Advances in Water Resources 28 (2005) 1351–1367

www.elsevier.com/locate/advwatres

Variable-density groundwater flow and solute transport inporous media containing nonuniform discrete fractures

Thomas Graf *, Rene Therrien

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

Received 10 September 2004; received in revised form 15 April 2005; accepted 15 April 2005

Available online 20 June 2005

Abstract

Variations in fluid density can greatly affect fluid flow and solute transport in the subsurface. Heterogeneities such as fractures

play a major role for the migration of variable-density fluids. Earlier modeling studies of density effects in fractured media were

restricted to orthogonal fracture networks, consisting of only vertical and horizontal fractures. The present study addresses the phe-

nomenon of 3D variable-density flow and transport in fractured porous media, where fractures of an arbitrary incline can occur. A

general formulation of the body force vector is derived, which accounts for variable-density flow and transport in fractures of any

orientation. Simulation results are presented that show the verification of the new model formulation, for the porous matrix and for

inclined fractures. Simulations of variable-density flow and solute transport are then conducted for a single fracture, embedded in a

porous matrix. The simulations show that density-driven flow in the fracture causes convective flow within the porous matrix and

that the high-permeability fracture acts as a barrier for convection. Other simulations were run to investigate the influence of frac-

ture incline on plume migration. Finally, tabular data of the tracer breakthrough curve in the inclined fracture is given to facilitate

the verification of other codes.

� 2005 Elsevier Ltd. All rights reserved.

Keywords: Numerical modeling; Fractured rock; Inclined fracture; Contaminant transport; Density; Instability; Buoyancy term

1. Introduction

In subsurface environments, contaminants are mainly

transported by groundwater. The transport pattern isgreatly affected by the physical parameters of both the

medium and the contaminant. The transport properties

of the fluid, such as viscosity and fluid density, can also

have a significant impact on contaminant transport. The

fluid density q (fluid mass per unit volume of fluid) can

not always be assumed uniform. It decreases with

increasing temperature, increases with increasing salinity

and increases with increasing pressure due to fluidcompressibility.

0309-1708/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.advwatres.2005.04.011

* Corresponding author.

E-mail address: thomas.graf.1@ulaval.ca (T. Graf).

Spatial variations of fluid density play an important

role in contaminant migration within various geological

media. If, for example, a fluid of high density overlies a

less dense fluid, the system is potentially unstable anddensity-driven flow may take place, which levels out

the density stratification and eventually stabilizes the

system. Additionally, if flow is transient, there are tem-

poral changes in density. Examples for density-driven

flow and transport can be found in many areas of sub-

surface hydrology, oceanography, meteorology, geo-

physics and hazardous waste disposal.

The safe disposal of hazardous chemicals is com-monly regarded as feasible in low-permeability geologi-

cal media at a depth of up to 1000 m [4]. In Canada, one

conceivable host rock for the repository of radioactive

waste is the crystalline metamorphic rock of the Cana-

dian Shield. Here, the groundwater at depths greater

Nomenclature

• scalar variables are denoted in normal italic

letters• vector variables are denoted in bold small

letters

• matrix variables are denoted in bold capital

letters

Latin letters

2b fracture aperture [L]

c solute concentration, expressed as relativeconcentration [–]

Dd free-solution diffusion coefficient [L2 T�1]

Dij coefficients of hydrodynamic dispersion ten-

sor [L2 T�1]

Dfrij hydrodynamic dispersion coefficient of the

fracture [L2 T�1]

f three-dimensional function, defined over a

surface S; f = f(x,y,z) [–]f global boundary flux vector [L2 T�1]

F function that defines the surface S in three

dimensions; F(x,y,z) = constant [–]

g acceleration due to gravity [L T�2]

g global body force vector [L2 T�1]

h0 equivalent freshwater head [L]

H hypotenuse of a rectangle [L]

H global conductance matrix [L T�1]i unit vector in x-direction [–]

I+, I� fracture–matrix interface [–]

j unit vector in y-direction [–]

J Jacobian matrix [–]

k unit vector in z-direction [–]

K0ij coefficients of hydraulic conductivity tensor

of freshwater [L T�1]

K fr0 hydraulic freshwater conductivity of the frac-

ture [L T�1]

‘v geometry of the model domain; v = x, y, z [L]

Lv geometry of a block element; v = x, y, z [L]

LTG characteristic length scale [L]

Nfe total number of fracture elements in the do-

main [–]

p vector that is normal to region R [–]

P dynamic pressure of the fluid [M L�1 T�2]Peg grid Peclet number [–]

qi Darcy flux [L T�1]

R vertical projection of S on a coordinate plane

[–]

Re Reynolds number [–]

s axis along a sloped tube [L]

S fluid mass matrix [L]

S surface, defined by the function F(x,y,z) =

constant [L2]SS specific storage [L�1]

Sop specific pressure storativity [M�1 L T2]

t time [T]

vi linear flow velocity [L T�1]

wi approximation function [–]

Greek letters

afl coefficient of the compressibility of the fluiddue to fluid pressure or hydraulic head varia-

tions [M�1 L T2]

afr fracture dispersivity [L]

al matrix longitudinal dispersivity [L]

am coefficient of the compressibility of the por-

ous medium due to fluid pressure or hydrau-

lic head variations [M�1 L T2]

at matrix transverse dispersivity [L]b ratio of saltwater density to freshwater den-

sity [–]

c maximum relative density [–]

dij Kronecker delta function [–]

gj indicator for the fracture face orientation [–]

j intrinsic permeability in a sloped tube [L2]

jij coefficients of the intrinsic permeability ten-

sor [L2]l dynamic viscosity of the fluid [M L�1 T�1]

m kinematic viscosity of the fluid [L2 T�1]

q fluid density [M L�3]

q0 reference fluid density [M L�3]

qr relative fluid density [–]

s factor of tortuosity [–]

u fracture incline [1�]/ porosity of the rock matrix [–]v global coordinates; v = x, y, z [L]�v local coordinates; �v ¼ �x; �y;�z [L]

Special symbols

o partial differential operator [–]

$ Nabla or divergence operator; rð Þ ¼ oð Þox þ

oð Þoy þ oð Þ

oz [L�1]

Sub- and superscripts

0 freshwater [–]

fe fracture element [–]

fr fracture [–]

i, j spatial or nodal indices [–]

n normal direction [–]

1352 T. Graf, R. Therrien / Advances in Water Resources 28 (2005) 1351–1367

than 800 m is a Ca–Na–Cl brine with dissolved solids

exceeding 100,000 mg l�1 [8]. Therefore, the density of

such deep fluids varies dramatically with solute concen-

tration. Acute safety questions regarding nuclear waste

T. Graf, R. Therrien / Advances in Water Resources 28 (2005) 1351–1367 1353

repository arise due to the presence of fractures in the

hard rock formations of the Canadian Shield. Fractures

have a great impact on the mass transport, because they

represent preferential pathways where accidentally re-

leased radionuclides might migrate at velocities that

are several orders of magnitude larger than within therock matrix itself. Clearly, it is of paramount impor-

tance to understand the movement of contaminants in

fractured media under the influence of fluctuating water

density.

In the last century, the foundations for modeling den-

sity-driven flow and transport were laid by Rayleigh

[20]. He mathematically formulated the principles that

govern the onset of instabilities caused by thermally in-duced fluid density variations in porous media. Vari-

able-density flow has been investigated experimentally

and numerically for half a century, beginning with the

experimental studies by Elder [6] and Taylor [29].

Improvements in computer performance have enabled

increasingly precise and complex simulations of density

effects, predominantly in porous media (Fig. 1a and b).

In homogeneous porous media, variable-density flowand transport have recently been studied in experimental

and numerical simulations in the fields of (i) convection

beneath salt lakes [25,26,35], (ii) seawater intrusion in

coastal aquifers [12,36], (iii) infiltration of leachates

Fig. 1. Different styles of geological media: (a) homogeneous porous medium

vertical and horizontal fractures and (d) fractured geological medium with no

In (a) and (b), the shades of grey represent hydraulic conductivity.

from waste disposal sites [9] and (iv) the analysis of

instability development [15,22,34].

In heterogeneous porous media, Schincariol and Sch-

wartz [21] were the pioneers in experimentally investigat-

ing density dependent flow and transport in layered and

lenticular media. They found that (i) the transport pat-tern in the layers is greatly sensitive to hydraulic conduc-

tivity and (ii) the heterogeneities in the lenticular

medium create relatively large dispersion that tends to

dissipate instabilities. Thus, Simmons et al. [27] infer

that heterogeneities play opposite roles in the generation

and growth of instabilities. On one hand, they are the

trigger mechanism for the onset of instabilities while

on the other hand, heterogeneities play the most impor-tant role in diminishing instabilities, once they are gen-

erated. Simmons et al. [27] showed that the style of

heterogeneity in a porous medium will greatly influence

the propagation of dense plumes, with disorganized het-

erogeneity tending to dissipate convection by mixing

and thus reducing plume instabilities. Prasad and Sim-

mons [18] confirmed this observation. They carried out

numerical variable-density transport simulations in aheterogeneous porous media. Prasad and Simmons

[18] used a modified form of the Elder [7] problem,

where the permeability was randomly distributed to

study the effects of heterogeneity on solute transport

, (b) heterogeneous porous medium, (c) fractured medium consisting of

nuniform fracture aperture, trace and orientation (modified from [27]).

1354 T. Graf, R. Therrien / Advances in Water Resources 28 (2005) 1351–1367

processes. Diersch and Kolditz [5] and Holzbecher [11]

provide an excellent overview of prior modeling work

in porous media, clearly showing that there is currently

a lack of models that take into account the crucial influ-

ence of fractures.

As opposed to the study of variable-density flow inporous media, laboratory experiments in fractured med-

ia have not yet been carried out. A ground-breaking

study in this field was published by Shikaze et al. [24]

who numerically simulated variable-density flow and

transport in discretely fractured media. They found that

vertical fractures of aperture as small as 50 lm signifi-

cantly increase contaminant migration relative to the

case where fractures are absent. Interestingly, it was alsoshown that dense solute plumes may develop in a highly

irregular fashion and are extremely difficult to predict.

However, Shikaze et al. [24] limited their studies to a

regular fracture network consisting of only vertical

and horizontal fractures, embedded in a porous matrix

(Fig. 1c). Therefore, the development of dense plume

instabilities in a discrete fracture of an arbitrary incline

remains unknown. Moreover, a network of fractureshaving irregular orientations will exhibit a different style

of heterogeneity, as opposed to regular fracture distribu-

tion, which may affect dense plume migration as sug-

gested by Simmons et al. [27].

The goal of this study is to investigate variable-den-

sity flow and transport in discretely fractured porous

media where the fracture network is lacking a regular

pattern and where fractures of an arbitrary slope can oc-cur (Fig. 1d). An existing model that solves 3D flow and

transport in discretely fractured porous media has been

modified to simulate density-dependent flow and trans-

port and to discretize discrete 2D fractures of any orien-

tation. Unlike previous studies [24], this work is not

limited to flow in orthogonal fracture networks but fo-

cuses on the growth of instabilities in an inclined frac-

ture, located in a low-permeability porous matrix.Thus, the present investigation will enable the numerical

simulation of variable-density transport in complex,

irregular fracture networks, which more closely resemble

nature.

2. Mathematical modeling

2.1. The FRAC3DVS model

FRAC3DVS is a saturated–unsaturated numerical

groundwater flow and solute transport model [30,31].

The governing equations for flow and transport are de-

rived from the continuum approach. A control volume

finite element method is used to spatially discretize the

flow and transport equations. The porous, low-perme-ability matrix is represented by regular three-dimen-

sional blocks and fractures of high permeability are

represented by two-dimensional rectangular planes.

Using undistorted finite elements allows an analytical

discretization of the governing equations by means of

elemental influence coefficient matrices [9,30]. Thus,

there is no need to numerically integrate. The solution

takes into account advective flow and transport, molec-ular diffusion, and mechanical dispersion in both the

fractures and the matrix.

In FRAC3DVS, vertical and horizontal fractures are

incorporated into the grid by superimposing two-dimen-

sional face elements onto the three-dimensional grid,

consisting of regular block elements. Two-dimensional

faces represent the fracture whereas three-dimensional

blocks denote the porous matrix. In order to fully cou-ple the fracture with the porous matrix, faces and blocks

share common nodes along the fracture walls. Thus,

nodes at fracture locations receive contributions from

both the block elements as well as from the fracture

faces. Furthermore, for these mutual nodes, both

hydraulic head and concentration at the fracture/matrix

interface are assumed to be equal. Therefore, it is not

necessary to explicitly calculate the exchange terms qnand Xn in Eqs. (14) and (16), respectively. This discrete

fracture approach has previously been applied by several

authors [28,23,30,24] and its description is, therefore,

not repeated here.

2.2. Model development

The model FRAC3DVS has been modified here toincorporate inclined fractures in the simulation of vari-

able-density flow and transport. Density variations

cause weak nonlinearities in the flow equation. In the

numerical model they are treated by means of a sequen-

tial iterative approach (SIA), also called Picard itera-

tion, which links the two governing equations for flow

and transport. This method alternately solves the two

governing equations during each time step until conver-gence is attained.

2.2.1. Discretizing inclined fractures

Similar to the integration of horizontal and vertical

(i.e. regular) fractures, inclined fractures are incorpo-

rated in the finite element grid by superimposing two-

dimensional fracture faces onto the block elements of

the porous matrix. While the six faces on the outer sur-face of the block elements can be used for the discretiza-

tion of regular fractures, for inclined fractures an

additional six inclined faces inside the blocks are also

available. Fig. 2 exhibits the orientation of the six in-

clined faces. Note that the two faces in the block on

the right of Fig. 2 are not inclined but vertical. However,

for convenience, they are considered as inclined as well

because they do not correspond to a boundary face ofthe block. Furthermore, it can be seen that all available

six element boundary faces and six inclined faces are

Fig. 2. Inclined faces in three-dimensional block elements.

T. Graf, R. Therrien / Advances in Water Resources 28 (2005) 1351–1367 1355

rectangular and undistorted. Using the control volume

finite element method, and assuming continuity of

hydraulic head and concentration at the common frac-

ture–matrix nodes, results in an unchanged connectivity

pattern for the 3D porous medium elements, irrespective

of the presence of fractures.

In FRAC3DVS, the location of three-dimensional in-

clined fractures is defined by two points and by a coor-dinate axis parallel to the fracture. The fracture shown

in Fig. 3 is parallel to the y-axis and defined by the

two points West (W) and East (E) that coincide with

the beginning and the end of a fracture. The fracture

nodes betweenW and E are selected using a simple least

distance criterion: for every node P that defines the frac-

ture, the distance of all three neighbor nodes of P to the

undiscretized fracture is calculated. The neighbor nodewhose distance to the fracture is the smallest is selected

as a fracture node and becomes point P for the next

step. Initially, P is identical to W. This process is re-

peated until the point P meets the end point E. Fig. 3

is an example of how an inclined fracture is discretized

in an irregular, relatively coarse grid. Note that the grid

is 3D with a unit thickness. Thus, the fracture is 2D and

defined by W and E and by its orientation (parallel tothe y-axis).

Inclined fractures are a combination of inclined, hor-

izontal and vertical faces. The mathematical formula-

tion of density effects in each of these three types of

fracture elements is described in the following section.

W

E

x

z

P 3

21

fracture in nature

discretized fracture

Fig. 3. Selecting two-dimensional elements of an inclined fracture.

2.2.2. Constitutive equations

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

defined by Frind [9] as

h0 ¼P

q0gþ z ð1Þ

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

gravitational acceleration and z [L] is the elevation above

datum. The transport variable is the dimensionless rela-

tive concentration, c, which varies between 0 and 1. It islinked with density through the linear relationship

qr ¼ cc ð2Þwhere qr is the dimensionless relative density, defined by

Frind [9] as

qr ¼qq0

� 1 ð3Þ

where q [M L�3] is the fluid density. The dimensionless

constant c is the maximum relative density given by

c ¼ qmax

q0

� 1 ð4Þ

where the assumption is made that the solute concentra-

tion of a fluid with the density q = qmax is cmax = 1. It is

also assumed that the impact of salinity on fluid viscos-

ity is negligible.Under variable-density flow conditions, the Darcy

flux, qi = /vi [L T�1], is a function of both the physical

flow variable, h0, and the chemical property, qr. The

Darcy flux in porous media can be completely expressed

in terms of freshwater properties [9]:

qi ¼ �K0ij

oh0oxj

þ qrgj

� �i; j ¼ 1; 2; 3 ð5Þ

where the assumption of a horizontal datum (i.e. oz/

oz = 1) is made and where gj [–] represents the directionof flow with gj = 0 in the horizontal directions and gj = 1in the vertical direction [9]. Assuming that fluid viscosity

does not depend on salt concentration allows the use of

the freshwater hydraulic conductivity of the porous

medium, K0ij [L T�1], given by [1] as:

K0ij ¼

jijq0gl0

ð6Þ

1356 T. Graf, R. Therrien / Advances in Water Resources 28 (2005) 1351–1367

where jij [L2] is the matrix permeability and where l0 [M

L�1 T�1] is the reference fluid viscosity. A three-dimen-

sional Cartesian coordinate system is assumed, where

the axes are co-linear with the principal directions of

anisotropy. With this assumption, cross terms (K0ij where

i5 j) in the hydraulic conductivity tensor vanish.The Darcy flux in differently oriented two-dimen-

sional fracture faces can be calculated using the follow-

ing form of the Darcy equation, presented by Bear [1]

for an inclined one-dimensional tube

qs ¼ � jl

oPos

þ qgozos

� �ð7Þ

where s [L] is the axis along the tube and where j [L2] is

the permeability of the porous material in the tube. The

fluid pressure in (7) can be written in terms of equivalent

freshwater head using relation (1). With (6) and becauseoz/os is the cosine of the slope, the Darcy flux in a frac-

ture element can be written as

qfri ¼ �K fr0

ohfr0oxj

þ qfrr gj cosu

� �i; j ¼ 1; 2 ð8Þ

where gj is 0 in the horizontal direction and 1 along the

fracture incline. The incline of a fracture face is given by

u with u = 0� for a vertical face and u = 90� for a hor-izontal face. In the case of flow within fractures, a local

two-dimensional Cartesian coordinate system is as-

sumed. The freshwater hydraulic conductivity of the

fracture, K fr0 [L T�1], is derived from the parallel plate

model as

K fr0 ¼ ð2bÞ2q0g

12l0

ð9Þ

where (2b) [L] is the fracture aperture.

The application of Darcy�s law in fractures (8) re-

quires that the Reynolds number be smaller than 1 [2].

2.2.3. Governing equations

The governing equations for flow and transport in the

3D porous blocks are described first, followed by the

flow and transport equations in open fractures.

The equation that describes three-dimensional vari-

able-density flow under saturated conditions is [9]:

o

oxiK0

ij

oh0oxj

þ qrgj

� �� �¼ SS

oh0ot

i; j ¼ 1; 2; 3 ð10Þ

where fluid sources and sinks are not considered. Thespecific storage SS [L�1] accounts for both matrix and

fluid compressibility and is defined as [9]:

SS ¼ q0gðam þ /aflÞ ð11Þwhere am [M�1 L T2] and afl [M�1 L T2] are the matrix

and fluid compressibility, respectively.

Neglecting adsorption, radioactive decay and solutesources/sinks, the equation that governs solute transport

is the advective–dispersive flow equation in three dimen-

sions [1]:

o

oxi/Dij

ocoxj

� qic� �

¼ oð/cÞot

i; j ¼ 1; 2; 3 ð12Þ

The coefficients of the hydrodynamic dispersion ten-

sor Dij [L2 T�1] are given by Bear [1] as

/Dij ¼ ðal � atÞqiqjjqj þ atjqjdij þ /sDddij i; j ¼ 1; 2; 3

ð13Þwhere al [L] and at [L] are the longitudinal and trans-

verse dispersivity, respectively, dij [–] is the Kronecker

delta function, s [–] is matrix tortuosity and Dd

[L2 T�1] is the free-solution diffusion coefficient.

Flow and transport in an open discrete fracture take

place in two dimensions. Therefore, the corresponding

governing equations are defined in a local 2D coordinatesystem. The equation that governs variable-density flow

in fractures has been presented by several authors

[3,28,24,31]:

ð2bÞ o

oxiK fr

0

ohfr0oxj

þ qfrr gj cosu

� �� �� Sfr

S

ohfr0ot

� �þ qnjIþ � qnjI� ¼ 0 i; j ¼ 1; 2 ð14Þ

where the last two terms represent normal components

of the fluid flux across the boundary interfaces (I+ and

I�) that separate the fracture and the porous matrix.

In the conceptual model, fractures are idealized astwo-dimensional parallel plates. Therefore, both the

freshwater head, hfr0 , and the relative concentration, cfr,

are uniform across the fracture width.

The specific storage in the fracture, SfrS [L�1], can be

derived from (11) by assuming that the fracture is

incompressible, such that am = 0, and by setting its

porosity to 1:

SfrS ¼ q0gafl ð15ÞShikaze et al. [24] provide the equation for transport

in a discrete fracture as

ð2bÞ o

oxiDfr

ij

ocfr

oxj

� �� qfri

ocfr

oxi� ocfr

ot

� �þ XnjIþ � XnjI� ¼ 0 i; j ¼ 1; 2 ð16Þ

where Dfrij [L

2 T�1] is the hydrodynamic dispersion coef-

ficient of the fracture, defined by [30]:

Dfrij ¼ ðafr

l � afrt Þ

qfri qfrj

jqfrj þ afrt jqfrjdij þ Dddij i; j ¼ 1; 2

ð17Þ

where afr [L] is the fracture dispersivity. The last two

terms in Eq. (16) represent advective–dispersive loss or

gain of solute mass across the fracture–matrix interfaces

I+ and I� [28].

(0, Ly, Lz)

3

Ly

Lz

Lx

x

z

y

Sxy

4

12

Afeϕ

z

y

x=

(Lx, Ly, Lz)

(Lx, 0, 0)(0, 0, 0)

S: F(x, y, z) = constant

Sxz

H

Fig. 4. Geometry of an inclined 2D fracture element in three

dimensions.

T. Graf, R. Therrien / Advances in Water Resources 28 (2005) 1351–1367 1357

2.2.4. Finite element formulation of the buoyancy term in

a fracture

Frind [9] provides the finite element formulation of

the 2D variable-density flow equation in porous media

in the absence of fractures. The elements used in this

study are two-dimensional vertical undistorted rectan-gles. However, if fractures are present and assuming

the common node approach, the hydraulic head at the

fracture/matrix interface is identical in both media such

that the exchange terms qnjIþ and qnjI� in Eq. (14) van-

ish. In this case, the 2D flow equation in a porous matrix

as given in [9] and the 2D flow equation in a discrete

fracture given by (14) are mathematically identical. As

a consequence, Frind�s finite element formulationfor variable-density flow in 2D matrix elements can

also be used for density-driven flow in 2D fracture

faces.

Following Frind�s derivation, the finite element for-

mulation for Eq. (14) can be written as a semi-discrete

global matrix system in the compact form

H � h0 þ S � oh0ot

þ g ¼ f ð18Þ

where superscript ‘‘fr’’ is dropped for clarity. In (18), H

[L T�1] is the conductance or stiffness matrix, S [L] is the

fluid mass matrix, g [L2 T�1] is the body force vector and

f [L2 T�1] is the boundary flux vector. Vector g repre-sents density effects and is, therefore, of special interest

in this study. See [9] for details on the definition the

other matrices and vectors.

If 2D fracture faces are assumed, each of the four

arrays in (18) can be written as the sum of all the Nfe

elemental arrays, Nfe being the total number of fracture

elements in the grid. Following this, the fracture body

force vector, g, can be expressed as

g ¼Xfe

gfe ð19Þ

where gfe [L2 T�1] is the body force vector, written at thefracture elemental level. In a two-dimensional quadrilat-

eral element, whose sides coincide with the local coordi-

nate axis �x and �z, the entries gfei [L2 T�1] of vector gfe are

calculated after [9] as

gfei ¼Z Z

Afe

K fr0 �q

fer cosu

owfei

o�zd�xd�z i ¼ 1; 2; 3; 4 ð20Þ

where �qfer is the average relative density in the fracture

element (face) fe, wfei [–] is the value of the 2D approxi-

mation function in face fe at node i and Afe [L2] is the

surface area of fe. The double integral in (20) represents

the general mathematical expression of buoyancy in a

2D fracture face of any three-dimensional orientation.

However, the discretized inclined fracture can be a com-

bination of horizontal, vertical and inclined two-dimen-

sional elements. Integration of Eq. (20) is required toobtain the finite formulation of density effects for frac-

ture elements with arbitrary orientations, in order to

fully account for density effects in the entire fracture.

The right side of (20) can be integrated in two differ-

ent ways. First, the fracture weighting function is de-

fined in terms of local coordinates in the usual way

(e.g. for node 1) as wfe1 ¼ ðL�x � �xÞðL�z � �zÞ=ðL�x � L�zÞ, where

L�x [L] and L�z [L] are the element dimensions in the �x- and�z-direction, respectively (Fig. 4). With the derivative of

wfe1 , the integration in (20) is identical to that presented

by [9], resulting in the following coefficient for node 1:

gfe1 ¼ �K fr0 �q

fer cosu

L�x

2ð21Þ

which can be obtained likewise for the other nodes 2, 3

and 4.

The second, more general method first assumes a ver-

tical face, which will be rotated to match its real inclined

position. In this case of a vertical face, the cosine in (20)

is unity and the local coordinate axes of the face, �x and �z,coincide with the global coordinates, x and z (Fig. 4)

such that:

gfei ¼Z Z

Afe

K fr0 �q

fer

owfei

ozdxdz i ¼ 1; 2; 3; 4 ð22Þ

Unlike in the first method, the density term is now de-

fined over a vertical fracture element. If the quasi-verti-

cal face is rotated back to its original inclined position,

the buoyancy term (20) has to be integrated over the en-tire 2D inclined element area for an arbitrary face orien-

tation in 3D. However, the weighting function as well as

its derivative are 2D functions that are defined over a

surface defined in 3D space. Thus, a surface integral

calculus problem has to be solved where the function

f ðx; y; zÞ ¼ K fr0 �q

fer ðowfe

i =ozÞ is defined over the surface S

of the fracture face.

Following [32], the integral of a function f(x,y,z) overa surface S in space, described by the function

p

x

Surface F(x, y, z) = constant

Vertical projection orshadow of on acoordinate plane

Syz

R

S

Fig. 5. Projection of S on a coordinate plane [32].

1358 T. Graf, R. Therrien / Advances in Water Resources 28 (2005) 1351–1367

F(x,y,z) = constant (Fig. 5), can be calculated by evalu-

ating a closely related double integral over the vertical

projection or shadow of S on a coordinate plane in

the formZ ZSf ðx; y; zÞdS ¼

Z ZRf ðx; y; zÞ jrF j

jrF � pj dA ð23Þ

where R is the shadow region on the ground plane be-

neath surface S and p is a vector normal to R. This sur-

face integral can be directly evaluated only if a 1:1

mapping of S in the xy- or in the xz-plane exists. Then,

R could be Sxy or Sxz because both projections yield the

same result. The projection of S in the xz-plane is con-

sidered because a pseudo-vertical fracture element was

assumed.From Fig. 4, equation F(x,y,z) = constant of the sur-

face S, which defines the plane of the two-dimensional

fracture element, can easily be derived as Lzy � Lyz = 0.

Thus, a 1:1 mapping of S in both coordinate planes ex-

ists and, as a consequence, the integral on the right-hand

side of (23) can be evaluated and we have

S : F ðx; y; zÞ ¼ Lzy � Lyz ð24ÞrF ¼ 0 � iþ Lz � j� Ly � k ð25Þ

jrF j ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2y þ L2

z

qð26Þ

and with p = j, the unit vector in y-direction

jrF � pj ¼ jrF � jj ¼ Lz ð27Þ

Therefore, Eq. (23) becomes

Z ZSf ðx; y; zÞdS ¼

Z ZSxz

f ðx; y; zÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2y þ L2

z

qLz

dxdz

ð28Þ

where dA = dxÆdz. With the function f ðx; y; zÞ ¼K fr

0 �qfer ðowfe

i =ozÞ, which must be integrated, the entries

of the elemental body force vector gfei from Eq. (20)

are given in the form

gfei ¼Z Lz

0

Z Lx

0

K fr0 �q

fer

owfei

oz

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

L2y

L2z

sdxdz i ¼ 1; 2; 3; 4

ð29Þ

The elemental approximation function wfei is always

formulated as a function of local coordinates rather

than global ones. Therefore, the integral in Eq. (29)

has to be evaluated in the local coordinates �v. The re-

quired coordinate transformation is a rotation around

the x-axis by angle u (see Fig. 4) and can be written inthe matrix form

1 0 0

0 cosu � sinu

0 sinu cosu

264

375 �

x

y

z

8><>:

9>=>; ¼

�x

�y

�z

8><>:

9>=>; ð30Þ

Thus, we can write the derivatives

o�zoz

¼ cosu ð31Þ

ozo�z

¼ cosu ð32Þ

The integral of Eq. (29) is rewritten in terms of local

coordinates by first substituting the derivative by means

of the chain rule, leading to

owfei

oz¼ owfe

i

o�z� o�zoz

ð33Þ

and, second, by adjusting the elementary volume

following

dxdz ¼ det J � d�xd�z ð34ÞHere, the Jacobian matrix J [–] collapses to the simple

1 · 1 matrix

J ¼ ozo�z

� �ð35Þ

with determinant detJ given by Eq. (32). According toFig. 4, the approximation function for node 1 may be

expressed in local coordinates as

wfe1 ð�x;�zÞ ¼

1

LxHðLx � �xÞðH � �zÞ ð36Þ

with the spatial derivative

owfe1

o�z¼ �x� Lx

LxHð37Þ

where H [L] is the hypotenuse of the occurring triangle

in the yz-plane, given as H ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2y þ L2

z

q. Now use can

be made of Eqs. (31)–(35) and (37) to rewrite Eq. (29).

The elemental body force vector entry gfe1 can be written

in local coordinates as

gfe1 ¼Z H

0

Z Lx

0

K fr0 �q

fer

�x� Lx

LxH

� �cos2u

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

L2y

L2z

s0@

1Ad�xd�z

ð38ÞFinally, a two-dimensional integration in Eq. (38)

directly yields the solution for node 1:

gfe1 ¼ �K fr0 �q

fer

LTG

2ð39Þ

T. Graf, R. Therrien / Advances in Water Resources 28 (2005) 1351–1367 1359

where the characteristic length LTG is a function of the

element geometry such that

LTG ¼ Lx �Lzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

L2y þ L2

z

q ð40Þ

Note that solution (39) is identical to (21) because

Lx ¼ L�x and because the second factor in (40) is the

cosine of the fracture face incline, cosu ¼ Lz=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2y þ L2

z

q.

The repetition of steps (36)–(39) for nodes 2, 3 and 4

yields the following final form of the elemental body

force vector gfe for arbitrarily inclined two-dimensional

fracture elements.

gfe ¼ K fr0 �q

fer

LTG

2

�1

�1

1

1

8>>><>>>:

9>>>=>>>;

ð41Þ

where LTG is given by Eq. (40). The characteristics of

this length scale are:

(1) For a vertical fracture element, the calculated den-

sity effects reach a maximum. Solution (41)

becomes identical to that in [9] because LTG

becomes Lx:

limLy!0

LTG ¼ Lx

(2) For a horizontal fracture element, no density

effects occur and solution (41) becomes zero

because LTG vanishes:

limLz!0

LTG ¼ 0

(3) For every arbitrarily inclined fracture element, the

magnitude of the density effect will exceed zero

and will be smaller than the density effect in a ver-

tical fracture element:

0 < LTG < Lx

(4) For an inclined fracture element, which is not par-allel to the x-axis as shown in Fig. 4, but to the y-

axis as shown in Fig. 3, the x- and y-dimensions in

Eq. (40) are simply switched:

LTG ¼ Ly �Lzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

L2x þ L2

z

q

2.3. Model verification

Variable-density flow and transport in porous media

was verified in two and three dimensions. All simula-

tions used implicit transport time weighting, as is com-

mon in other variable-density simulations, and fullupstream weighting as proposed by Folkovic and De

Schepper [10].

First, the Elder [7] salt convection problem was sim-

ulated to qualitatively test the model in two dimensions.

Kolditz et al. [14] point out that, for a coarse grid, the

central transport direction is downwards, whereas a fine

grid exhibits central upwelling. These observations were

confirmed by Prasad and Simmons [19] as well as in thepresent study using FRAC3DVS (Fig. 6).

Another code verification consisted in comparing the

Elder [7] results presented by Folkovic and De Schepper

[10] to those of FRAC3DVS. This verification is more

trustworthy than the one described in the previous par-

agraph because both numerical models (the Folkovic

and De Schepper [10] model and FRAC3DVS) use

the same numerical approach (control volume finite ele-ment method CVFE) as well as the same flow variable

(fluid pressure P). The governing flow equation for this

variable is given in the Appendix A. Folkovic and De

Schepper [10] carried out their numerical simulations

in the half domain of the symmetric Elder problem.

Interestingly, they found that an extremely fine grid

(32,768 nodes in the half domain) again exhibits central

downwelling, which was also discovered by Diersch andKolditz [5]. Their results are in very good visual

agreement with those from the FRAC3DVS model

(Fig. 7).

A new benchmark problem for variable-density

transport in 3D has been presented by Oswald and Kin-

zelbach [17]. This problem is based on the experimental

three-dimensional variable-density flow and solute

transport simulations in porous media conducted byOswald [16]. In these experiments, a 0.2 m · 0.2 m ·0.2 m closed box initially contained saltwater from the

bottom up to 8 cm, with the rest of the box filled with

freshwater. A constant freshwater recharge through

one upper corner of the box disturbed this stable layer-

ing of two miscible fluids. The concentration of the

mixed fluid versus time was measured at the discharging

open hole on the opposite side of the input location.Oswald [16] used two different initial concentrations

c01 = 0.01 (case 1) and c02 = 0.1 (case 2). The experimen-

tal results were numerically reproduced by Johannsen et

al. [13], who also present tabular data of the measured

concentrations versus time.

The FRAC3DVS model output was compared in

three dimensions with Oswald�s [16] experimental re-

sults, given in Johannsen et al. [13]. The physical param-eters given by Oswald and Kinzelbach [17] were used.

The first problem of the lower initial concentration

0.01 (case 1) was used because [13] showed that, in this

case, grid convergence is achieved with a relatively

coarse grid, whereas for case 2, the solution converged

only for a very fine grid, consisting of at least 274,625

grid points [13]. Good agreement between the experi-

mental results from [16], the numerical results from [5]and the FRAC3DVS model was obtained (Fig. 8). The

long-term results of this low density case more closely

2 years 2 years

4 years 4 years

10 years 10 years

Fig. 6. Results of the Elder problem for a coarse grid (left; 60 · 30 elements) and a fine grid (right; 120 · 40 elements) at 2, 4 and 10 yr simulation

time by Elder [7] [—; coarse grid], Kolditz et al. [14] [—; fine grid], Prasad and Simmons [19] [- �� -] and the present model [- -].

1 year 10 years

2 years 15 years

4 years 20 years

Fig. 7. Results of the Elder problem for an extremely fine grid (256 · 128 elements in the half domain) at 1, 2, 4, 10, 15 and 20 yr simulation time by

Folkovic and De Schepper [10] [—] and the present model [- -]. Shown are the 20%, 40%, 60% and 80% contours.

1360 T. Graf, R. Therrien / Advances in Water Resources 28 (2005) 1351–1367

resemble the experimental data than in [5]; however, dif-

ferences remain.

Variable-density flow in fractures was verified in a

two-step manner. First, the results of Shikaze et al.

0

0.01

0.02

0.03

0.04

0.05

0.06

0 50 100 150

time [min]

salt

mas

s fr

actio

n [%

] Experimental (Johannsen et al., 2002)

Numerical (the developed model)

Numerical (Diersch and Kolditz, 2002)

Fig. 8. Results of three-dimensional variable-density transport simu-

lations in porous media.

Fig. 10. Simulation geometry, boundary and initial conditions for

simulating density dependent flow and transport in fractured media.

Table 1

Model parameters used in fractured media studies

Parameter Value

Freshwater densitya (q0) 1000 kg m�3

Maximum fluid densitya (qmax) 1200 kg m�3

Fluid dynamic viscosityb (l) 3.545 · 104 kg m�1 yr�1

Fluid compressibilityb (afl) 4.42 · 10�25 kg�1 m yr2

Matrix compressibilityb (am) 2.51 · 10�24 kg�1 m yr2

Acceleration due to gravity (g) 9.75 · 1015 m yr�2

Tortuositya (s) 0.1

Matrix permeabilitya (jij) 10�15 m2

Matrix porosityc (/) 0.35

Matrix longitudinal dispersivitya (al) 0.1 m

Matrix transverse dispersivitya (at) 0.005 m

Fracture dispersivitya,d (afr) 0.1 m

Fracture aperturea (2b) 50 lmFree-solution diffusion coefficienta (Dd) 0.15768 m2 yr�1

a Ref. [24].b Ref. [14].c Ref. [9].d Ref. [30].

T. Graf, R. Therrien / Advances in Water Resources 28 (2005) 1351–1367 1361

[24] were used to verify density effects in vertical frac-

tures. With implicit transport time weighting, full up-

stream weighting of velocities and matrix porosity /= 0.35, which is used in [9], perfect agreement of the

output was obtained (results not shown).Second, variable-density transport in a fracture with

incline u = 45� was verified by comparing the results

from two different scenarios. In scenario 1, an inclined

fracture is discretized by only inclined faces. In a second

scenario, the inclined fracture consists of only vertical

and horizontal faces (Fig. 9), for which density effects

have already been successfully validated using the results

from [24]. In all simulations, we used a three-dimen-sional vertical slice with dimensions ‘x = 12 m,

‘y = 1 m and ‘z = 10 m as model domain. The left and

right boundaries are assumed to be impermeable,

whereas the top and bottom boundaries are specified

as constant equivalent head boundaries with zero con-

stant heads (Fig. 10). The contaminant source of con-

stant concentration, c = cL, overlies groundwater of

initial concentration, c = c0, where c0 = 0.0 < cL = 1.0.All simulations cover a time of 20 yr. Time step sizes

are kept constant at 0.2 yr. The input parameters for the

numerical simulations are shown in Table 1. These

parameters were held constant throughout all simula-

Fig. 9. Different discretization of the inclined fracture in scenario 1

and 2.

tions unless otherwise stated. It is assumed that the por-

ous matrix is isotropic and homogeneous throughout

and that the entire aquifer is completely saturated.

Grids of different discretization levels were generated

to investigate the adequate grid line density. The method

is called grid convergence study and it involves perform-

ing a simulation on successively finer grids. As the grid is

refined, the spatial discretization errors should asymp-totically approach zero, excluding computer round-off

errors.

Here, the grid at the lth level (l = 1,2, . . .) consists of480l2 identical square elements, which have the size

Dx = Dz = 1/2l m. Increasing grid levels correspond to

finer grids. Several simulations of scenario 1 were

performed at increasing grid levels and the tracer

breakthrough was monitored at the observation point(x = 6 m, z = 6 m) in the fracture as shown in Fig. 9.

Grid convergence was achieved for the grid of level 5

(Fig. 11), consisting of 12,000 square elements of size

Dx = Dz = 0.1 m. With this grid, the grid Peclet number

Peg = Dx/al becomes 1.0, satisfying the widely accepted

0

0.2

0.4

0.6

0.8

1 3 5 7 9 11 13 15

time [yr]

conc

entr

atio

n [--

]

grid5 grid4 grid3 grid2 grid1

Fig. 11. Scenario 1: grid convergence.

1362 T. Graf, R. Therrien / Advances in Water Resources 28 (2005) 1351–1367

criteria for neglecting numerical dispersion, Peg 6 2, as

well as oscillations, Peg 6 4.

It is remarkable that grid convergence was accom-

plished that easily, which is not obvious for convective

systems. This difficulty can be highlighted by consider-ing the Elder [7] problem, where the solute transport is

strongly controlled by convection. For this problem,

grid convergence is practically never achieved because

different qualitative results (i.e. central downwelling–

upwelling–downwelling) are obtained with different

spatial discretizations (i.e. coarse–fine–extremely fine).

Shikaze et al. [24] reached grid convergence by refining

the grid until the resulting concentration plots appearedunchanged (Shikaze, 2004, personal communication).

With the appropriate grid level 5, simulations of

scenario 1 and 2 were run. In order to account for the

longer path in the fracture of scenario 2, the fracture

velocities in this scenario were multiplied at each time

step with a correction factor. The ratio of the lengths

of the two fractures has to coincide with the ratio of

their average flow velocities, represented by the maxi-mum fracture velocity (see Fig. 13.4b). In the present

scenario 1

scenario 2

0

0.2

0.4

0.6

0.8

1

0

conc

entr

atio

n [--

]

Fig. 12. Results of the model verification for scenario 1 and 2: contour plots

observation point.

case of a 45�-inclined fracture, this ratio isffiffiffi2

p. The out-

put of the two simulations is shown in Fig. 12.

The model output from scenario 1 and 2 was objec-

tively compared by means of quantitative indicators

described by Prasad and Simmons [19]. The calculated

indicators are shown in Fig. 13. Using a quantitativeevaluation of model results also accounts for mass fluxes

(indicators 1 and 2) and for mass balance (5), in addition

to the conventional comparison of isochlors and break-

through curves (Fig. 12 left and right, respectively).

Figs. 12 and 13 clearly show that there is good agree-

ment between the two simulation results.

Fig. 13.4b can also be used to verify the Reynolds

number requirement, Re = qfrd/m < 1, for Darcy�s lawin fractures (Eq. (8)). The figure shows that the Darcy

flux in the fracture, qfr, does not exceed 10,000 m yr�1.

If the fracture aperture is chosen as the representative

microscopic length, d = (2b), and with the kinematic vis-

cosity given by m = l/q0, the Reynolds number is

0.01495. Thus, laminar flow as well as a linear relation-

ship between qfri and ohfr0 =oxj is ensured for all simula-

tions presented in Fig. 12 and Section 3.

3. Variable-density flow and transport in a single

inclined fracture

Variable-density flow simulations in a single fracture

have been conducted. For these simulations, the domain

geometry, as well as initial and boundary conditions areidentical to those shown in Fig. 10. All model parame-

ters correspond to those used by Shikaze et al. [24],

shown in Table 1. Similar to the code verification exam-

ples, constant time step sizes of 0.2 yr are used. Simula-

tions are first run for a 45�-inclined fracture, embedded

in a porous matrix. A demonstration of different buoy-

ancy effects in a single fracture of variable incline then

follows.

5 10 15 20

simulation time [yr]

scenario 1

scenario 2

(20% and 60%) at 8 yr simulation time and breakthrough curve at the

53.52 54.92

0

10

20

30

40

50

60

scenario 1 scenario 2tota

l mas

s tr

ansp

orte

d [y

r]

0

1

2

3

4

5

6

7

8

0 10 15 20

tota

lly s

tore

d m

ass

[kg]

0

2000

4000

6000

8000

10000

12000

0 10 15 20

max

. fra

c. v

el. [

m y

r-1]

2factor

(4b) (5)

0

2

4

6

8

10

0 10 15 20

Sher

woo

d nu

mbe

r [--]

0

0.01

0.02

0.03

0.04

0.05

0 10 15 20

simulation time [yr]

simulation time [yr]simulation time [yr]

simulation time [yr]

simulation time [yr]

max

. mat

rix v

el. [

m y

r-1] (4a)

0

2

4

6

8

10

0 10 15 20

pene

trat

ion

dept

h [m

] (3)

(2)(1)

5

5 5

5 5

Fig. 13. Results of the model verification with the quantitative parameters (1)–(5), applied to scenario 1 [—] and 2 [- -].

T. Graf, R. Therrien / Advances in Water Resources 28 (2005) 1351–1367 1363

For the first type of simulation, concentration dis-

tributions as well as velocity fields at 2, 4 and 10 yr

simulation time are shown in Fig. 14. The measured

concentrations versus time of this problem are given as

tabular data in Appendix B. After 2 yr, a convection cell

has formed on the left of the fracture near the source.

Flow along the fracture–matrix interface is mostly awayfrom the fracture. Note that the arrows do not represent

the magnitude of the velocity. Thus, solutes are trans-

ported from the fracture into the adjoining matrix

mainly by molecular diffusion and to a smaller degree

by advection. At 4 yr, a second cell has formed below

the fracture near the solute source. Both cells migrate

downwards into the aquifer. However, the cell above

the fracture has already moved further downwards be-cause it has been formed earlier. This more effective con-

vection cell forces the flow direction very close to the

fracture at the top of the domain to change directions.

As a consequence, the advective transport through the

upper boundary of the domain increases after 4 yr,

resulting in a rise of the Sherwood number as shown

in Fig. 13.1. Both convection cells are separated by the

high-permeability fracture, which, therefore, acts as a

barrier to convection. After 10 yr, some of the flow vec-

tors near the solute source have changed direction, thus

advectively transporting tracer from the matrix into the

fracture, and enhancing the buoyancy-induced flow

within the fracture. Therefore, convection in the porousmatrix appears to control the transport rate in an in-

clined fracture.

Several simulations were run with variable inclines

(u= 0�. . .70�) to investigate the transport behavior at

the limits of the fracture slope (0� and 90�). For all sim-

ulations, the grid had to be locally refined to ensure that

the fracture consists of only inclined faces. A value of

70� was the maximum incline used. The breakthroughcurves within the fracture at z = 6 m were monitored

and are shown in Fig. 15. The figure illustrates that,

for decreasing inclines, the observed buoyancy effect ap-

proaches the solution for a vertical fracture. Analo-

gously, the effect becomes less and less pronounced for

increasing inclines, finally being the solution for a

Fig. 14. Results of variable-density flow simulations after 2, 4 and 10 yr simulation time. Shown are the concentration distributions (left) and the

velocity field. Reddish colors represent a high solute concentration and blueish colors refer to low concentration.

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14

time [yr]

conc

entr

atio

n [--

]

vertical 10 20 30 40 45 50 60 70

45°

70°

vertical

Fig. 15. Results of variable-density flow simulations with a single

variably inclined fracture: breakthrough curves at z = 6 m in the

fracture.

1364 T. Graf, R. Therrien / Advances in Water Resources 28 (2005) 1351–1367

horizontal fracture. Clearly, the two breakthrough

curves for the vertical and (almost) horizontal fracturecases are the envelope functions for the family of curves,

which can be described by c = cu(t). Interestingly, the

difference in concentration between two scenarios at a

given time is large for inclines which exceed 45� (close

to horizontal). Conversely, the concentration difference

is much smaller for almost vertical fractures. In Fig.

15, this phenomenon is indicated at 9 yr by two double

arrows. In both cases, two scenarios are comparedwhere the difference of the fracture incline is 10�. Thisphenomenon can be understood by remembering that

the buoyancy term in the Darcy equation (8) is weighted

with the cosine of the slope. The cosine function changes

weakly for small arguments because its derivative, the

negative sine, almost vanishes. On the contrary, the

change of the cosine function is relatively large for argu-

T. Graf, R. Therrien / Advances in Water Resources 28 (2005) 1351–1367 1365

ments close to 90�. Thus, the weight of the buoyancy

term in the Darcy equation changes weakly for almost

vertical fractures and changes greatly for almost hori-

zontal ones.

4. Summary and conclusions

An existing numerical model was modified to study

variable-density flow and transport in fractured-porous

media with inclined fractures. A general expression of

the body force vector, accounting for density effects,

was derived for fractures of any incline. Verification of

the model against a new experimental 3D test case forvariable-density transport in porous media showed very

good agreement between the experimental and model re-

sults. Variable-density flow in fractures was then tested

using two scenarios with a single 45�-inclined fracture.

In the first scenario, the fracture was discretized by in-

clined 2D elements, while in the second case, the fracture

consisted of horizontal and vertical elements. The frac-

ture velocities in this staircase-fracture were correctedwith the ratio of the two fracture lengths from scenario

2 and 1, respectively, in order to account for the longer

path in fracture 2. In order to rigorously test the code,

not only isochlors were compared, but also the break-

through curve on an observation point within the frac-

ture, the penetration depth of the 60% contour, global

mass fluxes and maximum velocities. Good agreement

between the results from the two scenarios wasobtained.

Simulations of variable-density flow in a porous ma-

trix with a 45�-inclined fracture show that two convec-

tion cells form at different times. Both cells grow with

time and migrate downwards into the aquifer. The

high-permeability fractures appear as barriers to convec-

tion. The velocity field at later simulation times indicates

that the two convection cells cause solutes to be trans-ported by advection from the porous matrix into the

fracture. Thus, convection within the matrix seems to

control density-driven flow in the fracture.

Several simulations in discretely fractured porous

media with a single variably inclined fracture are also

presented. The simulated density effects in an inclined

fracture approach those in fractures at the limits of the

incline. As the fracture becomes steeper, the resultsapproach the ones gained from a vertical fracture.

Correspondingly, results from virtually horizontal and

perfectly horizontal fractures are very similar. This

asymptotical behavior showed that the density effects

are calculated correctly at the fracture incline limits.

The foundation for further numerical modeling of

variable-density transport in more complex fracture net-

works was laid. The model can be used to study the on-set as well as the subsequent development of instabilities

in a fracture network, without being limited to only ver-

tical and horizontal fractures. Furthermore, neglecting

the influence of temperature effects on fluid density,

long-term predictions of contaminant migration in

fractured subsurfaces can be made on a large spatial

scale.

Acknowledgements

We thank the Canadian Water Network (CWN) as

well as the Natural Sciences and Engineering Research

Council of Canada (NSERC) for financial support of

this project. Author T.G. wishes to acknowledge both

the International Council for Canadian Studies (ICCS)and the German Academic Exchange Service (DAAD)

for providing a Postgraduate Scholarship stipend. We

are grateful to M.D. Graf for checking and improving

the English phrasing of the manuscript. The construc-

tive comments of four anonymous reviewers are

greatly appreciated and have helped improve the

manuscript.

Appendix A. Governing equations for the fluid pressure P

In order to reproduce the Elder [7] results given by

Folkovic and De Schepper [10], the FRAC3DVS

model had to be modified such that the same flow vari-

able, the fluid pressure P, can be employed to simulate

variable-density flow and solute transport in porousmedia.

The governing equation for variable-saturated vari-

able-density flow in porous media is given by Voss [33]

and has under fully saturated flow conditions the

three-dimensional form

o

oxibK0

ij

1

goPoxj

þ qgj

� �� �

¼ qSop

oPot

þ /oqoc

ocot

i; j ¼ 1; 2; 3 ðA1Þ

where b = q/qmax [–] and Sop [M�1 L T2] is the specificpressure storativity, given in [33] as

Sop ¼ ð1� /Þam þ /afl ðA2Þ

All other parameters have been defined in the text.

Moreover, the pressure gradient form of Darcy�s law

is

qi ¼ � jij

loPoxj

þ qggj

� �i; j ¼ 1; 2; 3 ðA3Þ

Lacking externally applied solute mass sources,

which is the case in the Elder [7] problem, the governing

equation for solute transport does not need to beadapted to the new flow variable P, and can be taken

as given by Eq. (12).

1366 T. Graf, R. Therrien / Advances in Water Resources 28 (2005) 1351–1367

Appendix B. Results of scenario 1 as tabular data

Table B.1

Simulation results of scenario 1: concentration breakthrough at z = 6 m in the fracture

t [yr] c [–] t [yr] c [–] t [yr] c [–] t [yr] c [–] t [yr] c [–]

0.0 0.00000

0.2 0.00000 4.2 0.30192 8.2 0.66246 12.2 0.76073 16.2 0.803484

0.4 0.00000 4.4 0.33476 8.4 0.67039 12.4 0.76362 16.4 0.805039

0.6 0.00000 4.6 0.36559 8.6 0.67783 12.6 0.76640 16.6 0.806557

0.8 0.00000 4.8 0.39437 8.8 0.68482 12.8 0.76908 16.8 0.808038

1 0.00000 5 0.42114 9 0.69141 13 0.77166 17 0.809485

1.2 0.00000 5.2 0.44599 9.2 0.69762 13.2 0.77415 17.2 0.810899

1.4 0.00002 5.4 0.46903 9.4 0.70349 13.4 0.77656 17.4 0.812281

1.6 0.00014 5.6 0.49037 9.6 0.70903 13.6 0.77888 17.6 0.813633

1.8 0.00071 5.8 0.51015 9.8 0.71428 13.8 0.78113 17.8 0.814955

2 0.00264 6 0.52847 10 0.71925 14 0.78331 18 0.816250

2.2 0.00749 6.2 0.54547 10.2 0.72398 14.2 0.78542 18.2 0.817518

2.4 0.01724 6.4 0.56124 10.4 0.72847 14.4 0.78747 18.4 0.818759

2.6 0.03342 6.6 0.57590 10.6 0.73274 14.6 0.78946 18.6 0.819976

2.8 0.05652 6.8 0.58953 10.8 0.73681 14.8 0.79139 18.8 0.821169

3 0.08578 7 0.60222 11 0.74069 15 0.79326 19 0.822339

3.2 0.11961 7.2 0.61404 11.2 0.74440 15.2 0.79508 19.2 0.823487

3.4 0.15615 7.4 0.62507 11.4 0.74795 15.4 0.79685 19.4 0.824614

3.6 0.19362 7.6 0.63537 11.6 0.75135 15.6 0.79858 19.6 0.825719

3.8 0.23092 7.8 0.64500 11.8 0.75461 15.8 0.80025 19.8 0.826805

4 0.26720 8 0.65402 12 0.75773 16 0.80189 20 0.827872

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