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WATER RESOURCES RESEARCH, VOL. 45, W02414, doi:10.1029/2008WR007096, 2009

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ArticleModeling the coupled effects of pore space geometry and velocity
on colloid transport and retention
Scott A. Bradford,1 Saeed Torkzaban,2 Feike Leij,3 Jiri Šimůnek,2and Martinus T. van Genuchten4
Received 16 April 2008; revised 31 October 2008; accepted 2 December 2008; published 12 February 2009.

[1] Recent experimental and theoretical work has demonstrated that pore space geometry

and hydrodynamics can play an important role in colloid retention under unfavorable
attachment conditions. Conceptual models that only consider the average pore watervelocity and a single attachment rate coefficient are therefore not always adequate todescribe colloid retention processes, which frequently produce nonexponential profiles ofretained colloids with distance. In this work, we highlight a dual-permeability model
formulation that can be used to account for enhanced colloid retention in low-velocity
regions of the pore space. The model accounts for different rates of advective anddispersive transport, as well as first-order colloid retention and release in fast and slow
velocity regions of the pore space. The model also includes provisions for the exchange of
colloids from fast to slow regions in the aqueous phase and/or on the solid phase. A
sensitivity analysis performed with the dual-permeability model parameters indicated that
low rates of advective transport to low-velocity regions had a pronounced influence on
colloid retention profiles, especially near the inlet. The developed model provided a good
description of measured colloid breakthrough curves and retention profiles that werecollected for a variety of conditions.
Citation: Bradford, S. A., S. Torkzaban, F. Leij, J. Šimůnek, and M. T. van Genuchten (2009), Modeling the coupled effects of porespace geometry and velocity on colloid transport and retention, Water Resour. Res., 45, W02414, doi:10.1029/2008WR007096.

1. Introduction[2] Accurate simulations of the transport and retention ofcolloids in porous media is needed for a wide variety of
applications: to protect water supplies from pathogenic
microorganisms [Gerba et al., 1996; Loge et al., 2002;
Abbaszadegan et al., 2003] and a variety of colloid associated contaminants [Grolimund et al., 1996; Kim et al., 2003;
Chen et al., 2005; Šimůnek et al., 2006]; to quantify
processes of soil genesis, erosion, and aquifer and petroleum reservoir production [Khilar and Fogler, 1998; Mays
and Hunt, 2005]; and to improve engineered bioremediation
strategies [Mishra et al., 2001; Vidali, 2001]. Most models
for colloid migration in porous media consider only theaverage pore water velocity in the porous medium and asingle attachment rate coefficient to quantify interactions
between the colloids and the solid surfaces [Yao et al., 1971;
Rajagopalan and Tien, 1976]. These models predict thatcolloids move with the average pore water velocity andthat the profile of retained colloids in porous media will

1

US Salinity Laboratory, ARS, USDA, Riverside, California, USA.
Department of Environmental Science, University of California,
Riverside, California, USA.
3
Civil Engineering and Construction Engineering Management,
California State University, Long Beach, California, USA.
4
Department of Mechanical Engineering, Federal University of Rio deJaneiro, Rio de Janeiro, Brazil.
2

Copyright 2009 by the American Geophysical Union.
0043-1397/09/2008WR007096$09.00

decrease exponentially with distance. Filtration theory [Yaoet al., 1971] is commonly used in such models to predict theattachment rate coefficient. In this case, the rate of masstransfer of colloids to solid surfaces by sedimentation,interception, and diffusion is quantified from correlation
expressions [Logan et al., 1995; Tufenkji and Elimelech,
2004; Nelson and Ginn, 2005] that were determined fromsingle collector simulation results using the sphere-in-cellmodel. Interactions of colloids with the solid surface aresubsequently assumed to be controlled by physicochemical
forces that can be quantified using a theory developed
by Derjaguin, Landau, Verwey, and Overbeek (DLVO)[Derjaguin and Landau, 1941; Verwey and Overbeek, 1948].
[3] Filtration theory has proven to be a useful tool for
predicting colloid transport and retention in porous media
under conditions that are favorable for attachment [Yao et al.,
1971; Rajagopalan and Tien, 1976; Harvey and Garabedian,1991; Li and Johnson, 2005], for example when there is noenergy barrier to attachment of colloids in the primary
minimum of the DLVO interaction energy profile. Conversely under unfavorable attachment conditions (whenthere is an energy barrier to attachment in the primary
minimum), models that are based on filtration theory havecommonly been found inadequate to predict the amount ofretention, the shape of the deposition profile, and thedependence of colloid retention on velocity, solution chemistry, and grain and colloid size [Bradford et al., 2003, 2006,
2007; Tufenkji and Elimelech, 2005; Tong et al., 2005; Li
and Johnson, 2005; Li et al., 2005; Johnson et al., 2007a;
Torkzaban et al., 2007, 2008a, 2008b]. Variations in porous
media and/or colloid surface charge have frequently been

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invoked as a potential explanation for nonexponential deposition profiles, leading to the development of multiple deposition rate models to describe such behavior [Grolimund andBorkovec, 2001; Tufenkji et al., 2003]. Furthermore, mobile
colloids have sometimes been reported to be transported ata velocity that is higher than the average pore water velocity[Cumbie and McKay, 1999; Ginn, 2002].
[4] Filtration theory quantifies the rate of mass transfer ofcolloids to soil surfaces, and assumes that colloid retention iscontrolled by chemical interactions. In reality, colloid retention will depend on the forces and torques that act on colloids
near solid surfaces [Cushing and Lawler, 1998; Bergendahl
and Grasso, 2000; Torkzaban et al., 2007]. The mostsignificant forces that act on colloids near a solid surface
are thought to be due to adhesion (chemical interactions),hydrodynamics, and diffusion [Cushing and Lawler, 1998;
Johnson et al., 2007b; Bradford and Torkzaban, 2008].
Under favorable attachment conditions adhesive forces will
likely dominate colloid retention, as assumed by filtration
theory [Torkzaban et al., 2007]. Conversely, under unfavorable attachment conditions that are typical for most natural
subsurface environments and colloids, the influence of hydrodynamics and diffusion cannot be neglected [Cushing and
Lawler, 1998; Simoni et al., 1998; Bergendahl and Grasso,
2000; Dong et al., 2002; Torkzaban et al., 2007]. Lifting,sliding, and rolling are hydrodynamic mechanisms that cancause colloid removal from an interface [Hubbe, 1984;
Soltani and Ahmadi, 1994; Bergendahl and Grasso, 2000].
Rolling has been reported to be the dominant hydrodynamic
mechanism of detachment from solid surfaces under laminarflow conditions [Tsai et al., 1991; Bergendahl and Grasso,
1998, 1999].
[5] Colloids can interact with solid surfaces via the secondary minimum of the DLVO interaction energy profile underunfavorable attachment conditions [Franchi and O’Melia,
2003; Redman et al., 2004; Hahn and O’Melia, 2004; Hahn
et al., 2004; Tufenkji and Elimelech, 2005]. In this case,consideration of the balance between adhesive (due to thesecondary minimum) and hydrodynamic torques indicates that
low-velocity regions may be hydrodynamically and chemicallyfavorable for colloid retention [Bradford et al., 2007;
Bradford and Torkzaban, 2008; Torkzaban et al., 2007,2008a, 2008b]. During steady state fluid flow the hydrodynamic forces that occur near a single collector grain or in
porous media are highly dependent on the spatial location
along the solid surface. Specifically, lower-hydrodynamic
forces occur on a single collector grain near the front andrear stagnation points, whereas in porous media lower
velocities occur at grain-to-grain contacts, solid-air-water
triple points, and in the smallest regions of the pore space[Bradford et al., 2007; Bradford and Torkzaban, 2008;
Torkzaban et al., 2007, 2008b]. Indeed, recent experimental
evidence by Kuznar and Elimelech [2007] demonstrated
that colloids captured in the secondary energy minimum
can be translated along the collector surface via hydrodynamic forces and be retained in regions near the rearstagnation point.
[6] The pore space geometry may cause fluid streamlines
to separate from the grain surface, and to form low-velocity
regions near the gaps between grains [Davis et al., 1976;
Davis and O’Neill, 1977; Taneda, 1979]. In these locations
water does not mix with the bulk solution, but rather rotates

W02414in an infinite set of nested ring vortices. This type of
behavior of Stokes flow around two spheres in contact orclose to each other has theoretically been demonstrated[Davis et al., 1976; Davis and O’Neill, 1977] and experimentally visualized by Taneda [1979]. The literature also
indicates that pore-scale fluid flow is hydrodynamicallydisconnected from the bulk fluid flow (eddy zones) nearsolid-air-water triple points [Sheng and Zhou, 1992]. It
should be mentioned that low-velocity regions in porous
media have been used to explain a variety of observed
transport behavior without recognizing the exact systemhydrodynamics [Padilla et al., 1999; Toride et al., 2003].
Torkzaban et al. [2008b] recently demonstrated that theselow-velocity regions can have a very significant effect onbacteria retention in saturated systems, and can retain bacteria
in these locations even when the adhesive force is negligible.
[7] Fluorescent microscopy and X-ray microtomography
studies have also demonstrated that colloids accumulate innarrow regions of pore spaces near the contacts of irregularly shaped sand grains under unfavorable attachment
conditions [Bradford et al., 2005, 2006; Xu et al., 2006;
Li et al., 2006; Yoon et al., 2006; Gaillard et al., 2007]. In
these studies, pore-space constrictions served as locations
for colloid retention, whereas few colloids were immobilized far from the grain-to-grain contacts. Colloid retention
in the smallest regions of the pore space, such as thoseformed near grain to grain contact points, has been referred
to in the literature as straining [Hill, 1957; McDowell-Boyeret al., 1986; Cushing and Lawler, 1998; Bradford et al.,2006].
[8] The above literature indicates that, under unfavorable
attachment conditions, colloid retention will primarily occur
in the low-velocity regions of the porous media. It istherefore not surprising that models that consider only theaverage pore water velocity and a single attachment coefficient have been found to be inadequate to predict colloid
retention behavior in many instances. The objective of thiswork is to present an alternative model formulation that canbe used to account for different colloid retention mechanisms in the various regions of the pore space. Applications,
implications, and limitations of this model to characterize
colloid transport and retention will be demonstrated anddiscussed.

2. Pore-Scale Considerations[9] Once colloids are transported to the solid surface inporous media, colloid retention under unfavorable attachment conditions will depend on a balance of forces andtorques that act on colloids at this location [Cushing and
Lawler, 1998; Bergendahl and Grasso, 2000; Torkzaban etal., 2007]. Calculated locations on the solid surface that are
chemically and hydrodynamically favorable for the retention of colloids may be determined from a balance ofadhesive (Tadhesive; ML2T2; where M, L, and T denoteunits of mass, length, and time, respectively) and hydrodynamic (Thydrodynamic; ML2T2) torques that act on colloids atthe solid surface. Colloid retention only occurs on regionsof the solid surface where Tadhesive > Thydrodynamic. A
detailed description to calculate Tadhesive and Thydrodynamic
has recently been provided by Torkzaban et al. [2007,2008b]. In brief, Tadhesive for colloids attached in the

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BRADFORD ET AL.: COLLOID TRANSPORT AND RETENTION

W02414secondary minimum of the DLVO interaction energy profile
can be estimated using DLVO and Johnson, Kendall, andRoberts (JKR) theories [Johnson et al., 1971; Soltani and
Ahmadi, 1994; Bergendahl and Grasso, 2000; Torkzaban etal., 2007; Ahmadi et al., 2007], while Thydrodynamic may be
determined using available theory in the literature [Goldman
et al., 1967; O’Neill, 1968; Sharma et al., 1992].
[10] Application of the torque balance approach in natural
porous media is limited by our knowledge of the exact porescale geometry and hydrodynamics, which influence both
the rate of colloid mass transfer to solid surfaces and thehydrodynamic forces and torques acting on retained colloids. It is currently not computationally feasible to simulate
all of the full three-dimensional pore-scale water flow andcolloid transport processes at length-scales that are muchgreater than a few pores. Alternatively, simple geometries
are sometimes used to provide insight about flow andtransport processes that occur in natural porous media. In
a capillary tube analogy of a porous medium the pore-scale
velocity distribution is well known and the value ofThydrodynamic can be determined as [Goldman et al., 1967;
O’Neill, 1968; Sharma et al., 1992; Bergendahl and Grasso,
2000]
Thydrodynamic ¼ 14:29pmrc3@v
DP¼ 7:14prc3
ðR c  rc Þ
@r
Lct

ð1Þ

where m [M L1T1] is the viscosity, rc [L] is the colloid
radius, Rc [L] is the capillary tube radius, r [L] is the radial
direction from the solid surface, v [L T1] is the pore-scale
velocity vector, and DP/Lct [M L2 T2] is the gradient inpressure over the capillary tube length. When Tadhesive =Thydrodynamic the value of Rc in equation (1) can be solved todetermine a critical capillary tube radius for which colloid
retention will occur in equal and smaller sized tubes as
Rc ¼

Gadhesive Lctþ rc
7:14prc3 DP

ð2ÞAlternatively, the value of Rc may also be estimated from ageometric analysis. Herzig et al. [1970] suggested that whenseveral colloids arrive simultaneously at a pore then thevalue of Rc will become some multiple of the colloid radius.
[11] The volume fraction, g [], of pore space wherecolloid retention will occur can theoretically be determinedfrom Rc and from measured capillary pressure curves andresidual saturations. In this case, the capillary pressure curveis rewritten in terms of effective pore diameters using
Laplace’s equation by assuming that soil pores can beapproximated as a bundle of cylindrical capillaries that are
tortuous [e.g., Dane and Hopmans, 2002; Flint and Flint,2002]. If a measured capillary pressure curve is parameterized using the model of van Genuchten [1980], then g is
given as m2sap ng ¼ ð1  Srw Þ* 1 þ
þ Srw
rgRc

ð3Þ

where Srw [] is residual water saturation, ap[L1] is thereciprocal of the air entry pressure head, r [M L3] is thedensity of water, g [L T2] is the acceleration due to gravity,s [M T2] is the surface tension, and m [] and n [] are

W02414

fitting parameters. It should be mentioned that Bradford etal. [2006] used equation (3) and average measured capillarypressure-saturation parameters from various soil types toestimate values of g. Depending on soil type from 10 to 71,
10 to 86, and 10 to 92% of the pore space was found to besmaller than 0.1, 2, and 6 mm colloids, respectively[Bradford et al., 2006].
[12] Although the capillary tube analogy will undoubtedly have some significant limitations in describing the exact
pore-scale geometry, the hydrodynamics, and the adhesive
forces involved, equations (1) – (3) do provide useful insighton trends that can be expected for colloid retention in
porous media. For example, equations (2) and (3) indicatethat g is expected to increase for increasing adhesive forces(Tadhesive) and to decrease for increasing flow rates (pressure
gradient). This finding is consistent with recent pore-scale
simulations of colloid retention on single collector surfaces[Torkzaban et al., 2007]. Furthermore, the influence ofpore space geometry and colloid size can be related tomeasurable relationships in equation (3).

3. Dual-Permeability Model[13] At the column-scale the water flow and colloidtransport parameters are averaged over a representative
elementary volume (REV) to obtain effective parameters.
Because of this averaging process some pore-scale flow andtransport information is lost, such as the pore-scale distribution of the flow velocity and the exact location of colloid
retention. The implicit assumption of this averaging procedure is that the lost pore-scale information does not control
the flow and transport processes. The literature review andthe information presented above indicate that this assumption may be violated for colloid transport and retention in
porous media under unfavorable attachment conditions.
Below we present a REV scale model to better accountfor the observed pore-scale physics that was discussed
above.
[14] Dual-permeability models have commonly been used
to study preferential and nonequilibrium flow and solutetransport in structured soils and fractured rocks [Šimůnek etal., 2003; Gerke, 2006; Šimůnek and van Genuchten, 2008].
In this case, the dual-permeability model partitions the porespace into two regions that have fast (fracture) and slow
(matrix) rates of advective and dispersive transport ofsolutes. In contrast to previous work, we use the dualpermeability model herein to simulate the different colloid
retention mechanisms that occur in fast (larger pore spaces;
region 1) and slow (small pore spaces, dead end pores, andgrain-to-grain contact points: region 2) velocity regions ofhomogeneous porous media. The dual-permeability model
has not been previously used to gain insight into enhanced
colloid retention processes in low-velocity regions ofhomogeneous porous media. The approach is somewhat
analogous to multiphase flow and transport models thatpartition the pore space to regions accessible for the wetting(small pore spaces) and nonwetting (large pore spaces)
phases. The small pore spaces in the dual-permeability
model are assumed to maintain continuity by slow flow
adjacent to the solid phase, in crevice sites near grain-tograin contacts, and in small pores in the same way as thewetting phase in multiphase systems.

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Figure 1. A conceptual picture of the flow and colloidtransport processes that are simulated in the dual-permeability model. Model parameters are defined in the maintext.
[15] Figure 1 presents a conceptual picture of the flow
and colloid transport processes that are assumed to occur inthe dual-permeability model. As indicated above, colloids
that collide with solid surfaces in fast regions of the porespace experience different hydrodynamic forces than colloids in slow regions. The higher hydrodynamic forces inregion 1 act to remove colloids from the solid surface inregion 1, thus causing region 1 to be associated with lower
rates of colloid retention. Colloid exchange in the aqueous
phase may occur to and from ‘‘slow’’ water (region 2). In
addition, in this work we also consider the potential for
colloid exchange on the solid phase from fast to slow
regions due to either rolling or sliding of colloids on thesolid surface. The governing equations for water flow in the
dual-permeability model are well known and are available
in the literature [Gerke and van Genuchten, 1993a, 1993b;
Šimůnek and van Genuchten, 2008]. The corresponding
one-dimensional dual-permeability equations for localscale colloid transport and retention are as follows:

regions 1 and 2, qw1 and qw2 are the volumetric watercontents in regions 1 and 2 [], rb1 and rb2 are the bulk
densities in regions 1 and 2 [M L3], k1 and k2 [T1] are thefirst-order colloid retention rate coefficients in regions 1 and
2, kdet1 and kdet2 [T1] are the first-order detachment
coefficients in regions 1 and 2, J1 and J2 [Nc L2 T1] are
the total solute fluxes (sum of the advective and dispersiveflux) for colloids in regions 1 and 2, w [T1] is a coefficientfor colloid exchange between liquids in regions 1 to 2,
k12 [T1] is a coefficient for transfer of colloids from solid
phase region 1 to 2, and w is the ratio of the volume ofregion 2 to the total volume (volume of region 1 plus
volume of region 2). The term Gs [N L3 T1] accounts foraqueous phase mass exchange of colloids between regions 1
and 2.
[16] Equations (4) – (8) were written in terms of localscale mass balances of regions 1 and 2. To formulate theequations in terms of the total pore space, the mass balance
equations for regions 1 and 2 need to be multiplied by (1-w)
and w, respectively. The relationship between several
variables at the local-scale and total pore space is provided
below. Under steady state flow conditions, the total water
flux (qt; LT1) is defined as
qt ¼ ð1  wÞq1 þ wq2

ð4Þ

@ ðqw2 C2 Þ
@J2 Gs¼
  qw2 k2 C2 þ rb2 kdet2 s2
@t
@z
w

ð5Þ

@ ðrb1 s1 Þ
r k12 s1
¼ qw1 k1 C1  rb1 kdet1 s1  b1
1w
@t

ð6Þ

@ ðrb2 s2 Þ
r k12 s1
¼ qw2 k2 C2  rb2 kdet2 s2 þ b1
w
@t

ð7Þ

Gs ¼ wð1  wÞqw1 ðC2  C1 Þ

ð8Þ

where z [L] is distance, t [T] is time, C1 and C2 [Nc L3; Nc
denotes the number of colloids] are the liquid phase
concentrations of colloids in regions 1 and 2, s1 and s2[Nc M1] are the solid phase concentrations of colloids inð9Þ

where q1 (LT1) and q2 (LT1) are the local-scale water
fluxes in regions 1 and 2, respectively. Expressions for the
total water content and bulk density are written in an
analogous manner as equation (9). The total flux concentration of colloids (Ct, Nc L3) is given as [Šimůnek and van
Genuchten, 2008]
Ct ¼

wq2 C2 þ ð1  wÞq1 C1
wq2 þ ð1  wÞq1

ð10Þand the total solid phase colloid concentration (st, Nc M1)
asst ¼

@ ðqw1 C1 Þ@J1Gsþ
 qw1 k1 C1 þ rb1 kdet 1 s1
¼@z 1  w@t

W02414wrb2 s2 þ ð1  wÞrb1 s1
wrb2 þ ð1  wÞrb1

ð11Þ[17] The dual-permeability model outlined above has been
implemented into the HYDRUS-1D computer model[Šimůnek et al., 2007; Šimůnek and van Genuchten,
2008]. The code employs the Galerkin-type linear finite
element method for spatial discretization of the governing
differential equations, and a finite difference method toapproximate temporal derivatives. A Crank-Nicholson finite
difference scheme was used to solve the outlined transport
equations sequentially (with equations for region 1 solved
first). Complete details about the numerical techniques areprovided in the HYDRUS-1D technical manual [Šimůnek etal., 2007]. For the simulations discussed below, a third-typeboundary condition was used at the inlet, and a concentration gradient of zero was fixed at z equal to the outlet depth.
The initial concentration of the simulation domain was zero.
We note that the HYDRUS-1D code allows model
parameters to be fitted to the breakthrough curve and theretention profile simultaneously using a nonlinear leastsquares optimization routine based upon the LevenbergMarquardt algorithm [Marquardt, 1963].

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W024143.1. Model Parameters[18] In this paper we apply the dual-permeability model
to homogeneous porous media. Values of the volumetricwater content and bulk density were therefore assumed to be
the same for both regions 1 and 2, to be determined fromindependently measured values. Similarly, we assume thatthe dispersivity for both regions is equal and can be
determined from conservative solute tracer data.
[19] Equations (1) – (3) indicate that colloid retention is‘‘chemically and hydrodynamically’’ favorable in the smallest portions of the pore space (region 2). We therefore
assume that k2 can be determined from clean bed filtration
theory [Yao et al., 1971] as
k2 ¼

3ð1  qw Þ
hv
2d50

ð12Þ

where v [L T1] is the average total pore water velocity,
qw is the total volumetric water content [], and d50 [L]is the median grain diameter. The parameter h is definedas the ratio of the integral of the colloid flux that strikes
the collector to the rate at which particles flow toward thecollector. This parameter was determined using thecorrelation expression of Tufenkji and Elimelech [2004].
Since colloid retention is favorable in region 2, the valueof kdet2 is set equal to zero.
[20] Colloid retention is assumed to be ‘‘chemically andhydrodynamically’’ unfavorable in the larger portions of thepore space defined by region 1. Two different conceptual
approaches may be used to quantify colloid behavior inregion 1. The first approach assumes that ‘‘irreversible’’
colloid retention occurs in region 1 (for given experimental
conditions) as a result of chemical heterogeneity and surface
roughness, or because of the potential for colloids todiffusion over the DLVO energy barrier [Hubbe, 1984;
Simoni et al., 1998; Dong et al., 2002]. This ‘‘irreversible’’
colloid retention can be quantified with k1 when kdet1 is set
equal to zero. The approach also implies that the influenceof pore-scale hydrodynamics on colloid removal from thesolid surface in region 1 has already occurred, in which casethe value of k12 can be set equal to zero. The value of w
therefore reflects the lumped exchange of colloids fromregion 1 to 2 due to diffusion in the bulk solution and slow
advection of weakly associated colloids (secondary minimum) near the solid phase. In this case, k1 and w can be
determined by fitting to experimental data. We note herethat Gerke and van Genuchten [1993b] presented analternative model formulation for equation (8) to accountfor both diffusive and advective exchange between regions
1 and 2 under unsaturated conditions.
[21] The second approach considers all colloid retention
to be ‘‘reversible’’ in region 1, while the value of k1 isdetermined analogously to equation (12) (k1 = k2). With thisassumption the values of kdet1 and k12 are used to partition
weakly associated colloids (secondary minimum) back to
the aqueous phase or to the region 2 solid phase where theyare ‘‘irreversibly’’ retained, respectively. In this case, thevalue of w reflects diffusive exchange between regions 1
and 2. For advection dominated systems we assume that thevalue of w goes to zero, and that the values of kdet1 and k12
can be determined by fitting to experimental data.

W02414[22] Bradford et al. [2006] presented an approach toestimate the advective water flux to the smallest regions of
the pore space. In brief, the water flux in region 2 portionsof the pore space can be calculated as
wq2 ¼ qt

kr2
krw

ð13Þ

where krw [] is the water relative permeability and kr2 []is the water relative permeability to the low-velocity regionsof the pore space. The value of krw is typically determinedfrom capillary pressure data using a pore size distribution
model. In analogy to this approach, the value of kr2 can be
determined using the pore size distribution model of
Burdine [1953] by limiting the integration to only region 2
areas of the porous medium as follows:Zg
kr2 ðSw Þ ¼ Sw2 0 1
ZRðS Þ2 dShm i¼ Sw2 1  1  g 1=m

ð14Þ

2

RðS Þ dS
0

where Sw [] is the water saturation in the porous medium,
R [L] is pore radius distribution in the porous medium,
S [] is a dummy saturation variable of integration, and g
and m (m = 1 –2/n) were defined in equation (3). When wq2
and qt are both known then it follows that (1  w)q1 = qt 
wq2. Furthermore, the value of w in the dual-permeability
model is related to g as
w¼

ge
qw2

ð15Þ

where e[] is the porosity of the porous medium. For
saturated conditions the value of e = (1  w)qw1 + wqw2. Ifregions 1 and 2 have the same porosity, then e/qw2 = 1 andhence w = g.
[23] In summary, for steady state flow conditions inhomogeneous porous media the dual-permeability model
parameters can be measured independently or estimatedfrom experimental data. Depending on the conceptualmodel choice, three critical model parameters were identified: k1, w, and wq2; or kdet1, k12 and wq2.
3.2. Example Simulations[24] The sensitivity of dual-permeability model simulations to key parameters (w, wq2, qt, and k12) is explored inthis section. All of the simulations were conducted forsteady state water flow conditions. Breakthrough curves (ata depth of 10 cm) are plotted herein with the total relativeflux concentrations on the y axis and pore volume on the xaxis. The total relative flux concentration is defined as Ct/Ci
where Ci [Nc L3] is the colloid concentration of theinfluent tracer suspension. Final profiles of retained colloids
are plotted herein with the total normalized solid phase
colloid concentration (st/Nic) on the x axis and dimensionless distance from the column inlet on the y axis, where Nic[Nc] is the number of colloids in a unit volume of theinfluent colloid suspension. For these simulations values ofthe volumetric water content, bulk density, median grain

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W02414

Figure 2. Plots of (a) simulated breakthrough curves and (b) retention profiles when wq2 was 0.001 cm
min1, qt was 0.0918 cm min1, and w was 0.0001, 0.001, 0.01, and 0.1 min1. Other model parameters
are provided in the manuscript text.
size, and dispersivity were set equal to 0.334 cm3 cm3,
1.765 g cm3, 0.15 mm and 0.1 cm, respectively, to beconsistent with sandy soils. The colloid input pulse duration
was approximately two pore volumes. A summary of allmodel parameters that were employed in the simulations
below is provided in the figure captions.
[25] Figure 2 presents plots of simulated breakthroughcurves and retention profiles when wq2 = 0.001 cm min1,
qt = 0.0918 cm min1, and w was fixed at 0.0001, 0.001,
0.01, and 0.1 min1. The value of wq2 in these simulations
was selected to be less than 1% of qt to reflect low flow to
the smaller pore spaces as calculated by Bradford et al.
[2006] for sandy soils. The value of k1 was selected to
be 3% of the favorable attachment condition rate (k1 =0.03*k2) based on findings of Dong et al. [2002]. Noticethat increasing w produces lower effluent concentrations
and greater amounts of colloid retention. The value of w
also has a pronounced effect on the shape of the colloid
retention profile. When w is large, colloid retention becomesconsistent with the single region attachment/detachment
model, in which case regions 1 and 2 are approaching a well
mixed condition. In contrast, regions 1 and 2 are not well
mixed when w is low. The overall colloid retention profile isthen the sum of largely independent exponential profiles
from regions 1 and 2. In this case, colloid retention occurs
to a greater extent near the column inlet and produces anoverall nonexponential retention profile because of theslower rate of advection and the high rate of retention inregion 2.
[26] A sensitivity analysis was conducted to further
investigate the influence of the water flux on colloidtransport and retention. The water flux to low- and highvelocity regions is equal to wq2 and (1  w)q1, respectively.
Figure 3 presents plots of the simulated breakthrough curves
and retention profiles when qt was equal to 0.0918 cm
min1 and the value of wq2 was taken to be 0, 0.001,
0.002, 0.004, and 0.008 cm min1. For reasons discussed
above, the values wq2 were selected to be low (<10% of qt),

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Figure 3. Plots of (a) simulated breakthrough curves and (b) retention profiles when qt was equal to
0.0918 cm min1 and the value of wq2 was 0, 0.001, 0.002, 0.004, and 0.008 cm min1. Other model
parameters are provided in the manuscript text.
while k1 was set equal to 0.03*k2. The value of w wasselected to be 0.001 min1 based on parameter sensitivity
observed in Figure 2. Observe that changing the value of
wq2 had relatively little influence on the breakthroughcurves, but a pronounced effect on the shape of the retention
profiles (values of st/Nic at a depth of 0.1 cm were 0.28,
0.40, 0.52, 0.68, and 0.83 g1 when wq2 was 0.0, 0.001,
0.002, 0.004, and 0.008 cm min1, respectively). In
particular, higher values of wq2 produced greater colloid
retention at the column inlet that is not consistent with an
exponential distribution with depth. As mentioned in the
introduction, depth-dependent colloid retention profiles[Bradford et al., 2003, 2006, 2007; Tufenkji and Elimelech,
2005; Li and Johnson, 2005] and earlier breakthrough times[Ryan and Elimelech, 1996; Ginn, 2002] have frequently
been observed under unfavorable attachment conditions.
[27] Figure 4 presents plots of the simulated breakthroughcurves and retention profiles when wq2 was 0.001 cm
min1and qt was equal to 0.024, 0.046, 0.091, 0.181, and

0.361 cm min1. In these simulations values of wq2, k1 =0.03*k2, and w = 0.001 min1 were selected based onjustification provided earlier. Notice that higher values of qt
have a pronounced influence on the shape of the breakthrough curves and the retention profiles. Higher effluent
concentrations occur with increasing qt, while the shape ofthe retention profile becomes more uniform with depth. In
particular observe that less colloid retention occurs at thecolumn inlet with increasing qt. This prediction is consistent
with experimental observations that have shown decreasing
colloid retention near the column inlet with increasing
velocity [Bradford et al., 2006, 2007]. We note here thatclassical filtration theory predicts that the retention profile
will be exponential with distance regardless of the velocity.
[28] Increasing k1 will also impact the breakthroughcurves and retention profiles (data not shown). In particular,
for increasing k1 greater colloid retention will occur
throughout the profile, leading to lower effluent concentrations. When k1 and k2 are equal and wq2 is low then the

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Figure 4. Plots of (a) the simulated breakthrough curves and (b) retention profiles when wq2 was
0.001 cm min1and qt was equal to 0.024, 0.046, 0.091, 0.181, and 0.361 cm min1. Other model
parameters are provided in the manuscript text.

retention profile will approach results of a conventional
single-region attachment/detachment model.
[29] When exchange of colloids on the solid phase (rolling) is considered in the model, then different values for k1
and kdet1 need to be employed to accurately reflect theeffects of hydrodynamics on colloid retention. As discussed
in section 3.1, when rolling occurs we assumed that k1 isequal to k2 to be consistent with an initially high rate ofmass transfer to the solid-water interface. The values of kdet1
and k12 are then used to partition colloids on the solid
surface (region 1) back into solution (region 1) or to thesolid phase in region 2, respectively. At equilibrium thevalue of kdet1 = k1/(R1), where R [] is the retardation
coefficient. For determination of kdet1 we assumed a valueof R = 2.5 to allow sufficient time for rolling to occur andthis choice produces a slightly retarded breakthrough time
for the colloids. Figure 5 shows plots of the simulated
breakthrough curves and retention profiles when qt =0.0918 cm min1, wq2 = 0.001 cm min1, and k12 wasequal to 0.001, 0.01, 0.05, and 0.1 min1. Notice that the

breakthrough curves and the retention profiles are sensitive
to values of k12. Little colloid retention occurs when k12 issmall because of the low-mass transfer rate (rolling) toregion 2, causing the colloids in region 1 to move back into
the aqueous phase because of the high value of kdet1.
Conversely, increasing k12 leads to lower effluent concentrations and greater colloid retention throughout the profile(effects are not localized to the column inlet).
3.3. Application to Data
[30] This section investigates the ability of the dualpermeability model to describe experimental colloid breakthrough curves and retention profiles that were presented
by Bradford et al. [2007] for various solution chemistries(pH = 10 and ionic strength, IS), Darcy velocities, andcolloid (carboxyl modified latex microspheres) and grain(Ottawa quartz sand) sizes. Table 1 provides a summary of
the experimental conditions. DLVO calculations and batch
experiments conducted for these same colloids, porous
media, and solution chemistries indicated that conditions
were highly unfavorable for colloid attachment [Bradford

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Figure 5. Plots of (a) the simulated breakthrough curves and (b) retention profiles when qt was equal to
0.0918 cm min1, wq2 was equal to 0.001 cm min1, and k12 was equal to 0.001, 0.01, 0.05, and
0.1 min1. Other model parameters are provided in the manuscript text.

Table 1. Summary of Experimental Conditions and Dual-Permeability Model Parametersa
dcb (mm)d50 (mm)

ISc (mM)eqt (cm min1)l (cm)

wq2/qtk1 (min1)

2
RBTC

2
RRP

3
3
3
1.1
1.1
1.1
1.1
1.1
1.1
1.1

360
240
150
150
150
150
150
150
150
150

6
6
6
56
56
6
31
56
81
106

0.35
0.32
0.35
0.35
0.36
0.33
0.35
0.35
0.35
0.34

0.096
0.104
0.096
0.099
0.446
0.104
0.098
0.099
0.084
0.101

0.718
0.280
0.718
0.393
0.393
0.393
0.393
0.393
0.393
0.393

0.011
0.004
0.008
0.008
0.001
0.015
0.011
0.008
0.005
0.013

0.008
0.042
0.058
0.040
0.061
0.001
0.003
0.040
0.037
0.202

0.974
0.976
0.936
0.940
0.980
0.977
0.961
0.940
0.980
-

0.989
0.997
0.998
0.995
0.934
0.947
0.876
0.995
0.992
0.998

a
As used in the simulations presented in Figures 6 – 8, as well as the coefficient of linear regression for the goodness of the model fit to breakthrough2
2) and retention profile (RRP
) data.
curve (RBTCbHere is dc colloid diameter.
c
IS is ionic strength.

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Figure 6. Plots of (a) observed and simulated breakthrough curves and (b) retention profiles for 3 mm
colloids when qt was approximately 0.1 cm min1, the solution ionic strength was 6 mM, the pH was 10,
and d50 was equal to 360, 240, and 150 mm. Other model parameters are provided in the manuscript text
and in Table 1.
et al., 2007]. In brief, colloid transport experiments wereconducted by pumping a monodispersed colloid suspension
upward through a vertically oriented saturated column(4.8 cm inside diameter and 13 cm in length) at a steadypore water velocity. After a given colloid tracer pulse
duration was applied, a three-way valve was used to switch
to eluting solution of the same solution chemistry forseveral more pore volumes. Effluent samples were collected
and analyzed for colloid concentration using a Turner
Quantech Fluorometer (Barnstead/Thermolyne, Dubuque,IA). Following completion of the colloid transport experiments, the spatial distribution of retained colloids in eachpacked column was determined by excavating the sand into
tubes containing excess eluant solution, slowly shaking thetubes for 15 min, and measuring the concentration of thecolloids in the excess solution with the fluorometer. A
detailed discussion of the experimental conditions andprotocols is given by Bradford et al. [2002, 2007].
[31] Dual-permeability model parameters were fitted tothe colloid transport data discussed above. Information

presented in section 3.2 was used to constrain the modelfits. Figures 3 and 4 demonstrated that colloid retention near
the inlet boundary can be controlled by the value of wq2 that
was associated with high rates of irreversible retention toregion 2 when w is low. Consistent with these conditions,the value of wq2 was optimized with w = 0.0001 min1 (seeFigure 2), k2 was determined from equation (12), and kdet2 =0. The value of (1  w)q1 was determined as qt  wq2.
Figures 2 – 5 demonstrate that colloid retention away fromthe inlet boundary was controlled by k1 or k12 depending on
the conceptual model choice of ‘‘irreversible’’ or ‘‘reversible’’ retention in region 1, respectively. In the absence ofadditional information, we assumed ‘‘irreversible’’ colloid
retention in region 1 by optimizing the value of k1, andsetting k12 = 0 and kdet1 = 0. Other dual-permeability model
parameters were determined from measured properties orwere estimated. Values of the volumetric water content,bulk density, and dispersivity (l; L) were assumed to beequal for both regions 1 and 2, and were set equal tomeasured values in the various sands. Table 1 provides a

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Figure 7. Plots of (a) observed and simulated breakthrough curves and (b) retention profiles for 1 mm
colloids when the solution ionic strength was 56 mM, the pH was 10, d50 was 150 mm, and qt was around
0.1 and 0.45 cm min1. Other model parameters are provided in the manuscript text and Table 1.

summary of dual-permeability model parameters that wereused in the simulations discussed below, as well as thecoefficient of linear regression for the goodness of themodel fit to breakthrough curve (R2BTC) and retention profile(R2RP) data.
[32] Figure 6 presents plots of observed and simulated
breakthrough curves (Figure 6a) and colloid retention profiles(Figure 6b) for 3 mm colloids when qt was approximately
0.1 cm min1, the solution ionic strength was 6 mM, the pH
was 10, and d50 was equal to 360, 240, and 150 mm. The
dual-permeability model provided a good description ofboth the breakthrough curves (R2BTC > 0.94) and theretention profiles (R2RP > 0.99). The fitted value of wq2 waslow, less than 2% of qt (Table 1). Hence, small amounts offlow to low velocity regions with enhanced rates of colloid
retention provide one plausible explanation for nonexponential colloid retention profiles near the column inlet.
Information presented by Bradford et al. [2006] indicatesthat for a given value of w the value of wq2/qt will dependon the grain size distribution, with more uniform sandhaving higher values of wq2/qt. Fitted values of wq2/qt inTable 1 are consistent with this prediction. Colloid retention inregion 1 dominated the deposition profile shown in Figure 6bat dimensionless depths greater than around 0.2. Table 1 also
indicates that k1 increased with increasing d50. This finding
indicates that colloid retention in the higher-velocity regionsof the pore space increased with increasing solid surface area,
likely either due to attachment or colloid retention inducedby surface roughness. We note that alternative conclusions
may be reached if reversible colloid retention in region 1 isassumed (k12 and kdet2 greater than zero).
[33] Figure 7 presents plots of observed and simulated
breakthrough curves and colloid retention profiles forsmaller 1 mm colloids when the solution ionic strength
was 56 mM, the pH was 10, d50 was 150 mm, and qt wasaround 0.1 and 0.45 cm min1. The dual-permeability
model again provided a good description of the breakthrough curves and the colloid retention profiles. The

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Figure 8. Plots of (a) observed and simulated breakthrough curves and (b) retention profiles for 1 mm
colloids when d50 was 150 mm, qt was 0.1 cm min1, the pH was 10, and the solution ionic strength was
6, 31, 56, 81, and 106 mM. Other model parameters are provided in the manuscript text and Table 1.
transport data for the higher water velocity system yielded alower value of wq2/qt and a higher value of k1. A lower
value of wq2/qt with increasing velocity is expected basedon equation (2) (the pressure gradient and the velocity areproportional to each other). An increase in k1 with velocity
is also expected based on filtration theory predictions.
Hence, these observations both imply that the dualpermeability modeling approach accurately characterizes
the physics of the system.
[34] Figure 8 shows plots of observed and simulated
breakthrough curves and colloid retention profiles for
1 mm colloids when d50 was 150 mm, qt was around
0.1 cm min1, the pH was 10, and the solution ionicstrength was 6, 31, 56, 81, and 106 mM. The model
provides a good description of the breakthrough curves andthe colloid retention profiles. As the solution ionic strength
increased the value of k1 increased as expected becauseof greater attachment in the secondary minimum of the
DLVO interaction energy profile. The value of wq2, however,

did not vary in a systematic fashion as predicted byequation (2). This is likely due to nonunique parameter fits
and/or the assumed conceptual model of irreversible
retention in region 1. In reality, it may be possible to haveboth reversible and irreversible colloid retention in region 1.
[35] For comparison purposes the data presented in
Figures 6 –8 were also simulated using the conventional
first-order attachment-detachment model (single region)by optimizing the attachment (katt, T1) and detachment(kdet, T1) coefficients. Table 2 provides a summary offitted attachment-detachment model parameters, as well as
R2BTC and R2RP. A comparison of Tables 1 and 2 reveals thatthe dual-permeability model provided superior fits to bothbreakthrough curves and especially the retention profiles
when optimizing the same number of parameters. Thisfinding suggests that the dual-permeability modelingapproach provides a more physically realistic descriptionof the physics of the system than the conventional
attachment-detachment model. Another advantage of the

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Table 2. Summary of Experimental Conditions and Attachment-Detachment Model Parametersa
dcb (mm)d50 (mm)

ISc (mM)eqt (cm min1)l (cm)

katt (min1)

kdet (min1)

2
RBTC

2
RRP

3
3
3
1.1
1.1
1.1
1.1
1.1
1.1
1.1

360
240
150
150
150
150
150
150
150
150

6
6
6
56
56
6
31
56
81
106

0.35
0.32
0.35
0.35
0.36
0.33
0.35
0.35
0.35
0.34

0.096
0.104
0.096
0.099
0.446
0.104
0.098
0.099
0.084
0.101

0.718
0.280
0.718
0.393
0.393
0.393
0.393
0.393
0.393
0.393

0.007
0.042
0.057
0.039
0.051
0.002
0.003
0.039
0.040
0.285

1.5E-4
5.7E-4
3.1E-6
3.0E-4
3.1E-5
2.2E-3
1.6E-33.0E-4
9.3E-6
9.9E-7

0.956
0.974
0.917
0.956
0.981
0.987
0.982
0.956
0.973
-

0.610
0.735
0.773
0.761
0.948
0.142
0.471
0.761
0.999
0.998

2
a
As fitted to data shown in Figures 6 – 8, as well as the coefficient of linear regression for the goodness of the model fit to breakthrough curve (RBTC) and
2) data.
retention profile (RRPb
Here is dc colloid diameter.
c
IS is ionic strength.

dual-permeability model is that specific hypotheses about
the influence of hydrodynamics on colloid retention can betested.

4. Summary and Conclusions[36] Recent experimental and theoretical work has demonstrated that pore space geometry and hydrodynamics play
an important role in colloid retention under unfavorable
attachment conditions. In particular, hydrodynamic forces
have been demonstrated to funnel weakly associated colloids to grain-to-grain contacts and to hydrodynamicallyisolated low-velocity (eddy) regions. A balance of adhesiveand hydrodynamic torques acting on colloids in theselocations indicated that enhanced retention will occur inthese low-velocity regions. Consequently, conceptual models that consider the average pore water velocity in the
porous medium and a single attachment rate coefficient are
not always adequate to describe observed colloid retention
profiles, which are frequently nonexponential with distance.
[37] In this work, we highlighted a model formulationthat can be used to account for enhanced colloid retention in
low-velocity regions of the pore space. The model conceptualized the pore domain in terms of low- and highervelocity regions, and included provisions for advective
and dispersive transport and first-order colloid retention
and release in both regions. The model also considered
colloid exchange between low- and higher-velocity regionsof the pore space in the aqueous and solid (rolling) phases.
On the basis of literature evidence, we assumed thatenhanced colloid retention occurred in the low-velocity
regions of the pore space. With this assumption, a sensitivity analysis was performed with the dual-permeability
model parameters. Simulation results indicated that low
amounts of advective transport to low-velocity regions of
the pore space had a dramatic effect on the shape of theretention profile, especially near the inlet boundary. The
total water flow rate was also shown to have a significant
influence on the shape of the colloid retention profile near
the inlet boundary. In this case, higher water velocities were
found to produce less colloid retention near the inlet
boundary because greater amounts of colloids bypassed
the low-velocity regions. Both of these predictions are
consistent with experimental observations that have beenreported in the literature. Away from the inlet boundary,colloid retention was controlled by the deposition rate in thehigher-velocity region, and the aqueous and solid phaseexchange rates.
[38] Published colloid transport and retention data thatwere obtained for unfavorable attachment conditions for arange of colloid and sand sizes, water velocities, and solution
chemistries were described using the dual-permeability
model. Fitted model parameters exhibited systematic trends
with grain size, velocity, and solution chemistry. The dualpermeability model provided a plausible interpretation for
the experimental observations, and a reasonable approximation of the pore-scale physics controlling colloid retention
under unfavorable attachment conditions.
[39] Acknowledgments. This research was supported by the 206
Manure and Byproduct Utilization Project of the USDA-ARS, by a grantfrom NRI (NRI 2006-02541), and by an interagency agreement with the
EPA (IAG DW-12-92189901-0). Mention of trade names and company
names in this manuscript does not imply any endorsement or preferential
treatment by the USDA.

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S. A. Bradford, US Salinity Laboratory, ARS, USDA, 450 West Big
Springs Road, Riverside, CA 92507, USA. (scott.bradford@ars.usda.gov)
F. Leij, 43 Tortuga Cay, Aliso Viejo, CA 92656, USA.
J. Šimůnek and S. Torkzaban, Department of Environmental Science,
University of California, Riverside, CA 92521, USA.
M. T. van Genuchten, Department of Mechanical Engineering, Federal
University of Rio de Janeiro, Rio de Janeiro, RJ CEP 21945-970, Brazil.

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