446885/FULLTEXT01.pdf · Stress effects on solute transport in fractured rocks iii ABSTRACT The effect of in-situ or redistributed stress on solute transport in fractured rocks is

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TRITA-LWR PhD thesis 1060

ISSN 1650-8602

ISRN KTH/LWR/PHD 1060-SE

ISBN 978-91-7501-099-1

STRESS EFFECTS ON SOLUTE

TRANSPORT IN FRACTURED ROCKS

Zhihong Zhao

September 2011

Zhihong Zhao TRITA LWR PhD Thesis 1060

ii

© Zhihong Zhao 2011

PhD thesis

Group of Engineering Geology and Geophysics

Department of Land and Water Resources Engineering

Royal Institute of Technology (KTH)

SE-100 44 STOCKHOLM, Sweden

Reference to this publication should be written as: Zhao, Z (2011) ―Stress effects on solute transport in fractured rocks‖ TRITA LWR PhD 1060.

Stress effects on solute transport in fractured rocks

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ABSTRACT

The effect of in-situ or redistributed stress on solute transport in fractured rocks is one of the major concerns for many subsurface engineering problems. However, it remains poorly understood due to the difficulties in experiments and numerical modeling. The main aim of this thesis is to systematically investigate the influences of stress on solute transport in fractured rocks, at scales of single fractures and fracture networks, respectively.

For a single fracture embedded in a porous rock matrix, a closed-form solution was derived for modeling the coupled stress-flow-transport processes without considering damage on the fracture surfaces. Afterwards, a retardation coefficient model was developed to consider the influences of damage of the fracture surfaces during shear processes on the solute sorption. Integrated with particle mechanics models, a numerical procedure was proposed to investigate the effects of gouge generation and microcrack development in the damaged zones of fracture on the solute retardation in single fractures. The results show that fracture aperture changes have a significant influence on the solute concentration distribution and residence time. Under compression, the decreasing matrix porosity can slightly increase the solute concentration. The shear process can increase the solute retardation coefficient by offering more sorption surfaces in the fracture due to gouge generation, microcracking and gouge crushing.

To study the stress effects on solute transport in fracture systems, a hybrid approach combing the discrete element method for stress-flow simulations and a particle tracking algorithm for solute transport was developed for two-dimensional irregular discrete fracture network models. Advection, hydrodynamic dispersion and matrix diffusion in single fractures were considered. The particle migration paths were tracked first by following the flowing fluid (advection), and then the hydrodynamic dispersion and matrix diffusion were considered using statistic methods. The numerical results show an important impact of stress on the solute transport, by changing the solute residence time, distribution and travel paths. The equivalent dispersion coefficient is scale dependent in an asymptotic or exponential form without stress applied or under isotropic compression conditions. Matrix diffusion plays a dominant role in solute transport when the hydraulic gradient is small.

Outstanding issues and main scientific achievements are also discussed.

Key words: Fractured rocks; Solute transport; Stress effects; Coupled stress-flow-transport processes; Analytical solution; Particle mechanics model; Fracture surface damage; Discrete element method; Particle tracking method.


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ACKNOWLEDGEMENT

Swedish Nuclear Fuel and Waste Management Co. (SKB) is gratefully acknowledged for the financial support through the DECOVALEX-2011 project.

I wish to express my sincerest appreciations to my two supervisors, Dr. Lanru Jing and Prof. Ivars Neretnieks. Dr. Jing brought me into the international project, DECOVALEX-2011, and helped me in research and completion of this thesis during my three years‘ study at KTH. Prof. Neretnieks Ivars introduced me into the solute transport field. He helped me in clarifying the fundamental concepts and reviewed each manuscript of my papers.

Special thanks are given to Prof. Luis Moreno who is from the same division as Prof. Neretnieks, for providing the basic transport code and literatures, and valuable comments on some of my papers. I also like to thank Dr. Longcheng Liu for his transport phenomenon course that I attended and fruitful discussions.

By participating in the DECOVALEX-2011 project, I learnt a lot from many participating scientists and students from different countries, especially the Task C co-workers, Prof. John Hudson, Mr. Colin Leung, Dr. Jonny Rutqvist, Dr. Hokr Milan, Prof. Xiating Feng, Dr. Yuexiu Wu and Dr. Lingfang Shen. I like to thank them for letting me lead a joint paper with them for a journal publication.

I deeply appreciated the help and encouragement from all the people in LWR. Dr. Joanne Fernlund helped me to settle down when I arrived in Stockholm. Dr. Fuguo Tong picked me up at the airport and shared a lot of useful discussions with me. Mimmi Magnusson always showed up when I had Swedish problem. Additionally, I want to thank Zairis Coello, Solomon Tafesse, Katrin Grünfeld, Majid Noorian Bidgoli, Riccardo Vesipa, Imran Ali and Caroline Karlsson for a very friendly and helpful atmosphere in our corridor. Britt Chow and Aria Saarelainen are thanked for all kinds of department routines. I also thank other Chinese colleagues Sihong Wu, Ting Liu and Jingjing Yang for the nice relaxing talking in Chinese.

I am grateful to Rolf Christiansson from SKB for inviting and driving us to visit the Äspö underground laboratory and recommending me the SKB reports of shear test.

I am thankful to my former MSc supervisor Prof. Jin-an Wang from University of Science and Technology Beijing for encouragements all the time, and Dr Y.M. Cheng at the Hong Kong Polytechnic University for my research fellowship there before my study in Sweden.

I thank my parents and sister in China for love, and I always missed them in the past three years. Their endless support and understanding gave me energy to focus on my study.

Finally, the biggest thank you goes to my wife, Zhina Liu, for her never-ending love, encouragement and support all the time.


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LIST OF PUBLICATION

This thesis is to a large extent a summary of the following papers, which are appended at the end of the thesis and referred in the text of the thesis by their Roman numbers (I- IV).

I: Zhao Z, Jing L, Neretnieks I, Moreno L. 2011. Analytical solution of coupled stress-flow-transport processes in a single fracture. Computers and Geosciences. 37:1437-1449.

II: Zhao Z, Jing L, Neretnieks I. 2011. Shear effects on solute retardation coefficient in rock fractures: Insights from a particle mechanics model. International Journal of Rock Mechanics and Mining Sciences. (Under review).

III: Zhao Z, Jing L, Neretnieks I, Moreno L. 2011. Numerical modeling of stress effects on solute transport in fractured rocks. Computers and Geotechnics. 38:113-126.

IV: Zhao Z, Jing L, Neretnieks I. 2010. Evaluation of hydrodynamic dispersion parameters in fractured rocks. Journal of Rock Mechanics and Geotechnical Engineering. 2:243-254.

Some additional content in this thesis is from the parts of two manuscripts below, which are not appended.

V: Zhao Z, Jing L, Neretnieks I, Moreno L. Numerical investigation of the effects of local dispersion in single fractures on the macrodispersion at the fracture network scale. (Manuscript).

VI: A joint paper for Task C, DECOVALEX-2011 led by Zhao Z (under preparation).

The following publications were written or published during the period of my doctoral study, but are not appended after this thesis:

VII: Zhao Z, Jing L, Neretnieks I. 2010. Stress effects on nuclide transport in fractured rocks: a numerical study. In: Proceeding of European rock mechanics symposium (EUROCK2010), Lausanne, Switzerland, pp783-786.

VIII: Vesipa R, Zhao Z, Jing L. 2009. Estimating hydraulic permeability of fractured crystalline rocks using geometrical parameters. In: The 9th international conference on analysis of discontinuous deformation (ICADD9), Singapore, pp685-692.


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TABLE OF CONTENT

Abstract ........................................................................................................... iii

Acknowledgement ............................................................................................v

List of publication .......................................................................................... vii

Table of Content ............................................................................................. ix

1. Introduction ............................................................................................. 1

1.1. Background and DECOVALEX-2011 project .................................... 1

1.2. Research objectives and thesis layout ............................................... 2

2. Literature review ..................................................................................... 3

2.1. Fractured rocks characterization ....................................................... 3

2.2. Solute transport in fractured rocks .................................................... 4

2.2.1. Basic solute transport mechanisms ........................................... 4

2.2.2. Solute transport in single rock fractures .................................... 6

2.2.3. Solute transport in fracture network .......................................... 9

2.3. Stress effects on solute transport in fractured rocks ......................... 11

2.3.1. Mechanical process ................................................................... 11

2.3.2. Stress effects on solute transport............................................... 13

2.4. Outstanding issues and motivation of this thesis ............................ 14

3. Stress effects on solute transport in single fractures ............................. 17

3.1. Analytical model without damage of fracture surface ...................... 17

3.1.1. Conceptual model ..................................................................... 17

3.1.2. The closed-form solution for stress effects on solute transport in single fractures ................................................................... 18

3.2. Particle mechanics model with fracture surface damage ................. 26

3.2.1. Conceptual mechanisms of rock fracture surface damage ....... 26

3.2.2. Retardation coefficient model ................................................... 29

3.2.3. Particle mechanics model ......................................................... 31

3.3. Results for stress effects on solute transport in single fractures ...... 33

3.3.1. Without fracture surface damage case ...................................... 33

3.3.2. With fracture surface damage case ........................................... 41

3.4. Summary ........................................................................................... 51

4. Stress effects on solute transport in fracture network ........................... 53

4.1. Methodology ..................................................................................... 53

4.1.1. DEM model for coupled stress-flow simulation ....................... 53

4.1.2. Solute transport simulation ....................................................... 56

4.1.3. Code .......................................................................................... 61

4.2. Numerical experiment procedure .................................................... 62

4.2.1. Coupled stress-flow-transport modeling procedure ................. 62

4.2.2. Evaluation of macroscopic behaviors of solute transport in fractured rocks ....................................................................................... 64

4.3. Results ............................................................................................... 66

4.3.1. Flow results ............................................................................... 66


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4.3.2. Advection ................................................................................... 68

4.3.3. Hydrodynamic dispersion ......................................................... 73

4.3.4. Matrix diffusion ......................................................................... 82

4.4. Summary ........................................................................................... 84

5. Discussions ............................................................................................ 85

5.1. Outstanding issues of the analytical model...................................... 85

5.2. Three damaged zones of fracture surfaces undergoing shear ......... 86

5.3. Influence of hydraulic gradient and boundary conditions ............... 87

5.4. Comparison with SOLFRAC (Bodin et al., 2007) ............................ 89

5.5. Comparison with other DECOVALEX teams ................................. 90

6. Concluding remarks and future study ................................................... 93

6.1. Main conclusions and scientific achievements ................................ 93

6.2. Recommendations for future study .................................................. 94

References ....................................................................................................... 95


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1. INTRODUCTION

1.1. Background and DECOVALEX-2011 project

The coupled stress-flow-transport processes in fractured geological media have become a major research topic, due to the increasing demands for design, construction, operation and performance/safety assessments of underground radioactive waste repositories and other energy engineering projects and environmental protection issues, such as groundwater pollution and oil/gas reservoir engineering. For final repositories of spent nuclear fuel, the possible leaching of the radioactive nuclides and subsequent migration in the underground bedrock is always a major concern (Neretnieks, 1980). Besides advection with the fluid flow in fractured geological materials, the solutes in the groundwater system can also be retarded by other physical or chemical mechanisms with various significance, such as sorption on fracture surfaces, diffusion in and out of the rock matrix, chemical reactions between the solute and the rock matrix or the fracture walls, and decay.

It is well known that the in-situ stresses and their redistributions due to engineering perturbations must always be taken into account in underground engineering. The coupled stress-flow-transport issues are especially important for the performance and safety assessment of the repositories of radioactive wastes in deep fractured crystalline rocks, such as granite, because the connected fracture systems serve as the main pathways for groundwater flow and radioactive nuclide transport. The stresses can change the fracture apertures (by causing normal closure and shear dilation) matrix porosity and produce gouge infillings, then consequently change the flow paths or induce channeling for fluid flow. If the fluid contains nuclides or other solutes, the transport phenomenon would also be influenced by the mechanical processes. Therefore, it is expected that the stresses would play an important role in performance/safety assessments of radioactive waste repositories, so this subject should be properly investigated.

However, in the past, the mechanical, fluid flow and solute transport processes were most often studied separately in different branches of geosciences, and the combined mechanical effects on the fluid flow and solute transport processes have not been investigated systematically. Therefore, the current study will provide some basic insights for better understanding of the solute transport process in fractured rocks loaded by the stresses.

This study is a part of the international DECOVALEX-2011 project for numerical modeling of coupled thermo-hydro-mechanical-chemical (THMC) processes in geological media for performance and safety assessments of nuclear waste repositories, established since 1992 (Hudson et al., 2009). The DECOVALEX-2011 project consists of the following three tasks.

Task A: HMC processes in argillaceous rocks;

Task B: Coupled mechanical thermal loading of hard rocks;


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Task C: Characterization and modeling of THMC processes in fractures and fractured rocks.

This thesis contributes to the Task C of DECOVALEX-2011, aiming to investigate the state-of-the-art of numerical modeling approaches and techniques for characterization and modeling of the coupled HMC processes in both single fractures and fractured rock masses, with focus on crystalline rocks (Jing & Hudson, 2008; Neretnieks et al., 2009). The research conducted in this thesis was financed by SKB through DECOVALEX-2011.

1.2. Research objectives and thesis layout

Although general conceptual understanding of the importance of coupled stress-flow-transport processes in fracture networks is available, but reliable experimental investigations of such complex processes encounter many challenges, with few publications reported. The main technical difficulties are realizing fracture systems with realistic distributions of length, orientation and aperture, especially roughness of the fracture surfaces under well-controlled laboratory or in-situ test conditions. Mathematical modeling, therefore, becomes almost the only available tool for predicting the conceptual behaviors of such coupled processes at present.

Based on the above mentioned motivations and present understandings, the general objective of this thesis is to investigate the influences of stress/deformation on solute transport in fractured crystalline rocks by analytical and numerical methods, at the scales of both single fractures and complex fracture networks. Therefore, this thesis provides not only basic insights on the coupled stress-flow-transport processes in fractured geological media, but also useful guidance for future experimental studies.

The content of the thesis is organized as below.

DECOVALEX-2011 project research background and motivations are briefly introduced in Chapter 1.

Literature survey on modeling solute transport in fractured rocks as well as stress effects is presented in Chapter 2.

Stress effects on solute transport in a single rock fracture embedded in a porous matrix are studied in Chapter 3, for cases without and with damage along the fracture surfaces (Paper I and II), by analytical and particle mechanics method, respectively.

Stress effects on solute transport in a fracture network are studied in Chapter 4, including the macroscopic transport behaviors of advection, hydrodynamic dispersion and matrix diffusion (Paper III, IV and V).

Five important issues relating to the current study are discussed in Chapter 5.

In Chapter 6, the main conclusions and scientific achievements are highlighted. Based on the outstanding issues in the current study, some suggestions are proposed for future studies.


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2. LITERATURE REVIEW

2.1. Fractured rocks characterization

Generally there are three conceptual approaches to characterize fractured rocks. The first one is the discrete element method (DEM) model, which explicitly represents individual permeable fractures, their connectivity geometry, and the impermeable rock blocks. The discrete fracture network (DFN) method is a subset of DEM (Jing & Stephansson, 2007). The second is the equivalent (deterministic or stochastic) continuum (EC) approach, using single, dual or multiple continuum models. The third approach is to combine the above two into a hybrid model, which is the equivalent continuum model containing a relatively small number of (or statistical information on) discrete fractures (National Research Council, 1996; Selroos et al., 2002; Neuman, 2005; Bodin et al., 2007).

(1) The DEM/DFN models explicitly depict the geometry, contact mechanisms and permeability of fracture systems. Fractures are usually represented by one-dimensional (1D) line (or pipe) elements in two-dimensional (2D) models, or by 2D planar elements (in shape of circle, ellipse or polygon) in three-dimensional (3D) models (Long et al., 1982; Andersson & Dverstorp, 1987; Billaus et al., 1989; de Dreuzy et al., 2000; Huseby et al., 2001). Therefore, the DEM/DFN model is, in theory, more realistic representation of real fractured rocks, but its disadvantage is that very detailed information of fracture geometry (position, size, shape and orientation) and properties (stiffness, strength and permeability) is needed but difficult to be obtained. Consequently, it demands considerable computer memory and speed.

To avoid the difficulties that 3D DFN models exhibit, the fracture networks can be simplified into a network of connected channels, which connect the centers of the fractures through a point at the intersection line between the fractures (Cacas et al., 1990a, b). The above model was further simplified as a channel network (CN) model (Moreno & Neretnieks, 1993; Moreno et al., 2006), in which each channel member does not necessarily have the clearly identifiable physical features. Actually the individual channel members equivalently represent the stochastic properties of the flow paths, so the CN models need to be validated by field measurements. Although it has been demonstrated that the CN model can successfully predict the fluid flow and solute transport processes in fractured rocks, it also faces the calibration problems when mechanical process needs to be included in the modeling.

(2) The EC model assumes that the macroscopic behaviors of fractured rocks can be represented by that of an equivalent porous medium, where the effects of the fractures are implicitly represented in the equivalent constitutive models and other associated properties or parameters. Compared with DEM/DFN models, the EC models are more suitable for predicting the overall behaviors of fractured rocks at relatively large scales, and there are


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of three types of EC models basically, according to the sub-continua considered.

The single-continuum models for fractured rocks equivalently consider the average properties of both fractures and matrix, based on the principles of classical continuum theory (Berkowitz et al., 1988; McKay et al., 1997; Brikholzer et al., 1999; Doughty, 1999; Wang et al., 1999; Jackson et al., 2000). The dual-continuum models represent the fractures as secondary porosity contributors to the original porosity of rock matrix by breaking the porous media into blocks, and the fluid and mass transfer between the fractures and matrix was modeled as the exchange flux between the two continua (Barenblatt et al., 1960; Warren & Root, 1963). The dual-continuum models can be defined as either dual-porosity and dual-permeability models, with and without fluid exchange between two continua, respectively (Berkowitz, 2002). In triple (or multi)-continuum models, either the matrix is subdivided, or the fractures are divided into large (globally connected) and small (locally connected) fractures (Pruess & Narasimhan, 1985; Liu et al., 2003; Wu et al., 2004). The key issue of applying EC model is the determination of equivalent properties of fractured rocks.

(3) A kind of the hybrid models only considers a few interconnected large fractures within the equivalent continuum model of matrix and small fractures (Sudicky & Mclaren, 1992; Therrien & Sudicy, 1996; Tsang et al., 1996; Stafford et al., 1998; Lee et al., 2001; Jørgensen et al., 2004; Graf & Therrien, 2008). The other hybrid models are to use DFN models to estimate the effective properties of large EC model (Schwartz & Smith, 1988; National Research Council, 1996). The third hybrid models are to map the individual fractures onto a finite difference/element grids as conductive fields (Svensson, 2001a, b; Doughty et al., 2002; Reeves et al., 2008; Botros et al., 2008). This method is relatively easier to implement, because only one grid is required with respect to EC models, but the accurate mapping of fracture properties onto the mesh remains a challenging issue.

2.2. Solute transport in fractured rocks

2.2.1. Basic solute transport mechanisms

Solute transport in fractured geological media has been a major research topic in earth sciences, and is especially important for underground repositories for radioactive nuclear wastes and groundwater pollution in fractured reservoirs (Bodin et al., 2003a). Many books and review papers have provided the current knowledge about the solute transport mechanisms in fractured rocks (Bear et al., 1993; Sahimi, 1995; Adler & Thovert, 1999; Berkowitz, 2002; Bodin et al., 2003a, b; Neuman, 2005; MacQuarrie & Mayer, 2005). This thesis is specifically dedicated to the case where the permeability of the rock matrix is negligible compared with that of fractures, such as the granitic rocks in Sweden. In addition, the research mainly focuses on the single phase laminar flow under saturated and steady conditions.


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Figure 2.1 Conceptualized drawing of solute transport processes through a rock fracture (modified from Berglund & Selroos (2004)).

The general mechanisms of solute transport through fractured rocks mainly include (Fig. 2.1):

Advection: Solute moves with the fluid‘s bulk motion in the connected fracture networks. Advection in the rock matrix is usually negligible due to the high contrast of permeability between fractures and matrix (Bodin et al., 2003a). In most cases of practical interest in crystalline rocks, advection through fractures is the main and basic ―carrier‖ process for solutes in the groundwater (Berglund & Selroos, 2004).

Hydrodynamic dispersion: It describes the spreading of solutes during advection, due to the variations of fluid velocity with respect to the average velocity and molecular diffusion. In the cases with long travel distances or high fluid velocities, the contribution of molecular diffusion can be neglected.

Matrix diffusion: Solutes may migrate into the matrix micropores through the fracture surfaces by molecular diffusion, and the sorbing solutes will be adsorbed on the inner surfaces of micropores (Neretnieks, 1980, 1982).

Sorption: Solutes may attach to the fracture surfaces or infill materials by various physic-chemical interactions (for example, electrostatic or electromagnetic interactions), which are usually considered as instantaneous and reversible processes (Freeze & Cherry, 1979).

Chemical reaction: A wide range of chemical reactions, such as dissolution and precipitation, interactions among chemical species, radionuclide decay, organic reactions and volatilization, and/or biotransformation, may occur in fracture voids, on fracture surfaces and in the adjacent porous host rocks (Berkowitz, 2002).

Hydrodynamic dispersion

channeling

Decay and precipitation

Diffusion into micro-pores

(or cracks)

Sorption on fracture walls or gouge fillings

Rock matrix Water-bearing fracture


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2.2.2. Solute transport in single rock fractures

The general solute transport equation in a single fracture, considering the transport mechanisms cited above, can be formulated by the differential mass conservation equations below (Tang et al., 1981).

x z

c

bR

Dc

x

c

R

D

x

c

R

v

t

c

bz

me

f

ffff00

2

2

(2.1a)

zb c

z

cD

t

cm

ma

m 02

2

(2.1b)

This is the basic equation upon which most solute transport models are based, including two coupled one-dimensional equations: one for fracture (Eq. (2.1a)), and the other for rock matrix (Eq. (2.1b)). The coupling is provided by the continuity of concentration along the fracture surfaces, represented by the last term in the left side of Eq. (2.1a) (Tang et al., 1981; Bodin et al.,

2007). In the above equations, fc and mc are the solute

concentration in the fracture and matrix, respectively; t is time

variable; v is the fluid velocity along the fracture axis x; fD is the

longitudinal hydrodynamic dispersion coefficient in the fracture, which describes the spreading of a tracer pulse by local variations

in velocity and molecular diffusion; R is the retardation coefficient accounting for the solute sorption onto the fracture

surfaces; is the solute decay constant; eD is the effective

diffusion coefficient; b is the half aperture of fracture; z is the

coordinate perpendicular to the fracture axis; and aD is the

apparent diffusion coefficient. Neretnieks (1980) clarified the distinction between effective and apparent diffusion coefficients.

aD is usually used for describing non-stationary diffusion (Eq.

(2.1b)), whereas eD is used in the literature for flux calculation (Eq.

(2.1a)).

At the origin of the fracture, two types of boundary conditions are widely used. One is constant concentration boundary condition (Tang et al., 1981; Neretnieks, 1982),

0),0( ctxc f (2.2)

where 0c is the constant continuous source concentration at the

inlet (x=0). This boundary condition is widely used in the modeling of solute transport in single fractures, which corresponds to a solute that is dissolved in the fluid to a concentration determined by its solubility in practice (Moreno & Rasmuson, 1986). However, the solute is transported by both advection and hydrodynamic dispersion along the fracture axis, and the concentration at the locations close to the inlet can be significantly influenced by the boundary conditions when the value of Péclet number is small (hydrodynamic dispersion is important). In other words, non-realistic amounts of solute mass may be introduced


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into the fractures by dispersive fluxes at the inlet boundary when hydrodynamic dispersion is strong. In order to meet the mass conservation condition, the constant flux (Danckwerts‘) inlet boundary condition was proposed (Danckwerts, 1953),

),0(),0(0 txcx

Dtxvcvc fff

(2.3)

This condition corresponds to a constant flux at x=0 and there is

no hydrodynamic dispersion for 0x (Moreno & Rasmuson, 1986). The distinction and relation between the two boundary conditions were reviewed by Kreft & Zuber (1978), Van Genuchten & Parker (1984), Rasmuson (1986) and Moreno & Rasmuson (1986). There are other boundary conditions, which can be found in Gershon & Nir (1969) and Neretnieks (1980).

The usually used initial conditions are (Neretnieks, 1980; Tang et al., 1981),

0)0,( txc f (2.4)

0)0,,( tzxcm (2.5)

These conditions indicate that both the fracture and matrix are free of any solute initially.

Note that the solute concentration in the fracture fc in the

previous statement can be understood as the volume-averaged (or resident) concentration, which is the mass of solute per unit volume of the fluid contained in an elementary volume of the fracture at a given instant. There is another useful concentration definition in terms of the mass of solute per unit volume of fluid passing through a given cross section of the fracture during an elementary time interval, which is called flux-averaged (or flux) concentration (Kreft & Zuber, 1978; Parker & Van Genuchten, 1984). The relationship between the two concentrations is,

x

cDvcc

resident

f

f

resident

f

flux

f

(2.6)

where flux

fc and resident

fc are the flux and resident concentrations,

respectively. It was shown that both of the concentration definitions obey the classical advection-dispersion equation (ADE) of identical mathematical form for conservative solutes, but the boundary conditions for the two modes are not identically (Parker & Van Genuchten, 1984).

(1) Analytical models

Different analytical solutions for the solute transport model (Eq. (2.1), or simplified) under various initial and boundary conditions were derived by many investigators in the past. Among them, Tang et al. (1981) solved Eq. (2.1) under constant concentration boundary condition (Eq. (2.2)) by applying Laplace transform. Their solutions are in an integral form and can be evaluated by Gaussian quadrature at each point in space and time. The solutions neglecting hydrodynamic dispersion was given by Neretnieks


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(1980). The solutions without considering hydrodynamic dispersion and decay can be found in Grisak & Pickens (1981). Wallach & Parlange (1998, 2000) derived a solution considering a boundary layer that was formed along the interface between the fracture and the matrix. Rasmuson (1986) and Moreno & Rasmuson (1986) presented the analytical solutions for a transport model of rock fractures under constant flux boundary conditions.

Sudicky & Frind (1984), Cormenzana (2000), Sun & Buschec (2003) and Shih (2007) derived a series of analytical or semi-analytical solutions considering radioactive decay (two-member or three-member) under different boundary conditions. Berkowitz & Zhou (1996) derived three approximate analytical solutions for considering different fracture surface sorption models. Rasmuson & Neretnieks (1981) and Rasmuson (1984) presented the analytical models which treated the matrix block as spheres.

Generally, to derive the above analytical solutions, different assumptions or approximations have to be made. However, there are a few common and important assumptions widely adopted, as noted below.

The fracture surfaces were idealized as smooth parallel plates, therefore neglecting the surfaces roughness and tortuosity. Although this assumption is not appropriate with regard to the natural rock fractures, it still remains as the basis for many complicated transport modeling approaches in fractured rocks.

The matrix diffusion was assumed as a one-dimensional process perpendicular to the fracture axis. For larger diffusion coefficients, this assumption was verified (Tang et al., 1981). If the rock matrix is significantly heterogeneous or has smaller diffusion coefficients, this assumption should be carefully reviewed.

Fluid flow only occurred in fractures, and both the fracture and the matrix were saturated under steady state. This assumption is proper for the crystalline rocks, but for the rocks with large porosity it is questioned because of the possible fluid exchange between fracture and matrix.

Across the fracture section, the fluid flow velocity and solute concentration are assumed to be uniformly distributed. For fractures with aperture at micrometer scale, under relatively low hydraulic pressure gradient and at the time scale of performance/safety assessments for underground radioactive waste repositories, this assumption is reasonable.

(2) Numerical models

Numerical methods have the advantages of considering different complexities in fracture geometry and the exchanges of solutes between fractures and the rock matrix more efficiently and flexibly.

To consider the effects of rough fracture surfaces on the solute transport, the fracture plane was discretized into a mesh of square elements to which variable apertures were assigned, based on geostatistical methods (Moreno et al., 1988; Nordqvist et al., 1992;


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Ewing & Jaynes, 1995; Detwiler et al., 2000; James et al., 2005). Then numerical methods, for example the finite difference method, were used to compute the fluid flow velocity in each element, after which particle tracking method was employed to simulate the solute movements. James et al. (2005) presented the algorithms of particle tracking for considering the transport mechanisms including advection, dispersion and sorption onto fracture walls and colloid particles, as well as diffusion into and sorption onto the surrounding porous rock matrix. Based on the fracture mesh, the transport equations could also be numerically solved (Thompson & Brown, 1991a, b).

In addition, a conceptual variable-aperture channel model was proposed by Tsang & Tsang (1987), Tsang et al. (1988) and Tsang & Tsang (1991), in which transport through a single fracture and a connected fracture network was hypothesized to be represented by a limited number of tortuous and intersecting channels. Pot & Genty (2005) and Tan & Zhou (2008) used Lattice Gas Automata method and Lattice Boltzmann method to simulate the transport through a single fracture with variable aperture. Kumar (2008) developed a numerical model which included the matrix diffusion and nonlinear sorption in a single fracture based on the dual-porosity concept.

Kennedy & Lennox (1995) used a control volume model to investigate the significance of matrix diffusion in the direction parallel to the fracture axis. Their results showed that the assumption of the 1D matrix diffusion vertical to the fracture axis holds for most of the cases, but may become questionable for small fracture aperture or velocity.

2.2.3. Solute transport in fracture network

Based on the characterization methods of fractured rocks and the understanding of the solute transport in single rock fractures, there are three basic ways to simulate solute transport through the fractured rocks: analytical, (random walk) particle tracking and numerical methods.

(1) Analytical solutions

The most idealized fracture network model is a set of parallel fractures evenly situated in a porous rock matrix, and many analytical solutions were derived for this model under different boundary conditions (Barker, 1982; Sudicky & Frind, 1982; Davis & Johnston, 1984; Chen & Li, 1997; Robinson et al., 1998; West et al., 2004). Compared with the single fracture model in section 2.2.2, these models can consider the interactive effects between the neighboring fractures, and their results showed that the small fracture spacing make the solute migrate further along the fracture axis due to the limited storage capacity of the porous matrix. Note that this parallel fracture model is still far from the reality of fractured rocks, where the fractures are seldom parallel (not connected) and uniformly distributed.


10

(2) Particle tracking approachs

Particle tracking (PT) method was widely used to simulate transport in randomly distributed discrete fracture/channel network models (Schwartz et al., 1983; Smith & Schwartz, 1984; Andersson & Thunvik, 1986; Cacas et al., 1990; Moreno & Neretnieks, 1993; Berkowitz & Scher, 1997, 1998; Delay & Bodin, 2001; Bodin et al., 2007; Roubinet et al., 2010) or equivalent continuum models (Liu et al., 2000; Berkowitz & Scher, 2000; Hassan & Mohamed, 2003; Delay et al., 2005; Salamon et al., 2006).

Standard particle tracking methods, initially from statistical physics, do not solve the governing differential equation directly, and do not suffer from numerical dispersion. The idea of PT method to solute transport process is based on an analogy between the random walk equation and the Fokker-Planck equation for diffusion. The mass of solute is represented by a larger number of particles, which move in a flow field over discrete time steps by the transport mechanisms such as advection, dispersion, matrix diffusion and sorption. Therefore, the solution of the differential equations governing the movement of the solute transport can be approximated by tracking the motions of the particles in time and space. This is the basic algorithm of PT method, more details of the PT procedure for EC and DFN models can be found in the literature (see for example, Hassan & Mohamed, 2003; Bodin et al., 2007) or in later sections of this thesis.

When the PT method is applied to DFN models, a key issue is the mass transfer at the fracture intersections. There are three basic models existed currently (Bear et al., 1993). The first and most widely used one is the ‗complete (or full) mixing‘ assumption, in which molecular diffusion is assumed to be able to ensure the homogenization of the solute mass fluxes at the fracture intersections. In this way, the same concentration is observed at each outlet fracture segment, and the mass distribution is proportional to the relative discharge flow rates (Smith & Schwartz, 1984; Cacas et al., 1990b). Secondly, without mixing at the fracture intersection, it is called ‗stream-tube‘ model. In other words, this model assumes that the solute strictly follows the streamlines from the inflow segments to the outlet segments, so the mass distribution depends on the configuration of inlet and outlet fluxes at the fracture intersection (Robinson & Gale, 1990; Wels & Smith, 1994). The third one is ‗diffusional-mixing‘ model, developed by Park & Lee (1999). It is the intermediate case between the extreme cases of ‗complete mixing‘ and ‗no mixing at all (Stream-tube)‘, i.e. the solute is assumed to follow the streamlines, as well as to diffuse partially between the streamlines (Bodin et al., 2007).

There is another particle tracking based method called continuous time random walk (CTRW) introduced by Berkowitz & Scher (1997, 1998, 2000). It combines the advection and dispersion inextricably using a generalized master equation. The time steps


11

can be randomly operated during the procedure (Srinivasan et al., 2010).

(3) Numerical methods

Numerical approaches (commonly finite difference and finite element methods) were used extensively in solute transport simulation for EC models of fractured rocks in the past (Bibby, 1981; Huyakorn et al., 1983a, b; Berkowitz et al., 1988; Sudicky & Mclaren, 1992; Gerke & van Genuchten, 1996; Leo & Booker, 1996; McKay et al., 1997; Kischinhevsky & Paes-Leme, 1997). They impose some requirements of restrictive spatial discretizations to avoid the numerical dispersion or artificial oscillations. The numerical dispersion can be minimized by the total variance diminishing technique (Zheng & Wang, 1999), but the computational burden increases dramatically.

2.3. Stress effects on solute transport in fractured rocks

It is well known that the underground fractured rocks are always subjected to in-situ stresses, resulting from the weight of the overlying strata and the locked-in stresses of tectonic origin, and other possible natural or engineering perturbations due to excavation. All these factors may result in significant alterations in fracture properties (National Research Council, 1996). The deformation of matrix, displacement of fractures and possible damages induced by the in-situ stresses or other perturbations can change the matrix porosity and fracture apertures, which in turn change the solute storage capability of matrix and fluid conductivity of fractures.

2.3.1. Mechanical process

(1) Fractures

Changes in normal stress cause fractures to open or close, but small changes in shear stress may have little influences until the occurrences of significant shear displacements or failure (National Research Council, 1996). Effective stresses usually play an important role in the mobilized shear strength. In the physical world, the fracture surfaces are not planar, but rough and undulating. In their natural undisturbed states, fracture surfaces can be mated to produce the smallest aperture. However, when the fracture undergoes a degree of shearing, the asperities will ride up over each other, resulting in dilation and an increase in permeability (Barton et al., 1995; Yeo et al., 1998). The amount of dilation increases with increasing shear displacement, but decreases with increasing normal stress, if without damage of fracture surfaces.

However, the normal compression and shearing loadings may result in severe damage on fracture surfaces and produce gouge materials that reduce the permeability, as shown in Fig. 2.2 (Rogers, 2003). Microcrack development is localized within a narrow zone in the rock matrix close to the fracture surfaces at the same time. These microcracks are considered to be contact-induced tension cracks generated by stress concentrations at


12

contacting asperities. The density of these microcracks may also increase with increasing shear displacements and normal stress.

Basically, with respect to hydraulic permeability of fractures, three types of stress induced behavior can be clarified (Smart et al., 2001).

Only with normal displacement (open or closure), without engagement of the asperities, resulting simply in a high or low permeability conduit.

Dilation due to shear under either moderate normal stress or normal stiffness loading conditions, initially increase the hydraulic aperture, and then tending to generating a permeability barrier as worn gouge materials produced and accumulated, which may progressively fill the aperture space if they become immobile with fluid flow.

Shear under high compressive stress, creating a permeability barrier by severely damaging the skin layers of fractures.

Based on the mechanical behaviors that rock fractures exhibit, the constitutive models for fracture displacements without significant damage were developed in the past (Goodman, 1976; Bandis et al., 1983; Barton et al., 1985; Plesha, 1987; Amadei & Saeb, 1990; Jing et al., 1993, 1994; Rutqvist & Stephansson, 2003; Jaeger et al., 2007; Jing & Stephansson, 2007). However, the issues of fracture surface damage and gouge material production are rarely discussed in those early publications. In addition, accurate quantification of surface damage and gouge material generation is still difficult in laboratory test at present. Therefore, numerical methods (for example, particle mechanics models) are needed to predict the complex post-failure behaviors of fractures (Cundall, 1999; Wang et al., 2003; Jing & Stephansson, 2007; Park & Song, 2009; Asadi & Rasouli, 2010; Duriez et al., 2011).

(a) Before shear tests (b) After shear tests

Figure 2.2 Fracture surface damage of medium-grained metagranodiorite-granite under shear tests (Jacobsson, 2005).


13

(2) Matrix

Besides fractures, subsurface rocks are naturally filled with micropores saturated with groundwater or other fluids (e.g. air, gases and oil). The micropore fluid can have a significant influence on the mechanical behavior of a rock mass, either resulting in the state of stress closer to the failure surface or give rise to macroscopic elastic deformation of the rock (Jaeger et al., 2007).

The theory of poroelasticity, initially developed by Biot (1941), can account for the coupled hydromechanical behaviors between matrix deformation and pore pressure when the rodks are fully satureated, based on the idealization of the rock matrix as a solid material permeated with interconnected micropores. Under hydrostatic loadings, the porosity change of a poroelastic rock was derived by Zimmerman (1991),

nmbc dCCd 1 (2.7)

where is the matrix porosity; bcC is the effective bulk

compressibility; mC is the compressibility of the matrix materials;

and n is the normal effective stress. The matrix porosity change

to the volumetric strain is related as (Chin et al., 2000; Schutjens et al., 2004),

b

bd

1

1 (2.8)

where b is the bulk volumetric strain. The challenge of this

model is its laboratory calibration before application in the field, because the strain is sensitive to many factors, not only the stress state and path, but also the pressure, size and shape of the pores (Han & Dusseault, 2003).

The above discussion is limited in the context of rock elasticity, which is to say that no plasticity or failure was considered. Although no rocks are actually ―linearly elastic‖ over a wide range of stresses, this approximation is often quite useful and accurate, since many rocks behave linearly for incremental changes in stress (Jaeger et al., 2007).

2.3.2. Stress effects on solute transport

The influence of stress or engineering perturbations on permeability of single fractures or fractured rocks has been investigated by numerical modeling in the past two decades (see for example, Kelsall et al., 1984; Oda, 1986; Pusch, 1989; Bai & Elsworth, 1994; Zhang & Sanderson, 1996; Chen & Bai, 1998; Bai et al., 1999; Zhang et al., 1999; Smart et al., 2001). Important progress was made more recently by Min et al. (2004a, b) and Baghbanan & Jing (2008), based on discrete element method. Their results indicate that fracture permeability or equivalent permeability of fractured rocks, as well as flow patterns, may significantly change under different stress conditions. The sensitivity of permeability to stress in fractured rocks depends on


14

the connectivity of the conducting fracture networks, and the mechanical and hydraulic properties of the fractures. However, to the authors‘ knowledge there are only a few publications concerning the stress (or perturbations) effects on solute transport processes in single rock fractures (Jeong & Song, 2005; Koyama et al., 2006, 2008) and fractured rocks (Genty et al. 2011; Waber et al., 2011), even though many efforts have been made to understand different types of retardation mechanisms for solute or particle transport in fractured media. Modeling the coupled stress-flow-transport processes in fracture networks has been rarely attempted before.

The concept of stress effects on solute transport in fractured rocks can be qualitatively indicated in Fig. 2.3. The fracture aperture changes, possible damage and matrix porosity changes may occur due to the in-situ or redistributed stresses, and the former two phenomena can influence the fluid flow significantly. Consequently, the solute transport process can be impacted by the changes of fluid flow and matrix properties, because the different transport mechanisms such as advection, hydrodynamic dispersion, matrix diffusion and sorption are strongly dependent on the fluid flow conditions and rock properties. Fig. 2.3 gives the basic conceptual idea of stress effects on fluid flow and transport and a logical order of the modeling presented in this thesis.

2.4. Outstanding issues and motivation of this thesis

The basic principles and understanding of the individual process of stress, fluid flow and solute transport in fractured rocks are built, and a large number of analytical solutions and numerical codes are developed over the years to simulate solute transport in fractured rocks as reviewed above, however, there are many important issues still poorly understood or not yet resolved today, in terms of fundamental concepts, technical and numerical difficulties, as follows:

(1) For the conceptual models of fractured rocks, one of the main challenges is to properly represent the thousands or millions of fractures with significantly varied properties in an accurate and efficient way. The limited data with large uncertainties of the fractured systems add more difficulties to the reasonable modeling of stress, fluid flow and solute transport in underground rocks.

(2) For fluid flow and solute transport through single fractures, the effects of roughness of fracture surfaces and infilling materials were poorly understood and difficult to simulate. Many kinds of physical or chemical reactions between rocks and groundwater that may play some critical roles on transport are critically related to the fracture surface roughness, but experimental study is still limited and not able to provide scientific basis for reliable modeling. The laboratory or in-situ experiments and observations lag far behind the development of numerical modeling of the flow and transport in fractured rocks, so that many presently available models are poorly calibrated and verified.


15

(3) An important factor to improve the present modeling of fluid flow and solute transport through complicated fracture network is the complexity at the fracture intersection. The other critical issue is how to extend the understanding for single fractures to a large fracture system.

(4) The in-situ stress and other natural or human perturbation has been demonstrated to have a significant influence on the solute transport in fractured rocks by a few publications, which attracts increasing attention but still remains as rare investigations.

It is not possible to investigate all the mentioned outstanding issues above in this research. As pointed out in chapter 1, the main content of this thesis focuses on the stress effects on solute transport in fractured rocks, in respects of single fractures and large fracture network, as a 2D generic study. Under this framework, a detailed modeling procedure that combines the mechanical, hydraulic and chemical processes is developed to systematically investigate the underground engineering and environmental problems.

Figure 2.3 Conceptual view of the effects of in-situ or redistributed stresses on the solute transport in fractured rocks. The dashed line represents the mechanisms not considered in the present study.

In-situ or redistributed stresses

Fractured rocks Fracture aperture changes Matrix porosity change

Damage (wear) Fluid flow

Solute transport

Advection Hydrodynamic dispersion Matrix diffusion Sorption


16


17

3. STRESS EFFECTS ON SOLUTE TRANSPORT IN SINGLE

FRACTURES

As a basis to understand the stress effects on solute transport in fractured rocks, this chapter focuses on the solute transport behaviors in a single rock fracture impacted by the stresses. Firstly, a smooth parallel plate model is used to derive an analytical solution for the case without damage along the fractures‘ skin layers, and then a particle mechanics model is applied to study the effects of surface wear and damage on the solute retardation (Sorption).

3.1. Analytical model without damage of fracture surface

3.1.1. Conceptual model

The simplest mathematical model of a single rock fracture is the smooth parallel plate model, which assumes that the two walls of a fracture can be represented by two nominally smooth and parallel

surfaces (Snow, 1965), separated by a initial aperture of 2b (Fig. 3.1). Considering a single fracture embedded in a saturated porous rock matrix, with a solute source located at the origin of the

fracture (x=0), the solute then migrates with the fluid flow along the fracture (advection and hydrodynamic dispersion), and at the same time may also diffuse in and out of the porous matrix. Although the parallel plate assumption differs from the natural fractures with rough surfaces, it remains, however, the basis of many complicated modeling approaches for flow and transport processes at the local scale (Bodin et al., 2003b). It can provide the basic insights for fluid flow and solute transport through an individual rock fracture under different stress conditions. In addition, the parallel plate model is the necessary assumption for deriving a closed-form solution for such a complex problem of coupled stress-flow-transport processes in a single fracture, because such an analytical solution does not exist for a natural fracture of rough surfaces. This, however, does not prevent considerations of some mechanical effects of roughness on the mechanical component of the closed-form solution indirectly, as described in the mechanical constitutive models.

x

T D

kn

T: Tensile strength

kn: Nonlinear normal stiffness

D: Dilation

S: Slider

Ss: Shear strength

ks: Nonlinear shear stiffness

ks S Ss

z

0 2b

σn

σn

Shear

Shear

Figure 3.1 Schematic illustration of smooth parallel plate model for single fractures undergoing shear under normal stress.


18

(a) Compressive loading-unloading (b) Normal stress-normal displacement

Figure 3.2 Fracture normal closure under compression (after Bandis et al., 1983).

Besides the parallel plate model as noted above, the following basic assumptions were adopted as well, in order to derive an analytical solution.

The current model was defined in a 2D space, with the fracture aperture much smaller than its length.

The hydraulic aperture of the fracture was assumed to be equal to its mechanical aperture, and a residual aperture always existed even under high compression stress. This could be regarded as an indirect consideration of the effects of fracture surface roughness in reality.

The effect of fluid pressure on the variation of normal fracture displacement was assumed to be negligible, which indicated that the fracture surfaces always remained parallel during the stress/deformation process. This assumption is reasonable for groundwater flow under small hydraulic pressure gradients in subsurface rocks in practice.

The fracture aperture changes due to the normal closure or shear dilation were described by commonly adopted mechanical constitutive models of rock fractures in the field of rock mechanics, in order to simplify the derivation of the analytical model. This assumption is reasonable, since the scope of this study is the effects of stress on the transport behaviors, not the complicated stress/displacement processes in details.

Transverse dispersion within the fracture was neglected, compared with the longitudinal dispersion. This means that the solute concentration was assumed to be equal at the fixed fracture section.

The permeability of the porous matrix was sufficiently low to hinder fluid flow, but solute particles could move in and out of the rock matrix by molecular diffusion.

3.1.2. The closed-form solution for stress effects on solute transport in single fractures

With the above conceptualization and assumptions, explicit descriptions to the individual mechanical, hydraulic and transport processes in a single fracture can be obtained separately, and then

un

σn

un

σn


19

combined together to generate a closed-form solution for description of the coupled stress-flow-transport processes in a single rock fracture. When stresses are applied on the rocks, both the fractures and the rock matrix may move and/or deform, which cause the changes in fracture aperture and matrix porosity. These changes can consequently influence the fluid flow and solute transport in fractures, as well as the diffusion of solutes into and out of the rock matrices.

(1) Mechanical process

Normal closure model

It has been demonstrated that the normal closure of a rough rock fracture is typically nonlinear under compressive normal stress, i.e. the normal stiffness of a fracture increases with the increasing normal stress (Goodman, 1976; Swan, 1983; Cook, 1992; Rutqvist & Stephansson, 2003), and finally levels off to some asymptotic value at the large compressive normal stress (Jaeger et al., 2007). The most commonly applied fracture normal closure model is Bandis‘ hyperbolic function as shown in Fig. 3.2(b) (Bandis et al., 1983).

nmaxn

nnn

uu

uk

10 (3.1)

where 0nk is the initial normal stiffness at a prescribed stress state;

nu is the normal displacement of the fracture; and nmaxu is the

maximum normal displacement, approached asymptotically with

the increasing normal stress. The basic parameters 0nk and nmaxu

are determined by experiments (Barton et al., 1985). The normal closure of the fracture can be expressed by rearranging Eq. (3.1) as,

max0 nnn

nn

uku

(3.2)

It is noted here that those parameters in the Bandis‘ model depend on the stress paths of normal compressive loading-unloading cycles (Jing & Stephansson, 2007). Barton et al. (1985) suggested that the in-situ fractures probably behave in a manner similar to the third or fourth loading cycles (Fig. 3.2(a)).

Shear-dilation model

Fracture displacement during shear is inherently a coupled process, in which both normal and shear displacements occur (Jaeger et al., 2007). Fig. 3.3(a) shows the typical, although much idealized, shear stress-shear displacement behavior of a fresh rock fracture under constant normal stress conditions, during direct shear tests. During such tests, the most important mechanism is the so-called shear dilation, i.e., the increase of fracture aperture during shear motion due to the roughness of the fracture surfaces. Various empirical models have been suggested to simulate the dilation phenomenon in the past, with varying degrees of success. The simplest approach


20

is to use a dilation angle relating the normal and tangential displacements of a fracture:

dsd uu tan (3.3)

where du is the shear dilation; su is the shear displacement; and

d is the dilation angle. Note that we did not consider the dilation

angle varying with the normal stress and shear displacement in this study, but assumed that the dilation angle was a constant for deriving a closed-form solution.

A simplified model, as shown in Fig. 3.3(b), was adopted to approximate the behavior of rock fractures during shear (Fig. 3.3(a)), so that the analytical model can capture the main mechanical features without losing necessary generality. The peak

shear stress, p , is determined by the Mohr-Coulomb criterion,

characterized by a cohesion( C ) and a frictional angle ( ):

pnC tan (3.4)

where is the shear stress.

(a) Conceptual model (b) Simplified model

Figure 3.3 Mechanical behavior of fractures during direct shear test under constant normal stress (After Jing &

Stephansson, 2007). r is the residual shear stress, and dmaxu

is the maximum of shear dilation.

ucs

τr

τp

τ

us

udma

x

(I) (II)

us

(III) ud

d

τ

τp

τr

us

ud

us

udma

x

up


21

Before reaching the peak shear stress, the shear stiffness, sk , is

assumed to be constant. In the case of

p (3.5)

one has

psusign )( (3.6)

where ()sign is the sign function, which extracts the sign of the

shear displacement, i.e., the shear direction.

Dilation starts to occur at the onset of slip of the fracture, governed by the dilation angle. The accumulated dilation is limited

by a critical value of shear displacement, csu , corresponding to the

observation that crushing of asperities at large shearing would eventually prevent continued increase of dilation. In this way, a stepwise function is employed to describe the dilation during the three different stages (I, II and III in the Fig. 3.3(b)) of the shear displacement:

scs

cssp

spps

dcs

dsd

uu

uuu

kuu

u

uu

tan

tan

0

(3.7)

where pu is the shear displacement under the peak shear stress.

Hydraulic aperture

Experimental results typically show that an apparent residual

aperture, resb , exists in the rock fractures even under high

compressive stresses, when fracture appears to be nominally closed by compression (Witherspoon et al., 1979; Raven & Gale, 1985; Cook, 1992; Renshaw, 1995). Additionally, the residual aperture may vary with the sizes of the tested fracture specimens, which was not considered in this study. Therefore, the resultant hydraulic aperture can be expressed as,

dnnmaxres uuubb )(2 (3.8)

where nu and du are determined by Eqs. (3.2) and (3.7),

respectively.

Matrix porosity

Rock matrix porosity changes under the hydrostatic stresses and can be defined as a function of volume change of micropores in the rock matrix as Eq. (2.7) (Han & Dusseault, 2003). Note that

n is the effective compressive stress acting on the rock matrix. In

this study, it was assumed to be equal to the stress applied on the

fracture surfaces. Because mC is usually small enough to be

negligible, Eq. (2.7) can be simplified as (Han & Dusseault, 2003),

nbc dCd )1( (3.9)


22

In reality, bcC is stress dependent and can be measured by

laboratory tests. Based on experimental data, Zimmerman (1991)

derived an empirical relation for stress-dependent bcC as,

na

bc eaaC

3

21 (3.10)

where 1a ,

2a and 3a are constants determined from curve fitting

of the test data; the unit of n in Eq. (3.10) should be MPa.

Therefore, the stress-dependent porosity of rock matrix can be written as,

niananin ee

a

aa

i e

33

3

21

)1(1

(3.11)

where i is the initial matrix porosity at zero stress. This relation

links stress with rock porosity which, in turn, is related to the matrix diffusion process.

(2) Fluid flow process

Fluid flow in the conceptual model of rock fractures shown in Fig. 3.1 is represented by the well-known cubic law (Zimmerman & Bodvarsson, 1996). With a fracture of constant cross section along the x direction (fracture axis direction), one can argue that the fluid velocity is negligible in the y direction. With these

simplifications and the ‗‗no-slip‘‘ boundary condition, the relation between the flow rate and the hydraulic pressure gradient can be represented by the cubic law (Bear, 1972),

dx

dpbwQx

3

2 3

(3.12)

where xQ is the volumetric flow rate; w is the fracture width;

is the dynamic viscosity; and dxdp is the hydraulic pressure

gradient.

The average flow velocity at a steady state can be obtained by

dividing the flux, xQ , by the cross sectional area, wb2 ,

dx

dpb

wb

Qv x

32

2

(3.13)

This equation links the change of flow velocity in fracture with the stress-dependent aperture, and it also transfers the effects of stress to the advection in the fracture, as well as other transport mechanisms.

(3) Solute transport process

Three transport models under two inlet boundary conditions are considered here, in order to examine the stress effects on different transport mechanisms.

Transport model I

The simplest solute transport model in a single fracture is the one only considering the advection in the flow direction, matrix


23

diffusion perpendicular to the flow direction and sorption in the porous matrix, but without dispersion and decay. Considering the mass balance of solute in the fracture and in the micropores of the matrix, respectively, one can obtain the following simplified differential equations of solute transport from Eq. (2.1) (Neretnieks, 1982):

x z

c

bR

D

x

cv

t

c

bz

m

f

eff00 (3.14a)

zb

z

cD

t

c ma

m 02

2

(3.14b)

Equation (3.14a) is readily extended to include also sorption on the fracture surface and on fracture infill. This will be discussed in a

later section. The effective diffusion coefficient, eD , which is

related to the pore diffusion coefficient, mD , and the rock matrix

porosity, , in the following form,

me DD (3.15)

Neretnieks (1980) defined the pore diffusion coefficient as

)( 2* DDm , where *D is the molecular diffusion coefficient

in the fluid, is the pore constrictivity (<1), and is the tortuosity, which is a function of rock matrix porosity. Because the constrictivity and tortuosity are difficult to measure and to be separated experimentally, an empirical relation of Archie‘s law (Archie, 1942) is often used to relate the effective diffusion coefficient to the porosity,

m

e DD * (3.16)

The exponent m can take the values between 1.3 and 4 for different rocks, and is often taken to be 1.6 for granites (Ruffet et

al., 1995). Therefore, one has 6.1*DDe and

6.0*DDm .

The apparent diffusion coefficient, aD , is defined as,

mm

ea

K

DD

(3.17)

where m is the bulk density of the matrix, and mK is the

distribution coefficient that was defined by Freeze & Cherry (1979) as the mass of solute adsorbed per unit volume of matrix divided by the concentration of solute in the solution. The term (

mmK ) is called rock capacity, m . The volume sorption

coefficient, mmK , is equal to zero if no sorption occurs. The

value of rock capacity is then equal to that of porosity ; i.e.,

ma DD for the conservative solute transport.

Without detailed derivation, we present only the solutions of Eq. (3.14) below, under the boundary condition of Eq. (2.2) and initial


24

conditions of Eqs. (2.4) and (2.5) (Carslaw & Jaeger, 1959; Neretnieks, 1980):

5.0

5.0

0

)(2 w

a

ewftt

D

D

b

tErfc

c

c (3.18a)

5.0

5.05.0

0

)(2 w

aa

ewm ttD

bz

D

D

b

tErfc

c

c (3.18b)

where wt is water residence time, and is equal to vxtw / . Note

that this transport model without including the longitudinal hydrodynamic dispersion has the identical solutions under either the constant concentration or flux boundary conditions, because the two of them become the same.

Transport model II

When considering the hydrodynamic dispersion and sorption on the fracture surface, the governing equation for the solute transport in the fracture then becomes,

x z

c

bR

D

x

c

R

D

x

c

R

v

t

c

bz

m

f

ef

f

ff

f

f00

2

2

(3.19)

The hydrodynamic dispersion coefficient can be expressed as *DvD Lff , where Lf is the dispersivity in the direction of

the fracture axis (Bear,1972). Detwiler et al. (2000) proposed a

complex theoretical model of fD dependent on the Péclet

number ( Pe ), but the first one was used here for simplicity. Together with Eq. (3.14b), one obtains the following solutions for

fDxt 22 4 :

lfaf

e

ff

fd

DxtDD

D

b

xErfc

D

vx

D

v

c

c

5.0222

2

22

222

0 )]4([2

1

416exp

2exp

2

(3.20a)

lf

aaf

e

ff

m dDxt

D

bz

DD

D

b

x

ErfcD

vx

D

v

c

c

5.022

2

2

22

222

0 )]4([2

4

16exp

2exp

2 (3.20b)

where tDxl f2 .

Transport model III

The solutions for the governing equations of Eqs. (3.19) and (3.14b) under the constant flux boundary condition (Eq. (2.3)) were derived by Rasmuson (1986) and Moreno & Rasmuson (1986):


25

dydDxtDD

D

b

xErfc

D

vx

D

vx

D

vx

D

v

D

v

c

c

faf

e

x lfffff

f

5.0222

2

22

222

0

)]4([2

1

4

16exp

2exp

2expexp

(3.21a)

dydDxt

D

bz

DD

D

b

x

Erfc

D

vx

D

vx

D

vx

D

v

D

v

c

c

f

aaf

e

x lfffff

f

5.022

2

2

22

222

0

)]4([2

4

16exp

2exp

2expexp

(3.21b)

Compared with Eq. (3.20), it can be found that the solutions of transport model II were included in the above solutions for transport model III.

(4) Solutions for coupled stress-flow-transport processes

Combining Eqs. (3.8) and (3.13), the average velocity of groundwater for a steady-state flow in a single fracture is obtained as,

dx

dpuuubv dnnmaxres

2)(

12

1

(3.22)

As shown in Fig. 3.4, substituting the updated half aperture b of the fracture (Eq. (3.8)), the mean fluid velocity v obtained from Eq. (3.22), and the effective and apparent diffusion coefficients (Eqs. (3.16) and (3.17)) due to the changed matrix porosity represented by Eq. (3.11) into the three transport models presented above (Eqs. (3.18), (3.20) and (3.21)), one can have the solutions for the coupled stress-flow-transport process in the fracture-matrix system as shown in Fig. 3.1, and examine the effects of stress on solute transport in a single fracture. The complete expression was not presented here to save space.

To evaluate the integrals in the transport models II and III (Eqs. ((3.20) and (3.21)), the Gauss-Legendre quadrature technique with 100 Gauss points was used. It was found that the numerically significant portion generally extends over a smaller range, even though the integrand theoretically extends to infinity (Tang et al., 1981). In this way, before the integration, a prior scanning procedure was carried out to determine the actual integration range. The equations can also be numerically integrated directly by Mathmatica or other commercial computation software.


26

Figure 3.4 Derivation of the analytical solution for stress effects on solute transport in single fractures.

3.2. Particle mechanics model with fracture surface damage

3.2.1. Conceptual mechanisms of rock fracture surface damage

According to the fundamental microscopic wear mechanisms between two rough surfaces in contact, as defined in contact mechanics of solids, the failure/wear/damage processes can have four mechanisms basically: abrasion, adhesion wear, corrosion and surface fatigue (Belem et al., 2009). The observed behaviors of rock fracture surface damage during shear tests under different normal loading conditions show that the main wear micro-mechanisms involved are abrasion, adhesion wear, crushing (of the worn particles previously produced) and microcracking in the matrix forming the fractures (Fig. 3.5) (Haggert et al., 1992; Pereira & de Freitas, 1993).

Abrasion occurs when the contacting asperities, on the opposite two fracture surfaces of a rock fracture, plough through each other to generate gouge particles of relatively larger sizes (two-body abrasion or three-body abrasion in terms of contact mechanics) during a shear process (Fig. 3.5(b)).

Adhesion wear represents the generation of smaller gouge particles due to more gradual breakage of asperities in the intimate contacts (Fig. 3.5(c)).

The previously generated gouge particles could be further crushed (or comminuted) into smaller particles with the increasing shear displacement or normal stress, which may change the gouge particle size distribution significantly (Fig. 3.5(d)).

niananin ee

a

aa

i e

33

3

21

)1(1

dnnmaxres uuubb )(2

Stresses change fracture aperture and matrix porosity

m

e DD *mm

ea

K

DD

dx

dpbv

3

2

*DvD Lff

Cubic law Archie’s law

Flow velocity Longitudinal dispersion

coefficient

Effective diffusion

coefficient

Solute transport model I, II and III (Eqs. (3.18), (3.20) and (3.21))

Closed-form solutions for coupled stress-flow-transport processes in single fractures

Apparent diffusion

coefficient


27

(a) Sketch of rock fracture during a direct shear test under constant normal stress

(b) Abrasion

(c) Adhesion wear (d) Crushing of gouge particles

(e) Microcrack development

Figure 3.5 Schematic views of wear process in rock fractures during shear

displacement. vb is the half distance between the upper and lower reference

plane. (b)–(e) conceptually explain the wear mechanisms associated with the shear process.

Two-body Three-body

σ

n

Shear

Shear

Upper fracture profile

Lower fracture profile

Upper reference plane

Lower reference plane

x

z

σn

2bv H0


28

Figure 3.6 A microscopic view of the solute transport in a sheared rock fracture. The solutes may adsorb onto the outer surfaces of the abrasive, adhesive and crushed particles during the migration. Abrasive particles with larger sizes were assumed not to move with the flowing fluid, but the adhesive and crushed particles may be either mobile or immobile since of their relatively smaller sizes.

The stress concentration at the close contact area can initiate and propagate the microcracks into the intact rocks in either tensile or shear modes (Fig. 3.5(e)), an especially important phenomenon for rock fractures.

The influence of surface damage on the mechanical or transport behaviors of rock fracture depends on the roughness of the fractures and the loading conditions (Sun et al., 1985). Other mechanisms such as corrosion and surface fatigue are coupled or long-term processes and little known for rock fractures, so not considered in the present study. For rock fractures, usually the sizes of gouge particles caused by abrasion are more likely to be larger and can behave as immobile particles.

Pereira & de Freitas (1993) classified the wear particles in the rock fractures generated during rotary shear tests into four groups according to their size: coarse (0.6-2.0 mm), medium (0.2-0.6 mm), fine (0.06-0.2 mm) and silt/clay (<0.06 mm). They found that the percentage of coarse particles decreases and the percentages of fine and silt/clay particle increases with shear displacement. Amitrano & Schmittbuhl (2002) observed that the sizes of gouge particles range from 2 μm to 2 mm, following a power law distribution in a granite shear band. For those small wear particles, which may be suspended in the groundwater and migrate with the fluid flow because of their small gravitational settling, are called mobile gouge particles. In contrast, with increasing size and weight, some gouge particles that may settle down or be captured by the fracture surfaces with small fluid velocities, are called immobile gouge particles. All of the gouge particles can adsorb sorbing solutes and affect their migration in different ways.

Abrasive particles Adhesive or crushed particles Solutes

Microcracks


29

Sorption of solutes onto the surfaces of immobile gouge particles results into extra retardation of solute transport. However, if the solutes are adsorbed onto the mobile particles, the retardation could be reduced, thus increasing the potential spreading of solutes (Oswald & Ibaraki, 2001). The sorption capacity per mass of the gouge particles is expected to be at least as strong as in the undisturbed rock matrix (André et al., 2009a, b). In addition, their results also show that the disturbed zone of microcracks (or microstructures) close to fracture surfaces due to the stress concentration near the contacting asperities (Conrad & Friedman, 1976; Teufel, 1981, 1987), not only increase the extra available sorption surfaces, but also provide more inner surfaces for matrix diffusion and sorption of solute into the micropores of the host rocks. The sizes of gouge particles caused by adhesion wear and crushing are usually smaller than those of abrasive gouge particles by one or two orders of magnitude, and such particles can behave like colloids (Fig. 3.6). Therefore, some of them are mobile and can migrate with fluid flow. The immobile gouge particles may stay in the fractures over long time periods and can retard the solute migration.

3.2.2. Retardation coefficient model

Both the immobile gouge particles and the microcracks can provide additional surfaces for solute sorption in the fractures, so as to increase the fracture sorption capacity. The microcracks mentioned here refer to those in the nearby rock matrix but connected to the fracture space and filled with solutions, i.e. the solutes can directly come into contact with the surfaces of these microcracks. As for the microcracks in the matrix but far away from the fractures, solutes in the fractures cannot contact them directly, but, the surfaces of those microcracks will eventually be reached by the solutes through molecular diffusion. They are not considered in the current model, because they are related to the

mechanism of matrix diffusion. By using abS , cS , adS and coS to

denote the new surface area generated by abrasive particles, microcracks, adhesive particles and crushed particles, the total

extra sorption surface area, sorS , generated during shear is,

cocoadadcabsor SSSSS (3.23)

where ad and co are defined as the immobile portion of the

adhesive and crushed particles, respectively, because the mobile gouge particles are assumed to be flushed away in a short time by fluid flow.

A retardation coefficient, R , can be defined as (Wels & Smith, 1994),

b

KR

f1 (3.24)

where fK in unit of m is the equilibrium distribution coefficient

of solute with the fracture surface; , usually ≥ 1, is a


30

dimensionless coefficient representing the extra available sorption surfaces other than reference planes shown in Fig. 3.5(a), due to the fracture surface roughness.

Assuming that the same equilibrium distribution coefficients for gouge particles and microcracks as that for fracture surfaces, the impacts of extra surface added by wear and damage can be

incorporated into the dimensionless coefficient . 1 represents an idealized fracture with the assumed two smooth and

planar, sorbing surfaces in an open fracture of average aperture 2b (Wels & Smith, 1994). With the extra fracture surface area due to roughness, infilled minerals (can be treated as worn particles) and

microcracks, one has 1 in the form of,

wl

S

wl

S

f

sor

f

actualwt

22 (3.25)

where t and w are the surface increase factors due to the

roughness or tortuosity of fracture surface and shear process,

respectively; actualS is the real surface area of the fracture; fl and

w represent the fracture length and width, respectively. With Eq. (3.24), the retardation coefficient influenced by the wear process can be also obtained as,

b

K

b

KR

fwtf )(11

(3.26)

where w can be further divided into ab , c , ad and co to

denote the surface increase factors resulting from abrasion, microcracking, adhesion wear and crushing, respectively.

wl

SSSS

f

cocoadadcabcoadcabw

2

(3.27)

Using b

KR

f1 as the reference retardation coefficient for

smooth parallel plate fractures to divide Eq. (3.27) leads to the retardation coefficient increase factor with respect to the idealized smooth parallel plate model, given as,

wl

SSSS

wl

S

b

Kb

K

b

Kb

K

R

RA

f

cocoadadcab

f

actual

wt

f

fwt

f

fwt

22

)(

1

)(1

0

(3.28)

The use of the approximation K f

b>>1 can be justified when there

is noticeably surface sorption to be considered, which is the main theme for the study of abrasion effects. It can be seen that the retardation coefficient increase factors due to the roughness and shear process are approximately equal to the surface increase

factors, i.e. ttA and wwA , where tA and wA are the


31

retardation coefficient increase factors due to the roughness and shear process, respectively.

3.2.3. Particle mechanics model

The present modeling focuses on the mechanisms of abrasion and microcracking. As for adhesion wear and crushing, their impacts on solute retardation coefficient are not included here, but briefly discussed in the later sections, due to the lack of experimental data, available modeling tools and limitation of computational power at present.

Both the abrasive particles produced by the cut or plough of interlocked asperities, and the microcracks developed in the disturbed zones are caused by complex processes that have not been quantitatively investigated by laboratory or field experiments so far, to the author‘s best knowledge. Therefore a generic numerical model was adopted to investigate their impacts on solute transport in rock fractures, because of the important role they may play on solute retardation processes.

(a) Intact rocks (b) DEM model (c) PFC model

Figure 3.7 A conceptual view of intact rocks represented by bonded particles in PFC models. Each bond contact is described by the following five micro parameters: normal and shear stiffness, friction coefficient and tensile and shear strengths (Potyondy & Cundall, 2004).

(a) Before shear (b) After shear

Figure 3.8 Abrasive particles (the dark particles) produced after shear in the PFC model of fracture sample. The white and gray particles represent the intact rock matrix and fracture surfaces, respectively.

PFC2D 3.10

Itasca Consulting Group, Inc.Minneapolis, MN USA

Step 5240 14:05:48 Thu Jan 20 2011

View Size: X: 4.683e-002 <=> 5.817e-002 Y: -5.000e-003 <=> 5.000e-003

Ball

Wall

PFC2D 3.10

I t asca C onsult ing Group, I nc.Minneapolis, MN U SA

St ep 106340 14: 07: 01 Thu Jan 20 2011

View Size: X: 4. 616e-002 <=> 5. 884e-002 Y: -5. 000e-003 <=> 5. 000e-003

B al l

Wal l

Bond


32

The particle flow code, PFC, was chosen as a tool to numerically study the process of abrasive particle generation and microcrack evolution during a shear process of a typical rock fracture. Basically, a PFC model represents the rock matrix as an assembly of rigid two-dimensional circular disks or three-dimensional spheres bonded in the normal and shear directions at all contacts, which possess finite normal and shear stiffness and tensile and shear strengths as shown in Fig. 3.7 (Potyondy & Cundall, 2004). The fracture model shown in Fig. 3.8(a) is represented by a group of particles (in gray color) with zero bond strength and friction coefficient, which separate the intact rocks into two parts, similar to a freshly created tensile crack of very small aperture (Cundall, 1999). Without the repetition of the basic assumptions and laws of particle flow method that can be found in the literature (see for example, Jing & Stephansson, 2007; Itasca, 2008), the methodology of modeling the effects of abrasion and microcrack evolution during shear on the solute retardation is focused below. Note that the 2D PFC models used in this study refer to a thin slice of rock samples, so that the width of the model can be understood as the same magnitude of the disk radius in millimeter. This assumption may or may not represent a strict theoretical plane strain model, but is needed to facilitate the gouge particle surface calculation for retardation analysis.

(1) Particle generation by abrasion

When a particle representing fracture surfaces in the PFC model loses all its contacts with the neighbor particles during shear, it is regarded as an abrasive gouge particles cut out or ploughed out by the opposite asperities (dark particles in Fig. 3.8(b)). The whole outer surfaces of all abrasive gouge particles were considered to be available sorption surfaces. In the present formulation of PFC, the previously free abrasive particles may re-contact with other particles during the shear process due to normal closure and shear displacement. It is challenging and unnecessary to accurately monitor the reconnected particles and calculate its available sorption surface for a retardation coefficient model for the purpose of present study, so is was assumed that half of the whole surface of a reconnected abrasive particle was active to adsorb the solutes, because the other half of their surfaces were needed to contact the fracture surfaces closely. The surface increase factor in respect to the area of reference planes due to abrasion can be determined by,

f

reabab

f

abab

l

rr

wl

S

2

)2(

2

(3.29)

where abr is the radius of the abrasive particles; rer is the radius of

the particles re-contacted the fracture surfaces; ab is a shape

coefficient defined as the actual particle surface area divided by the surface area of a smooth disk with the same radius, because PFC can only consider smooth particle surfaces but gouge particle surfaces of rocks are rough in reality. The generation of abrasive


33

particles (fully loosing contacts during the direct shear test) can be continuously monitored, including their positions and other geometrical information, after which the new surfaces available for solute sorption provided by them can be estimated.

(2) Microcrack development

When the corresponding component of contact force exceeds either the tensile or shear strengths, the contact bond fails. The space between the two particles connected by the failed contact previously was treated as a microcrack (Fig. 3.9). In this study, each microcrack was assumed to be formed by two smooth

parallel surfaces whose length cL and aperture cb are shown in

Fig. 3.9(b). According to the contact failure modes (the normal or shear bond strength is exceeded), the microcracks can be classified as tensile and shear cracks, respectively. In this study, the microcracks located in the zones close to the surfaces in the upper

and lower halves of the fracture sample with a thickness of 3 vb (

vb denotes the half distance of the two parallel reference planes in

Fig. 3.5(a)) were assumed to be directly connected to the fracture surfaces, and their surface area served as sorption surfaces. The corresponding zones are called severely damaged zones hereafter. This phenomenon is further explained in a later section.

Compared with the area of reference planes, the surface increase factor from the microcracks can be calculated as,

f

cc

f

cc

f

cc

l

L

l

L

wl

S

2

2

2 (3.30)

where c is a roughness coefficient defined as the surface area of

real microcrack divided by the area of the smooth planar surfaces.

The surface increase factor c can be further divided into ct and

cs , representing the contributions from tensile and shear

microcracks, respectively.

csct

f

csctc

f

ccc

l

LL

l

L

)( (3.31)

3.3. Results for stress effects on solute transport in single fractures

3.3.1. Without fracture surface damage case

The analytical model was applied for a case without considering fracture surface damage to demonstrate the effect of stress on the solute transport of a rock fracture in a porous rock matrix. The sensitivity of different transport mechanisms, such as advection, matrix diffusion, and longitudinal hydrodynamic dispersion, to the normal closure, shear dilation and stress-dependent matrix porosity, respectively, were tested. The material properties adopted are given in Table 3.1.


34

(a) microcracks formed during shear (b) conceptual model of microcracks (Itasca,

2008)

Figure 3.9 Microcracks development represented by the failed contacts. (a) The short gray line segments indicate shear microcracks, and the short black line segments are tensile microcracks. (b) The microcracks are represented by a pair of smooth parallel plate surfaces with length Lc and aperture 2bc.

Table 3.1 Material properties of groundwater, rock matrix and fracture.

Material Parameters value

Groundwater Hydraulic pressure gradient, dxdp (Pa/m) 10

Dynamic viscosity, (Pa·sec) 1.0×10-3

Intact rock

Initial matrix porosity, 0.01

Empirical constants, 1a ,

2a , 3a (×10

-4, ×10

-4, 1) 0.82, 5.35, 0.12

Cementation factor, n 1.6

Fractures

Normal stiffness at zero stress state, 0nk (GPa/m) 434

Shear stiffness, sk (GPa/m) 434

Dispersivity along fracture, L (m) 0.5

Diffusivity in water, *D (m

2/s) 1.6×10

-9

Frictional angle, (°) 24.9

Cohesion, C (MPa) 0

Critical shear displacement for dilation, csu (mm) 3

Shear dilation angle, d (°) 5.0

Residual aperture, resb (μm) 5

Maximum normal displacement, maxnu (μm) 25

PFC2D 3.10


St ep 107840 11: 45: 29 Thu Feb 10 2011

View Size: X: 4. 270e-002 <=> 5. 512e-002 Y: -5. 450e-003 <=> 4. 344e-003

B al l

FISH funct i on crk_i t em

𝑟𝑐𝑟𝑎𝑐𝑘 = 𝑅𝐴 + (𝑅𝐵 − 𝑅𝐴)𝑅𝐴 + 𝑏𝑐

𝑑

B

A

2bc

d Lc


35

For larger longitudinal dispersivity of L =1 m, the longitudinal

hydrodynamic dispersion increased the solute concentration in the fracture or in the rock matrix at the initial state under the constant concentration inlet boundary (Figs. 3.10(a) and (c)), which is in line with the results shown by Tang et al. (1981). However, under constant flux boundary conditions, the relative solute concentration estimated by transport model III was smaller than that from transport model I in the locations closer to the inlet

(x<3 m), due to the influence of boundary condition. After that part, longitudinal hydrodynamic dispersion increased the solute concentration in the fracture (Fig. 3.10(a)). The same phenomenon exists in the matrix for the constant flux inlet boundary as well (Fig. 3.10(c)). Three transport models gave similar concentration

profiles for smaller longitudinal dispersivity of L =0.1 m (Figs.

3.10(b) and (d)), which indicates that the longitudinal dispersion plays a negligible role in this case.

αL=1 m αL=0.1 m

(a) (b)

(c) (d)

Figure 3.10 Comparison of concentration profiles among the different transport models under the initial conditions without stress applied. Two longitudinal dispersion coefficients (αL=1 m and 0.1 m) were assumed to study the sensitivity of the transport model to the inlet boundary conditions.

F o r m a t r i x

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.2

0.4

0.6

0.8

1.0

Re

lative

co

nce

ntr

atio

n c

/c0

Distance into matrix (m)

Model I

Model II

Model III

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.2

0.4

0.6

0.8

1.0

Re

lative

co

nce

ntr

atio

n c

/c0


Model I

Model II

Model III

F o r f r a c t u r e

0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0

Model I

Model II

Model III

Re

lative

co

nce

ntr

atio

n c

/c0

Distance along fracture (m)0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0

Model I

Model II

Model III

Re

lative

co

nce

ntr

atio

n c

/c0

Distance along fracture (m)


36

(1) Normal closure effect by pure normal compression

Fig. 3.11 shows the concentration profiles along the fracture and in

the rock matrix (x=2 m) at a time scale of 10 years, under two normal stresses of 5 and 10MPa, respectively. The results illustrate that the solute concentrations both along the fracture and in the rock matrix decreased with the increasing normal stress, for all three transport models used. For transport model III, the concentration at the inlet decreased with increasing normal stress. The fracture‘s normal closure reduced the transport activity in the fractured rocks. However, the effect of hydrodynamic dispersion becomes more significant during the process of increasing normal stress, and it increased the solute concentration along the fracture and especially in the rock matrix. In Figs. 3.11(c) and (d), the relative concentration from the transport model III was higher than that from transport model I, indicating indirectly an important role of dispersion in the fractures. Without considering the hydrodynamic dispersion, it was almost impossible for solutes to diffuse into rock matrix under compressive stress of 10 MPa at this time scale of 10 years.

5 MPa 10 MPa

(a) (b)

(c) (d)

Figure 3.11 Concentration profiles along the fracture and into the matrix (x=2 m) at the short time scale (t=10 years), under different normal stress conditions.

F o r m a t r i x

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.2

0.4

0.6

0.8

1.0

Re

lative

co

nce

ntr

atio

n c

/c0


Model I

Model II

Model III

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.2

0.4

0.6

0.8

1.0

Re

lative

co

nce

ntr

atio

n c

/c0


Model I

Model II

Model III


0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0

Model I

Model II

Model III

Re

lative

co

nce

ntr

atio

n c

/c0


0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0

Model I

Model II

Model III

Re

lative

co

nce

ntr

atio

n c

/c0



37

Fig. 3.12 presents the same information as in Fig. 3.11, except that it is at a longer time scale of 50 years. With the longer time, the solute concentrations both along the fracture and in the rock matrix further increased (compared with the results at the time scale of 10 years), and the effects of stress become more obvious. However, the general trend remained the same. Fig. 3.12(c) shows that the relative concentration estimated by transport model I without dispersion became higher than that of transport model III after a longer time, but the relative concentration of transport model I was still smaller than that of transport model III, as shown in Fig. 3.12(d). Considering the relative concentration in the

fracture at x=2 m, transport model I gave a higher concentration value in Fig. 3.12(a) than that of transport model III, and inverse in Fig. 3.12(b). It was concluded that larger concentrations in the fracture induced higher concentrations in the rock matrix, and after a long enough time the concentration of transport model I would be higher than that of transport model III, both in fracture and in matrix.

5 MPa 10 MPa

(a) (b)

(c) (d)

Figure 3.12 Concentration profiles along the fracture and into the matrix (x=2 m) at the long time scale (t=50 years), under different normal stress conditions.

F o r m a t r i x

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.2

0.4

0.6

0.8

1.0

Re

lative

co

nce

ntr

atio

n c

/c0


Model I

Model II

Model III

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.2

0.4

0.6

0.8

1.0

Re

lative

co

nce

ntr

atio

n c

/c0


Model I

Model II

Model III


0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0

Model I

Model II

Model III

Re

lative

co

nce

ntr

atio

n c

/c0


0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0

Model I

Model II

Model III

Re

lative

co

nce

ntr

atio

n c

/c0



38

Figs. 3.13 and 3.14 show the influence of the normal stress on

breakthrough curves for the fracture (x=2 m) and the matrix (x=2

m, z=0.1 m), respectively. Under the initial condition of no stress, the breakthrough curves for models I and II overlapped very much after longer time, which indicates that the influence of hydrodynamic dispersion was negligible. However, at the beginning period of transport, relative concentration increased more quickly with longitudinal dispersion under constant concentration inlet conditions. Due to less solute mass introduced under constant flux inlet conditions, the relative concentrations of transport model III were lower than the others after a long time. The normal stress decreased the fracture aperture and flow rate, and in turn the rate of solute concentration in the fracture and the rock matrix. When a normal stress of 5 MPa was applied, the hydrodynamic dispersion in transport model II increased the rate of solute concentration in the fracture and the rock matrix compared with transport model I. The same phenomena occurred for transport model III after the transport started immediately, but its relative concentration was lower after a long time.

From the above results, it can be seen that the fracture aperture closure induced by normal stress plays an important role in the flow and transport in fractured rocks. It is expected that the extent of fracture closure would become smaller with further increase of the normal stress, because the normal stiffness increases with the increasing normal stress. In this way, the impact of normal stress would decrease under high normal stresses, especially when fracture aperture is close to its residual aperture value. The influence of hydrodynamic dispersion becomes more significant, when the fluid velocity (or fracture aperture) becomes smaller.

Figure 3.13 Comparison of breakthrough curves for the fracture at x=2 m, between the initial case with zero normal stress (solid lines) and the case under the normal stress of 5 MPa (dash lines).

0 20 40 60 80 100 120 140 160

0.0

0.2

0.4

0.6

0.8

1.0

Model I

Model II

Model III

Re

lative

co

nce

ntr

atio

n c

/c0

Time (years)


39

Figure 3.14 Comparison of breakthrough curves for the matrix (x=2 m, z=0.1 m), between the initial case with zero normal stress (solid lines) and the case under the normal stress of 5 MPa (dash lines).

0.1 mm 0.2 mm

(a) (b)

(c) (d)

Figure 3.15 Concentration profiles along the fracture and into the matrix (x=2 m) at the short time scale (t=10 years), under different shear displacements.

0 20 40 60 80 100 120 140 160

0.0

0.2

0.4

0.6

0.8

1.0

Model I

Model II

Model III

Re

lative

co

nce

ntr

atio

n c

/c0

Time (years)

F o r m a t r i x

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.2

0.4

0.6

0.8

1.0

Re

lative

co

nce

ntr

atio

n c

/c0


Model I

Model II

Model III

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.2

0.4

0.6

0.8

1.0

Re

lative

co

nce

ntr

atio

n c

/c0


Model I

Model II

Model III


0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0

Model I

Model II

Model III

Re

lative

co

nce

ntr

atio

n c

/c0


0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0

Model I

Model II

Model III

Re

lative

co

nce

ntr

atio

n c

/c0



40

(2) Shear dilation effect

To examine the effect of shear displacement of the fracture, the normal stress was fixed at 5 MPa, and the shear displacement was increased gradually to produce shear dilation. From Eqs. (3.4) and (3.7), the fracture aperture would start to increase by shear dilation,

when shear displacement su became larger than 5.3 mm. Fig. 3.15

shows the concentration profiles along the fracture and in the rock

matrix (x=2 m) at a time scale of 10 years, with the shear displacement of 0.1 and 0.2 mm, respectively. The influence of increasing shear displacement was just the inverse of that of increasing normal stress. The shear dilation increased fracture aperture, then flow rate, and finally the rate of solute concentration in the fracture and the rock matrix. The influence of inlet boundary conditions became less significant when large shear dilation occurred, because the advection became more significant since of the increasing fluid velocity.

(3) Effect of matrix porosity change

5 MPa 10 Mpa

(a) (b)

(c) (d)

Figure 3.16 Comparison of concentration profiles for the fracture and matrix (x=2 m) between the case with (dash lines) and without (solid lines) considering the matrix porosity change under normal stress conditions, at the short time scale (t=10 years).

F o r m a t r i x

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.2

0.4

0.6

0.8

1.0

Re

lative

co

nce

ntr

atio

n c

/c0


Model I

Model II

Model III

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.2

0.4

0.6

0.8

1.0

Re

lative

co

nce

ntr

atio

n c

/c0


Model I

Model II

Model III

0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0

Re

lative

co

nce

ntr

atio

n c

/c0


Model I

Model II

Model III

0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0

Re

lative

co

nce

ntr

atio

n c

/c0


Model I

Model II

Model III


41

Under normal stress conditions, the influences of matrix porosity change on transport mechanisms were investigated by comparison between the results with and without considering matrix porosity changes, under the compressive normal stress of 5 MPa and 10 MPa, respectively (Fig. 3.16). For the concentration profiles along the fracture, if considering the matrix porosity change under normal stresses, the concentration along the fracture increased significantly, compared with the case of constant matrix porosity. Under higher normal stress, this difference became more obvious (Fig. 3.16(a) and (b)). The decreasing matrix porosity caused by increasing the normal stress, inducing decreasing diffusion coefficients, made the particles more difficult to diffuse into the rock matrix. The concentration in the rock matrix also increased with decreasing matrix porosity, due to the increasing normal stress, compared with the case with constant matrix porosity (Fig. 3.16(c) and (d)). The possible reason was that larger contaminant concentration gradient at the fracture wall tended to push more particles into the matrix pores.

3.3.2. With fracture surface damage case

(1) Model setup and shear procedure

Two synthetic fracture samples were assumed here to demonstrate the effects of fracture surface damage during the direct shear tests on the solute retardation coefficient in single fractures. One was a macroscopically relatively smooth planar fracture with roughness at microscopic scale (Fig. 3.17(a)), and the other was a rough rock fracture with a much simplified triangular saw-tooth profile (Fig. 3.17(b)). A total of 20000 particles were packed to form the fracture samples with a length of 100 mm and a height of 50 mm. The particles in the zone near the fracture had smaller size (radius of about 0.2mm) than those in the rest of the sample (radius of about 0.3 mm). The particle size chosen was in the range of mineral grain size distribution of typical crystalline granite. The basic specimen-genesis procedure used in this study can be found in Poryondy & Cundall (2004) and Itasca (2008). All of the model parameters are listed in Table 3.2, most of which were obtained from the literature (Cundall, 1999; Poryondy & Cundall, 2004; Park & Song, 2009).

Table 3.2 Micro parameters for the intact rocks of demonstration fracture samples.

Parameter (unit) Value

Particle density (kg/m3) 2600

Bond contact normal stiffness (GPa) 10

Bond contact shear stiffness (GPa) 3.0

Bond contact friction coefficient (-) 0.6

Bond contact normal strength, mean (MPa) 20

Bond contact normal strength, standard deviation (MPa) 5.0

Bond contact shear strength, mean (MPa) 20

Bond contact shear strength, standard deviation (MPa) 5.0


42

(a) A relative smooth fracture sample

(b) A rough saw-tooth fracture sample

Figure 3.17 Two PFC models for synthetic fresh tensile fractures.

Figure 3.18 Results for the relative smooth fracture sample under a constant normal stress of 1.0 MPa during shear. The left axis represents the number of abrasive particle, tensile microcrack or shear microcrack within the severely damaged

zone with thickness of 3𝒃𝒗 . The right axis represents the percentage of the number of tensile microcracks and shear microcracks, respectively, within the severely damaged zone

with thickness of 3 𝒃𝒗 over the corresponding total microcracks developed in the whole sample.

0 1 2 3 4 5

0

30

60

90

120

150

Left Y axis Right Y axis

Abrasive particle

Tension microcrack Tension microcrack

Shear microcrack Tension microcrack

Shear displacement (mm)

Nu

mb

er

0

20

40

60

80

100

Pe

rce

nta

ge

(%

)

15°

50 mm

100 mm


43

Figure 3.19 Surface increase factors due to abrasion (𝜶𝒂𝒃),

microcracking (with tensile ( 𝜶𝒄𝒕 ) and shear ( 𝜶𝒄𝒔 ) microcracks) and total retardation coefficient increase factor versus the shear displacement.

Under the different constant normal stress conditions with a servo control algorithm (Itasca, 2008), the PFC models of the two fracture samples were sheared, in a shear displacement interval of about 0.5 mm, for 10 steps in total, so as to make the upper block of the samples move in the right direction. A FISH code (FISH is a programming language embedded within PFC) was developed to monitor the positions and geometries of the abrasive gouge particles and microcracks. The crushing of abrasive particles was not considered, but the re-contacted abrasive particles were

approximately estimated. Note that both 𝜆𝑎𝑏 and 𝜆𝑐 were assumed to be 1.0 for simplicity.

(2) Results with shear under a constant normal stress

Here, the smooth fracture sample under a constant normal stress of 1.0 MPa was took as an example. In Fig. 3.18, the increasing number of abrasive particles and (tensile and shear) microcracks located in the severely damaged zones with shear displacement are shown, and more tensile microcracks were generated in this thin zones close to the fracture. The number of microcracks in the severely damaged zones increased, but the percentage of the microcracks in severely damaged zones over the total generated microcracks in the model decreased. This indicates that the microcracks in the intact rocks also increased with the shear displacement. The corresponding surface increase factors from abrasion and microcracking increased with the shear displacement (Fig. 3.19). The total retardation coefficient increase factor increased during the same process as a result, in which the contributions from the abrasion and tensile microcracks to the

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Su

rfa

ce

in

cre

ase

fa

cto

r

Shear displacement (mm)

ab

ct

cs

A


44

increasing retardation coefficient were larger than that from the shear microcracks.

(3) Results with shear under variable normal stresses

Both the relatively smooth fracture model and the rougher saw-tooth shaped model were simulated to be sheared till a maximum displacement of about 5.0 mm, subject to three different constant normal stresses of 1.0 MPa, 5.0 MPa and 10.0 MPa, respectively.

Figs. 3.20 and 3.21 show that the evolution of abrasive particles and microcracks after a shear displacement of about 5.0 mm with increasing normal stresses, for the relatively smooth and saw-tooth fracture models, respectively. Compared with the cases under smaller normal stress, the number of abrasive particle decreased, but the number of microcracks increased considerably, for both the tensile and shear microcracks (Table 3.3 and 3.4). The main reason for the decreasing abrasive particles was due to the fact that the aperture of the chosen fracture models had limited space to move freely without re-establishing contacts with other particles, so that the purely abrasive particles (free particles with no contacts at all with any other particles) decreased with increasing normal stress. The abrasive particles can also be re-attached to the fracture surfaces to form a part of the fractures again in such model. The abrasive particles can also be crushed into smaller particles in reality but this effect could not be considered in the current study.

With the increasing normal stress, the thickness of the damage zone increased, which is indicated by the decreasing percentage of microcracks in the shallow zones close to the fracture surfaces (Table 3.3 and 3.4). The contact force in the model was larger under higher normal stress and consequently generated more microcracks. From the results, it is found that for most of the

cases, a zone with a thickness of 3 vb , measuring form the fracture

axis towards the depth of rock matrix, contained more than 50% of the generated microcracks. This is the basis of our assumption that the microcracks in this zone were connected to the fracture surfaces, and provided more sorption surfaces. Other microcracks located beyond this zone were considered to influence the matrix porosity and inner sorption, which can further increase the residence time of solutes diffusing into the rock matrix.

For the fracture model of the saw-tooth profile, the microcracks developed mainly along the two contact sides of teeth, and the abrasive particles mostly located at other sides of the teeth, due to the fact that the shear dilation made the initially full contact area of the sample be reduced and restrict to the contact sides only (Fig. 3.21).

Fig. 3.22 shows the distribution of microcracks versus the depth into the matrix for the fracture model of the saw-tooth profile. Generally most of the microcracks were generated in a thin layer of severely damaged zone close to the fracture surfaces, and the number of microcracks decreased with the depth into matrix. However, the parts closest to the fracture (0-0.8 mm) had fewer microcracks compared with its neighbor area deeper in the matrix


45

(0.8-1.6 mm), because the dissipated energy by shear was spent mostly on generating loose abrasive particles.

Consequently, the retardation coefficient increase factors shown in Table 3.3 and 3.4 considering the effects of abrasion and microcracking phenomena increased with the normal stress during the shear test.

(a) n =1.0MPa

(b) n =5.0MPa

(c) n =10.0MPa

Figure 3.20 The abrasive particles and microcracks for the relative smooth fracture sample after a shear displacement of about 5mm under different normal stresses. Dark disks represent abrasive particles, and black and red lines indicate the tensile and shear microcracks, respectively. A more clear view of a small rectangular zone is displayed alongside the models to indicate more details.

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46

(a) n =1.0MPa

(b) n =5.0MPa

(c) n =10.0MPa

Figure 3.21 The abrasive particles and microcracks for the saw-tooth fracture after a shear displacement of about 5mm under different normal stresses. The color conventions are the same as Fig. 3.20.

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47

Table 3.3 Results for the relative smooth fracture sample after a shear displacement of about 5mm with increasing normal stresses.

𝜎𝑛

(MPa)

Abrasive particle Tension microcrack (< 3𝑏𝑣) Shear microcrack (< 3𝑏𝑣) Retardation coefficient increase factor

𝐴 Number

𝛼𝑎𝑏 Free* Recontacted** Number 𝛼𝑐𝑏 Percentage* (%) Number 𝛼𝑐𝑠 Percentage* (%)

1 72 0 0.502 143 0.654 72 47 0.214 56 1.370

5 58 14 0.444 481 2.220 74 166 0.750 56 3.414

10 34 38 0.364 676 3.120 69 315 1.428 51 4.912

Free* means the generated abrasive particles without being re-contacted.

Recontacted** represents the previous abrasive particle re-established contacts with other free abrasive particles or with the fracture surfaces during shear.

Percentage* represents the percentage of the number of tensile microcracks and shear microcracks, respectively, in the severely damaged zone with thickness of 3𝑏𝑣 over the corresponding total microcracks developed in the whole model.

All of them have the same meaning in the following tables hereafter.

Table 3.4 Results for the saw-tooth shaped fracture sample after a shear displacement of about 5mm with increasing normal stresses.

𝜎𝑛

(MPa)

Abrasive particle Tension microcrack (< 3𝑏𝑣) Shear microcrack (< 3𝑏𝑣) Retardation coefficient increase factor

𝐴 Number

𝛼𝑎𝑏 Free* Recontacted** Number 𝛼𝑐𝑏 Percentage* (%) Number 𝛼𝑐𝑠 Percentage* (%)

1 60 0 0.423 213 0.964 63 75 0.332 48 1.719

5 56 4 0.403 359 1.620 53 214 0.954 38 2.977

10 39 21 0.335 498 2.256 34 363 1.608 22 4.119


48

Figure 3.22 Microcrack distribution in a thin slice (a black parallelogram shown in Fig. 3.21(c)) with the distance from the fracture surface towards the rock matrix. The percentage represents the number of microcrack over total microcracks in the area.

Table 3.5 Parameters defining an example fracture to consider the adhesion wear influence.

Parameter (unit) Value Reference

Fracture length 𝑙 (m) 0.1 This study

Fracture width 𝑤 (m) 0.001 This study

Half distance between reference planes 𝑏𝑣 (m) 4.0×10-4 This study

Fracture space ratio 𝛽 0.5 This study

Dimensionless wear coefficient/Characteristic length 𝐾𝑎𝑑 𝐿𝛾 (m-1) 500 Leong & Randolph (1992)

Asperity hardness 𝐻 (GPa) 1.0 Nix & Gao (1998)

Scieszka & Jadi (1998)

(4) Other issues

Adhesion wear effects

The adhesion wear is usually assessed by the volume of gouge materials formed (Leong & Randolph, 1992; Wang & Scholz, 1994). Queener et al. (1965) showed that the process of adhesion wear consists of a transient (nonlinear) part and a steady (linear) part, and the adhesive contribution from steady wear is generally small compared with that from transient wear, and is negligible for most cases. Based on Archard‘s wear law, the increment of gouge

wear volume adV produced in sheared fractures for an increment

of shear displacement su is given as,

0 1 2 3 4 5 6

0

5

10

15

20

25

30

Mic

rocra

ck p

erc

en

tag

e (

%)

Depth into the matirx (mm)

Damaged zone

Severely damaged

zone

Wear zone


49

f

fnad

s

ad

s

ad

L

V

HK

du

dV

u

V

(3.32)

where adK is a dimensionless wear coefficient; n is the normal

stress imposed on the fracture surfaces; H is the asperity

hardness; fV is the volume by which the fracture surfaces depart

from a perfectly flat surface (an indirect measure of current surface

roughness); fL is a characteristic length considering the scale

effects, as well as to make adK dimensionless (Leong & Randolph,

1992). In other words, the term of fad LK can be considered to

reflect the combined dependence on the material roughness and

fracture scales. fV can be expressed as adf VVV 0 , where 0V is

the initial volume by which the initial fracture surfaces depart from a perfectly flat plane (Leong & Randolph, 1992). Eq. (3.32) can be rearranged as,

f

adnad

s

ad

L

VV

HK

du

dV 0

(3.33)

Thus one has,

f

snadad

L

u

HKVV

exp10 (3.34)

Eq. (3.34) indicates that the volume of the adhesive gouge materials in the sheared fractures increases with the normal stress and shear displacement, but decreases with increasing asperity hardness. Due to the lack and complexity of detailed geometrical information of fracture surface geometry, one simple method to

approximately estimate the initial volume 0V is proposed in Fig.

3.5(a). By introducing a fracture space ratio , the initial volume

0V is determined by,

vlwbV 20 (3.35)

where is defined as the volume by which the fracture surfaces

depart from a perfectly flat surface divided by the total volume between the two reference planes (Fig. 3.5(a)).

With the parameters in Table 3.5, the adhesive gouge volume loss was about 4 mm3 according to Eqs. (3.34) and (3.35). If assuming that the adhesive gouge particles have a uniform shape of thin disks with radius of 2 μm, and the same thickness as the fracture model width, the total surface area of the adhesive gouge particles is about 4 m2. Therefore, if we take the extreme case that all of these adhesive gouge particles are immobile, then the retardation coefficient increase factor due to adhesive wear will be 40 according to Eq. (3.28), indicating an important influence of adhesive wear. However, there is a portion of mobile adhesive gouge particles that is probably flushed away by fluid flow after


50

being produced during shear, and they are not of uniform size. Usually, these particles have a similar size distribution like the colloid ranging in an interval of 0.01-10μm, so they could be considered as colloid-like particles, as studied by the model proposed by Bake & Pitt, jr (1996). The accuracy of the above analytical model for adhesion wear largely depends on the

estimation of the initial volume 0V and other associated

parameters. The approximate method proposed here to compute

the volume 0V did not consider the actual contact area ratio

between the two fracture surfaces, and the measurements and calibrations of the model parameters always remain to be a challenging issue.

Crushing of the abrasive particles

Although experimental results show that the size reduction of abrasive particles is an important process during the shear test even under small normal stress (Pereira & de Freitas, 1993; Scieszka & Jadi, 1998), PFC code in the present formulation is not able to capture the crushing (comminution) or fragmentation of the abrasive particles as shown in Fig. 3.5(d), because of the rigid disk assumption and limited computational power. A three-dimensional lattice solid model was used to simulate the joint gouge evolution during shear process by supercomputers (Abe & Mair, 2005; Mair & Abe, 2008). Both the experiments and simulations showed an increasing portion of abrasive particles that were crushed or comminuted into much smaller ones, with increasing normal stress and shear displacement. However, even under relatively high normal stresses and shear displacements, there still exist some abrasive particles that are not crushed, especially when shear dilation is significant in reducing active contact area. In this study, the abrasive particles predicted by the PFC model for the two fracture models in the previous sections were uncrushed abrasive particles. Even though a few theoretical models for fragmentation phenomenon may offer some insights into the crushing processes in fractures (See for example, Allegre et al., 1982; Turcotte, 1986; Sammis et al., 1987), it still remains an unclear issue due to lack of available mathematical models and experimental evidence support. For simplicity, if the abrasive particles were crushed into smaller particles similar to the adhesive particles mentioned before, in a uniform shape of thin disks with radius of 2 μm, then one abrasive particle of radius of 0.2 mm could be crushed into 10 000 small particles. The total surface area of these small crushed particles from one original abrasive particle would be about 0.13m2, compared with the surface area of 0.012 m2 of the original abrasive particle. Those 10000 small particles could result in a retardation coefficient increase factor of 1.3 for the fracture geometry shown in Table 3.5, for an extreme case that all of them are immobile. This indicates that the crushing process may have important impacts on the solute retardation. In reality, the particles are not 2D disks, but are 3D polyhedra, so the surface increase could be larger.


51

Shear dilation effects

The retardation coefficient is inversely dependent on the average fracture aperture, because of the larger surface-area-to-volume

ratio for a fracture with smaller b (Wels & Smith, 1994). It is well

known that the dilation, du , is usually associated with the shear

displacement, besides the normal closure of fracture under the

normal stress, nu . Therefore, the Eq. (3.24) can be modified as

Eq. (3.36) below to consider the effect of fracture aperture changes due to shear dilation.

2/)(1

nd

f

uub

KR

(3.36)

Then the retardation coefficient decrease factor due to the shear dilation is given by,

2/)(

2/)(

1

2/)(1

ndf

nd

f

f

nd

f

uub

b

b

K

uub

K

b

K

uub

K

B

(3.37)

From the PFC model simulation performed in this study, the values of fracture dilation were obtained. For the relative smooth fracture sample, after a shear displacement of about 5 mm, the dilation of 0.36 mm, 0.21 mm and 0.12 mm was obtained for the constant normal stresses of 1.0 MPa, 5.0 MPa and 10.0 MPa, respectively. Taking an average value of the half fracture aperture

as 0.15mm, the retardation coefficient decrease factors, B, due to the dilation were then 0.45, 0.60 and 0.71, under the constant normal stresses of 1.0 MPa, 5.0 MPa and 10.0 MPa, respectively. This indicates that the fracture aperture changes due to shear dilation can play an important role on the solute retardation behavior in single fractures. However, it should be remembered that a decrease in aperture has an inversely square impact on the fluid velocity.

3.4. Summary

For the single fractures, the stress effects on the solute transport could be conceptually estimated by an analytical model that linked the coupled stress-flow-transport processes, for the case without surface damage. An expression for the solute retardation coefficient in the fractures were derived considering fracture surface damage, and a particle mechanics model was used to simulate the gouge generation and microcrack development. The results obtained in this chapter not only provide the basic insights into various stress effects on solute transport in a single fracture, but also are helpful to understand the influence of stress on solute transport in a fracture system that is presented in next chapter.


52


53

4. STRESS EFFECTS ON SOLUTE TRANSPORT IN FRACTURE

NETWORK

Based on the understanding of solute transport and stress effect in the single rock fractures, this chapter presents a study on the stress effects on solute transport in a macroscopic fractured rock. The algorithms of modeling the advection, matrix diffusion and hydrodynamic dispersion in single fractures by particle tracking method are presented first, followed by the macroscopic solute transport behaviors both without and with stress influences.

4.1. Methodology

A number of general assumptions on rock matrix, fractures and fluid were adopted in this study. (1) The problem was defined in a 2D space, with a plane-strain assumption (without considering the gravity) for stress analysis and fully saturated steady-state for fluid flow analysis. (2) The rock matrix was assumed to be impermeable, and the fluid was conducted by connected fractures only. (3) Individual fractures were idealized as the smooth parallel plate model, following the cubic law between the flow rate and hydraulic gradient. (4) The hydraulic apertures of fractures were equal to their mechanical apertures with constant initial values. (5) Fluid was incompressible. (6) Complete mixing occurred at the fracture intersections for solute transport simulations. (7) Rock matrix was a linear, isotropic and homogeneous elastic material with constant porosity. (8) The particles mentioned in ‗particle tracking‘ can be thought of as small solute molecules that can move by diffusion and they are sufficiently small (about 0.3 nm) not to be strained in even the narrowest fractures.

Fig. 4.1 shows a flowchart for modeling the coupled stress-flow-transport processes in fracture networks. For the coupled stress-flow simulations, a 2D DEM code Universal Distinct Element Code (UDEC) (Itasca, 2004), was used to obtain the stress and deformation results of the fracture system (i.e. the fluid flow pathway changes), the changes of aperture, and flow velocity in all individual fractures contributing to the overall flow pattern. The details of this approach can be found in Itasca (2004), Min et al. (2004) and Baghbanan & Jing (2008). In order to maintain the integrity of methodology, the basic principles for modeling coupled stress-flow processes using UDEC was briefly presented below, followed by a detailed description of the algorithms for solute transport by particle tracking method.

4.1.1. DEM model for coupled stress-flow simulation

(1) Mechanical process

When stress is applied on fractured rocks, both rock matrix and fractures will deform or be displaced, governed by the equations of motions of the rock bloack and constrained by the specified constitutive models and material parameters for rock matrix and fractures, and the initial and boundary conditions. In this study, the rock blocks, treated as linear, isotropic, homogeneous and elastic materials, were internally discretized into triangle finite-


54

difference elements, whereas fracture displacements were determined by relative motions of the contacting blocks (that define the fracture concerned) in both normal and tangential direction of the fracture. The deformation of the rock matrix (blocks) is described simply by theory of elasticity with two elastic

constants, Young‘s modulus (E) and Poisson‘s ratio (υ). The constitutive behaviors of fractures are described in more detail below.

Figure 4.1 Flow chart for coupled stress-flow-transport simulation in fractured rocks.

Fracture network regularization

Fracture network geometrical properties

(Fracture density, position, length, orientation and aperture) from field mapping

Distinct element method (DEM) model

Stress-deformation simulation

Fluid flow simulation

Particle transport simulation

Monte Carlo approach or deterministic models

Constitutive models/properties

Stress boundary conditions

Hydraulic pressure boundary conditions

Transferring coupled stress-flow results

Introducing particles at inlet boundaries

Macroscopic solute transport

results in fractured rocks

Fluid flow field results

DEM model after deformation

UDEC

PTFR


55

σn

σn3

σn2

σn1

un1 un2 un3 un (-)

(+)

(+)

Unloading/reloading

1 ks

Shear displacement

Shear

str

ess

d

Shear displacement

Dila

tion

ucs

(a) Normal direction (b) Shear direction

Figure 4.2 Mechanical behavior model of the rock fracture (adapted from Itasca (2004)).

Figure 4.3 Schematic diagram for ‘domain’ and particle distribution at fracture intersection.

In the normal direction of a pair of parallel nominal planar fracture surfaces, the constitutive model (stress-displacement relation) was

assumed to be governed by a nonlinear stiffness, nk , in form of a

simplified Barton–Bandis model as shown in Fig. 4.2(a),

nnn uk (4.1)

In the tangential (shear) direction of the fracture, the response is

characterized by a constant shear stiffness, sk , before the

occurrence of shear dilation. The shear stress, , is limited by a combination of cohesion and friction angle, according to a Mohr–Coulomb criterion (Eq. (3.4)). Additionally, in the normal direction, the shear dilation occurs at the onset of slip (non-elastic sliding) of the fracture (Fig. 4.2(b)). Dilation is governed by the Coulomb slip model through a specified dilation angle. The accumulated dilation is generally limited by either a high normal

stress level or by a critical shear displacement, csu . Dilation is a

Domain

①

②

③

④

⑤

in

i

outi

outQ

QQP )(

①-⑤ Contacts

Inlet flow

Outlet flow


56

function of the direction of shearing. Dilation increases if the shear displacement increment is in the same direction as the total shear displacement, and decreases if the shear increment is in the opposite direction.

(2) Fluid flow process

To obtain the flow field under different stress conditions, the mass continuity equation is solved at fracture intersections through an iterative scheme in combination with the prescribed hydraulic boundary conditions. In UDEC, a concept of ‗domain‘, which is the region of space between blocks defined by the contact points, is used to compute the fluid pressure distribution (Fig. 4.3). The fluid flow is then governed by the pressure gradient between the adjacent domains. The flow rate per unit width of fractures follows the cubic law (Eq. (3.12)). The current hydraulic aperture is

computed by Eq. (3.8). A residual value resb and a maximum value

maxb were assumed for apertures to improve the efficiency of

calculation, below or beyond which mechanical displacement does not affect the hydraulic conductivity of fractures (Itasca, 2004).

4.1.2. Solute transport simulation

Once a steady-state flow field through a fracture network is obtained, using UDEC under a set of prescribed stress and hydraulic boundary conditions, the particle movement in the fracture network can be traced for all particles through all their possible paths, basically as an advection process based on the known flow velocity in each fracture or fracture segment. At first, a large number of particles are injected at a time, either in each inlet fracture along the upstream boundary or at the middle of upstream boundary, dependent on the problem definition. For the first case, the number of particles at each injected fracture is proportional to the local flow rate in the fracture, which is equivalent to injecting a tracer solution with constant concentration. Then the particles would be transported by the flowing fluid in fracture networks, following the fluid flow paths (connected fractures). According to the assumption of complete mixing at fracture intersections, the particles are fully mixed with fluid and with each other, so that each particle randomly chooses any one of the outlet fractures connected to that intersection with a probability proportional to its flow rate. The domain representing an intersection consisting of five fractures (two inlet fractures and three outlet fractures in Fig. 4.3) was took for example to illustrate the procedure of simulating this random process. The probability of a particle leaving the domain through a certain effluent fracture depends on the ratio of the outlet flow rate in each outlet fracture over total inlet (or outlet, they are equal) flow rate of the domain (see the equation in Fig. 4.3). This is the way of a particle leaving the domain and entering the next fracture segment, which is repeated until the particle is traced through the whole model, i.e. it reaches one of the outlet boundaries of the fracture network model.


57

(1) Advection

In each fracture segment (‗contact‘), the water residence time, i

wt ,

is calculated by,

i

ii

wQ

Vt (4.2)

where iV is the volume of fracture segment i , i.e. the product of

the fracture length iL , the width iW and the aperture ib ; iQ is

the flow rate calculated using Eq. (3.12). For 2D cases, iW was

assumed to be a unity value of 1m.

The total water residence time, j

wt , for a particle (or the flow path

it passed) j , is the sum of the water residence time of this particle

in all the fracture segments it has passed.

i

i

w

j

w tt i i

i

Q

V (4.3)

(2) Hydrodynamic dispersion

Based on the comparison and reviews by Gershon & Nir (1969), Kreft & Zuber (1978), Van Genuchten & Parker (1984), a summary of expressions of the classical solutions widely used in the literature for the classical ADE (Eq. (4.4)) in single fractures is given below.

02

2

x

cD

x

cv

t

c f

f

ff (4.4)

Semi-infinite fracture under constant concentration boundary condition (Ogata & Banks, 1961)

tD

vtxErfc

D

xv

tD

vtxErfc

c

txc

fff

f

2exp

22

1),(

0

(4.5)

It represents the relative concentration in a fracture where the solutes exist at the inlet boundary at a prescribed constant

concentration. Besides the advection flux 0vc , there is a dispersive

flux )),0(( xtcD ff , which is the reason why solute mass

does conserve (Van Genuchten & Parker, 1984, Abdel-Salam & Chrysikopoulos, 1994). So far it is generally agreed that this boundary condition is not applicable for larger dispersion coefficients, and the correct one should be the constant flux boundary condition, as shown below.

Semi-infinite fracture under constant flux boundary condition (Gershon & Nir, 1969)

tD

vtx

tD

vt

D

vtxv

tD

vtxErfc

D

xv

tD

vtxErfc

c

txc

ffffff

f

4

)(exp

)(1

2exp

2

1

22

1),( 2

0

(4.6)


58

Finite fracture under constant flux boundary condition

In reality, the fractures are finite in length, extending form 0x

to flx . It is assumed that the boundary condition at the inlet

boundary is constant flux inlet boundary, and the boundary condition at the outlet of the fracture is,

0),(

x

tlc ff (4.7)

which indicates that assumption of continuous concentration at

flx (Danckwert, 1953; Van Genuchten & Parker, 1984). The

solute concentration at the outlet boundary ( flx ) is (Bastian &

Lapidus, 1956),

1

22

2

2

20 42exp

4

)()(

)sin(21

),(

n

nf

ff

f

f

f

f

f

fn

fnfnfftD

D

tv

D

vl

D

vl

D

vll

ll

c

tlc

(4.8)

where the eigenvalues n are the positive roots of

f

ff

f

f

fnfnvl

Dvl

D

vlll

2)(

4)cot( .

To simulate the hydrodynamic dispersion by particle tracking method, a random number uniformly distributed in the interval of [0, 1] is used to replace the relative concentration in Eq. (4.5), and the actual particle travel time through the single fracture can be obtained by solving Eq. (4.9).

tD

vtxErfc

D

xv

tD

vtxErfcR

fff 2exp

22

11

0 (4.9)

With a large number of particles injected passing though that fracture, it can statistically represents the local longitudinal dispersion effects on the particle travel time. Nauman (1981) showed the probability of a particle introduced at the inlet first

arriving at the outlet during the time t and dtt is,

tD

vtx

tD

xdttf

ff4

)(exp

4)(

2

3 (4.10)

where )(tf is the probability density function of residence time of

a particle through a single fracture. The cumulative distribution function of residence time is then derived as,

tD

vtxErfc

D

xv

tD

vtxErfcdttftF

fff

t

2exp

22

1)()( (4.11)

which is fully the same as Eq. (4.5). It also means the ratio of the output solute over the input solute with time, i.e. breakthrough curve. By the injection method, Eq. (4.11) can be used to


59

statistically produce the particle travel time through each single fracture for pulse injection conditions too.

(3) Matrix diffusion

The matrix diffusion can cause an increase in the particle residence

time, jt , compared with the water residence time, j

wt , because the

real solute particles (atom, ion or molecule) can randomly enter the stagnant water in the rock matrix by molecular diffusion and reside there for some time before they again move out to the flowing water in the same or other fractures by diffusion. Each particle will

always arrive later than the water residence time j

wt in the flow

path actually, and they will then have a distribution of residence time, instead of a constant water residence time. When there is no dispersion, the Residence Time Distribution (RTD) of a particle (flow path) can be determined by Eq. (4.12), which shows the

effluent concentration, jc , for flow path j where the water

residence time is j

wt , under a step injection of solute with inlet

concentration, 0c , at time zero (Retrock, 2004).

2/1

0 )(2

1j

w

j

j

qj

tt

MPG

Q

AErfc

c

c j

w

j tt (4.12)

where Erfc is the complementary error function defined as

x

y dyexErfc22

; qA is the contact surface between the

flowing water and the rock, which is called the flow wetted surface

(FWS). The entity of j

q QA , which is the ratio of FWS over the

flow rate in the flow path, determines the interaction between the

rock and the particle. In other words, a large value of j

q QA

allows a particle to diffuse into the saturated stagnant pore water in the rock matrix with more opportunities and thus give the particle more time to reside in rock matrix than if it only moves with the flowing water in fractures. Some solutes especially cations (positively charged solute particles) can be adsorbed on the (negatively charged) surfaces of micropores in the rock matrix, which further retards their migration. In this way, particles have additional residence time in the rock matrix, other than the pure

water residence time through the fractures. The value of j

q QA

is determined by,

i i

ii

i i

i

q

j

q

Q

WL

Q

A

Q

A 2 (4.13)

In Eq. (4.13), the ratio of j

q QA is the sum of i

i

q QA in each

fracture i passed by the particle, where i

qA is the FWS in fracture

i . The value of ‗2‘ in Eq. (4.13) represents the effect of two

fracture surfaces of each fracture, and iW is assumed to be a unity


60

value of 1 m as computing the water residence time in Eq. (4.2). Even though Eq. (4.12) was originally derived for a single fracture, Moreno et al. (2006) demonstrated that Eq. (4.12) is valid for a

flow path consisting of a multitude of fractures. The term MPG (Material Property Group) in the present application is assumed to be a collective constant containing information about effective

diffusion coefficient in the matrix pores eD , sorption coefficient

dK and matrix porosity (Eq. (4.14), Retrock, 2004).

d

e

KDMPG

)1(1 (4.14)

For non-sorbing species, 0dK . More background for matrix

diffusion is given in Retrock (2004).

Based on Eq. (4.12) that shows the cumulative probability density distribution of residence time for a flow path, the rejection method (Yamashita & Kimura, 1990; Moreno & Neretnieks, 1993; Gylling et al., 1999; Delay & Bodin, 2001; Bodin et al., 2007) can be used to incorporate the effects of matrix diffusion into the particle tracking. By choosing random numbers, R, from the uniform

interval [0, 1] to equal to the values of 0cc j

f in Eq. (4.12), the

stochastic total residence time distribution for this flow path can be determined by Eq. (4.15).

2/1

1

0)(2

1][

j

w

j

j

q

tt

MPG

Q

AErfcR (4.15a)

2

21

0

1 2][4

1

j

qj

w

j

Q

AMPG

RErfctt (4.15b)

0.0

0.2

0.4

0.6

0.8

1.0

c/c

0

Time Figure 4.4 Schematic diagram for computation of average residence time for certain concentration.

2 3 Average breakthrough curve

j 1


61

When evaluating the ensemble residence time distribution for all particles, one way is to use Eq. (4.15b) with one random number for each flow path and calculate the ratio of particles exiting over total number of particles as concentration at specified time (Bodin et al., 2007), and the other way is to use Eq. (4.12) for each flow path at specified time and add the results to obtain the arithmetic mean value of concentrations of all the flow paths, with calculated

j

wt and j

q QA from Eqs. (4.11) and (4.13). The latter approach

in further explained in Fig. 4.4. If ploting Eq. (4.12) with time, it can be obtained that the residence time distribution (breakthrough curve) for each particle that travels through any flow path ‗ j ‘ to

the effluent point (dash lines in Fig. 4.4). In this way, one has the individual breakthrough curve for each particle. Then the average

0cc j

f considering the ensemble of total particles at any specified

time t can be obtained by Eq. (4.16).

N

j

j

w

j

qN

j

j

tt

MPG

Q

AErfc

Nc

c

Nc

c

1

2/1

1 00 )(2

111 (4.16)

If j

wtt , one has 00 cc j

f .

Marching through the specified time interval, the average breakthrough curve for all interacting particles can be obtained.

Additionally, for a concentration of 0cc j

f , the residence time

t can be obtained by rearranging the above equation. It may be

noted that two random processes are involved in the technique proposed by Yamashita & Kimura (1990). Firstly, the flow path ‗ j ‘

has been randomly selected, because the particle moving with the flowing water randomly chooses the effluent fracture segment at each intersection. The second random component is to determine the (random) residence time for the particle travelling along flow path ‗ j ‘. This procedure is faster than summing N times of Eqs

(4.12), one for each flow path ‗ j ‘ for the range of time of interest.

After comparing the above two approaches for considering matrix diffusion, it was found that the difference of computing time between these two procedures is only a few seconds for the transport simulation presented in the later sections, but they gave quite similar results. The most time consuming part of transport simulation is tracking particles with flowing water through the fracture network, i.e. the first random process.

4.1.3. Code

Based on an earlier computer code called ‗CHAN3D‘ developed by Moreno and his co-workers (Moreno & Neretnieks, 1993; Gylling et al., 1999) for tracking particle movement through regular channel networks, a new code called PTFR for 2D random irregular fracture network geometries was developed and integrated with the code UDEC. Adopting the same concepts of ‗contact‘ and ‗domain‘ as that used in the UDEC data structure, the mechanical and fluid flow results obtained by UDEC


62

calculation can be input into PTFR directly. For performing the particle tracking for transport simulation, the solute particles can be injected either along the inlet boundary or at the center of the inlet boundary. Besides advection and matrix diffusion, hydrodynamic dispersion can also be simulated.

4.2. Numerical experiment procedure

4.2.1. Coupled stress-flow-transport modeling procedure

Min et al. (2004a) and Baghbanan & Jing (2007) investigated the scale-dependent equivalent permeability of fracture networks, and showed that an acceptable Representative Elementary Volume (REV) scale is above 5 m for the fracture systems with constant apertures or above 20 m for the fracture systems with correlated apertures, for a specific set of fracture properties based on site investigation results. In this study, small square DEM models of a side length of 5 m or 10 m was extracted from the center of the original parent model of fracture system with a side length of 20m, based on the same fracture network model data as used in Min et al. (2004a) and Jing & Hudson (2008). Fig. 4.5 shows the DEM models of constant aperture of 30 μm before and after the fracture system regulation. The material properties are shown in Table 4.1.

(a) Before regulation (b) After regularization

Figure 4.5 DEM Model of fracture network with constant aperture and a side length of 5m.

Particles input

Top

Bottom

Lef

t

Rig

ht

Particles in

put

(a) Horizontal hydraulic (b) Stress condition (c) Vertical hydraulic

gradient gradient

Figure 4.6 Stress conditions and hydraulic conditions.


63

Table 4.1 Material properties of intact rock and fractures.

Material Parameters value

Intact rock

Elastic modulus, E (GPa) 84.6

Poisson’s ratio, υ 0.24

Material property group, MPG (ms-0.5

) 1×10-8

Fractures

1nu (μm) 15

1n (MPa) 4

2nu (μm) 20

2n (MPa) 10

3nu (μm) 25

3n (MPa) 30

Shear stiffness, sk (GPa/m) 434

Friction angle, (º) 24.9

Dilation angle, d (º) 5

Cohesion, C (MPa) 0

Critical shear displacement for dilation, csu (mm) 3

Initial aperture 0b (μm) 30

Minimum aperture value, resb (μm) 5

Maximum aperture value, maxb (μm) 50

Note: 1nu , 1n , 2nu , 2n , 3nu , 3n are parameters control fracture mechanical

behavior in normal direction (Fig. 4.2(a)).

The numerical experiments for coupled stress-flow–transport simulation were performed step by step following the modeling logic displayed in Fig. 4.1, after building the DEM model. Firstly, boundary stresses as shown in Fig. 4.6(b) were applied to generate deformed models for fluid flow analysis, and two cases of stress boundary conditions were considered. In the first case, a constant vertical compressive stress of 5 MPa was specified at the top and bottom boundaries, and the horizontal compressive stresses, varying from 5 MPa to 20MPa, were applied at the left and right boundaries, in order to induce shear/dilation in the fractures. In the second case, a constant horizontal compressive stress of 5 MPa was specified at the left and right vertical boundaries, and the vertical compressive stresses, varying from 10 MPa to 20MPa, were applied at the top and bottom boundaries. The stress ratio is

defined as K=horizontal:vertical stresses, and K=0 represents the free state that both horizontal and vertical stresses are zero.

After completion of the stress/deformation calculations, groundwater flow through the deformed DEM models was simulated under the specified hydraulic boundary condition as illustrated in Fig. 4.6(a) and (c). The hydraulic gradient was fixed at 1×104 Pa/m, which was equivalent to a water head gradient of 1.0 m/m. Two sets of hydraulic conditions were simulated in order to evaluate different flow and transport behaviors in macroscopically


64

horizontal and vertical directions, respectively. A much smaller hydraulic gradient was used to compare the effects of matrix diffusion under different hydraulic boundary conditions.

When the fluid flow simulations were finished, the fracture system geometry and fluid flow information for all fractures were transferred to PTFR. For each model, a large number of particles were injected along or at the middle of the main inflow boundaries, i.e. the right boundary for the horizontal flow case and the top boundary for the vertical flow case, respectively (Fig. 4.6(a) and (c)). Particles were collected from the other three outlet boundaries. The calculations were carried out for varying number of input particles ranging from 6000 to 20000, in order to investigate the effect of the number of particles on the breakthrough curves. The results demonstrated that 8000 particles are more than adequate for the study since they yield the same breakthrough curves as that using more particles.

4.2.2. Evaluation of macroscopic behaviors of solute transport in fractured rocks

(1) Advection

The breakthrough curves (history of solute concentration at the outlet boundaries), average residence time and travel distance and particle distribution were used to analysis the macroscopic advection behaviors.

(2) Hydrodynamic dispersion

Hydrodynamic dispersion coefficient matrix, which describes the spreading of a solute pulse caused by local variations in groundwater flow velocity and molecular diffusion, usually defined by the equation below for isotropic geological porous media (Salamon et al., 2006):

Ivv

vvD mTTL )()( D (4.17)

where L and

T are the longitudinal dispersivity (or called

dispersion length) and transverse dispersivity, respectively; v is

the magnitude of the average fluid flow velocity vector.

To avoid the confusion between dispersion length and travel distance of particles, the term of longitudinal dispersivity is used, representing the same meaning as dispersion length in other literatures. In practice, it is almost impossible to measure the equivalent hydrodynamic dispersion coefficients defined in Eq. (4.17), even though the fluid flow velocity field could be obtained. The main reason is that the dispersivity actually represents the geometrical and topological effects of fracture network configurations, but how to evaluate the effects of fracture system geometry of fractured rocks on the dispersivity remains a challenge not resolved.

Compared with permeability, the hydrodynamic dispersion is not only related to the fracture network geometry, but also dependent on the fluid velocity field (hydraulic conditions). Therefore, it


65

depends on the stress/deformation processes of the fracture system as well.

The Péclet number is a widely used measure to quantify the hydrodynamic dispersion (Rasmuson & Neretnieks, 1981). The dimensionless Péclet number in fracture rocks can be defined as,

LDvxPe (4.18)

where x is the average travel distance in flow direction; v is the

average flow velocity of fluid in the flow direction; and LD is the

longitudinal hydrodynamic dispersion coefficient for fracture network. High values of the longitudinal dispersion coefficient yield low Péclet numbers. When molecular diffusion is negligible, one has,

vD LL (4.19)

Combining Eqs.(4.18) and (4.19), the Péclet number can be expressed as,

LαxPe (4.20)

In field tracer tests, Pe is often found to vary from 1 to 100 in fractured rocks (Gelhar, 1993).

The hydrodynamic dispersion coefficients of the fracture network are determined with a statistical description of the spread of a swarm of particles with time. If the spatial particle distribution behaves essentially as a moving and spreading Gaussian curve over an adequately long time (or travel distance), the values of the hydrodynamic dispersion matrix and dispersivity can be determined using the following equations (Grubert, 2001):

)()()()(2

1lim tttt rrrr

tD

t

(4.21)

)()()()()(2

1lim tttt

trrrr

rα

t

(4.22)

where )(tr is the spatial position of the particles, represents

the center of the swarm of reference particles at time t , and α is the dispersivity tensor. Substituting Eqs. (4.22) into (4.20), the Péclet number can be determined by,

)()()()(

)()(2lim

tttt

ttPe

rrrr

rr

t

(4.23)

Note that Eqs.(4.22) and (4.23) are correct only for cases with negligible molecular diffusion.

Another method to determine the hydrodynamic dispersion matrix was proposed by Lee et al. (2007). The main idea is to fit the analytical (continuous) breakthrough curves to the discrete data obtained from particle tracking method by trials of different values

of hydrodynamic dispersion matrix D .

The method proposed by Grubert (2001) was used because of its simplicity. After a steady state flow field in the fracture network model is obtained, a sufficiently large number of reference


66

particles are injected to the central point of the upstream inlet boundary simultaneously. The particles then move randomly with the flowing groundwater in the fracture network model. During this process, one can monitor the positions of all particles within

the fracture network model. At any time t , one can have a statistical description of the spread of a swarm of reference particles, with the center of the swarm computed whenever needed. The components of the hydrodynamic dispersion matrix

can be obtained by using Eq. (4.21). Once the values of xxD , yyD

and xyD (or yxD ) in the matrix D are determined, the principal

values of the hydrodynamic dispersion coefficients at time t and their directions can be derived in the following forms (Way & Mckee, 1981):

2

4)(

2

22

11

xyyyxxyyxxDDDDD

D

(4.24)

2

4)(

2

22

22

xyyyxxyyxxDDDDD

D

(4.25)

22 4)()(

2tan

xyyyxxyyxx

xy

DDDDD

D

(4.26)

where 11D and

22D are the major and minor hydrodynamic

dispersion coefficients, respectively; is the angle between the

major axis 11D and the horizontal direction.

By repeating the above procedure over the prescribed number of

time step ( t ), the values of the hydrodynamic dispersion coefficients can be obtained as a function of time, or the distance that the reference particles have traveled.

(3) Matrix diffusion

Besides the breakthrough curves, another important parameter is

QAq , because it determines the additional residence time induced

by matrix diffusion, if the diffusion and sorption properties of the rocks are known as constant (Moreno et al., 2006). As Neretnieks (1983) indicated, the mean residence time for interacting tracers is

unbounded, so the 5.0t and wtt 5.0 were computed to reflect the

importance of matrix diffusion.

4.3. Results

4.3.1. Flow results

Fig. 4.7 shows the flow pattern changes with increasing stress ratio. The flow results are presented here for understanding the solute transport results in the following subsections. In general, the hydrostatic stress condition (with equal boundary stress values in both the horizontal and vertical directions) made all the flow rates in fracture networks decrease, whereas, under the larger stress difference between two discretions, a few major channels


67

consisting of fractures undergoing significant shear dilation dominated the flow pattern, with the main flow direction parallel with direction of larger boundary stresses. If the main flow direction is perpendicular to the direction of the larger boundary stresses, the channeling followed the direction of larger boundary stresses, instead of main flow direction. Thus, the preferred solute transport paths would be expected to change as well.

(a) K=0:0 (b) K=5:5

(c) K=10:5 (d) K=5:10

(e) K=15:5 (f) K=5:15

(g) K=20:5 (h) K=5:20

Figure 4.7 Flow rates distributions under horizontal (from right to left as Fig. 4.6(a), left) and vertical (from top to bottom Fig. 4.6(c), right) hydraulic gradient, with increasing stress difference. Thickness of the line represents the magnitude of flow rates. Each single line indicates the flow rate of 1×10-9m3/sec and the flow rates smaller than that value are not shown. K represents the ratio of horizontal stress over vertical stress.


68

4.3.2. Advection

(1) Macroscopically horizontal flow case

Fig. 4.8 shows the breakthrough curves for conservative tracers with the varying stress ratio under a horizontal hydraulic gradient of 1×104 Pa/m (1m/m), and Table 4.2 summarizes the general solute transport simulation results. Note that the breakthrough curves account for all the particles that exited through the left, top and bottom boundaries. The numbers of particles that exited from different boundaries are compared in Table 4.2. According to the flow results (Fig. 4.7), the explanation for transport results and variations of breakthrough curves is given below, as an attempt to investigate the stress effects on pure advection in fractured rocks.

When compressive stresses of MPa were applied (K=5:5), the breakthrough curve shifted to the right direction, indicating a longer travel time of one order of magnitude from the stress-free

state (K=0). The reason is that almost all the fractures experienced normal closures under isotropic compressive stresses, which resulted in decreasing flow rates in most of the fractures, and caused increasing residence time in turn. The average particle travel length did not change significantly, but the average water residence time increased by about a factor of 5 with respect to that

with K=0.

Figure 4.8 Breakthrough curves for non-interacting tracers with increasing stress ratio, under horizontal hydraulic gradient of 1×104Pa/m (1m/m).

10-1

100

101

102

103

104

105

0.0

0.2

0.4

0.6

0.8

1.0

K=0

K=5:5

K=10:5

K=5:10

K=15:5

K=5:15

K=20:5

K=5:20

c0/c

Time (hour)

c/c

0


69

When the stress ratio K became 15:5 or 5:15, the breakthrough curves started to move backwards. The main reason is that many particles chose the paths exiting from top and bottom boundaries (Figs. 4.7(e) and (f)), which resulted in the average travel length decreasing drastically (Table 4.2). To be more specific, over 1/3

particles exited from bottom boundary for the case of K=15:5, and almost 80% particles exited from top boundary for the case of

K=5:15, with an average travel length of 1.64m. This change made the breakthrough curve shift backwards in a big stride. The possible reason for such big changes is the shear dilations starting to occur for a few larger fractures oriented with critical angles, so that most particles selected these fractures consequently with a larger probability.

The breakthrough curve for stress ratio K=20:5 continued to move backwards (Fig. 4.8), because of the three obvious horizontal flow channels formed under this stress condition (Fig. 4.7(g)). About 70% particles exited from the left boundary, with the normal average travel length. The breakthrough curve for

K=5:20 overlapped with the one for K=5:15 for the part under

0cc =0.6, even beyond the breakthrough curve for K=0. Those

fractures having larger shear dilations would dominate the flow pattern and provide the preferable particle travel paths (Tsang & Neretnieks, 1998). Therefore, the main mechanisms generating

changes for these two cases are different. For the case of K=20:5, the stresses decreased individual particle residence time by rising

flow rates. However, for the case of K=5:20, the stresses decreased individual particle residence time by providing much shorter particle travel paths.

Fig. 4.9 shows the spatial distribution of the particle collection and flow rates normalized with respect to the maximal flow rate in the outlet fractures along the left boundary, as an example to investigate the channeling phenomenon with increasing stress ratio. Basically, fractures with larger flow rates attracted more

particles. Before K=10:5, at least 70% particles followed the main

flow direction and exited from the left boundary. After K=15:5, only a few fractures carried a dominated number of particles.

(2) Macroscopically vertical flow case

Similar work was done with the same hydraulic gradient of 1×104

Pa/m (1m/m) but in the vertical direction. Fig. 4.10 shows that the general trend of breakthrough curve shifting with varying stress ratio is similar to that with a main horizontal flow direction as shown in Fig. 4.8, with minor differences as described below.

When the main flow direction was parallel with the direction of

larger boundary stresses, from K=5:15, the breakthrough curve moved backwards very much, because some large fractures oriented in sub-vertical directions had much larger flow rates due to shear dilation.

For the case where the main flow direction was perpendicular to the direction of larger boundary stresses, the breakthrough curve


70

did not shift backwards too much, even when K=20:5 (Fig. 4.10). The reason could be that almost 90% particles chose much longer and more meandering travel paths, and exited from the left

boundary. The average travel length for K=20:5 was 7.20 m (Table 4.3), which was much larger than the model side length of 5 m.

(a) K=0 (b) K=5:5

(c) K=10:5 (d) K=15:5

(e) K=20:5

Figure 4.9 Particle and normalized flow rate distribution in fractures intersecting the left vertical boundary under horizontal hydraulic gradient of 1×104Pa/m. Particle percentage represent the number of particles exiting from each fracture over total particle numbers. The flow rate is normalized with respect to the maximal flow rate in the outlet fractures along the whole boundary.

-2 -1 0 1 2

0

5

10

15

20

25

Particle percentage

Normalized flow rate

Left boundary

Pa

rtic

le p

erc

en

tag

e

0.0

0.3

0.6

0.9

1.2

No

rma

lize

d flo

w ra

te-2 -1 0 1 2

0

5

10

15

20

25

Particle percentage


Left boundaryP

art

icle

pe

rce

nta

ge

0.0

0.3

0.6

0.9

1.2

No

rma

lize

d flo

w ra

te

-2 -1 0 1 2

0

5

10

15

20

25

Particle percentage


Left boundary

Pa

rtic

le p

erc

en

tag

e

0.0

0.3

0.6

0.9

1.2

No

rma

lize

d flo

w ra

te

-2 -1 0 1 2

0

5

10

15

20

25

Particle percentage


Left boundary

Pa

rtic

le p

erc

en

tag

e

0.0

0.3

0.6

0.9

1.2

No

rma

lize

d flo

w ra

te

-2 -1 0 1 2

0

5

10

15

20

25

Particle percentage


Left boundary

Pa

rtic

le p

erc

en

tag

e

0.0

0.3

0.6

0.9

1.2

No

rma

lize

d flo

w ra

te


71

Table 4.2 Summary of transport results as increasing stress ratio for conservative tracers, under horizontal hydraulic gradient of 1×104Pa/m (or 1m/m).

Stress ratio (K)

Average results Left boundary Top boundary Bottom boundary

L (m) tw (hour) L (m) N tw (hour) L (m) N tw (hour) L (m) N tw (hour)

0 (0:0) 5.43 4.06 6.78 6484 5.06 2.26 623 1.55 1.87 1893 1.44

1 (5:5) 5.52 19.54 6.98 6593 24.63 1.83 703 5.88 1.38 1704 5.51

2

10:5 5.39 21.19 6.51 6976 25.34 1.81 548 6.94 1.43 1476 6.87

5:10 6.33 36.87 7.78 6732 45.48 1.61 874 8.07 2.25 1394 13.34

3

15:5 4.76 14.08 6.27 5123 21.63 2.99 617 7.68 2.71 3260 3.41

5:15 2.88 9.08 9.31 771 53.23 1.64 7104 1.12 6.29 1125 29.04

4

20:5 5.23 4.95 6.03 6398 5.73 4.02 1599 2.49 2.10 1003 3.86

5:20 2.44 5.49 10.75 414 45.01 1.53 5732 1.08 3.06 2854 8.55

Note: L is the average travel distance; tw is the mean water residence time; and N represents the number of particles exiting from the corresponding the boundary.


72

Table 4.3 Summary of transport results as increasing stress ratio for conservative tracers, under vertical hydraulic gradient of 1×104Pa/m (or 1m/m).

Stress ratio (K)

Average results Bottom boundary Left boundary Right boundary

L (m) tw (hour) L (m) N tw (hour) L (m) N tw (hour) L (m) N tw (hour)

0 (0:0) 5.78 3.66 6.24 7709 3.92 3.92 812 2.80 1.45 479 0.81

1 (5:5) 5.97 17.95 6.37 7917 19.02 4.10 670 14.13 1.33 413 3.64

2

10:5 6.33 34.09 6.86 7557 36.48 4.64 990 28.50 1.33 453 6.38

5:10 5.65 17.97 6.02 8088 18.99 3.56 485 13.94 0.99 427 3.02

3

15:5 6.69 40.18 7.95 4696 45.61 6.17 3422 40.44 1.96 882 10.20

5:15 3.65 6.00 5.70 4456 11.29 1.37 114 2.57 1.64 4430 0.77

4

20:5 7.20 16.98 9.41 232 32.41 7.49 8145 17.06 2.63 623 10.32

5:20 4.10 4.51 5.55 5692 6.63 2.68 51 8.30 4.57 3257 0.73

Note: L is the average travel distance; tw is the mean water residence time; and N represents the number of particles exiting from the corresponding the boundary.


73

Figure 4.10 Breakthrough curves for non-interacting tracers with increasing stress ratio, under vertical hydraulic gradient of 1×104Pa/m (1m/m).

4.3.3. Hydrodynamic dispersion

(1) A verification example

Since there are no available experimental results for hydrodynamic dispersion in fracture systems, the accuracy and ability of the proposed method was verified by comparison with the theoretical or empirical scale-dependent hydrodynamic dispersion models proposed in literature, without considering the longitudinal dispersion in single fractures. The commonly accepted and widely used scale-dependent hydrodynamic dispersion models are in asymptotic (Eq. (4.27)) or exponential (Eq. (4.28)) forms (Pickens & Grisak, 1981; Yates, 1992; Huang et al., 1996), given by,

mLL DXnnvD ])(1[ (4.27)

mLL DaxvD )]exp(1[ (4.28)

Where n is the characteristic half length, which can be represented

by the mean travel distance corresponding to 2L ; a is a

constant.

Fig. 4.11 shows the evolution of the shapes and sizes of the particle swarm with increasing travel time, without longitudinal dispersion in the individual fractures in the PTFR model. At the initial stage (or when the swarm is close to the injection point), the size of the swarm was small and did not show evident Gaussian spreading behavior, due to the lack of a proper spatial averaging within a short travel time (distance) (Schwartz & Smith, 1988). Without stochastic properties, the estimated hydrodynamic dispersion coefficient fluctuated (Fig. 4.12). Gradually, the size of the swarm expanded with longer travel time (distance), and the

10-1

100

101

102

103

104

105

0.0

0.2

0.4

0.6

0.8

1.0

K=0

K=5:5

K=10:5

K=5:10

K=15:5

K=5:15

K=20:5

K=5:20

c0/c

Time (hour)

c/c

0


74

shape of the swarm became an approximate ellipse during this transport process. This behavior demonstrates that particle movement in the fracture network model can be described by a Gaussian probability distribution. Therefore, Eq. (4.21) was employed to determine the equivalent hydrodynamic dispersion coefficients. Fig. 4.12 shows the calculated equivalent horizontal

hydrodynamic dispersion coefficient xxD , with increasing travel

distance. The discrete values of xxD was well fitted by an

exponential curve, demonstrating that the hydrodynamic dispersion calculated from the DEM model described a trend of a scale-dependent hydrodynamic dispersion coefficient as predicted by the empirical model in Pickens & Grisak (1981).

Figure 4.11 Spreading of reference particles during transport through the fracture network with constant aperture and a side length of 10 m.

Swarm center (4.52, -0.12) after 1000s


-5 -4 -3 -2 -1 0 1 2 3 4 5

-5

-4

-3

-2

-1

0

1

2

3

4

5

Y (

m)

X (m)


-5 -4 -3 -2 -1 0 1 2 3 4 5

-5

-4

-3

-2

-1

0

1

2

3

4

5

Y (

m)

X (m)

-5 -4 -3 -2 -1 0 1 2 3 4 5

-5

-4

-3

-2

-1

0

1

2

3

4

5

Y (

m)

X (m)

Particle

input


75

The calculated equivalent vertical hydrodynamic dispersion

coefficient ( yyD ) with increasing travel distance is plotted in Fig.

4.13. Compared with xxD , yyD did not show significant scale-

dependent behavior. After a period of fluctuation, the values of

yyD tend to be stable, together with xxD .

Figure 4.12 Horizontal hydrodynamic dispersion coefficient

xxD with increasing travel distance.

Figure 4.13 Vertical hydrodynamic dispersion coefficient yyD

with increasing travel distance.

0 1 2 3 4 5 6 7

10-7

10-6

10-5

10-4

Base case

Variance case

Dyy (

m2/s

)

Travel distance (s)Travel distance (m)

0 1 2 3 4 5

10-6

10-5

10-4

Dxx (

m2/s

)

Time (s)Travel distance (m)


76

Figure 4.14 Péclet number with increasing travel distance.

(a) αL/Lf = 1/2 (b) αL/Lf = 1/5

(c) αL/Lf = 1/10 (d) αL/Lf = 1/20

Figure 4.15 Spatial distribution of particle swarm after t=15000s for different local dispersivity correlated with fracture length.

0 1 2 3 4 5 6 7 8 9 10

-5

-4

-3

-2

-1

0

1

2

3

4

5

Y (

m)

Travel distance (m)0 1 2 3 4 5 6 7 8 9 10

-5

-4

-3

-2

-1

0

1

2

3

4

5

Y (

m)

Travel distance (m)

0 1 2 3 4 5 6 7 8 9 10

-5

-4

-3

-2

-1

0

1

2

3

4

5

Y (

m)

Travel distance (m)0 1 2 3 4 5 6 7 8 9 10

-5

-4

-3

-2

-1

0

1

2

3

4

5

Y (

m)

Travel distance (m)

0 1 2 3 4 5 6 7

0

10

20

30

40

50

60

Pe

cle

t n

um

be

r

Travel distance (m)

base case

Variance case


77

Figure 4.16 Equivalent hydrodynamic dispersion coefficient changes with local longitudinal dispersivity correlated with fracture length.

The results for the Péclet number are shown in Fig. 4.14. Compared with Figs. 4.12 and 4.13, even though the hydrodynamic dispersion coefficients reached stable values after certain travel distance, the Péclet number still increased with travel distance. This indicates that the influence of hydrodynamic dispersion on the transport reduces with travel time or distance,

because the horizontal hydrodynamic dispersion coefficient xxD

became stable.

(2) Effect of local hydrodynamic dispersion

In fact, the local dispersivity of individual fractures largely depends on their geometries, and it may vary from different fractures in orders of magnitude. In order to study the scale-dependent local dispersivity, a linear relationship between the local dispersivity and the fracture length was assumed as below,

constl f

L

(4.29)

The local dispersion coefficient varies in different fractures according to its length. By varying this ratio from 1/2, 1/5, 1/10 to 1/20, the spatial distribution of reference particle and the estimated macroscopic hydrodynamic dispersion coefficients are compared in Figs. 4.15 and 4.16, respectively. Compared with the case without considering the local dispersion in single fractures, the particle swarm size and center do not exhibit big differences. Except that the estimated macroscopic hydrodynamic dispersion

coefficients for fL l =1/2 were obviously larger than other cases,

the estimated macroscopic hydrodynamic dispersion coefficients

did not change significantly with increasing value of fL l . Even

0 2000 4000 6000 8000 10000 12000 14000 16000

1E-5

1E-4

Dxx (

m2/s

)

Travel time (s)

L/L

f

1/2

1/5

1/10

1/20

No dispersion


78

so, the estimated macroscopic hydrodynamic dispersion

coefficients for fL l =1/20 did not overlap fully with that only

considering advection in single fractures, compared with the case in Fig. 4.11. It indicates that the local dispersion may play certain roles in a few fractures with longer fracture length. The results presented here is in agreement with the macrodispersion in heterogeneous porous media shown by Beaudoin et al. (2010). The local longitudinal dispersivity increase the longitudinal macrodispersion coefficient, but to a small extent.

(3) Stress effects on macroscopic hydrodynamic dispersion

Fig. 4.17 shows a comparison of shapes and sizes of the particle

swarms with the varying stress ratios at t =60000 s. Under

conditions of isotropic stress (K=5:5) or K=10:5, the swarm of reference particles, moved much more slowly, but showing a Gaussian behavior. They retained the general shape of approximate ellipses during the movement. The reason is that all the fractures underwent normal closure under these stress conditions, which resulted in almost uniformly decreasing fracture apertures and flow rates. For the same travel distance, it needs 5

times more time than that in the condition with K=0. By comparison of the magnitudes of horizontal hydrodynamic dispersion coefficient at the same travel distance, it was found that they decreased by 5 times as well (Fig. 4.18(a)). It can be concluded that the decreasing hydrodynamic dispersion resulted from the decreasing flow rates. This effect could also be demonstrated by the similar longitudinal dispersivity values of 1.2 and 1.3, respectively, before and after stress applied. Therefore, for the fracture network with approximately uniform apertures, fracture aperture plays a minor influence on the longitudinal dispersivity. For the scale-dependent hydrodynamic dispersion coefficients (

xxD and yyD ), they increased steadily before travel distance of 2

m, after which they leveled off to some asymptotic values (Figs.

4.18(a) and (b)). By fitting the discrete values of xxD using

exponential curve, it is found that the behavior of hydrodynamic dispersion could still be described by a scale-dependent hydrodynamic dispersion coefficient in an exponential form (Eq. (4.26)).

After K=15:5, however, the shape of reference particle swarm became more irregular (Figs. 4.17(c) and (d)), induced by the influence of channeling of the fluid flow due to the shear dilations under higher stress ratios. The percentage of fractures with larger aperture than initial aperture of 30μm increased (Fig. 4.19). A large number of particles followed a few big channels, consisting of fractures with larger shear dilation. The values of horizontal

hydrodynamic dispersion coefficient ( xxD ) increased drastically,

and then decreased. The reason for the decreasing equivalent hydrodynamic dispersion was that a part of reference particles following the big channels exited the DFN models. Neretnieks


79

(1983) showed that channeling would make the equivalent hydrodynamic dispersion coefficients increase with the travel

distance, which is in accord with our results. The values of yyD

also increased with increasing travel distance, compared to the

nearly constant values of yyD under the stress-free state (K=0).

Fig. 4.20 shows that the Péclet number changes with travel distance under different stress ratios. When the isotropic stress applied, the Péclet number increased compared with the condition

of K=0, because of the decreasing hydrodynamic dispersion coefficients. With continuously increasing the stress ratio, the Péclet number decreased, in contrast to the increasing hydrodynamic dispersion coefficient with increasing stress ratio. This is in line with the fact that low Péclet number corresponds to high hydrodynamic dispersion effect. To try to explain the stress influences on the hydrodynamic dispersion, the aperture distribution under different stress ratios is shown in Fig. 4.19. The peak of the aperture distribution shifted in the direction of small magnitude of aperture, as stress ratio increased. A large number of fractures continued to close, but the portion of fractures undergoing dilations increased, which is the reason for the channeling in a few dominant flow paths caused by stress.

(a) K=5:5 (b) K=10:5

(c) K=15:5 (d) K=20:5

Figure 4.17 Spreading of reference particles during transport through the fracture network at t=60000s.

-5 -4 -3 -2 -1 0 1 2 3 4 5

-5

-4

-3

-2

-1

0

1

2

3

4

5

Y (

m)

X (m)

Swarm center (-0.85, -1.57)

-5 -4 -3 -2 -1 0 1 2 3 4 5

-5

-4

-3

-2

-1

0

1

2

3

4

5

Y (

m)

X (m)


-5 -4 -3 -2 -1 0 1 2 3 4 5

-5

-4

-3

-2

-1

0

1

2

3

4

5

Y (

m)

X (m)


-5 -4 -3 -2 -1 0 1 2 3 4 5

-5

-4

-3

-2

-1

0

1

2

3

4

5

Y (

m)

X (m)



80

(a) xxD

(b) yyD

Figure 4.18 The hydrodynamic dispersion coefficients changing with increase stress ratio for case considering longitudinal hydrodynamic dispersion in individual fractures.

0 1 2 3 4 5 6

10-7

10-6

10-5

K=0:0

K=5:5

K=10:5

K=15:5

K=20:5

Dyy (

m2/s

)

Travel distance (s) (m)

0 1 2 3 4 5 6

10-6

10-5

10-4

K=0:0

K=5:5

K=10:5

K=15:5

K=20:5

Dxx (

m2/s

)

Travel distance (s) (m)


81

Figure 4.19 Fracture aperture distribution with increase stress ratio. For the initial case without stress applied (K=0:0), the fracture aperture is constant of 30μm.

Figure 4.20 Péclet number changing with increase stress ratio for case considering longitudinal hydrodynamic dispersion in individual fractures.

1.0x10-5

2.0x10-5

3.0x10-5

4.0x10-5

5.0x10-5

0.0

0.1

0.2

0.3

0.4

0.5

0.6

K=5:5

K=10:5

K=15:5

K=20:5F

req

ue

ncy

Aperture (m)

0 1 2 3 4 5 6

0

10

20

30

40

50

60

K=0:0

K=5:5

K=10:5

K=15:5

K=20:5

Pe

cle

t n

um

be

r

Travel distance (m)


82

4.3.4. Matrix diffusion

The simulations of advection were done under a hydraulic gradient of 104 Pa/m (1 m/m). However, in reality the hydraulic gradient at a site for radioactive waste repositories, such as in Sweden, is most likely several orders of magnitude smaller. According to Eq. (3.12),

the flow rate Q in a fracture is just scaled by the hydraulic

gradient dxdp . In this way, the flow rate can be simply decreased

by a factor of 1000 in each fracture to obtain new flow field under a much decreased hydraulic gradient of 10Pa/m, based on the previous case of 104Pa/m. Then the solute transport simulations considering matrix diffusion were performed for the lower hydraulic gradient condition. Only the non-sorbing species were

considered in this study, so dK =0. Solute particle sorption on the

inner surfaces of the rock matrix can be easily added by changing

the material property group, MPG , defined by Eq. (4.14). With a

matrix porosity of =0.316% and an effective pore diffusion

coefficient of 10-11 m2/s, the MPG has an approximate constant

value of 10-8 ms-0.5. Thus the property of QAq for each particle

path j would determine how much increase in particle residence

time is caused by matrix diffusion (Eq. (4.15b)).

Table 4.4 summarizes the average values characterizing the particle tracking results for two different flow directions, vertical and horizontal. The breakthrough curves, shifting with increasing stress ratio, are shown in Figs. 4.21 and 4.22. From these results, the following features were observed.

Table 4.4 Summary of transport results considering matrix diffusion with increasing stress ratio, under hydraulic gradient of 10Pa/m (or 0.001m/m).

Stress ratio (K)

horizontal hydraulic gradient vertical hydraulic gradient

Aq/Q (s/m) t0.5 (year) t0.5/tw Aq/Q (s/m) t0.5 (year) t0.5/tw

0 9.67×1011

4.76 10.3 8.82×1011

3.87 9.26

1 8.90×1012

368 265 8.10×1012

257 126

2

10:5 1.13×1013

596 246 1.83×1013

1310 336

5:10 1.88×1013

1600 380 9.23×1012

358 175

3

15:5 7.83×1012

412 256 2.24×1013

2030 442

5:15 4.58×1012

387 373 3.02×1012

115 168

4

20:5 2.15×1012

104 184 7.41×1012

444 229

5:20 2.58×1012

191 305 2.01×1012

69.4 135


83

Figure 4.21 Breakthrough curves for interacting tracers with increasing stress ratio, under horizontal hydraulic gradient of 10 Pa/m (0.001 m/m).

Figure 4.22 Breakthrough curves for interacting tracers with increasing stress ratio, under vertical hydraulic gradient of 10 Pa/m (0.001 m/m).

101

103

105

107

109

1011

0.0

0.2

0.4

0.6

0.8

1.0

K=0

K=5:5

K=10:5

K=5:10

K=15:5

K=5:15

K=20:5

K=5:20

c0/c

Time (hour)

c/c

0

101

103

105

107

109

1011

0.0

0.2

0.4

0.6

0.8

1.0

K=0

K=5:5

K=10:5

K=5:10

K=15:5

K=5:15

K=20:5

K=5:20

c0/c

Time (hour)

c/c

0


84

The relations between the residence time 5.0t (or QAq ) and

stress conditions were similar to that of advection cases. The

residence time 5.0t (or QAq ) increased during stress ratio K

varying from 0 to 10:5 or 5:10 at first, after which it decreased with

further increasing K.

Larger values of wtt 5.0 indicate that matrix diffusion became

the dominated transport mechanism under this lower hydraulic pressure condition. From Table 4.4, it could also be observed that matrix diffusion had a more significant influence on interacting tracers under condition that the main flow direction was perpendicular to the direction of larger boundary stresses, in which direction shear dilation occured.

The trends of the breakthrough curve shifting with increasing stress ratio were similar between interacting (Figs. 4.21 and 4.22) and conservative tracers (Figs. 4.8 and 4.10). However, the shapes of breakthrough curves became less steep and with much longer tails. A small fraction of particles, meandering

through more tortuous flow paths with extremely large QAq ,

resided within the matrix for a much longer time.

4.4. Summary

The basic numerical DEM models and algorithms of simulating the stress/deformation and fluid flow in fractured rocks were presented, followed by a transport code PTFR developed to study the stress effects on solute transport in same fractured rocks models. Based on the results in Chapter 3, the macroscopic behaviors of advection, hydrodynamic dispersion and matrix diffusion in a fracture network and impacts of stresses were analyzed. The stress has an important effect on the fluid flow and solute transport through fracture network, through its influences on the fracture aperture change and matrix deformation. For application of the present approach in practice, the proper and realistic stress and hydraulic boundary conditions should be used based on in-situ investigations. As a preliminary attempt to study the stress effects on solute transport in fractured rocks, some limitations of current model and results are presented in next chapter.


85

5. DISCUSSIONS

5.1. Outstanding issues of the analytical model

Both in the analytical model and in the particle tracking algorithms, the groundwater velocity and solute concentration are assumed to be uniform across the fracture section, following the parallel plate model for a fracture with a constant aperture. For fractures with variable apertures, the above assumption may not be valid. However, for fractures with aperture of micrometer scale, under relatively low hydraulic pressure gradient and steady flow conditions and at the time scale of performance/safety assessments for underground radioactive waste repositories, this assumption is still acceptable for such conceptual analysis for evaluating the general behaviors of rock fractures and fracture networks.

The present study is based on some empirical mechanical models for stress/displacement behaviors of rock fractures, so its validity largely depends on the range of validity of those empirical models and associated parameters. Actually, the main contribution of this study is to highlight the importance of the stress effects on the coupled flow-transport processes in fractured crystalline rocks. Any other mechanical models can be used as long as they can consider the changes of fracture aperture and matrix porosity under stress, explicitly in compact mathematical function forms, to meet the requirements of special rocks or sites, and the resultant equations can be integrated for closed-form solution.

There are a large number of material parameters and empirical constants in the present models. How to obtain reasonable values for them from laboratory or in-situ experiments remains to be a difficult problem and no attempt has been made to test coupled stress-flow-transport processes in a single rock fractures or fracture networks with quantitative representations of transport mechanisms, variables and parameters, so far, due to experimental difficulty. The uncertainties in measuring the material properties have an important influence on the final assessments of radioactive waste repositories, and these properties and parameters always vary among different sites.

The above outstanding issues also exist in the 2D DEM models for coupled stress-flow-transport processes in fracture networks.

For the transport models used for deriving the analytical solutions, adsorption on the walls of fracture and within the matrix pores, and the radioactive decay were not included. More details for these retardation mechanisms can be seen in literature (e.g. Tang et al., 1981; Sudicky & Frind, 1984; Cormenzana, 2000; Sun & Buschec, 2003). The basic mechanical and flow models developed in this thesis can be directly coupled with those retardation mechanisms without major mathematical difficulties. The present model can also be further extended to a system with multiply parallel fractures or under other initial and boundary conditions, with the known analytical transport solutions (Sudicky & Frind, 1982; West et al., 2004).


86

5.2. Three damaged zones of fracture surfaces undergoing shear

Fig 5.1 shows a schematic view of three damaged zones of fracture skin layers during shear, with various effects on solute transport processes, according to the gouge generation and microcrack distribution.

(1) Wear zone: It represents the gouge generation zone immediately adjacent to the void space of the fracture. The free abrasive gouge particles are most likely to be generated due to the damage of the contacted or interlocked asperities from two opposite surfaces of a fracture. The colloid-like particles are generated from adhesion wear and crushing process in this zone. Note that in reality the shape of the gouge particles may be more irregular instead of circular disks. Considering such particle shapes will increase the computational complexity (not theoretical difficulty) considerably. On the other hand, for conceptual simulation such simplification is still useful.

The abrasive and immobile colloid-like particles increase the retardation coefficients by providing extra sorption surfaces for solutes. The mobile colloid-like particles can transport the adsorbed solutes with the flowing fluid, decreasing the potential for matrix diffusion and sorption on the matrix pores, and in turn the retardation coefficients. However, the mobile colloid-like particles would eventually be flushed away, but the retardation by immobile colloid-like particles may persist.

(2) The severely damaged zone: It represents the region where the intensive microcracks develop due to stress concentrations. The conceptual model in this study shows that at least over 50% microcracks created in this zone, in which they are assumed to be open microcracks connected with the fracture surfaces. The microcracks in this zone are so close to the fracture surfaces so that they provide much more adsorption surfaces.

Figure 5.1 A schematic view of the wear and damaged zones in a sheared fracture. The possible wear occurs at the contacted asperities along the fracture surfaces. In the severely damaged zones located closer to the fracture surfaces, microcracks were intensively developed, so the solutes could directly reach their surfaces. In the damaged zones slightly further away from fracture surfaces, microcracks were generated, but the solutes have to reach there by molecular diffusion.

Wear zone

Severely damaged zone

Damaged zone


87

(3) The damaged zone: It indicates the region where the microcracks still exist but less densely. The microcracks in this zone can enhance the porosity or inner surfaces of the rock matrix, and the solutes may reach these microcracks only through the matrix diffusion. Beyond this zone, the rock remains largely undisturbed and not influenced by shear.

5.3. Influence of hydraulic gradient and boundary conditions

To help discussing the importance of matrix diffusion, Fig. 5.2 compares two sets of breakthrough curves, corresponding to conservative (without matrix diffusion) and interacting (with matrix diffusion) cases, under hydraulic gradients of 1×104 Pa/m and 10 Pa/m, without stress, respectively. For the high pressure gradient case, the two breakthrough curves overlap each other

below the ratio of 0cc =0.8, indicating that about 80% particles

were not influenced by matrix diffusion under this higher hydraulic gradient. For the case when the hydraulic gradient decreased to 10 Pa/m, the two breakthrough curves for conservative and interacting tracers differ considerably from each other for the

whole time period. The ratio of wtt 5.0 is 19.8 (Table 4.4), which

means matrix diffusion has become an important factor for particle transport (Neretnieks, 1980; Maloszewski & Zuber, 1993; Maloszewski et al., 1999; SR-Can, 2006). Another finding is that matrix diffusion makes the breakthrough curves with much longer tails, indicating that a small number of particles might stay in the micropores of rock matrix for an extremely long time. For longer travel distances, the impact of matrix diffusion becomes

increasingly stronger and the ratio wtt 5.0 increases drastically as

estimated by Eq. (4.15).

In the previous sections, a region where the water and solutes may exit from the sides that are aligned in the direction of the hydraulic gradient was modeled. In Fig. 5.3, comparisons are made between the case with closed lateral boundaries and that with open lateral boundaries. The differences are clearly seen, especially when the main flow direction is perpendicular to the direction of larger boundary stresses with significant effects of shear dilation. The particles that can exit through open lateral sides on average have shorter travel distances than those that exit through the main outlet boundary. This causes the very early breakthrough of some particles for the case with open lateral sides, which is the first difference.

Under some stress conditions, the shear-dilation causes preferable flow and transport pathways to form. Then groundwater and solutes may then move through different pathways than what the overall gradient suggests. This results in the particles exiting perpendicular to the main gradient direction. In this case, the difference between two boundary conditions is significant. For

example, when K=20:5 under vertical hydraulic gradient, the breakthrough curve for the case with closed lateral sides lie at the right side of that with open lateral sides with a large distance. This


88

indicates that particles take much longer time to exit through the bottom boundary. The whole exercise demonstrates the importance of the choice of boundary conditions for modeling stress-flow-transport problems.

Figure 5.2 Comparison of breakthrough curves between non-interacting tracers and interacting tracers under different hydraulic gradient.

Fig 5.3 Comparison of breakthrough curves between the case with permeable lateral boundaries and the case with impermeable lateral boundaries under different stress conditions.

10-2

10-1

100

101

102

103

0.0

0.2

0.4

0.6

0.8

1.0

K=0

K=20:5

K=0

K=20:5

K=20:5

K=20:5

c0/c

Time (hour)

c/c

0

10-2

100

102

104

106

108

1010

0.0

0.2

0.4

0.6

0.8

1.0

Hydraulic gradient

10MPa/m

1E104MPa/m

Noninteracting tracers

Interacting tracers

c0/c

Time (hour)

c/c

0

10 Pa/m

104 Pa/m


89

5.4. Comparison with SOLFRAC (Bodin et al., 2007)

Delay and Bodin (2001) and Bodin et al. (2007) developed a software called SOLFRAC to simulate solute transport in fracture networks by particle tracking method, but they used an approximate methodology of computing the residence time in single fractures including advection and dispersion. They assumed that the residence time distribution was lognormal for Péclet

number larger than 10 ( 10/1fLf l ), with a mean vl f and

a variance 32 2 vlD ff , and the residence time for a particle

through the single fractures was determined by (Delay and Bodin, 2001),

)1ln(

1lnexp 22

22

Zt (5.1)

where Z is a Gaussian random number of zero mean and unit variance. Under this case, the ratio of output of material flux over the input material flux, or cumulative distribution function of

residence time )(tF is

)1ln(2

)1ln()ln(

2

1)(

22

22

tErfctF (5.2)

For the fractures where Péclet numbers was less than 10, the assumption of a lognormal travel time distribution in these segments may yield inaccurate results, so they proposed an empirical correction to preserve accuracy of the particle tracking method (Bodin et al., 2007)).

Figure 5.4 Comparison of cumulative distribution of residence time considering dispersion between PTFR and SOLFRAC.

100 1000 10000

0.0

0.2

0.4

0.6

0.8

1.0

L/L

f=1/2 Eq. (15)

L/L

f=1/10 Eq. (15)

L/L

f=1/2 Eq. (13)

L/L

f=1/10 Eq. (13)

F(t

)

Travel time (s)

(5.2)

(5.2)

(4.11) (4.11)


90

Fig. 5.4 shows the cumulative distribution of residence time between the method used in this thesis (Eq. (4.11)) and SOLFRAC

(Eq. (5.2)) under different ratio of fLf l . The cumulative

distributions of residence time obtained from both methods are

the same for 10/1fLf l , but there are some slight differences

when 10/12/1 fLf l . This is in agreement with the

statement of Péclet numbers in Delay and Bodin (2001) and Bodin et al. (2007). However, the method proposed in present study can be applicable for any cases without corrections, although the price to be paid is that Eq. (4.9) has to be solved.

5.5. Comparison with other DECOVALEX teams

Two other teams also worked for Task C of DECOVALEX-2011 project, using either DFN (discrete fracture network) or EC (equivalent continuum) models to simulate flow and transport processes in fractured rocks. The research team from Imperial College London (IC) carried out the simulations using DFN code NAPSAC, based on a direct representation of the discrete fractures making up the flow-conducting DFN model as defined in Task C. The University of Liberec Czech (TUL) team of Czech Republic used a finite volume code FLOW123D to simulate flow and transport in fractured rocks, based on a system of combining 3D continuum, 2D discrete fracture network and a network of 1D fracture intersection. Both NAPSAC and FLOW123D codes evaluated the stress and displacements for each fracture empirically and independently, by simple stress (force) projection without solution of the equations of motion of the block system defined in Task C. More details of the methodologies used by various teams can be found in the DECOVALEX2011 Task C report and a joint paper (Paper VI).

The 2D BMT model of Task C, DECOVALEX-2011, totally contained 7797 fractures in a square region with side length of 20 meters, and with the correlation between fracture length and aperture. Here, only the comparisons of advection results between the three teams are presented. When K=0 (no stress applied), the breakthrough curves given by each team were quite close, whatever the flow direction or boundary conditions were adopted (see for example, Fig. 5.5(a)). This demonstrated that the method of stress-flow-transport simulations developed in this thesis was supported by different teams using various approaches when stress effects were excluded.

When considering the stress effects, IC and the models developed in this thesis (represented as KTH in the figures afterwards) predicted a general trend of shift of breakthrough curves in the longer time direction with increasing stress ratios. In contrast, the set of results TUL_1 showed a significant backward shift of breakthrough curves, when K=25:5 for horizontal flow (Fig. 5.5). Note that TUL_1 and TUL_2 represent two sets of results with assumed high and low shear stiffness values of fractures after failure. Similarly, both IC and KTH predicted the increasing


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average residence time with increasing stress ratios, but TUL predicted decreased average residence time decreased when K=25:5 for horizontal flow. It was demonstrated that the stresses play an important role on the solute transport in complex fractured rocks, and that the constitutive model of fractures and the proper stress calculation methods could be the possible reasons for the divergences among three teams‘ results. Fig. 5.6 shows an example of the stress concentrations in a small area of the KTH model when K=5:5, which cannot be considered in IC and TUL models using stress (force) projection.

Because a few small fractures located in the boundary area were removed in KTH model, the possible paths of particle exiting along the lateral boundaries decreased. Therefore, the beginning parts of breakthrough curves obtained by KTH were slightly lower than that of breakthrough curves obtained by IC and TUL (Fig. 5.5).

(a) K=0 (b) K=5:5

(c) K=15:5 (d) K=25:5

Figure 5.5 Comparison of breakthrough curves for conservative tracers exiting from all the three outlet boundaries with increasing stress ratio, under horizontal hydraulic gradient of 104 Pa/m.

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Figure 5.6 Stress field of a small zone in Tacks C model obtained by UDEC under stress ratio of K=5:5.


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6. CONCLUDING REMARKS AND FUTURE STUDY

6.1. Main conclusions and scientific achievements

In the previous chapters, the approaches and results of research on stress effects on solute transport through individual fractures and fractured rocks are presented, respectively. Although the solute transport process in fracture networks has been studied for a long time with many publications, the main novelty of this thesis is to investigate the tress effects on the solute transport behaviors systematically by analytical and numerical methods, for the first time. The results demonstrate that the stresses may have a significant effect on the fluid flow field and the solute transport processes, which is an important issue that needs to be taken into account in performance and safety assessments of radioactive waste repositories in fractured crystalline rocks. The main conclusions are summarized below.

(1) For individual fractures, the stress may change the fracture apertures and matrix porosity or damage the fracture surfaces and skin layers, depending on the stress conditions. For the case without significant surface damage, an analytical model as developed in this thesis can be used for conceptual understanding, and it shows that fracture‘s normal closure decreases the solute concentrations both along the fracture and into the rock matrix. However, the shear induced dilation increases the corresponding solute concentrations. With decreasing matrix porosity under compression, the concentrations both along the fracture and into the rock matrix increase.

(2) If there is serious surface damage occurring along fracture skin layers, the newly generated gouge materials and microcracks in the damaged layers provide more sorption surface area. A solute retardation model for the fractures undergoing shear was derived, considering the surface damage mechanisms of abrasion, microcracking, adhesion wear and gouge particle crushing. Simulated by a particle mechanics model, the effects of fracture surface damage on the solute retardation coefficient can be estimated numerically for deepening our understanding of the damage effects on transport.

(3) A code called PTFR was developed to simulate the solute transport in 2D irregular fracture systems using the particle tracking method, based on the coupled stress-flow results from a DEM model. By tracking the particle migration following the flowing fluid (advection) the solute travel paths can be determined, and then the hydrodynamic dispersion and matrix diffusion were considered using a statistic method (rejection method). The algorithm of hydrodynamic dispersion is a new development, but the methods used for modeling advection and matrix diffusion are the classical methods from the literature.

(4) The main effect of stress is through its influence on fracture aperture change that changes fluid flow velocity in turn. This change will then affect the travel path and residence time of solute particles, no matter what retardation mechanisms are considered.


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The influence of matrix diffusion is more significant when the hydraulic gradient is small and/or travel distance is longer, and its influence will increase with decreasing hydraulic gradient. Hydrostatic compression stress decreases the magnitudes of hydrodynamic dispersion coefficients, but the magnitudes of hydrodynamic dispersion coefficients may not reach a constant value in the same travel distance under conditions of significant channeling.

6.2. Recommendations for future study

Although this thesis provides some insights for understanding the stress effects on solute transport in fractured crystalline rock, it employed some assumptions to simplify the problems, thus opened some new windows for continued improvements in future studies.

(1) The 2D model was adopted mainly for its simplicity since our work is basically generic in nature, due to the fact that no reliable experimental investigation under laboratory or field conditions can be made at present with adequate confidence for testing fractured rocks. The laboratory or in-situ experiments and 3D modeling are strongly recommended, because they reflect the real situations.

(2) The isolated fractures and fracture dead-ends cannot be considered at present, due to current limitations of DEM methods. These fractures will not influence the stress/deformation and fluid flow simulations, but the specific area of the surfaces of such isolated fractures and dead-ends in rock matrix could be important for sorption, matrix diffusion and possible chemical reactions. Therefore, proper representations of their effects are suggested to be included in the future solute transport simulations.

(3) The particle mechanics model can be developed in the future to more realistically and efficiently investigate the effects of fracture surface damage on the other transport mechanisms, such as advection, hydrodynamic dispersion and matrix diffusion, by considering different particle shapes, size distribution and contact models. In addition, the effects of fracture surface damage on fluid flow through single fractures should be investigated first.

(4) The hydraulic pressure may significantly influence the mechanical deformation process in rock matrix, so it should be included in the future mechanical (constitutive) models. Pressure solution may also change the fracture aperture and induce some gouge particles, so this effect should also be studied in future.

(5) Based on the methodologies of solute transport simulation by particle tracking methods developed in this thesis, more solute migration mechanisms such as decay and diffusion into stagnant water and into a matrix series could be included by similar algorithms considering hydrodynamic dispersion and matrix diffusion.

(6) In reality, most of fractured media may or may not always be saturated, so the partially saturated conditions should also be investigated.


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446885/FULLTEXT01.pdf · Stress effects on solute transport in fractured rocks iii ABSTRACT The effect of in-situ or redistributed stress on solute transport in fractured rocks is