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Particle tracking approach for transport in three-dimensional discrete

fracture networks

Article  in  Computational Geosciences · October 2015

DOI: 10.1007/s10596-015-9525-4

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Comput GeosciDOI 10.1007/s10596-015-9525-4

ORIGINAL PAPER

Particle tracking approach for transportin three-dimensional discrete fracture networksParticle tracking in 3-D DFNs

Nataliia Makedonska1 · Scott L. Painter1,2 ·Quan M. Bui1,3 ·Carl W. Gable1 · Satish Karra1

Received: 3 November 2014 / Accepted: 28 August 2015© Springer International Publishing Switzerland (outside the USA) 2015

Abstract The discrete fracture network (DFN) model is amethod to mimic discrete pathways for fluid flow through afractured low-permeable rock mass, and may be combinedwith particle tracking simulations to address solute trans-port. However, experience has shown that it is challenging toobtain accurate transport results in three-dimensional DFNsbecause of the high computational burden and difficultyin constructing a high-quality unstructured computationalmesh on simulated fractures. We present a new particletracking capability, which is adapted to control volume(Voronoi polygons) flow solutions on unstructured grids(Delaunay triangulations) on three-dimensional DFNs. Thelocally mass-conserving finite-volume approach eliminatesmass balance-related problems during particle tracking. Thescalar fluxes calculated for each control volume face bythe flow solver are used to reconstruct a Darcy velocityat each control volume centroid. The groundwater veloc-ities can then be continuously interpolated to any pointin the domain of interest. The control volumes at fractureintersections are split into four pieces, and the velocity isreconstructed independently on each piece, which resultsin multiple groundwater velocities at the intersection, one

� Nataliia Makedonskanataliia@lanl.gov

1 Computational Earth Science Group, Los Alamos NationalLaboratory, Los Alamos, NM 87545, USA

2 Present address: Environmental Science Division, Oak RidgeNational Laboratory, Oak Ridge, TN 37831, USA

3 Present address: Applied Mathematics, Statistics,and Scientific Computation Program, University of Maryland,College Park, MD 20742, USA

for each fracture on each side of the intersection line. Thistechnique enables detailed particle transport representationthrough a complex DFN structure. Verified for small DFNs,the new simulation capability enables numerical experi-ments on advective transport in large DFNs to be performed.We demonstrate this particle transport approach on a DFNmodel using parameters similar to those of crystalline rockat a proposed geologic repository for spent nuclear fuel inForsmark, Sweden.

Keywords Discrete fracture network · Subsurface flow ·Numerical modeling · Control volume method ·Advective transport · Particle tracking

1 Introduction

Modeling flow and solute transport in sparsely fracturedrock remain a significant challenge [29]. Among the mul-tiple conceptual approaches available [13, 20, 29, 30, 40,48], the discrete fracture network approach is conceptuallyappealing for its direct use of site-specific observations onfracture geometry and its avoidance of a representative ele-mentary volume assumption. The DFN approach (e.g., [6,12, 15, 25, 39]) attempts to represent, in a statistical sense,discrete groundwater flow paths by direct stochastic simula-tion of geometry and properties of individual fractures usinginformation from site characterization activities. DFN mod-eling is of interest in a wide range of applications, but isparticularly well suited to modeling flow in fractured for-mations that have negligible flow in the background matrix.Detailed applications to the Aspo Hard Rock Laboratory[9, 37], Laxemar [47], Forsmark [20, 41–43], and Olkiluoto[38] sites for potential nuclear waste repositories clearlydemonstrate the value of the DFN approach in practice.

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The typical steps in representing transport in a DFN-based modeling approach would involve stochastically gen-erating the DFN, solving for flow, and then solving fortransport. As is true for subsurface transport in general, thechallenge of minimizing numerical dispersion in advection-dispersion equation-based transport simulations [3, 45]motivates the use of particle-based methods as alternativesto solving the system of equations resulting from discretiz-ing the coupled advection-dispersion and matrix diffusionequations.

Implementation of particle tracking algorithms on thefully unstructured meshes that arise from DFN modeling issensitive to how the flow solution is obtained. Finite elementmethods are common in production codes because of theirgeometrical flexibility [2, 11, 14, 16, 19]. However, experi-ence has shown that particles can become stuck in elementsassociated with numerically stagnant regions [17, 18, 31]either due to poor mesh quality or, in the case of a Galerkinfinite element solution, due to lack of local mass conser-vation. Stuck particles limit the usefulness of flow fieldprovided by finite element methods for particle trackingsimulations [18]. To avoid stuck particles, Cacas et al. [6]and Dershowitz and Fidelibus [12] developed networks ofone-dimensional pipe segments. The pipe-network approx-imation leads to fast computations with no stuck particles,but the three-dimensional structure of fracture connectiv-ity is significantly simplified and many details of the fullythree-dimensional particle motion are lost. Research codesbased on mixed hybrid finite elements, which are locallymass conserving in contrast to the Galerkin finite elementmethod, have recently grown in popularity [10, 26, 27]and presumably should be superior for particle tracking ona DFN. An alternative to mixed hybrid finite elements isthe control volume (or finite-volume) method [46], whichis locally mass conserving by construction and is wellsuited for multiphase flow simulations. Motivated by theneed for locally mass-conserving flow solutions on high-quality unstructured meshes which need to be compatiblewith existing high-performance flow codes [24] based onthe control volume method, Hyman et al. [22] presenteda DFN meshing approach targeting control volume flowcodes. Their method ensures a high-quality Voronoi con-trol volume computational mesh on fracture planes andunions of Voronoi polygons at fracture intersections. Con-trol volume codes based on the two-point flux method (e.g.,PFLOTRAN [24]) do not provide velocity fields directly,and additional work is required to extend the approach toinclude particle tracking [22].

Here we present a method for performing particle track-ing using control volume flow solutions obtained from DFNgrids using the feature rejection algorithm for meshing(FRAM) developed by Hyman et al. [22]. In this paper, wefocus on the representation of advective transport because

it has the fundamental control on both the downstreammovement of solutes and the exchange of solutes betweenfractures and porous matrix (e.g. [9, 34]). The DFN genera-tion and meshing capability of FRAM makes it possible touse existing subsurface flow codes, which provide the scalarfluxes on each control volume face. From the obtainedfluxes, the Darcy velocity is reconstructed for each vertexin the network using an approach described previously [36].We focus here on the reconstruction of velocities on fractureintersections, on problematic configurations associated withfracture boundaries, and the subsequent tracking of parti-cles in the reconstructed flow field. The particle trackingalgorithm on DFNs is described in Section 2. Verificationtests and demonstration examples of the particle trackingalgorithm are provided in Sections 3 and 4, respectively,followed by discussion and conclusions in Section 5.

2 Methodology

2.1 Computational mesh characteristics

A wide variety of lengths, stochastic location, and orien-tation in space, coupled with the complexity of the inter-sections between fractures, are typical characteristics ofthree-dimensional DFNs. These characteristics create diffi-culties in velocity reconstruction and particle transport on anunstructured computational mesh. Additionally, poor meshquality contributed by a high aspect ratio, a high ratio of thelongest to the shortest side of triangular cells, degrades thequality of the numerical solution.

Regardless of whether the mesh is generated for use ina control volume or finite element method for flow calcu-lation, the major meshing issues are associated with patho-logical geometry that leads to high-aspect-ratio cells. Toovercome the meshing difficulties, Hyman et al. [22] pro-posed the FRAM approach to generate Delaunay triangula-tions on stochastically generated DFNs. The FRAMmethodworks by rejecting fracture configurations that would createa pathological mesh element. We refer the reader to Hymanet al. [22] for detailed explanations of the DFN generationand meshing technique. Figure 1 shows an example of athree-dimensional stochastically generated DFN using theFRAM algorithm.

We point to mesh characteristics that are being usedin the particle tracking algorithm for transport modelingin the current work: (1) mesh faces (triangle edges) con-form to fracture/fracture intersection lines; (2) edges that areincident on a fracture intersection line are shared by fourtriangles, one on each side of each of the intersecting frac-tures; (3) the mesh is fine near fracture intersections andcoarse away from fracture intersections, thus reducing themesh size (total number of vertices and cells) in order to

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Fig. 1 Example of stochastically generated discrete fracture net-work with 5033 individual fractures in a domain of size1000 m × 1000 m × 1000 m. Each fracture in the network is displayedin a single color. The DFN is generated according to the DFN stochas-tic parameters representative of the natural repository site Forsmark,Sweden (Table 2)

reduce computational cost; (4) the mesh cells meet stringentcriteria on the cell shape and overall mesh quality to ensurestability and accuracy of the physical solution.

We focus on planar fractures here, but note that non-planar fractures can be easily accommodated in everythingpresented here; the approach in that case would be to repre-sent non-planar fractures with multiple planar sub-fractures.As we are not considering flow in the matrix here, it is nec-essary that the DFN be composed of one or more connectedclusters that connect an inflow to an outflow boundary forflow to occur.

Figure 2 shows a close-up view of several intersectionlines of a few fractures with fracture intersection conform-ing Delaunay triangulations (black solid lines). Mesh ver-tices display different features depending on their locationon the mesh and are classified as follows:

• Interior is a vertex located inside a fracture; it does notbelong to a fracture boundary or fracture intersection.

• Exterior is a vertex that resides on the fracture exteriorboundary.

• Interior interface is a vertex located on a line of inter-section between fractures.

• Exterior interface is a vertex that not only belongs to anintersection line but also is located on a boundary of one(or both) of the intersecting fractures.

In the last two cases, interface vertices are considered to becommon to both intersecting fractures and belong to bothfractures simultaneously.

Since the goal of the meshing is to obtain the flow solu-tion using the two-point flux-based control volume methodfollowed by solute transport modeling, accuracy is achieved

by deriving Voronoi control volume cells from the Delau-nay triangulation. The triangulation of each fracture meetsthe Delaunay triangulation criteria with the added constraintthat no boundary triangles have obtuse angles incident onthe exterior boundary [28]. As a result, each fracture isdivided into control volume cells formed by perpendicularbisectors between adjacent vertices in an underlying trian-gulation of the cell centers (Fig. 2). This perpendicularityattribute ensures accuracy of the evaluated mass flux usingthe two-point-based control volume method.

2.2 Flow velocity field reconstruction

The particle simulation requires the flow velocity to bedefined at all points of the simulation domain. Use of a flowsolution obtained on an unstructured control volume grid isnot straightforward because the control volume flow solu-tion does not provide a continuous velocity field. Instead, itprovides a set of scalar quantities that are approximationsto the normal component of Darcy flux integrated over eachedge of each control volume cell. Thus a reconstruction of acontinuous velocity field from the scalar quantities obtainedfrom flow solver is required.

Recently, Painter et al. [36] developed an approach forreconstruction of a velocity field using flow solution (andthe corresponding Darcy fluxes) obtained on unstructuredcontrol volume grids. The approach uses an unconstrainedleast square method on an interior cell when the cell cen-ter is an interior vertex. A linearly constrained least squaremethod is applied to reconstruct velocities on boundarycells for exterior vertices that have Neumann boundary con-ditions. The Darcy velocities thus obtained can then becontinuously interpolated to any point in the domain ofinterest. Using two-dimensional tests, they demonstratedcorrect reproduction of uniform corner-to-corner flow on afully unstructured grid. This approach by Painter et al. [36]is extended and tested for the first time here for transportmodeling in DFNs.

2.2.1 Flow velocity reconstruction on fracture intersections

The main difficulties in DFN transport modeling comeinto play at fracture intersections. In the FRAM approachused here, control volume cells centered on the interiorand exterior vertices are two-dimensional planar polygonsin three-dimensional space. The control volume cells cen-tered on interface vertices (e.g., at fracture intersections)are three-dimensional objects formed from the union of twopolygons in different planes. Moreover, in the fracture inter-sections, the flow behavior is more complicated than that atinterior or exterior vertices. For example, the main flow cango through the intersection and continue in the same fracturein the same direction or change its direction and proceed in

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exterior verticesinterior inteface v.exterior interface v.

Fig. 2 A few intersecting fractures showing conforming Delaunaytriangulations (black solid lines). The Delaunay triangulation edgesconform to the fracture intersection without distorting the lines ofintersection of the fracture geometry. The Voronoi control volumes(colored solid lines shown in zoom in figure) are common to both

intersecting fractures along the line of intersections, forming three-dimensional objects. Four types of vertices are distinguished on thecomputational grid: interior (vertices of Delaunay triangulation), exte-rior, interior interface, and exterior interface vertices. Note that themesh resolution is fine near intersections and coarse away fromfracture intersections

the intersecting fracture, or as is usually the case, the flowis split by some percentage between the fractures. There-fore, flow velocity information is needed on each side of theintersection and on each fracture.

In order to provide the necessary flow information atfracture intersections, the technique schematically presentedin Fig. 3 is developed. First, the control volume cell onthe intersection (Fig. 3b) is split into two two-dimensionalpolygons, each of them corresponding to the part of the con-trol volume on one of the intersecting fractures (Fig. 3c).Second, each obtained control volume polygon is furtherdivided into two parts along the intersection line (Fig. 3d).Thus, the control volume at the intersection is split into fourpolygons. Afterward, each of the four polygons is used toreconstruct flow velocity in the same way as an interior ver-tex. Figure 3e shows the reconstructed flow velocities onthe interface vertices accompanied with velocities on theinterior vertices of neighboring triangular cells. The veloc-ity vectors on the interface vertices show flow direction oneach of the four fracture portions adjacent to the intersectionline: the blue part of fracture 1 shows outgoing flow fromthe intersection line; the purple part of fracture 1 and yel-low and green parts of fracture 2 indicate flow toward thefracture intersection.

Figure 4 illustrates the implementation of this techniqueand shows the flow field of an individual fracture extractedfrom the DFN of Fig. 1. The vertices are colored by theirrespective pressure magnitude, where low pressure corre-sponds to dark blue colors. The black arrows representreconstructed flow field velocities, which are always in theplane of a fracture. There are 10 fracture intersection lineson the fracture providing locations for flow to pass throughthe fracture. Flow is coming in from some intersections andgoing out through others. Two frames, zoomed in on inter-section lines with flow outgoing from intersection and flow

incoming into intersection, are also shown. As a result ofthe dividing of control volume cells into two polygons alongthe intersection line, two flow velocity vectors are asso-ciated with each interior interface vertex. In addition, twoflow velocity vectors are associated with the same vertexon the intersecting fracture, which is out of plane in figure.Thus, each vertex on a fracture intersection has four recon-structed velocities, one for each side of the intersection oneach fracture.

The procedure for splitting a control volume also allowsthe effects of pumping or injection wells to be represented.Specifically, by splitting a control volume that containsa specified groundwater sink associated with a pumpingwell, two velocities become associated with the node andboth of these will be directed toward the extraction node.Thus, the procedure of Painter et al. [36], which consid-ered source-free regions only, is extended. Pumping wellsare not further addressed here but have been demonstratedin [23].

2.2.2 Special cases on fracture boundaries

Figure 5 shows three special configurations where our basicalgorithm must be modified; these cases all involve exter-nal boundary vertices. The modifications are necessary toavoid the relatively rare situation where nonphysical behav-ior is observed in the particle motion, either exiting no-flowboundaries or getting stuck or nearly stuck in a stagnantflow region in a region free of true physical sinks.

Figure 5a shows a zoom in frame on a circular-shapefracture boundary, where a discretized round perimeterforms obtuse exterior angles. The approach used for flowvelocity reconstruction [36] implies no flow outside of frac-tures, and on fracture boundaries, flow is specified alongstraight boundaries. However, in the case of an obtuse angle

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(a) (b)

(c)

(e) (d)

Fracture 1

Fracture 2

Fig. 3 Illustration of the technique applied to interface control vol-ume cells for the flow velocities reconstruction. a The interior interfacevertex shown on the intersection of two fractures, fracture 1 (blue)and fracture 2 (green), belongs to both fractures. Each single three-dimensional control volume cell b is used for the purposes of obtainingthe flow solution. The control volume cell on the intersection is splitinto two two-dimensional polygons c, one for each of the intersectingfractures. Those polygons are further divided along the intersectionline; in d, the blue and purple polygons belong to fracture 1, and theyellow and green polygons belong to fracture 2. The flow velocities arereconstructed on each of the split polygons, with the result e being thatfour velocities are associated with each interior interface vertex alongfracture intersection lines

on a boundary, a reconstructed flow velocity can point out-side of the fracture (red arrow in Fig. 5a). To ensure physicalcorrectness in such cases (i.e., no mass is lost across ano-flow boundary), the velocity vector is simply redirectedalong the boundary line on the downstream side of thevertex, keeping the same velocity magnitude.

The same procedure is applied to a velocity vectoron external vertices with Dirichlet boundary conditions(Fig. 5b). Velocities of those vertices are calculated as inte-rior vertices, since they belong to either an inflow or outflowboundary of the domain. At the vertices on the intersectionof a Dirichlet and Neumann boundary condition, the no-flow constraint on the velocity reconstruction is not appliedand the velocity vector may point outside of a fracture. Insuch cases, we simply post-process the velocity to direct italong the fracture boundary edge.

Specified no-flow boundary conditions on fractureboundary vertices constrain the direction of flow to be along

the boundary lines. However, in rare cases, the flow veloc-ities may be reconstructed to be antiparallel and pointingtoward each other (Fig. 5c). There are two preconditionsfor such special case: (a) the boundary triangular cell isvery close to fracture intersection, but none of the triangu-lar edges belong to the intersection line; (b) the fluid flow isrunning toward the intersection from both sides of the inter-section line (Fig. 5c, green arrows show flow direction). Itis seen that this situation can artificially and significantlyslow down particle movement if a particle would attempt

1.93

1.94

1.95

1.96

1.97

1.98

P [MPa]

Fig. 4 The flow field of an individual fracture extracted from a DFN(previously shown in Fig. 1) is viewed down the axis normal to thefracture plane. The vertices are colored by their corresponding pressurevalue: low pressure is indicated by blue colors, and high pressure isindicated by red colors. The black arrows represent the reconstructedflow field velocities. Detailed views of fracture intersections are shownin zoom in fragments: on the top figure, flow is toward the intersection,while on the bottom figure flow comes into the shown fracture fromthe intersecting fracture and moving in both directions away from theintersection line. Note that two velocities are shown for each interiorinterface vertex. There are two more corresponding velocities on theintersecting fracture that is not in the plane of the image (not shownhere)

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Neumann b.c.

intersection line flow direction

Neumann b.c.

In - flow boudary

(a)

(b)

(c)

Fig. 5 Three pathological cases are considered for flow velocitieson fracture boundaries: a obtuse angle on a fracture perimeter; b ona fracture corner, where the external vertex represents Neumann andinflow (or outflow) boundary conditions; c antiparallel velocity vec-tors, located on the boundary and close to an intersection line with theflow toward the intersection

to transit through this boundary triangular cell. To avoidunphysical artifact as in this case, the following procedure isapplied. First, the boundary triangular cell with two bound-ary vertices and antiparallel velocity vectors is identified.The next step is to detect the closest vertex that belongsto the fracture intersection line. It is not necessary that theintersection vertex be a part of the boundary triangular cell;most likely, it belongs to one of the neighboring cells. Thelast step is to redirect one of the antiparallel velocity vectorsalong the edge that connects the boundary and intersectionvertices toward the intersection line, keeping its originalmagnitude.

Although our model domain is three dimensional, allfractures are planar objects in the domain. During the veloc-ity reconstruction process and subsequent particle trackingsimulation, each fracture is viewed as a two-dimensionalpolygon. It significantly reduces computational cost, and itis also convenient due to the fact that particles travel alonga two-dimensional polygon in a three-dimensional space. Inorder to map three-dimensional fracture coordinates to two-dimensional x − y space, the rotational matrix, MR , givenby

MR = cosϕI + sinϕ[n]× + (1 − cosϕ)n ⊗ n, (1)

is applied to all vertices in each fracture of the DFNprior to any numerical calculations. In Eq. 1, n is a three-dimensional normal vector of a fracture, [n]× is the crossproduct matrix of the fracture’s normal vector, [n]× =⎡⎣

0 −nz ny

nz 0 −nx

−ny nx 0

⎤⎦ . n ⊗ n is the tensor product of the

normal, n ⊗ n =⎡⎣

n2x nxny nxnz

nxny n2y nynz

nxnz nynz n2z

⎤⎦ . I is the identity

matrix, and ϕ is an angle that the fracture forms with thex −y plane. A particle’s instantaneous velocity and its localtwo-dimensional positions are defined using MR of the cur-rent fracture. Once the particle passes from one fractureto another, its new positions are calculated based on therotational matrix of the new fracture, where the particle iscurrently positioned.

2.3 Particle tracking procedure

Once the discrete representation of the flow velocity field isdefined and every vertex on the DFN mesh is assigned flowvelocity vectors, a particle’s instantaneous velocity can bedetermined at any point in the DFN.

The standard barycentric interpolation approach [7] isapplied to determine velocity at any location within a cell. Inplanar two-dimensional space, if r1, r2, and r3 are the posi-tions of vertices of the triangular cell (where the particle is

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located at the current time step), υ1, υ2, and υ3 are veloci-ties at those vertices, then the particle’s velocity υ at someposition r in the triangular cell can be interpolated as

υ(r) = �i=1,2,3λiυi , (2)

where(λ1λ2

)= T −1(r − r3). (3)

Here, T is a 2 × 2 matrix that has r1 − r3 as the firstcolumn and r2 − r3 as the second column with λ3 = 1 −(λ1 + λ2).

To define time step, Δt , of each particle movement, thesame approach of barycentric interpolation is applied:

Δt = �i=1,2,3λi

√Aci

|υi | , (4)

where λi is a weight value calculated by Eq. 3, Aci is thearea of the control volume cell whose center is i vertex,υi is a reconstructed velocity at i vertex. Taking unstruc-tured grids into account, where the control volume cell areavaries along a fracture, Eq. 4 provides a non-uniform timestep adapted to each control volume and satisfies stabilitycriteria.

Because the particle’s velocity, v, and time step, Δt , areavailable at any point in the domain, it is straightforward todefine the particle’s new position. A first-order predictor-corrector method with adaptive time step control is appliedin the current particle tracking algorithm. For every timestep, the new particle position is defined in two steps, wherethe first predictor step is as follows:

rp(t + �t) = r(t) + υ(r(t))�t. (5)

Then, using Eq. 2, the new particle’s velocityυ

(rp(t + �t)

)is evaluated. The particle is moved to its

new position calculated by the corrector step using thevelocity calculated from the predictor step:

r(t + �t) = r(t) + υ(rp(t + �t)

)�t. (6)

This predictor-corrector technique prevents particlesfrom reaching the edge of a fracture with no-flow boundaryconditions.

The particle movement through an unstructured grid onthe DFN requires determination of the triangular cell wherethe particle resides at any time step. In our model, an edge-crossing test is performed every time step. The particle’smovement inside one fracture is straightforward: after defin-ing the triangular cell currently occupied by the particle,the particle’s instantaneous velocity and time step are calcu-lated by Eqs. 2 and 4, respectively, followed by Eqs. 5 and6, which determine the new particle position.

When the particle gets into the cell adjacent to an inter-section line, the distance between the current position and

intersection line is computed. If the distance is smaller thanthe particle’s movement during the last time step, then onepredictor step (Eq. 5) is done and a test on crossing the inter-section line is performed. If satisfied, the particle is stoppedat the line representing the fracture intersection.

Being located on the intersection line implies that the par-ticle is facing three triangular cells (excluding the originalcell with flow incoming to the intersection), which belongto two intersecting fractures. The next important step is tomake a decision about which cell will be occupied by theparticle at the next time step. The downstream cell is chosenrandomly with probability proportional to the outgoing flux.For example, if there are two cells with outgoing flow, andthe mass flux interpolated to these cells is qe1 and qe2, thencell 1 is chosen if u ≤ |qe1||qe1|+|qe2| , where u is a generated ran-

dom number, uε(0, 1). Conversely, if u >|qe1||qe1|+|qe2| , then

cell 2 is chosen. The chosen triangular cell dictates the nextfracture where the particle continues to travel.

3 Verification tests

The four-fracture DFN shown in Fig. 6 is chosen as afracture network for verification purposes. The flow andtravel time solution for this network can be calculated usingequivalence to an electrical circuit [4], where fracture per-meability plays the role of conductance of electric current.In this case, particle travel times through the DFN may beestimated analytically and compared to numerical results.

Three horizontal rectangular fractures are connectedthrough one vertical fracture (Fig. 6). Pressure boundaryconditions are applied on −x and x faces of the domain, 2and 1 MPa, respectively. Fluid flow goes from the left to theright side of the domain and obeys Darcy’s law:

Q = −kA�P

μL, (7)

where Q is a fluid discharge, k is a fracture permeability,A is a cross-sectional area, �P is applied boundary pressuredrop between right and left sides of the domain, μ is a fluidviscosity, and L is a length of fluid path. Darcy’s law forfluid flow is analogous to Ohm’s law for electrical circuit,leading to

Q = −�P

Rt

, Rt = μL

kA. (8)

To calculate total flow in the DFN, the equivalent resistornetwork (Fig. 6, insert) is used. The total resistance Rt in thenetwork is

Rt = R1 + (R24 + R4) · (R23 + R3)

R24 + R4 + R23 + R3. (9)

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Once Q is computed, Darcy flux q, and flow velocityv = q/n, where n is porosity, are calculated analytically,thus obtaining the analytical value for particle travel time.

In our model, fluid viscosity is μ = 1.002 [Pa s], crosssection A is constant all over the DFN, A = 0.01 [m2].Length, L, of R1, R3, and R4 paths is 0.9 [m]; length of R23

and R24 paths is 0.2 [m]. There are two tests of flow solu-tion and particle tracks are performed in the DFN. In thefirst test, the fracture permeability k is constant for everyfracture in the DFN: k1 = k2 = k3 = k4 = 2.5 × 10−13

[m2]. In the second test, fracture 3, the bottom fracture inthe domain, is giving higher permeability k3 = 2.5× 10−12

[m2], and k1 = k2 = k4 = 2.5 × 10−13 [m2].The numerical results of steady-state flow solutions are

shown in Fig. 7 for both tests. In the first test (Fig. 7a),fluid pressure gradually decreases from the left to the rightside of the domain; it does not show significant differ-ence between top and bottom fractures. In the second case(Fig. 7b), as a result of higher permeability on fracture3, the pressure on it is lower than on fracture 4. There-fore, in the second test, the pressure gradient is significantlylower on the right side of the domain than in the firstcase.

The particle transport numerical experiments are per-formed for both tests. Initially, all particles are placed on theinflow boundary side, on the left edge of fracture 1 (Fig. 6),moving toward fracture 2. When they reach the first inter-section, particles choose one of two paths either through thetop (fracture 4) or bottom (fracture 3). Figure 8 shows par-ticle trajectories of 50 particles. In the first test (Fig. 8a),both flow pathways are equal in their ability to flow; there-fore, particles are equally apportioned between two paths.The distribution is significantly different in the second test(Fig. 8b), where the bottom path is more likely for flowthan the top one. However, there is a small probability ofchoosing the way through fracture with lower permeability;therefore, we observe a few particles traveling through thetop fracture of the DFN.

The numerically measured and analytically calculatedvalues of particle travel time for both considered tests aregiven in Table 1. This good comparison between numericaland analytical results in the above verification test allows usto use our algorithm to particle transport modeling in a largeDFN similar to natural sites.

4 Numerical results on a demonstration example

4.1 Stochastically generated DFN, similar to a naturalrepository site

In this section, we demonstrate the results of advectivetransport modeling in subsurface fracture networks on a

stochastically generated DFN based loosely on a well-characterized repository site at Forsmark, Sweden. TheForsmark area is located in northern Uppland within themunicipality of Osthammar, about 120 km north of Stock-holm. The area consists of crystalline bedrock that has beenaffected by both ductile and brittle deformation. The ductiledeformation has resulted in large-scale, ductile high-strainbelts and more discrete high-strain zones. Tectonic lenses,in which the bedrock is fractured but less affected by ductiledeformation, are enclosed between the ductile high-strainbelts.

The DFN parameters used here (Table 2) are simplifiedfrom those given in [44] and applied in a three-dimensionaldomain of size 1000 m × 1000 m × 1000 m. Three setsof fractures with circular shape are oriented according to aFisher distribution [21]:

f (θ) = κ · sin θ · eκ·cos θ

eκ − e−κ, (10)

where θ is the deviation of the fracture pole orientation fromthe mean orientation and the parameter κ > 0 is the concen-tration parameter. The concentration parameter quantifiesthe degree of clustering; values approaching zero representa uniform distribution on the sphere and large values implysmall average deviations from the mean direction.

The fracture sizes at Forsmark obey a truncated powerlaw distribution with lower and upper cut-off radii of cir-cular fractures, R0 and Ru, respectively. The power lawdistribution is sampled by first generating uniformly a

Fracture 1, k

Fracture 4, k

Fracture 3, k1

2

3

4

In-flow boundary

Fracture 2, k

Out-flow boundary

Out-flow boundary

Fig. 6 Discrete fracture network of four fractures is used for the verifi-cation test of the particle tracking method. Constant pressure boundaryconditions are applied to x faces of the domain, and flow goes from theleft to right side of the DFN. Domain size is 1.8 m × 1.0 m × 0.65 m.Fracture aperture is constant and equal to 0.01 m. The insert showsan electrical circuit that is an analog to the DFN for purposes ofcalculating the verification target values of flow and particle traveltimes

Comput Geosci

Fluid Flow Pressure [MPa]

1.05

1.95

1.50

(a)

(b)

Fig. 7 The steady-state flow solution obtained for the four fracturesDFN used in the two verification tests (Fig. 6). Light red colors cor-respond to high pressure. a Pressure solution obtained for DFN whereall fractures are designated with equal permeability value, k1 = k2 =k3 = k4 = 2.5 × 10−13 m2. b Pressure solution obtained for thesame DFN, where fracture 3 is prescribed higher permeability, k3 =2.5 × 10−12 m2, k1 = k2 = k4 = 2.5 × 10−13 m2

random number u over the open interval (0.0, 1.0). Thefracture radius, R, is then calculated as

R = R0 ·[1 − u + u ·

(R0

Ru

)a]−1/a

. (11)

Fracture transmissivity, σ , is estimated using a power lawrelationship of a correlated transmissivity model [44]

log(σ ) = log(γ · Rβ), (12)

with parameters γ = 1.6 × 10−9, β = 0.8. The frac-ture aperture, b, is partially correlated to fracture size andcalculated using the cubic law (e.g., [1])

σ = b3

12. (13)

Three fracture sets are represented in the DFN. Thestochastic parameters for the three fracture sets are given inTable 2. Figure 1 shows one of the DFN realizations (thelower cut-off of fracture size is R0 = 15 m), where eachfracture is shown by a different color. Note that we considerclusters of fractures connected to each other and each clusteris connected to domain faces. Isolated clusters and isolatedfractures are removed at the end of the fracture-generatingprocedure since they would not participate in flow in themodel domain.

4.2 Steady-state flow solution

The control volume method is used here to obtain thegroundwater flow solution. Applied to flow, the control vol-ume method solves conservation equations for water masson each Voronoi control volume cell. Applied to trans-port simulation, the control volume method solves the massconservation equation on each computational cell, whichis a necessary requirement to avoid local mass stagnationand artificial sources and sinks due to local errors in massconservation.

Our DFN modeling capability has been adapted to workwith the finite element heat and mass flow (FEHM) simula-tor [49] as well as with the massively parallel PFLOTRANsimulator [24]. FEHM and PFLOTRAN are able to accom-modate the fully unstructured grid, which is locally twodimensional within each fracture. The link between ourDFN capability and PFLOTRAN makes it possible to per-form flow simulations on large DFN grids by utilizing state-of-the-art high-performance parallel computing hardware.Additionally, the complex multiphase flow and equations ofstate in FEHM and PFLOTRAN can also be applied to DFNsimulations.

A flow solution on a DFN with statistical propertiessimilar to the Forsmark site (shown in Fig. 1) using PFLO-TRAN is presented in Fig. 9. The transmissivity distributionin the DFN (Fig. 9a) is based on fracture size power lawdistribution and is used for solving the steady-state fully

(a)

(b)

Fig. 8 Colored lines on four fracture verification tests show particletrajectories. a All the fractures are given the same permeability value,and approximately the same number of particles travel through fracture3 as through fracture 4. b Fracture 3 (bottom fracture) is given higherpermeability. As a result, more particles travel through the bottom pathof the DFN

Comput Geosci

Table 1 Numerical and analytical results of particle travel time in verification tests

Travel time Travel time Error Mass fraction per Mass fraction per Error

Analytical [s] Numerical [s] [%] path Analytical path Numerical [%]

Test 1 1.802 · 103 1.801 · 103 0.055 0.50 0.496 0.806

0.50 0.504 0.793

Test 2, path 1 1.025 · 103 1.017 · 103 0.786 0.905 0.908 0.33

Test 2, path 2 2.926 · 103 2.923 · 103 0.102 0.095 0.092 3.26

saturated flow to obtain pressure (see Fig. 9b). In this exam-ple, constant pressure boundary conditions are applied tothe z−faces of the domain. All the vertices of the DFNlocated on the bottom face (−z) of the domain are giventhe lowest pressure value, 1.0 MPa, while all the ver-tices on the top face of the domain (+z) are given thehighest pressure, 2.0 MPa. No-flow boundary conditionsare applied on the remaining boundary faces of the DFN.These boundary conditions result in flow from the bot-tom to the top of the domain. In this calculation, gravityis not considered, for simplicity. In Fig. 9b, red color rep-resents high pressure, while blue color corresponds to lowpressure.

4.3 Particle tracking results of numerical experiments

In this section, three numerical experiments of particletracking on the large DFN in Fig. 1 are performed. In eachexperiment, the starting positions of particles are chosen dif-ferently. The particle travel time is recorded during eachnumerical experiment. The travel time distribution behav-ior analysis is of high interest for understanding subsurfaceflow and contaminant transport.

In the first numerical experiment of particle transportthrough the DFN, all particles were uniformly distributedinitially on a small region (20 m × 20 m) in the centerof the inflow bottom boundary. This experiment allows usto observe how fast particle trajectories diverge from theirinitial common location (Fig. 10a) due to high topologicalvariety of the DFN and the probabilistic approach used onfracture intersections. Final positions of particles are cap-tured at the outflow boundary and plotted with their initialpositions in Fig. 10b. A wide dispersion orthogonal to thedominant direction of the pressure gradient is observed,along with channeling, with higher particle density on largefractures. This is explained by the higher transmissivityvalue of large fractures, which provide the main flow. There-fore, once a particle reaches a large size fracture, it is moreprobable for the particle to continue moving on this highertransmissive fracture.

In the second experiment, particle starting positions arerandomly chosen in all the fracture cells of the entireDFN domain. These particles are released in the steadyflow field and travel toward the outflow boundary. Twothousand particle trajectories are shown in Fig. 10c. Thetravel time cumulative distributions of both experiments are

Table 2 DFN parameters used in simulations of the Forsmark repository site, Sweden [43]

Set Orientation distribution: Size distribution: Fracture density

Fisher power law

Mean Mean Conc. a Ru, m R0, m Number of fractures

trend plunge κ in 1 km3

1. (NS) 90.0◦ 0.0◦ 21.7 2.5 560 15 2093

20 1019

25 583

2. (NE) 135.0◦ 0.0◦ 21.5 2.7 560 15 2000

20 919

25 503

3. (HZ) 360.0◦ 90.0◦ 8.2 2.38 560 15 7711

20 3887

25 2285

Comput Geosci

(b)Fluif Flow Pressure [MPa]

2.00

1.75

1.50

1.25

1.00

(a)Transmissivity [m /s]

2.786e-07

2.128e-07

1.470e-07

8.119e-08

1.539e-08

2

Fig. 9 a Transmissivity distribution and b steady-state flow pressuresolution are shown on the DFN similar to Forsmark repository site(previously shown in Fig. 1)

shown in Fig. 11. The case where the particles are ran-domly seeded in the domain shows larger time variance.Due to a uniformly random choice of the starting position,the proportion of particles close to the outflow boundary isalmost equal to those that are far from the outflow boundary.Therefore, some particles have short travel time and shorttravel distance; some particles contributed to the long tailof the travel time distribution, indicating long trajectorieswith convoluted paths. Also, since particles are distributedthroughout the entire DFN, some particles may have initialpositions in a location where the flow velocity is extremelysmall.

In the third numerical experiment, all particles are ini-tially equally distributed over all fracture edges on theinflow boundary of the domain. The complementary cumu-lative distributions (exceedance probabilities) of travel timeare plotted in Fig. 12a for 10 realizations of each DFN setwith R0 = 15 m, R0 = 20 m, and R0 = 25 m, where

every realization contains around 4 × 106 control volumecells on thousands of fractures. The measured travel timeto the outflow boundary is also weighted by flow fluxes at

200 400 600 800 1000

200

400

600

800

1000

X [m]

Y [m

]

0

Particles initial positions

Particles ending positions

(a)

(b)

(c)

Fig. 10 a Five thousand particle trajectories starting from one regionin the middle of the inflow boundary side diverge quickly and fill upthe domain; their initial positions at inflow (red) and final positionsat outflow (blue) boundaries are shown in b. c Two thousand particletrajectories are shown for the case when each particle starting positionis randomly chosen over all fracture cells over the entire DFN

Comput Geosci

Fig. 11 Complementary cumulative distribution (CCDF) of measuredparticle travel time through DFN for two numerical experiments: parti-cles start from the inflow boundary (magenta) and particles start fromrandomly chosen cells (blue)

particle starting points (Fig. 12b). The particle travel timedistributions are seen to follow a power law.

The significant difference in the distribution tail isobserved between R0 = 15 m and the two other cases,R0 = 20 m and R0 = 25 m. As we include the fractureswith smaller sizes, the fracture density in DFN signifi-cantly increases. In particular, increasing the number ofrepresented small fractures increases the number of frac-tures that have only one intersection with the main fracturenetwork. In other words, those fractures produce a sce-nario superficially similar to “dead end” in two-dimensionalmodels. However, in three-dimensional DFNs, when a par-ticle arrives at the intersection of a fracture, it may makean arc path on the intersecting fracture and return to theoriginal fracture through a different location on the samefracture intersection. Therefore, there is no “dead end”scenario observed in the three-dimensional fracture net-work model. Taking into account that a smaller fractureindicates a smaller transmissivity value (Eq. 12), parti-cles passing through them typically take longer times,which creates the tail of the travel time distribution(Fig. 12).

Particle tracking is notable for its computational effi-ciency in large detailed transport simulations. Particle track-ing on unstructured meshes is less common than particletracking on structured meshes, and suffers from additionalcomputational overheads associated with the computational

geometry calculations needed to keep track of the mov-ing particle’s position relative to the unstructured mesh.Although a detailed investigation would be needed to fullyquantify the computational cost relative to particle trackingon structured grids, our experience with particle trackingon large DFN networks (see Table 3) suggests that it isefficient and highly competitive with alternative solutionmethods.

100

10-2

10-4

10-6

10-8

10-10

10-12

1410

-

Travel time [years]

100

102

104

106

101010

8

Travel time [years]

100

102

104

106

101010

8

100

10-2

10-4

10-6

10-8

10-10

10-12

1410

-

(a)[m][m]

R = 15 R = 20 R = 25 [m]

R = 15 [m]R = 20 [m]R = 25 [m]

Exc

eeda

nce

prob

abili

tyE

xcee

danc

e pr

obab

ility

(b)

Fig. 12 Complementary cumulative distribution of particle traveltimes: a uniformly weighted at starting positions; b flux weighted atstarting positions. Here, 5× 106 particles participate in the travel timestatistical curve

Comput Geosci

Table 3 Two examples of particle tracking simulations with measured CPU run time on a single core

Number of Total number Total number of Number of particles System CPU User’s CPU

fractures of Voronoi triangular participated in time time

in DFN control volume cells elements the simulation

4 530 1017 515,000 0.217 s 4:32.52

5464 4,229,477 8,607,073 515,000 4.663 s 3:13:22.59

5 Discussion and conclusions

The DFN approach has emerged as an important methodfor evaluating transport of radionuclides and other contam-inants in sparsely to moderately fractured rock. Compu-tational workflows established in nuclear waste disposalresearch (e.g., [9, 34, 41, 43]) involve tracking of ground-water tracers in the complex DFN flow fields to calculatea set of potential transport pathways followed by transportcalculations on the computed pathways to assess radionu-clide migration including matrix diffusion and sorption. Theparticle tracking approach introduced here addresses thecalculation of the transport pathways using DFN flow fieldsproduced by finite-volume flow codes. This work extendsa recently introduced FRAM strategy [22] for meshingDFN models to include the entire computational workflowrequired in field-scale applications of contaminant trans-port in discretely fractured rock. Our finite-volume-basedworkflow compatible with the FRAM meshing strategy isan alternative to the existing DFN tools based on the finiteelement method. The numerical verification experimentsummarized here demonstrated the accuracy of the method.Additionally, in simulations using realistic large-scale DFNswith thousands of fractures and millions of computationalcells, all released particles eventually exit the system. It isimportant to note that this lack of numerical sinks is incontrast to experience with traditional finite element codes,which either suffer from mass loss due to particles becom-ing stuck in numerical sinks or require heuristic equivalentpipe-network approximations to avoid mass loss issues.Mixed hybrid finite element approaches are also capableof avoiding stuck particles, and the approach presentedhere should be regarded as an alternative to that approach.The major advancement due to the finite-volume approachpresented here is that it is compatible with the mature, open-source, and highly parallel flow code PFLOTRAN [24], thusenabling large-scale DFN transport simulations on parallelarchitectures.

Particle tracking in DFN-derived flow fields is envi-sioned as being of interest in several contexts. For example,the method is expected to be of use for visualizing potentialtransport pathways. In the context of establishing transportpathways for subsequent analyses of radionuclide transport

with matrix retention, the approach is strictly limited to sit-uations where the flow field is appropriately approximatedas steady, the radionuclides may be treated as trace elementsthat do not significantly alter the groundwater chemistry,and transverse dispersion within fractures may be neglected.Relaxing the latter constraint is one direction for futureresearch. The particle tracking method used here allowsfor dispersion on unstructured meshes [35] but needs tobe extended to treat the fracture intersections before it canbe used on DFNs. Another important direction for futureresearch is to extend the approach to transient flow fields.Since the flow solver can provide the Darcy fluxes on eachcontrol volume face at every time step, the procedure ofvelocity reconstruction can be applied and Darcy velocitiescan be updated at each time step. This mechanism allows usto model advective transport in transient flow.

The approach presented here is verified using a simplefour-fracture DFN for which analytical flow and trans-port solutions can be deduced. It is also demonstrated onrealistic large-scale DFNs with thousands of fractures andmillions of computational cells. As a result of high topo-logical diversity and the probabilistic nature of particlerouting at fracture intersections, wide dispersion of parti-cle trajectories orthogonal to the dominant flow directionis observed. Measured particle travel times tend to followa power law distribution over a wide range of travel time,consistent with earlier studies of particle tracks on DFN [5,8, 32, 33]. The determined slope exponent value is greaterthan 1, which demonstrates non-Gaussian behavior of theanomalous transport of the considered DFN cases.

Acknowledgments We thank Terry Miller, Tsung-Lin Hsieh, andAhinoam Pollack for their technical support, and Dr. Andrew Framp-ton for fruitful discussions. We thank the U.S. Department of EnergyUsed Fuel Disposition Campaign for supporting this work.

References

1. Adler, P.M., Thovert. J.-F., Mourzenko, V.V.: Fractured porousmedia. Oxford University Press (2012)

2. AMEC: ConnectFlow technical summary, release 10.4. AMECreport AMEC/ENV/CONNECTFLOW/15 (2012)

Comput Geosci

3. Ataie-Ashtiani, B., Lockington, D.A., Volker, R.E.: Numeri-cal correction for finite-difference solution of the advection—dispersion equation with reaction. J. Contam. Hydrol. 23(1), 149–156 (1996)

4. Bear, J.: Dynamics of fluids in porous media, p. 702. AmericanElsevier Publishing company Inc. (1972)

5. Berkowitz, B., Scher, H.: Anomalous transport in random fracturenetworks. Phys. Rev. Lett. 79(20), 4038 (1997)

6. Cacas, M.C., Ledoux, E., de Marsily, G., Tillie, B., Barbreau, A.,Durand, E., Feuga, B., Peaudecerf, P.: Modeling fracture flow witha stochastic discrete fracture network: calibration and validation 1,The flow model. Water Resour. Res. 26(3), 479–489 (1990)

7. Coxeter, H.S.M.: Introduction to geometry. John Wiley and Sons,216–221 (1969)

8. Cvetkovic, V., Frampton, A.: Solute transport and retention inthree-dimensional fracture networks. Water Resour. Res. 48(2),W02509 (2012)

9. Cvetkovic, V., Painter, S.L., Outters, N., Selroos, J.-O.: Stochasticsimulation of radionuclide migration in discretely fractured rocknear the Aspo hard rock laboratory. Water Resour. Res. 40(2),W02404 (2004)

10. De Dreuzy, J.-R., Meheust, Y., Pichot, G.: Influence of fracturescale heterogeneity on the flow properties of three-dimensionaldiscrete fracture networks (DFN). JGR 117, B1120 (2012)

11. Dershowitz, W.S., Lee, G., Geier, J., Foxford, T., LaPointe, P.,Thomas, A.: FracMan Version 2.6. Interactive discrete feature dataanalysis, geometric modeling, and exploration simulation, userdocumentation. Report 923–1089, Golder Associates Inc. Seattle,Washington (1998)

12. Dershowitz, W.S., Fidelibus, C.: Derivation of equivalent pipe net-work analogues for three-dimensional discrete fracture networksby the boundary element method. Water Resour. Res. 35(9), 2685–2691 (1999)

13. Eikimo, B., Lie, K.-A., Eigestad, G.T., Dahle, H.K.: Discontinu-ous Galerkin methods for advective transport in single-continuummodels of fractured media. Adv. Water Resour. 32(4), 493–506(2009)

14. Elsworth, D.: A hybrid boundary element-finite element analysisprocedure for fluid flow simulation in fractured rock masses. Int.J. Numer. Anal. Method Geomech. 10, 569–584 (1986)

15. Erhel, J., de Dreuzy, J.-R., Poirriez, B.: Flow simulation in three-dimensional discrete fracture networks. SIAM J. Sci. Comp.31(4), 2688–2705 (2009)

16. FracMan and Mafic software: http://www.fracturedreservoirs.com/SolutionsMM.asp

17. Frampton, A., Cvetkovic, V.: Numerical and analytical modelingof advective travel times in realistic three-dimensional fracturenetworks. Water Resour. Res. 47(2), W02506 (2011)

18. Geier, J.: Investigation of discrete-fracture network conceptualmodel uncertainty at Forsmark. Swedish Radiation Safety Author-ity, Stockholm (2011)

19. Hartley, L., Cox, I., Holton, D., Hunter, F., Joyce, S., Gylling,B., Lindgren, M.: Groundwater flow and radionuclide transportmodeling using connectflow in support of the SR Can assessment.Swedish Nuclear Fuel and Waste Management Co., Stockholm(2004)

20. Hartley, L., Joyce, S.: Approaches and algorithms for ground-water flow modeling in support of site investigations and safetyassessment of the Forsmark site, Sweden. J. Hydrol. 500, 200–216(2013)

21. Hogg, R.V., Craig, A.T.: Introduction to mathematical statistics.Macmillan, New York (1978)

22. Hyman, J.D., Gable, C.W., Painter, S.L., Makedonska, N.: Con-forming Delaunay triangulation of stochastically generated three

dimensional discrete fracture networks : a feature rejection algo-rithm for meshing strategy. SIAM Sci. Comp. 36(4), A1871–A1894 (2014)

23. Karra, S., Makedonska, N., Viswanathan, H.S., Painter, S.L.,Hyman, J.D.: Effect of advective flow in fractures and matrix dif-fusion on natural gas production. Water Resour. Res., under review(2015)

24. Lichtner, P.C., Hammond, G.E., Lu, C., Karra, S., Bisht, G., Mills,R.T., Kumar, J., Andre, B.: PFLOTRAN: a massively parallel flowand reactive transport model for surface and subsurface processes,Los Alamos National Laboratory Technical Report No: LA-UR-15-20403, Los Alamos National Laboratory, Los Alamos, NM(2015)

25. Long, J.C.S., Remer, J.S., Wilson, C.R., Witherspoon, P.A.:Porous media equivalents for networks of discontinuous fractures.Water Resour. Res. 18(3), 645–658 (1982)

26. Martinez, M.J.: Comparison of Galerkin and control volume finiteelement for advection–diffusion problems. Int. J. Numer. MethodsFluids 50(3), 347–376 (2006)

27. Maryska, J., Severyn, O., Vohralik, M.: Numerical simula-tion of fracture flow with a mixed-hybrid FEM stochastic dis-crete fracture network model. Comput. Geosci. 8, 217–234(2004)

28. Murphy, M., Gable, C.W.: Strategies for nonobtuse boundaryDelaunay triangulations. 7th International Meshing RoundtableProceedings, pp. 309–320. Sandia National Laboratory, Albu-querque (1998)

29. Neuman, S.P.: Trends, prospects and challenges in quantifyingflow and transport through fractured rocks. Hydrogeol. J. 13, 124–147 (2005)

30. Nick, H.M., Matthai, S.K.: Comparison of three FE-FV numeri-cal schemes for single-and two-phase flow simulation of fracturedporous media. Transp. Porous Med. 90(2), 421–444 (2011)

31. Outters, N., Shuttle, D.: Sensitivity analysis of discrete fracturenetwork model for performance assessment of Aberg. R 00-48.Swedish Nuclear Fuel and Waste Management Co., Stockholm(2000)

32. Painter, S.L., Cvetkovic, V., Selroos, J.-O.: Transport and retentionin fractured rock: consequences of a power-law distribution forfracture length. Phys. Rev. E 57(6), 6917 (1998)

33. Painter, S.L., Cvetkovic, V., Selroos, J.-O.: Power-law veloc-ity distributions in fracture networks: numerical evidence andimplications for tracer transport. Geophys. Res. Lett. 29, 1676(2002)

34. Painter, S.L., Cvetkovic, V., Mancillas, J., Pensado, O.: Timedomain particle tracking methods for simulating transport withretention and first-order transformation. Water Resour. Res. 44(1),W01406 (2008)

35. Painter, S.L.: User’s Manual for Walkabout Version 1.0. LosAlamos National Laboratory Technical Report No: LA-UR-11-01952, Los Alamos National Laboratory, Los Alamos, NM (2011)

36. Painter, S.L., Gable, C.W., Kelkar, S.: Pathline tracing on fullyunstructured control-volume grids. Comput. Geosci. 16(4), 1125–1134 (2012)

37. Poteri, A.: Retention properties of flow paths in fractured rock.Hydrogeol. J. 15, 1081–1092 (2009)

38. Poteri, A., Nordman, H., Pulkkanen, V.-E., Smith, P.: Radionu-clide transport in the repository near-field and far-field. PosivaReport 2014-02, Posiva Oy, Helsinki, Finland (2014)

39. Robinson, P.C.: Connectivity, flow and transport in network offractured media. D.Phil Thesis. Oxford University (1984)

40. Selroos, J., Walker, D., Storm, A., Gylling, B., Follin, S.: Compar-ison of alternative modeling approaches for groundwater flow infractured rock. J. Hydrol. 257, 174–188 (2002)

Comput Geosci

41. Selroos, J.O., Painter, S.L.: Effect of transport-pathway simpli-fications on projected releases of radionuclides from a nuclearwaste repository (Sweden). Hydrogeol. J. 20(8), 1467–1481(2012)

42. Selroos, J.O., Follin, S.: Overview of hydrogeological site-descriptive modeling conducted for the proposed high-levelnuclear waste repository site at Forsmark, Sweden. Hydrogeol. J.22(2), 295–298 (2014)

43. SKB: Radionuclide transport report for the safety assess-ment SR-Site. SKB TR-10-50, volumes I–III. SwedishNuclear Fuel and Waste Management Co., Stockholm(2010)

44. SKB: Long-term safety for the final repository for spent nuclearfuel at Forsmark, SKB TR-11-01. Swedish Nuclear Fuel andWaste Management Co., Stockholm (2011)

45. Van Genuchten, M.T.H., Gray, W.G.: Analysis of some dis-persion corrected numerical schemes for solution of the trans-

port equation. Int. J. Numer. Methods Eng. 12(3), 387–404(1978)

46. Versteeg, H.K., Malalasekera, W.: An introduction to computa-tional fluid dynamics: the finite volume method. Pearson Educa-tion (2007)

47. Wahlgren, C.H., Curtis, P., Hermanson, J., Forssberg, O., Oehman,J., Fox, A., Juhlin, C.: Geology laxemar. Site descriptive modelingSDM-Site Laxemar. Swedish Nuclear Fuel and Waste Manage-ment Co., Stockholm (2008)

48. Zidane, A., Firoozabadi, A.: An efficient numerical model formulticomponent compressible flow in fractured porous media.Adv. Water Resour. 74, 127–147 (2014)

49. Zyvoloski, G.A.: FEHM: A control volume finite element codefor simulating subsurface multi-phase multi-fluid heat and masstransfer. Los Alamos National Laboratory Technical Report No:LA-UR-07-3359, Los Alamos National Laboratory, Los Alamos,NM (2007)

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