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A test case for the simulation of three-dimensional variable-density flow and solute transport in discretely-fractured porous media

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A test case for the simulation of three-dimensional variable-density ﬂowand solute transport in discretely-fractured porous media
Thomas Graf
a,
*
, René Therrien
b
a
Center of Geosciences, Georg-August-University Göttingen, Goldschmidtstraße 3, 37077 Göttingen, Germany
b
Département de Géologie et Génie Géologique, Université Laval. Ste-Foy, Québec G1K 7P4, Canada
a r t i c l e i n f o
Article history:
Received 16 November 2007Received in revised form 16 June 2008Accepted 1 July 2008Available online 11 July 2008
Keywords:
Test case3DDensityFractureNumerical model
a b s t r a c t
A test case has been developed for three-dimensional simulations of variable-density ﬂow and solutetransport in discretely-fractured porous media. The simulation domain is a low-permeability porousmatrix cube containing a single non-planar fracture. The initial solute concentration is zero everywhere.A constant solute concentration is assigned to the top of the domain, which increases near-top ﬂuid den-sity andinduces downward density-drivenﬂow. Thetest caseis therefore comparable todownwelling of a dense brine below a saline disposal basin or a waste repository. Numerous ﬁngers and distinct convec-tion cells develop early in the fracture but the ﬁngers later coalesce and convection becomes less appar-ent. To help test other variable-density ﬂow and transport models, results of the test case are presentedboth qualitatively (concentration contours and velocity ﬁelds) and quantitatively (penetration depth,mass ﬂux, total mass stored, maximum fracture and matrix velocity).
2008 Elsevier Ltd. All rights reserved.
1. Introduction
Groundwater density varies as a function of ﬂuid temperature,solute concentration and, to a lesser extent, ﬂuid pressure. Spatialvariations of ﬂuid density play an important role in contaminantmigration. When, for example, a high-density ﬂuid overlies a lessdense ﬂuid, unstable density-driven ﬂow may occur, which damp-ens or eliminates the density stratiﬁcation and eventually stabi-lizes ﬂow. Transient ﬂow can also induce temporal changes inﬂuid density [21]. Examples of density-driven ﬂow and solute
transport can be found in many areas of subsurface hydrology,oceanography,meteorology,geophysics,hazardouswastedisposal,and geothermal reservoirs [10,15,23,28].Numerical models used to simulate variable-density ﬂow andsolutetransportingroundwatermustbetestedtoensurethattheyrepresent the required physical processes and are numerically rig-orous. Model testing relies in part on
test cases
, which are well-de-ﬁned analytical, numerical, laboratory or ﬁeld results [35,47].A test case possesses a physical and mathematical character,such as nonlinearity, but has a simpliﬁedgeometry such that com-parable numerical solutions of accepted quality are available [1,7].Theevaluationofnewnumerical groundwatermodelstypicallyre-liesoninternalconsistencytestsincludingmassbalanceindicatorsand external tests such as comparison with other numerical mod-els. Clearly, the more test cases available for model testing, themore rigorous the testing process.Several test cases exist for variable-density ﬂow in homoge-neous porous media [5,8,20,35,36,39,42,47]. In contrast, few test
cases exist for variable-density ﬂow in discretely-fractured porousmedia. Results presented by Shikaze et al. [41] can be used to testdense plume migration in a set of vertical fractures by comparingisohalinesfor givensimulationtimes. Variable-densityﬂowresultsin orthogonal fracture networks presented by Graf and Therrien[17] are equally useful. The inclined-fracture problem introducedby Graf and Therrien [15] can be used in both a qualitative (usingisohalines) and a quantitative (using detailed information onbreakthrough curves, mass ﬂuxes, maximum velocities, etc.) man-ner to rigorously benchmark variable-density ﬂow in an inclinedfracture embedded in a porous matrix. Caltagirone [5] has pre-sented an analytical solution for the onset of convection in homo-geneousmedia.Thesolutioncanbeappliedtoverticalandinclinedfractures by assuming a homogeneous fracture of constant aper-ture and by introducing a cosine weight to account for fracture in-cline [14]. Although the solution of Caltagirone [5] is the only
available analytical solution for variable-density ﬂow applied to abox type domain of ﬁnite dimensions, it is not truly an analyticalsolution,butratherusesanalyticalprocedurestoderivethecriticalRayleigh number for different aspect ratios in order to determineconditions that govern stable and unstable states of convectionfor box type domains. The solution presented by Caltagirone [5]also forms the basis for newly proposed benchmark problems pre-sented by Weatherill et al. [47].
0309-1708/$ - see front matter
2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.advwatres.2008.07.003
*
Corresponding author.
E-mail addresses:
tgraf@gwdg.de(T.Graf),rene.therrien@ggl.ulaval.ca (R.Therrien).
Advances in Water Resources 31 (2008) 1352–1363
Contents lists available at ScienceDirect
Advances in Water Resources
journal homepage: www.elsevier.com/locate/advwatres
Existingtestcasesforvariable-densityﬂowinfracturedrockare2D test problems [5,15,17,41,47]. Test cases using results from
either physical or laboratory experiments for variable-density ﬂowin fractured rock do not exist. Solute transport through fracturedrock has previously been investigated in two laboratory studies,where fractures were embedded either in an impermeable [31]or in a porous [22] matrix. The physical model used in the twostudies was built from a horizontally oriented sheet of plexiglas,into which a series of fractures was cut. A tracer solution contain-ing dye and salt was injected and the tracer was transported by animposed background ﬂow. Conducting a similar laboratory exper-iment in 2D(and possibly3D) fracturedmedia where density vari-ations are signiﬁcant remains a major challenge.To date, a robust and well accepted (numerical) test case forvariable-density ﬂow in fractures oriented in 3D and embeddedin a 3D porous matrix does not exist. We therefore propose herea new test problem for variable-density ﬂow in a non-planar frac-ture embedded in a 3D porous matrix. The test case is developedwith the HydroGeoSphere model [15,44,45], which solves 3D var-
iable-density ﬂowandsolutetransport indiscretely-fractured por-ousmedia.Withthemodel, variable-densityﬂowis describedforarock matrix containing a single non-planar fracture [16,19]. The
fracture is represented by nine planar facets and discretized withtriangular fracture elements. The transient test case describesadvection, mechanical dispersion, molecular diffusion in both thefracture and the porous matrix, as well as fracture–matrix diffu-sion.Weprovidedetaileddataonmassﬂuxes,solutebreakthroughcurves,concentrationcontoursandsolutepenetrationdepthtoen-surerigorousobjectivetestingofanewmodel.Wealsogivedetailson the numerical methods used to ensure further rigorousness of thenewtestcase. Thenewtestproblemallowsnumerical compar-ison for other variable-density ﬂow models.
2. Numerical model
2.1. The HydroGeoSphere model
HydroGeoSphere is a numerical 3D variable-density, variably-saturatedgroundwaterﬂowandmulti-componentsolutetransportmodel for fractured porous media, and is based on the FRAC3DVSmodel. The HydroGeoSphere model applies the control volume ﬁ-nite element (CVFE) method to the ﬂow equation, the Galerkin ﬁ-nite element method with full upstream weighting to the solutetransport equation, and a ﬁnite difference scheme to time deriva-tives [15,44]. It is assumed that 2D fracture elements and 3D ma-
trix elements share common nodes in the 3D grid. Thus,hydraulic heads and concentrations are assumed identical alongthe fracture–matrix interface.The HydroGeoSphere model applies the ﬁrst level of the Ober-beck–Boussinesq (OB) approximation to discretize the ﬂow andtransport equations [4,21,27,34]. The OB assumption reﬂects the
degree to which density variations are accounted for. Level 1 of the OB approach considers density effects only in the buoyancyterm of the momentum equations (Darcy equations) and neglectsdensity in the ﬂuid and solute mass conservation equations. Thisassumptioniscorrectif spatialdensityvariationsaresmallrelative
Nomenclature
Latin letters
(2
b
) [L] fracture aperture
A
[L
2
] surface area
b
[L
2
T
1
] boundary ﬂux vector
c
[–] relative solute concentration
d
[–] microscopic length
D
d
[L
2
T
1
] free-solution diffusion coefﬁcient
D
ij
[L
2
T
1
] hydrodynamic dispersion
f
[L
3
T
1
] groundwater ﬂux vector
g
[L T
2
] acceleration due to gravity
g
r
,
h
[–] grid at discretization levels
r
and
h
g
[L
2
T
1
] buoyancy vector
h
0
[L] equivalent freshwater head
H
[L T
1
] conductance or stiffness matrixI
+
, I
[–] fracture–matrix interface
K
0
ij
[L T
1
] freshwater hydraulic conductivity of porous matrix
K
fr0
[L T
1
] freshwater hydraulic conductivity of fracture
L
v
[L] geometry of porous matrix element
v
=
x
,
y
,
z Pe
g
[–] grid Peclet number for triangular fracture element
q
[L T
1
] Darcy ﬂux
Q
[T
1
] groundwater ﬂux
Re
[–] Reynolds number
S
[L] ﬂuid mass matrix
S
S
[L
1
] speciﬁc storage
t
[T] time
v
[L T
1
] linear ﬂow velocity
V
[L
3
volume
w
[–] approximation function
x
[L] space
Greek letters
a
[M
1
L T
2
] matrix compressibility
a
l
[L] longitudinal dispersivity
a
t
[L] transverse dispersivity
b
[M
1
L T
2
] water compressibility
c
[–] maximum relative density
d
ij
[–] Kronecker delta function
g
j
[–] indicator for ﬂow direction
h
[–] temporal discretization level
j
ij
[L
2
] permeability of porous matrix
l
[M L
1
T
1
] water viscosity
q
[M L
3
] water density
q
max
[M L
3
] maximum water density
r
[–] spatial discretization level
s
[–] factor of tortuosity
/
[–] matrix porosity
u
[1
] fracture incline
X
[M M
1
T
1
] advective–dispersive–diffusive solute ﬂux
Sub- and superscripts
0 [–] reference ﬂuide [–] porous matrix elementfe [–] fracture elementfr [–] fracture
i
,
j
[–] spatial indicesI [–] nodal indexn [–] normal direction
Special symbols
o
[–] partial differential operator
D
[–] difference
T. Graf, R. Therrien/Advances in Water Resources 31 (2008) 1352–1363
1353
to density [21,33]. Although the OB assumption ‘‘has not been
completely justiﬁed” [26], it has been adapted in a number of var-
iable-density ﬂow studies (e.g. [9,15,27,41]). Referring to the OB
assumption,level 1, Joseph[26] alsostates‘‘that thereisnospecialreason besides our lack of proofs to doubt the validity of the non-linear OB-equations”.Density variations cause weak nonlinearities in the ﬂow equa-tion. Thus, groundwater ﬂow and solute transport are physicallycoupled. In the numerical model, the coupled system of equationsis solved by Picard iteration. The spatiotemporally discretized ma-trixequationsaresolvedusingtheWATSITiterativesolverpackagefor general sparse matrices [6] and a conjugate gradient stabilized(CGSTAB) acceleration technique [38]. Governing equations of groundwater ﬂow and solute transport are presented below.
2.2. Governing equations 2.2.1. Equations for the 3D porous matrix
The following equations describe variable-density ﬂow and sol-ute transport in 3D porous media [3,21]:
o
q
i
o
x
i
¼
S
S
o
h
0
o
t i
¼
1
;
2
;
3
ð
1
Þ
oo
x
i
/
D
ij
o
c
o
x
j
q
i
c
¼
o
ð
/
c
Þ
o
t i
;
j
¼
1
;
2
;
3
ð
2
Þ
where
q
i
[L T
1
] Darcy ﬂux
S
S
[L
1
] speciﬁc storage
h
0
[L] equivalent freshwater head
/
[–] matrix porosity
D
ij
[L
2
T
1
] hydrodynamic dispersion
c
[–] solute concentration, expressed as relative concentration
The speciﬁc storage
S
S
is deﬁned as [12]:
S
S
¼
q
0
g
ð
a
þ
/b
Þ ð
3
Þ
where
q
0
[M L
3
] freshwater density
g
[L T
2
] gravitational acceleration
a
[M
1
L T
2
] matrix compressibility
b
[M
1
L T
2
] ﬂuid compressibility
The coefﬁcients of the hydrodynamic dispersion tensor are gi-ven by [3]:
/
D
ij
¼ ð
a
l
a
t
Þ
q
i
q
j
j
q
j þ
a
t
j
q
j
d
ij
þ
/
s
D
d
d
ij
i
;
j
¼
1
;
2
;
3
ð
4
Þ
where
a
l
[L] longitudinal dispersivity
a
t
[L] transverse dispersivity
d
ij
[–] Kronecker delta function
s
[–] matrix tortuosity
D
d
[L
2
T
1
] free-solution diffusion coefﬁcient
2.2.2. Equations for the 2D fracture
The variable-density ﬂowand solutetransport equations for 2Ddiscrete fractures are [21,41,44]:
ð
2
b
Þ
o
q
fr
i
o
x
i
S
frS
o
h
fr0
o
t
( )
þ
Q
n
j
I
þ
Q
n
j
I
¼
0
i
;
j
¼
1
;
2
ð
5
Þð
2
b
Þ
oo
x
i
D
fr
ij
o
c
fr
o
x
j
q
fr
i
c
fr
o
c
fr
o
t
þ
X
n
j
I
þ
X
n
j
I
¼
0
i
;
j
¼
1
;
2
ð
6
Þ
where (2
b
) [L] is fracture aperture, and the last two terms in eachequation denote components of ﬂuid ﬂux and solute mass ﬂux nor-mal to fracture–matrix interfaces I
+
and
I
. Speciﬁc storage in thefracture can be derived from Eq. (3) by assuming that the fractureis incompressible, such that
a
=0, and by setting its porosity to 1[15]:
S
frS
¼
q
0
g
b
ð
7
Þ
Coefﬁcients of the hydrodynamic dispersion tensor of the frac-ture are deﬁned by [44]:
D
fr
ij
¼ ð
a
frl
a
frt
Þ
q
fr
i
q
fr
j
j
q
fr
j þ
a
frt
j
q
fr
j
d
ij
þ
D
d
d
ij
i
;
j
¼
1
;
2
ð
8
Þ
2.2.3. Darcy ﬂux
The variable-density Darcy ﬂux,
q
i
, is a function of freshwaterhead,
h
0
, and solute concentration,
c
. Darcy ﬂuxes for matrix andfracture are given by [3]:
q
i
¼
K
0
ij
o
h
0
o
x
j
þ
c
c
g
j
i
;
j
¼
1
;
2
;
3
ð
9
Þ
q
fr
i
¼
K
fr0
o
h
fr0
o
x
j
þ
c
c
fr
g
j
cos
u
!
i
;
j
¼
1
;
2
ð
10
Þ
where
K
0
ij
[LT
1
] freshwater hydraulic conductivity of rock matrix
K
fr0
[LT
1
] freshwater hydraulic conductivity of fracture
g
j
[–] indicator for ﬂow direction:
g
j
= 0 in horizontal directions;
g
j
= 1 otherwise
c
[–] maximum relative density,
c
= (
q
max
/
q
0
)
1
q
max
[M L
3
] maximum ﬂuid density
u
[
] fracture incline:
u
= 0
for a vertical fracture;
u
= 90
for ahorizontal fracture
and where the assumption is made that solute concentrations of aﬂuid with densities
q
=
q
max
and
q
=
q
0
are
c
=1 and
c
=0, respec-tively. It is also assumed that ﬂuid viscosity is constant [15,41].Freshwater hydraulic conductivities are given by [3]:
K
0
ij
¼
j
ij
q
0
g
l
0
i
;
j
¼
1
;
2
;
3
ð
11
Þ
K
fr0
¼ ð
2
b
Þ
2
q
0
g
12
l
0
ð
12
Þ
where
j
ij
[L
2
] permeability of the rock matrix
l
0
[M L
1
T
1
] freshwater viscosity
2.3. Numerical formulation of buoyancy
In the present study, variable-density ﬂow is simulated in a 3Dporous rock matrix (represented by hexahedral 3D elements) con-taininganon-planar2Dfracture(representedbytriangular2Dele-ments). Variable-density ﬂow in a 3D rock matrix and inrectangular 2D fracture elements has previously been simulated[15]. However,rectangular2Delementsareinadequatetodescribea non-planar inclined fracture in 3D [19]. Therefore, the HydroGe-
oSphere model has been modiﬁed here to incorporate triangularfracture elements (Fig. 1) in the study of variable-density ﬂowand transport in 3D fractured rock. The enhanced model is thenused to deﬁne the new 3D test case.Triangular elements are generated in the model such thattheir nodes must correspond to nodes of 3D hexahedral ele-ments. With that constraint, a total of 56 different triangularelements, of different incline, can be generated for the 8-nodehexahedral element shown in Fig. 1a. For example, the triangu-lar element shown in Fig. 1a, whose nodes are 2–7–5, is oneof 56 possible triangular elements contained in a hexahedralelement.
1354
T. Graf, R. Therrien/Advances in Water Resources 31 (2008) 1352–1363
ApplicationoftheCVFEmethodtovariable-densityﬂowEqs.(1)and (5) gives semi-discrete global matrix-equations for the porousmedium and fractures, respectively:
H
h
0
þ
S
o
h
0
o
t
þ
g
¼
b
ð
13
Þ
H
fr
h
fr
0
þ
S
fr
o
h
fr
0
o
t
þ
g
fr
¼
b
fr
ð
14
Þ
where
H
,
H
fr
[L T
1
] conductance or stiffness matrices
S
,
S
fr
[L] ﬂuid mass matrices
g
,
g
fr
[L
2
T
1
] buoyancy vectors
b
,
b
fr
[L
2
T
1
] boundary ﬂux vectors
Vector
g
and
g
fr
can be written as
g
¼
X
e
g
e
ð
15
Þ
g
fr
¼
X
fe
g
fe
ð
16
Þ
where
g
e
[L
2
T
1
] buoyancy vector of matrix element e
g
fe
[L
2
T
1
] buoyancy vector of fracture element fe
2.3.1. Numerical formulation of buoyancy in 3D hexahedral matrixelements
According to Frind [12], nodal entries of
g
e
are calculated as
g
eI
¼
Z
V
e
K
0
ij
c
c
e
o
w
eI
o
z
d
V
e
I
¼
1
;
2
;
. . .
;
8
ð
17
Þ
where
V
e
[L
3
] volume of matrix element e
c
e
[–] average solute concentration in e
w
eI
[–] value of the 3D approximation function in e at node I
Using usual approximation functions for 3D hexahedral ele-ments [24] gives the following buoyancy vector:
g
e
¼
K
0
ij
c
c
e
L
x
L
y
4
f
1
1
1
1 1 1 1 1
g
T
ð
18
Þ
where
L
x
and
L
y
[L] are sizes of element e in
x
- and
y
-direction,respectively.
2.3.2. Numerical formulation of buoyancy in 2D triangular fractureelements
According to Frind [12] and Graf and Therrien [15], nodal en-
tries of
g
fe
are calculated as
g
feI
¼
Z
A
fe
K
fr0
c
c
fe
cos
u
o
w
feI
o
z
d
A
fe
I
¼
1
;
2
;
3
ð
19
Þ
where
A
fe
[L
2
] surface area of fracture element fe
c
fe
[–] average solute concentration in fe
w
feI
[–] value of the 2D approximation function in fe at node I
z
[L] local
z
-axis of fe
Usingusualapproximationfunctionsfor2Dtriangularelements[24], we have derived the following buoyancy vector:
g
fe
¼
K
fr0
c
c
fe
cos
u
12
x
3
x
2
x
1
x
3
x
2
x
1
8><>:9>=>;
ð
20
Þ
where
x
I
[L] is the value of the local
x
-coordinate of node I.
2.4. Veriﬁcation of numerical buoyancy formulation
Wesimulatedvariable-densityﬂowinvariablyinclinedfracturesconsistingoftriangularfractureelementsandcomparedtheresultswiththosepresentedbyGrafandTherrien[15] forrectangular ele-ments.Theporousmatrixisdiscretizedbyhexahedral3Delements.Themodeldomainisaverticalsliceofunitthicknessanddimensions12 m
10 m. Model domain, initial and boundary conditions andﬂow/transporttimeweightingschemesusedinthesimulationscon-ductedhereareidenticaltothoseusedbyGrafandTherrien[15].Fig.
2 shows results from three different inclines and demonstratesagreementofresultswhendiscretizinganinclinedfracturebytrian-gular(thisstudy)andrectangular[15] fractureelements.
3. The 3D test case
3.1. Problem design
The model domain for the test case is a cubic box of a sidelengthequal to 10 m. The box represents a porous matrix and con-tains a single non-planar fracture. The fracture is represented bynine planar facets as shown in Fig. 3. Fracture geometry and facetvertex locations are given in Tables A.1 and A.2.Lateral boundaries are assumed to be impermeable and the topand bottom boundaries are assigned a constant hydraulic head
h
0
=0. A constant concentration
c
=1 is assigned to the top bound-ary, and all other boundaries are assigned a zero-dispersive ﬂuxboundary condition [41]. The initial condition for ﬂow is
h
0
=0,andfortransport
c
=1onthedomaintopand
c
=0everywhereelse[14,17,41].Thetotalsimulationtimeis10years.Fullyimplicittime
a b
Fig. 1.
Geometry and node numbering conventions for (a) a 3D porous matrix element containing eight nodes (1
8) and (b) a 2D fracture element containing three nodes(
1
3). The local
x
-axis (
x
) is horizontal, points to the right and fracture nodes are numbered counterclockwise.
T. Graf, R. Therrien/Advances in Water Resources 31 (2008) 1352–1363
1355
weighting is used for both ﬂow and transport [13,15]. Model
parameters for the test case are summarized in Table A.3.ThePicarditerationisterminatedwhenoneoftwoconvergencecriteria is met. The two convergence criteria are the maximumnumber of iterations, set to 100, and the maximumchange in con-centration and hydraulic head between iterations, set to 0.05. InSection 3.4 we examine the sensitivity of the results to Picard con-vergence criteria.
3.2. Spatial discretization
As pointed out by [25], investigating grid convergence (pre-
sented in Section 3.4) requires a hierarchy of uniformly reﬁnedgrids. Therefore, we did not use locally reﬁned grids.Here, the 3D grid at spatial discretization level
r
(
r
= 1,2,
. . .
,11) consists of (10
r
+1)
3
grid nodes and (10
r
)
3
iden-tical hexahedral porous matrix elements, which have a uniformsize
D
x
=
D
y
=
D
z
=1/
r
m. Increasing grid levels correspond toﬁner grids. The locations of fracture facet vertexes (Table A.2) havebeenchosensuchthat thelocationof vertexesandsegmentsofthenine fracture facets is preserved for all grid levels. For each level
r
,triangularfractureelementsthatrepresentthenon-planarfracturehave been selected prior to conducting the simulation. Therefore,changing the grid does not modify the location, shape and orienta-tion of the non-planar fracture.
3.3. Time discretization
The 3D test case is simulated for increasing levels of time dis-cretization
h
(
h
=1,2, 3).Toguaranteenumericalstability,thesolu-tion process is completed using a variable time-step proceduresimilar to the one outlined by [11,44,48]. After obtaining the solu-
tionattimelevel
L
,thefollowingtime-stepsizeiscomputedusing:
ð
D
t
Þ
L
þ
1
¼
c
ð
h
Þ
max
I
j
c
L
þ
1I
c
L
I
jð
D
t
Þ
L
ð
21
Þ
where (
D
t
) [T] is time-step size and
c
*
(
h
) [–] is the maximumchange in relative solute concentration during a single time-step.To ensure further numerical stability, the maximum multiplier of time-step sizes is 2.0, and the maximum time-step size is 1 month.The valueof
c
*
(
h
) is a functionof timediscretization level andgivenby:
c
ð
h
Þ ¼
0
:
2
12
ð
h
1
Þ
ð
22
Þ
Increasing time discretization levels give smaller values of
c
*
(
h
)and smaller time-step sizes. Variable time-stepping allows the useof increasingly larger time-step sizes when there are only smallchanges in concentration. Thus, CPU time is reduced withoutdecreasing numerical accuracy. Similarly, drastic concentrationchanges increase time step sizes. It can be shown that the numer-ical stability of a simulation using adaptive time-stepping isidentical to the stability of a simulation using very small constanttime-step sizes. The Courant-criterion is satisﬁed in simulationswith adaptive time-stepping if the initial time-step size chosen isvery small.
3.4. Numerical accuracy
A grid convergence study is carried out to determine the neces-sary discretization level in space and time. The study involves per-forming a simulation on successively ﬁner grids and smaller timestep sizes. As discretization levels
r
and
h
increase, the discretiza-tion errors should exclude computer round-off errors and asymp-totically approach zero. In addition to the grid convergencestudy, the Picard convergence criterion is examined to guaranteefurther numerical accuracy.We deﬁne
g
r
,
h
here as the grid of spatial discretization level
r
and time discretization level
h
. Simulations were carried out atincreasing grid levels, and time-variable solute mass ﬂux
f
([L
3
T
1
] when using dimensionless relative concentration)through the domain top was observed. In the grid convergencestudy,wechosemassﬂuxesastherepresentativequantitybecausemass ﬂux is an integral parameter that does not yield informationat a speciﬁc point in the grid. We denote the mass ﬂuxes using
g
r
,
h
at 1 yr,
. . .
,10 yr by
f
1
r
;
h
,
. . .
,
f
10
r
;
h
and deﬁne:
f
r
;
h
:
¼ f
f
1
r
;
h
;
. . .
;
f
10
r
;
h
g
T
2
R
10
ð
23
Þ
We use the
L
2
-norm to estimate the error
e
r
+1/2,
h
[L
3
T
1
] betweensolutions corresponding to increasing spatial discretization levels:
e
r
þ
1
=
2
;
h
¼ k
f
r
þ
1
;
h
f
r
;
h
k
N
ð
24
Þ
Likewise, theerrore
r
,
h
+1/2
[L
3
T
1
]betweensolutionscorrespondingto increasing time discretization levels is
00.20.40.60.810 5 10 15 20
time [yr]
s o l u t e c o n c e n t r a t i o n [ - ]
45°60°30°
trianglesrectangles
Fig. 2.
Simulated breakthrough curve at
z
=6 mto verify variable-density ﬂow in asingle fracture of variable incline using triangular (this study) and rectangular [15]fracture elements.
x - d i r e c t i o n [ m ]
0246810
y - d i r ec t io n [ m ]
0 2 4 6 8 10
z - d i r e c t i o n [ m ]
0246810
5
1
4 987 6 3 21
23456789
10
Fig. 3.
Geometry of the triangulated non-planar fracture for the test case. Circlednumbers are facet IDs and numbers in italic are facet vertex IDs.1356
T. Graf, R. Therrien/Advances in Water Resources 31 (2008) 1352–1363

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