Environ. Sci. Technol. 2003, 37, 2242-2250

Modeling Colloid Attachment,
Straining, and Exclusion in
Saturated Porous MediaSCOTT A. BRADFORD,* JIRKA SIMUNEK,MEHDI BETTAHAR,MARTINUS TH. VAN GENUCHTEN, AND
SCOTT R. YATES
George E. Brown, Jr., Salinity Laboratory, USDA, ARS,
450 West Big Springs Road, Riverside, California 92507-4617

A conceptual model for colloid transport is developedthat accounts for colloid attachment, straining, and exclusion.
Colloid attachment and detachment is modeled using firstorder rate expressions, whereas straining is describedusing an irreversible first-order straining term that is depth
dependent. Exclusion is modeled by adjusting transportparameters for colloid-accessible pore space. Fitting
attachment and detachment model parameters to colloidtransport data provided a reasonable description of effluent
concentration curves, but the spatial distribution ofretained colloids at the column inlet was severely
underestimated for systems that exhibited significant
colloid mass removal. A more physically realistic descriptionof the colloid transport data was obtained by simulating
both colloid attachment and straining. Fitted straining
coefficients were found to systematically increase withincreasing colloid size and decreasing median grain size.
A correlation was developed to predict the straining
coefficient from colloid and porous medium information.
Numerical experiments indicated that increasing the colloid
excluded volume of the pore space resulted in earlier
breakthrough and higher peak effluent concentrations as
a result of higher pore water velocities and lower residence
times, respectively. Velocity enhancement due to colloid
exclusion was predicted to increase with increasing exclusion
volume and increasing soil gradation.

Introduction
Knowledge of colloid transport and fate is required to assesscontamination potential and to protect drinking watersupplies from pathogenic microorganisms (1), to develop
engineered bioaugmentation and bioremediation strategies(2), to microbially enhance oil recovery (3), and to quantify
colloid-facilitated transport of many organic and inorganic
contaminants (4). Colloid transport theory has been derived
from a combination of convection-dispersion-retardation
theory for solute transport and colloid attachment theory (5,6). Traditional attachment theory employs irreversible firstorder kinetic attachment of colloids (7) that predicts an
exponential decrease in colloid concentration with distancein porous media. Complex correlations have been developed
between experimental attachment coefficients and various
physical and chemical variables (8, 9). These correlations
predict an optimum particle size for transport for a given* Corresponding author phone: (909)369-4857; fax: (909)342-4963;
e-mail: sbradford@ussl.ars.usda.gov.
2242 9 ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 37, NO. 10, 2003

aqueous-porous medium system (10). Smaller particles are
predicted to be removed more efficiently by diffusivetransport and larger particles by sedimentation and interception.
A constant first-order attachment rate coefficient istypically used to describe colloid transport under initial orclean-bed conditions and when the fractional surface coverage of the porous media is small. Studies investigating thetransport of microorganisms, however, have shown that the
spatial distribution of retained microorganisms is not always
consistent with a unique value of the attachment coefficient(11-16). Furthermore, attachment coefficients are reported
to increase due to filter ripening (17-19) or decrease due tofilling of favorable attachment sites (20-22). Predictedattachment coefficients have also been found to be underestimated when repulsive forces exist between the colloids
and porous media (12, 23). Previous studies suggest that soil
surface roughness (24, 25), charge heterogeneity (21), andvariability in colloid characteristics (26) may be responsible
for these discrepancies.
Some of the discrepancies between colloid transport data
and attachment theory may also be due to the fact that colloid
attachment theory does not account for straining. Strainingis the trapping of colloid particles in down-gradient porethroats that are too small to allow particle passage (6). The
magnitude of colloid retention by straining depends on both
colloid and porous medium properties. Straining occurs when
colloids are retained in pores that are smaller than somecritical size. Colloid transport may still occur in pores thatare larger than this critical size. Few studies have examined
the influence of soil pore size distribution characteristics on
colloid straining. Sakthivadivel (27, 28) and Herzig et al. (29)developed geometric relations between the effective diameterof colloids and soil grain size distribution characteristics topredict mass removal by straining. Matthess and Pekdeger(30) generalized this rule to porous media made up of a
distribution of grain sizes. Bradford et al. (31) suggested thatmeasured capillary pressure curves and residual saturations
can also be used to provide an estimate of the volume fractionof the pore space that has pores less than some critical pore
diameter for straining.
Colloid transport data from Harvey et al. (32) and Bradford
et al. (31) indicate that straining was more pronounced than
predictions based upon the criteria given by Matthess and
Pekdeger (30). Harvey et al. (32) observed increasing colloid
retention with increasing colloid size and hypothesized that
migration of clay particles was responsible for this straining
behavior. Bradford et al. (31) reported that effluent colloid
concentration curves and the final spatial distribution ofcolloids retained by the porous media is highly dependent
on the colloid size and soil grain size distribution. Relative
peak effluent concentrations decreased, and the mass of
colloids retained at the column inlet increased when thecolloid size increased and the soil median grain size
decreased. These observations were attributed to increased
straining of the colloids. Bradford et al. (31) also examinedthe experimental colloid transport data using a traditional
first-order attachment and detachment model. The observed
removal trends were not consistent with traditional attachment theory. In contrast to the criteria given by Matthessand Pekdeger (30), the reported residual saturation values
for the experimental soils employed by Bradford et al. (31)indicate that 2-30% of the pore space contain small pores(less than 10 µm) that may have contributed to the observed
straining behavior. We are unaware of any models that
explicitly account for colloid straining.
10.1021/es025899u Not subject to U.S. copyright. Publ. 2003 Am. Chem.Soc.
Published on Web 04/08/2003

A conceptual model is presented herein to account for
both colloid attachment and straining. Colloid attachment
is described using traditional attachment theory, whereas
colloid straining is quantified using an irreversible first-orderstraining factor that is depth dependent. The model also
accounts for colloid exclusion (charge or size) by consideringcolloid accessible pore space and pore sizes. The ability ofthis model to describe the experimental colloid transport
data of Bradford et al. (31) is tested and correlations for model
parameters are developed. The calibrated model and correlations are then used to predict effluent concentration
curves and the final spatial distribution of colloids inexperimental systems.

TheoryThe aqueous phase colloid mass balance equation is written
herein as∂(θwC)
str) - 3‚JT - Eattsw - Esw∂t(1)

where C [N L-3; N and L denote number and length,
respectively] is the colloid concentration in the aqueous
phase, t [T] is time, θw [-] is the volumetric water content,
JT [N L-2 T-1] is the total colloid flux (sum of the advective,-3 T-1] and Estr
dispersive, and diffusive fluxes), and Eattsw [N Lsw
-3
-1
[N L T ] are the colloid mass transfer terms between the
aqueous and solid phases due to colloid attachment andstraining, respectively. In the absence of colloid inactivation
or degradation, a generalized term for Eattsw is given by

Eatt
sw )

∂(FbSatt)) θwkattψattC - FbkdetSatt∂t(2)

where Fb [M L-3; M denotes mass] is the soil bulk density,
Satt [N M-1] is the solid-phase concentration of attachedcolloids, katt [T-1] is the first-order colloid attachment
coefficient, kdet [T-1] is the first-order colloid detachment
coefficient, and ψatt [-] is a dimensionless colloid attachment
function. The value of ψatt equals 1 for clean-bed conditions.
Other functional forms for ψatt have been proposed in theliterature to account for colloid blocking, ripening, andnonexponential spatial distributions (21, 25, 33).
For the work reported in this manuscript we will employ
traditional clean-bed colloid attachment theory by setting
the value of ψatt equal to one. In this case, colloid attachment
theory can be incorporated into the katt term as follows (7)katt )

3(1 - θw)
ηRνw
2d50

(3)

where η [-] is the collector (porous medium) efficiency, R [-]is the colloid sticking efficiency, d50 [L] is the median porous
medium grain diameter, and vw [L T-1] is the pore watervelocity. The collector efficiency accounts for colloid removal
due to diffusion, interception, and gravitational sedimentation (7). The value of η is frequently calculated using thefollowing correlation written in terms of dimensionlessvariables (7, 9)
15/8η ) 4As1/3NPe-2/3 + AsN1/8+ 0.00338AsNG1.2NR-0.4 (4)Lo NR

where As [-] is the Happel correction factor, NPe [-] is thePeclet number, NLo [-] is London-van der Waals attractive
forces number, NR [-] is the interception number, and NG [-]is the gravitational number. The value of R represents thefraction of particles colliding with the porous media thatremain attached and therefore reflects the net effect ofrepulsive and attractive forces between colloids and solid
surfaces. The value of R is usually derived from experimental
breakthrough curves (34, 35) or from fitted values of katt andcalculated values of η (36, 37), although theoretical approaches have also been proposed to predict R (35).
Colloid attachment theory has been used to predict colloidtransport behavior for a wide variety of colloid and porous
media characteristics. Little attention has been payed to
colloid and porous media parameter ranges that may limit
applicability of the attachment model, because colloid
straining has been assumed a priori to be insignificant (Estrsw
is set equal to 0 in eq 1). The following simple and flexible
strform for Esw is utilized in our study to account for straining

Eswstr )∂(FbSstr)) θwkstrψstrC∂t(5)

where kstr [T-1] is the straining coefficient, ψstr [-] is adimensionless colloid straining function, and Sstr [N M-1] isthe solid-phase concentrations of strained colloids. The
dependence of kstr on colloid and porous medium size will
be demonstrated later in this manuscript.
Data from Bradford et al. (31) indicates that straining is
a strong function of distance. A depth-dependent power law
function for ψstr will therefore be utilizedψstr )()

d50 + z -β
d50

(6)

where β [-] is a fitting parameter that controls the shape ofthe colloid spatial distribution, and z [L] is the down gradientdistance from the porous medium inlet or some soil texture
discontinuity in the medium. The value of d50 is used here
as a surrogate parameter for pore length, while the quantity(d50 + z)/d50 is related to the number of pore lengths.
Equations 5 and 6 assume that colloids are accessible to all
soil pores at the column inlet or at interfaces of soil textural
discontinuities (ψstr equals 1 when z ) 0). Straining near thecolumn inlet will cause some pores to become dead-ends,
thus restricting mobile colloids to only the larger continuous
pore networks. The number of dead-end pores is hypothesized to decrease with increasing distance because size
exclusion and/or limited transverse dispersivity tend to keep
colloids within the larger networks (bypassing smaller pores).
Hence, straining is greatest at the column inlet and then is
expected to decrease with increasing distance. Additional
pore-scale studies are planned to test this conceptual modelof straining.
Size exclusion may influence the transport behavior of
colloids by limiting the mobility of particles to the larger
pores (12, 38). If the electrostatic forces between colloids
and porous media are repulsive, then the colloid may also
be excluded from locations adjacent to solid surfaces due tocharge exclusion. Charge exclusion of many anions has been
well documented (39, 40). In that case only a portion of thepore space will be accessible to mobile colloid particles. If
colloids are excluded from the smaller pore spaces, thenthey will likely sample the more conductive ranges of thepore water velocity distribution, and hence be transported
faster than a conservative solute tracer (41-43). To accountfor both size and charge exclusion behavior, values of θw andthe Darcy water velocity, qw [L T-1], that occur in eqs 1-3
and 5 are replaced with the colloid-accessible volumetricwater content, θcw [-], and Darcy water velocity, qcw [L T-1],
respectively. The values of θcw and qcw are defined here asθcw ) θw - ∈γ(7)

qwkrcw
krw

(8)

qcw )

VOL. 37, NO. 10, 2003 / ENVIRONMENTAL SCIENCE & TECHNOLOGY 9 2243

where  [-] is the porosity, γ [-] is the water saturation thatis not accessible to mobile colloids, krw [-] is the water relative
permeability, and krcw [-] is the colloid accessible water relative
permeability. The ratio of qcw and θcw yields the colloid
accessible pore water velocity, vcw [L T-1]. The value of krw
is typically determined from capillary pressure data using apore size distribution model (44, 45). In analogy to thisapproach, we determine the value of krcw using the pore size
distribution model of Burdine (44) by adjusting its limits ofintegration to reflect the accessible pore space to colloids as∫ R(x) dx
S >γ∫ R(x) dx
Sw

krcw(Sw) ) S2w

2

γ

1

2

w

(9)

0

where Sw [-] is the water saturation, R [L] is the pore size, andx [-] is a dummy saturation variable of integration. When Sw
is less than γ then the value of krcw is set equal to zero. The
value of γ can be theoretically related to a critical pore radius
for exclusion using Laplace’s equation of capillarity and acapillary pressure curve.
The proposed colloid transport model discussed above
was implemented into the HYDRUS-1D computer code (46).
HYDRUS-1D is a finite element model for simulating water,
heat, and multiple reactive solute movement in onedimensional variably saturated porous media. The program
numerically solves the Richards equation for saturatedunsaturated water flow and Fickian based advection dispersion equations for heat and solute transport. The code iscoupled to a nonlinear least squares optimization routine
based upon the Marquardt-Levenberg algorithm (47) tofacilitate the estimation of transport parameters fromexperimental data and comes with a graphical user interface
to manage input data, nodal discretization and editing,parameter allocation, problem execution, and visualizationof results.

Model Applications
This section examines the ability of the proposed colloidtransport model to describe the experimental transport dataof Bradford et al. (31). Four different sized fluorescent colloids(Interfacial Dynamics Company, Portland, OR 97224) were
employed in this study, with diameters (dp [L]) equal to 0.45,
1.0, 2.0, and 3.2 µm. These microspheres had carboxyl surfacefunctional groups grafted onto latex particle surfaces by the
manufacturer to create a negatively charged hydrophobic
colloid surface with a particle density of 1.055 g cm-3 and an
air-water contact angle of 115.2 degrees.
Aquifer material used for the soil column experiments
consisted of various sieve sizes of Ottawa sand (99.8% quartz
which is negatively charged at a neutral pH. U.S. Silica,
Ottawa, IL 61350). The Ottawa sands will be designatedherein, from coarser to finer, as 2030, 3550, MIX, and 70110.
These porous media were selected to encompass a range insoil grain size distribution characteristics. Specific propertiesof the 2030, 3550, MIX, and 70110 sands, include mediangrain size of 0.74, 0.36, 0.24, and 0.15 millimeters anduniformity index of 1.21, 1.88, 3.06, and 2.25, respectively.
The aqueous phase chemistry of the tracer, resident, andeluant solutions utilized in the soil column experiments waschosen to create a stabilized monodispersed suspension withthe selected colloids. The initial resident and eluant solutions
consisted of 0.001 M NaCl with its pH buffered to 6.98 using
NaHCO3 (5 × 10-5 M). The colloid-conservative tracer solution
consisted of 0.001 M NaBr with its pH buffered to 6.73 using
NaHCO3 (5 × 10-5 M) and the initial colloid concentration(Ci). The value of Ci was 4.24 × 1011, 3.86 × 1010, 4.85 × 109,
and 1.18 × 109 particles per liter for the 0.45, 1.0, 2.0, and 3.2µm colloids, respectively.
2244 9 ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 37, NO. 10, 2003

TABLE 1. Soil Column Properties (Column Length, L; Porosity,
E; and Darcy Water Velocity, qw) and the Recovered Effluent(ME), Soil (MS), and the Total Colloid Mass Fraction (MBT)soil type dp (µm) qw (cm min-1)
2030
2030
2030
2030
3550
3550
3550
3550
MIX
MIX
MIX
MIX
70110
70110
70110
70110

0.45
1.00
2.00
3.20
0.45
1.00
2.00
3.20
0.45
1.00
2.00
3.20
0.45
1.00
2.00
3.20

0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.11
0.11
0.11
0.11
0.11
0.11
0.11

E

L (cm)

ME

MS

MBT

0.37
0.38
0.37
0.36
0.34
0.34
0.34
0.34
0.33
0.33
0.34
0.34
0.34
0.34
0.35
0.36

13.2
13.4
12.2
13.1
12.7
12.7
12.8
12.8
12.4
12.5
12.7
12.5
12.7
12.7
13.0
13.0

0.90
0.83
0.47
0.44
0.84
0.87
0.26
0.34
0.96
0.84
0.12
0.12
0.66
0.46
0.07
0.03

0.11
0.08
0.37
0.64
0.15
0.10
0.52
0.73
0.22
0.18
0.97
1.25
0.32
0.43
0.96
0.88

1.01
0.91
0.84
1.08
0.99
0.97
0.78
1.07
1.18
1.02
1.09
1.37
0.98
0.89
1.03
0.91

We refer to Bradford et al. (31) and the Supporting
Information for a detailed description of the experimental
materials, protocols, and data analysis. In brief, the colloidconservative tracer solution was pumped (Darcy velocity of∼0.1 cm min-1) upward through columns (∼13 cm in length)
packed with water saturated sand for approximately 75 min,after which a three-way valve was used to switch the pumped
solution to the eluant for another 175 min. Colloid concentration in the column effluent and retained on the media
were determined by fluorescence measurements. Table 1
summarizes relevant properties of the column experimentsthat we examined. To account for mass balance errors in theexperimental data, the concentration of colloids retained in
the soil is multiplied by one minus the colloid mass balancein the effluent divided by the colloid mass balance in thesoil. This approach assumes that the colloid mass balanceerror is associated primarily with the solid phase.
Four different model formulations will be considered: M1- attachment and detachment (i.e., eqs 1 and 2); M2 straining (i.e., eqs 1, 5, and 6); M3 - attachment, detachment,
and straining (i.e., eqs 1-6); and M4 - attachment, detachment, straining, and exclusion (eqs 1-9). Models M1 andM2 assume that colloid retention occurs exclusively byattachment or straining, respectively, whereas M3 and M4
assume that both colloid attachment and straining may occur.
For models M3 and M4 the attachment coefficient is predictedfrom eqs 3 and 4 and a fixed value of R. The value of R wasdetermined from transport experiments conducted using thesame porous medium and 0.45 µm carboxyl colloids; little
or no straining occurs in these systems. The detachment
rate coefficient used for M3 and M4 was obtained by fitting
the colloid effluent concentration curves. All models (M1,
M2, M3, and M4) employed the same value of the colloidhydrodynamic dispersivity, λH [L], for a particular colloidporous medium system. This λH value was obtained by fitting
to the colloid effluent concentration data. Simulated transport
behavior was obtained by fitting specific model parameters
to the effluent concentration curves and the spatial distribution of retained colloids. Tables 2-4 summarize the valuesof the fitted M1, M2, and M3 model parameters, respectively.
Statistical parameters for the goodness of fit are also included,
such as the coefficient of linear regression (r2) and thestandard error (SE).

Model M1
Figure 1a presents the observed and M1 model effluent
concentration curves (relative colloid effluent concentration,
C/Ci, versus pore volume) for several representative colloid

FIGURE 1. Observed and model M1 fitted effluent concentration
curves (Figure 1a) and spatial distribution of retained colloids (Figure1b) for several representative colloid transport experiments indicatedin the legend. The first number in the legend indicates the soil type(cf. Table 1), and the second number is the colloid size in µm.
transport experiments that encompass a range in transport
behaviors. Notice that the M1 simulation fits describe theexperimental data fairly well. Figure 1b presents the observedand M1 model normalized colloid number (N/Ni - numberof colloids/initial number of colloids in effluent) per gramof dry soil as a function of normalized distance from thecolumn inlet for several of the colloid transport experiments.
The M1-simulated decrease in the colloid concentration withdepth is generally much more gradual than that shown by
the experimental data. This is especially true for systems inwhich significant colloid mass was retained at or near thesoil surface. For example, compare the difference between
M1 simulated and experimental 2 µm colloids in MIX sandwith 0.45 µm colloids in 70110 sand. Although the effluent
concentration curves in Figure 1a were adequately describedusing a fitted first-order attachment and detachment coefficient, Figure 1b suggests that this M1 modeling approach
does not describe the observed spatial colloid distribution
well for systems that exhibit significant colloid mass retention,
presumably due to straining in these systems. For the model
to be deemed “correct”, both effluent and retention data
must be simulated accurately.
Table 2 also presents calculated values of η according to
eq 4. The value of η, for a particular soil type, is maximum
for the smallest (0.45 µm) colloids and tends to achieve aminimum value for the 2.0 µm colloids. The attachment
coefficients given in Table 2, however, tend to reach amaximum value for the largest colloids. To compensate forthis deviation, values of R (as determined from eq 3 using
the values of η and katt given in Table 2) vary with colloid sizeand soil type. Recall that R theoretically accounts for the net
effect of repulsive and attractive forces between colloids andsolid surfaces. Since the electrostatic forces between the

FIGURE 2. Observed and model M2 fitted effluent concentration
curves (Figure 2a) and spatial distribution of retained colloids (Figure2b) for several representative colloid transport experiments indicatedin the legend.
various sized carboxyl colloids and Ottawa sands are similar
in magnitude, the scatter in calculated values of R (2 ordersof magnitude) reflects the inability of this approach to predict
the experimental data. Ryan and Elimelech (12) also noted
serious discrepancies between calculated sticking efficiencies
and experimentally measured values. This limitation may
occur in part because straining was not taken into accountby attachment theory.
Other models that have been developed to describe timedependent attachment behavior are not considered in thepresent study. It should be noted, however, that blocking
implies that colloid retention decreases with increasing
attached phase concentration and therefore produces lessmass retention at the column inlet than clean-bed behavior.
This is certainly not the case for our data. Conversely, ripening
implies that colloid retention increases with increasing
attached phase concentration and therefore produces moremass retention at the column inlet than clean-bed behavior.
A second consequence of ripening is a decrease in colloid
effluent concentration with increasing time of colloid addition. The experimental data examined in this manuscript
do not support this second factor. In addition, the carboxyl
colloids are strongly negatively charged and therefore exhibit
a repulsive force between colloids that should inhibit
ripening.

Model M2
Figure 2a shows observed and M2 model effluent concentration curves for several representative colloid transport
experiments. Similar to the results in Figure 1, the M2
simulation describes the experimental effluent data fairly
well. Figure 2b presents the observed and M2 model
normalized colloid numbers per gram of dry soil as a functionof normalized distance from the column inlet for several
VOL. 37, NO. 10, 2003 / ENVIRONMENTAL SCIENCE & TECHNOLOGY 9 2245

TABLE 2. Fitted (katt and kdet Were Fitted to Effluent and Retention Data) and Calculated (η - from Eq 4; and r - from Eq 3)
M1 Model Parameters
soil typedp (µm)λH (mm)katt × 103 (min-1)

SE × 103 (katt)kdet × 103 (min-1)

SE × 103 (kdet)

η × 102r × 102

r2

2030
2030
2030
2030
3550
3550
3550
3550
MIX
MIX
MIX
MIX
70110
70110
70110
70110

0.45
1.00
2.00
3.20
0.45
1.00
2.00
3.20
0.45
1.00
2.00
3.20
0.45
1.00
2.00
3.20

1.9
2.4
1.1
2.4
1.5
1.6
4.9
2.9
1.0
1.8
3.9
3.6
1.7
0.7
3.0
3.6

3.7
2.9
20.1
19.3
5.5
3.6
37.2
30.2
3.4
5.6
67.1
72.0
12.9
22.4
73.2
101.5

0.1
0.5
0.6
0.6
0.2
0.1
1.0
3.1
0.2
0.3
12.5
10.9
0.6
1.1
12.0
62.4

2.9
0.0
0.9
0.4
1.9
0.6
0.5
0.0
6.9
1.3
0.0
0.0
0.8
0.6
0.0
0.1

0.4
1.7
0.3
0.2
0.4
0.4
0.4
0.4
0.9
0.4
0.9
1.0
0.4
0.4
0.8
3.4

2.9
1.8
1.8
2.9
4.8
3.0
2.6
3.5
6.3
3.9
3.1
4.0
8.0
5.2
4.3
5.1

3.6
4.6
29.8
17.6
1.4
1.5
17.6
10.6
0.4
1.1
16.7
13.4
0.8
2.1
8.5
9.9

1.00
0.94
0.94
0.94
0.99
1.00
0.83
0.58
1.00
0.99
0.66
0.74
0.95
0.86
0.85
0.67

TABLE 3. Fitted (kstr and β) M2 Model Parametersa
soil type dp (µm)

kstr × 103 SE × 103(min-1)(kstr)β

SE (β) RMSE

r2

2030
2030
2030
2030
3550
3550
3550
3550
MIX
MIX
MIX
MIX
70110
70110
70110
70110

4.4
4.1
44.6
33.1
36.5
18.9
94.8
906.8
9.2
21.3
989.4
427.5
115.5
271.9
338.0
2160.2

0.08
0.00
0.21
0.14
0.43
0.36
0.21
0.78
0.34
0.28
0.57
0.44
0.42
0.47
0.31
0.55

0.13
0.05
0.04
0.04
0.08
0.06
0.04
0.05
0.20
0.13
0.04
0.06
0.04
0.11
0.04
0.03

1.00
0.95
0.96
0.95
1.00
1.00
0.87
0.97
0.99
0.99
0.93
0.89
0.98
0.94
0.93
0.94

0.45
1.00
2.00
3.20
0.45
1.00
2.00
3.20
0.45
1.00
2.00
3.20
0.45
1.00
2.00
3.20

2.5
0.8
7.0
5.0
12.9
5.3
21.6
185.1
7.8
14.1
149.9
58.5
23.1
146.1
59.9
433.6

0.0
7.1
48.5
8.0
90.2
83.3
30.8
91.5
0.0
0.0
77.1
54.6
97.5
69.0
59.8
79.3

a The same value of λ that is given in Table 2 was employed in M2
Hsimulation.

colloid transport experiments. The observed spatial distribution of colloids is more consistent with this model ascompared to the M1 (Figure 1b) simulations. The reduction
in error associated with the observed and (M2 compared toM1) simulated colloid spatial distributions is quantified inTable 3 using the relative mean square error (RMSE). RMSE
is defined as the percentage of the difference in the colloid
spatial distribution mean square error (MSE) of M1 and M2
divided by the MSE of M1. Hence, the RMSE values in Table
3 indicate that M2 exhibited a significant reduction in thesimulated colloid spatial distribution error as compared toM1 (as much as 97%). Although M2 does a good job ofdescribing both the effluent concentration curves and thefinal spatial distribution of colloids, it is unlikely that strainingis the exclusive mechanism controlling mass removal. Batch
and soil column experiments suggest that colloid attachment
may also play a role (31). Hence, the improved description
may be attributed in part to the increased flexibility (i.e.,
more adjustable parameters) of the M2 model.

Model M3
Figure 3a presents the observed and M3 model effluent
concentration curves for several representative colloidtransport experiments. Similar to the M1 and M2 simulations,notice that the M3 simulations describe the effluent datawell. Figure 3b presents the observed and M3 model
normalized colloid number per gram of dry soil as a function
2246 9 ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 37, NO. 10, 2003

FIGURE 3. Observed and model M3 fitted effluent concentration
curves (Figure 3a) and spatial distribution of retained colloids (Figure3b) for several representative colloid transport experiments indicatedin the legend. In this case, attachment was predicted using eqs
2-4 with the sticking efficiency determined from the 0.45 µm colloidtransport experiment in a particular porous medium and straining
parameters (β and kstr) were fitted to the data.
of normalized distance from the column inlet for several ofthese colloid transport experiments. The observed spatial
distribution of colloids is accurately described by accounting
for both colloid attachment and straining. Fitted andpredicted M3 model parameters are given in Table 4, alongwith the percentage of total colloid mass retention via
straining. The reduction in error associated with the observedand (M3 compared to M1) simulated colloid spatial distributions is quantified as before with RMSE values. The RMSE
values in Tables 3 and 4 are quite similar, indicating that
M3 also exhibited a significant reduction in the simulated

TABLE 4. Fitted (kstr and β) and Predicted (katt) M3 Model Parametersa
soil typedp (µm)katt × 103 (min-1)

kdet × 103 (min-1)

kstr × 103 (min-1)

SE × 103 (kstr)β

SE (β)%MRstr

RMSE

r2

2030
2030
2030
2030
3550
3550
3550
3550
MIX
MIX
MIX
MIX
70110
70110
70110
70110

0.45
1.00
2.00
3.20
0.45
1.00
2.00
3.20
0.45
1.00
2.00
3.20
0.45
1.00
2.00
3.20

3.8
2.3
2.4
4.0
5.5
3.4
3.0
4.1
3.1
1.9
1.5
2.0
12.7
8.0
6.5
7.7

3.0
0.1
1.1
0.4
2.2
0.7
0.9
0.3
4.5
1.0
0.6
0.4
1.4
1.2
0.7
2.4

0.0
2.0
44.6
31.7
9.8
28.0
96.2
814.1
5.4
83.0
1152.4
795.5
76.2
301.1
367.5
1853.0

0.0
0.5
7.4
5.6
61.6
47.2
22.9
208.5
38.0
57.6
220.4
165.0
110.8
208.1
69.2
334.9

0.00
0.00
0.25
0.18
0.85
1.00
0.23
0.78
1.00
0.68
0.60
0.50
0.76
0.57
0.34
0.58

290.47
0.05
0.04
0.04
1.50
0.65
0.04
0.06
2.17
0.19
0.04
0.04
0.42
0.15
0.04
0.06

0.0
46.9
89.8
80.9
8.6
18.6
93.4
92.3
7.5
71.5
98.9
98.4
20.0
75.7
93.0
97.8

0.0
7.7
52.4
9.7
18.3
60.6
32.0
89.5
0.0
28.8
77.6
80.1
44.1
52.6
61.1
77.1

1.00
0.95
0.96
0.95
1.00
1.00
0.88
0.97
1.00
0.99
0.93
0.96
0.97
0.93
0.93
0.94

a The value of k
det comes from Bradford et al. (31).

TABLE 5. Fitted (kstr) and Predicted (katt) M3 Model Parameters When β ) 0.43
soil typedp (µm)katt × 103 (min-1)

kdet × 103 (min-1)

kstr × 103 (min-1)

SE × 103 (kstr)

β

RMSE

r2

2030
2030
2030
2030
3550
3550
3550
3550
MIX
MIX
MIX
MIX
70110
70110
70110
70110

0.45
1.00
2.00
3.20
0.45
1.00
2.00
3.20
0.45
1.00
2.00
3.20
0.45
1.00
2.00
3.20

3.8
2.3
2.4
4.0
5.5
3.4
3.0
4.1
3.1
1.9
1.5
2.0
12.7
8.0
6.5
7.7

3.0
0.1
1.1
0.4
2.2
0.7
0.9
0.3
4.5
1.0
0.6
0.4
1.4
1.2
0.7
2.4

0.1
10.4
87.3
79.4
1.4
1.7
242.7
220.5
2.2
29.3
560.1
632.4
16.0
153.4
560.8
1133.3

65.8
327.1
564.1
330.2
82.6
0.0
2110.0
575.4
0.0
78.4
411.4
274.1
144.2
303.0
280.1
191.9

0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43

10.3
9.0
23.3
-18.8
13.3
20.1
6.3
72.0
0.0
25.6
73.8
82.5
29.5
57.3
62.7
76.5

1.00
0.95
0.95
0.94
0.99
1.00
0.87
0.88
1.00
0.99
0.91
0.95
0.96
0.92
0.93
0.92

colloid spatial distribution error compared to M1 (as muchas 90%).
Notice in Table 4 that the percentage of mass retention
by straining (%MRstr) tends to increase with increasing colloid
size. The percentage of mass retention via attachment follows
from the mass balance. Hence, the percentage of massretention by attachment decreases with increasing colloid
size. For a given colloid size, the percentage of colloid massretention by straining also tends to increase with decreasing
soil median grain size. Although the percentage of massretention by attachment is low for some of the total soilcolumns, as distance increases the relative importance of
attachment increases in comparison to straining. For example, attachment accounted for 0.5 and 31.7% of theretained colloid mass at the column inlet and outlet,
respectively, for the 3.2 µm colloids in 3550 sand. Eventually,
colloid attachment may become the dominant mechanism
for mass removal. Straining is believed to be diminished withincreasing distance down gradient from the column inlet
because size exclusion and/or limited transverse dispersivity
tend to restrict colloids to the larger pore networks (bypassing
smaller pores).
The M3 simulations shown in Figure 3a,b employ adifferent value of β for each colloid transport data set.
Considering the shape similarity of the spatial distributionof retained colloids in systems that exhibit straining (cf. Figure3b), we assume below that a single value of β may be
employed to characterize straining behavior. For this purpose,the value of β was set equal to 0.43; i.e., the average β value

for the 2.0 and 3.2 µm colloids listed in Table 4. With thisassumption, the value of kstr (M3 model) was again fitted toeach colloid transport data set. Figure 4a,b shows theobserved and M3 simulated fits to representative colloidtransport data when β is set equal to 0.43. Notice in Figure
4a,b that the effluent concentration curves and the spatial
distribution of retained colloids, respectively, are described
well with the M3 model using β ) 0.43. Table 5 presents thefitted values of kstr and the associated statistical parameters
for the goodness of fit. The RMSE values in Table 5 tend tobe slightly lower than those given in Table 4 but still indicate
a significant reduction in error when using M3 with fixed β) 0.43 compared to M1.
Figure 5 presents a plot of fitted kstr values (when β ) 0.43)
as a function of experimental dp/d50 ratios. Notice that thevalue of kstr systematically increases with increasing dp/d50.
This behavior is consistent with straining behavior; i.e.,
increasing with increasing colloid size and decreasing mediangrain size. The following power function correlation between
kstr and dp/d50 was established:kstr ) 269.7( )dp 1.42d50r2 ) 0.93(10)The correlation prediction is shown in Figure 5 as a solidline. The above correlation, in conjunction with a fixed valueof 0.43 for β and eqs 1, 5, and 6, now provides a first
approximation to predict the influence of straining on colloidtransport. We emphasize that straining will likely depend
VOL. 37, NO. 10, 2003 / ENVIRONMENTAL SCIENCE & TECHNOLOGY 9 2247

FIGURE 4. Observed and model M3 fitted effluent concentration
curves (Figure 4a) and spatial distribution of retained colloids (Figure4b) for several representative colloid transport experiments indicatedin the legend. In this case, the attachment coefficient was predicted,
a fixed value of β ) 0.43 was employed (cf. eq 6), and only kstr was
fitted to the data.

FIGURE 6. Observed and predicted M3 effluent concentration curves(Figure 6a) and spatial distribution of retained colloids (Figure 6b)
for several representative colloid transport experiments indicatedin the legend. Here straining behavior was predicted using eqs 5
and 6 with β set equal to 0.43 and kstr determined from eq 10.
Attachment was predicted using eqs 2-4 with the sticking efficiency
determined from the 0.45 µm colloid transport experiment in aparticular porous medium.

TABLE 6. Predicted (katt and kstr) M3 Model Parameters

FIGURE 5. A plot of fitted straining coefficient (kstr) values whenβ ) 0.43 as a function of experimental dp/d50 ratios (colloid diameter/
median grain size diameter).
also on other factors that are currently not included in thecorrelation, such as soil grain size uniformity, aqueous phasevelocity, initial colloid concentration, colloid distribution,water content, experimental scale, colloid surface charge,
aqueous phase chemistry, and solid-phase mineralogy.
Additional experimental studies are needed to assess theinfluence of these factors on straining.
Figure 6a,b presents the observed and predicted effluent
concentration curves and spatial distributions of retained
colloids, respectively, for several representative colloidtransport experiments. Here straining behavior was predictedusing eqs 5 and 6 with β set equal to 0.43 and kstr determinedfrom eq 10. Attachment was predicted using eqs 2-4 with
2248 9 ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 37, NO. 10, 2003

soil typedp(µm)

katt × 103(min-1)

kdet × 103(min-1)

kstr × 103(min-1)

β

r2

2030
2030
2030
2030
3550
3550
3550
3550
MIX
MIX
MIX
MIX
70110
70110
70110
70110

0.45
1.00
2.00
3.20
0.45
1.00
2.00
3.20
0.45
1.00
2.00
3.20
0.45
1.00
2.00
3.20

3.8
2.3
2.4
4.0
5.5
3.4
3.0
4.1
3.1
1.9
1.5
2.0
12.7
8.0
6.5
7.7

3.0
0.1
1.1
0.4
2.2
0.7
0.9
0.3
4.5
1.0
0.6
0.4
1.4
1.2
0.7
2.4

7.9
24.4
65.2
126.9
20.6
63.9
170.8
332.5
36.6
113.5
303.5
591.0
71.3
221.1
591.0
1151.0

0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43

1.00
0.94
0.95
0.75
0.99
0.95
0.77
0.78
0.98
0.92
0.72
0.95
0.95
0.93
0.93
0.92

the sticking efficiency determined from the 0.45 µm colloidtransport experiment in a particular porous medium. The
agreement between observed and predicted colloid transport
is not always satisfactory. The model predictions do, however,
capture the general trends in the observed data, notably
increasing colloid retention with increasing colloid size anddecreasing median grain size. Table 6 summarizes theparameter values used in the predictions and the statisticalparameters for the goodness of the model description.

oocysts (4.5-5.5 µm) were transported 1.19 and 1.37 times
faster than chloride in coarse-textured sand, whereas Harvey
et al. (50) reported that bacteria (0.5-0.7 µm) traveled 1.25
faster than bromide in a sandy glacial outwash aquifer. In
these cases, Figure 8 implies that γ would range between
0.16 and 0.51 depending upon the porous medium type.
In general, smaller values of γ are probably more realistic
since higher values of γ are also associated with greater colloid
mass removal. Because the predicted velocity enhancement
is less for smaller values of γ, greater travel distances may
be required to accurately identify this exclusion phenomenon.
Hence, larger-scale experiments may be better suited toinvestigate exclusion behavior than soil column studies.

Supporting Information Available
FIGURE 7. A plot of M4 simulated 3.2 µm colloid effluent
concentration curves in 2030 sand for various values of γ (the water
saturation that is not accessible to mobile colloids).

Additional information on materials and methods. This
material is available free of charge via the Internet at http://
pubs.acs.org.

Literature Cited

FIGURE 8. A plot of vcw/vw (colloid pore water velocity/averagepore water velocity) as a function of γ for various hypothetical(saturated) porous media. The value of n in the figure legend
corresponds to the van Genuchten (48) capillary pressure curveparameter (m ) 1-2/n).

Model M4
The colloid transport data presented by Bradford et al. (31)
did not exhibit earlier colloid breakthrough than bromide.
Numerical experiments were therefore conducted to assess
the sensitivity of model output to exclusion (charge and/orsize). Figure 7 presents M4 simulated 3.2 µm colloid
effluent concentration curves in 2030 sand (the relevant
van Genuchten (48) capillary pressure curve parameters, aren ) 6.875 with m ) 1-2/n) for various values of γ. Noticethat increasing γ leads to earlier colloid breakthrough andhigher peak colloid concentrations as a result of an increasein the colloid pore water velocity and a decrease in the colloid
residence time, respectively.
The predicted velocity enhancement (the ratio of the porewater velocity for colloids and a conservative solute, vcw/vw)
for a particular porous medium as a function of γ can beeasily calculated from eqs 7 and 8 and the analytical solution
for eq 9 (49). The values of vcw/vw as a function of γ for various
hypothetical (saturated) porous media are shown in Figure
8. The value of n in the figure legend corresponds to the van
Genuchten (48) capillary pressure curve parameter (m ) 1-2/n) and is related to the capillary pressure curve slope at the
inflection point. A higher value of n corresponds to a morenarrow pore size distribution. Hence, Figure 8 indicates thatvelocity enhancement is expected to increase in more gradedsoils and for larger values of γ. If the degree of velocity
enhancement is known from experimental studies, thenFigure 8 can also be used to estimate a value of γ. For example,
Harter et al. (43) reported that Cryptosporidium parvum(1) Bitton, G.; Harvey, R. W. In. New Concepts in Environmental
Microbiology; Mitchell, R., Ed.; Wiley-Liss: New York, 1992; pp103-124.
(2) Wilson, J. T.; Leach, L. E.; Henson, M.; Jones J. N. Ground Water
Monit. Rev. 1986, 6, 56-64.
(3) MacLeod, F. A.; Lappin-Scott, H. M.; Costerton, J. W. Appl.
Environ. Microbiol. 1998, 54, 1365-1372.
(4) Puls, R. W.; Powell, R. M. Environ. Sci. Technol. 1992, 26, 614621.
(5) de Marsily, G. In Quantitative hydrogeology: groundwaterhydrology for engineers; Academic Press: Orlando, FL, 1986.
(6) McDowell-Boyer, L. M.; Hunt, J. R.; Sitar, N. Water Resour. Res.
1986, 22, 1901-1921.
(7) Logan, B. E.; Jewett, D. G.; Arnold, R. G.; Bouwer, E. J.; O’Melia,
C. R. J. Environ. Eng. 1995, 121, 869-873.
(8) Yao, K.-M.; Habibian, M. T.; O’Melia, C. R. Environ. Sci. Technol.
1971, 5, 1105-1112.
(9) Rajagopalan, R.; Tien, C. AIChE J. 1976, 22, 523-533.
(10) Tobiason, J. E.; O’Melia, C. R. J. Am. Water Works Assoc. 1988,
80, 54-64.
(11) Swanton, S. W. Adv. Colloid Interface Sci. 1995, 54, 129-208.
(12) Ryan, J. N.; Elimelech, M. Colloids and Surfaces A: Physicochem.
and Eng. Aspects 1996, 107, 1-56.
(13) Song, L. F.; Elimelech, M. J. Colloid Interface Sci. 1994, 167,
301-313.
(14) Song, L. F.; Johnson, P. R.; Elimelech, M. Environ. Sci. Technol.
1994, 28, 1164-1171.
(15) Bhattacharjee, S.; Ko, C.-H.; Elimelech, M. Langmuir 1998, 14,3365-3375.
(16) Elimelech, M.; Nagai, M.; Ko, C. H.; Ryan, J. N. Environ. Sci.
Technol. 2000, 34, 2143-2148.
(17) Tobiason, J. E. Colloids Surfaces 1989, 39, 53-77.
(18) Darby, J. L.; Lawler, D. F. Environ. Sci. Technol. 1990, 24, 10691079.
(19) Tobiason, J. E.; Vigneswaran, B. Water Res. 1994, 28, 335-342.
(20) Hunt, J. R.; Hwang, B.-C.; McDowell-Boyer, L. M. Environ. Sci.
Technol. 1993, 27, 1099-1107.
(21) Johnson, P. R.; Elimelech, M. Langmuir 1995, 11, 801-812.
(22) Ko, C.-H.; Elimelech, M. Environ. Sci. Technol. 2000, 34, 36813689.
(23) Vaidyanathan, R.; Tien, C. Chem. Eng. Sci. 1991, 46, 967-983.
(24) Kretzschmar, R.; Barmettler, K.; Grolimund, D.; Yan, Y.-D.;
Borkovec, M.; Sticher, H. Water Resour. Res. 1997, 33, 11291137.
(25) Redman, J. A.; Grant, S. B.; Olson, T. M.; Estes, M. K. Environ.
Sci. Technol. 2001, 35, 1798-1805.
(26) Bolster, C. H.; Mills, A. L.; Hornberger, G. M.; Herman, J. S.
Water Resour. Res. 1999, 35, 1797-1807.
(27) Sakthivadivel, R. Theory and mechanism of filtration of noncolloidal fines through a porous medium; Rep. HEL 15-5;
Hydraulic Engineering Laboratory, University of California:
Berkeley, CA 1966.
(28) Sakthivadivel, R., Clogging of a granular porous medium bysediment; Rep. HEL 15-7; Hydraulic Engineering Laboratory,
University of California: Berkeley, CA, 1969.
(29) Herzig, J. P.; Leclerc, D. M.; LeGoff, P. Ind. Eng. Chem. 1970, 62,
129-157.
VOL. 37, NO. 10, 2003 / ENVIRONMENTAL SCIENCE & TECHNOLOGY 9 2249

(30) Matthess, G.; A. Pekdeger, Survival and transport of pathogenic
bacteria and viruses in groundwater, in Ground Water Quality;
Ward, C. H., Giger, W., McCarty, P., Eds.; John Wiley: New York,
1985.
(31) Bradford, S. A.; Yates, S. R.; Bettahar, M.; Simunek, J. Water
Resour. Res. 2002, 38(12), 1327, doi: 10.1029/2002WR001340.
(32) Harvey, R. W.; Kinner, N. E.; MacDonald, D.; Metge, D. W.;
Bunn, A. Water Resour. Res. 1993, 29, 2713-2721.
(33) Adamczyk, Z.; Siwek, B.; Zembala, M.; Belouschek, P. Adv. Colloid
Interface Sci. 1994, 48, 151-280.
(34) Pieper, A. P.; Ryan, J. N.; Harvey, R. W.; Amy, G. L.; Illangasekare,
T. H.; Metge, D. W. Environ. Sci. Technol. 1997, 31, 1163-1170.
(35) Ryan, J. N.; Elimelech, M.; Ard, R. A.; Harvey, R. W.; Johnson,
P. R. Environ. Sci. Technol. 1999, 33, 63-73.
(36) Bales, R. C.; Hinkle, S. R.; Kroeger, T. W.; Stocking, K. Environ.
Sci. Technol. 1991, 25, 2088-2095.
(37) Redman, J. A.; Grant, S. B.; Olson, T. M.; Hardy, M. E.; Estes, M.
K. Environ. Sci. Technol. 1997, 31, 3378-3383.
(38) Ginn, T. R. Water Resour. Res. 2002, 38(4), 1041, doi: 10.1029/
2001WR000865.
(39) Gvirtzman, H.; Gorelick, S. M. Nature 1991, 352, 793-795.
(40) Ginn, T. R. A brief review of bacterial transport in natural porous
media; PNL-10876; UC-402; Pacific Northwest National Laboratory: Richland, WA, 1995.
(41) Reimus, P. W. The use of synthetic colloids in tracer transport
experiments in saturated rock fractures; LA-13004-T; Los Alamos
National Laboratory: Los Alamos, NM, 1995.

2250 9 ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 37, NO. 10, 2003(42) Cumbie, D. H.; McKay, L. D. J. Contam. Hydrol. 1999, 37, 139157.
(43) Harter, T.; Wagner, S.; Atwill, E. R. Environ. Sci. Technol. 2000,
34, 62-70.
(44) Burdine, N. T. Trans. Am. Inst. Min. Metall. Pet. Eng. 1953, 198,
71-78.
(45) Mualem, Y. Water Resour. Res. 1976, 12, 513-522.
(46) Simunek, J.; Sejna, M.; van Genuchten, M. Th. The HYDRUS-1D
software package for simulating the one-dimensional movementof water, heat, and multiple solutes in variably saturated media-Version 2.0; IGWMC - TPS - 70; International Ground Water
Modeling Center, Colorado School of Mines: Golden, CO, 1998.
(47) Marquardt, D. W. J. Soc. Indust. Appl. Math. 1963, 11, 431-441.
(48) Van Genuchten, M. Th. Soil Sci. Soc. Am. J. 1980, 44, 892-898.
(49) Van Genuchten, M. Th.; Leij, F. J.; Yates, S. R. The RETC codefor quantying the hydraulic functions of unsaturated soils; EPA/
600/2-91/065; U.S. Environmental Protection Agency: Ada, OK,
1991.
(50) Harvey, R. W.; George, L. H.; Smith, R. L.; LeBlanc, D. R. Environ.
Sci. Technol. 1989, 23, 51-56.

Received for review June 19, 2002. Revised manuscript received February 25, 2003. Accepted March 7, 2003.
ES025899U