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Transport in Porous Media ISSN 0169-3913Volume 87Number 2 Transp Porous Med (2011)87:503-523DOI 10.1007/s11242-010-9698-1

Shear-Induced Flow Channels in a SingleRock Fracture and Their Effect on SoluteTransport

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Transp Porous Med (2011) 87:503–523DOI 10.1007/s11242-010-9698-1

Shear-Induced Flow Channels in a Single Rock Fractureand Their Effect on Solute Transport

Victor Vilarrasa · Tomofumi Koyama ·Ivars Neretnieks · Lanru Jing

Received: 18 November 2008 / Accepted: 6 December 2010 / Published online: 22 December 2010© Springer Science+Business Media B.V. 2010

Abstract The effect of mechanical shearing on fluid flow anisotropy and solute transportin rough rock fractures was investigated by numerical modeling. Two facing surfaces of arock fracture of 194 mm×194 mm in size were laser scanned to generate their respectivedigital profiles. Fluid flow through the fracture was simulated using a finite element code thatsolves the Reynolds equation, while incremental relative movement of the upper surface wasmaintained numerically to simulate a shearing process without normal loading. The motionof solute particles in a rough fracture undergoing shear was studied using a particle trackingcode. We found that shearing introduces anisotropy in fracture transmissivity, with a greatlyincreased flow rate and particle travel velocity in the direction perpendicular to the shearingdirection. Shear-induced channels yield a transport behavior in which advection dominatesin the direction parallel with shear and dispersion dominates in the direction perpendicular toshear. The shear-induced flow channels not only increase the flow connectivity, but also thetransport connectivity in the direction perpendicular to shear. This finding has an important

V. Vilarrasa (B)GHS, Institute of Environmental Assessment and Water Research (IDAEA),Spanish National Research Council (CSIC), 08034 Barcelona, Spaine-mail: victor.vilarrasa@upc.edu

V. VilarrasaGHS, Department of Geotechnical Engineering and Geosciences,Technical University of Catalonia (UPC-Barcelona Tech),08034 Barcelona, Spain

T. KoyamaDepartment of Urban Management, Graduate School of Engineering, Kyoto University, Kyoto, Japan

I. NeretnieksDepartment of Chemical Engineering, Royal Institute of Technology (KTH), Stockholm, Sweden

L. JingDivision of Engineering Geology and Geophysics, Royal Institute of Technology (KTH),Stockholm, Sweden

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impact on the interpretation of the results of coupled hydromechanical and tracer transportexperiments for measurements of hydraulic and transport properties of rock fractures.

Keywords Rock fracture · Shear displacement · Coupled hydro-mechanical behavior ·Transport connectivity · Particle tracking

1 Introduction

Underground excavations in fractured rock masses modify the stress field and induce defor-mations. This affects the hydraulic parameters of fractures, which alters flow and transport.While an increase in normal loading closes fractures, reducing its transmissivity, shear dis-placement tends to not only open fractures due to dilation, but also, more importantly, gen-erate continuous channels that increase its transmissivity (Barton et al. 1985; Rutqvist andStephansson 2003). This is of crucial importance for isolation of waste (e.g., nuclear waste)in low-permeability fractured media (Malkovsky and Pek 2009).

Coupled stress-flow processes in rock fractures play a significant role in understandingthe behavior of fluid flow and solute transport in fractured rocks. Coupled shear-flow testshave been commonly performed under either parallel or radial flow conditions, in which theflow direction is either parallel with the shear direction (Olsson and Barton 2001; Lee andCho 2002; Mitani et al. 2005), or radial from the center of the fracture specimen (Gentieret al. 1996; Esaki et al. 1999; Hans and Boulon 2003). This is mainly due to limitations setby experimental difficulties. While the fracture can be successfully sealed for flow experi-ments in the direction parallel with shear, the seal is damaged for flow experiments in thedirection perpendicular to shear. Unfortunately, direct measurements and visualizations ofaperture, transmissivity, and solute transport paths inside the tested rock fracture samples aregenerally not possible, especially during shear. Numerical simulations may help us to over-come some of these drawbacks. Despite these limitations, experiments show that flow rateand hydraulic conductivity of fractures increase with increasing shear displacement becauseof shear dilation, regardless of the shear direction (Kim et al. 2004). In addition, a moreheterogeneous, anisotropic, and conductive flow behavior has been observed in the directionperpendicular to the shear displacement, both in the laboratory with shear displacements upto 2 mm (Yeo et al. 1998), and numerically with shear displacements up to 20 mm (Koyamaet al. 2004, 2006). This finding is important for the fundamental understanding of coupledhydromechanical behavior of rough rock fractures. It also has great relevance to the enhance-ment and validation of numerical models and codes for coupled hydromechanical processesin fractured rocks, especially discrete element methods (Hart and St John 1986; Kim et al.1999; ITASCA Consulting Group 2000, 2003; Jing et al. 2001) and discrete fracture network(DFN) models (Herbert 1996; Dershowitz et al. 1998), in which fluid flow in fractures playsa central role in the processes of stress/deformation and heat/mass transport.

Coupled hydromechanical processes and, in particular, changes in the flow field due toshearing will affect solute transport. Solute transport can be studied using continuous timerandom walk (Berkowitz and Scher 1997; Noetinger et al. 2001; Cortis and Birkholzer 2008),multi-rate mass transfer (Haggerty and Gorelick 1995; Carrera et al. 1998), and the advec-tion–dispersion equation (ADE) (Neuman 1984). The ADE can fail to simulate correctlythe tailing of solute transport because of the non-Fickian behavior of dispersive transport(Carrera 1993; Berkowitz et al. 2006). However, this drawback can be overcome by knowingmicroscopic variations of water velocity (Molz et al. 1983; Güven et al. 1985). The ADEallows us to gain knowledge about the relative importance between advection and dispersion

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through the Péclet number. This equation has been widely used to interpret transport in roughrock fractures (e.g., Neretnieks et al. 1982; Tsang et al. 1988; Thompson 1991; Thompsonand Brown 1991; Kennedy and Lennox 1995; Wendland and Himmelsbach 2002).

The aperture variation within a rough rock fracture results in a heterogeneous fluid velocityfield that influences solute transport. The aperture field of fractures can be artificially gener-ated using geostatistical or fractal methods (Brown 1987; Moreno et al. 1990; Wendland andHimmelsbach 2002; Jeong and Song 2005; Goloviznin et al. 2007) or measured by scanningreal fractures with a laser (Lee and Cho 2002; Fardin 2003; Sharifzadeh et al. 2004; Koyamaet al. 2006). The latter permits the use of realistic aperture fields that depict the rough natureof fractures. Roughness in fractures forces fluid to follow the least resistance paths. Thisphenomenon is known as the channeling effect (Bodin et al. 2003). Channeling is linked toconnectivity. Since connectivity is a process-dependent concept (Knudby and Carrera 2005),flow and transport connectivity have to be evaluated separately. Hence, the flow field hasto be obtained, and then solute transport can be solved using a particle tracking method(e.g., Moreno et al. 1988, 1990; Grindrod and Lee 1997; Jeong and Song 2005).

Overall, hydromechanical couplings and solute transport in rough rock factures have beenwidely investigated independently. For instance, the effect that variable aperture during ashearing process has on fluid flow has been examinated by Plouraboue et al. (2000) andAuradou et al. (2005). Similarly, solute transport has been studied in rough rock fractures(e.g., Cvetkovic et al. 1999; Detwiler et al. 2000). However, the effects of shear deformationon solute transport have not been investigated.

This article aims to investigate the effects of shear on both the fluid flow and solutetransport processes in rock fractures through numerical modeling. We use a real rock frac-ture geometry for modeling the flow (through a 2D finite element method (FEM) code) andtransport behavior (through a particle tracking code), with a numerically simulated shear-ing process without normal loading. This is important for understanding the fundamentalbehavior of fractured rocks in terms of coupled hydromechanical processes, especially whenapplications for simulating radionuclide transport processes for underground nuclear wasterepositories are considered.

2 Aperture and Transmissivity Evolution of a Fracture During Shearing

We used the rock rough fracture from Fardin (2003) and the numerically simulated fluid flowduring a shearing process from Koyama et al. (2006). Since details are available in thesestudies, we refer the interested reader to them for specific information.

Rock fractures typically have some initial anisotropy in aperture/transmissivity fieldsbefore shear (US National Research Council 1996; Adler and Thovert 1999). Shearinginduces additional anisotropy, which is controlled by the surface roughness characteristics,shearing history, and direction of shear. High-strength concrete fracture replicas producedfrom a natural rough rock fracture were used in this study. The lower blocks of these replicashave a cross-sectional area of 280 mm×280 mm, and their upper blocks have a cross-sec-tional area of 194 mm×194 mm. The upper surface is smaller than the lower one to maintaina constant nominal contact area between the two fracture surfaces during shearing (Koyamaet al. 2006). The digitized aperture field was divided into 6,400 (80×80) square elements ofan edge length of 2.425 mm. The same mesh resolution was kept during the simulated shearprocess.

The geometrical aperture of the fracture specimen is defined as the separation distancebetween its two surfaces in the direction perpendicular to the mean plane of the fracture. The

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aperture was determined numerically at a much larger number of points than the numberof FEM elements used for flow simulations. The aperture fields were created for the initialstate (before shear) and for each shear displacement interval. The opposing points on thetwo surfaces were changed with the shift of the upper surface, and apertures values at all themeasuring points were recalculated. To simulate a translational shearing process, the digitalreplica of the upper surface was continuously displaced tangentially with a 1 mm interval, upto a total shear displacement of 20 mm.

The aperture fields (maps) for each interval of shearing (Fig. 1) were created using thefollowing technique. The digital images of the upper and lower surfaces of the fracture replicawere superimposed with some contact points (at least three) to determine the initial positionsof the fracture surfaces at the start of the shear simulation. Then, shear displacement is appliedwithout normal loading. The upper surface was relocated numerically until new contact points(at least three) were found, avoiding penetration between the upper and the lower surfaces,for each shear displacement increment. This numerical simulation of the shear process wascompared with a mechanical direct shear test on the same sample under a normal loading of1 MPa. The evolution of the numerical mean aperture is in good agreement with the dilationof the laboratory test (Koyama et al. 2006). Crushing of the weakest fracture asperities inthe contact areas due to shearing and gouge formation are not considered. These, togetherwith a high normal stress, could reduce the transmissivity of the fracture (Chen et al. 2000).However, the important physical processes affecting solute transport are well represented.

After the aperture field is known, the transmissivity, T , of each FEM element [L2T−1] canbe calculated, assuming that the cubic law is valid locally within the elements (Witherspoonet al. 1980; Jeong and Song 2005; Crandall et al. 2010), from

T = ρgb3

12μ, (1)

where ρ is the fluid density [ML−3], g is the gravitational acceleration [LT−2], b is themean local fracture aperture (which may change with shear displacement) [L], and μ isthe fluid viscosity [ML−1T−1]. The density and viscosity of water at 20◦C were taken asρ = 998.2 kg/m3 and μ= 1.002×10−3 Pa s, and the gravitational acceleration is taken asg = 9.807 m/s2. The local transmissivity of each element is assumed to be isotropic.

Note that the local cubic law is not assumed valid for the entire fracture, but for elementswith small local relative roughness and with limited inertial effects (Oron and Berkowitz1998). Furthermore, the fracture apertures of this study are much larger than those found todiffer from the cubic law (Sisavath et al. 2003).

3 Numerical Methods for Simulations of Fluid Flow and Solute Transport

3.1 Numerical Method for Fluid Flow Simulations

The Reynolds equation can be used instead of the full Navier–Stokes equation to describeflow in fractures when viscous forces dominate inertial forces. This occurs for low flow ratesand a fracture surface geometry that does not vary too abruptly (Yeo et al. 1998)

∇ · (T ∇h) + Q = 0, (2)

where h is the hydraulic head [L], and Q is the source/sink term [LT−1] (representing,e.g., fluid withdrawal or injection by pumping and is positive when fluid is flowing into thefracture). Equation 2 is solved applying the FEM method.

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The validity of the Reynolds equation is a frequently asked question when fluid flow inrock fractures is concerned. Some studies solve the Navier–Stokes equations using FEM ina real fracture geometry, for flow (Zimmerman et al. 2004) and transport (Cardenas et al.2007). However, the Reynolds equation is commonly used for flow in fractures due to itssimplicity (Zimmerman et al. 1991; Zimmerman and Bodvarsson 1996; Prat et al. 2002). Itis typically valid for cases with slowly varying smooth roughness, low flow rate, and smallReynolds number (Yeo et al. 1998). The numerical models chosen here comply with theseassumptions.

Two types of boundary conditions were considered for simulating fluid flow through thefracture: unidirectional flow in the x-direction (parallel with the shear direction) and in they-direction (perpendicular to the shear direction). Fixed hydraulic heads were specified atthe inlet and outlet boundaries in the assumed flow direction as 1.0 and 0.0 m, respectively.The other two boundaries were specified with no-flux conditions.

3.2 Numerical Method for Solute Transport Simulations

Mass balance of the solute leads to the advection–dispersion equation (ADE) (Bear 1972)

∂c

∂t= −V · ∇c + ∇ · (D∇c). (3)

where c is the volumetric concentration [ML−3], t is time [T], V = q/φ is the actual velocity[LT−1], q is the volumetric flux [LT−1], φ is porosity, which can be set equal to 1 for flowin a single fracture (Wendland and Himmelsbach 2002) [–], and D is the dispersion tensor[L2T−1] which is a symmetric positive semi-definite tensor.

The volumetric flux in each element is given by Darcy’s law

q = −ρgb2

12μ∇h. (4)

The solution of Eq. 2 gives the hydraulic head in all the fractures. Then, the velocity fieldis calculated using Eq. 4. Once the velocity field is known, transport of particles is calculatedusing a particle tracking technique. Particles are initially placed at one edge of a FEM ele-ment. They follow the velocity vector of each element. Thus, the new position of a particleis given by the direction of the velocity vector of this element (see Fig. 2). The travel timeinside each element is given by the magnitude of the velocity vector of the element as

�t ij =

∣∣∣xi+1

j − xij

∣∣∣

|V| (5)

where �t ij is the travel time of particle j in the tracking step i [T],

∣∣∣xi+1

j − xij

∣∣∣ is the travel

distance of particle j inside the element corresponding to the tracking step i [L], and |V| isthe magnitude of the velocity vector of the element corresponding to the tracking step i forparticle j [LT−1].

The total particle travel time was calculated as the sum of the residence time of all theelements passing through by each particle as

t j =m

∑

i=1

�t ij , (6)

where t j is the travel time of particle j [T], and m is the number of tracking steps forparticle j .

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Fig. 2 An example of particletracking using a grid of 3 × 3elements

flow velocity

particle pathparticle path

1 2 3

The random movement of the solute particles due to diffusion and retardation mecha-nisms, such as sorption, matrix diffusion, or decay, were not taken into account. However,mechanical dispersion appears naturally because the local fluctuations of water velocitieswith respect to its mean value are known in detail.

The breakthrough curves were analyzed by fitting them to a 1D analytic solution of theADE, which reads (De Marsily 1986)

c (x, t)

C0= 1

2

[

erfc

(x − V t

2√

DLt

)

+ exp

(V x

DL

)

erfc

(x + V t

2√

DLt

)]

, (7)

where V is the effective tracer velocity [LT−1], DL is the longitudinal dispersion coefficient[L2T−1], and C0 is the inlet concentration [ML−3].

Equation 7 applies for a smooth fracture of constant aperture with uniform fluid velocityand no dispersion. When applying this equation to a rough rock fracture, the fluctuations in thevelocity field with respect to the effective tracer velocity V , which is constant, are accountedfor by DL. Both the effective tracer velocity V and the dispersion coefficient DL are takenas adjustable parameters to fit the breakthrough curves (Thompson 1991; Thompson andBrown 1991). The dispersion coefficient resulting from Eq. 7 cannot be used for predictionof transport with the ADE because of the non-Fickian behavior of transport. However, theseparameters can be used for estimating the Péclet number (Pe) [–] as (Tsang et al. 1988)

Pe = V L

DL, (8)

where L is the fracture sample length [L].The Péclet number compares advection (first term in the right-hand side of Eq. 3) with

dispersion (second term in the right-hand side of Eq. 3). This gives a measure of the rela-tive degree of channeling in a fracture undergoing shear when fluid flows parallel with orperpendicular to shear.

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x

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q1

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q79

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=80

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Fig. 3 Calculation method for the number of particles that are introduced at the inlet boundary for the particletransport in x- and y-directions (Option Two)

Transport connectivity is estimated using a transport connectivity indicator proposed byKnudby and Carrera (2005). They use the ratio between mean and early arrival times as atransport connectivity indicator, defined as

CT = tAVE

t5, (9)

where tAVE is the mean arrival time [T], and t5 is the time at which 5% of the particles hasarrived at the outlet [T].

3.3 Two Options for Choosing the Number of Particles at the Inlet Boundary

Under in situ subsurface conditions, solute concentration or particles may most likely fol-low the paths of higher hydraulic conductivity, rather than be uniformly distributed. Thismeans that the concentrations in fractures may be closely correlated to the fluid flow rates(Abelin et al. 1991). On the other hand, evenly distributed initial tracer concentration (or par-ticle numbers) is usually injected at the inlet boundaries in laboratory tracer tests (Hakamiand Barton 1990; Wendland and Himmelsbach 2002) and numerical studies (Thompson andBrown 1991; Zhou et al. 2006). This enables an easier control of the initial particle injectionconditions at the inlet. This condition can be performed by injecting the same amount of fluidwith fully mixed tracer concentration at evenly distributed injection points along the inletboundary, using a parallel array of syringes (Hakami and Barton 1990).

Hence, two options for particle distributions at the inlet boundaries were investigated inthis article: a uniform distribution (Option One), and a non-uniform distribution weightedwith the flow rates (Option Two). Figure 3 shows the method used to determine the number ofparticles at the inlet boundary in both the x- and the y-directions for Option Two. The numberof particles introduced in each element along the inlet boundary, ni , is proportional to theflow rate of the element, qi , obtained from the FEM flow calculations. The inlet boundariesfor particle injection chosen in this study are the right-hand side boundary for the model withflow in the x-direction (parallel with the shear direction), and the bottom boundary for themodel with flow in the y-direction (perpendicular to the shear direction).

As there are 80 elements along the inlet boundaries (fracture edges), 80 particles wereinjected along the inlet boundary. Thus, one particle was injected into each element along the

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1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

0 5 10 15 20

Shear Displacement, mm

Flow

Rat

e, m

3 /sec

Flow in the x-directionFlow in the y-direction

Fig. 4 The flow rates along the edges x = 0 for flow in the x-direction (parallel with the shear direction) andat edge y =0.194 m for flow in the y-direction (perpendicular to the shear direction), respectively (Koyamaet al. 2006)

boundary for Option One. The fluid flow rates of the 80 FEM elements at the inlet boundarieswere calculated from FEM flow simulations at each shear displacement increment, and werethen used for calculating the number of particles to be injected in each FEM element at theinlets for Option Two.

4 Results

4.1 Fluid Flow Patterns During Translational Shearing

Figure 4 shows the mean flow rates calculated at two cross sections. One corresponds to theflow rate in the x-direction, and the other to the flow rate in the y-direction (Koyama et al.2006). The flow rate for the flow in the y-direction (perpendicular to shear) is much higherthan that in the x-direction (parallel with shear), by about a factor of three. This is becauseof shear in the x-direction, which induces fast flow channels with much increased flow ratein the y-direction. This figure clearly illustrates the limitations of the traditional shear-flowtests. When flow and shear directions are parallel, the significant effect of shear dilation onflow in the direction perpendicular to shear direction cannot be captured, and therefore isoften neglected. This shortcoming may lead to significant underestimation of the anisotropyof fluid transmissivity of rock fractures during shearing processes (Yeo et al. 1998).

Figure 5a and b displays the evolution of the transmissivity field, superimposed with thecontours of flow streamlines (particle paths), for flow in the x- and y-directions at four sheardisplacements, 0, 1, 5, and 20 mm, in the x-direction. The particles paths vary with increasingshear, clustering in higher transmissivity regions (channels) and bypassing the lower trans-missivity regions (contact areas). These channels are oriented in the y-direction, i.e., thedirection perpendicular to shear.

It should be noted that a high transmissivity region (fast flow channel) exists on the frac-ture sample before shearing starts, close to the x = 0 boundary (the left-hand side boundaryof the fracture, see Fig. 1). It was caused partly by damages on the fracture sample duringsample preparation, not during the shearing process. This initial anisotropy in aperture and

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Fig. 5 Transmissivity evolution and the change of particle paths for the unidirectional fluid flow a parallelwith and b perpendicular to the translational shear direction, as shown by the dotted arrows. The values in thelegend indicate the order of magnitude

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transmissivity fields affected flow and solute transport as shown in the results presented inthe following sections, especially for small shear displacements.

4.2 Solute Transport During Translational Shear

Figure 6 shows the particle number distributions injected along the inlet boundaries at dif-ferent shear displacements for particle injection Option Two (particle number weighted withflow rates). Figure 6a (flow parallel with shear) displays clearly almost uniform distributionsof particles at the inlet when shear displacement is small. Two clear new contact areas at theinlet boundaries (around element number 50 and 62 in the last figure of Fig. 6a) were formedat large shear displacement (20 mm), but the particle concentration was not very clustered inthe rest of the inlet. The small flow rate in the elements closest to the contact area attractedless than 0.001% of the particles. Hence, the number of particles in such very low flow rateareas (representing contacts) can be neglected. When using Option One (uniform distributionof particles), an equal number of particles is forced to enter the elements at the inlets even ifthey present a very low transmissivity, which causes delays in travel time.

Figure 6b clearly shows the process of concentration of particle in channels at the inletwhen the flow is perpendicular to shear. Two phenomena should be noted: (1) a significantinitial channel for flow and solute transport at the left-hand side of the fracture which attractsmany particles before and at the beginning of shear (because of the damage caused duringsample preparation as mentioned above), and (2) a progressive formation of a major newshear-induced flow channel in the middle of the sample (around element number 38). Thelatter was already formed at 5 mm shear displacement (the third figure of Fig. 6b), whichattracted an increasing number of particles with increasing shear displacement. Two mainchannels attracted the vast majority of particles after 20 mm shear displacement. One placedin the left-hand side region, and the other one in the middle of the sample, as illustrated inthe last figure of Fig. 6b.

Figure 7 displays the breakthrough curves for the solute transport in the x- and y-directionsat several shear displacements, which were calculated and compared for the two options ofparticle injection at the inlet. The difference between the two options is small when flow isparallel with shear except at 20 mm shear displacement (Fig. 7a). In contrast, the differencebetween the two options is significant in the direction perpendicular to shear (Fig. 7b). BothFigs. 7a and b show that more particles travel faster to the outlet boundary using OptionTwo (with the numbers of particles injected at the inlet weighted with flow rates). The break-through curves shift slightly to the left without changing their shape and inclination withflow parallel with shear (Fig. 7a). This means that the mean tracer velocity in the x-directionbecomes slightly higher, but the dispersivity remains almost constant. The long tails of thebreakthrough curves at shear displacement of 20 mm with Option One (not-weighted) doesnot appear with Option Two (bottom figure of Fig. 7a). This is because some particles gettrapped in the very low-transmissivity regions that form for large shear displacement at theinlet when the particles are uniformly distributed. On the other hand, much more particlesreach the outlet boundary with much less travel time, traveling through the fast shear-inducedflow channels, when injecting the particles weighted with the flow rates for flow perpendic-ular to shear (Fig. 7b). As a result, the breakthrough curves shift significantly to the left andbecome more steep. This implies a significant increase in the mean tracer velocity.

Figure 8 shows the mean values and standard deviations of the travel time for flow in thex- and y-directions. Both the mean and the standard deviation of travel time decrease withshear displacement for both options of particle injection at the inlet with flow in both x- and

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1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79

Fig. 6 The number of particles at inlet boundary for the particle transport (Option Two): a flow in thex-direction and b flow in the y-direction, at several shear displacements (0, 1, 5 and 20 mm)

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Fig. 8 The comparison of mean values and standard deviations of the travel time during translational shear interms of different flow directions. Letter x indicates results with flow in the x-direction (parallel with the sheardirection), and letter y indicates the results with flow in the y-direction (perpendicular to the shear direction).The two different particle number options injected at the inlet boundary are Option One: not-weighted withflow rate and Option Two: weighted with flow rate

y-directions. This indicates a general trend of increasing particle travel velocity with increas-ing shear displacement. An exception to this trend occurs for the non-weighted particleinjection at the inlet (Option One) for 15 and 20 mm shear displacements with flow parallelwith shear. This may be caused by significant delays in travel time of a small number of par-ticles that are injected in low-transmissivity areas (see also the long tails of the breakthroughcurves in the last figure of Fig. 7a). Overall, the mean particle travel time in the directionperpendicular to shear is much smaller (about a factor of 3) than that in the direction par-allel with shear, for both options of particle injection at the inlet, because of the dominantshear-induced channeling effect in the direction perpendicular to shear.

Transport connectivity can be calculated from the information contained in Figs. 7 and 8.We focus on the values of the transport connectivity indicator CT (Eq. 9) for Option Two(weighted) because it is the most realistic option of injection. CT values are around 8 in bothdirections of flow before shear. Once shear starts, CT becomes smaller because the meantravel time is reduced significantly. CT acquires values around 1.4 and 4.0 in the directionparallel with and perpendicular to shear, respectively, for shear displacements greater or equalto 5 mm.

4.3 Péclet Number

Figure 9 displays two examples of curve-fitting against 1-D analytical solution (Eq. 7) atseveral small shear displacements (0, 1 and 2 mm), for flow parallel with (Fig. 9a) and per-pendicular to (Fig. 9b) the shear direction. This fitting was used for back-calculations of thePéclet number (Eq. 8).

Figures 10 and 11 show the results of the calculated Péclet numbers considering the twooptions of particle injection at the inlet: non-weighted (Option One) and weighted (OptionTwo) with flow rates, in the x- and y-directions, respectively. As shown in Fig. 10 (flow par-allel with shear), the calculated Pe values increase continuously from the start of shearing,reach a peak at a shear displacement of about 3 mm, and then decrease without stabilizing forincreasing shear displacement. One interpretation for this behavior may be that the gradualincrease of the initially widely distributed contact areas may become more and more con-centrated and clustered into fast flow and particle travel channels, starting from the 3 mm

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shear displacement and continued to grow with shear (cf. Figs. 1, 5). Note that the Pe valuesfor the non-weighted injection option are always smaller than for the weighted injectionoption. Thus, the non-weighted injection option presents more dispersion. This is becausesome particles get trapped in low-transmissivity areas at the inlet.

Figure 11 shows the calculated Pe values for flow perpendicular to shear. The magnitudeof these Pe values is one order of magnitude smaller than those for flow parallel with shear(Fig. 10). This indicates that dispersion is much significant for flow perpendicular to shear.The Péclet number stabilizes (Pe=2) at a shear displacement of 5 mm when the number ofparticles injected at the inlet is weighted with flow rate (Option Two). However, the Pécletnumber does not stabilize and present greater values when the number of injected particles atthe inlet was not weighted with flow rate (Option One). This may be due to the difference inthe flow rates between the two channels (see Fig. 5b), which produces a higher dispersivity

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when the injected particles are weighted with the flow rate than when they are uniformlydistributed.

5 General Discussions

The flow rate perpendicular to shear is much higher than the one that is parallel with shear(Fig. 4). This reduces significantly the solute travel time when water flows perpendicularto shear (Fig. 7). This cannot be measured with shear-flow tests in the laboratory becausesamples cannot be effectively sealed in the direction perpendicular to shear. Thus, this findingis relevant for risk analysis (Bolster et al. 2009) of long-term nuclear waste disposal.

Gouge formation can reduce the permeability of the fracture. However, its effect is typi-cally significant after relatively large shear displacement (7–8 mm) and produces a stabiliza-tion of the fracture permeability (Lee and Cho 2002). While the flow rate increases by twoorders of magnitude within the first 7–8 mm of shear displacement, it increases only by halfan order of magnitude between 7–8 and 20 mm of shear displacement (Fig. 4). Thus, solutevelocity may be slightly overestimated for shear displacements larger than 7–8 mm becausegouge formation is neglected in this study. The absence of normal loading also overestimatesthe aperture fields. However, the comparison between the mean aperture of the numericalshear process and the normal displacement measured from the shear test of the same sample,with 1 MPa of normal stress, shows an overestimation of less than 0.5 mm, which representsless than a 10% overestimation (Koyama et al. 2006).

We have investigated two options of injecting particles at the inlet. The injection of evenlydistributed particles provides an easier experimental control, but may underestimate the effectof channeling on travel velocity. Alternatively, particles may be injected weighted with theflow rate at the inlet. We have been able to numerically simulate this option because we havea detailed geometrical information of the fracture.

The number of particles introduced at the inlets of the sample simulated in this arti-cle is 80. This number can be increased by many times without any numerical difficulty.

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Such increase would make the breakthrough curves smoother, but would not introduce sig-nificant changes in the essential behavior of the breakthrough curves and would not affectthe evaluation of shear displacement effects on fluid flow and solute transport processes.

The streamline particle tracking adopted in this study is applicable when advective–dis-persive transport is dominant over diffusion. The characteristic diffusion time, tcD , can bedetermined from

tcD = t ′ L2

Dw, (10)

where L is the length of the sample (0.194 m), Dw is the diffusivity of water (= 10−9 m2/s)and t ′ is the dimensionless time of diffusion. This dimensionless time equals 0.1 when diffu-sion begins to have an influence on solute transport and 1.0 when the process is completed.Hence, the time at which diffusion starts to become significant is t ′ = 0.1.

Figure 8 shows that the maximum mean particle travel times for advective transport are452 and 472 s (before shear displacement starts), for flow parallel with and perpendicular tothe shear direction, respectively. From Eq. 10, the characteristic diffusion time is 3.8×106 s,which is much larger than the above-mentioned maximum mean particle travel time obtainedin this study. Thus, the transport simulations are in a pre-asymptotic regime where advection–dispersion still dominates diffusion. Therefore, our particle tracking method is applicable inthis study, because the entire solute transport process happens before diffusion starts to act.

Figure 12 shows a schematic interpretation of the channeling effect on the breakthroughcurves for the transport in the x- and y-directions, i.e., parallel with and perpendicular tothe shear direction, respectively. This figure displays a simplified flow channel geometry forhighlighting the basic concepts presented in this article. Two main flow channels, an initialone located close to the x =0 boundary and another induced by shear in the middle of thefracture sample, are oriented perpendicular to shear (see Fig. 5). Therefore, the geometry ofthe flow channel can be simplified as shown in the top left figure of Fig. 12. With this ideali-zation, all particles will reach the outlet boundary at the same time when fluid flows parallelwith shear. Thus, the breakthrough curve has only one stepwise change (top right figure ofFig. 12). On the other hand, there are four different idealized transmissivity areas when thefluid flows perpendicular to shear. Two of them have high conductivities, coinciding with fastchannels. The other two have low transmissivity, coinciding with contact areas. This leadsto four stepwise changes on the particle breakthrough curve (bottom left figure of Fig. 12).Hence, the breakthrough curve in this case becomes flatter (more dispersion) than the casewith flow parallel with shear. However, the particles traveling through the fast shear-inducedchannels (perpendicular to shear) will have a smaller travel time than those crossing thechannels and contact areas (parallel with shear). The bottom right figure of Fig. 12 shows themechanisms for the change in the shape of the breakthrough curves for this idealized situa-tion. The breakthrough curve becomes flatter in shape when dispersivity dominates (lowerPéclet number), and shifts leftward when tracer velocity increases, i.e., advection increases(higher Péclet number). The solute transport in the direction perpendicular to shear has flatterbreakthrough curve shapes, yielding larger variance in travel time, and thus smaller Pécletnumber (one order of magnitude lower than those for flow parallel with shear). Therefore,dispersion is more significant when the flow is perpendicular to shear than when it is parallelwith shear. The Péclet numbers evaluated in this study have the same order of magnitude asreported in experimental and numerical tracer tests in rock fractures (Table 1).

The transport connectivity indicator CT (Knudby and Carrera 2005) shows that transportconnectivity is similar in the x- and y-directions before shear starts. However, transport con-nectivity becomes higher (by a factor of three) in the direction perpendicular to shear. This

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Table 1 Péclet numbers, Pe, in experimental and numerical tracer tests in rock fractures

Case-test Sample length (cm) Rock type Pe range

Neretnieks et al. (1982) 30 Granite 8–27Moreno et al. (1985) 27 Granite 9–80Moreno et al. (1988) Dimensionless Stochastic generated fracture 2–40Thompson (1991) 100 Stochastic generated fracture 3–50This study 19.4 Granite (numerical) 0.4–89

is due to the fact that dispersion dominates for flow in this direction, yielding faster firstarrival times (because of the shear-induced channels) and a long tailing, which increases themean travel time. In contrast, the tail is short for flow parallel with shear, in which advectiondominates and the breakthrough curve is steep, yielding a very similar early arrival times andmean travel time.

6 Conclusions

Flow and solute transport in a fracture of variable aperture subjected to shear was studiednumerically. The fracture aperture maps were obtained from the superposition of the digital-ized surfaces of a real rock rough fracture. Shearing was simulated without normal loading,obtaining good agreement with the measurements of normal displacement of the same frac-ture in a shear test performed in the laboratory under a normal stress of 1 MPa. No asperitydegradation or gouge formation was considered, which may overestimate the flow rate. Theflow in the fracture was simulated using the Reynolds equation, and the local transmissiv-ity was calculated assuming that the cubic law is valid locally. Particle tracking following

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streamlines was used to trace out flow paths and to characterize the particle residence timedistributions.

The results show that shear displacement can lead to an increase in the magnitude offlow rates and solute velocity in the direction perpendicular to shear in comparison withthe direction parallel with shear displacement by a factor of three. This suggests that sheardisplacements strongly influence the flow rate and solute transport travel times in fracturedrocks. This also complements the results of conventional laboratory coupled shear-flow tests,which are limited to unidirectional flow in the direction parallel with the shear displacementbecause of sealing limitations.

Shear induces a channeling effect in the direction perpendicular to shear. Solute parti-cles have to cross the fast channels and the low-permeability areas for flow parallel withshear. This yields an advection-dominated transport with a one stepwise breakthrough curve.However, particles travel at different velocities through the channels when flow is perpendic-ular to shear. This leads to a large variance in particle travel time. Thus, dispersion dominates,as indicated by the Péclet numbers, which are one order of magnitude lower than those forflow parallel with shear. Last, but not least, transport connectivity becomes higher in thedirection perpendicular to shear due to the shear-induced channels in this direction.

Acknowledgments V.V. would like to acknowledge the financial support provided by the Spanish Ministryof Science and Innovation (MIC) (“Formación de Profesorado Universitario” program), and the “Colegiode Ingenieros de Caminos, Canales y Puertos—Catalunya”. The authors thank the Swedish Nuclear PowerInspectorate (SKI) for the financial support for T.K.’s Ph.D studies at Royal Institute of Technology (KTH),Stockholm, Sweden. The authors are very grateful to Professor R. W. Zimmerman, J. Carrera, and D. Bolster,for their fruitful and in-depth discussions and their continuous efforts in enhancing the scientific presentationof this article.

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