Simulating Conservative Tracers in Fractured
Till under Realistic Timescales
by M.F. Helmke1, W.W. Simpkins2, and R. Horton3
Abstract
Discrete-fracture and dual-porosity models are infrequently used to simulate solute transport through fractured unconsolidated deposits, despite their more common application in fractured rock where distinct flow regimes are hypothesized. In this study, we apply four fracture transport models—the mobile-immobile model
(MIM), parallel-plate discrete-fracture model (PDFM), and stochastic and deterministic discrete-fracture models
(DFMs)—to demonstrate their utility for simulating solute transport through fractured till. Model results were
compared to breakthrough curves (BTCs) for the conservative tracers potassium bromide (KBr), pentafluorobenzoic acid (PFBA), and 1,4-piperazinediethanesulfonic acid (PIPES) in a large-diameter column of fractured till.
Input parameters were determined from independent field and laboratory methods. Predictions of Br BTCs were
not significantly different among models; however, the stochastic and deterministic DFMs were more accurate
than the MIM or PDFM when predicting PFBA and PIPES BTCs. DFMs may be more applicable than the MIM
for tracers with small effective diffusion coefficients (De) or for short timescales due to differences in how these
models simulate diffusion or incorporate heterogeneities by their fracture networks. At large scales of investigation, the more computationally efficient MIM and PDFM may be more practical to implement than the threedimensional DFMs, or a combination of model approaches could be employed. Regardless of the modeling
approach used, fractures should be incorporated routinely into solute transport models in glaciated terrain.
Introduction
Hydrogeologists have long considered fractures in
till to act as preferential flowpaths and facilitate rapid
transport of contaminants. The bulk hydraulic conductivity (Kb) of fractured till is commonly 1 to 3 orders of
magnitude greater than the hydraulic conductivity of the
till matrix (Km) (Freeze and Cherry 1979; Helmke 2003).
Fracture porosity (nf) is frequently 1 to 4 orders of magnitude less than the total porosity (nT) of the till (Jørgensen
1Corresponding
author: Department of Geology and
Astronomy, West Chester University, West Chester, PA 19383;
(610) 436-3565; Fax (610) 436-3036; mhelmke@wcupa.edu
2Department of Geological and Atmospheric Sciences, Iowa
State University, Ames, IA 50011; (515) 294-7814; fax (515) 2946049; bsimp@iastate.edu
3Agronomy Department, Iowa State University, Ames, IA
50011; (515) 294-7843; fax (515) 294-3163; rhorton@iastate.edu
Received June 2003, accepted February 2005.
Copyright ª 2005 National Ground Water Association.
doi: 10.1111/j.1745-6584.2005.00129.x
and Spliid 1992; McKay et al. 1993a). The combined
effect of increased Kb and decreased effective porosity
(ne, nf in a fractured medium) may result in calculated
fluid velocities up to 200 m/d under a unit hydraulic gradient (Jørgensen and Spliid 1992; McKay et al. 1993a).
Because of this, the potential for fractures to rapidly
transmit contaminants through till is well documented
(Grisak and Pickens 1980; Jørgensen and Spliid 1992;
McKay et al. 1993b).
Fractures are also well documented in till throughout
the world and constitute a ubiquitous feature of these deposits. In the United States, till fractures have been documented in Iowa (Kemmis et al. 1992; Helmke 2003),
Wisconsin (Connell 1984; Simpkins and Bradbury 1992),
and Ohio (Brockman and Szabo 2000). Fractures in till
have also been reported throughout Canada (Keller et al.
1988; McKay et al. 1993a) and Denmark (Klint and
Gravensen 1999).
Despite strong evidence that fractures control solute
transport in till, relatively few ground water studies in
this material include fractures in models (e.g., Grisak and
Vol. 43, No. 6—GROUND WATER—November–December 2005 (pages 877–889)
877
Pickens 1980; McKay et al. 1993b; Jørgensen et al. 1998,
2003). Numerous studies have employed fracture models
to simulate solute transport through fractured rock (e.g.,
Bear et al. 1993), and these models are readily available
and well documented. We suspect, therefore, that it is the
perceived complexity of obtaining input parameters and
not a lack of model availability that deters their application to fractured till. The objective of this study is to demonstrate the use of two major classes of models—the
mobile-immobile model (MIM) and discrete-fracture
model (DFM)—to illustrate how these models may be
applied to fractured till. The input requirements, computational efficiency, and relative merits and weaknesses
of these models are discussed, and the accuracy of model
simulations are tested by comparing model output with
laboratory-derived breakthrough curves (BTCs) from
a large-diameter column of fractured till. By demonstrating the application of these models, we hope to inspire
future researchers to apply these methods to fractured unlithified materials at the field scale.
Model Descriptions
Models that simulate solute transport through fractured media differ from porous media models because
water flow through a fracture is typically orders of magnitude faster than within the matrix and solute storage is
greater in the matrix than in the fracture. This dichotomy
between rapid advection and efficient storage causes fractured systems to react sensitively to changes in flow rate
and input concentration. Most fracture transport models
simulate advection in the fractures and diffusive
exchange between the fractures and the matrix. These
models differ primarily by how they represent the geometry of the fracture system and solve the problem mathematically. Some models represent fracture networks
using a simplified geometry of orthogonal or parallel
plates (e.g., FRACTRAN, Sudicky and McLaren 1998).
Other models specify fracture geometry explicitly by
three-dimensional (3D) sets of fractures (e.g., FracMan/
MAFIC, Dershowitz et al. 1994 or Frac3DVS, Therrien
et al. 2000), while still others disregard fracture geometry
entirely (e.g., CXTFIT, Toride et al. 1999). Once the
fracture geometry and boundary conditions are specified,
the models are used to calculate solute concentration in
the fractures and matrix in space and time.
The selection of which model to apply to a particular
application depends on the information available and the
type of information desired from the model. A simple
model such as the MIM may be easy to construct and
efficient but may not provide useful information in cases
where fracture spacing is large with respect to the scale
of investigation. On the other hand, an extremely complex model including a 3D network of thousands of fractures may produce similar results as a simpler model and
may be computationally prohibitive at large scales. A
third alternative is to use a combination of models by
placing a few large fractures explicitly into a model,
simulating smaller sets of fractures using the MIM, and
solving both systems simultaneously.
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M.F. Helmke et al. GROUND WATER 43, no. 6: 877–889
Values of model input parameters were determined
or estimated a priori in this study using methods independent of the model simulations. This approach was
selected to test the ability of the models to be used as predictive tools. An alternate approach would be to fit the
models to the BTCs by adjusting parameters until the
best fit was achieved. Unfortunately, previous studies
have demonstrated that this approach can lead to nonunique combinations of input values (Parker and van
Genuchten 1984). Therefore, model simulations were
conducted in the forward mode during this study.
The laboratory experiment was conducted in one
dimension under steady-state flow conditions to minimize
the number of unknown variables. Moreover, only conservative (nonreacting and nondegrading) tracers were used.
However, all the models discussed in this study may be
expanded to the third dimension and can be modified
to include sorption, degradation, and production under
transient-flow conditions.
Mobile-Immobile Model
The MIM simulates a dual-porosity medium as two
regions: one in which fluid is moving and the other where
fluid is stagnant. When applied to a fractured medium,
the MIM represents fractures as the mobile region and
the matrix as the immobile region. Advection and dispersion occur exclusively in the mobile region, and the
immobile region is a sink that stores the solute. The MIM
simulates exchange between the mobile and the immobile
regions (matrix diffusion) as a first-order process (Coats
and Smith 1964). Early versions of the MIM included
only advection and dispersion in the mobile region and
first-order exchange between the mobile and the immobile regions. The MIM was later expanded to include
sorption (van Genuchten and Wagenet 1989) and firstorder degradation and production (Toride et al. 1993),
although only conservative tracers will be considered in
this study.
The MIM has been used widely by soil physicists to
model solute transport through soil containing macropores. The model is ideal because the great density of
macropores within the top meter of soil precludes explicit
knowledge of pore geometry. An added benefit is the
model’s computational efficiency, which allows it to be
used in the inverse mode to predict input parameters from
experimental data (Parker and van Genuchten 1984).
The MIM includes two governing equations, one for
the fracture (mobile region, Equation 1) and one for the
matrix (immobile region, Equation 2):
nf
@cm
@ 2 cf
@cf
¼ nf D f 2 2 q
2 aðcf 2 cmat Þ
@t
@z
@z
ð1Þ
@cmat
¼ aðcf 2 cmat Þ
@t
ð2Þ
nmat
where cf and cmat are solute concentrations in the mobile
and immobile regions, nf is the fracture porosity, nmat is
the porosity within the matrix available to diffusion, t is
the time, Df is the fracture dispersion coefficient, z is the
location along the flowpath, q is the Darcy flux, and a is
the first-order mass-transfer coefficient between the
fracture and the matrix (Coats and Smith 1964). Estimates of a may be obtained using the relation
a¼
aDe nmat
l2
ð3Þ
where a is a shape factor, De is the effective diffusion
coefficient, and l is a characteristic length (Parker and
Valocchi 1986). For a system of equally spaced, parallel
fractures separated by prismatic slabs, a may be set to 3
and l is one-half the fracture spacing (Sudicky 1990).
Semianalytical solutions to Equations 1 and 2 were
developed by van Genuchten and Wagenet (1989), and
Toride et al. (1993) for one-dimensional flow. Numerical
solutions of the MIM have also been developed in two
and three dimensions using finite-element (Sudicky
and McLaren 1998; Therrien et al. 2000) and finitedifference (Zheng and Wang 1999) methods. The computer program CXTFIT (Toride et al. 1999) was used in
the forward mode to simulate the MIM BTCs presented in
this study.
Discrete-Fracture Models
DFMs require that the location, shape, orientation,
size, aperture, and hydraulic and solute transport properties of each fracture be specified explicitly as input. We
constructed three DFMs in this study, each with a unique
approach for representing the fracture network observed
in the till column. The first approach represented the fracture network as a system of parallel-plates (parallel-plate
discrete-fracture model [PDFM]), with spacing equal to
the fracture spacing measured in the field. The second
approach was to create a 3D network of fractures with
orientation, size, and location statistically similar to those
measured in the field (stochastic DFM). A third approach
was to reconstruct a virtual network of fractures identical
in orientation, size, and position to those identified when
the till column was dissected at the end of the experiment
(deterministic DFM).
The regular geometry of the PDFM allows the entire
system to be simulated as a single fracture with one-half
of a matrix block on either side (Sudicky and Frind 1982).
This simplified system results in a computationally efficient model and minimizes the number of required input
parameters. Unlike the MIM, the dimensions of the fracture must be specified during construction of the PDFM,
including fracture aperture (2b) and the spacing between
fractures (2B). The PDFM is perhaps the most widely
used model to simulate solute transport in fractured till. It
was used in the forward mode to simulate chloride BTCs
from a fractured till column in Canada (Grisak et al. 1980)
and chloride and pesticide transport through large columns of till in Denmark (Jørgensen et al. 1998). These
studies demonstrated that the PDFM produced simulated
BTCs that closely resembled laboratory-derived BTCs.
The PDFM was also used to simulate bromide transport at
the field scale during a trench-to-trench test conducted in
Canada (McKay et al. 1993b).
Models of ground water flow and solute transport
through fractured rock typically represent fractures as 3D,
discrete planar features. These models have been used for
development of well fields in fractured media (Jones et al.
1999), oil and gas reservoir engineering (Dershowitz et al.
1994), and evaluation of sites for disposal of high-level
nuclear waste (Anna 1998). They are well suited for simulating flow through fractured rocks because of the large
contrast in K between fractures and the rock matrix and
the low density of fractures encountered in rock.
Similar to the MIM, DFMs require governing equations for the fracture (Equation 4), matrix (Equation 5),
and diffusive exchange between the two pore regions
(Equation 6) given by:
@cf
@cf
@ 2 cf G
1 vf
2 Df
1 ¼0
@t
@z
@z2 b
@cmat
@ 2 cmat
2 De
¼0
@t
@x2
@cmat G ¼ 2nmat De
@x b
ð4Þ
ð5Þ
ð6Þ
where z and x are distances along the fracture in the
direction of flow and into the matrix normal to the fracture, respectively; G is the diffusive flux across the
fracture/matrix interface (Sudicky and Frind 1982). Unlike the MIM, however, the aforementioned governing
equations treat matrix diffusion as a second-order, Fickian
process in the dimension normal to the fracture plane.
This is considered to be more accurate than the first-order
approach for solutes with low diffusion coefficients, at
short timescales, or for systems with larger fracture spacing (Harrison et al. 1992). Equation 4 represents onedimensional advection and dispersion; however, a 3D
network of fractures may be simulated using this equation by orienting the z-axis in the direction of flow as a
3D vector.
Solutions to Equations 4 through 6 have been achieved for a system of parallel plates (PDFM) using semianalytical (Sudicky and Frind 1982), finite-element
(Sudicky 1989, 1990; Sudicky and McLaren 1998) methods. The computer program FRACTRAN (Sudicky and
McLaren 1998) was used to simulate the PDFM BTCs in
this paper. The stochastic and deterministic DFMs were
solved using the program MAFIC (Miller et al. 1997),
which uses particle tracking instead of solving the system
of equations produced by Equations 4 through 6. This
approach moves virtual particles through the fracture network in discrete timesteps, and the distance each particle
moves is a function of the advective velocity. Dispersion
and matrix diffusion are simulated by moving each particle at the end of a timestep according to stochastic random functions. The principal advantage of the particletracking approach is that it is relatively simple to program. However, studies in fractured rock suggest that at
least 50,000 particles are required to produce realistic
BTCs (Herbert et al. 1992), so the approach tends to be
computationally demanding.
Site Description, Column Preparation, and
Tracer Experiments
The study site is located within the Walnut Creek
watershed, 6 km south of Ames, Iowa, in the Des Moines
M.F. Helmke et al. GROUND WATER 43, no. 6: 877–889
879
Lobe landform region (Figure 1). The surficial deposit at
the site is the Alden Member till of the Dows Formation,
deposited 14 to 12.5 ka during the late Wisconsinan (Prior
1991; Eidem et al. 1999). The Alden Member is a massive, basal till with a bulk density (qb) of ~1.7 Mg/m3.
The Alden Member is classified as a loam, containing
~40% sand, 45% silt, and 15% clay. Previous investigations at the site revealed that the till is extensively
fractured (Eidem et al. 1999).
A 4-m-deep trench was excavated using a backhoe to
provide access to the till. The trench was carved using
a bench and tier method to provide multiple faces for
fracture mapping and to ease column collection. Fractures were identified as planes with iron oxide staining or
as leached zones in the till. Fractures were mapped using
sheets of clear acetate on both vertical and horizontal
faces in the trench, and fracture strike and dip was measured using a Brunton compass. The excavation revealed
that the till contains numerous subhorizontal and subvertical fractures from ground surface to the base of the pit.
Fracture spacing ranged from <2 cm near the surface to
~4.6 cm at a depth of 4 m. The most prominent fractures
were observed below 3-m depth where the till is partially
weathered. At this depth, the fractures were stained reddish brown (Munsell color: 10YR 5/8), in contrast to the
olive-brown (Munsell color: 2.5Y 5/4) till matrix.
Fractures mapped near the base of the excavation pit
were dense and preferentially oriented (Figure 2). Fracture spacing at a depth of 3.3 m was 0.043 m. Other
measures of fracture intensity were 643 fractures/m3
(P30), 23.3 m/m2 (P21), and 24.4 m2/m3 (P32) at this
depth, as estimated using the ISIS module of the program
FracMan (Dershowitz et al. 1994). Analysis of fracture
strike and dip revealed the presence of two fracture sets—
both predominantly vertical and striking northeastsouthwest. The first fracture set followed a Fisher distribution with a fracture pole trend of 326.0, plunge of
16.1, and Fisher dispersion (k) of 6.13. The second fracture set displayed a trend of 124.5, plunge of 10.1, and
a k of 4.85.
Figure 1. Map of Iowa showing the location of the study site
within the Des Moines Lobe landform region (after Prior
1991).
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M.F. Helmke et al. GROUND WATER 43, no. 6: 877–889
Figure 2. Plan-view map of fractures observed at a depth of
3.3 m at the site. Fractures are predominantly subvertical in
orientation at this depth. Trend of fractures is from northeast to southwest.
An intact column of till, 43 cm in diameter and
45 cm in length, was carved from the basal step of the
excavation trench at a depth of 3.3 to 3.75 m (Figure 3)
using a shovel and putty knife. Iron-stained fractures
were prominently visible along the sides of this column
(Figure 3). The cylindrical shape of the column was
maintained using a level and a section of polyvinyl chloride (PVC) pipe as a guide. A 61-cm-long piece of PVC
with an interior diameter (ID) of 45.7 cm was placed over
the column, leaving a 1- to 1.5-cm void between the column and the pipe. This annulus between the till and the
casing was sealed using paraffin wax to prevent side-wall
flow (Grisak et al. 1980). After the wax cooled (~8 h), a
putty knife was used to separate the column from its base.
The column was then winched from the trench, and then
2-mm-thick disks of high-density polyethylene (HDPE)
were placed at the column ends to prevent moisture loss
during transport to the laboratory.
The physical properties of the till column were consistent with previous studies of the Alden Member till.
The soil texture was a loam, with a particle-size percentage
Figure 3. Photograph of the till column (43-cm diameter
and 45-cm length) prior to encasement in the field. Subvertical, iron-stained fracture surfaces are prominent. Putty
knife for scale.
of 48.2% sand, 37.0% silt, and 14.8% clay (determined
by the sieve-and-pipette method). The qb of the column
was 1.83 Mg/m3. Total porosity of the column was 29.6%
(determined gravimetrically), resulting in a total pore
volume (PV) of 0.0172 m3.
In the laboratory, the ends of the column were carefully scraped with a putty knife to eliminate smear zones.
The resulting length of the column was 40 cm. A 5-mmthick layer of Ottawa Sand was placed at each end of the
column and held in place by the HDPE disks. Perforated
tubes (3-mm-ID HDPE) were fed through the sand to
provide fluid access to the sand packs. Plywood pistons
(19-mm thick and 43-cm diameter) were added to each
end and sealed with silicone caulking. The ends and the
walls of the column were mechanically compressed to
60 kPa to approximate the lithostatic stress at 3.5 m.
Although care was exercised to minimize desaturation of
the column, it is possible that some of the larger pores
drained during excavation and transport. To reduce the
chance of entrapped air, the column was slowly resaturated by upward flow for 7 d. Once saturation was complete, the Kb of the column was determined to be 6.8 3
1028 m/s using Darcy’s Law.
Three conservative solutes were used as tracers:
potassium bromide (KBr), pentafluorobenzoic acid
(PFBA), and 1,4-piperazinediethanesulfonic acid (PIPES)
disodium salt. These tracers were selected because they
do not sorb or undergo biodegradation (Jaynes 1993;
Moline et al. 1997) and because they possess differing
aqueous diffusion coefficients (D0). Differences in the
morphology of their BTCs would indicate matrix diffusion and provide evidence of fracture flow. The diffusion
coefficients in pure aqueous solution of Br and PFBA at
25C are 1.8 3 1029 and 7.6 3 10210 m2/s, respectively
(Bowman and Gibbens 1992). The D0 of PIPES has not
been determined experimentally; however, the calculated D0 using the Stokes-Einstein equation is 4.1 3
10210 m2/s (Helmke et al. 2004).
Tracer solutions were introduced into the columns
under a constant unit hydraulic gradient using a Mariotte
bottle. In situ ground water spiked with a 0.5-mM concentration (C0) of KBr, PFBA, and PIPES was passed
through the column for a period of 70 d (equivalent to 3.4
PVs). This concentration is equivalent to 39.95 mg/L Br,
106.04 mg/L PFBA, and 167.69 mg/L PIPES. The resulting density ratio of the solution was ~1.0003 with respect
to water, which is far less than the density ratio of 1.2
reported to cause density-driven flow in fractured systems
(Shikaze et al. 1998). Although an upward gradient was
applied (to prevent desaturation at the column base),
ground water flow was, in effect, downward because the
column was inverted in the laboratory. The temperature of
the column was maintained at a constant 12C to simulate
in situ conditions. Effluent samples were passed through
a 0.2-lm filter immediately upon collection and stored
at 4C until analyzed at the end of the experiment. Concentrations of Br, PFBA, and PIPES were determined by
ion chromatography. Analytical precision for Br (±0.63
mg/L), PFBA (±1.14 mg/L), and PIPES (±2.65 mg/L)
was determined using replicates of spiked samples.
Input Parameters, Model Construction,
and Evaluation
Fracture Networks
The three DFMs (PDFM, stochastic DFM, and deterministic DFM) required that fractures be placed explicitly
into each model, satisfying the fracture properties measured in the field (Table 1). The PDFM was the simplest
case, containing only one fracture placed within a matrix
slab of thickness 2B (Figure 4a).
The stochastic DFM was constructed of sets of fractures statistically similar to those recorded in the field
using the enhanced Baecher approach (Baecher et al.
1977) and the program FracMan (Dershowitz et al. 1994).
The advantage of using a stochastic network is that it can
place fractures within a model in areas where direct field
measurements are unavailable. Many realizations of the
stochastic DFM may be constructed, which allows the
model to perform uncertainty analyses using Monte Carlo
simulations (Doe 1997). Each fracture in the stochastic
DFM was placed randomly in space (a Poisson point process) and then expanded sequentially in the appropriate
orientation (following the Fisher distribution) to the
desired size (mean radius 7.9 cm, standard deviation 5.7
cm as determined from fracture maps). Intersecting fractures were truncated at a frequency equal to the termination percentage recorded in the field (35.5%). Fractures
were placed into the model until the fracture intensity
(P32) of the model equaled the P32 derived from the fracture map (24.4 m2/m3). The resulting stochastic DFM
contained 103 fractures and 1884 triangular elements
(Figure 4b).
For cases where fracture geometry and location is
known, each fracture may be placed explicitly within
a model to construct a deterministic DFM. Working with
till provides a unique opportunity to map fractures in three
dimensions because it can be cut with simple hand tools.
In this study, the column was dissected in horizontal 5-cm
sections at the end of the experiment. Fractures observed
at each interval were mapped and then joined in three
dimensions to recreate the ‘‘true’’ fracture network of the
column. The final fracture network included 2077 triangular elements (Figure 4c). In practice, a DFM may include
both stochastically generated and deterministic fractures.
Hydraulic Properties
The hydraulic properties of the models in this study
were specified to satisfy the Kb of the column (6.8 3
1028 m/s). Because the experiments were conducted
under a unit hydraulic gradient, q was equal to Kb. Fracture transmissivities (Tf) of the stochastic and deterministic DFMs were adjusted until the model-simulated Kb
matched the observed Kb of the column. In studies of
fractured rock, the Tf distribution may be determined
experimentally using borehole packer tests (Doe 1997).
This approach may be impractical in till due to the small
fracture spacing, although it is a potential area for future
research. Therefore, we assumed that Tf followed a lognormal distribution as reported in the literature (Dershowitz
1994). This resulted in a mean Tf of 6.6 3 1029 m2/s with
M.F. Helmke et al. GROUND WATER 43, no. 6: 877–889
881
Table 1
Input Parameters Used in the MIM, PDFM, and Stochastic and Deterministic DFMs
Parameter
Fracture Network
Fracture intensity
Fracture spacing, 2B
Length of fractures per area, P21
Number of fractures per volume, P30
Area of fractures per unit volume, P32
Fracture orientation
Set 1
Set 2
Fracture radius (stochastic DFM)
Fracture termination
Hydraulic Properties
Bulk hydraulic conductivity, Kb
Darcy flux, q
Fracture transmissivity, Tf
Stochastic DFM
Deterministic DFM
Fracture aperture, 2b
MIM and PDFM
Stochastic DFM
Deterministic DFM
Fracture porosity, nf
Fracture velocity, vf
Dispersion
Longitudinal dispersivity, aL
Diffusion
Effective diffusion coefficient, De1
Br
PFBA
PIPES
Exchange coefficient, a
Br
PFBA
PIPES
Matrix porosity, nmat
Br
PFBA
PIPES
1Converted
Value
Source
0.043 m
23.3 m/m2
643 fractures/m3
24.4 m2/m3
Field measurements
Fracture maps
FracMan
FracMan
Trend 326.0, plunge 16.1, Fisher k 6.13
Trend 124.5, plunge 10.1, Fisher k 4.65
Mean, l ¼ 7.9 cm; standard deviation, r ¼ 5.7 cm
35.5%
Field measurements/ISIS
Field measurements/ISIS
Fracture maps
Fracture maps
6.8 3 1028 m/s
6.8 3 1028 m/s
Darcy’s Law
Darcy’s Law
l ¼ 6.6 3 1029 m2/s, r ¼ 2.0 3 1028 m2/s
l ¼ 5.2 3 1029 m2/s, r ¼ 1.6 3 1028 m2/s
Adjusted to match Kb
Adjusted to match Kb
1.6 3 1025 m
l ¼ 1.8 3 1025 m, r ¼ 1.9 3 1025 m
l ¼ 1.8 3 1025 m, r ¼ 1.9 3 1025 m
0.038%
1.8 3 1024 m/s
Cubic law (Equation 7)
Cubic law (Equation 7)
Cubic law (Equation 7)
Equation 8
Equation 9
0.05 m
Neretnieks et al. 1992
4.3 3 10210 m2/s
2.6 3 10210 m2/s
1.3 3 10210 m2/s
Helmke et al. 2004
Helmke et al. 2004
Helmke et al. 2004
7.5 3 1027 1/s
4.3 3 1027 1/s
1.7 3 1027 1/s
Equation 3
Equation 3
Equation 3
26.8%
25.2%
21.4%
Helmke et al. 2004
Helmke et al. 2004
Helmke et al. 2004
from 23C to 12C using the Stokes-Einstein equation.
a standard deviation of 2.0 3 1028 m2/s for the stochastic DFM, and a mean Tf of 5.2 3 1029 m2/s with a standard deviation of 1.6 3 1028 m2/s for the deterministic
DFM.
Figure 4. PDFM (a), stochastic DFM (b), and deterministic
DFM (c) representations of the till column.
882
M.F. Helmke et al. GROUND WATER 43, no. 6: 877–889
Hydraulic aperture (2b) was estimated using the
cubic law (Snow 1969):
1
Kb 12l2B 3
2b ¼
ð7Þ
qg
where l is the water viscosity, q is the water density, and
g is the acceleration due to gravity. Equation 7 may be
used for a system of equally spaced fractures. The resulting 2b for this study was 1.6 3 1025 m. Hydraulic aperture may also be estimated by replacing Kb2B in
Equation 7 with fracture transmissivity (Tf, assuming the
matrix is impermeable). The cubic law assumes that the
walls of a fracture are smooth and parallel, which is
unlikely to exist in nature. Moreover, the cubic law estimates hydraulic aperture, which may deviate from transport aperture (Shapiro and Nicholas 1989; Tsang et al.
1991). Despite these deficiencies, the cubic law is
commonly used to provide a first-cut estimate of aperture
in fractured media.
Fracture porosity was estimated by assuming that the
fractures are parallel and equally spaced. For such a system, nf may be obtained from (Sudicky 1990):
nf ¼
2b
2B
ð8Þ
The resulting estimate of nf for this till was 0.038%.
This nf was in turn used to estimate a vf (required by the
DFMs) of 1.8 3 1024 m/s by:
vf ¼
q
nf
ð9Þ
Dispersion
The dispersion coefficient is normally determined
empirically by fitting a model to BTC data. Therefore,
independent estimates of Df were unavailable for the
till evaluated in this study. For this reason, we estimated
Df by:
D f ¼ aL v f 1 D 0
ð10Þ
where aL is the longitudinal dispersivity and D0 is the
aqueous diffusion coefficient of each compound. Dispersivity was assumed to be 0.05 m in this study, which
corresponds with dispersivities determined in fractured
rock at this scale (Neretnieks et al. 1982). The use of D0
in Equation 10 assumes an absence of tortuosity in the
fracture, which is suspect. However, the rapid vf causes
mechanical dispersion to dominate, and Df is likely to be
insensitive to D0.
Matrix Diffusion
Estimates of De for Br, PFBA, and PIPES were determined for this till by conducting radial diffusion experiments (Helmke et al. 2004). Measurements of De at 23C
were 5.8 3 10210, 3.5 3 10210, and 1.7 3 10210 m2/s for
Br, PFBA, and PIPES, respectively. The Stokes-Einstein
equation was used to modify the values of De to correct
for the temperature difference between 23C and 12C,
which resulted in corrected De values of 4.3 3 10210,
2.6 3 10210, and 1.3 3 10210 m2/s. These values are less
than the D0 for each compound, which indicates that
tortuosity affects matrix diffusion in this till.
Diffusion is sensitive not only to the diffusion coefficient but also to the porosity available to diffusing compounds in the matrix. During the radial diffusion
experiments, it was determined that the ‘‘effective diffusive
porosity’’ was slightly different for Br, PFBA, and PIPES
(26.8%, 25.2%, and 21.4%, respectively; Helmke et al.
2004). All these values were slightly less than nT (29.6%)
and demonstrated that nmat may be a function of not only
the medium but also the solute. Similar results have been
reported in other tills (van der Kamp et al. 1996).
Statistical Evaluation of Models
Model goodness of fit was evaluated using the modified index of agreement (d1) of Willmott et al. (1985).
The parameter d1 has been used as a model-selection criterion in water resources investigations and is defined as
n
P
d1 ¼ 1:0 2 P
n
jOi 2 Pi j
i¼1
ð11Þ
ðjPi 2 Oj 1 jOi 2 OjÞ
i¼1
where O and P are the observed and model-simulated
data, respectively, O is the mean of observed values, and
n is the number of observations (Legates and McCabe
1999). The value of d1 varies from 0 to 1, with 1 indicating a perfect fit between the simulated and the observed
data. Although d1 may be interpreted in a similar fashion
as the coefficient of determination (R2), d1 is considered
superior because it is less sensitive to outliers and proportional differences than R2.
Results and Discussion
Breakthrough Curves
Detectable concentrations (C/C0 > 0.02) of the three
tracers were observed in the column effluent after 4.7 d,
resulting in a first-arrival velocity of at least 0.085 m/d
(Figure 5). Such a rapid velocity indicates that this till
would act as a poor barrier to contaminant migration.
Moreover, this transport rate is faster than the time estimated for one PV to pass through the column (19.9 d),
which indicates that preferential flowpaths (presumably
fractures) are responsible for this rapid advection. Breakthrough (defined as the time for C/C0 to reach 0.5) of Br,
PFBA, and PIPES was achieved after 19.2, 18.9, and
13.8 d, respectively, resulting in center-of-mass velocities
of 0.021, 0.021, and 0.029 m/d. All three solutes resulted in
breakthrough times earlier than the time estimated for one
PV (19.9 d), providing additional evidence that fracture
flow controlled solute transport and the shape of the BTCs.
Separation of the BTCs of the three tracers suggests
that matrix diffusion was an influential process during
transport. The concentration of PIPES increased more
rapidly than Br or PFBA during the first 30 d of the
experiment. This separation was likely due, in part, to the
lower De of PIPES compared to Br and PFBA. A similar
separation between PIPES and Br was observed in BTCs
produced from a column of fractured saprolite from Tennessee (Moline et al. 1997). Grisak et al. (1980) observed
that calcium increased in concentration more rapidly than
chloride during a laboratory experiment using an intact
column of fractured till. The authors attributed the separation of calcium and chloride to differences in their De values, which were 5 3 10211 and 1.9 3 10211 m2/s,
respectively. Differences in nmat may also have contributed to the separation of the BTCs. The low nmat of
PIPES may have caused the concentration between the
fracture and the matrix to reach local equilibrium faster
than PFBA or Br, which, when combined with the lower
De, could have caused the greater separation between the
PIPES and the Br BTCs than the separation between
PFBA and Br.
Solutes reached a C/C0 of 0.98 or greater after
a period of ~50 d. Clearly, it would be inappropriate to
use the fracture velocity calculated by the cubic law (15.3
m/d) and assume advection only (i.e., plug flow), which
M.F. Helmke et al. GROUND WATER 43, no. 6: 877–889
883
Table 2
Goodness of Fit for the MIM, PDFM, and Stochastic
and Deterministic DFMs Grouped by Tracer as
Determined by the Modified Index of Agreement, d1
Model
MIM
PDFM
Stochastic DFM
Deterministic DFM
Figure 5. BTCs for Br, PFBA, and PIPES from the tracer
experiments. BTCs simulated by the MIM, PDFM, stochastic DFM, and deterministic DFM are shown in a, b, c, and d,
respectively.
would result in a predicted C/C0 of 1.0 after only 38 min.
There are several explanations for this discrepancy,
including (1) some process is serving to retard migration
through the fractures (such as matrix diffusion); (2) solutes are being advected through a pore network with
a broad distribution of velocity (i.e., mechanical dispersion); or (3) a combination of both processes is affecting
the BTCs. Regardless of the underlying process, this
effect would likely be more pronounced at greater transport distances and would cause a significant lag time
between changes in boundary conditions and solute concentration downgradient of the source area.
Evaluation of Model Simulations
The goodness-of-fit analysis demonstrated that both
the MIM and the DFM approaches performed reasonably
well, simulating the observed BTCs (Figure 5; Table 2).
The goodness-of-fit values (d1) ranged from 0.72 to 0.96
884
M.F. Helmke et al. GROUND WATER 43, no. 6: 877–889
Br
PFBA
PIPES
0.90
0.91
0.94
0.95
0.84
0.85
0.94
0.95
0.72
0.81
0.96
0.95
for all models and tracers (Table 2). The simplest model,
the MIM, simulated most of the BTCs (Figure 5a), with
the best match occurring with the Br BTC (d1 ¼ 0.90).
The PFBA and PIPES simulations, however, overpredicted observed solute concentration during the initial
phase of the experiment, causing their goodness of fit to
decrease (d1 ¼ 0.84 and 0.72, respectively; Table 2). Values of C/C0 for modeled PFBA and PIPES BTCs appear
to increase instantly to values of 0.14 and 0.40, respectively. However, closer inspection reveals that these concentrations were predicted within 1 h, which corresponds
with the calculated fracture velocity of 15.3 m/d. The
MIM’s inability to accurately simulate BTCs at short
timescales for solutes with low De values may be a result
of its first-order approximation of matrix diffusion. This
effect would be more pronounced for systems with large
fracture spacing, short timescales, and tracers with small
De values. Despite its limitations, the MIM reproduced
the overall morphology of the three BTCs and would
likely be applicable at the field scale, where transport distances are large with respect to fracture spacing. In addition, the model’s disadvantages should be weighed
against the advantages of fewer input parameters and the
model’s computational efficiency (CXTFIT simulated
each BTC in <1 CPU second).
The morphology of the BTCs produced by the
PDFM simulations was comparable to the observed data
(Figure 5b). The PDFM predicted the Br BTC well (d1 ¼
0.91) but was less effective for PFBA and PIPES (d1 ¼
0.85 and 0.81, respectively). Similar to the MIM, the
PDFM simulations of Br, PFBA, and PIPES overpredicted their concentration during the first 30 d of the
experiment. Unlike the MIM simulations, the predicted
concentration rose gradually and thus was more similar to
the observed BTCs. The PDFM was more difficult to
assemble than the MIM because it required specifying the
spatial location of the fracture explicitly within the
model. Once constructed, however, simulations required
only 3 CPU seconds to complete.
The stochastic and deterministic DFMs produced
BTCs that most closely matched the observed BTCs of
all model approaches investigated (stochastic DFM d1 ¼
0.94, 0.94, and 0.96; and deterministic DFM d1 ¼ 0.95,
0.95, and 0.95 for Br, PFBA, and PIPES, respectively),
although the breakthrough times were ~5% to 10% earlier
than observed. The fits of the 3D DFMs were generally
superior to those from MIM and PDFM simulations. For
example, the d1 value for the MIM prediction of the
PIPES BTC was 0.72, which was the poorest of all the d1
statistics, vs. d1 values of 0.96 and 0.95 for the stochastic
and deterministic DFMs, respectively. The generally
superior results of the 3D DFMs may be a function of
a more realistic fracture geometry. Alternatively, this may
have been a result of representing heterogeneities by the
lognormal distribution of fracture transmissivity. If this is
this case, the MIM and PDFM might also be improved by
including advective heterogeneities. Differences between
the stochastic and the deterministic DFMs were slight,
which indicates that the stochastic fracture network was
an adequate surrogate for the actual fracture network for
this particular till. Accuracy entails a cost in this case,
however. The stochastic DFM included 103 fractures
within a cube only 40 cm on a side, and the deterministic
DFM included 2077 triangular elements. Ten million particles were required to generate the BTCs for both models. The simulation process required ~8 h on a personal
computer with a 1-GHz CPU—a much longer time than
either the MIM or the PDFM simulations.
A stochastic analysis of residuals using the KruskalWallis test (Conover 1980) revealed that there was no statistically significant difference between the models for
simulations of Br (at the 0.05 significance level). This
lack of a significant difference suggests that features unique
to each modeling approach do not significantly affect the
results for this compound. There was a statistically significant difference (at the 0.05 significance level), however,
between simulations of the PFBA and PIPES BTCs. This
is confirmed by visual inspection of the BTCs, which
show that differences between models are more pronounced for compounds with smaller De values (Figure 5). In situations where De is small or where fracture
spacing is large, the first-order approximation of matrix
diffusion used by the MIM may fail; hence, a secondorder approach should probably be used (Sudicky 1990).
Moreover, in cases where matrix diffusion governs the
shape of a BTC, matrix diffusion may obscure the lesser
effect of fracture geometry and orientation. For tracers
such as PFBA and PIPES that are less susceptible to
matrix diffusion, oversimplification of the fracture network in the model may result in unrealistic transport
times and parameters.
As discussed previously, the models were conducted
in the forward mode to test their ability to predict BTCs
using independently determined input parameters. It
should be noted that these models may be fitted to the
observed BTCs with d1 values of 0.99 or greater by adjusting aL, De, and nf. Such optimization can lead to nonunique and unrealistic input parameters (Parker and van
Genuchten 1984). Nonetheless, parameter optimization
has been employed in studies of fractured till for cases
where reliable estimates of input parameters are unavailable or impractical to obtain. The PDFM and MIM were
used in the inverse mode to simulate Br BTCs in fractured
tills from Denmark (Jørgensen et al. 2003). Both the
PDFM and the MIM were successfully fit to the observed
BTCs. Further tests using multiple flow rates, however,
demonstrated that only the PDFM was capable of simulating BTCs under increased gradients. The Denmark
study illustrates the importance of testing models at
various temporal scales if parameter optimization is employed and underscores the rationale for running models
in the forward mode for this study.
One of the surprising results of this study was the
ability of all four models to predict the Br BTC with
accuracy, despite the diversity of models employed. We
suggest that Br was in relative equilibrium between the
fractures and the matrix at the timescale selected for this
study, which was not the case for PFBA or PIPES (these
would require longer timescales to reach this state).
Under such conditions, the differences between the models become insignificant as the influence of macroscopic
dispersion becomes obscured by matrix diffusion.
Although this would seem to be a special case, it is likely
to be the common condition for deeper tills (>3-m depth)
that are characterized by longer residence times and less
transient boundary conditions. The unit gradient and 70-d
timescale employed by this study were selected to best
represent in situ conditions. Using unnatural hydraulic
gradients upsets this balance to reveal differences
between model approaches. For example, Jørgensen et al.
(2003) observed differences between the MIM and the
PDFM when gradients of 7.09 and 28.3 were applied to
a column of similar depth to the one evaluated in this
study. This effect was reproduced in this study by increasing the gradient to 10 and rerunning the MIM and PDFM
models. At this elevated flow rate, the PDFM and MIM
predict Br breakthrough times (time to reach a C/C0 of
0.5) of 6 h and 4 min, respectively. This clearly shows
that the models diverge at high flow rates using similar
input parameters. Conversely, the PDFM and MIM agree
closely when using the input parameters of Jørgensen
et al. and a gradient of 1. These simulations suggest that
under longer timescales and realistic gradients, differences
between model approaches may become unimportant.
Parametric Analysis
A parametric analysis was conducted to determine
how sensitive the MIM was to four input parameters (De,
aL, nf, and nmat; Figures 6a through 6d, respectively). The
goal of this exercise was to identify the most influential
solute transport processes and determine the relative precision required of input parameters by adjusting input parameters and comparing the resulting BTCs. Simulated
BTCs were compared to the observed Br BTC, using the
parameters listed in Table 1 as a ‘‘base case.’’
The parametric analysis revealed that the MIM is
sensitive to both diffusion and dispersion under the basecase conditions. The best fit was achieved by increasing
De to D0 (1.8 3 1029 m2/s), and the worst fit occurred
when De was reduced by 1 order of magnitude (4.3 3
10211 m2/s). Under the assumptions of this model, simulations are sensitive to De, which suggests that matrix diffusion is a dominant process. Although the model
produced a superior fit when De was increased to D0,
doing so would neglect the effect of tortuosity and contradicts the results from diffusion cell experiments (Helmke
et al. 2004). Longitudinal dispersivity was varied from
0.005 to 0.5 m. Increasing aL to 0.5 m resulted in the
poorest fit, and the best fit occurred when aL was reduced
M.F. Helmke et al. GROUND WATER 43, no. 6: 877–889
885
Figure 7. Modified index of agreement (d1) representing
goodness of fit of the MIM to the Br BTC for various combinations of the effective diffusion coefficient (De) and dispersivity (aL). The shape of the d1 surface indicates a
nonunique relationship between diffusion and dispersion.
Figure 6. Parametric analysis of the MIM and Br BTCs
comparing variable effective diffusion coefficient (De, a), dispersivity (aL, b), matrix porosity (nmat, c), and fracture
porosity (nf, d). The special case of nf ¼ total porosity (nT)
represents the system as an EPM.
to 0.005 m. Setting aL below 0.005 m resulted in no additional improvement of fit.
Using the MIM in the inverse mode to estimate De
and aL would result in nonunique values under the basecase conditions (Figure 7). When dispersion is set to zero,
the model fits the observed data well (d1 ¼ 0.98) when De
is set to 7.4 3 10210 m2/s. Under these conditions,
the model is insensitive to dispersion until aL exceeds
0.005 m. Above this value, De may be adjusted for any aL
up to 0.05 m to produce d1 values of 0.95 or greater. This
nonunique relationship between matrix diffusion and dispersion has been encountered by previous investigators
(Brusseau et al. 1994; Toride et al. 1999). Fitting additional parameters would likely be less unique and therefore less informative, suggesting that the input parameters
should be estimated independently of model fitting to
avoid this problem.
The MIM appears to be moderately sensitive to
changes in nmat (Figure 6c). Increasing nmat from the
base-case value of 26.8% to 29.6% (nT) changes the
886
M.F. Helmke et al. GROUND WATER 43, no. 6: 877–889
shape of the BTC only slightly. However, reducing nmat to
a value <1% causes the fractures and matrix to reach
rapid equilibrium, which produces BTCs that predict
early breakthrough. Although the effect of nmat is small
over the typical range of tills (10% to 30%), its relationship with diffusion should not be overlooked.
Modifying nf has only a small effect on the BTC
even when increased by 3 orders of magnitude. When nf
is set equal to nT, the MIM behaves as an equivalent
porous medium (EPM) and fits the observed BTC
remarkably well (d1 ¼ 0.97, Figure 6d). This raises an
important question: why would the EPM model predict
results so similar to a DFM in this study? Our explanation
of this phenomenon is that the solutes were advected primarily through the fracture network and that rapid diffusion coupled with closely spaced fractures caused
fracture/matrix equilibrium to occur in a matter of weeks.
Such a process would generate BTCs that appear similar
to those produced by an EPM, and matrix diffusion would
appear to behave as dispersion. This conceptual model of
the flow system was confirmed by a dye-trace study that
revealed dye following fracture traces in this column
upon dissection (Helmke et al. 2005). Other researchers
have suggested that for long timescales and large transport distances, fractured tills may behave as an EPM for
the same reason (McKay et al. 1993b, 1998). The separation of BTCs by the three compounds provides further
evidence that the column in this study did not behave
strictly as an EPM.
The experimental setup was designed to represent an
idealized solute transport scenario of steady-state, onedimensional flow with a single constant-concentration
boundary condition. This design was selected to reduce
the number of unknown variables in the experiment.
Modifications to the experimental design might have resulted in greater sensitivity to model input parameters
such as the effective diffusion coefficient. For instance,
a variety of flow rates could have been applied or flow
could have been interrupted during the experiment
(Brusseau et al. 1994), which might have reduced the
nonunique nature of dispersion vs. diffusion. Rinsing the
column to evaluate tailing or using a pulse injection rather
than a continuous source might have provided additional
information pertaining to the solute transport processes
involved.
Application of fracture models to the field scale
would require that the models be run in three dimensions
under transient conditions. It is possible that in this scenario, the 3D nature of the stochastic or deterministic
DFM would prove more applicable to the field scale.
Alternatively, including heterogeneities in the MIM, or
combining the MIM and select 3D discrete fractures,
might produce superior results.
Conclusions
Four modeling approaches (MIM, PDFM, stochastic
DFM, and deterministic DFM) were used to simulate solute transport through fractures in a laboratory column of
till. Goodness-of-fit analysis demonstrated that all modeling approaches employed were reasonable predictors of
the BTCs, yet reflected the apparent differences between
the model approaches. Differences between the models
were not significant when simulating Br transport, suggesting that more elaborate models do not necessarily
produce results that are more accurate under the flow
conditions employed by this study. Simulations involving
PFBA and PIPES, on the other hand, revealed that the 3D
DFMs were superior to the MIM and PDFM due to the
MIM and PDFM predicting more rapid transport of
PFBA and PIPES than observed during the initial portions of the experiment. We think that the first-order
approximation of diffusion employed by the MIM may
produce inaccurate results if the rate of diffusion is low.
This phenomenon should be more apparent in cases
where fracture spacing is large and/or timescales are
small. The MIM and PDFM approaches might also benefit by including heterogeneities using a lognormal transmissivity distribution as employed by the 3D DFMs.
In cases where De is particularly small (e.g., PIPES),
goodness-of-fit differences between the modeling approaches might also result from how the models incorporate fracture orientation and geometry. The MIM and
PDFM assume that fractures provide flowpaths that are
oriented in the direction of ground water flow, which may
be overly simplistic. The more realistic orientation and
geometry of fractures in the 3D DFMs produce flowpaths
that are slightly longer than those predicted using a simplified geometry, which would effectively increase residence
time. The lognormal fracture transmissivity assigned to
these models may also have caused the more favorable
fits. However, these effects might be masked by matrix
diffusion over longer timescales.
Efficiency of the models and their potential application in field situations also differ. The MIM and PDFM
require fewer input parameters and are computationally
efficient, with run times <3 s in most cases. The 3D
DFMs may be more compelling in depicting reality but
are more difficult to construct and are computationally
less efficient (run times >8 h per BTC) than the MIM or
PDFM. More importantly, further investigations are
required to test and compare these methods at the field
scale. At larger scales, the MIM and PDFM will likely
prove more practical to implement than the 3D DFMs
due to the greater computational effort required by simulating large numbers of fractures. On the other hand, in
cases where fracture spacing is large with respect to the
investigation scale, or during very short timescales,
DFMs are likely to produce superior results than the simplified MIM or PDFM approaches. It is for this reason
that 3D DFMs have been favored for simulations of solute transport through fractured rock (Doe 1997). An alternative approach would be to use the MIM or PDFM to
represent small-scale fractures, such as those identified in
this study, and use a 3D DFM to simulate large fractures
in the model. This method would take advantage of the
relative strengths of these models and would be particularly useful at the field scale.
The model simulations indicated that the close fracture spacing of the till allowed diffusive equilibrium to
occur between the fractures and the matrix over a relatively short time period (several weeks). This effect
caused the system to behave similar to an EPM, even
though a dye-trace study showed that flow occurred primarily through fractures, and halos surrounding the fractures provide evidence of rapid matrix diffusion (Helmke
et al. 2005). A parametric analysis revealed that the MIM
was capable of reproducing the Br BTC in the EPM mode
(nf set to nT). However, differences between the BTCs
using solutes with different Des indicate that diffusion is
a controlling process. Nonetheless, investigations might
consider using an EPM approach in these deposits for
large spatial or temporal scales. However, for short timescales or situations where fracture spacing is large with
respect to the scale of investigation, EPM assumptions
would be inappropriate. Additionally, for cases where
boundary conditions are transient (e.g., recharge events or
remediation systems), these systems would remain in
a state of constant disequilibrium and would require models that simulate diffusion explicitly.
Regardless of the modeling approach used, fractures
should be incorporated routinely into solute transport
models in glaciated terrain. Recent advances in computer
codes and hardware, and the development of independent
methods for obtaining input parameters (Helmke et al.
2004) make this approach much easier than it has been in
the past. Depending on the scale of investigation, various
combinations of these models could be employed, which
would take advantage of the benefits of DFMs yet simulate denser fracture sets using computationally efficient
methods.
Acknowledgments
This research was funded by grants from the American Geophysical Union (Horton Grant), the Association
of Ground Water Scientists and Engineers, the Geological
Society of America and its Hydrogeology Division,
Sigma Xi, and the U.S. EPA through an Interagency
M.F. Helmke et al. GROUND WATER 43, no. 6: 877–889
887
Agreement DW12036252 to the Agricultural Research
Service. The authors wish to thank Ed Sudicky and Rob
McLaren at the University of Waterloo for the use of
FRACTRAN, and Tom Doe and Bill Dershowitz at Golder
Associates for access to the FracMan/MAFIC software
package. We thank Phil Jardine and Gerilynn Moline
for the suggestion of using PIPES as a tracer, and Allen
Shapiro, Tom Doe, and two anonymous reviewers for
their extensive, critical, and insightful contributions to
the manuscript.
References
Anna, L.O. 1998. Preliminary three-dimensional discrete fracture model of the Topopah Spring Tuff in the Exploratory
Studies Facility, Yucca Mountain Area, Nye County, Nevada.
USGS Open-File Report 97-834. Denver, Colorado: USGS.
Baecher, G.B., N.A. Laney, and H.H. Einstein. 1977. Statistical
description of rock properties and sampling. In Proceedings of the 18th U.S. Symposium on Rock Mechanics,
American Institute of Mining Engineers, ed. F.D. Wang,
and G.B. Clark, 5C1–8. Keystone, Colorado: A.A. Balkema.
Bear, J., C. Tsang, and G. de Marsily. 1993. Flow and Contaminant Transport in Fractured Rock. San Diego, California:
Academic Press Inc.
Bowman, R.S., and J.S. Gibbens. 1992. Difluorobenzoates as
nonreactive tracers in soil and ground water. Ground Water
30, no. 1: 8–14.
Brockman, C.S., and J.P. Szabo. 2000. Fractures and their distribution in the tills of Ohio. Ohio Journal of Science 100,
no. 3–4: 39–55.
Brusseau, M.L., Z. Gerstl, D. Augustijn, and P.S.C. Rao. 1994.
Simulating solute transport in an aggregated soil with the
dual-porosity model: Measured and optimized parameter
values. Journal of Hydrology 163, no. 1–2: 187–193.
Coats, K.H., and B.D. Smith. 1964. Dead-end pore volume and
dispersion in porous media. Society of Petroleum Engineers
Journal 4, no. 1: 73–84.
Connell, D.E. 1984. Distribution, characteristics, and genesis of
joints in fine-grained till and lacustrine sediments, eastern
and northwestern Wisconsin. M.S. thesis, Department of
Geology and Geophysics, University of Wisconsin, Madison.
Conover, W.J. 1980. Practical Nonparametric Statistics, 2nd ed.
New York: John Wiley and Sons.
Dershowitz, W., G. Lee, J. Geier, S. Hitchcock, and P.R. La Pointe.
1994. FracMan Version 2.4—Interactive Discrete Feature
Data Analysis, Geometric Modeling, and Exploration Simulation. Redmond, Washington: Golder Associates Inc.
Doe, T. 1997. Understanding and solving fracture flow problems.
Water Engineering Management 144, September: 36–43.
Eidem, J.M., W.W. Simpkins, and M.R. Burkart. 1999. Geology,
groundwater flow, and water quality in the Walnut Creek
watershed. Journal of Environmental Quality 28, no. 1: 60–69.
Freeze, R.A., and J.A. Cherry. 1979. Groundwater. Englewood
Cliffs, New Jersey: Prentice-Hall.
Grisak, G.E., and J.F. Pickens. 1980. Solute transport through
fractured media: 1. The effect of matrix diffusion. Water
Resources Research 16, no. 4: 719–730.
Grisak, G.E., J.F. Pickens, and J.A. Cherry. 1980. Solute transport through fractured media: 2. Column study of fractured
till. Water Resources Research 16, no. 4: 731–739.
Harrison, B., E.A. Sudicky, and J.A. Cherry. 1992. Numerical
analysis of solute migration through fractured clayey deposits into underlying aquifers. Water Resources Research
28, no. 2: 515–526.
Helmke, M.F. 2003. Studies of solute transport through fractured till in Iowa. Ph.D. diss., Department of Geological
and Atmospheric Sciences, Iowa State University, Ames,
Iowa.
888
M.F. Helmke et al. GROUND WATER 43, no. 6: 877–889
Helmke, M.F., W.W. Simpkins, and R. Horton. 2005. Fracturecontrolled nitrate and atrazine transport in four Iowa till
units. Journal of Environmental Quality 34, no. 1: 227–236.
Helmke, M.F., W.W. Simpkins, and R. Horton. 2004. Experimental determination of effective diffusion parameters in the
matrix of fractured till. Vadose Zone Journal 3, no. 3: 1050–
1056.
Herbert, A.W., G.W. Lanyon, J.E. Gale, and R. MacLeod. 1992.
Discrete fracture network modelling for phase 3 of the
Stripa project using NAPSAC. In In Situ Experiments at the
Stripa Mine, OECD Nuclear Energy Agency and Swedish
Nuclear Fuel and Waste Management Company, Paris:
Organisation for Economic Co-operation and Development
219–235.
Jaynes, D.B. 1993. Evaluation of fluorobenzoate tracers in surface soils. Ground Water 32, no. 4: 532–538.
Jones, M.A., A.B. Pringle, I.M. Fulton, and S. O’Neill. 1999.
Discrete fracture network modeling applied to groundwater
resource exploitation in southwest Ireland. In Fractures,
Fluid Flow and Mineralization, ed. K.J.W. McCaffey,
L. Lonegran, and J.J. Wilkinson, 83–103. London: Geological Society, Special Publications no. 155.
Jørgensen, P.R., T. Helstrup, J. Urup, and D. Seifert. 2003. Modeling of non-reactive solute transport in fractured clayey till
during variable flow rate and time. Contaminant Hydrology
68, no. 3–4: 193–216.
Jørgensen, P.R., L.D. McKay, and N.H. Spliid. 1998. Evaluation
of chloride and pesticide transport in a fractured clayey till
using large undisturbed columns and numerical modeling.
Water Resources Research 34, no. 4: 539–553.
Jørgensen, P.R., and N.H. Spliid. 1992. Mechanisms and rates of
pesticide leaching in shallow clayey till. In European Conference on Integrated Research for Soil and Sediment Protection and Remediation, 11. Maastricht, The Netherlands:
MECC.
Keller, C.K., G. van der Kamp, and J.A. Cherry. 1988. Hydrogeology of two Saskatchewan tills, I. Fractures, bulk
permeability, and spatial variability of downward flow.
Journal of Hydrology 101, no. 1–4: 97–121.
Kemmis, T.J., E.A. Bettis III, and G.R. Hallberg. 1992. Quaternary geology of Conklin Quarry. Iowa Department of Natural Resources Guidebook Series no. 13. Iowa Department
of Natural Resources, Iowa City, Iowa.
Klint, K.E.S., and P. Gravensen. 1999. Fractures and biopores in
Weichselian clayey till aquitards at Flakkebjerg, Denmark.
Nordic Hydrology 30, no. 4–5: 267–284.
Legates, D.R., and G.J. McCabe Jr. 1999. Evaluating the use of
‘‘goodness-of-fit’’ measures in hydrologic and hydroclimatic model validation. Water Resources Research 35,
no. 1: 233–241.
McKay, L.D., D.J. Balfour, and J.A. Cherry. 1998. Lateral chloride migration from a landfill in a fractured clay-rich glacial deposit. Ground Water 36, no. 6: 988–999.
McKay, L.D., J.A. Cherry, and R.W. Gillham. 1993a. Field
experiments in a fractured clay till: 1. Hydraulic conductivity and fracture aperture. Water Resources Research 29, no.
4: 1149–1162.
McKay, L.D., J.A. Cherry, and R.W. Gillham. 1993b. Field
experiments in a fractured clay till: 2. Solute and colloid
transport. Water Resources Research 29, no. 12: 3879–
3890.
Miller, I., G. Lee, and W. Dershowitz. 1997. MAFIC Version
1.6—Matrix/Fracture Interaction Code with Heat and Solute Transport. Redmond, Washington: Golder Associates
Inc.
Moline, G.R., C.R. Knight, and R. Ketcham. 1997. Laboratory
measurement of transport processes in a fractured limestone/shale saprolite using solute and colloid tracers. Geological Society of America Abstracts with Programs 29,
no. 6: 370.
Neretnieks, I., T. Eriksen, and P. Tähtinen. 1982. Tracer movement in a single fissure in granitic rock: Some experimental
results and their interpretation. Water Resources Research
18, no. 4: 849–858.
Parker, J.C., and A.J. Valocchi. 1986. Constraints on validity of
equilibrium and first-order kinetic transport models in
structural soils. Water Resources Research 22, no. 3: 399–407.
Parker, J.C., and M.Th. van Genuchten. 1984. Determining
transport parameters from laboratory and field tracer experiments. Virginia Agricultural Experiment Station Bulletin 84, 1–97.
Prior, J.C. 1991. Landforms of Iowa. Iowa City, Iowa: University
of Iowa Press.
Shapiro, A., and J. Nicholas. 1989. Assessing the validity of the
channel model of fracture apertures under field conditions.
Water Resources Research 25, no. 5: 817–828.
Shikaze, S.G., E.A. Sudicky, and F.W. Schwartz. 1998. Densitydependent solute transport in discretely-fractured geologic
media: Is prediction possible? Journal of Contaminant
Hydrology 34, no. 3: 273–291.
Simpkins, W.W., and K.R. Bradbury. 1992. Groundwater flow,
velocity, and age in a thick, fine-grained till unit in southeastern Wisconsin. Journal of Hydrology 132, no. 1–4:
283–319.
Snow, D.T. 1969. Anisotropic permeability of fractured media.
Water Resources Research 5, no. 6: 1273–1289.
Sudicky, E.A. 1990. The Laplace transform Galerkin technique
for efficient time-continuous solution of solute transport in
double-porosity media. Geoderma 46, no. 1–3: 209–232.
Sudicky, E.A. 1989. The Laplace transform Galerkin technique:
A time-continuous finite element theory and application to
mass transport in groundwater. Water Resources Research
25, no. 8: 1833–1846.
Sudicky, E.A., and E.O. Frind. 1982. Contaminant transport in
fractured porous media: Analytical solutions for a system
of parallel fractures. Water Resources Research 18, no. 6:
1634–1642.
Sudicky, E.A., and R.G. McLaren. 1998. FRACTRAN User’s
Guide. An Efficient Simulator for Two-Dimensional, Saturated Groundwater Flow and Solute Transport in Porous or
Discretely-Fractured Porous Formations. Ontario, Canada:
Groundwater Simulations Group, Waterloo Centre for
Groundwater Research, University of Waterloo.
Therrien, R., E.A. Sudicky, and R.G. McLaren. 2000. User’s
Guide for NP 3.49. A Preprocessor for FRAC3DVS 3.49:
An Efficient Simulator for Three-Dimensional, SaturatedUnsaturated Groundwater Flow and Chain-Decay Solute
Transport in Porous or Discretely-Fractured Porous Formations. Ontario, Canada: Groundwater Simulations Group,
Waterloo Centre for Groundwater Research, University of
Waterloo.
Toride, N., F.J. Leij, and M.Th. van Genuchten. 1999. The
CXTFIT code for estimating transport parameters from laboratory or field tracer experiments version 2.1. Research
Report 137. Riverside, California: U.S. Salinity Laboratory,
USDA, ARS.
Toride, N., F.J. Leij, and M.Th. van Genuchten. 1993. A
comprehensive set of analytical solutions for nonequilibrium solute transport with first-order decay and
zero-order production. Water Resources Research 29, no. 7:
2167–2182.
Tsang, C.F., Y.W. Tsang, and F.V. Hale. 1991. Tracer transport
in fractures; analysis of field data based on a variableaperture channel model. Water Resources Research 27, no.
12: 3095–3106.
van der Kamp, G., D.R. Van Stempvoort, and L.I. Wassenaar.
1996. The radial diffusion method 1. Using intact cores to
determine isotopic composition, chemistry, and effective
porosities for groundwater in aquitards. Water Resources
Research 32, no. 6: 1815–1822.
van Genuchten, M.Th., and R.J. Wagenet. 1989. Two-site/tworegion models for pesticide transport and degradation: Theoretical development and analytical solutions. Soil Science
Society of America Journal 53, no. 5: 1303–1310.
Willmott, C.J., S.G. Ackleson, R.E. Davis, J.J. Feddema, K.M.
Klink, D.R. Legates, J. O’Donnell, and C.M. Rowe. 1985.
Statistics for the evaluation and comparison of models.
Journal of Geophysical Research 90, no. 5: 8995–9005.
Zheng, C., and P.P. Wang. 1999. MT3DMS: A modular threedimensional multispecies model for simulation of advection, dispersion and chemical reactions of contaminants in
groundwater systems; documentation and user’s guide.
Contract Report SERDP-99-1. Vicksburg, Mississippi: U.S.
Army Engineer Research and Development Center.
M.F. Helmke et al. GROUND WATER 43, no. 6: 877–889
889