASS STRENGTH AND THREE- DIMENSIONAL EXTENSION HYDRO-MECHANICAL COUPLING APPLICATION TO LARGE SCALE PERFORMANCE A SSESSMENT (PA) ANALYSIS REFERENCES...12 v

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8 1 INTRODUCTION The equivalent continuum approach is a common modeling method used in the fields of rock mechanics and hydrogeology, especially for large-scale problems (Sitharam 2001, Long et al.1982). It assumes that fluid flow and deformation behavior of fractured rock masses can be described by principles of continuum mechanics, as long as its constitutive relations and associated properties can be properly established. On the other hand, the discrete approach, represented mostly at present by the Distinct Element Method (DEM) and Discontinuous Deformation Analysis (DDA) for mechanics (Hart 1993, Jing 1998) and Discrete Fracture Network (DFN) analysis for hydraulics (Herbert 1996), considers that the rocks are assemblies of individual blocks defined by fracture systems. The interactions between the rock blocks and fractures (interfaces between blocks) are the main factors affecting the mechanical behaviour and the connected fracture networks provide the main pathway of fluid. Because blocks and fracture systems are explicitly represented more in detail in the discrete models, the approach is better equipped to investigate the small-scale behavior of fractured rocks. However, the disadvantage of discrete approach is that representations of individual blocks and fractures in large scales, for instance km scale, put large demands on computer memory and speed. In this case, the equivalent continuum 1km 500m Formation type 2 100m Formation type 1 Vertical fault zone 100m Detailed model area 50m 100m m m 5m 10m Repository block Sea level approach has the advantage in representing the overall behavior of fractured rocks for problems of large scales, while the effects of the fractures are implicitly contained in the equivalent constitutive models and associated properties. This practical usefulness made the equivalent continuum approach enjoy the wide application, especially for large scale problems. However, the conditions for applicability of equivalent continuum approach to fractured rock mass remain to be an open question to rock mechanics and hydrogeology application (La Pointe 1993, Aydan 1995, Neuman 1997). Given the fact that many important underground facilities such as nuclear waste repository are constructed in rock conditions that inevitably contain large number of fractures, it is of great importance to rigorously investigate the hydraulic and mechanical behaviour of fractured rock mass in equivalent and averaged senses. The content of the thesis is mainly part of a benchmark test (called BMT2) of the ongoing international co-operative research project DECOVALEXIII 1 and BENCHPAR 2. The purpose of BMT2 is to develop numerical methodologies of homogenization and upscaling of coupled hydro-mechanical properties of fractured rocks and their impacts on large-scale transport analysis for performance and safety assessments of potential underground radioactive waste repositories in fractured rocks (DECOVALEX III, 2000). Fig. 1 shows the schematic view of the reference problems for BMT2 where hypothetical repository is located in about 500 m deep level. Domain of interest consists of three main rock formations (Formation 1, a collective name of low permeability zones, Formation 2, a collective name of upper low permeability zones, and a fault zone), and each was regarded as geological unit of statistical homogeneity in terms of fracture system geometry and lithology. Of three rock formations, only the Formation 1 is concerned with this thesis. The problem to be tackled in the BMT2 has the Not to Scale Fig. 1 Reference problem for BMT2 with two formations and fault zone. Note that it is not to scale (DECOVALEX III 2000). 5 km 1 Acronym of international project named DEvelopment of Coupled models and their VAlidation against EXperiments in nuclear waste isolation. 2 Acronym of an European Project: BENCHmark tests and guidance on coupled processes for Performance Assessment of nuclear waste Repositories 1

9 Ki-Bok Min TRITA LWR LIC 2007 Parameter Upscaling Large scale PA analysis Determination of equivalent hydraulic and mechanical properties at REV DEM code (UDEC) Stochastic Input parameter Large scale THM coupled analysis with equivalent properties FEM code (ROCMAS ) Fig. 2 Strategy for large-scale equivalent continuum approach. Shaded area is the scope of thesis work. scale of up to 5 km and it involves complex Thermo-Hydro-Mechanical coupled processes. Therefore current capability of discrete approach cannot cope with this large scale, complex coupled processes. The general outline of the strategy for large scale analysis with upscaling and homogenization of properties are presented in Fig. 2. The thesis work concerns the parameter upscaling for the future application to large scale problems. As a first step for final Thermo-Hydro- Mechanically coupled analysis, this thesis aims to establish a systematic methodology for upscaling and homogenization of hydraulic and mechanical properties of fractured rock masses. A unified technique for the determination of hydraulic and mechanical properties is established with the multiple realization of DFN. A systematic evaluation of appropriateness of equivalent continuum approach with the given site characterization information is investigated. Two conditions are used as criterion for the justification of the equivalent continuum approach; firstly, a representative elementary volume (REV) must exist for a certain problem in order that a basis of statistical equivalence between the sampled rock masses and numerical models can be established, secondly, the derived equivalent properties must be represented by tensors to be used for the constitutive equations for continuum analysis. Those conditions are suggested by Long (1982) for flow analysis for the application to equivalent porous medium. Given the fact that the both permeability and mechanical compliance have tensor quantity with only different ranks, it is possible to extend this condition to the problems of mechanics. A series of numerical experiments were conducted using the distinct element method on multiple realizations of discrete fracture networks (DFN) models at different scales. The data used for this analysis is from the site characterization of Sellafield, Cambria, UK undertaken by nirex (Nirex 1997). For fluid analysis, the assumption is that fluid flow occurs only through the fracture and the flow through matrix is neglected. The flow rate calculation in each fracture is based on cubic law and mass continuity equation is described in the intersecting point of fractures for the flow analysis of DFN. For mechanical analysis, fractured rock masses are treated as completely anisotropic elastic material and the fracture constitutive model adopts the constant stiffness model, which linearly relate the loading and displacement in the fracture. For both hydraulic and mechanical analysis the twodimensional Distinct Element Method, UDEC (Itasca 2000) is used with the assistance of independent fracture generating program developed for this study. This thesis is consists of three papers. PAPER A: Ki-Bok Min, Lanru Jing, Ove Stephansson, 2002 Determination of the permeability tensor of fractured rock masses based on stochastic REV approach Published in : ISRM regional symposium, 3 rd Korea-Japan Joint symposium on rock engineering, Seoul, Korea, Vol. 1, pp PAPER B: Ki-Bok Min, Lanru Jing, Ove Stephansson 2

10 Table 1 Parameter for determination of equivalent hydraulic and mechanical properties (Paper B, C). Intact rock Elastic modulus Poisson s ratio (GPa) Fractures set Dip/Dip Fisher Normal Shear Fracture Hydraulic direction constant (K) stiffness stiffness density* Aperture (GPa/m) (GPa/m) (m -2 ) (µm) 1 8/ / / / / / / / *: Fracture density is calculated from the cumulative number of fractures with minimum and maximum cutoff trace lengths. **: Hydraulic aperture value is calculated from Coupled Shear Flow test. Fracture system characterization and evaluation of the equivalent permeability tensor of fractured rock masses using a stochastic REV approach Submitted to Hydrogeology Journal PAPER C: Ki-Bok Min, Lanru Jing Numerical determination of equivalent compliance tensor of elasticity for fractured rock masses using the Distinct Element Method Submitted to International Journal of Rock Mechanics and Mining Sciences All three papers are based on the same data sets provided from a benchmark test of DECOVALEX III and BENCHPAR projects. Paper A is concerned with the development of homogenization and upscaling approach for deriving equivalent hydraulic properties of fractured rocks of BMT2, using the stochastic DFN technique. It starts with the establishment of the conditions for determination of proper REV sizes and the equivalent permeability tensor, but without considering the necessity of for a measure of confidence for the approximation. The work is more rigorously developed in Paper B, which provides a more comprehensive coverage on the fracture system characterization, justification of the REV approach, conditions for establishment of permeability tensor and the necessary measures for confidence of approximation-the coefficient of variation and the prediction error. Paper C extends the approach and concepts developed in Paper A and Paper B to the process of mechanical deformation, with a distinct contribution to derive an equivalent elastic compliance tensor of the 4 th order rank using the general theory of anisotropic elasticity. The common ground for the papers is the REV concept and generation of multiple realizations of DFN models for DEM or DFN analysis. The relative merits and shortcomings of the basic concept and approaches are probes in detail in the papers, based on a relatively in-depth literature survey. A verification of the DEM approach for deriving equivalent mechanical properties is conducted against existing closedform solutions with regular and persistent sets of perpendicular fractures, with the results showing very good agreements. The verification is not performed for hydraulic analysis since the DFN approach is proven in practice. 2 DISCRETE FRACTURE NETWORK (DFN) ANALYSIS 2.1 Data used for analysis The fracture data used for this work is taken from the site characterization results of the Sellafield area, Cumbria, England, undertaken by United Kingdom Nirex Limited (Nirex 1995, 1997). From the site investigation, four sets of fractures are identified and the orientations of fracture sets are shown to follow Fisher distributions. Table 1 shows the basic information about the fracture system, and properties of intact rock and fractures used in this paper. The data set shows highly fractured rock condition with high fracture density and the highly dispersed pattern with low Fisher constant. The fracture trace lengths are characterized by fractal scaling laws as given by N = 4 L D (1) 3

11 Ki-Bok Min TRITA LWR LIC 2007 Table 2 Complete list of generated model and its numerical experiments. H: Hydraulic analysis, M: Mechanical analysis (Paper B, C) Geometry Side length of model (m) DFN1 H/M H/M H/M H/M H/M H/M H/M H/M H/M H/M H H DFN2 H/M H/M H/M H/M H/M H/M H/M H/M H/M H/M H H DFN3 H/M H/M H/M H/M H/M H/M H/M H/M H/M H/M H H DFN4 H/M H/M H/M H/M H/M H/M H/M H/M H/M H/M H H DFN5 H/M H/M H/M H/M H/M H/M H/M H/M H/M H/M H H DFN6 H/M H/M H/M H/M H/M H/M H/M H/M H/M H/M H H DFN7 H/M H/M H/M H/M H/M H/M H/M H/M H/M H/M H H DFN8 H/M H/M H/M H/M H/M H/M H/M H/M H/M H/M H H DFN9 H/M H/M H/M H/M H/M H/M H/M H/M H/M H/M H H DFN10 H/M H/M H/M H/M H/M H/M H/M H/M H/M H/M H H DFN11~ DFN50 H H H H H DFN1_ H/M H/M H/M H/M H/M H/M H/M H/M H/M H H H DFN1_60 H/M H/M H/M H/M H/M H/M H/M H/M H/M H H H DFN1_90 H/M H/M H/M H/M H/M H/M H/M H/M H/M H H H DFN1_120 H/M H/M H/M H/M H/M H/M H/M H/M H/M H H H DFN1_150 H/M H/M H/M H/M H/M H/M H/M H/M H/M H H H Comments Additional 40 DFNs Rotated models from DFN1 where D is the fractal dimension and N is the number of fractures greater than a given fracture length L (m) per unit area (m 2 ). Fractal dimension (D) from the site characterization was 2.2 ± 0.2 and 2.2 is used for this study. From the fracture density distribution minimum and maximum cutoff lengths is chosen as 0.5 m and 250 m, respectively. 2.2 Discrete Fracture Network (DFN) An independent stochastic discrete fracture network generation program is developed for the generation of realistic fracture networks. The methodology for the DFN program is largely based on the approach suggested by Priest (1993) and Jang et al. (1996). Multiple generation of DFN is conducted through random number generation, the location of fractures are generated by poisson process, and the orientation and trace lengths are simulated based on cumulative probability density function through Monte Carlo Simulation. An algorithm to consider the boundary effect is implemented in the program. This effect is caused by the fact that the centers of generated fractures may lie outside the DFN models, depending mainly on their trace-length. It is suggested in this study that the gap between the analysis model and parent network model should be larger than the half of the maximum cut-off trace lengths (Paper A, B). Based on this suggestion, the parent network for the DFN models was decided as 0 m 0 m so that the gap length between analysis model (maximum 10 m 10 m, in this study) and parent netwrok is larger than half of the maximum trace length of 250 m. From each of the ten large parent networks models twelve smaller DFN analysis models are extracted with varying sizes from 0.25 m 0.25 m to 10 m 10 m as the geometrical models to investigate the effects of scale on the hydraulic and mechanical properties. In order to check whether the calculated properties have a tensor characteristic or not, one of the generated DFN models are rotated at the interval of degrees in the clockwise direction for the evaluation of its tensor quantity. Table 2 presents the complete list of generated models for this study. In total, 120 models are used for the scale dependency investigation and 60 models are used for tensor quantity investigation. 3 METHODOLOGY The technique to conduct the hydraulic numerical experiment are relatively standardized while the mechanical numerical experiments still contains some of hurdles since its parameters involved increase rapidly when it comes to consideration of anisotropy. In what follows, brief introduction of methodology for the calculation of hydraulic and mechanical properties are presented with more focus on mechanical part. More detailed coverage can be found in the attached papers. 4

12 P1 P2 Y P2 P1 where k pq is the permeability tensor in the rotated axis and α pi and α pi are the direction cosines. The ellipse equation of directional permeability in the direction of pressure gradient is given by (Bear, 1972). 2 2 x y + = 1 (4) 1/ k 1/ k x y Fig. 3 Generic boundary conditions for calculation of permeability tensor. P1 and P2 indicate the hydraulic pressures (Paper A, B). 3.1 Hydraulic analysis (Paper A, B) Generalized Darcy s law for anisotropic and homogeneous porous media is used for the calculation of equivalent permeability tensors (Bear, 1972), kij P Qi = A µ x j (2) where Q i = flow rate; A = the cross-section area of the DFN model; k ij = permeability tensor; µ = dynamic viscosity; and P = hydraulic pressure applied. Fig. 3 shows the boundary conditions for calculation of permeability tensors. Flow rates in x- and y- directions were measured with the constant hydraulic pressure gradient in the x- direction and the same calculation was conducted with the constant hydraulic pressure gradient in the y-direction. Complete component of permeability tensors can be obtained from the two boundary conditions. Besides the experiments on ten series of DFN, a series of models (DFN1) were then rotated at every degrees in the clockwise direction for calculation of directional permeability in different direction to compare the average permeability and numerically calculated directional permeability values, which would indicate whether the calculated directional permeability in the pertinent regions can be represented as tensors. The transformation of permeability tensor in a rotated Cartesian system in two-dimension is performed by the rotation mapping operations given as follows, k = k α α (3) pq ij pi qj where k x and k y are principal permeabilities in the direction of pressure gradient x and y, respectively. 3.2 Mechanical analysis (Paper C) General stress and strain relationship can be expressed as (Ting 1996), ε S S S S S S σ x x ε y S21 S22 S23 S24 S25 S26 σ y ε z S31 S32 S33 S34 S35 S σ 36 z = λyz S41 S42 S43 S44 S45 S46 τ yz λ S S S S S S τ xz xz λ xy S61 S62 S63 S64 S65 S τ 66 xy (5) where matrix S ij is called the compliance matrix and is the contracted form of the fourth order tensor S ijkl. The symbols of ε i and λ ij (i, j = x, y, z) denote normal and shear strains, and symbols of σ i and τ ij (i, j = x, y, z) denote normal and shear stresses, respectively. The compliance matrix can be described explicitly by giving the physical meaning of each element as functions of Young s moduli, Poisson ratios and shear moduli, etc (Lekhnitskii 1962). The transformation of compliance tensor is associated with four direction cosines as, S = β β β β S (6) ijkl im jn kp lp mnpq where S ijkl and S mnpq are compliance tensors in the transformed and the original axes, respectively, and β im is direction cosines representing rotational transformations. Since the shear strain and stress component associated with z-direction can be removed for two-dimensional analysis, Eq. (5) can be reduced to the following equation. 5

13 Ki-Bok Min TRITA LWR LIC 2007 y y+ y y+ y xy x x x x x x y y+ y y+ y B.C.(1) B.C.(2) B.C.(3) Fig. 4 Generic boundary conditions for calculation of compliance matrix(paper C). xy ε S S S S σ x x ε y S21 S22 S23 S26 σ y = ε z S31 S32 S33 S σ 36 z λ xy S61 S62 S63 S τ 66 xy (7) Assuming a plane strain condition and ignoring the effect of out-of-plane fractures, eventually only three components are independent in each row of the matrix. Therefore three linearly independent stress boundary conditions are enough to determine all components in the matrix shown in Eq. (7). Fig. 4 illustrates the three linearly independent boundary conditions (BC1, BC2 and BC3) used to 1 ν yx ν η x, yz η xxz, η x, xy zx E x E y E z G yz G xz G xy ν xy 1 νzy ηy, yz ηy, xz η y, xy ε x Ex Ey Ez Gyz Gxz Gxy σ x ν yz 1 ηz, yz ηzxz, η ε y ν xz z, xy σ y ε E x Ey Ez Gyz Gxz G z xy σ z = λyz η yz, x ηyz, y η yz, z 1 µ yz, xz µ yz, xy τ yz λ xz Ex Ey Ez Gyz Gxz G τ xy xz λ xy ηxz,,,, 1 τ, xy x η xz y ηxz z µ xz yz µ xz xy E x Ey Ez Gyz Gxz G xy η xy, x η xy, y η xy, z µ xy, yz µ xy, xz 1 Ex Ey Ez Gyz Gxz G xy : cannot be determined in 2D UDEC simulation : can be determined beforehand from E z =E intact, ν zx =ν zy = ν intact, and symmetry condition : 0 due to the parallel fracture along z-axis in UDEC Fig. 5 Conceptualization for the determination of compliance matrix (Paper C). produce the two-dimensional compliance matrix in Eq. (7). BC1 consists of biaxial normal stresses and BC2 is created by sequentially increasing the normal stress in the y-direction. Similarly, BC3 is created by sequentially superimposing the shear stress increments over the stress conditions of final BC2. An investigation on the response between the stresses and strains shows that linearity of each boundary condition is rather acceptable (Paper C). The conceptual principle for the determination of compliance matrix in two-dimensional plane strain condition is shown in Fig. 5 and this conceptualization can be further compared with the numerical solution calculated from Eq. (7) and the comparison can be used for the accuracy checking of the UDEC modeling. For calculation of the average strain values in the domain, eleven parallel reference lines within each square model are set in both x and y direction in ten intervals with the same distance. Strain components are then calculated from displacement values according to the following strain-displacement relationship λ = ( u + u ) (8) ij i, j j, i where u i (i = x, y) is the displacement components along the sampling lines. Fig. 6 presents the UDEC verification on the orthogonally fractured rock with different fracture shear to normal stiffness ratio (K). The results show that the UDEC simulation technique can be successfully used for property determination of anisotropic rock mass. 6

14 Normalised elastic modulus RESULTS Hydraulic properties - Permeability tensors (Paper A, B) Fig. 7 shows one example result of calculated permeability tensor elements k xx at different scales from the ten random realizations (from DFN1 to DFN10). The variation of calculated permeability components become smaller as the model size increases, and the permeability values maintain constant ranges of values after a certain size. At the side length of 2 m (about two times of the mean trace-length and below which 95 % of fractures have their trace-lengths) drastic reduction of variance can be observed in the figures. The mean values of k xx with further increasing model size do not show significant change, indicating approach on a REV. 0 K0.1 K0.2 K0.5 K1.0 K2.0 K5.0 Fig. 6 Verification of UDEC modeling results against analytical solution for a fractured rock mass with two orthogonal sets of fractures. Points of symbols correspond to numerical results with different K ratio values and the lines are the analytical solutions. Y X 4.2 Mechanical properties - Compliance matrix (Paper C) Fig. 8 presents the normalized elastic moduli in x-directions with ten DFN realizations of increasing side lengths from 0.25 m to 8 m scales with the K ratio (ratio of fracture shear stiffness to normal stiffness) of 1.0. Overall trend is similar to that of permeability value as seen in Fig. 7. At the side length of the models less or equal to 1 m, the ranges of the values are notably larger than other bigger side length scales. This scattered data implies that the model cannot be described as statistically homogeneous because one cannot assume a stable range of possible properties. However, the scattering of the results clearly narrows down with increase of the side lengths, and points to the possibility of existence of a REV with the normalized elastic modulus about 43 % of intact rock. The results of the Poisson s ratio also show similar pattern and the values converge to about 0.1 and 0.35 for K ratio of 1.0 and 0.2, respectively (Paper C). 4.3 Comparison between average and calculated properties in rotated model (Paper B, C) Considering that both permeability and compliance matrix have tensor characteristics from Eqs.(3) and (6), they can be determined in any transformed axis once the compliance matrix in original axis and the angle of rotation is known. This relation is used as a major tool in evaluating whether the constructed numerical models behaves like continuum or not. Numerical experiments are performed on the rotated models in order to compare the average tensor, which is 3.0E Permeability, k xx (m 2 ) 2.5E E E E E-14 DFN1 DFN2 DFN3 DFN4 DFN5 DFN6 DFN7 DFN8 DFN9 DFN10 Normalized elastic modulus DFN1 DFN2 DFN3 DFN4 DFN5 DFN6 DFN7 DFN8 DFN9 DFN10 0.0E Side length of square model (m) Side length of square model (m) Fig. 7 Variation of directional permeability in the x- direction with the increase of side lengths of square models (Paper A, B). Fig. 8 Variation of elastic modulus in the x- direction with the increase of side lengths of square models (K=1.0) (Paper C). 7

15 Ki-Bok Min TRITA LWR LIC 2007 determined by averaging six tensors of rotated models in a reference axis (x- and y-axis), with the calculated values from numerical experiments. Here, average permeability tensor is defined as, k 1 N r ij kpqαipα jq r= 1 = (9) N where k ij is average permeability tensor, N denotes the number of rotations, k r pq is the calculated permeability in each rotated models. And average compliance tensor is similarly defined as, S ij = 1 N N r= 1 S r pq q pi q qj (10) r where S are the calculated compliance pq matrixes at rotated p-q axes, and the q pi and q qj are the matrix of direction cosines of the rotation defined in Appendix II of Paper C. The results in Fig. 9 show that the permeability variation does not conform to an ellipse for models of side length < 5 m, which means that the continuum application cannot be justified at such scales. As the side length of the model increases, the tendency of forming an ellipse by the numerical data becomes stronger. Fig. 10Fig. 10 presents the average and calculated curves of the normalized elastic moduli with º intervals. Like in the case of permeability, numerical results from the small scale models do not match well with the averaged values. The properties do not have a tensor quantity at smaller scales. However, as the size of model increases the numerical results match increasingly well with the average values. At the size of 5 and 7 meters of side lengths, the numerical values and the average trace curves agrees very well, indicating that the property at these sizes has a fourth-order tensor quantity. These results support that fractured rock mass can be modeled by applying continuum mechanics principle to a certain degree of accuracy. 4.4 Evaluation of acceptable error in determining REV (Paper B, Paper C) 0 5x x10 6 3x x10 6 1x x x x x10 6 1x x x x x10 6 1x x10 6 2x10 6 3x10 6 4x10 6 5x Side length 0.25 m 120 1x10 6 2x10 6 3x10 6 4x10 6 5x Side length 0.5 m 120 1x10 6 2x10 6 3x10 6 4x10 6 5x Side length 1 m 120 5x x10 6 3x x10 6 1x x10 6 2x10 6 3x x10 6 5x Side length 2 m 5x10 5x x10 6 4x x x x10 6 2x10 6 1x10 6 1x x10 6 1x10 6 2x10 6 2x x x x10 6 4x x x Side length 5 m Side length 8 m Average 1/K 1/2 Calculated 1/K 1/ Fig. 9. Approximation of equivalent permeability tensor with increasing model size (expressed in 1/K 1/2 (θ), Paper B). 8

16 Elastic modulus (GPa) Elastic modulus (GPa) Angle (degrees) (a) Angle (degrees) (b) Elastic modulus (GPa) Elastic modulus (GPa) Elastic modulus (GPa) (c) Angle (degrees) Elastic modulus (GPa) (d) Angle (degrees) (e) Angle (degrees) (f) Angle (degrees) Predicted by average tensor Measured by numerical experiment Fig. 10 Comparison between predicted and measured elastic moduli (Ex) in rotated axes. Predicted 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 (Paper C). The Coefficient of variation and mean prediction error are defined as measures for a quantitative evaluation of errors in determining hydraulic and mechanical REV. Coefficient of variation, defined as the ratio of standard deviation to mean values at the given sizes, presents the variation of property and it is calculated on ten DFN realization of models. Prediction error for permeability (EPp i ) can be defined as (Paper B), 2 N 1 EPp1 = N EPp r= 1 N 1 = N r = 1 r ( k 11 k11 ) k 11 r ( k 22 k22 ) k 22 (11) where EPp i is the prediction error of permeability tensor in the i-direction (i = x, y), k ij r is the average permeability tensor, k ij is permeability values from numerical experiments. N denotes the number of cases for rotation (six rotation cases for this study with 0º, º, 60º, 90º, 120º and 150º, respectively). Note that, in evaluating the prediction error, influence of nondiagonal components of permeability tensor are omitted for simplicity. Prediction error for compliance matrix is defined as followings (Paper C), N 2 j= 1 r ij s sij 1 r= 1 j= 1 EPci = (12) 2 N s ij where EPc i is prediction error of compliance matrix in i direction (i = x, y), s ij is the average compliance tensor, s r ij is compliance tensor from numerical experiments, Note that, in evaluating 9

17 Ki-Bok Min TRITA LWR LIC 2007 Coefficient of Variation (%) REV below 20% of acceptable variation kxx kyy REV below 10% of acceptable variation Coefficient of Variation (%) REV below 10% of acceptable variation Ex, K=1.0 Ey, K=1.0 Poisson's ratio, K=1.0 Ex, K=0.2 Ey, K=0.2 Poisson's ratio, K=0.2 REV below 5% of acceptable variation Side length of square model (m) (a) Side length of square model (m) (b) Fig. 11 Coefficient of variation with increasing side lengths. (a) permeability (Paper B), (b) elastic modulus and Poisson s ratio (Paper C). the compliance tensor error, influence of shear stress on normal strain is omitted for simplicity and z-directional strain is not included in error evaluation. Prediction error for compliance matrix intended to represent the error involved in the normal strain predictions in each direction. Fig. 11 presents the coefficient of variation versus scale for permeability and mechanical properties. Trends of coefficient of variations indicate the decrease of variation with the increase of model size. From this evaluation, the hydraulic REV size can be decided as 5 m (about five times of mean trace length) and 8 m with the acceptable variation of 20 % and 10 %, respectively. The mechanical REV has 3 m and 8 m scale with the acceptable variation of 10 % and 5 % (Table 3). Investigation of mean prediction error suggests 5 m scale of hydraulic REV and 2 m scale of mechanical REV with 5 % of acceptable error (Paper B,C). 5 CONCLUDING REMARKS 5.1 REV, equivalent permeability and compliance matrix (Paper B, C) The hydraulic and mechanical properties to be determined may have very wide range below a certain scale. Results in this study (Fig. 7 & Fig. 6) clearly demonstrate the scale dependency and existence of REV for the fractured rock mass with the given fracture data. Example of determined hydraulic and mechanical properties are as follows. At permeability tensor at 8 m scale, kxx kxy = 10 (m ) kxy k yy (13) At 6 m scale with the ratio of shear stiffness to normal stiffness of fracture 1.0, Table 3 REV determination based on coefficient of variation and mean prediction error (Paper B, C). Coefficient of Variation (Scale dependency) Mean prediction error (Tensor quantity evaluation) Acceptable Level (%) Hydraulic REV Mechanical REV 20 5 m 5 m 1 m 1 m 10 8 m 8 m 3 m 3 m 5 > 10 m 10 m 6 m 6 m m 0.5 m 2 m 2 m 5 5 m 5 m 2 m 2 m 2 >10 m 10 m 6 m 6 m 10

18 S S S S S21 S22 S23 S26 = S31 S32 S33 S 36 S61 S62 S63 S (1/ Pa) (14) One of the important results from the study is that the effect of scale and individual fracture properties can be readily understood by the series of numerical experiments. In both hydraulic and mechanical analysis, beyond the side length of 2-3 m, the scattering range of results becomes notably smaller and constant. Note that mean trace length of fracture is 0.92 m, 95 % of fractures are less than 2 m in trace length and 99 % of fractures are less than 5 m in trace length according to the fractal nature of fracture distribution. This concentration of smaller fractures is perhaps an important factor for the size of final REV. Many factors will have an influence on the REV, namely fracture properties, constitutive relation, distributions of trace length, orientation and locations of fractures. When the statistics of fractures are homogeneous, it is probable that REV can exist in certain scales as demonstrated by this study. However, existence of REV for fractured rock masses needs site specific investigation, especially regarding effects of fracture size and deformation characteristics. 5.2 Appropriateness of equivalent continuum approach From the comparison between predicted properties and measured properties in the rotated DFN, the appropriateness of equivalent continuum approach is investigated. It is shown that in small scales, like less than 2 m scale, there can be some error when the fractured rock is approximated as continuum. However, after a certain scales, the hydraulic and mechanical properties in rotated axes can be calculated with acceptable accuracy (Table 3). This demonstrates the fractured rock larger than some scale is homogeneous in equivalent senses and fractured rock mass may behave like continuum after certain scales. The method presented in this thesis gives a rigorous justification of equivalent continuum approach and its condition of validity. 5.3 Comparative study about the character of hydraulic and mechanical properties of fractured rock The variations of permeability and elastic moduli calculated from the numerical experiments show similar trend with the increase of side length of DFN model. It was clearly seen that the fractures are playing significant role in both hydraulic and deformation behaviour of fractured rock. It is noted that the variation of hydraulics is bigger than the mechanics. This is because, in DFN analysis, the aperture of fracture is one single pathway for the fluid flow. Therefore the flow response to the stochastic realization or sampling of DFN is rather sensitive. On the other hand, the mechanical properties are sum of the fracture deformation and intact rock deformation. Therefore, mechanical property is relatively less sensitive than that of hydraulic response. This feature demands that the acceptable variation of hydraulics may be set to a bigger level than that for the mechanics due to more variable nature of hydraulics. 6 RECOMMENDATION FOR FURTHER RESEARCH 6.1 The effect of fracture constitutive model, rock mass strength and threedimensional extension The constitutive equation of fractures used for this study adopted a constant stiffness model. However, extensive laboratory test demonstrate that the relation is basically hyperbolic (Bandis et al, 1983). Thus a more realistic fracture constitutive model with hyperbolic relations will give a more realistic rock mass behavior, leading to stress dependent compliance matrix. Some researches have been conducted on the strength of fractured rock (Bhasin 1998, Pouya 2001). The technique developed here will be readily extended to strength problems with multiple realization and consideration of anisotropy. Two-dimensional analysis performs analysis by converting the three-dimensional fracture information to two-dimensional form with 11

19 Ki-Bok Min TRITA LWR LIC 2007 apparent dip angle. It assumes that the direction of strike is toward in-plane direction. Although, the current computing capacity for threedimensional analysis may not enough for this study, which involve maximum 5000 blocks in the region, it is obvious that the burden imposed on three-dimensional analysis will soon be removed with the advance of computer technology. Thus the three-dimensional application would be also possible. 6.2 Hydro-Mechanical coupling In this study, constant hydraulic aperture was used to focus on the evaluation of the impact of geometry on hydraulic properties. This condition should be interpreted as stress free condition without mechanical loading. The stress/deformation processes of the fractures are usually ignored in the DFN models, simply due to the complexity of the numerical techniques and extraordinary computational efforts needed including this study. Although simple estimates concerning the effects of in-situ stresses on fracture aperture variations have been used in DFN models, e.g. in the NAPSAC code (Herbert, 1996), proper representation of the coupled stress/flow process in fractures is needed to estimate more realistic permeability in in-situ condition. Considering that Hydro-Mechanical coupling with realistic fracture network is not well investigated, hydro-mechanical coupling simulation of DFN models will give a valuable contribution to the understanding of complex behaviour in fractured rock masses 6.3 Application to large scale Performance Assessment (PA) analysis Recalling that the objectives of this study, following step after this upscaling analysis is to perform equivalent continuum analysis on largescale problems. The results presented in this thesis show a possibility to evaluate the uncertainties involved in the equivalent continuum approach and the determined properties give a range of values. The outcome from this study should be incorporated in application of continuum analysis, i.e. stochastic continuum analysis. Also rather general approach adopted in this study with the consideration of anisotropy, scale effect, multiple realization and tensor quantity evaluation made it possible to evaluate the uncertainty related to the assumption of isotropy, homogeneity and continuum application to fractured rock mass. Ultimately, numerical determination of equivalent property with multiple realization can be completed when systematic continuum analysis is applied with close link to the parameter determination methodology. 7 REFERENCES Aydan Ö, Jeong GC, Seiki T, Akagi T. 1995, A comparative study on various approaches to model discontinuous rock mass as equivalent continuum, In: Mechanics of Jointed and Faulted Rock, Rossmanith (ed.), p Bandis SC, Lumsden AC, Barton NR. 1983, Fundamentals of rock joint deformation, Int J Rock Mech Min Sci, vol.20(6), p Barton N. 2002, Some new Q-value correlations to assist in site characterization and tunnel design, Int J Rock Mech Min Sci, vol.39(2),p Bhasin R, Hoeg K. 1998, Numerical modelling of block size effects and influence of joint properties in multiply jointed rock, Tunnelling and Underground Space Technology, vol.13(2),p Bear J, 1972, Dynamics of fluids in porous media, Dover, p.764. DECOVALEX III secretariat, 2000, TASK 3, BMT2 protocol, version 6.0, (unpublished report). Hart RD. 1993, An introduction to distinct element modeling for rock engineering. In: Comprehensive Rock Engineering, J. A. Hudson (Ed. in-chief), Vol.2, Pergamon Press, Oxford, p Herbert AW. 1996, Modelling approaches for discrete fracture network flow analysis. In: Stephansson et al. eds., Coupled Thermo- Hydro-Mechanical Processes of Fractured Media, Developments in Geotechnical Engineering, vol. 79, Elsevier, Amsterdam, p Itasca Consulting Group Inc. 2000, UDEC user s guide, Minnesota. Jang HI, Chang KM, Lee CI. 1996, Groundwater flow analysis of discontinuous rock mass with probabilistic approach, J. of Korean 12

20 Society for Rock Mech, vol.6, p.-38 (in Korean). Jing L. 1998, Formulations of discontinuous deformation analysis for block systems. Engineering Geology, vol.49, p La Pointe PL, Wallmann PC and Follin S. 1996, Continuum modeling of fractured rock masses: Is it useful?, In Barla (eds), Eurock 96, Rotterdam, Balkema, p Lekhnitskii SG. 1963, Theory of elasticity of an anisotropic elastic body, Holden Day, Inc. San Francisco, p.404. Long JCS, Remer JS, Wilson CR, Witherspoon PA. 1982, Porous media equivalents for networks of discontinuous fractures, Water Resour Res, vol.18(3), p Nirex, 1995, Geotechnical studies at Sellafield, Executive summary of NGI/WSA work from , Nirex Report 801. Nirex, 1997, Evaluation of Heterogeneity and Scaling of Fractures in the Borrowdale Volcanic Group in the Sellafield Area, Nirex Report SA/97/028. Neuman SP. 1987, Stochastic continuum representation of fractured rock permeability as an alternative to the REV and fracture network, 28 th US symposium on Rock Mechanics, Tucson, p Pouya A, Ghoreychi M. 2001, Determination of rock mass strength properties by homogenization, Int J Numer Anal Meth Geomech, vol.25, p Priest SD. 1993, Discontinuity analysis for rock engineering, Chapman & Hall, London, p.473. Sitharam TG, Sridevi J, Shimizu N. 2001, Practical equivalent continuum characterization of jointed rock masses, Int J Rock Mech Min Sci, vol. 38(3), p Ting TCT. 1996, Anisotropic Elasticity Theory and Applications, Oxford University Press, p

21 Ki-Bok Min TRITA LWR LIC

22 PAPER A Ki-Bok Min, Lanru Jing, Ove Stephansson, 2002 Determination of the permeability tensor of fractured rock masses based on stochastic REV approach Published in: ISRM regional symposium, 3 rd Korea-Japan Joint symposium on rock engineering, Seoul, Korea, Vol. 1, pp

23 PAPER B Ki-Bok Min, Lanru Jing, Ove Stephansson Fracture system characterization and evaluation of the equivalent permeability tensor of fractured rock masses using a stochastic REV approach Submitted to Hydrogeology Journal

24 PAPER C Ki-Bok Min, Lanru Jing Numerical determination of equivalent compliance tensor of elasticity for fractured rock masses using the Distinct Element Method Submitted to International Journal of Rock Mechanics and Mining Sciences