Variable-density groundwater flow and solute transport in irregular 2D fracture networks

$Page 1: Variable-density groundwater flow and solute transport in irregular 2D fracture networks$
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Advances in Water Resources 30 (2007) 455–468

Variable-density groundwater flow and solutetransport in irregular 2D fracture networks

Thomas Graf *, Rene Therrien

Departement de Geologie et Genie Geologique, Universite Laval, Ste-Foy, Que., Canada G1K 7P4

Received 16 November 2005; received in revised form 25 April 2006; accepted 2 May 2006Available online 3 July 2006

Abstract

Numerical simulations of variable-density flow and solute transport have been conducted to investigate dense plume migration forvarious configurations of 2D fracture networks. For orthogonal fractures, simulations demonstrate that dispersive mixing in fractureswith small aperture does not stabilize vertical plume migration in fractures with large aperture. Simulations in non-orthogonal 2D frac-ture networks indicate that convection cells form and that they overlap both the porous matrix and fractures. Thus, transport rates inconvection cells depend on matrix and fracture flow properties. A series of simulations in statistically equivalent networks of fractureswith irregular orientation show that the migration of a dense plume is highly sensitive to the geometry of the network. If fractures in arandom network are connected equidistantly to the solute source, few equidistantly distributed fractures favor density-driven transport.On the other hand, numerous fractures have a stabilizing effect, especially if diffusive transport rates are high. A sensitivity analysis for anetwork with few equidistantly distributed fractures shows that low fracture aperture, low matrix permeability and high matrix porosityimpede density-driven transport because these parameters reduce groundwater flow velocities in both the matrix and the fractures.Enhanced molecular diffusion slows down density-driven transport because it favors solute diffusion from the fractures into the low-per-meability porous matrix where groundwater velocities are smaller. For the configurations tested, variable-density flow and solute trans-port are most sensitive to the permeability and porosity of the matrix, which are properties that can be determined more accurately thanthe geometry and hydraulic properties of the fracture network, which have a smaller impact on density-driven transport.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Numerical modeling; Density; Fracture networks; Convection

1. Introduction

Deep disposal of hazardous waste in low-permeabilitygeological media represents an alternative to other formsof storage. The salinity of groundwater generally increaseswith depth and at depths below 800 m, groundwater is aNa–Ca–Cl brine [10,35]. Consequently, the density of suchdeep fluids cannot be assumed uniform as is commonlydone for groundwater in surficial aquifers. Spatial and tem-poral variations of fluid density play an important role forsolute transport within various geological media. If, forexample, a fluid of high density overlies a less dense fluid,

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doi:10.1016/j.advwatres.2006.05.003

* Corresponding author. Tel.: +1 418 656 2131; fax: +1 418 656 7339.E-mail address: [email protected] (T. Graf).

the system is potentially unstable and density-driven flowmay take place, which levels out the density stratificationand eventually stabilizes the system.

Designs of deep repositories for waste must account forthe potential presence of fractures in low-permeability rockformations. Fractures can have a great impact on masstransport, because they represent preferential pathwayswhere dissolved solutes migrate at velocities that are sev-eral orders of magnitude higher than within the surround-ing rock formation. An understanding of the influence ofdensity variations in fractured geological formations is nec-essary for designing deep repositories. Variations in tem-perature also affect the fluid density and should beincluded for a complete assessment of fluid flow near deeprepositories. However, the present study neglects the

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Nomenclature1

Latin letters

(2b) fracture aperture [L](2B) fracture spacing [L]c solute concentration, expressed as relative con-

centration [–]C solute concentration, expressed as volumetric

mass [M L�3]Cmax maximum solute concentration [M L�3]Dd free-solution diffusion coefficient [L2 T�1]g acceleration due to gravity [L T�2]h0 equivalent freshwater head [L]K0

ij coefficients of hydraulic conductivity tensor offreshwater [L T�1]

K fr0 hydraulic freshwater conductivity of the fracture

[L T�1]Peg grid Peclet number [–]qi Darcy flux [L T�1]t time [T]vi linear flow velocity [L T�1]Xp dimensionless sensitivity of parameter p [–]

Greek letters

al matrix longitudinal dispersivity [L]at matrix transverse dispersivity [L]

afr fracture dispersivity [L]c maximum relative density [–]Cfluid source and sink term of fluid mass [M L�3 T�1]Csolute source and sink term of solute mass [M L�3 T�1]gj indicator for the fracture face orientation [–]jij coefficients of the intrinsic permeability tensor

[L2]l dynamic fluid viscosity [M L�1 T�1]q fluid density [M L�3]q0 reference fluid density [M L�3]qmax maximum fluid density [M L�3]s tortuosity [–]/ porosity of the rock matrix [–]u fracture incline [�]

Special symbols

o partial differential operator [–]$ nabla or divergence operator [L�1];

rðÞ ¼ oðÞox þ

oðÞoy þ

oðÞoz

Sub- and superscripts

0 reference fluid [–]fr fracture [–]i, j spatial indices [–]

456 T. Graf, R. Therrien / Advances in Water Resources 30 (2007) 455–468

influence of temperature on density and only focuses ongroundwater flow driven by differences in fluid salinity.

Variable-density fluid flow and transport has been stud-ied in detail in various subsurface environments. In homo-geneous porous media (Fig. 1a), variable-density flow hasbeen studied by a large number of authors (e.g.,[15,16,9,3,5,11,26,32,40,21,33,7,8]). Schincariol and Sch-wartz [25] presented the first study of density-driven flowin heterogeneous porous media (Fig. 1b). They found that(i) the transport pattern in the layers is greatly sensitive tohydraulic conductivity and (ii) the heterogeneities in the len-ticular medium create relatively large dispersion that tendsto dissipate instabilities. Thus, Schincariol et al. [27,28]inferred that heterogeneities play opposite roles in the gen-eration and subsequent growth of instabilities. On onehand, heterogeneities initially perturb a plume while onthe other hand, once instabilities are generated, heterogene-ities dampen their growth on probably all spatial scales.Simmons et al. [34] showed that the style of heterogeneityin a porous medium will greatly influence the propagationof dense plumes, where heterogeneities can dissipate con-vection by mixing and thus reduce plume instabilities.

Fractured porous media exhibit a different style of het-erogeneity compared to granular porous media. Studies

1 The use of symbols for main variables is consistent throughout theentire text. All symbols represent scalar variables denoted in normal italicletters.

by Murphy [20], Malkovsky and Pek [17,18] and Shi [29]showed that two-dimensional convective flow with a rota-tion axis normal to the fracture plane can occur within ver-tical fractures. These studies, however, did not representfractures as discrete planes but as vertical high-permeabil-ity fault zones. Shikaze et al. [31] numerically simulatedvariable-density flow and transport in discretely fracturedmedia. They found that vertical fractures with an apertureas small as 50 lm significantly increase solute transport rel-ative to the case where fractures are absent. Interestingly, itwas also shown that dense solute plumes may develop in ahighly irregular fashion and are extremely difficult to pre-dict. However, Shikaze et al. [31] limited their studies toa regular fracture network consisting of only vertical andhorizontal fractures, embedded in a porous matrix(Fig. 1c). The development and propagation of denseplume instabilities in a network of fractures having irregu-lar apertures, traces and orientations, such as the exampleshown in Fig. 1d, therefore remains to be studied.

In numerical simulations of variable-density flow, spa-tial dimensionality is a key factor that influences the flowpattern in the finite element grid. In nature, convection infractures is physically possible. However, representing adiscrete fracture by one-dimensional segments, as doneby Shikaze et al. [31], completely inhibits numerically sim-ulating convection within such 1D elements. In verticaltwo-dimensional representations of the flow domain, con-vection cells with rotation axes normal to the 2D grid

(a) (b) (c) (d)

Fig. 1. Different styles of geological media: (a) homogeneous porous medium, (b) heterogeneous porous medium, (c) fractured medium consisting ofvertical and horizontal fractures and (d) fractured geological medium with nonuniform fracture aperture, trace and orientation. In (a) and (b), the shadesof grey represent hydraulic conductivity (figure modified from [34]).

T. Graf, R. Therrien / Advances in Water Resources 30 (2007) 455–468 457

can be numerically simulated. In this case, the 2D domaincan represent a vertical porous layer [15,16,5,9,39], a tiltedporous layer [3,39], a highly permeable fault zone [20,29] ora discrete fracture of arbitrary incline [12]. A fully three-dimensional representation of the model domain allowsfree convection to occur in multiple ways. This representa-tion has been applied to aqueous systems [2], highly perme-able fault zones [17,18] and porous media [6,1,7].

To investigate variable-density flow and transport inirregular fracture networks, the FRAC3DVS model[37,38], which solves 3D variably saturated flow andmulti-component solute transport in discretely fracturedporous media, has been modified by Graf and Therrien[13] to simulate density-driven flow in nonuniform frac-tures. In this paper, the modified version of FRAC3DVSis applied to study plume migration in irregular fracturenetworks. Similar to Shikaze et al. [31], we will representdiscrete 1D fractures within a vertical 2D porous medium.With this constraint of spatial dimensionality, fractureinterconnectivity in the third dimension is not consideredsuch that groundwater flows in each 1D fracture in onlyone direction. However, unlike previous studies [31], thiswork is not limited to flow in orthogonal fracture net-works. This paper investigates how irregular 2D fracturenetworks will favor or impede variable-density flow. Wealso assess the sensitivity of such complex systems toparameter uncertainties.

2. Mathematical modeling

2.1. The FRAC3DVS model

FRAC3DVS is a 3D saturated-unsaturated numericalgroundwater flow and multi-component solute transportmodel [37,38] that has been modified by Graf and Therrien[13] to solve variable-density flow and transport using theequivalent freshwater formulation. A control volume finiteelement method is used to spatially discretize the flow andtransport equations. The porous low-permeability matrix isrepresented by regular three-dimensional blocks and frac-tures of high permeability are represented by two-dimen-sional rectangular planes. Fluid flow and transportequations are solved for both porous matrix and fractures.Assuming undistorted finite elements allows an analytical

discretization of the governing equations by means of ele-mental influence coefficient matrices [11,37], avoiding theneed to numerically integrate. The solution takes intoaccount advection, molecular diffusion and mechanical dis-persion in both the fractures and the matrix.

Vertical, horizontal and inclined 2D fractures are incor-porated into the 3D grid by superimposing two-dimen-sional fracture elements onto three-dimensional regularblock elements. Faces and blocks share common nodesalong the fracture walls. Thus, nodes at fracture locationsreceive fluid and mass contributions from both the blockelements as well as from the fracture faces. Furthermore,for these mutual nodes, both hydraulic head and concen-tration at the fracture/matrix interface are assumed to beequal, ensuring continuity of these variables between frac-tures and matrix. This discrete fracture approach has pre-viously been applied by several authors [36,30,37,31] andits description is not repeated here.

The variable-density flow and solute transport equationsare coupled. Small density variations cause weak nonlin-earities in the flow equation. In the numerical model, theflow and transport equations are coupled with a sequentialiterative approach (SIA), also called Picard iteration. Thisapproach alternately solves the two governing equationsduring each time step until convergence is attained.

2.2. Governing equations

Variable-density fluid flow and solute transport aredescribed by the following two equations, respectively:

oð/qÞot¼ �r � ð/qvÞ � Cfluid ð1Þ

and

oð/CÞot

¼ �r � J� Csolute; ð2Þ

where / is the dimensionless matrix porosity, q [M L�3] isthe fluid density, v [L T�1] is the average fluid velocity vec-tor, C [M L�3] is the solute concentration, t [T] is time and$ [L�1] is the divergence operator. The solute mass flux, J

[M L�2 T�1], is the sum of advective and dispersive/diffu-sive mass fluxes. Sources and sinks of fluid and solute massare represented by Cfluid and Csolute [both M L�3 T�1],respectively.

Table 1Model parameters used in fractured media studies

Parameter Value

Reference densitya,b,c (q0) 1000 kg m�3

Maximum fluid densityb,c (qmax) 1200 kg m�3

Fluid dynamic viscosity (l) 1.124 · 10�3 kg m�1 s�1

Acceleration due to gravity 9.80665 m s�2

Tortuosityb (s) 0.1Free-solution diffusion coefficientb (Dd) 5 · 10�9 m2 s�1

Matrix permeabilityb (jij) 10�15 m2

Matrix porositya (/) 0.35Matrix longitudinal dispersivityb (al) 0.1 mMatrix transverse dispersivityb (at) 0.005 mFracture dispersivityb,d (afr) 0.1 mFracture aperture (2b) Problem dependent

a Frind [11].b Shikaze et al. [31].c Elder [9].d Therrien and Sudicky [37].


The model uses the equivalent freshwater head, h0 [L],defined as

h0 ¼P

q0gþ z; ð3Þ

where P [M L�1 T�2] is the dynamic fluid pressure, q0

[M L�3] is the reference fluid density, g [L T�2] is the grav-itational acceleration and z [L] is the elevation above da-tum. Using equivalent freshwater head, the Darcy flux,q = /v [L T�1], in both porous and fractured media is givenin scalar form by [11,13]:

qi ¼ �K0ij

oh0

oxjþ ccgj

� �; i; j ¼ 1; 2; 3; ð4Þ

qfri ¼ �K fr

0

oh0

oxjþ ccgj cos u

� �; i; j ¼ 1; 2; ð5Þ

where u is the incline of the fracture with u = 0� indicatinga vertical fracture and u = 90� indicating a horizontal frac-ture. Assuming constant fluid viscosity, the freshwaterhydraulic conductivities, K0

ij and K fr0 [both L T�1], of both

media are given by

K0ij ¼

jijq0gl

; ð6Þ

K fr0 ¼ð2bÞ2q0g

12l; ð7Þ

where jij [L2] is the matrix permeability, g [L T�2] is grav-itational acceleration, l [M L�1 T�1] is fluid viscosity and(2b) [L] is fracture aperture. The dimensionless relativeconcentration, c = C/Cmax, varies between 0 and 1 for den-sities q [M L�3] that vary between q0 for freshwater andqmax for saltwater. The direction of flow is represented bygj [–] with gj = 0 indicating the horizontal direction whilegj = 1 indicates the non-horizontal (vertical and inclined)direction 11,13]. The dimensionless constant c is the maxi-mum relative fluid density and is given by

c ¼ qmax

q0

� 1: ð8Þ

We applied the first level of the Oberbeck–Boussinesq (OB)approximation 22,4] to discretize Eqs. (1) and (2). The OBassumption reflects the degree to which density variationsare accounted for. Level 1 of the OB approach considersdensity effects only in the buoyancy term of the momentumequations (4) and (5) and neglects density in the fluid andsolute mass conservation equations. This assumption isgenerally correct because spatial density variations, $q,are commonly minor relative to density, q. More detailson the discretization of Eqs. (1) and (2) by the control vol-ume finite element method are presented in [37,38,13] arenot repeated here.

3. Illustrative examples

3.1. Model design used in all examples

The domain for the variable-density flow and transportsimulations is a two-dimensional vertical slice of unit thick-

ness in the y-direction, and with lengths ‘x and ‘z in the x-and z-directions, respectively. The solute concentration isinitially equal to zero and the top of the domain is assumedto be a solute source with constant concentration, creatinga potentially unstable situation. All other boundaries areassigned zero dispersive flux for transport. The left andright boundaries are assumed to be impermeable for flow,whereas the top and bottom boundaries are constant equiv-alent head boundaries with zero constant heads. The topand bottom boundary conditions for flow generate ahydraulic gradient equal to zero across the domain, whicheliminates forced convection. Therefore, the only solutetransport mechanism is free convection due to spatial andtemporal density variations.

All simulations cover 10 years and time step sizes werekept constant at one month. Table 1 summarizes the inputparameters for the numerical simulations. These parame-ters were held constant throughout all simulations unlessotherwise stated. It is further assumed that the porousmatrix is isotropic and homogeneous and that the entiredomain is completely saturated.

3.2. 2D networks of orthogonal small and large fractures

This section focuses on the role of small fractures withinan orthogonal network of large fractures to determine ifsmall fractures enhance mixing and, thus, impede density-driven transport in large fractures. A similar phenomenonhas been demonstrated in heterogeneous porous media bySchincariol and Schwartz [25] and Simmons et al. [34],who have shown that the large dispersion in horizontallenses dissipates instabilities. It remains unclear if, in frac-tured media, horizontal fractures reduce vertical plumemigration by dispersive and advective lateral mixing, simi-lar to the mixing process in heterogeneous porous media.

For the simulations presented in this section, the 2Ddomain has dimensions ‘x = 15 m · ‘z = 9.375 m and con-sists of 64,000 square elements of size Dx = Dz =


0.046875 m (Fig. 2). This spatial discretization wasobtained from successive grid refinement until simulatedconcentration contours remained unchanged. The largefractures are horizontal and vertical and have a uniformaperture, (2b), equal to 50 lm and a constant spacing,(2B), equal to 1.875 m. All simulations contain largeorthogonal fractures. For a given simulation, small frac-tures have uniform aperture, (2b)*, and spacing, (2B)*.The influence of small fractures is examined by varyingtheir aperture and spacing for a series of simulations. Val-ues of 1, 5, 10 and 25 lm are used for aperture and spacingratios (2B)/(2B)* equal to 2, 3, 4 and 5 are chosen. Simula-tions are first run for cases where small fracture are hori-zontal and where their aperture and spacing ratio isvaried according to the values mentioned above. In a sec-ond step, simulations are run with small horizontal andvertical fractures.

Fig. 3 shows concentrations in the domain for a basecase without small fractures. Fingers form in all vertical

large fractures small fractures

(2 )*B(2 )B

Fig. 2. Geometry of the orthogonal fracture network consisting of largefractures (bold lines) and small fractures (thin lines).

Fig. 3. Concentration distribution at 3 years in a network having onlylarge orthogonal fractures of aperture (2b) = 50 lm and spacing(2B) = 1.875 m. Concentration contours are 0.1, 0.5 and 0.9.

fractures, counterbalanced by upward flow in the porousmatrix. This base case is the reference for comparison withfurther simulations, where small fractures are included inthe network.

3.2.1. Simulations with small horizontal fracturesThe influence of small horizontal fractures on vertical

plume migration is discussed first. For all cases, the resultsof simulations with small fractures of any aperture (1, 5, 10and 25 lm) and spacing ratio (2, 3, 4 and 5) are identical tothe base case (Fig. 3) and are not shown here. The eight fin-gers that develop in the vertical fractures of the base casedo not change in shape if small horizontal fractures arepresent.

An intriguing result is that small horizontal fractureswith apertures of up to half the aperture of large orthog-onal fractures do not impact vertical plume migration.Dispersion in small horizontal fractures does not appearto disturb the established convective pattern of downwel-ling in the large fractures and upwelling in the porousmatrix. Even the simulation where (2b)* = 25 lm andratio = 5, and where the equivalent hydraulic conductiv-ity of the system is 1.3010 · 10�7 m s�1, hence 23%higher than that of the base case (1.0586 · 10�7 m s�1)does not show any influence of small horizontal fractures.Small horizontal fractures neither favor nor impededense plume transport in networks of large orthogonalfractures.

The simulations demonstrate that the convective patternin the studied network is largely controlled by verticalplume movement (fractures: downwelling – matrix: upwell-ing). Adding small horizontal fractures enhances theanisotropy of the system. However, the simulations showthat horizontal large fractures, and to a much smallerextent the rock matrix, are sufficient to horizontally con-duct groundwater between the downwelling zones (frac-tures) and upwelling zones (matrix). Additional smallhorizontal fractures increase the bulk horizontal hydraulicconductivity but not sufficiently to modify fluid exchangebetween adjacent vertical fractures.

3.2.2. Simulations with small horizontal and vertical

fractures

In a second step, small vertical fractures are added to thesmall horizontal fractures, while maintaining the same net-work of larger orthogonal fractures. The simulations withsmall fractures of aperture (2b)* = 1 and 5 lm and anyspacing ratio (2, 3, 4 and 5) are identical to the base case(Fig. 3) and are not shown here. Adding fractures with(2b)* = 5 lm and ratio = 5 increases the equivalent hydrau-lic conductivity of the system by 0.34%, which explains thatresults are identical to the base case.

Results of the simulations where (2b)* = 10 and 25 lmare quite different and shown in Figs. 4 and 5, respectively.Unlike results mentioned in the previous paragraph, theresults where (2b)* = 10 and 25 lm are not intuitive. Figs.4 and 5 indicate that for flow systems with even ratios

Fig. 4. Concentration distribution at 3 years in a network containing large vertical fractures (bold lines) and small horizontal and vertical fractures (thinlines) of aperture (2b)* = 10 lm. The (2B)/(2B)* ratio is (a) 2, (b) 3, (c) 4 and (d) 5. Concentration contours are 0.1, 0.5 and 0.9.

Fig. 5. Concentration distribution at 3 years in a network containing large vertical fractures (bold lines) and small horizontal and vertical fractures (thinlines) of aperture (2b)* = 25 lm. The (2B)/(2B)* ratio is (a) 2, (b) 3, (c) 4 and (d) 5. Concentration contours are 0.1, 0.5 and 0.9.


(2 and 4), there is upwelling in some of the large verticalfractures. On the other hand, for odd ratios (3 and 5),

results resemble the base case with downwelling in all 8 ver-tical large fractures.

Table 2Equivalent hydraulic conductivities of flow systems with horizontal andvertical small fractures in percent above base case equivalent conductivity

Small fracture aperture Spacing ratio

2 3 4 5(a) (b) (c) (d)

(2b)* = 10 lm (Fig. 4) 0.61 1.47 2.08 2.93(2b)* = 25 lm (Fig. 5) 9.59 22.9 32.5 45.8

σ2

σ1

σ3

σ1 < σ2 < σ3

ϕ

Fig. 6. Conjugated system of two fracture families, showing the principaldirections of normal stress, ri [M L�1 T�2].


Computed equivalent hydraulic conductivities for theflow systems shown in Figs. 4 and 5 are compared tothe base case equivalent conductivity. Table 2 shows thatthe 2 flow systems with downwelling in all 8 fractures (band d) have considerably different hydraulic conductivitiesand that the 2 systems with 2 or 4 fingers (a and c) areequally different. Therefore, the equivalent hydraulic con-ductivity cannot be used to predict fingering.

The location of small vertical fractures appears to con-trol fingering. Some locations seem to favor vertical plumetransport (a and c). In these cases, fingering occurs only inselected fractures where solutes are transported at a fasterrate than in the base case. Other positions appear toimpede density-driven transport (b and d). In those cases,fingering takes place in all 8 fractures and transport ratesare smaller than in cases (a) and (c). Keeping aperture con-stant for vertical fractures but changing their location withrespect to the larger fractures drastically changes verticaltransport of a dense plume.

3.3. 2D networks of non-orthogonal fractures

As opposed to the mutually orthogonal system pre-sented above, the network here is composed of fracturesthat intersect at different angles. The dimensions of the2D vertical domain is ‘x = 12 m and ‘z = 10 m. Successivegrid refinement for a single inclined fracture case revealedthat a nodal spacing of 0.1 m is appropriate in each coor-dinate direction [13]. With a grid Peclet numberPeg = Dx/al = 1.0, the widely accepted criteria for neglect-ing numerical dispersion, Peg 6 2, as well as oscillations,Peg 6 4, are satisfied.

Three random fracture networks were generated, eachconsisting of 50 fully connected fractures. Every networkis randomly generated using the same geostatistical distri-butions of fracture trace, aperture and orientation. Frac-ture traces are assumed to be log-normally distributed[19] between 2 and 10 m, with a mean value equal to1.4 m and standard deviation equal to 0.4 m. Fractureapertures follow an exponential distribution between 150and 250 lm [24]. The probability of aperture (2b) decreasesfrom 150 to 250 lm by the factor exp(�k Æ (2b)) wherek = 9000 lm�1. The aperture is assumed to be constantfor a single fracture. Furthermore, the fractures areassumed to be of tectonic origin, leading to a conjugatedsystem of two fracture families (Fig. 6). Therefore, fractureorientations (u) follow a two-peak Gaussian distribution

[14] with peaks at �30� and +30�, where 0� is the verticaldirection, and with the assumed standard deviation,r = 15�. The assumption of a conjugated system ofmultiple fracture families [14] applies to any type of frac-tured rock, either sedimentary (clay) or plutonic rock(granite).

Our approach to generate random fractures was to, first,assign fracture trace, aperture and orientation to each frac-ture. Second, each fracture was randomly placed into asmaller window of the 2D domain, to avoid that a fractureextends beyond the model domain. This approach honorsthe orientation and trace of each fracture but implies thatfracture density is greater in the middle of the 2D domainthan close to the boundaries. With our approach, we gen-erated the three networks shown in Fig. 7. Otherapproaches use a fracture generation window that is muchlarger than the simulation domain to avoid fracture cluster-ing in the middle of the domain. In that case, the fracturesthat extend beyond the model domain are truncated.

We did not conduct Monte-Carlo simulations but rantransient simulations with each individual fracture networkand observed the output at 4 years. For the three statisti-cally equivalent fracture networks, completely differentbehavior was observed depending on the spatial locationof the high-permeability fracture zones. In Fig. 8a, down-wards solute transport is minor. In this case, velocitiesare low and molecular diffusion dominates solute trans-port. However, in Fig. 8c, a large amount of mass has beentransported into the domain. In that case, velocities arehigh and advection and mechanical dispersion prevail.

The different degree of instability can be judged bothsubjectively by visual inspection (Fig. 8) and objectivelyusing measurable characteristics. An example of a measur-able characteristic is the penetration depth of some isoha-line, for example the 60% contour or the total massstored as proposed by Prasad and Simmons [23]. The60% contour migrates faster into the aquifer for the net-work (c) compared to network (a). Correspondingly, net-work (a) stores much less solute at any time than system(c). The temporal evolution of the penetration depth andtotal mass (Fig. 9) objectively confirms the subjectiveassessment of the degree of instability of the three exam-ples. Thus, the results shown in Figs. 8 and 9 show that,although the three networks are statistically similar,

Fig. 7. Three stochastic fracture networks, defined by an exponentialaperture distribution, a log-normal trace distribution and a two-peakGaussian orientation distribution.

Fig. 8. Simulated concentration at 4 years for 3 different fracturenetworks. The number of equidistantly distributed fractures near thesource is 6, 3 and 1 for figures (a), (b) and (c), respectively.


density variations lead to significantly different solutemigration.

The formation of instabilities is restricted to the highlypermeable fracture zones. The fracture network dictatesthe number of fingers, which is 6, 3 and 1 for cases (a),(b) and (c), respectively. The number of instabilities coin-cides with the number of equidistantly distributed fracturesnear the source. These fractures represent heterogeneitiesof the vertical conductivity and, therefore, disturb the flowfield. Transient results showed that the disturbances growin time to ultimately form instabilities. In conclusion, sys-tems with low fracture density near the domain top butwith at least one fracture near the source have the greatestpotential to be highly unstable (case c).

Figs. 7 and 8 also suggest that the system is only weaklysensitive to the aperture of the fractures near the source. Incases (b) and (c), the aperture of the near-source fracturewhere solutes are mainly transported falls in the lower limit(about 170 lm) of the aperture range (150–250 lm). Yet,solutes are transported through networks (b) and (c) at sig-nificantly different rates (Figs. 8 and 9).

Clearly, numerous equidistantly distributed fracturesimpede density-driven transport, whereas few fracturesclose to the solute source favor density-driven transport.In contrast to density-driven transport in porous media,the number of instabilities in fractured media does notchange with time. Simmons et al. [33] demonstrated that

0

2

4

6

8

10

pen

etra

tio

n d

epth

[m

]0

5

10

15

20

25

30

0 2 4 6 8 100 2 4 6 8 10time [yr] time [yr]

tota

l mas

s st

ore

d [

kg]

a

bc

bca

Fig. 9. Penetration depth of the 60% isochlor and total stored mass for the three simulations presented in Fig. 8.

Fig. 10. Velocity vectors in the (a) matrix and (b) fractures for thenetwork shown in Fig. 7c after 0.5 years. The velocity field is highlyirregular and complex convection cells form, one cell is highlighted. Somefractures that are close to the solute source are contaminated from below.


in a sandy aquifer, the number of fingers decreases withtime because large fingers increase, decreasing the numberof small fingers. In fractured media, however, this is not thecase because the location of the fingers is strongly con-trolled by the geometry of the fracture network.

The simulations further show that plume developmentin irregular fractured media is also influenced by the forma-tion of convection cells. Fig. 10a displays the highly non-uniform velocity field for network (c) after 0.5 years. Theflow system is characterized by complicated convection,where several convection cells form. The dark horizontaland vertical pattern is due to the presence of supplementarygrid lines, which have been added to ensure fracture con-nectivity. Fig. 10b shows a selected streamline of the veloc-ity field. The convective nature of the system generatesdownward flow in the dominant fracture at the top rightcorner and upward flow in other fractures. Thus, onlyone instability develops, because the solute in the two frac-tures close to the surface does not originate from the sourceat the top of the system, but rather from below.

3.4. Sensitivity analysis

Simulations have been conducted to assess the impact ofparameter uncertainties on variable-density flow in a com-plex fracture network. The fracture network shown inFig. 7a was chosen because it allows to examine the behav-ior of the network as a whole, rather than the behavior offew individual fractures that dominate the system as is thecase for the network shown in Fig. 7c. For this sensitivityanalysis, results are compared to the base case scenariowhose result is shown in Fig. 8a.

All simulations are examined with respect to the varia-tion of one dependent variable n that characterizes the sys-tem, with respect to variations in another parameter p. Theterms nlow, norg and nhigh denote the value of this dependentvariable for simulations where parameter p is low, unmod-ified and high compared to the base case. These values of pare denoted by plow, porg and phigh, respectively. A dimen-sionless sensitivity of the model parameter p, Xp, is evalu-ated using the following equation [41]:

X p ¼on=norg

op=porg: ð9Þ

According to Zheng and Bennett [41], the partial derivativeof the dependent variable, n, with respect to the inputparameter, p, is normalized by the original value of the var-iable, norg, and the parameter, porg.


The choice of the range over which to vary the inputparameter is subjective. If parameter changes (i.e., pertur-bations) are too small, computer round-off errors may con-ceal differences in the computed dependent variable. On theother hand, perturbations that are too large may yield inac-curate sensitivities, especially if the relation between thedependent variable and the perturbed parameter is nonlin-ear. Here, a uniform perturbation size of 5% is applied assuggested by Zheng and Bennett [41]. In order to visualizeparameter sensitivity, additional simulations were carriedout with much wider ranges of the input parameters.

A first series of eight simulations was carried out for themathematical sensitivity analysis. The penetration depth ofthe 30% contour at 4 years is assumed to adequately char-acterize this simulation and it is used as the dependent var-iable, n. The sensitivity of the penetration depth withrespect to perturbations in the dispersivity, free-solutiondiffusion coefficient, fracture aperture, matrix permeabilityand matrix porosity was analyzed by lowering and increas-ing original values given in Table 2 by 5%. Fig. 11 showsthe calculated sensitivities for each input parameter.

A second series of eight simulations was run for visual-ization purpose, where much larger parameter perturba-tions were used as shown in Table 3. The fractureapertures are distributed exponentially. Fig. 12 shows theresult of this visual sensitivity test at 4 years.

All simulations indicate that large diffusion coefficient,small fracture aperture, low matrix permeability and largematrix porosity even out plume migration, while changes indispersivity do not impact density-driven transport. Notethat these results only apply to variable-density transportin the 2D fracture networks used here. In 3D, groundwatercan flow in fractures in multiple directions. This reduces the

2.13

0.98

-0.49

0.0

-3.28

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

Matrix porosity

Matrix permeability

Fracture aperture

Free-solution diffusion coefficient

Dispersivity

Dimensionless sensitivity

Fig. 11. Dimensionless sensitivity of model parameters in variable-densityflow simulations in order from least (top) to most (bottom) sensitive.

Table 3Model parameter modifications only used for visualization purpose in the sen

Parameter Low value

Dispersivitya (a) 0.05 mFree-solution diffusion coefficient (Dd) 5 · 10�10 m2 s�1

Fracture apertureb (2b) 50–150 lmMatrix permeability (jij) 5 · 10�16 m2

Matrix porosity (/) 0.25

a All dispersivities (al, at and afr) were varied by factors 1/2 and 2.b Exponentially distributed.

influence of the rock matrix and makes 3D systems less sen-sitive to matrix permeability and porosity. The followingparagraphs explain the findings for 2D in more detail.

Dispersivity does neither favor nor impede density-dri-ven flow in 2D fracture networks (Fig. 11). Results withlow and high dispersivity values are identical to the basecase (Fig. 8a) and are not shown. While in fractures, flowvelocity and therefore mechanical dispersion is high, dis-persive mixing along fractures appears to be controlledby matrix diffusion. If, for example, dispersivity is high, atemporary increase in solute concentration at the solutefront leads to a steeper concentration gradient from thefracture into the porous matrix and, thus, to increasedmatrix diffusion. Therefore, the solute front does notadvance significantly faster. In the low-permeability porousmatrix, mechanical dispersion plays a minor role due tolow flow velocities.

The free-solution diffusion coefficient directly impactsdiffusion rates. If diffusion is high, enhanced diffusive mix-ing reduces plume instability. However, diffusion has nodirect influence on bulk velocities and convection and itssensitivity was found to be relatively minor. Furthermore,a high diffusion coefficient leads to high diffusion from thefractures into the adjacent porous matrix, which is oftentermed ‘‘loss of tracer’’. Therefore, the fractures aredepleted in solutes resulting in less efficient buoyancy withinthe fractures. Conversely, a low diffusion coefficient leads toless matrix diffusion and, thus, the concentration gradientsas well as the concentrations in the fractures remain high,resulting in high diffusive as well as buoyancy-driven trans-port within the fractures. Thus, solutes migrate further intothe porous matrix in the high diffusion case but deeper intothe aquifer in the low diffusion case (Fig. 12a).

Convection is important to control variable-density flowin fractured porous media. Therefore, the system is sensi-tive to fracture aperture and matrix permeability, whichare both affecting the flow velocity in convection cells.The dependency of fracture flow velocity on aperture isquadratic and, thus, stronger than the linear relationshipbetween matrix flow velocity and matrix permeability. Inthe example, however, the volume fraction of the fracturesis much smaller than that of the porous matrix. Conse-quently, uncertainties in the knowledge of fracture aperture(sensitivity of 0.98; Fig. 12b) have less impact than uncer-tainties in the knowledge of matrix permeability (sensitivityof 2.13; Fig. 12c).

sitivity analysis of variable-density flow

Original value High value

0.1 m 0.2 m5 · 10�9 m2 s�1 5 · 10�8 m2 s�1

150–250 lm 250–350 lm1 · 10�15 m2 2 · 10�15 m2

0.35 0.45

Fig. 12. Visual sensitivity of input parameters at 4 years. Shown are solute concentrations for lower (left) and higher (right) values of the followingparameters: (a) free-solution diffusion coefficient, (b) fracture aperture, (c) matrix permeability and (d) matrix porosity.


Increased matrix permeability destabilizes the systembecause of higher computed Darcy fluxes in the matrix(Eqs. (4) and (7)). Fig. 10 shows that in discretely fracturedmedia, the downward transport of solutes in the high-per-meability fracture zones is counterbalanced by upwardflow in both the matrix and in other fractures. Thus, rapidupward flow in the matrix enables rapid downward flow in

the fracture. Therefore, the magnitude of the groundwatervelocities in the porous matrix controls the transport ratein fractures with greater matrix permeability leading tohigher transport rates in fractures (Fig. 12c).

The impact of matrix permeability was further studiedfor a case where the rock matrix is essentially impermeable.An additional simulation with jij = 1 · 10�20 m2 was


performed and the result at 4 years showed that concentra-tion contours near the solute source are completely hori-zontal (results not shown). This indicates that the systemremains stable, meaning that there is (i) no convection inthe entire domain, (ii) no advective transport and that(iii) solutes are only transported by molecular diffusion.In conclusion, density-driven transport is completelyrepressed in rock formations of extremely low matrix per-meability even though the rock is fractured.

The impact of matrix porosity changes on variable-den-sity transport was found to be similar to the case where thefluid density is constant. The latter has been investigated bySudicky and McLaren [36], who showed that in discretelyfractured porous formations, the bulk travel distance isinversely proportional to porosity. This trend could be con-firmed with the presented variable-density flow simula-tions. In the base case scenario, the 30% penetrationdepth after 4 years is 6.1 m. If porosity is increased by5%, this depth is 5.5 m, giving a ratio of 1.11. This valueis close to the inverse porosity ratio, 0.3675/0.35 = 1.05.Thus, decreasing the porosity results in more rapid solutemigration within fractures. This last statement holds forboth the constant-density case [36] as well as for variable-density flow (Fig. 12d). Matrix porosity affects both trans-port mechanisms, hydrodynamic dispersion and advectivetransport. The average flow velocity is calculated asvi = qi//, thus, a smaller porosity results not only in lessattenuation of the plume [36], but also in greater advectivetransport. The high sensitivity of the matrix porosity, eval-uated as �3.28, expresses its twofold control on denseplume transport.

4. Summary and conclusions

A numerical model has been used to study dense plumemigration in 2D orthogonal and irregular fracture net-works. All simulations presented here were run in a verticaltwo-dimensional domain consisting of one layer of 3D por-ous matrix blocks. In this vertical slice, fractures aredescribed by 2D faces, which is essentially a 1D representa-tion of fractures. This constraint in spatial dimensionalityimplies that all following conclusions (i) neglect convectionwithin fractures and (ii) imply that groundwater cannotflow in multiple directions in each fracture.

Simulations in orthogonal networks of large and smallaperture fractures demonstrate that small fractures donot slow down vertical plume migration. Dispersive mixingin small fractures was not found to play a key role. Vari-able-density flow in a network consisting of fractures of dif-ferent apertures is dominated by the convective patternestablished in the network of large fractures. The simula-tions also show that fractures smaller than 10 lm haveno impact on density-driven flow in large fractures of aper-ture equal to 50 lm.

Different, yet statistically equivalent networks have alsobeen investigated. Monte-Carlo simulations were not con-ducted but we ran transient simulations with three individ-

ual fracture networks. The results suggest that themigration of a dense plume in an irregular fracture networkis highly sensitive to the geometry of the network. Thus,exact knowledge of the fracture distribution is crucial toreliably predict the migration of a dense plume in a nonuni-form fracture network. The study also shows that high-per-meability fracture zones can both favor and impededensity-driven transport. On one hand, few equidistantlydistributed fractures near the solute source favor density-driven transport. On the other hand, numerous fracturesclose to the solute source have a stabilizing effect.

The formation of unstable fingers is limited to fracturelocations representing zones of high permeability. Convec-tion cells develop, which transport the solutes quickly intothe aquifer. The convection cells include a combination offractures and the porous matrix. In most fractures, the flowdirection is downward while it is upward in the matrix andin some other fractures. Thus, groundwater velocities in thefractures and the matrix significantly control the transportrate within convection cells.

The sensitivity analysis revealed that high matrix perme-ability, low matrix porosity and large fracture apertures pro-mote density-driven transport in a fracture network. In allthree cases, flow velocities become large leading to strongconvection. Interestingly, the sensitivity test also demon-strates that large diffusion rates even out plume migration.When matrix diffusion is high, a large amount of solutes istransported from the high-permeability fracture zones intothe low-permeability porous matrix. There, groundwatervelocities and transport rates are typically low.

The low sensitivity of the diffusion coefficient suggeststhat plume migration through a complex fracture networkdoes not depend on the tracer characteristic, provided thatthe density contrast remains unchanged. Plume transport isstrongly affected by the matrix permeability and porosity.However, both matrix properties are relatively easy to mea-sure, such that the fracture network itself is the crucialunknown of the system. Therefore, unprecise informationabout the network geometry remains the limiting factorof reliable numerical simulations. Hence, this study alsoshowed that dense plume transport in irregular 2D fracturenetworks is close to impossible to predict. This finding isidentical to prior results for regular 2D networks [31].

The model used here for variable-density flow and solutetransport simulations is numerically stable and has beenfully verified. However, it has not been validated becauseappropriate field data are currently lacking. As a result,the input data has been selected to be representative ofactual fracture networks but other combinations of inputparameter could be chosen. For example, exact fracturelocations and fracture interconnectivity are hard to mea-sure in the field but were shown to have a crucial impacton plume migration. Also, a number of other random frac-ture generators exists that use different approaches to gen-erate fractures. As a consequence of having to assumeinput data, the 2D results presented thus far only allowan analysis and interpretation in a conceptual way.


In summary, the simulations presented here indicatethat

(1) In an orthogonal network of large and small frac-tures, fractures smaller than 10 lm have no impacton the established convective pattern in large frac-tures of aperture equal to 50 lm. Small fractures donot stabilize vertical plume migration by enhanceddispersive mixing.

(2) Convective flow in irregular, yet statistically equiva-lent networks proved to be highly sensitive to thegeometry of the network. Thus, unknown fractureinterconnectivity can be a limiting factor of reliablenumerical simulations.

(3) If fractures of a random network are connected equi-distantly to the solute source, few equidistantly dis-tributed fractures favor density-driven transport. Onthe other hand, numerous fractures have a stabilizingeffect.

(4) Convection cells in complex fracture networks over-lap both the porous matrix and fractures. Thus,transport rates in convection cells depend on matrixand fracture flow properties.

(5) High matrix permeability, low matrix porosity andlarge fracture apertures favor variable-density flow.In all three cases, flow velocities become large, lead-ing to strong convection. This finding supports theprevious statement.

(6) Large diffusion rates even out plume migration incomplex fracture networks by matrix diffusion.

Acknowledgements

We thank the Canadian Water Network (CWN) as wellas the Natural Sciences and Engineering Research Councilof Canada (NSERC) for financial support of this project.Author TG wishes to acknowledge both the InternationalCouncil for Canadian Studies (ICCS) and the GermanAcademic Exchange Service (DAAD) for providing a Post-graduate Scholarship stipend. The constructive commentsof six anonymous reviewers are greatly appreciated andhave helped improve the manuscript.

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Variable-density groundwater flow and solute transport in irregular 2D fracture networks