Engineering Geology 116 (2010) 166–177

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Engineering Geology

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Application of fracture network model with crack permeability tensor on flow andtransport in fractured rock

Jian-Bang Pan, Chen-Chang Lee, Cheng-Haw Lee ⁎, Hsin-Fu Yeh, Hung-I LinDepartment of Resources Engineering, National Cheng Kung University, Tainan, Taiwan 701

⁎ Corresponding author. Tel.: +886 6 2757575x6283E-mail address: leech@mail.ncku.edu.tw (C.-H. Lee).

0013-7952/$ – see front matter. Crown Copyright © 20doi:10.1016/j.enggeo.2010.08.007

a b s t r a c t

a r t i c l e i n f o

Article history:Received 2 March 2010Received in revised form 19 July 2010Accepted 4 August 2010Available online 20 August 2010

Keywords:FracturesFracture network modelCrack permeability tensorIngrowth-decay transportLan-Yu islandTaiwan

Very efficient and practical models that use a crack tensor in terms of fracture geometry parameters for fieldrock mass are proposed. A framework of the crack tensor approach combined with the identifying fracture-controlled characteristics was successively used in a hypothetical hydro-geological field for determining thedynamics of flow and transport of nuclides in fractured rock.The modeling results and site-specific data of Lan-Yu Island in Taiwan reveal the characteristics associatedwith the geometry of fracture networks under the prescribed boundaries. The simulated fracture networkswere transformed into a grid-based simulation procedure under consideration of the delaying effect inmatrix owing to particle migration. Synthetic flow results indicate that the flowing vector is very sensitive tofracture intensity (P32); high fracture intensity leads to high velocity. Heterogeneity generated by fractures'field will result in the shortest and the longest transmission distance of 12-fold difference. While taking intoconsideration of the ingrowth-decay transport, the plumes of the daughter product were relatively tortuousand anomalous to predict.

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1. Introduction

Fractured formation, which is the most common and generalmaterial in rocks, makes it difficult for hydro-geologists, geologists,engineers, and scientists to strictly quantify and characterize theflowing mechanism induced by spatial heterogeneity (Berkowitz,2002; Neuman, 2005). Geomedia disposal is viewed as the bestmethod for disposing of radioactive waste. One of the most seriousproblems is the isolation of the radioactive waste from the biosphere.Groundwater flow through geological discontinuities is believed to bethe most significant mechanism of radionuclide migration. Theoret-ical models and experimental studies have contributed to theunderstanding of flow and solute migration fields within fractureplanes, including the characterization of aperture variation, intersec-tions among various phases, and the transfer between fractures andthe rock matrix.

The analysis of subsurface fluid flow and mass transport is oftencomplicated by the fact that rockmasses, especially at shallow depths,may contain various systems of cracks or failure surfaces. The rockmass can therefore be regarded as an assemblage of intact rock blocksthat are separated by discontinuities. These discontinuities may befaults, joints, fissures, or fractures; the word “fracture” is used here asthe general term. In hydraulic modeling, each solid block can beconsidered as a continuous porous medium that is separated from the

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other blocks by fractures with various hydraulic behaviors. Thus, theflow through such a complex system depends on the magnitudes ofthe permeability in each of the two regimes. However, thepermeability of the intact rock is relatively low; fractures are regardedas more permeable than the rock matrix by several orders ofmagnitude. Consequently, fractures are the major path for fluids toflow (Priest, 1993; Lee and Farmer, 1993; National Research Council,2001; Berkowitz, 2002; Neuman, 2005).

When classified according to how heterogeneity is incorporated inthe model structure, fluid flow and solute transport fall into one ofthree groups: discrete network models, equivalent continuummodels, and hybrid models (National Research Council, 2001). Itcould be done via standard approaches such as by increasing thepermeability of the porous medium to account for the fracture system(or double-porosity models), or by using discrete stochastic fracturenetwork models.

In the discrete fracture network model, such as Baecher's model(Baecher et al., 1977; Dershowitz, 1984; Dershowitz and Einstein,1988; Ivanova, 1998) each fracture is placed in a space centered onPoisson points and oriented randomly according a specified distribu-tion. It is then extended to define a stochastic process for thetermination of the fracture shape and size. The usual process is tobeginwith unbounded Poisson planes and to superimposed on them aseries of Poisson lines or the lines of intersection between theunbounded Poisson planes; this is called the enhanced Baecher model(Dershowitz, 1984; Dershowitz and Einstein, 1988; Ivanova, 1998).The main advantage of the discrete fracture model is that it allows theinspection of the influence of individual fracture parameters on flow.

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167J.-B. Pan et al. / Engineering Geology 116 (2010) 166–177

Its disadvantage is that the detailed definition of fracture geometryand the hydraulic properties of individual fractures are uncertain.Researchers more often consider the orthogonal fracture network dueto its simplicity. In the discrete channel model is analogous to thediscrete fracture model except that the fractures are viewed aschannels. In the continuum orthogonal fracture mode, the fractureparameters are assumed to behave macroscopically. The formationhas either isotropic or non-isotropic properties in homogenousmedia.The phenomena (such as permeability tensor or crack tensor in thepaper) were represented correctly through the continuum equivalentso that three principal values of phenomena properties were coaxialto orthogonal axial. Long et al. (1982) suggested that if a sufficientnumber of fractures exist within a rock mass, then the rock massbehavior can be simulated using an equivalent porous medium.

Oda (1985) proposed a model for establishing the fracturenetwork to predict rock mass permeability. The fractures are regardedas a series of penny-shaped cracks. The crack geometry can generallybe regarded as a function of fracture orientation, fracture trace length,and fracture aperture. The permeability tensor is viewed as a uniquefunction of the crack tensor, and permeability principal axes areviewed as being coaxial with those of the crack tensors. In situmeasurements of the fracture size and aperture are difficult. Thefracture size may be obtained by mapping and measuring the fracturetrace length, but this method is time-consuming. In addition, thefracture trace length depends on the shape and size of the fracture, theangle between the fracture plane and outcrop plane, and themeasurement scale of the sampling window. Hence, learning howto estimate the representative values of fracture geometric para-meters (fracture orientation, size, and aperture) is important. It is thefollowing work to describe the network formed by fractures. Severalresearchers have used fracture continuum (FC) models to numericallysimulate flow and transport in fracturedmedia (Neuman, 1987;Wide´nand Walker, 1999; Svensson, 2001a,b; Ando et al., 2003; Langevin,2003; Pohlmann et al., 2004; McKenna and Reeves, 2006; Reeveset al., 2008). Instead of representing discrete fractures as continuousline elements in 2-D or planes in 3-D, FC models are based on theconversion of discrete fractures, or more commonly fracture zones, topermeability structures on a model grid. This conceptualization issupported by field observations that rock volumes are often intersectedby a few dominant fractures (Neuman, 2005). Often, the choice ofusing the discrete fracture network (DFN) or fracture continuum (FC)model depends on scale; DFN models are favored at smaller scaleswhere they remain computationally feasible and FC models are usedfor larger-scale models. Comparisons between DFN and FC modelshave shown that both methods are equally capable of capturing keyaspects of flow and transport in fractured crystalline rock masses(Svensson, 2001b; Selroos et al., 2002; Ando et al., 2003). Theassembled procedures to do such a task is called fracture networkmodel. After themodel is built, it is coupled to study permeability. In thepresent study, the focus is to determine hydraulic properties and also torealize in flow and transport to deal with different applicationcircumstances.

A framework from field data is integrated with a numericallyspatial distribution of fracture parameters in an equivalent fracturenetwork model which is combined with discrete fracture networksand the concept of fracture tensors in a fractured rock mass. Based onthe crack patterns and the statistical geometric characteristics of thefractured rock, the proposed model can be used to estimate the cracktensor and to predict the permeability tensor. The study area, apotential disposal site of low-level and intermediate-level radioactivewaste at Lan-Yu island (Fig. 1) composed of moderately fracturedandesite (Deng et al., 1995) is utilized on performance in this idea. Ahypothetical hydro-geological field is proposed to quantify andcharacterize the behavior of flow and transport of parent–daughterradionuclide for the assessment of transport behavior under thisframework.

2. Methodology

2.1. Study approach

The crack tensor approach is commonly used in geotechnical,environmental, and civil engineering fields, especially in the estima-tion of permeability or conductivity under a rock mass (Wang et al.,2002; Bluma et al., 2005). Firstly, the discrete fracture model countedon a statistical characterization of the rock mass (Lee et al., 2007). Thegeneration of the fracture networkwas based on realistic distributionsof geometric parameters, including fracture orientation, size, inten-sity, and location, as well as hydro-geological parameters, includingtransmissivity and transport aperture (Lee et al., 1996). A three-dimensional fracture network model was generated using thestochastic realization of polygonal fractures when the crack tensorapproach (Oda, 1985) was applied to assemble the fracture clustersfrom the discrete fracture network model. The numerical heat andmass transport tool (Zyvoloski et al., 1988; Zyvoloski et al., 2008) wasused to model the flow under the prescribed head condition withinthe particle tracking for the spatial distribution of particles. Finally,the spatial and temporal concentrations of decay nuclides wereinvestigated to assess possible contaminated zones.

2.2. Calculation of permeability and crack tensor

The presence of cracks can induce anisotropy in the permeabilitytensor for a cracked soil or rock. Cracks or fractures produce a highpermeability in the predominant direction of cracking. The coefficientof permeability of the cracked soil or rocks in the direction normal topredominant cracking will be small. The permeability's spatialvariability and anisotropy has attracted much attention to fracturedrock and petroleum studies (e.g., Snow, 1969; Hsieh et al., 1985;Teimoori et al., 2005).

If a fractured rock is treated as a homogeneous, anisotropic porousmedium, the permeability tensor, obeying Darcy's law, can beformulated as (Bear, 1972):

Vi = − gμKij

∂ϕ∂xj

=gμKij Jj ð1Þ

where Vi is the apparent flow velocity, Jj is the hydraulic gradient, andKij is the permeability tensor.

If a fractured rock mass is assumed to have an impermeable matrixand groundwater is assumed to only flow through the fractures, thenthe apparent flow velocity can be defined as:

Vi = ∫V cð ÞV

cð Þi

dV cð Þ

Vvð2Þ

where V(c) is the volume of the fractures and VI(c) is the local velocityfor each fracture. If the fracture is considered as a parallel plate modelwith an aperture e and groundwater moves by laminar flow in thefractures, obeying the cubic law, then the mean velocity VI(c) can beformulated as (Snow, 1969):

V cð Þi =

112

gμe2J cð Þ

i : ð3Þ

Each fracture has a void volume equal to (π/4)r2e. The total voidvolume dV(c) associated with the fracture geometry is given by:

dv cð Þ =πr2e4

dN: ð4Þ

Fig. 1. Location of Lan-Yu island and the sampling area.(Satellite picture obtained from Google Earth).

168 J.-B. Pan et al. / Engineering Geology 116 (2010) 166–177

Substituting Eqs. (1), (5), and (4) into Eqs.(2) and (3) produces anequivalent permeability tensor Kij for the fracture system:

Kij =112

Pkkδij−Pij� �

: ð5Þ

In this case, Pij and Kij are both symmetric, having principal valuesin the principal directions. Since Kij is a function of Pij, its principal axesare coaxial with those of Pij, with the major permeability in the minorprincipal direction of Pij. If i= j in Eq. (5), the scalar hydraulicconductivity Ks is given by:

Ks =g3μ

112

Pii: ð6Þ

In this study, the rock masses are assumed to be composed of aseries of fractures and an impermeable matrix. The crack geometrycan generally be regarded as a function of fracture orientation f,fracture size (area) a, and fracture aperture e. In addition, the unitvector f is normal to the fractures that are orientated inside a smallsolid angle dω around f and the diameters and the apertures rangefrom a to a+da and from e to e+de, respectively. E(f,a,e)dωdadeis the probability of the unit normals of (f,a,e) fractures. In this case,E(f,a,e) is defined over the entire solid angle ω. If m(v) fractures arelocated within the flow region of volume Vv, the frequency of theprobability of (f,a,e) fractures whose centers are located within Vv,then let dN be a number of (f,a,e) fractures, to estimate the total offractures,

dN = m vð ÞE f ; a; eð Þdωdade: ð7Þ

The crack tensor Pij can be written as:

Pij = ρ∫em

0∫am

0∫ωae3fi fjE f ; a; eð Þdωdade ð8Þ

where ρ is the volume density of fractures, defined as m(v)/Vv.Assuming that the random variables f, a, and e are statisticallyindependent of one another, E(f,a,e)=E(f) E(a) E(e), where E(f), E(a),and E(e) are the density functions of f, a, and e, respectively, then:

Pij = ρ∫em

0e3E eð Þde∫am

0a2E að Þda∫

wfi fjE fð Þdω: ð9Þ

Since fractures generally occur in clusters or sets, the densityfunction of E(f) for each set can be obtained in the form of a possibledistribution with the goodness-of-test on fracture orientations asdescribed above. However, it is difficult to obtain reliable data for thedensity functions E(e) and E(a) for each set. It is a time-consumingwork, especially for measuring the fracture apertures in situ (Snow,1969). To overcome this difficulty, fracture aperture in the presentstudy is estimated from laboratory tests; the value is constant under afixed normal stress. Since the fracture orientation f, fracture size(area) a, and fracture aperture e are independent, Eq. (9) can be givenas:

Pij = ∑Nf

k=1Pkij = ∑

Nf

k=1e3ρ⟨a⟩

kNkij ð10Þ

Nkij = ∫

ωninjE nð Þdω =

∑l2 ∑lm ∑ ln∑ml ∑m2 ∑mn∑nl ∑nm ∑n2

24

35 1Rk

ð11Þ

⟨a⟩k = ∫am

0aE að Þda k = 1;2;…nð Þ ð12Þ

where Rk is themagnitude of the directional cosine for the kth set.Nf isthe number of fracture sets. Nij or Nij

k is a symmetric, second-ranktensor which can be calculated from density function E(n).

169J.-B. Pan et al. / Engineering Geology 116 (2010) 166–177

3. Parameter estimation of fractures

3.1. Mean orientation of fracture sets

Fractures generally occur as fracture sets or clusters. The meanorientation of each cluster is required to estimate various engineeringproblems. Determining the pattern of a fractured network is anessential task for evaluating the contaminant transport in fracturedrock. The statistical description of orientation data in the past wasoften expressed as orientations of fracture values of normal in termsof spherical co-ordinates (θ, ϕ, 1) or direction cosines (l, m, n). Themean orientation of each cluster is then taken as the orientation of theresultant vector R, which is determined by adding all of the fracturepoles:

R = ∑l2 + ∑m2 + ∑n2h i

ð13Þ

where l=sinϕcosθ, m=sinϕ sinθ, and n=cosϕ. The angles (θ and ϕ)are related to the dip angle φ, and the dip direction α, by θ=2π−α(0≤θ≤2π) and ϕ=φ (0≤θ≤π/2). It should be noted that, beforeEq. (13) can be used in estimating the mean orientation for each set,poles of fractures for each set must be identified.

Therefore, a delineation of the fracture data is necessary before themean direction of each fracture set can be calculated. Methods fordelineating fracture sets were developed by Shanley and Mahtab(1976) (using one-level mode analysis) and by Miller (1983) (usingcontingency table analysis). Both models set up the sphere of radius rat each point (each pole of the fracture) on a stereonet. All of thecenters of the sphere containing κ or more observations are labeled as“dense points.” Finally, all of the non-dense points are added toadjoining points until every point is part of the cluster. Parameters r(radius or degree) and κ (center density) are found by optimizing theobjective function in order to illustrate the clustering system. Thevalue of r is generally selected to be between 10° and 20°. Theadvantage of the objective function is that, within a site, all data in acluster can be analyzed using a rational and standardized system.However, clustering is generally related to the method of objectivefunctions used as various means of clustering fractures as waysto represent fracture orientation distribution graphically. Severalmodels, such as Fisher distribution or Bingham distribution, canprovide better fits for asymmetric and girdle orientation data. TheFisher distribution is a valuable model for fitting fracture orientationdata due to its simplicity and flexibility (Priest, 1993). More detailedexplanations of the corresponding procedures of objection functionsfor delineating a cluster set of fracture orientations can be found in ourprevious research (Lee et al., 1995). Knowing the fracture sets candetermine the dominant orientation of fractures which is very usefulfor planning and designing the principal direction of subsurfacestructures.

4. Fracture frequency, mean spacing, and fracture intensity

When the fracture orientation is delineated as sets or clusters, themean fracture orientation for each set can be estimated from the polesof each set, as shown in Eq. (13). In order to estimate the principlefrequency for each set, Priest and Hudson (1981) showed that thefracture frequency λθ along a scanline can be represented as:

λθ = ∑Nf

i=1λi cos θi ð14Þ

where λi is the true frequency of the i-th fracture set along thedirection of its normal, θi is the acute angle between the scanline andthe mean normal to the i-th set, and Nf is the number of fracture sets.

Since thefield data of the fracture orientation are generally scattereddue to geological complexity, the deviationmust be reduced to estimatethe solution of true frequency of each fracture set. Karzulovic andGoodman (1985) established amethodwith a least square technique toobtain the true frequency of fracture sets of the rock mass. In theirmodel, when there are Sf non-parallel scanlines, it is possible to obtainthe true frequency of each of the Nf fracture sets. Their approachproduced the following result, expressed in matrix form:

A½ � λi½ � = B½ � ð15Þ

where [A] is a symmetricalmatrix withNf byNf. The term Akl is definedas the value of [A] in row k and column l and is given by:

Akl = ∑Sf

j=1cos δkl cos δlj ð16Þ

where Sf are non-parallel scanlines, δkj and δlj are acute anglesbetween the j-th sampling line and normal to sets k and l, respectively.The matrix [B] has one column and A rows. The term Lk in row k of [B]is given by:

Bk = ∑Sf

j=1λj cos δkj ð17Þ

where λj is the apparent frequency of the j-th scanline. The solutionvector [λi] in Eq. (15) can is obtained from Eq. (17) by applying anappropriate numerical solution for simultaneous equations.

Based on the above process, the true frequency of each fracture setcan be estimated. This method can be used if the mean direction ofeach set is known. It is possible to use more scanline surveys than thenumber of fracture sets in situ. If the true frequency of each fractureset is determined, then the mean spacing of each fracture set (Si) canbe obtained using:

Si =1λi

: ð18Þ

The P32 of a rock mass is used to define the fracture intensity. P32 isdefined as the total area of fractures per unit volume.

P32 =Af

Vtð19Þ

where Af is the total area of the fractures and Vt is the total volume of theregion being measured. P32 is dimensional (in units of 1/m), but is scaleindependent and invariant with respect to the distribution of fracturesize. It can therefore beused to classify a given rockmass andhence is thepreferredmeasure of fracture intensity (Dershowitz andEinstein, 1988).And knowing the fracture intensity is aimed for the vulnerability of rockmass and preventing some important structures form these areas.

5. Fracture aperture

Numerical models have been developed to describe aperturedeformation (Barton et al., 1985). A hyperbolic function was suggestedfor describing the relationship between stress and closure or opening offractures in terms of joint wall compressive strength (JCS), jointroughness coefficient (JRC), and normal stress. Barton et al. (1985)showed that anempirical relationship between themechanical apertureem and the hydraulic aperture ec can be expressed in empirical form as:

ec =e2m

JRC2:5 ð20Þ

where em and ec are inmicro-meters and JRC is in a rangeof 0 (smoothestsurface) to 20 (roughest surface). Eq. (20) is valid only if ec≤em.

170 J.-B. Pan et al. / Engineering Geology 116 (2010) 166–177

Since the effective stress is normal to the plane of the fracture, theamount of normal closure primarily depends on the mechanicalaperture em and the roughness of the fracture surface represented byJRC and JCS. JCS is the unconfined compressive strength of wall rockestimated from Schmidt hammer test. JRC can also be easily estimatedfrom tilt tests, preferablywith directmeasurement (Barton et al., 1985),using:

JRC =αo−ϕo

rlog JCSð Þ�

σ no

ð21Þ

where σno=γhucos2α, γ is the rock density, hu is the thickness of theupper block, ϕr

ois the residual friction angle of intact rock, and α is thetilt angle. The empirical value em is given by (Bandis et al., 1983):

em =JRC5

0:2σc

JCS−0:1

� �ð22Þ

where σc is the uniaxial compressive strength. Substituting Eq.(22)into Eq.(20) gives the hydraulic aperture:

ec =1

JRC2:5

JRC5

0:2σc

JCS−0:1

� �� �2: ð23Þ

Knowing the hydraulic aperture is aimed for the assessment onflow's influence on each section of the fractured rock.

6. Generation of enhanced Baecher's model

The three-dimensional model shown in Figure 2 (using FracMancode developed by Golder's Associate) was applied in the presentstudy. The choice of a stochastic conceptual model for a three-dimensional fracture system involved specifying (1) the individualcharacteristics of the fracture such as shape, size, orientation,

Fig. 2. Example of a three-dimensional stochastic fracture network generated within a100 (m) cube.

aperture, and transmissivity variation (2) and the arrangement ofthe fractures in space, defined by the type of point process theirlocations followed (Baecher model), and the way they truncate eachother. When dealing with a three-dimensional model, many of thesechoices must be made somewhat arbitrarily because only 1- or 2-Dsamples were taken from the 3-D system of the rock mass. A detaildescription and analysis of fracture parameter sampling and variationcan be found elsewhere (Dershowitz, 1984; Dershowitz and Einstein,1988). For all models, fracture termination is specified by thetermination probability PT, which is the probability that a fracturewill terminate, given an intersection with another fracture.

PT =NT

NFið24Þ

where NT is the number of T intersections for fractures in the presentset with fractures from previously generated sets, and NFi is thenumber of fractures in the set having intersections with fractures frompreviously generated sets.

This form of termination disturbs the distribution of fracture size,since portions of fractures are “discarded” if they are beyond theintersection where termination occurs. This is only a problem for hightermination percentages (Dershowitz et al., 2004). In the presentstudy, the enhanced Baecher's model was used to obtain the morerealistic representation of robustness in fractures at intersections withpre-existing fractures.

7. Flow and transport simulation

Using a continuum approach (Oda, 1985; Lee et al., 1995), thepermeability of fracture networks was translated onto a grid ofequally-sized cells, and FEHM (finite element heat and mass transfercode developed by Los Alamos National Laboratory and could bedownloaded by signing some agreement with DOE of USA) was usedto solve for fluid flow in both the fracture network and the lesspermeable matrix assigned to cells not occupied by fractures. Thisresulted in a discrete network solution which is a set of linearequations of pressures at intersections.

FEHMwas modeled within a non-isothermal, multiphase flow andtransport code that simulated the flow of water and air and thetransport of heat and solutes in two- and three-dimensional saturatedor partially saturated heterogeneous media. It also included compre-hensive reactive geochemistry and transport modules, and particletracking capability. Fractured media can be modeled using thecontinuum approach used in the present study, discrete fracture,dual porosity, or dual permeability approaches (Zyvoloski et al.,2008). The conservation of fluid mass can be written as:

∂Amass

∂t + ∇⋅ ̅f mass + qmass = 0 ð25Þ

where:

Amass ϕρf→

mass ρ v→

Amass fluid mass per unit volume (M/L3)f→

mass fluid mass flux (M/L3t)qmass fluid mass source (M/L3t)ϕ porosity in the systemρ fluid density (M/L3).

The velocity of a fluid can be expressed using Darcy's law:

v→

= − kμ∇ P−ρ g

→� �

ð26Þ

Fig. 3. Pole diagram of fracture orientation at the Lan-Yu site on a Schmidt net. Numbersindicates the set number.

171J.-B. Pan et al. / Engineering Geology 116 (2010) 166–177

where:

P fluid pressure (M/Lt2)μ dynamics of viscosity of the fluid (M/Lt)k permeability (L2)g→ acceleration due to gravity (L/t2).

The above equations are for constant-density water, so the resultsshould be converted to a head solution rather than a pressure solution.Using the reference density ρ=1000(kg/m3), g→ = 9:8 m=s2

� �, and

elevation z (reference to mean sea level), pressure was converted tohead using h=z+P/(ρg). The conservation of the solute equation isexplicitly coupled to the fluidflow field. The conservation equation for asingle component, non-reactive solute in a constant density fluid in asaturated medium is:

∂As

∂t = ∇⋅ ρϕDs∇Csð Þ−∇⋅ f→

s −qc ð27Þ

where:

As ϕCsρf→

s ρCs v→

As solute mass storage per unit total volume for aqueouscomponent liquid concentration

Cs concentration of solute (mole/M)Ds dispersion coefficient (L2/t)fs→

dvective mass flux of soluteqc solute source or sink (moles/L3t).

8. Model construction and application

A disposal site of intermediate radioactive waste in freshly fracturedandesite located in the southeast part of Lan-Yu Island, about 49 milessoutheast of Taitung City in the southeast part of Taiwan, was used toverify the proposed method (see Fig. 1). Investigations of the fracturegeometric parameters, such as the fracture sets, the mean orientation,the frequency, the spacing, and the trace length, were conducted on thefields listed in Table 1. Four sets of fracture orientationwere representedat the sampling area shown in Figure 3 (marked in red). Not only thesurface survey on geology but also the hydrogeology experiments weremade here (Lee et al., 1996). The transmissivity was estimated as8.9×10−4 m/s where the upper and lower bound of transmissivitydistributions was 3.81×10−4 m/s and 2.27×10−5 m/s, respectively.The average hydraulic conductivity of 8.04×10−5 m/swas converted toa permeability of 1.02×10−10 m2 at 20 °C. The domain in the 3-D

Table 1Input parameters considered in this study (Chen et al., 2001).

Set-1 Set-2 Set-3 Set-4

(Trend, plunge) (41.3,16.6) (311.2,29.6) (133.9,20.3) (213.2,19.6)Orientation distribution Fisher Fisher Fisher FisherDispersion 5 5 5 5Average fracture length (m) 0.355 0.441 0.322 0.308Fracture length distribution Lognormal Lognormal Lognormal LognormalFracture length standarddeviation (cm)

0.093 0.159 0.094 0.189

Average fracture aperture 1.91E−4 1.91E−4 1.91E−4 1.91E−4Fracture aperturedistribution

Lognormal Lognormal Lognormal Lognormal

Fracture aperturestandard deviation

1.82E−4 1.82E−4 1.82E−4 1.82E−4

Fracture intensity (PVA) 0.356 0.034 0.365 0.850

P32 is the areal intensity variant with respect to the distribution of fracture size(Dershowitz, 1984).

fracture network was assumed to be 10 m×10m×10m. The referenceparameters are shown in Table 1.

A general study of enhanced Baecher's model was proposed todetermine the importance of variability on structures based on spatialintersections of fractures. Simulated within a 10 m cubic-block regionof a 1% gradient from upstream to downstream flow was used tomimic a small portion of andersite rock mass. Generic boundaryconditions were chosen to obtain a globally unidirectional flow forcedfrom one vertical face to the opposite face by assigning a constanthead boundary condition at the faces, as shown in Figure 4. Otherboundaries were sealed by a no-flow condition in order to lower thesignificance of the boundary effects. Then, the crack tensor approachcombined with discrete fractures (Oda, 1985) was decided with thetopological characteristics of the fracture-clustered area of a specifiedvolume, while other volumes not coincident with fractures weretreated as a matrix of low permeability on the magnitude of 10−13

Fig. 4. Modeling domains with no-flow conditions on upper, lower, front and backboundaries where the flow is forced by a gradient of 1%.

Fig. 5. Fracture intensity (P32) distribution.

Fig. 6. Permeability distribution in rectangular x, y, and z directions (units are m2).

172 J.-B. Pan et al. / Engineering Geology 116 (2010) 166–177

(m2). Finally, the grids were assigned to the mesh and numericallysolved using the finite element tool (FEHM) to get the subsequentflow dynamics and transport behavior.

9. Results and discussion

Results presented here correspond to the characteristics of fracturedistribution and features of first-order decay particles of the half-yearlife at the outlet of the 10×10×10 m fractured block for the fracture-induced flow pattern and hydraulic head distribution difference.Conditions considered in this study are a unidirectional head gradientof 1×10−2 m/m (1%) along one direction of the block, and a panel ofparticles upstream from the middle of the block.

The simulated fracture intensity distribution is shown in Figure 5.The distribution was directly numerically transferred from Figure 4via the crack tensor approach. Figure 6(a), (b), and (c) shows thepermeability distributions in three rectangular directions (Kx, Ky, andKz, respectively). The reference system of 1605 planes containsapproximately 8000 elements, allowing for a detailed and realisticrepresentation of flow and transport within a very complex system. Asolid view of the flow field is shown in Figure 7. The figure shows howthe complexity of the flow field is represented in a detailed mannerusing the proposed finite element approach. Homogeneous matrixproperties are considered here but the extension to heterogeneouscases is not considered; that is, the matrix is thought to have a singlepermeability of 10−13 (m2).

From Figure 7, the overall directional flow is consistent within theassumed boundary conditions (the flow is universally from upstreamto downstream). Figure 8 shows that the flow moves arbitrarily,especially as it moves through the higher fracture density distribution.However, a lot of clusters of higher fracture intensity (P32) lead to alarge vector distribution. Moreover, the much more connected porespace may be supplied within more chance for flow, on the contrary itis also shown that the vector will evade from the matrix in theimplicitly of lower fracture intensity.

One thousand non-reactive particles were released as a panel at(4.5, 9.0, 4.5), as shown in Figure 9. The overall spatially distributedparticles are shown in Figure 10 and Figure 11 for cross-sections atZ=5 and X=5, respectively. In Figure 10, the plane is depicted as a

Fig. 7. Pressure head distribution of the modeling domain.

Fig. 9. Releasing panel of particles in modeling domain.

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horizontal spreading of particles and two very interesting pointsaddressed here one point is that the vanity of space was the placewhere almost few particles want to visit and the other point is that theparticles' moving trend is alignedwithmore right-hand side than left-hand side. In Figure 11 the particles seem to follow a general path;they almost follow a regular pattern of moving straight downward.These spreading of particles are generally according to the directionalflow but the moving-late particles do not. The blank space ispurposely regarded as relatively less transmissible media of matrix.

Looking on the migration path shown in Figure 12, the path line iscalculated as the fastest-moving and the slowest-moving particles.The pattern in Figure 12(c) is different from that in Figure 12(a)because in the beginning, the slowest chooses the way that is filledwith matrix and it has to spend more time to pass the relativelystagnant zone. In Figure 12(b) and (d), the patterns are almost the

Fig. 8. The fracture intensity distribution

same because the directional flow may contribute. The shortest andthe longest transmission distance of 12 times the difference under theconsideration of non-reactive particles. From Fig. 13, comparison oftwo modes of discrete fracture network model (DFN), convectionalequivalent porous model (convectional EQM) with the mode was toldthat the early arriving time happened in DFN and moved quickernearly one order of magnitude because the particles in DFN did travelonly in fractures while the others both considered the matrix'scontribution to transport. From Figure 13, it was also clearly seen thatour model not only caught the early traveling time but also took thematrix's effect in consideration.

The decaying particles are introduced to represent the phenomenaof the decay and in-growth of reactive particles. Figure 14(a) and (b)shows the plumes of parent product A and daughter product B,respectively. Parent product A, which has a half-year half life withoutadsorption, was released as a fixed concentration at the same location

associated with the velocity vector.

Fig. 10. Particles' spreading at Z=5.

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in Figure 9. Following the distributed concentration of Figure 14, theplumes of A are alignedwithin the dominant flowing direction such asflow in Figure 11, but in Figure 14(b) the plumes of B are moving quitetotally different. The relative concentration of B is smaller than that ofA because of the effect on decay, and simultaneously the higherconcentration of B is not obviously under the same way that parent Adid. A possible reason for this anomalous transport is that the complexstructures of fracture intersections give rise to a non-continuous

Fig. 11. Particles' spr

connection of void space and the inherent decay from the parentproduct.

10. Conclusion

An approach for developing an applicable model from a site-specific discrete fractured network was demonstrated for a smallsynthetic block on a 10×10×10 m scale. The initial model included

eading at X=5.

Fig. 12. The fastest and slowest movement of particles. (a) Fastest movement of particles at Z=5, (b) Fastest movement of particles at X=5, (c) Slowest movement of particles atZ=5 and (d) Slowest movement of particles at X=5.

Fig. 13. Comparison of these three different models.

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1605 fractures associated with complex patterns and the heteroge-neity of assigned continuum grids. A theoretical estimated perme-ability methodwas developed to deal with the complexity of fracturesin fractured rocks. The system was simplified using a crack tensor incombination with a discrete fracture network. This idea can beextended to a framework that couples groundwater flow andcontainment transport in fractured rock mass.

The results reveal the characteristics of the geometrical connectionof fracture networks under the prescribed flow field. The simulatedfracture networks were transformed into a grid-based simulationprocedure under consideration of the delaying effect in matrix owingto particle migration. Synthetic flow results indicate that the flowvector is very sensitive to fracture intensity (P32); a high fractureintensity (P32) leads to a high velocity. Dynamics of particles can beaddressed in some significant points that the space discrepancy maypose some fracture-clustered heterogeneity on transport as well asthe ratio of 12 times differences between the longest and shortestparticles' traveling distance. This effect will be extremely sensitive todecay-ingrowths transport because the plumes of daughter productare relatively tortuous and anomalous to predict.

The ability of the system simplified to major transport pathsproves that lower fracture intensities (background fractures) have aminor effect on the flow and transport. This result is closely relatedwith low connectivity of the larger fractures for the particular blockand is not expected for a more connected system where one wouldinfer that (1) increase onflowproperties of blocks and (2) consequentlyincrease at least the contact surface with the matrix zones and as

Fig. 14. Particles within decay. (a) Plumes of parent product A and (b) Plumes ofdaughter product B.

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such the retention properties of the block, found on Christophe et al.(2008).

Although using a three-dimensional approach is difficult in realworld applications, tentative studies on simple academic cases andgeometries can still provide valuable insights, following, for instance,the line of Rubin et al. (1999) or Wu et al. (2004). This integratedframework can be used to determine the benefit of using the cracktensor approach to capture the characteristics or non-uniformity ofthe geologic structure.

Acknowledgements

The authors would like to thank "Final LLW and HLW DisoposalTeam" at the Institute of Nuclear Energy Research in Taiwan forhelping generate the fracture networks and flow simulation and also

financial support coming from National Research Council under thecontract of NSC-98-3114-E-007-015.

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