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Water Resources Research
RESEARCH ARTICLE
10.1002/2014WR016566
Key Points:
 A model of colloid release withtransients in water saturation
 The model provided a good
description of published data
 Numerical experiments were
performed to understand factors
controlling release

Correspondence to:
S. A. Bradford,
Scott.Bradford@ars.usda.gov
Citation:
Bradford, S. A., Y. Wang, S. Torkzaban,
and J. 
Simůnek (2015), Modeling therelease of E. coli D21g with transientsin water content, Water Resour. Res., 51,
3303–3316, doi:10.1002/
2014WR016566.
Received 17 OCT 2014
Accepted 7 APR 2015
Accepted article online 14 APR 2015
Published online 8 MAY 2015

Modeling the release of E. coli D21g with transients in watercontent2
Scott A. Bradford1, Yusong Wang2, Saeed Torkzaban3, and Jiri Simůnek
1

US Salinity Laboratory, USDA, ARS, Riverside, California, USA, 2Department of Environmental Sciences, University ofCalifornia, Riverside, California, USA, 3CSIRO Land and Water, Glen Osmond, South Australia, Australia

Abstract Transients in water content are well known to mobilize colloids that are retained in the vadosezone. However, there is no consensus on the proper model formulation to simulate colloid release duringdrainage and imbibition. We present a model that relates colloid release to changes in the air-water interfacial area (Aaw) with transients in water content. Colloid release from the solid-water interface (SWI) is modeled in two steps. First, a fraction of the colloids on the SWI partitions to the mobile aqueous phase and airwater interface (AWI) when the Aaw increases during drainage. Second, colloids that are retained on the AWI
or at the air-water-solid triple line are released during imbibition as the AWI is destroyed. The developedmodel was used to describe the release of Escherichia coli D21g during cycles of drainage and imbibition
under various saturation conditions. Simulations provided a reasonable description of experimental D21g
release results. Only two model parameters were optimized to the D21g release data: (i) the cell fractionthat was released from the SWI (fr) and (ii) the cell fraction that partitioned from the SWI to the AWI (fawi).
Numerical simulations indicated that cell release was proportional to fr and the initial amount of retention
on the SWI and AWI. Drainage to a lower water content enhanced cell release, especially during subsequent
imbibition, because more bacteria on the SWI were partitioned to the AWI and/or aqueous phase. Imbibition to a larger water content produced greater colloid release because of higher ﬂow rates, and more
destruction of the AWI (smaller Aaw). Variation in the value of fawi was found to have a pronounced inﬂuenceon the amount of cell release in both drainage and imbibition due to changes in the partitioning of cells
from the SWI to the aqueous phase and the AWI.

1. Introduction
Transients in water content commonly occur in the vadose zone as a result of inﬁltration and redistribution,
evapotranspiration, and ﬂuctuations in the water table height and/or surface water levels (waves, tides, andriver stage). Considerable amounts of research have demonstrated that retained colloids in soils can beremobilized by sudden changes in water content [Saiers et al., 2003; Saiers and Lenhart, 2003; Auset et al.,2005; Gao et al., 2006; Zhuang et al., 2007; Shang et al., 2008; Cheng and Saiers, 2009; Engstrom et al., 2015].
Released colloids may be contaminants of concern such as pathogenic microorganisms [Zhang et al., 2012;
Russell et al., 2012] or nanoparticles [Chen et al., 2008], or can facilitate the transport of many adsorbed contaminants [Cheng and Saiers, 2010]. Consequently, an understanding and ability to predict the release of colloids with transients in water content are needed to protect water resources and human health.

C 2015. American Geophysical Union.
V

All Rights Reserved.

BRADFORD ET AL.

Column-scale studies have been conducted to investigate colloid retention and release during unsaturated
conditions [Cherrey et al., 2003; Gao et al., 2004; Chen and Flury, 2005; Cheng and Saiers, 2009; Shang et al.,2008; Zhuang et al., 2007, 2009]. Colloid retention has commonly been reported to increase with a decreasein water content under steady state unsaturated ﬂow conditions [Wan and Wilson, 1994; Schafer et al., 1998;
Gargiulo et al., 2008; Torkzaban et al., 2006a, b, 2008]. Conversely, a diversity of results has been reported forcolloid release during transients in water saturation, and there is no concensus on the relative importanceof drainage and imbibition. Some studies have attributed colloid release only to imbibition [Russell et al.,2012], whereas others have observed colloid release during both drainage and imbibition [Zhuang et al.,2009; Cheng and Saiers, 2009; Zhang et al., 2012]. Greater colloid release has commonly been observed during imbibition [Saiers and Lenhart, 2003; Auset et al., 2005; Gao et al., 2006; Zhuang et al., 2007], but drainage
has also been reported to produce greater colloid release [Cheng and Saiers, 2009]. These discrepancies are

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likely due to differences in the initial conditions, the saturation dynamics, and variations in colloid and soil
properties (hydrophobicity, size, and shape) and solution chemistry [Aramrak et al., 2014; Wang et al., 2014].
Pore-scale studies have been conducted to better understand mechanisms inﬂuencing the release of colmez-Sua
rez et al., 2001; Gao et al., 2006; Lazouskaya and Jin, 2008,
loids during drainage and imbibition [GoLazouskaya et al., 2011, 2013; Sharma et al., 2008; Zevi et al., 2012; Aramrak et al., 2013]. In water-wet porous
media, air invades the soil pore spaces during water drainage starting from the larger pores then to successively smaller regions of the pore space. The air-water interfacial area (Aaw) increases as receding water ﬁlms
cover drained portions of the solid-water interface (SWI). These water ﬁlms are thin, with reported thickness
ranging from a few nm up to 100 nm [Wan and Tokunaga, 1997; Shang et al., 2009; Chatterjee et al., 2012;
Lazouskaya et al., 2013]. In addition to electrostatic and van der Waals interactions from the SWI and airwater interface (AWI), a strong capillary force will act on retained colloids on the SWI if the water ﬁlm thickness (wf) is smaller than the colloid diameter (dc) [Schafer et al., 1998; Saiers and Lenhart, 2003; Lazouskaya
et al., 2011]. Consequently, colloids that are retained on the SWI experience different forces and torques as
wf decreases during drainage [Lazouskaya et al., 2013]. The colloids will begin to experience interactionenergies arising from both the SWI and AWI as wf decreases, which will alter the force and torque balance
at a particular location, and may mobilize some of the colloids from the SWI to the aqueous phase or partition colloids from the SWI to the AWI [Wang et al., 2014].
Destruction of the AWI occurs as the air-water-solid (AWS) contact line advances from smaller to larger porespaces during water imbibition. This process produces a decrease in Aaw, expansion of water ﬁlms, and elimination of AWS triple lines that may result in mobilization of colloids from the AWI to the aqueous phase[Auset et al., 2005; Gao et al., 2006; Chen et al., 2008; Cheng and Saiers, 2009]. Results demonstrate that colloid release is typically much more efﬁcient during imbibition than drainage [Lazouskaya et al., 2008, 2011;
Aramrak et al., 2011; Wang et al., 2014]. The amount of colloid release during imbibition has been reported
mez-Suarez et al., 2001; Aramrak et al., 2013], the water ﬁlm
to depend on the colloid size and shape [Gomez-Suarez et al., 2001], the velocity of the
thickness [Wan and Tokunaga, 1997], the surface tension [Goadvancing AWI [Saiers and Lenhart, 2003], and the hydrophobicity of the solid surface and colloid [Lazouskaya et al., 2013].
The above information strongly indicates that the AWS triple line and Aaw play critical roles in colloid releaseduring transient water content conditions. Conversely, existing mathematical models that simulate colloid
release during drainage and imbibition do not explicitly consider all relevant exchange process and interfaces. For example, the models of Saiers and Lenhart [2003], Cheng and Saiers [2009], and Russell et al. [2012]
only consider release of colloids from the SWI to the aqueous phase. In this case, the kinetic release rate is
assumed to be proportional to the pore water velocity [Saiers and Lenhart, 2003] or temporal changes in thewater content during imbibition [Russell et al., 2012] or drainage and imbibition [Cheng and Saiers, 2009].
However, these models did not consider colloid release from the SWI to the AWI, accumulation of colloidson the AWI, or release of colloids from the AWI to the aqueous phase. Zhang et al. [2012] employed a linear
equilibrium model to account for virus interactions at the AWI. The retardation coefﬁcient was a function
Aaw that increased as the water content decreased. However, this model did not explicitly consider colloid
release from the SWI to the AWI.
The objective of this research is to improve our ability to predict colloid release with transients in water saturation. In particular, we have developed a model that explicitly considers colloid release from the SWI tothe aqueous phase and the AWI, colloid accumulation at the AWI, and colloid release from the AWI to theaqueous phase as a result of temporal changes in the Aaw during transient water content conditions. The
developed model was subsequently used to simulate colloid release data from Wang et al. [2014] over awide range of water saturation conditions. Results provide valuable insight on the effects of water saturation dynamics on colloid release.

2. Experimental Information
Wang et al. [2014] presented breakthrough and release curves for Escherichia coli D21g under transientwater saturation conditions. Experimental details are given in this publication, but will be brieﬂy highlighted
below. Ottawa sand that is 120 mm in size was cleaned using a salt cleaning method [Bradford and Kim,
2010] to remove trace amounts of clay. A 5 mM NaCl solution at pH 5 5.8 was prepared for the resident,

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Table 1. Summary of Experimental Conditions for Breakthrough (Phases 1 and 2) and Release (Phase 3) Experimentsa
Figure #Phase

Cell Suspension PVs

2
2
2
3
3
3
4a
4a
4a
4a
4a
4a
4a
4a
4a
4a
4a
4a
4a
4b
4b
4b
4b
4b
4b
4b

1, 2
3
3
1, 2
3
3
1, 2
3
3
3
3
3
3
3
3
3
3
3
3
1, 2
3
3
3
3
3
3

2.2

a

Cycle

Step

1
1

I1
I2

1
1

D1
I1

1
1
1
2
2
2
3
3
3
4
4
4

D1
I1
I2
D2
I3
I4
D3
I5
I6
D4
I7
I8

1
1
2
2
3
3

D1
I1
D2
I2
D3
I3

Ending Saturation

Top Flux (mL/min)

Bottom Pressure (cm)

0.64
0.71
0.82
1
0.36
0.71
1
0.34
0.71
0.83
0.35
0.71
0.82
0.34
0.69
0.80
0.35
0.71
0.82
1
0.57
0.83
0.36
0.86
0.27
0.86

6.5
21.3
39.1
45.6
0
24.3
42.6
0
23.5
41.2
0
23.5
41.2
0
23.5
41.2
0
23.5
41.2
42.7
0
42.7
0
42.7
0
42.7

290
290
290
0
290
290
0
290
290
290
290
290
290
290
290
290
290
290
290
0
290
290
290
290
2100
2100

1.6

3.1

1.6

D# denotes drainage number (#); I# denotes imbibition number (#).

tracer, and eluting solutions during the transport experiments. The inﬂuent cell concentration (Co) was
equal to approximately 108 cells mL21. Zeta potentials of the sand and cells, and the corresponding interaction energy for this solution chemistry were calculated.
The sand was wet packed into a 13.2 cm diameter by 22 cm long column to a height of 20 cm. The porosity(e) of the packed column was about 0.34. Several pore volumes (PVs) of 5 mM NaCl solution were pumped
to the top of the column at a steady Darcy velocity to allow the sand to equilibrate with the solution. A step
pulse of E. coli D21g suspension was then injected at the top of the column (Phase 1) followed by continued
ﬂushing with the cell-free eluting solution at the same velocity and solution chemistry (Phase 2). Saturated
conditions were maintained in the column by keeping a small layer of ponding at the column top and azero pressure head at the bottom boundary. Steady state water ﬂow conditions at a selected water saturation were achieved by reducing the inﬂow rate at the column top and increasing the suction at the bottom
boundary with a hanging water column until a unit hydraulic head gradient was obtained (the column was
equipped with tip tensiometers at depths of 5, 10, and 15 cm from the column top to measure the hydraulichead).
Phase 3 consisted of various drainage and/or imbibition conditions to study the release of D21g. Drainage
was conducted with no ﬂow at the column top while maintaining a negative pressure at the bottom of thecolumn. Imbibition was initiated by increasing the ﬂow rate at the column top and/or increasing the bottom
boundary pressure. Cell concentrations in the column efﬂuent (Phases 1–3) were monitored over time witha spectrophotometer. The total water saturation in the column was continuously monitored with an electronic balance.
Table 1 provides a summary of experimental conditions for D21g breakthrough (Phases 1 and 2) and release(Phase 3) experiments.

3. Mathematical Model
3.1. Water Flow
Variably saturated water ﬂow is described using Richards’ equation [Richards, 1931] in the verticaldirection as

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Table 2. Fitted Parameter Values for the Unsaturated Soil Hydraulic Property Model of van Genuchten [1980], and the Pearson’s Correlation Coefﬁcient (R2)aFigure #hrhsa (cm21)nl

Ks (cm min21)

R2

2
3
4a
4b

0.03
0.00
0.03
0.00

0.34
0.34
0.36
0.34

0.012
0.016
0.013
0.016

2.69
3.25
5.07
3.25

0.00
0.66
0.00
0.66

0.30
0.23
0.31
0.23

1.00
1.00
1.00
1.00

a
hr denotes the residual water content, hs denotes the saturated water content, a denotes the reciprocal of the air entry pressure, ndenotes the pore size distribution parameter, with m 5 1 2 1/n, l denotes the pore-connectivity parameter, and Ks denotes the saturated
hydraulic conductivity.



@h @
@h
@K ðhÞ
5
K ðhÞ
2
;
@t @z
@z
@z

(1)

where z [L; L denotes units of length] is the distance in the vertical direction, t [T; T denotes unit of time] isthe time, h [L3L23] is the volumetric water content, K [LT21] is the unsaturated hydraulic conductivity, and h[L] is the pressure head. The parameters for the unsaturated soil hydraulic properties were obtained byinverse optimization to experimental outﬂow information. Note that these parameters were constant during
subsequent transport and release simulations. Table 2 provides a summary of the soil hydraulic property
parameters, along with the Pearson’s correlation coefﬁcient (R2). The value of R2 was always greater than
0.99. This agreement indicates that the water contents and pore water velocities in the column were accurately captured by the model.
3.2. Colloid Transport in the Aqueous Phase
Colloid transport in the aqueous phase is described using the advective dispersion equation that includes
terms for exchange to/from the SWI and AWI@hw C @
@C
@qw C
(2)
5
hw D
2
2Esw 2Eaw ;
@t
@z
@z
@z
where C [NL23; N denotes number] is the colloid concentration in the aqueous phase, D [L2T21] is thehydrodynamic dispersion coefﬁcient for colloids, qw [LT21] is the Darcy water velocity, Esw [NL23T21] is theexchange term between the aqueous phase and the SWI, Eaw [NL23T21] is the exchange terms between the
aqueous phase and the AWI, and hw [L3L23] is the volumetric water content that is accessible to colloids.
The ﬁrst and second terms on the right-hand side of equation (2) account for the dispersive and advective
ﬂuxes of the colloids, respectively. The dispersivity (D/v; where v [LT21] is the pore water velocity) was set
equal to 0.1 cm based on published tracer results [Wang et al., 2013]. It should be mentioned that hw isassumed to be equal to h in this work. However, it is possible that hw may sometime be less than h becauseof size and/or ion exclusion. Simůnek et al. [2006] discusses the determination of hw for this situation.
3.3. Colloids at the Solid-Water Interface
The solid phase mass balance equation for colloids and Esw (equation (2)) are given as@S
5hw wsw ksw C2qb krs S2Eswa ;@t(3)

Esw 5hw wsw ksw C2qb krs S2ð12fawi ÞEswa ;

(4)

qb

where S [N M21; M denotes unit of mass] is the solid phase colloid concentration, qb [M L23] is the bulk density, ksw [T21] is the retention rate coefﬁcient to the SWI, krs [T21] is the steady state release rate coefﬁcientfrom the SWI, wsw is a dimensionless blocking function on the SWI, Eswa [NL23T21] is the colloid exchange
term from the SWI due to water drainage, and fawi is the fraction of colloids released from the SWI that partitions to the AWI. Note that the right sides of equations (3) and (4) are not equal because only a fraction of
Eswa partitions to the aqueous phase (e.g., 1 2 fawi). As discussed below, the complementary fraction of Eswa
partitions to the AWI (deﬁned by fawi).
The ﬁrst and second terms on the right-hand side of equations (3) and (4) account for colloid retention anddiffusive release, respectively, during Phases 1 and 2. The Langmuir model assumes that wsw is a linear function of S that is given as [Adamczyk et al., 1994]

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Table 3. Summary of Mass Balance Information, Model Parameters, and Pearson’s Correlation Coefﬁcient for Observed and Simulated
Breakthrough Curves (Phases 1 and 2)
Figure #
a

2
3
4a
4b
a

IS (mM)

Mass Recovery (%)

ksw (min21)

krs (min21)

Smax/Co (cm3 g21)kaw (min21)

R2

5
5
5
5

19.3
38.5
33.2
38.2

0.027
0.079
0.089
0.080

0.001
0.002
0.003
0.003

0.606
0.328
0.810
0.280

0.0025

0.98
0.98
0.92
0.98

Water saturation was 0.64.

wsw 512

S
;
Smax

(5)

where Smax [NM21] is the maximum colloid concentration on the SWI. The parameters ksw, krs, and Smax
were obtained by inverse optimization to breakthrough curve (BTC) data under steady state ﬂow and constant saturation conditions (Table 3). This information determines the initial colloid concentration on the
SWI as a function of distance for subsequent release during Phase 3. Values of ksw and krs were set to zero
during Phase 3 because their functional dependency on hw was unknown, and it was not possible to experimentally quantify the simultaneous retention and release separately. Consequently, Eswa reﬂects the net
effects of retention and release on the SWI during Phase 3, but it was equal to zero during Phases 1 and 2.
3.4. Colloids at the Air-Water Interface
In addition to the SWI, colloid retention and release may also occur on the AWI under unsaturated conditions. The mass balance equation for colloids at the AWI and Eaw (equation (2)) may be described as@Aaw C5hw kaw C1fawi Eswa 2Era ;@t(6)

Eaw 5hw kaw C2Era ;(7)

where Aaw [L2L23] is the total air-water interfacial area per unit volume, C [NL22] is the colloid concentration
retained on the AWI, kaw [T21] is the retention rate coefﬁcient on the AWI, and Era [NL23T21] is the exchange
term for colloids on the AWI to the aqueous phase due to destruction of the AWI during imbibition. Notethat the right sides of equations (6) and (7) are not equal because only a fraction of Eswa partitions to the
AWI.
The value of Aaw in equations (6) and (7) accounts for the area of interfaces between bulk ﬂuids (e.g., thespeciﬁc interfacial area) and the area of wetting ﬁlms in drained portions of the pore space. It may be quantiﬁed as [Bradford and Leij, 1997] asð
1 hs
Aaw ðhÞ5
Paw ðhÞdh;
(8)
raw h
where raw [M T22] is the air-water surface tension, Paw [M L21 T22] is the capillary pressure, and hs is the saturated water content. Consequently, information about the capillary pressure-saturation curve for a porous
media can be used to determine Aaw(h). Figure 1 shows a representative plot of Aaw as a function of h forthe sand employed in the study of Wang et al. [2014]. The value of Aaw increases with a decrease in h andapproaches the geometric surface area of the porous media (As, L2L23) as h goes to 0 [Leverett, 1941]. Noteðh Þthat equation (8) indicates that dAdhaw is equal to 2 Paw
raw .
The ﬁrst term on right-hand side of equations (6) and (7) accounts for colloid retention to the AWI during
Phases 1 and 2. The value of kaw is expected to be a linear function of Aaw that is accessible to colloids inthe aqueous phase [Kim et al., 2008]. Diffusive detachment was assumed to be negligible because strong
capillary forces act on colloids at the AWI [Schafer et al., 1998]. Blocking was also neglected because it was
not possible to uniquely determine its effects on both the SWI and AWI, and the observed blocking behavior in Wang et al. [2014] was similar under saturated and unsaturated conditions.
Experimental mass balance information was used to constrain the optimization of SWI (ksw, krs, and Smax)
and AWI (kaw) retention parameters to the BTC under steady state, unsaturated conditions (Table 3). In particular, the amount of retention on the AWI was estimated as the difference in colloid retention in saturated

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700

Aaw (cm2 cm-3)

and unsaturated conditions.
Note that we do not attempt600
to differentiate various mechanisms of colloid retention in
500
unsaturated systems (e.g., ﬁlm
400
straining, attachment to the
AWI, and retention at the AWS
300
triple line). Rather, this informa200
tion was used to determine theinitial colloid concentration
100
that was associated with the0
SWI and AWI as a function of
0
0.1
0.2
0.3
0.4
distanceforsubsequentθrelease during Phase 3. The
value of kaw was set to zero
Figure 1. A representative plot of the air-water interfacial area (Aaw) as a function of water
during Phase 3 for similar reacontent (h) for the sand employed by Wang et al. [2014]. The value of Aaw was calculatedusing equation (8).
sons as ksw and krs that are discussed above. Consequently,
Eswa and Era reﬂect the net effects of retention and release on the SWI and AWI, respectively, during Phase3, but they were equal to zero during Phases 1 and 2.
3.5. Transient Exchange Terms
The above model indicates that values of Eswa (equations (3), (4), and (6)) and Era (equations (6) and (7))
need to be determined in order to simulate the release of colloids during transient water content conditions. Each of these issues will be discussed below.
The amount of colloids that will remain solely on the SWI after drainage is given as
S5Si FND ðhÞ1Si ð12fr Þð12FND ðhÞÞ:

(9)

21

Here Si [N M ] is the initial solid phase concentration of retained colloids before drainage, FND is the fraction of the pore space that contains colloids that has not been drained, and fr is the fraction of colloids thatis released from drained portions of the SWI. The value of fr is expected to be high when dc > wf and theadhesive force is low. Conversely, fr will be much lower in regions with dc < wf or when the adhesive force isstrong. It should be mentioned that Si may vary spatially with depth, the water saturation history, and sizeof the pore space (enhanced retention has been observed in grain-grain contacts and large scale surface
roughness locations). Initially, we assume that retained colloids are uniformly distributed on the SWI at aparticular location, but may vary with depth. Equation (9) indicates that S is directly related to Si anddecreases with a decrease in FND (drainage to a lower water content).
Information about As and Aaw(h) (equation (8)) can be used to determine FND as
FND ðhÞ5

As 2Aaw ðhÞ;
As 2Aaw ðhi Þ(10)

where hi [L3L23] is the initial volumetric water content during colloid deposition. Note that h < hi andAaw(h) > Aaw(hi) during drainage. Equation (10) provides a nonlinear description of FND that decreases from
1 to 0 as h decreases. Alternatively, a simple linear approximation of FND can be obtained from h as
FND 

h
:
hi

(11)

An equilibrium expression for colloid release during drainage can be derived from equation (9) by taking
the partial derivative of S with respect to time. When using the nonlinear description of FND given by equation (10) the value of Eswa is given as






@FND
@h
Si
dAaw @h@h
Ho 2
Eswa 5qb fr Si
5qb fr
Ho 2
:
(12)
@t
As 2Aaw ðhi Þ dh @t@t@t
The value of Eswa for a linear description of FND (equation (11)) is given as

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Table 4. Summary of Model Parameters and Pearson’s Correlation Coefﬁcient for D21g Release Experiments (Phase 3)
Figure #
2
3
3
4a

4b

a

Equation

IS (mM)

Cycle

fr

fawi

R2a

14
12
13
13
13
13
13
13
13
13

5
5
5
5
5
5
5
5
5
5

1
1
1
1
2
3
4
1
2
3

0
1.80
0.92
0.78
0.47
0.21
0.10
0.75
0.75
0.75

0
0.61
0.61
0.62
0.34
0.6
0.07
0.48
0.48
0.48

0.68
0.93
0.93
0.85

0.82

A single value of R2 was determined for all drainage and/or imbibition cycles.

Eswa 5
qb fr Si @h
@h
Ho 2
:
hi @t
@t

(13)

The Heaviside functions in equations (12) and (13) were used to turn release on during drainage. Cheng andSaiers [2009] and Russell et al. [2012] proposed an expression similar to equation (13) to describe colloid
release from the SWI to the aqueous phase during drainage and/or imbibition. Conversely, Eswa was partitioned in this work between the aqueous phase and the AWI during drainage using fawi in equations (4) and(6), respectively. Only the portion of Eswa that enters that aqueous phase can be transported during drainage. Values of fr and fawi were obtained by inverse optimization to the release data during Phase 3 (Table 4).
Colloid mobilization due to a destruction of the AWI is modeled during imbibition as 
dAaw @h@h
Era 52C
Ho
:
dh @t
@t

(14)

If a constant concentration of colloids on the AWI is assumed, then changes in Aaw during imbibition produce release given by equation (14). Entrapped air was not considered in equation (14) because of themany additional complexities and model parameters in hysteretic systems.
Zhang et al. [2012] accounted for release from the AWI to the aqueous phase during imbibition using an
expression that was proportional to @h@t . However, these authors did not consider the fraction of Eswa that partition to the AWI during drainage (equations (12) and (13)), and can be released during subsequent imbibition using equation (14).
Note that colloid release with transients in water content was initiated in equations (12–14) by changes in
Aaw or h with time. The AWS contact line advances as Aaw increases and h decreases during water drainage.
Similarly, destruction of the AWI occurs as the AWS contact line advances from smaller to larger pore spaces@Aawduring water imbibition. Hence, @h@t and @t can be thought of as an approximation for the rate of change inthe triple contact line. The amount of release depends on this rate, as well as the initial conditions (Si inequations (12) and (13), and C in equation (14)) and the removal efﬁciency (fr).
3.6. Solution of Governing Equations
The above equations were implemented into the HYDRUS-1D model [Simůnek et al., 2008]. Richards equation (equation (1)) was solved using time-dependent water ﬂux and pressure head boundary conditions at
the top and bottom boundaries, respectively, and an initial uniform water saturation. The advective dispersion equation (equation (2)) was solved using a time-dependent solute ﬂux boundary condition at the top
boundary, a zero dispersive ﬂux at the bottom boundary, and an initial condition of no colloids. The
HYDRUS-1D model includes a nonlinear least square ﬁtting routine to determine model parameters by optimizing model output to experimental data.

4. Results and Discussion
4.1. Breakthrough Curves
Wang et al. [2014] presented and discussed saturated and unsaturated D21g BTC data under various saturation conditions. Table 3 provides ﬁtted retention model parameters (Phases 1 and 2) and the R2 value for

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I1

1

I2

0.5

0.8

C/Co

0.4

0.6

Data
Model
Sw

0.3
0.2

0.4
0.2

0.1
0.0

0

100

200

300

400

Average Sw

0.6

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the goodness of model ﬁt. The
model description of the BTCs
was very good (R2 > 0.92). The
value of Si was determined at
the end of the steady state,
breakthrough curve simulation,and then used for simulationsof D21g release during Phase 3
that will be discussed below.

0500

4.2. Evaluation of Era
Figure 2 presents observed
and simulated D21g transport
Figure 2. An example of observed [Wang et al., 2014] and simulated D21g transport (Phases
under steady state, unsatu1 and 2) and release (Phase 3) from the AWI during imbibition. Initial deposition occurred
rated ﬂow when the water satunder unsaturated conditions (around 0.64 water saturation), and then the sand was saturated using two imbibition sequences (denoted by I1 and I2). See Tables (1–4) for experiuration was around 0.64 duringmental details and model parameters.
Phases 1 and 2. This ﬁgure also
shows the release behavior
when the sand was subsequently saturated using two imbibition sequences during Phase 3. The releasemodel provided a reasonable description of the observed data for the ﬁrst imbibition step even though norelease model parameters were ﬁtted during Phase 3 (Table 4). Note that the model predicts that only Era(equation (14)) contributed to cell release because Eswa 5 0. Consequently, simulated release only occurredfrom the AWI and was initiated by destruction of the AWI during imbibition. The agreement between thesimulation and experimental release provides a validation of equation (14) during the ﬁrst imbibition step.

Time (min)The release behavior for the second imbibition step was not described as well as the ﬁrst imbibition step. The
model predicts that all of the cells on the AWI will be released when the sand is completely saturated. The
second predicted release pulse is smaller than the ﬁrst because of a smaller change in Aaw (Figure 1). Deviation between the model prediction and experimental data in the second imbibition step occurs for several
reasons. First, the experimental data never go to complete water saturation because of air entrapment. Our
water ﬂow simulation matches this experimental data, and therefore never reaches complete water saturation. Consequently, we also have incomplete removal from the AWI during imbibition in our simulation
results. Second, equation (14) was derived under the assumption of a constant colloid concentration on the
AWI. In reality, the colloid concentration on the AWI may increase during imbibition [Keller and Auset, 2007].
Equation (14) may therefore need to be modiﬁed to account for such second order processes by making C
an increasing function of hw, but this was not attempted due to a lack of detailed experimental information.

D1

1

I1

C/Co

0.8

0.8
Data

0.6

Eq. [12]Eq. [13] 0.4
Sw

0.4
0.2
0.0

0.6

0.2

0

10

20

30

40

Time (min)50

60

Average Sw

1.0

0

Figure 3. An example of observed [Wang et al., 2014] and simulated D21g release during adrainage and imbibition cycle. Phases 1 and 2 were conducted under saturated conditions,and then the sand was drained and imbibed during Phase 3 by adjusting the boundary conditions at the bottom and top of the column. Simulation results are shown for models that
employed equation (12) or equation (13). D# and I# denote the drainage and imbibition
number (#), respectively. See Tables (1–4) for experimental details and model parameters.

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4.3. Evaluation of Eswa
Figure 3 presents observed
and simulated release (Phase3) of D21g during a single
drainage and imbibition cycle.
In this case, Phases 1 and 2
were conducted under saturated conditions, and initialretention of the cells therefore
only occurred on the SWI.
Release from the SWI during
Phase 3 was initiated by drainage when the bottom boundary condition was lowered to
290 cm and the water ﬂux atthe top was set to zero. Release
was also induced during imbibition when the water ﬂux was

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changed from 0 to 24.3 mL min21 at the column top. Two simulations are shown in the ﬁgure. One
employed the nonlinear expression for FND to determine Eswa (equations (10) and (12)), whereas the otherused the linear formulation for FND to quantify Eswa (equations (11) and (13)). Both models predict transfer
of cells from the SWI to the aqueous phase and the AWI during drainage (Eswa in equations (12) and (13) is>0) because of a decrease in FND. Cells that partitioned from the SWI into the aqueous phase were transported with ﬂowing water during drainage (equations (2), (4), (12), and (13)). Conversely, cells that partitioned from the SWI into the AWI were only released when the AWI was destroyed during imbibition(equations (7–14)). Transport of cells with the AWI was not considered in the model (equation (6)) becauseof the lack of experimental information to quantify this process and associated parameters.
The experimental release pulses of D21g were accurately simulated (R2 5 0.93) using both models (equations (12) and (13)) when values of fr and fawi were ﬁtted during Phase 3. The excellent agreement between
the experimental data and simulations provides strong support for the conceptual model employed inequations (1–14). Recall that fr accounts for the total amount of release from the SWI during drainage,
whereas fawi indicates the fraction of cells that partitioned to the AWI. The value of fawi 5 0.61 when using
models based on equation (12) or equation (13) and this indicates that most of the released cells from the
SWI partitioned to the AWI. Conversely, the value of fr was higher when using equation (12) (fr 5 1.8) thanequation (13) (fr 5 0.92). This high value of fr 5 1.8 reﬂects uncertainty in the initial estimate of As (equation(10)). In this case, equation (13) is preferred over equation (12) for simulating the transient release behavior
because both models gave an equal description of the data (R2 5 0.93), and equation (13) only requires
information on readily accessible h and hi values (rather than more difﬁcult quantities of Aaw(h), Aaw(hi), andAs). We therefore employ equation (13) for the remaining simulations presented in this work.
Figure 4a presents observed and simulated D21g release (Phase 3) with repeated cycles of similar amountsof water drainage and imbibition (two steps). The peak efﬂuent concentration in the release pulses
decreased with increasing numbers of drainage and imbibition cycles. This trend reﬂects a decrease in theinitial amount of cells on the SWI at the start of a drainage and imbibition cycle. However, the release model
still did not provide an adequate description of the data when considering only a single value of fr and fawi
(data not shown). Separate values of fr and fawi had to be ﬁtted to each drainage and imbibition cycle inorder to achieve good overall agreement between the data and model (R2 5 0.85). Fitted values of fr (from0.78, to 0.47, to 0.21, and then to 0.10) and fawi (from 0.62, to 0.34, to 0.06, and then to 0.07) rapidly
decreased with increasing number of drainage and imbibition cycles. This decrease in cell release efﬁciency
from the SWI and partitioning to the AWI likely reﬂects variations in cell accessibility and/or the strength ofcell adhesion. All SWIs exhibit roughness to varying extents. Consequently, cells that are retained on an easily accessible region of the grain surface (e.g., a smooth surface) will be more readily removed by the AWI
than cells in less accessible regions (e.g., the pits found on rough surfaces). Nanoscale roughness and/or
chemical heterogeneity may also produce variability in the strength of cell adhesive [Bradford and Torkzaban, 2013]. Cells that exhibit a weaker adhesive force will be easily removed by the ﬁrst drainage and imbibition cycle, whereas successive cycles will remove fewer cells because of their stronger adhesive force.
Figure 4b presents an example of observed and simulated D21g release during three continuous drainage
and imbibition cycles when the column was successively drained to lower water saturations of 0.57, 0.36,
and 0.27. The peak concentration and release amounts were observed to decrease for each drainage andimbibition cycle, but to a lesser extent than that shown in Figure 4a. In further contrast to Figure 4a, only asingle value of fr 5 0.75 and fawi 5 0.48 needed to be ﬁtted to the entire release data set to achieve goodoverall agreement between the data and model (R2 5 0.82). These observations indicate that drainage tolower water saturations promoted greater amounts of cell release from the SWI, with relatively stable valuesof fr and fawi at different water saturations. This may be explained by an increased accessibility to Si at lower
water saturations (equations (9–14)) and a relatively constant water ﬁlm thickness in drained portions of thepore space.
4.4. Numerical Experiments
Numerical experiments were conducted to better highlight the inﬂuence of fr, fawi, and initial and boundary
conditions on cell release. Phases 1 and 2 were conducted under saturated conditions, and the initial retention of the cells therefore only occurred on the SWI. The same values of Si with distance were employed inall simulations (determined using parameter values given in Table 3 for Figure 3). Release was initiated

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D1

1
0.8

C/Co

1.0

0.6
0.4

0.5
(a)
0.0

C/Co

1.0

0

100 200 300 400 500 600 700

D1 I1

0.2
0

Time (min)

Data
D2

I2

Model
D3

Sw
I3

1

0.8

0.8

0.6

0.6

0.4

0.4
(b)

0.2
0.0

0

Average Sw

Data
Model
Sw
I1-2 D2 I3-4 D3 I5-6 D4 I7-8

100

200

300

Time (min)

400

0.2
500

Average Sw

1.5

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0

Figure 4. An example of observed [Wang et al., 2014] and simulated D21g release withrepeated cycles of water drainage and imbibition. Phases 1 and 2 were conducted undersaturated conditions, and then the sand was repeatedly drained and imbibed during Phase
3 by adjusting the boundary conditions at the bottom and top of the column. In Figure 4a
similar amounts of drainage and imbibition occurred with each cycle, whereas in Figure 4b
the column was successively drained to lower water saturations of 0.57, 0.36, and 0.27. D#
and I# denote the drainage and imbibition number (#), respectively. See Tables (1–4) for
experimental details and model parameters.

during Phase 3 by a drainage
and imbibition cycle. In thiscase, the bottom boundary
condition was lowered to
290 cm and the water ﬂux atthe top was set to zero duringdrainage. Unless otherwise
noted, the water ﬂux waschanged from 0 to 24.3 mL
min21 at the column top during imbibition.
Figure 5a presents simulationsof D21g release during a single
drainage and imbibition cycle
when fawi 5 0.5 and values of fr
were varied (1, 0.75, 0.5, and0.25). Equations (12) and (13)indicate that Eswa is proportional to fr. The drainage andimbibition release curves andthe total amount of release
were therefore scaled with fr.
The value of fr is expected to
be strongly dependent on thestrength of the adhesive interaction. Indeed, release pulses
during drainage and imbibition
have been shown to decrease
with an increase in the solution
ionic strength [Cheng andSaiers, 2009; Zhuang et al.,2009; Wang et al., 2014].

The value of Eswa is also proportional to Si (equations (12)and(13)).
Consequently,
increasing values of Si will
scale the drainage and imbibition release curves in a similar manner to fr in Figure 5a. This observation indicates that the amount of colloid release will be highly sensitive to the initial conditions.
Greater values of Si are expected with longer input pulse durations and concentrations, greater adhesive interactions (higher ionic strength), and lower hydrodynamic forces (lower water velocities) [Bradford et al., 2009; Sasidharan et al., 2014]. In addition, ﬁner soil textures are commonly observed to
produce greater values of Si [Liang et al., 2013]. Wang et al. [2014] observed that higher input pulse
durations produced much greater amounts of D21g release during a drainage and imbibition cycle,
and that these effects were more pronounced during imbibition.
Equation (14) indicates the amount of release from the AWI during imbibition (Era) is proportional to theinitial value of C. Consequently, larger initial values of C are expected to produce greater amounts ofrelease during subsequent imbibition events in a similar manner to Figure 5a. Under steady state unsaturated ﬂow conditions the amount of colloid retention has been observed to increase with a decrease in hw[Wan and Wilson, 1994; Schafer et al., 1998; Gargiulo et al., 2008; Torkzaban et al., 2008]. This increase inretention has been attributed to enhanced retention at the AWI and AWS triple line [Schafer et al., 1998;
Auset et al., 2005; Gao et al., 2006; Chen et al., 2008; Cheng and Saiers, 2009]. It is therefore logical to anticipate greater amounts of colloid release will occur from the AWI when colloids are initially deposited atlower hw.

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1

C/Co

I1

D1

0.8
0.6

10.1002/2014WR016566

fr

(a)

0.25

0.4

0.50
0.75

0.2
0

1.00

0

20

40

60

Time (min)
1.50

C/Co

I1

D1

1.25

fawi

1.00
(b)

0.75

1.00
0.75
0.50
0.25
0.00

0.50
0.25
0.00

0

50

100

150

Time (min)Figure 5. Simulated D21g release during a drainage and imbibition cycle for various valuesof fr (fawi 5 0.5 in Figure 5a) and fawi (fr 5 1 in Figure 5b). Phases 1 and 2 were conductedunder saturated conditions, and then the sand was drained and imbibed during Phase 3 byadjusting the boundary conditions at the bottom and top of the column in a similar mannerto Fig. 3. D# and I# denote the drainage and imbibition number (#), respectively.

Figure 5b presents simulationsof D21g release during asingle drainage and imbibition
cycle when the drainage
time was 100 min, fr 5 1 andvalues of fawi were varied (0,0.25, 0.5, 0.75, and 1). The valueof fawi had a dramatic inﬂuenceon the drainage portion of therelease curve, with increasing
fawi producing lower concentrations because less of the D21g
mass partitioned into the aqueous phase. As expected (equation (14)), increases in fawi
caused an increase in D21g
release during imbibition due
to destruction of the AWI (peakrelative concentrations ranged
from 1 to 1.4). However, imbibition also yielded a cell release
pulse even when fawi 5 0 due
to an increase in the water ﬂux
that completed the drainage
release pulse in the aqueous
phase. Apparently, an increasein the water ﬂux and destruction of the AWI produced similarly shaped release pulses
during imbibition.

It should be mentioned thatthe total mass of D21g
released to the aqueous phaseduring the drainage and imbibition cycle was found to decrease with an increase in fawi (Figure 5b). Higher
values of fawi partition more of the released cells from the SWI into the AWI. As mentioned previously (cf.
Figure 2), not all of the cells were released from the AWI during imbibition because the optimized match
between experimental and simulated water outﬂow (Table 1) did not allow for complete water saturation
due to air entrapment. This information suggests that the AWI may act as a sink to retain cells that are
released from the SWI. The above effects of fawi on colloid release are expected to be more pronounced forhydrophobic and/or larger colloids that partition more readily to the AWI. The AWI and fawi are also
expected to play a more important role in ﬁner textured soils because of their larger interfacial area [Bradford and Leij, 1997]. Additional research and model development is needed to fully resolve all of theseissues.
Figure 6a presents simulations of D21g release during a single drainage and imbibition cycle when the
duration of the drainage phase was varied (25, 50, 75, and 100 min). Similar to Figure 4b, increasing theduration of the drainage phase produced lower ﬁnal water contents (average h 5 0.18, 0.15, 0.14, and 0.13).
Greater amounts of D21g release occurred with increasing drainage time because more of the pore space
was drained. However, the concentration of released cells only gradually increased during the drainage
phase because the water ﬂow rate rapidly decreased with decreasing h. The enhancement of cell releasewith drainage time was mainly apparent during subsequent imbibition. In particular, increasing the drainage time produced a systematic increase in the concentration of released cells during subsequent imbibition. This observation indicates a strong sensitivity of release behavior to the drainage time and watercontent, and provides an explanation for differences in the relative importance of drainage and imbibition

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1.2

D1

1.0

C/Co

0.8
0.6

10.1002/2014WR016566

I1
DrainageTime100

release pulses reported in theliterature [Saiers et al., 2003;
Chen and Flury, 2005; Cheng
and Saiers, 2009; Zhuang et al.,2009; Zhang et al., 2012].

75

C/Co

Figure 6b presents simulationsof D21g release during a sin0.4
50
gle drainage and imbibition(a)cycle when the water ﬂux at
0.2
25
the
column
inlet
was
0.0
decreased during the imbibi0
50
100
150
200
tion phase (0.75, 0.5, 0.25, and
0.1 3 24.3 mL min21). Note
Time (min)that the area under the break1.2
through curves in Figure 6b
D1
I1
Imbibition
does not correspond with the
1.0
Flux
release mass, since the ﬂuxes
0.8
0.75x
are different for each curve.
Decreasing the water ﬂux dur0.6
0.50x
(b)
ing the imbibition phase produced lower ﬁnal water
0.4
0.25x
contents (average h 5 0.28,
0.2
0.26, 0.23, and 0.19), but did
0.10x
not inﬂuence the release0.0
behavior during drainage. The
0
100
200
300
400
peak cell release concentraTime (min)tion increased with increasingwater ﬂux because more of
Figure 6. Simulated D21g release during a drainage and imbibition cycle for various values
the AWI was destroyed atof drainage time (25, 50, 75, and 100 min in Figure 6a) and imbibition ﬂuxes (0.1, 0.25, 0.5,
21higher h and this produced
and 0.75 3 24.3 mL min in Figure 6b). Phases 1 and 2 were conducted under saturated
conditions, and then the sand was drained and imbibed during Phase 3 by adjusting thegreater amounts of cellboundary conditions at the bottom and top of the column in a similar manner to Figure 3.
release (equation (14)). ThisD# and I# denote the drainage and imbibition number (#), respectively.
prediction is consistent withthe experimental observations shown in Figure 2, as well as results reported in the literature [Shang et al.,2008].

5. Summary and Conclusions
A novel mathematic model was developed to simulate colloid release during transients in water content
which commonly occur in the vadose zone. The model predicts that a fraction of the retained colloids on the
SWI (fr) can partition to the AWI (fawi) and aqueous phase (1 2 fawi) as h decreases and Aaw increases duringdrainage. Colloid release from the AWI or AWS triple line to the mobile water phase occurred during imbibition as Aaw was destroyed. The colloid release pulse during drainage and imbibition tended to reﬂect theamount of retained colloids that were partitioned from the SWI into the aqueous phase and AWI, respectively.
This model was employed to describe the release of E. coli D21g under transient water content conditions[Wang et al., 2014]. Simulations provided a reasonable description of experimental D21g release duringdrainage and imbibition, and this provides strong support for the implemented conceptual model. Simulation results indicated that the value of fr and fawi decreased with increasing numbers of drainage and imbibition cycles at a similar water content, because remaining cells were less accessible (pore geometry) toreceding water ﬁlms or they were more strongly held to the SWI (heterogeneity). However, repeated cyclesof drainage and imbibition to successively lower water contents produced similar values of fr and fawi due
to decreases in the water ﬁlm thickness and increases in Aaw at lower water contents, and/or enhanced cellretention in smaller regions of the pore space.

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Numerical experiments were conducted to investigate the inﬂuence of model parameters, initial conditions,and saturation dynamics on colloid release. Drainage and imbibition release curves were demonstrated to
be proportional to fr, and the initial amount of retention on the SWI and AWI. Variations in the value of fawi
were found to have a pronounced inﬂuence on both drainage and imbibition release pulses due to changes
in the partitioning of cells from the SWI to the aqueous phase and the AWI. Higher values of fawi produced
lower amounts of total recovery in the aqueous phase because air entrapment led to incomplete removalfrom the AWI. An increase in the drainage time produced lower h and enhanced colloid release, especiallyduring imbibition, because more of the colloids on the SWI were transferred to the AWI and/or aqueous
phase. Decreases in the imbibition ﬂow rate produced lower values of h and smaller peak concentrations in
the imbibition release pulse because of lower ﬂow rates and greater Aaw.
Additional research is warranted to study the effects of transients in water content on colloid release for various soil textures, colloid types and sizes, and colloid hydrophobicity. The model presented in this work is
expected to be a powerful tool to help understand and quantify such release processes.

Acknowledgments
This research was supported by the
214 Manure and Byproduct Utilization
Project of the USDA-ARS. Mention of
trade names and company names inthis manuscript does not imply any
endorsement or preferential treatmentby the USDA. All data can be provided
upon request from the corresponding
author.

BRADFORD ET AL.

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