Journal of Contaminant Hydrology 159 (2014) 57–66

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Journal of Contaminant Hydrology
journal homepage: www.elsevier.com/locate/jconhyd

Estimation and upscaling of dual-permeability model
parameters for the transport of E. coli D21g in soils withpreferential ﬂow
Yusong Wang a,⁎, Scott A. Bradford b, Jiří Šimůnek aab

Department of Environmental Sciences, University of California, Riverside, CA 92521, United States
USDA, ARS, Salinity Laboratory, Riverside, CA, United Statesa r t i c l e

i n f oArticle history:
Received 13 October 2013
Received in revised form 25 January 2014
Accepted 31 January 2014
Available online 7 February 2014
Keywords:
Microorganism
Transport
Preferential flow
Dual-permeability modela b s t r a c t
Dual-permeability models are increasingly used to quantify the transport of solutes andmicroorganisms in soils with preferential flow. An ability to accurately determine the model
parameters and their variation with preferential pathway characteristics is crucial for
predicting the transport of microorganisms in the field. The dual-permeability model with
optimized parameters was able to accurately describe the transport of E. coli D21g in columns
with artificial macropores of different configurations and lengths at two ionic strength levels(1 and 20 mM NaCl). Correlations between the model parameters and the structural geometryof the preferential flow path were subsequently investigated. Decreasing the macropore length
produced a decrease in the apparent saturated hydraulic conductivity of the macropore
domain and an increase in the mass transfer between the macropore and matrix domains.
The mass transfer coefficient was also found to be dependent on the configuration of thepreferential flow pathway. A linear superposition approach was used to estimate field-scale
preferential transport behavior for hypothetical fields with different amounts and configurations of macropores. Upscaling procedures were numerically investigated to predict thisfield-scale transport behavior from column-scale parameters. The upscaling method provided
a satisfactory prediction of the field results under the tested scenarios. This information will be
useful in assessing the risks of microbial transport due to preferential flow.
© 2014 Elsevier B.V. All rights reserved.

1. Introduction
Surface and groundwater contamination by pathogenic
microorganisms has been reported across the United States,
which put the public's health at risk (Abbaszadegan et al.,
2003; Borchardt et al., 2003; USEPA, 1997). Water entering
groundwater has to pass through porous media in the vadoseand saturated zones in the subsurface. Soil passage, such as
riverbank filtration, dune recharge, infiltration basins andtrenches, and sand filters, is also used to treat surface water tomeet demands for drinking water (Schijven and Hassanizadeh,2000). Therefore, an accurate understanding of the transport and⁎ Corresponding author. Tel.: +1 951 823 9431.
E-mail address: Ywang032@ucr.edu (Y. Wang).
0169-7722/$ – see front matter © 2014 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.jconhyd.2014.01.009

fate of pathogens in porous media is needed to protect our
drinking water resources.
Many studies have been conducted to quantify the influenceof physical (size of the microbe and the porous medium, microbe
concentration, water velocity, water content and surface roughness) and chemical (surface chemistry of the microbe and soil,
and aqueous solution pH, ionic strength, and chemical composition) factors on microorganism transport in homogeneous
porous media (Bradford et al., 2006; Chen and Walker, 2007;
Dong et al., 2002; Hendry et al., 1999; Mccaulou et al., 1995; Mills
et al., 1994; Yee et al., 2000). However, field experiments havefrequently revealed that preferential pathways are a major
contributor to the overall transport of microbes because they are
typically strongly retained in the soil matrix (Abu-Ashour et al.,
1994; Bales et al., 1989; Jiang et al., 2010).

58

Y. Wang et al. / Journal of Contaminant Hydrology 159 (2014) 57–66

The occurrence of preferential flow is found to be the rulerather than the exception in most field soils (Flury et al.,
1994; Singh and Kanwar, 1991), which may occur as a resultof plant roots, burrowing earthworms, cracks, or natural
structural heterogeneities (Beven and Germann, 1982; Cey etal., 2009; Madsen and Alexander, 1982; Unc and Goss, 2003;
Wollum and Cassel, 1978). Preferential flow has been
identified as a major problem hampering accurate predictions of contaminant transport in soils because of difficulty inquantifying the physical and chemical complexities of the soil
matrix and macropore system (Šimůnek et al., 2003).
The study of preferential flow and transport throughartificial macropores provides an opportunity to overcome
many modeling challenges in natural soils because themacropore geometry and hydraulic properties can be welldefined and controlled in a repeatable manner (Arora et al.,2011, 2012; Castiglione et al., 2003; Guzman et al., 2009;
Pivetz and Steenhuis, 1995). Artificial macropores can besystematically created by leaving small cylindrical openings
in repacked columns (Castiglione et al., 2003; Pivetz andSteenhuis, 1995) or by packing different sized sands togenerate layers and/or lens with contrasting hydraulicconductivities (Bradford et al., 2004; Fontes et al., 1991;
Morley et al., 1998; Saiers et al., 1994; Wang et al., 2013,2014). Controlled preferential flow and transport studies
make it possible to isolate and identify factors that have thegreatest influence on preferential flow and transport, andprovide valuable information on parameterization that isneeded to help adapt numerical models to complex natural
systems.
Several approaches have been developed to simulate
preferential flow and transport (Köhne et al., 2009a,b). One
method is to construct a 2D/3D simulation profile byexplicitly defining the heterogeneities or the preferential
pathways of the domain (e.g., Schelle et al., 2013). Wang etal. (2013, 2014) successfully applied Hydrus 2D/3D tosimulate microorganism transport in columns with artificial
macropores. A map of the structural geometry must be
known in order to conduct the simulation. Some lab-scale
mapping methods have been developed to accurately
measure pore sizes and connectivity of macropore-type
preferential pathways (Anderson et al., 2003; Monga et al.,2007), but this is extremely difficult, if not impossible, in thefield.
Other physically based models, such as dual-porosity anddual-permeability models, mostly divide the soil into two or
multiple domains, and assume uniform flow within each flow
domain. Both dual-porosity and dual-permeability models
assume that the porous medium consists of two interacting
regions, one associated with the inter-aggregate, macropore,or fracture system, and one comprising intra-aggregate pores
inside soil aggregates. While dual-porosity models assume
that water in the matrix is stagnant, dual-permeability
models allow for water flow in the matrix as well. The mass
exchange between the two regions normally is assumed to belinear (Leij and Bradford, 2009). Dual-permeability models
are increasingly used for analysis of preferential transport
both on the laboratory column-scale (e.g., Allaire et al., 2002;
Allaire-Leung et al., 2000; Castiglione et al., 2003; Greco,
2002; Gwo et al., 1995, 1996) and on the plot- or field-scale(e.g., Andreu et al., 1994; Jarvis et al., 1991; Kohler et al.,

2001; Larsson and Jarvis, 1999a,b). However, the difficulty indetermination of the many dual-permeability model parameters largely limits the application of these models (Köhne etal., 2009a,b).
Stochastic models have also been employed to simulate
preferential flow and transport in soils. A variety of stochastic
methods have been used to describe solute transport inheterogeneous flow fields (e.g., Dagan, 1984; Freyberg, 1986;
Gelhar and Axness, 1983; Kabala and Sposito, 1991; Russo,
1991; Sposito and Barry, 1987). In a simplified stochastic
modeling approach the field may be viewed as a series of
independent stream tubes with their hydraulic conductivity
and/or other selected parameters described using probability
density functions (PDFs) (Dagan, 1993; Jury and Roth, 1990).
However, no methodology has been developed to directly
characterize the PDF, especially when preferential flow pathways are presented. Kung et al. (2005) proposed an indirect
method to determine pore-size spectrum of macropore-type
preferential pathways, but quantification of PDFs remains achallenge for the application of stochastic models to preferential flow and transport.
The objectives of this study were to link the parameters ofthe dual-permeability model with geometry information of
the macropore, and then investigate methods to upscale
behavior from the column- to field-scales. In particular, thedual-permeability model was employed to simulate bromide
and microorganism transport in columns with preferential
pathways of different lengths and configurations. Correlations were then established between macropore geometry
information (length, configuration) and parameters of thedual-permeability model. A linear superposition approach
was used to estimate field-scale preferential transport
behavior for hypothetical PDFs. Upscaling procedures werenumerically investigated to predict the field-scale transport
behavior.

2. Experiment information
Transport experiments for E. coli D21g were conducted incolumns with artificial preferential flow pathways as described by Wang et al. (2013, 2014). Detail information about
the column experiments can be found in these publications,and will be briefly summarized below. Two types of Ottawa(quartz) sand with the median grain size of 120 and 710 μm(referred to as fine sand and coarse sand below) were used in
the column experiments after being cleaned using a saltcleaning method (Bradford and Kim, 2010) to eliminate any
background interference from clay particles. A 13.2 cm
diameter by 22 cm long plexiglass column with an 18 μm
nominal pore size polyester membrane at the bottom wasused in the experiment. The fine sand was wet packed in thecolumn with a plastic tube (with outside diameter of
1.14 cm) in the center. Then the coarse sand and fine sandwere filled into the hole created by pulling out the plastic
tube from the column to make the desired preferential flow
path as described in Fig. 1. Various NaCl and NaBr solutions atselected ionic strength (IS = 1 and 20 mM) with pH = 5.8
were prepared for the experiment. These IS levels wereselected to create a range of adhesive conditions between the
microorganisms and sand.

Y. Wang et al. / Journal of Contaminant Hydrology 159 (2014) 57–66

Type 0Type I

Type II

Type III

59

Type IV

Fig. 1. Axi-symmetric representation of the five types of lens structures (column center is on the left hand side) studied in this research: Type 0—homogeneous
fine sand column with no lens, Type I—one lens through the whole column, Type II—one lens opened to the bottom boundary, Type III—one lens opened to the top
boundary, and Type IV—a discontinuous lens opened to both boundaries (diagonal pattern represents coarse sand and point pattern represents fine sand).

Solutions were delivered onto the top of the column at a
steady flow rate using a rain simulator, and the bottom
boundary pressure was maintained at 0 cm by a hanging
water column. The column was kept saturated during theexperiment by maintaining a shallow depth (b5 mm) of
ponding at the surface boundary. The sand in the column was
equilibrated by flushing the column with two pore volumes(PVs) of a selected NaCl solution (phase 0) before initiatinga microbial transport experiment. Then, several PVs of
NaBr solution containing E. coli D21g at a concentration of∼1.0 × 108 cells/mL were introduced into the column at aconstant rate and IS (phase I), followed by NaCl solution at thesame flow rate and IS as in phase I until the effluent microbe
concentration returned to a baseline level (phase II). Effluent
samples were continuously collected during the transport
experiment at selected intervals using a fraction collector, andthen analyzed for Br and microbe concentrations.
3. Numerical modeling
Dual-permeability models describe preferential flowusing one equation for flow in the matrix and one for flowin the macropore system, with the two flow regions coupled
by a water transfer term:
!
∂h f
∂
Γ
¼
−K f − w
Kf
wf
∂t
∂z
∂z

∂θ f


∂θm
∂
∂h
Γw
¼
K m m −K m −
1−w f
∂t
∂z
∂z

ð1aÞ

macropore and matrix system, hf –hm. Thus, Γw equals 0 undersaturated, steady-state flow conditions.
Modified convection–dispersion equations were used todescribe transport and deposition of microorganisms in boththe macropore and matrix regions:∂ θf C f∂t

ð1bÞ

where the subscript “m” represents the matrix, the subscript“f ” defines a property of the faster macropore system, θ [−] isthe water content, h is the pressure head [L, L denotes theunits of length], K denotes the saturated hydraulic conductivity [LT−1, T denotes the units of time], wf is the fraction of
total soil occupied by the macropore system (0 b wf b 1), andΓw is the water transfer term (T−1). Γw is assumed to be
proportional to the pressure head difference between the

∂J f
∂z

−E f −Γswf∂ðθm C m Þ∂JΓs¼ − m −Em −
1−w f
∂z
∂t

ð2aÞ

ð2bÞ

where C [NcL−3; Nc denotes the number of microbes] is the
microbe concentration in the aqueous phase, J [NcL−2 T−1] isthe microbe flux (sum of the advective and dispersive fluxes),
the mass transfer rate between the macropore and matrix
regions Γs is given as

 

#
Γ s ¼ kfm 1−w f θm C f −C m þ Γ w C

ð3Þ

where kfm is a first-order diffusive mass transfer coefficient(T− 1), C# is equal to Cm for Γw N 0 and Cf for Γw b 0, and E is themass transfer function from the aqueous phase to/from thesolid–water interface as:
E¼



¼−

∂ðρSÞ
¼ θψkatt C−ρkdet S∂tð4Þ

where S [NcM−1; M denotes units of mass of soil] is the microbe
concentration on the solid phase, kdet [T−1] is the microbe
detachment rate coefficient, katt [T−1] is the microbe attachment rate coefficient, and ρ [ML−3] is the bulk density. The
parameter ψ [-] accounts for time and concentration dependent
blocking using a Langmuirian approach as (Adamczyk et al.,1994):
ψ ¼ 1−

S
Smax

ð5Þ

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Y. Wang et al. / Journal of Contaminant Hydrology 159 (2014) 57–66

where Smax [Nc M−1] is the maximum solid phase concentration of microbes.
The simulations were carried out with the HYDRUS-1D
code (Šimůnek et al., 2008). A constant positive pressure
head was applied at the column inlet as a shallow depth of
ponding was maintained at the surface of the column, and aconstant zero pressure head was employed at the outlet for
the water flow boundary. A third-type boundary condition
was employed at the column inlet and a zero concentration
gradient was used at the outlet for these simulations. Someof the model parameters were determined experimentallyor using a nonlinear least squares optimization routine in
HYDRUS-1D by the same method as described in ourprevious studies (Wang et al., 2013, 2014). In brief, Km, θm,
θf, and wf were directly measured, and retention parameters(katt, kdet, and Smax) were optimized in HYDRUS-1D with datafrom homogeneous column experiments. Some other parameters (e.g., Kf and kfm) were also determined using a nonlinear
least squares optimization routine in HYDRUS-1D.
4. Results and discussion
Results from the transport experiments for E. coli D21g incolumns with preferential flow paths of different lengths andconfigurations (Wang et al., 2013, 2014) were used toinvestigate the relationship between the geometry propertiesof the preferential path and parameters of the dual-permeability
model. Breakthrough data of bromide was first used tocharacterize parameters for water flow before looking into thetransport of E. coli D21g.
4.1. Transport of bromide
The transport of bromide depended on the following
parameters during saturated, steady-state flow conditions:saturated hydraulic conductivities of the matrix (Km) and themacropore (Kf), saturated water content of the matrix (θm)
and the macropore (θf), dispersivities of the matrix (λm) andthe macropore (λf), the ratio of the volumes of the macropore
domain and the total column system (wf), and the masstransfer rate at the interface between matrix and macropores(kfm). The saturated hydraulic conductivity of the matrix, Km,
was considered as a constant for all cases based on the value inthe homogeneous column. However, due to variation caused bypacking, Km was determined by the arrival time of the bromide
pulse from the matrix for each column experiment (Wang et al.,2013). We set the dispersivities of the matrix and the macropore
to the values obtained from homogeneous column experimentsof fine sand and coarse sand, respectively (Wang et al., 2013).
The porosities of both matrix and macropore were set tothe measured values in homogeneous column experiments.
The parameter wf was calculated based on the volume of themacropore and the total column with Type I lens configuration(cf. Fig. 1), and the same value was used for other lenses withdifferent lengths and configurations. The values of theseparameters are listed in Table 1.
Since Km and wf were set as constants, Kf was the main
parameter which determined the relative concentration ofbromide in the first pulse that was associated with preferential
transport. When the length of the lens decreased, the relative
contribution of preferential transport decreased and arrival
time of the preferential transport was delayed. Consequently,
Kf decreased with a decrease in the length of the lens for all
three types of lens configurations (Table 1). For the same lens
length, the value of Kf for the Type IV lens configuration was
smaller than that of Type II and Type III. This was consistent
with the observation that the Type IV lens configuration had alater arrival and a smaller amount of preferential transportcompared to Type II and Type III lens configurations.
Larger dispersion was reported for Type IV lens configuration compared to Type II and Type III configurations (Wanget al., 2014), which indicated greater mixing at the interface
between the matrix and the macropore domains. The greater
mixing at the interface for Type IV lens configuration
was reflected by the larger mass transfer coefficient in the
dual-permeability model as shown in Table 1. The masstransfer coefficient for Type IV configuration was almost
independent of the length of the lens. However, the masstransfer coefficient increased with a decrease in the lengthfor Type II and Type III lens configurations. Fig. 2 shows
selected examples of observed BTCs for bromide andcorresponding simulations with parameters in Table 1.
Table 1 indicates that the macropore geometry had a great
effect on both the hydraulic conductivity of the macropore(Kf) and the mass transfer coefficient between the macropore
and matrix (kfm). Recall that the experimental macropore
consisted of a specific length and location of coarse textured
sand lenses that were embedded in the center of the matrix
sand. Conversely, the simulated macropore geometry in the
dual permeability model was the same for all configurations,and was equal to the column length. Consequently, thehydraulic properties of the simulated macropore changed

Table 1Model parameters obtained from previous studies and optimized saturated conductivities of the fracture, Kf, and mass transfer coefficients for bromide, kfm, andtheir 95% confidence intervals.
Lens configuration

Lens length (cm)Km (cm/min)θmθfλm (cm)λf (cm)wf

Kf (cm/min)

kfm (min−1)

Type I
Type II
Type II
Type II
Type II
Type III
Type III
Type IV
Type IV
Type IV

20
19
18
16
10
19
16
19
18
16

0.31
0.32
0.30
0.32
0.34
0.32
0.32
0.32
0.32
0.32

0.36
0.36
0.36
0.36
0.36
0.36
0.36
0.36
0.36
0.36

0.36
0.36
0.36
0.36
0.36
0.36
0.36
0.36
0.36
0.36

0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10

0.55
0.55
0.55
0.55
0.55
0.55
0.55
0.55
0.55
0.55

0.00746
0.00746
0.00746
0.00746
0.00746
0.00746
0.00746
0.00746
0.00746
0.00746

10.9 ± 0.5
8.3 ± 0.7
7.8 ± 0.7
6.2 ± 0.7
4.0 ± 0.7
9.7 ± 0.9
6.7 ± 1.1
8.1 ± 0.8
7.1 ± 0.8
4.2 ± 0.7

(1.40 ± 0.7) ∗ 10−4
(1.37 ± 1.2) ∗ 10−4
(2.35 ± 1.0) ∗ 10−4
(2.82 ± 1.5) ∗ 10−4
(5.91 ± 3.1) ∗ 10−4
(1.60 ± 1.0) ∗ 10−4
(2.86 ± 2.0) ∗ 10−4
(5.80 ± 1.7) ∗ 10−4
(5.40 ± 2.1) ∗ 10−4
(5.27 ± 2.3) ∗ 10−4

Y. Wang et al. / Journal of Contaminant Hydrology 159 (2014) 57–66

1.2
1

C/C0

At low IS (1 mM), when there was little retention of E. coli
D21g in both the matrix and lens, forward simulations had alater arrival time for transport in the matrix and a lower
concentration for transport in the preferential path. Thisphenomenon was attributed to size exclusion by Wang et al.
(2014). The saturated water content of the macropore (θf)
and matrix (θm) were therefore optimized to account for the
influence of size exclusion on the transport data of E. coli
D21g at low IS. The mass transfer coefficient kfm was also
optimized, and the retention parameters were obtained fromhomogeneous experiments. A summary of model parameters
is given in Table 2. Smaller θm and θf were obtained from
E. coli D21g compared to bromide. This observation isconsistent with the assumption that size exclusion increased
the transport velocity of E. coli D21g by constraining cells tofaster flow domains and larger pore networks than bromide(Wang et al., 2014). The mass transfer coefficients of E. coli
D21g at low IS were quite close to that of bromide for thesame lens configuration. Fig. 3 shows an example of observed
BTCs for E. coli D21g at IS = 1 mM and simulations withparameters in Tables 1 and 2.
At high IS (20 mM), forward simulations tended to havehigher concentrations for the breakthrough from the preferential path. Table 2 presents the mass transfer coefficient
kfm that was optimized to better describe the transport ofE. coli D21g at high IS, and the retention parameters thatwere obtained from the homogeneous experiments. The
mass transfer coefficients for E. coli D21g at high IS weresignificantly greater than that for bromide for the same lensconfiguration and length. Similar to the trend with bromide,
the mass transfer coefficient increased with a decrease in thelens length, and the Type IV lens configuration had a greatertransfer rate than Type II and Type III lens configurations of
the same length. Examples of observed BTCs for E. coli D21g
at IS = 20 mM and simulations with parameters in Tables 1
and 2 are given in Fig. 4.

Type II 19 cm
Type II 19 cm model

0.8

Type IV 16 cm

0.6Type IV 16 cm model

0.4
0.2
0
0

10

20

30

40

50

60

70

61

80

Time (min)Fig. 2. Observed and simulated BTCs of bromide from columns of Type II
19 cm and Type IV 16 cm.

with the experimental length and location of the macropore.
In particular, the value of Kf decreased with decreasing lengthof the experimental macropore because greater amountsof the flow occurred in the matrix material which had asmaller hydraulic conductivity. A decrease in lens length also
produced a corresponding increase in kfm. For the same lens
length, the Type IV lens configuration tended to produced
higher values of kfm than the Type II and III lenses, because ofdifferences in the number of interruptions (lens–matrix
interfaces) to the preferential flow path (cf., Fig. 1). Eachinterruption tended to slow down water flow and transport,
and enhance the exchange process. This explains why theType IV lens configuration was associated with smaller Kf andlarger kfm, than Type II and III lenses.

4.2. Transport of E. coli D21g
After obtaining the information required to describe waterflow and transport of the conservative tracer, the transport ofE. coli D21g should be well quantified with the retention
parameters of E. coli D21g (microbe attachment rate coefficient, katt; microbe detachment rate coefficient, kdet; maximum
solid phase concentration of microbes, Smax) obtained fromhomogeneous experiments under the same conditions by
Wang et al. (2013) (Table 2). However, forward simulations
based on these parameters had relatively poor agreement withmost of the experimental results.

5. Upscaling approach
5.1. Virtual ﬁeld
In order to investigate the ability of the dual-permeability
model to simulate the transport of E. coli D21g at larger scales,
we generated virtual fields with a distribution of preferential

Table 2Retention parameters for the matrix used in simulations (kdet for the matrix and katt, Smax, and kdet for the macropore were set as 0 as they were extremely small),and optimized saturated water content of the fracture (θf) and matrix (θm), the ratio of saturated conductivities optimized from E. coli D21g to that optimized
from bromide for the fracture and matrix (θm/θ⁎m and θf/θ⁎f, respectively), and mass transfer coefficients, kfm, for E. coli D21g at 1 and 20 mM, and their 95%
confidence intervals.
Lens configuration

Lens length (cm)IS (mM)

katt

Smax

θm

θm/θ⁎mθf

θf/θ⁎fkfm (min−1)

Type II
Type III
Type IV
Type I
Type II
Type II
Type II
Type IV
Type IV
Type IV

19
19
19
20
19
18
16
19
18
16

1
1
1
20
20
20
20
20
20
20

0.00040.00040.0004
0.2
0.2
0.2
0.2
0.2
0.2
0.2

0.01
0.01
0.01
3.5
3.5
3.5
3.5
3.5
3.5
3.5

0.327 ± 0.002.333 ± 0.003.339 ± 0.003
NAa
NAa
NAa
NAa
NAa
NAa
NAa

0.91
0.93
0.94
NAa
NAa
NAa
NAa
NAa
NAa
NAa

0.313 ± 0.03
0.265 ± 0.02
0.263 ± 0.02
NAa
NAa
NAa
NAa
NAa
NAa
NAa

0.87
0.74
0.73
NAa
NAa
NAa
NAa
NAa
NAa
NAa

(1.97 ± 1.2) ∗ 10−4
(2.98 ± 1.1) ∗ 10−4
(5.80 ± 1.6) ∗ 10−4
(4.40 ± 0.38) ∗ 10−4
(5.27 ± 0.62) ∗ 10−4
(1.28 ± 0.15) ∗ 10−3
(2.31 ± 0.89) ∗ 10−3
(6.39 ± 0.58) ∗ 10−4
(2.96 ± 0.61) ∗ 10−3NAaa

Not available due to the low concentration in collected effluent.

62

Y. Wang et al. / Journal of Contaminant Hydrology 159 (2014) 57–66

1

Type IV 19cm

C/C0

0.8
0.6
0.4
0.2
0
0

10

20

30

40

50

60

70

Time (min)Fig. 3. Observed and simulated BTCs of E. coli D21g at IS = 1 mM fromcolumn of Type IV 19 cm.

pathways. The virtual fields consisted of independent “stream
tubes” that were run in parallel (e.g., no interaction between thestream tubes). Transport within each individual stream tube wasconsistent with the results of a particular column experimentwith selected macropore length and configuration, and solution
ionic strength (Figs. 3 and 4; Table 2). The virtual field wascomprised of selected fractions of each column type. Table 3
shows some examples of virtual fields with their fractions, fi, ofeach column unit. The fractions of all column units wererandomly allocated with the same probability to cover a wide
range of possible situations.
The stream tube model does not consider lateral exchange
between stream tubes. This assumption is justified in our virtual
fields because all of the stream tubes have the same matrix
properties and simulations demonstrate that the water flow at

C/C0

0.18

Type I and II

0.16

Type I 20 cm

0.14Type I 20 cm model

0.12

Type II 19 cmnX

f i qi C i ðt Þ
hqi C i ðt Þi i¼1
C ðt Þ ¼
¼
n
X
hqi i
f i qi

0.1

where C⁎(t) is the combined flux averaged effluent concentration of the virtual field at time t; fi is the fraction of unit i; qi is theflow rate of unit i; and Ci(t) is the effluent concentration of unit i
at time t. Another advantage of the stream tube model is that thevariance in C⁎(t) may also be calculated as (Toride et al., 1995):n
X

Type II 18 cm model

0.06

Type II 16 cm

0.04

Type II 16 cm model

0
0

10

20

30

40

50

60

70

Time (min)0.12Type IV
Type IV 19 cm

0.1

Type IV 19 cm model

0.08

Type IV 18 cm
Type IV 18 cm model

0.06
0.04
0.02
0
0

10

20

30

40

50

Time (min)Fig. 4. Observed and simulated BTCs of E. coli D21g at IS = 20 mM.

2

f i ðqi C i ðt Þ−hqi C i ðt ÞiÞ2

σ ¼ i¼1Type II 18 cm

0.08

60

ð6Þ

i¼1

Type II 19 cm model

0.02

C/C0

the outside boundary of the stream tube is essentially 1D in thevertical direction (Wang et al., 2013). In this work, we use thedual-permeability model to account for flow and transport
within a given stream tube (column). The dual-permeability
model considers lateral exchange between the matrix andmacropore and two advective fluxes within a given stream
tube (column); see Eqs. (1a)–(3). Consequently, the inclusion of
the dual-permeability model into the stream tube method is anadvantage because it provides one way to account for lateralexchange, as well as variations in advection.
Simulations were carried out for each column unit with IS
equaled to 1 and 20 mM and the same boundary conditions
as for the laboratory experiments. The duration of thebacterial input suspension was 24 min to be consistent withthe laboratory experiments. The model parameters for eachcolumn unit were given in Tables 1 and 2. The saturated
hydraulic conductivity of the macropore domain was set
equal to that of the fine sand matrix in the homogeneouscolumn (Type 0), so there is no preferential flow path in thisunit. The mean solute concentration at the bottom boundary
for an entire virtual field is given by the ensemble average ofthe local concentrations in all stream tubes as described by
the following equation:

hqi i2

ð7Þ

where σ is the standard deviation.
After obtaining the BTC of a virtual field with Eq. (6), we
estimated a new set of dual-permeability model parameters⁎ were fitted to the field
for the virtual field. Only Kf⁎ and kfm
BTC as all other parameters were the same for all units(Table 3). Figs. 5 and 6 show the ensemble and simulated
BTCs of the virtual fields listed in Table 3 at IS equal to 1 and20 mM, and standard deviation for each BTC. At IS = 1 mM,
the ensemble BTCs were very similar to the typical BTC fromlaboratory experiments as shown in Fig. 3, and differences
between the virtual fields were very small. Similar to what
has been observed in transport experiments conducted at
IS = 20 mM in Fig. 4, early breakthrough from the preferential path was observed while no significant breakthroughcame from the matrix. The amount of breakthrough from thepreferential path was positively correlated with the fractionsof units with long lenses (Type I, Type II 19 cm, and Type IV
19 cm) as these units could transport more cells than theothers.

63

It should be emphasized that virtual fields are a simplified
representation of the natural environment. Nevertheless, thestream tube model has often been used to approximate flow
and transport at the field scale (e.g., Dagan, 1984; Freyberg,
1986; Gelhar and Axness, 1983; Kabala and Sposito, 1991;
Russo, 1991; Sposito and Barry, 1987) and available experimental data are generally consistent with this approach(Dean and Foran, 1992; Evans and Owens, 1972; Guzman etal., 2009). Furthermore, the estimated mean and variance inmicrobial concentrations are needed for risk assessment.
Consequently, the outlined approach provides a relatively
simple means to account for observed preferential flow
and transport processes in virtual fields. Application of ourdual-permeability stream tube approach to real field data
would certainly be a worthwhile objective, but beyond thescope of the current manuscript.
5.2. Parameter upscaling

C/C0⁎ were obtained in the
The field-scale values of K⁎f and kfm
previous section by fitting the dual-permeability model to the
combined effluent concentrations. In this section we investigate
ways to predict these parameters from properties of individual
stream tubes. As the properties of the matrix were identical for
all column units, they were the same for the virtual field as well.
Simple averaging of hydraulic parameters has been used as one
way of upscaling (Fleckenstein and Fogg, 2008; Renard and De

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

Field 5
Field 6
Field 7
Field 8Field 5 Model
Field 5 upcaling

0

10

20

30

40

50

60

Time (min)0.07
Field 50.06

Field 60.05

Field 70.04

Field 8

σ

.21
.21
.07
.22
.03
.02
.24
7.4 ± 0.6
(2.6 ± 0.9) ∗ 10−4
.999
.17
.21
.05
.07
.08
.22
.20
7.1 ± 0.1
(2.9 ± 0.2) ∗ 10−4
1.000
.17
.25
.02
.22
.05
.01
.28
7.1 ± 0.1
(2.4 ± 0.2) ∗ 10−4
1.000
.32
.13
.12
.26
.06
.02
.09
5.8 ± 0.2
(1.6 ± 0.1) ∗ 10−4
.999
.18
.07
.13
.17
.18
.12
.15
7.5 ± 0.5
(9.5 ± 1.2) ∗ 10−4
.991
.24
.20
.15
.22
.07
.08
.04
8.7 ± 0.3
(7.8 ± 0.8) ∗ 10−4
.997

.22
.05
.07
.14
.18
.11
.23
7.0 ± 0.5
(10.4 ± 1.6) ∗ 10−4
.989

.08
.16
.15
.19
.22
.07
.13
7.9 ± 0.6
(9.0 ± 1.5) ∗ 10−4
.990

Field 7
Field 6
Field 5
Field 4
Field 3
Field 2

0.03
0.02
0.01

Type 0Type I
Type II 19 cm
Type II 18 cm
Type II 16 cm
Type IV 19 cm
Type IV 18 cm
K⁎f (cm/min)k⁎fm (min−1)
R2

0

fi

Field 1

IS = 20 mM

Table 3
Examples of virtual fields with their fractions (fi) for each unit, estimated model parameters, and R2.

IS = 1 mM

Field 8Y. Wang et al. / Journal of Contaminant Hydrology 159 (2014) 57–66

0

10

20

30

40

50

60

Time (min)Fig. 5. BTCs of E. coli D21g at 1 mM from virtual fields (calculated using Eq. (6)),
and simulations with inversely estimated (model) and upscaled (upscaling; using
Eqs. (9) and (11)) model parameters, and standard deviation σ of BTCs fromvirtual fields as a function of time. Not all simulations are plotted as they mimicvirtual field data.

64

Y. Wang et al. / Journal of Contaminant Hydrology 159 (2014) 57–66

0.12

Field 1
Field 1 model
Field 1 upscaling
Field 2
Field 2 model
Field 2 upscaling
Field 3
Field 3 model
Field 3 upscaling
Field 4
Field 4 model
Field 4 upscaling

0.1

C/C0

0.08
0.06
0.04
0.02
0
0

10

20

30

40

50

60

Time (min)
0.06
0.05

Field 1

0.04

Field 2σField 30.03

Field 4

0.02
0.01
0
0

10

20

30

40

50

60

Time (min)Fig. 6. BTCs of E. coli D21g at 20 mM from virtual fields (calculated using
Eq. (6)) and simulations with inversely estimated (model) and upscaled(upscaling; using Eqs. (9) and (11)) model parameters, and standard
deviation σ of BTCs from virtual fields as a function of time.

Marsily, 1997; Wen and Gómez-Hernández, 1996). Different
techniques for averaging relevant parameters can be applied,such as arithmetic, geometrical, and harmonic means. Several
methods were investigated to predict the upscaled parametersK⁎f and k⁎fm for the virtual fields as follows:nX

f i K f ;iK f  ¼ i¼1n
X

ð8Þ
fi

i¼1
n
X

K f  ¼ i¼1nX

2

f i K f ;ið9Þ
f i K f ;i

i¼1
n
X

f i kfm;ikfm  ¼ i¼1 n
X

ð10Þ
fi

i¼1

n
X

kfm  ¼

i¼1
n
X
i¼1

fi
fi

:

ð11Þ

kfm;i

Eqs. (8) and (10) assume that an upscaled parameter can be
determined as a simple linear combination of local parameters.
Conversely, Eqs. (9) and (11) assume a nonlinear relationshipbetween field and column-scale parameters. Since the flux
through the macropore and the preferential transport of
microbes are both correlated with Kf, Eq. (9) relates Kf⁎ to a
⁎ is related
flux weighted average of Kf. Eq. (11) considers that kfm
to the harmonic mean of kfm. This relationship seems plausible
because larger values of kfm produce greater mass transfer at thematrix–macropore interface, and thus higher microbe retention
in the matrix and less transport of microbes through themacropore.
Table 4 lists the upscaled model parameters obtained
from Eqs. (8)–(11) for the virtual fields listed in Table 3. The⁎ matchedsimulated BTCs with Eq. (9) for Kf⁎and Eq. (11) for kfm
the BTCs of the virtual fields the best (R2 N 0.98) (Figs. 5 and 6).
Similar upscaling procedures were tested for virtual fields
generated with different hydraulic conductivities for thematrix and macropore, and for various fractions of the totalsoil occupied by the macropore system. Eqs. (9) and (11) again
provided a good description of the field-scale BTCs. Consequently, the upscaled model parameters given by Eqs. (9) and(11) provided a reasonable prediction of the transport of
microorganism in the considered virtual fields.
6. Conclusions
Dual-permeability models were able to describe thetransport of bromide and E. coli D21g at both high and low
IS conditions when optimized parameters were used. However, forward simulation of E. coli D21g based on informationfrom transport experiments with E. coli D21g in homogeneous columns and from transport experiments with bromide in columns with preferential flow could not describe
some features of transport of E. coli D21g at both high andlow IS.
Inverse estimation indicated strong correlations between
the parameters of the dual-permeability model (e.g., hydraulic conductivity of the macropore, mass transfer coefficient)and the configuration and length of the preferential path. In
general, 1) the hydraulic conductivity of the macropore
decreased with a decrease in lens length; 2) the mass transfer
coefficient increased with a decrease in lens length, and TypeIV lens configuration produced greater mass transfer rate;and 3) E. coli D21g had a similar mass transfer coefficient asbromide at low IS, but the mass transfer coefficient was much

Table 4
Upscaled model parameters obtained by different methods.

K⁎f (cm/min)k⁎fm (min−1)

Eq. (7)
Eq. (8)Eq. (9)Eq. (10)

Field 1

Field 2

Field 3

Field 4

6.5
8.8
7.8 ∗ 10−4
9.3 ∗ 10−4

5.9
7.7
14.0 ∗ 10−4
14.6 ∗ 10−4

6.3
7.9
12.5 ∗ 10−4
11.7 ∗ 10−4

7.3
8.3
13.3 ∗ 10−4
9.6 ∗ 10−4

Y. Wang et al. / Journal of Contaminant Hydrology 159 (2014) 57–66

greater for E. coli D21g at high IS compared to a low IS andthat of bromide.
The fitted dual-permeability model parameters that wereobtained at the column-scale were subsequently used to
predict the BTCs of virtual fields are composed of preferential
flow paths of various lengths and configurations using alinear superposition approach. The field-scale BTCs weresuccessfully described using the dual-permeability model.
Methods to predict the upscaled dual-permeability model
parameters from local column parameters were then investigated. Eqs. (9) and (11) provided an accurate prediction of
the transport of E. coli D21g in the virtual fields. However, thestudied cases were still very limited compared to the possible
situations in the field. For instance, the dual-permeability
model predicts only two peaks (one from preferential flow
and one from matrix flow). If the duration of the microorganism application is very short and the arrival times fromdifferent local preferential flow pathways vary a lot, theensemble BTCs for the field could show multiple peaks. In
this case, stochastic models may perform better than the
dual-permeability model. Despite these acknowledged limitations, the dual-permeability model and upscaling approachpresented herein appear to be viable first approximation
to determine the effects of preferential flow on microbial
transport at the field-scale under saturated conditions.

Acknowledgments
This research was supported in part by the USDA, ARS, NP
214, and Multistate Research Funded project W-2188. The
USDA is an equal opportunity provider and employer.

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