Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 4.7.3 - Colloid Retention in Saturated Porous Media under Unfavorable Surface Conditions
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Permanent Link: http://ufdc.ufl.edu/UF00102023/00117
 Material Information
Title: 4.7.3 - Colloid Retention in Saturated Porous Media under Unfavorable Surface Conditions Colloidal and Suspension Dynamics
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Qiu, C.Q.
Gao, H.
Han, J.
Jin, Y.
Wang, L.-P.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: colloid retention
classic filtration theory
secondary energy minimum (SEM)
hydrodynamic interaction
 Notes
Abstract: Colloid transport and retention in saturated porous media is a complex phenomenon that is governed by pore-scale flow field, Brownian fluctuations, and a number of colloid-surface and colloid-colloid physicochemical interaction forces. Here we develop a computational approach to study the transport and retention of colloids under unfavorable attachment conditions where interactive surfaces are like charged and a repulsive energy barrier exists. A twodimensional flow model is used to represent a small region of a porous medium containing multiple grains. A direct Lagrangian particle-tracking is used to simulate colloid transport by solving the equation of motion for each colloid using a Runge-Kutta method with variable time steps. Sufficiently small time steps are used in zones where fluid velocity and DLVO forces change rapidly. Simulation results show that as long as the primary energy barrier is high enough, colloids cannot overcome the primary energy barrier and deposition at the primary energy minimum does not occur. For unfavorable cases where the secondary energy minimum (SEM) exists, retentions can occur within the SEM region. Colloids trapped in SEM tend to follow the near surface flow and move along a collector surface slowly towards the rear stagnation zones. The rate of retention at SEM increases with increasing depth of SEM until it reaches a maximum around a SEM depth of −10 kT to −15 kT , where k and T are Boltzmann constant and temperature, respectively. Most interestingly, the retention at the SEM regon becomes reversible if SEM is shallow, with a portion of trapped colloids escaping the SEM at later times due to Brownian fluctuations. The retention or attachment efficiency was found to reach 100% when the SEM depth is at least 15kT . The simulated attachment efficiency was found to be less than the previous Maxwell model prediction, although the qualitative trend is similar.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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Bibliographic ID: UF00102023
Volume ID: VID00117
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: 473-Qiu-ICMF2010.pdf

Full Text
ICMF 2010, Tampa, FL, May 30- June 4, 2010


Colloid Retention in Saturated Porous Media under Unfavorable Attachment
Conditions


C.Q. Qiu*, H. Gao*, J. Hant, Y. Jint and L.-P. Wang*

Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, USA
t Department of Plant and Soil Sciences, University of Delaware, Newark, DE 19716, USA
qiuqm@udel.edu, hgao@udel.edu, yjin@udel.edu, and lwang@udel.edu
Keywords: colloid retention, classic filtration theory, secondary energy minimum (SEM), hydrodynamic interaction




Abstract

Colloid transport and retention in saturated porous media is a complex phenomenon that is governed by pore-scale
flow field, Brownian fluctuations, and a number of colloid-surface and colloid-colloid physicochemical interaction
forces. Here we develop a computational approach to study the transport and retention of colloids under unfavorable
attachment conditions where interactive surfaces are like charged and a repulsive energy barrier exists. A two-
dimensional flow model is used to represent a small region of a porous medium containing multiple grains. A direct
Lagrangian particle-tracking is used to simulate colloid transport by solving the equation of motion for each colloid
using a Runge-Kutta method with variable time steps. Sufficiently small time steps are used in zones where fluid
velocity and DLVO forces change rapidly. Simulation results show that as long as the primary energy barrier is high
enough, colloids cannot overcome the primary energy barrier and deposition at the primary energy minimum does
not occur. For unfavorable cases where the secondary energy minimum (SEM) exists, retentions can occur within
the SEM region. Colloids trapped in SEM tend to follow the near surface flow and move along a collector surface
slowly towards the rear stagnation zones. The rate of retention at SEM increases with increasing depth of SEM until
it reaches a maximum around a SEM depth of -10 kT to -15 kT, where k and T are Boltzmann constant and
temperature, respectively. Most interestingly, the retention at the SEM region becomes reversible if SEM is shallow,
with a portion of trapped colloids escaping the SEM at later times due to Brownian fluctuations. The retention or
attachment efficiency was found to reach 100% when the SEM depth is at least 15kT. The simulated attachment
efficiency was found to be less than the previous Maxwell model prediction, although the qualitative trend is similar.


Introduction

When colloid particles of micrometer or sub-micrometer
size enter a soil medium saturated with water, they can
be transported by the microscale water flow through the
soil pores typically of dimension 100 pm to 1 mm. At
the same time, colloids interact with grain surface and,
under so-called favorable conditions where there is an
attractive electrostatic force between the colloids and
the grain surface, they can easily deposit on the soil
grain surface and become immobile. Such deposition
or retention of colloids is also driven in part by the van
der Waals attractive force acting on a colloid when it
is near a grain surface. The electrostatic force and the
van der Waals force are usually described together in
terms of Derjaguin-Landau-Verwey-Overbeek (DLVO)
energy potential (Derjaguin and Landau 1941; Verwey
and Overbeek 1948). Traditionally, the classic filtration


theory (CFT, e.g., (Yao et al. 1971)) has been success-
fully used to quantify the rate of retention for favorable
soil surface conditions and it predicts that the concentra-
tion of colloids decays exponentially with distance as a
result of the first-order retention kinetics. The transport
and retention of colloids in the subsurface environment
determine the distribution and penetration of colloids in
groundwater, and are of importance to groundwater con-
tamination and remediation treatments. In this paper, we
shall limit our discussions to a saturated porous medium
where water fills the pore space completely.
For most natural subsurface environment, however,
both the grain surface and colloids tend to be nega-
tively charged and the electrostatic interaction force is
then repulsive (the van der Waal force is still attrac-
tive). The net DLVO force can be repulsive at certain
small gap distances. If the repulsive energy barrier is
strong enough, a colloid may not break the barrier to






contact the grain surface (i.e, primary deposition). Out-
side this primary barrier, if the van deer Waals force
dominates, there may exist a negative energy well im-
mediately outside the primary energy barrier, known as
the secondary energy minimum (SEM). Colloids could
be retained by this secondary energy minimum without
direct contact with the grain surface, and they become
immobile as long as the strength of the secondary en-
ergy minimum is maintained. That is one of the rea-
sons that the classical filtration theory becomes inaccu-
rate when comparing to experiment results (Martin et al.
1996; Baygents et al. 1998; Bolster et al. 1999; Red-
man et al. 2001) under this so-called unfavorable con-
dition. An important characteristics of retention by the
secondary energy minimum is that such retention is re-
versible, namely, colloids can be released back to the
bulk flow when the medium is flushed with deionized
water (Hahn et al. 2004; O'Melia et al. 2004; Franchi
et al. 2003) or with low ionic strength solution remov-
ing the secondary energy minimum (Redman et al. 2004;
Tufenkji et al. 2004). Shen et al. (2007) observed contin-
uous colloidal reentrainment by Brownian motion with-
out chemical and hydrodynamic disturbances.
For unfavorable surface conditions, the classical fil-
tration theory becomes inaccurate when comparing to
experimental results (Martin et al. 1996; Baygents et al.
1998; Bolster et al. 1999; Redman et al. 2001). The re-
tention at the SEM is not considered in the classical fil-
tration theory. When the DLVO force creates a large pos-
itive primary energy barrier, most colloids cannot over-
come it to reach the primary energy minimum.
In this paper, we focus on the colloidal retention un-
der unfavorable attachment conditions with a large pri-
mary energy barrier but at the same time an apprecia-
ble secondary energy minimum. Numerical simulations
are used to investigate the dynamics of colloids near a
grain surface, under the coupled influence of the DLVO
force, hydrodynamic force, and Brownian fluctuations.
A direct Lagrangian particle-tracking model is used to
simulate colloid transport and retention in a model 2D
porous channel that is partially filled with cylindrical
glass beads as shown in Fig. 1. This flow model was
previously solved in Gao et al. (2008). An interesting
feature to be discussed here is the possibility of retained
colloids near the secondary energy minimum to return
to the bulk flow due to Brownian thermal fluctuations,
if the depth of the secondary energy minimum is below
15kT. Here kT is a measure of energy level for thermal
fluctuations, where k is the Boltzmann constant and T
is the medium temperature. This dynamic reversibility
occurs even when the solution conditions are unaltered,
due to the competition of the attractive DLVO force near
the secondary energy well and the random Brownian
force. The competition is modulated by the hydrody-


ICMF 2010, Tampa, FL, May 30- June 4, 2010


namic drag force. Simulation results show that the depth
of the secondary energy well is crucial in determining
the colloidal retention ratio. This dynamical reversibility
has previously been observed indirectly in experiments
by Hahn et al. t 21 '4); O'Melia et al. t 21 '4); Franchi et
al. (2003); Redman et al. (2 114); Tufenkji et al. (2 114),
but our numerical simulations provide direct evidence
and can quantify the degree of reversibility. It is impor-
tant to recognize that the associated colloid-grain surface
interactions take place at a gap distance of the order of
100 nm, accurate observations of colloid trajectories in
experiments could be rather difficult. To better investi-
gate this dynamically reversible retention, a simple the-
oretical model to predict the attachment efficiency will
also be discussed in this paper.


Flow simulation

A two-dimensional channel partially filled with fixed
glass beads is used to model soil porous media. Fig. 1
shows one periodic length of the flow model where
seven cylindrical glass beads are placed between two
channel walls. This 2D flow model is used to mimic
a slice of 3D channel of a 0.8mmx0.8mm square cross
section, packed with glass beads (0.20mm in diameter)
used in the experimental study of Han (2008). The di-
ameter of the cylinders is set to 0.12 mm, and the 2D
model has a porosity of 0.734 that is identical to the
porosity of the 3D experimental model.
The slow viscous flow through this porous channel
was solved previously by Gao et al. (2008), using two
very different methods: a mesoscopic lattice Boltzmann
equation and macroscopic Navier-Stokes solver. They
showed that the two methods yielded an excellent agree-
ment of the simulated flow details. Since it is very diffi-
cult to accurately measure experimentally the local flow
in microscale porous media, this cross validation of nu-
merical flow fields provides a necessary starting point
for the simulation of colloid transport. In this paper, we
simply adopt the steady-state flow field already solved
by Gao et al. (2008), assuming the presence of colloids
at very low volume fraction does not affect the flow.


Lagrangian Tracking of Colloids

With the flow field being established, each colloid is re-
leased into the porous channel at the inlet, with its cross
channel (x) position selected randomly assuming a uni-
form probability distribution. The initial velocity of the
colloid is set to the local fluid velocity at the release lo-
cation. The rate of release at the inlet corresponds to
a bulk concentration of colloids at Ippm. Here 1 ppm
indicates that the colloid mass concentration is 1 mg/1




ICMF 2010, Tampa, FL, May 30- June 4, 2010


0


Figure 1: A 2-D geometric model of the porous medium
used for colloid transport study. The elemental flow do-
main consists of 7 cylinders in a channel and the peri-
odic extension of the domain provides a porous channel
of any length in the flow (y) direction.


or a number concentration of 1810 per mm3 of the so-
lution. It is assumed that the colloids do not affect the
flow field. Since the density of colloids (1055 kg/m3) is
very close to the density of the solution (1000 kg/m3),
gravity and buoyancy force cancel each other; thus they
are not considered in our colloid tracking model.
The motion of a colloid is solved more conveniently
in directions normal and tangential to the nearest surface
of a grain or channel wall according to


4 5 ) 6 ) ( 7



1 2
1: 1 ( 2 ) ( 3 1
./ ', .,. .


-2000 | I I I I I
0.00 0.02 0.04 0.06 0.08 0.10
ha/a
L i I I


(b)
-5

.-10

g -15

a2
4 -20
6


dVn
m-
dt
m-
dt


Fdrag + F F Brown (1)
Fin in i,n 1


Fdrag FBrown
it i,t


where Vi,n and V,t [m s 1] are the colloid velocity
components, t [s] is time FV [N] is the DLVO inter-
action force acting on the i-th colloid with the nearest
collector or another nearby colloid. F~ are Fda9
are Stokes drag force in the normal and tangential direc-
tions, respectively. F~ron" and F~ row denote Brown-
ian force. The normal direction is defined as the surface
normal pointing into the fluid. Each colloid trajectory
is updated by adding the deterministic part (due to drag
force and colloidal force) and Brownian motion part to-
gether. Determnistic part is updated with a 4-th order
Runge-Kutta scheme. Brownian force is added into col-
loidal trajectory using the Euler scheme after the deter-
ministic forces have been added.
Surface interactions of a colloid with a collector
surface or another nearby colloid include electrostatic,
Lifshitz-van der Waals and Lewis acid/base interactions:

Fc = FDL + FLW + FAB (3)


0.20
hpWa1,


Figure 2: (a) Energy profiles at different solution and
surface conditions. (b) The zoom-in view of energy pro-
files around SEM. Two vertical lines at gap distance of
50nm and 150nm mark threshold distances to define the
SEM retention. 50nm defines separation distance where
SEM retention occurs for ionic strengths from 0.01M,
0.03M, 0.05M,0.1M, 0.3M. 150nm defines the separa-
tion distance where SEM retention at ionic strength of
0.001M.


where F DL is the electrostatic double layer force,
FLw is the Lifshitz-van der Waals force, and FAB is
the Lewis acid/base force. Each force is obtained from
its corresponding interaction energy (Liang et al. 2007).
The detail expressions for these interaction forces are
given in Gao et al. (2010).
The DLVO energy profiles for the cases considered
in this study are shown in Fig. 2. We include a case
indicating favorable surface condition with the electro-


5000
(a)
4000

3000

A
S2000

a 1000

o o
-1000
-1000


favorable
IS=0.30M
IS=O.1OM
IS=0.05M
IS=0.03M
----IS=O.O1M
- IS=0.001M




ICMF 2010, Tampa, FL, May 30- June 4, 2010


static interaction taken from 0.1 M but it is assumed that
the electrostatic interaction is attractive. The assumed
values of the zeta potentials can be found in Gao et al.
(2008, 2010). It is noted that the primary energy barrier
is above 1000 KT for all unfavorable cases. The depth
of the secondary energy minimum varies from -1 kT to
-30 kT for the unfavorable cases considered.
At a very small particle Reynolds number (Rep < 1),
the drag force acting on the colloid is assumed to follow
Stokes' law (Heimenz et al. 1997). When colloids move
close to a collector surface, this drag force is modified
due to the local hydrodynamic interaction of the colloid
with the nearby collector surface and this modification
is handled by hydrodynamic correction factors (Brenner
1961; Goren 1970; Gorenet al. 1971; Goldman et al.
1967). The Brownian force is also modified as a result
of the collector surface. Correction factors to account
for the impact of nearby collector surface were defined
in Spielman et al. (1972). In our simulations, the drag
forces are described as


Sdrag
F n = 6/tac f2Ui,.


Vf,
i)


(4)

(5)


Fd ra a 67G ac (f3Ui,t Vi,t)
f4


where p is the solution viscosity, a, is the colloid ra-
dius, uia, are uit [ms 1] are the normal and tangen-
tial fluid velocity components at the colloid location.
The physical significance of the hydrodynamic correc-
tion factors fl, f2, f3, f4 has been discussed in Brenner
(1961); Goren (1970); Gorenet al. (1971); Goldman et
al. (1967). In our simulations, we use the following ex-
plicit forms (Gao et al. 2010)


fi = 1.0 0.443 exp(-1.299h)
-0.5568 exp(-0.327~ 75)
f2 = 1.0 + 1.455 exp(-1.2596h)
+0.7951 exp(-0.56h7050)
f3 1.0- 0.487 exp(-5.423h)
-0.5905 exp(-37.83h50)


f4 1.0 0.35 exp(-0.25h)
-0.40 exp(-10.0h) (9)

where h = is gap distance between a colloid and
a,
nearest grain surface normalized by the colloid radius.
Fig. 3 shows the values of fl, f2,f3,f4 as a function of
h.
The Brownian force away from the grain surface is
specified as Ff = (Ff, FB), where each component is
an independent Gaussian random variable of zero mean
and the following standard deviation

/127vpackT
aB = p, = dt (10)
V~ dt


(6)


'-7


4.0
a-


S3.0
o


U 2.0



0
S1.0
-


I I


0.0 -fI I I . I . I . I I I
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
hgadapp

Figure 3: Hydrodynamic correction factors at different
normalized gap distance h h/a, h is gap distance, ac
is colloid radius.


200 -



100 -

k -

0-



-100


i-


0.30 0.40


hg.aap


Figure 4: Comparison of Brownian force and DLVO
forces at different ionic strengths. The DLVO force at
the favorable surface condition is also plotted for com-
parison. All the forces are normalized by drag force at
the mean flow or Darcy velocity (Gao et al. 2008).


where dt is the time step size, T is the temperature (as-
sumed to be 293 K), k 1.38 x 10-23 J/K is the Boltz-
mann constant. The hydrodynamic corrections near a
surface can be found in Gao et al. (2010).
Fig. 4 shows DLVO forces at various solution ionic


f4
f3

f2


------


favorable
FBrownian
modified FBrownian
IS=0.30M
IS=0.10M
IS=0.05M
IS=0.03M
----IS=0.01M
- IS=0.001M




ICMF 2010, Tampa, FL, May 30 June 4, 2010


strengths (IS) as a function of the normalized gap dis-
tance. Also shown are Brownian force and the DLVO
force for favorable surface condition. The DLVO force
changes rapidly at small gap distances. The magnitude
of Brownian force without modification by the hydro-
dynamic correction factor is comparable to the magni-
tude of DLVO forces. The modified Brownian force
is slightly larger than DLVO forces at different ionic
strengths under unfavorable conditions. When the ionic
strength is larger than 0.1M, numerical simulations show
that no colloid entered in the secondary energy mini-
mum (SEM) could escape to return to the bulk flow (see
Fig. 8 below), although the Brownian force is a little
larger than DLVO forces. This might be due to the fact
that the Brownian force changes direction from one time
step to the next, while the DLVO force is always in the
normal direction of collector and is attractive in the SEM
region. Thus, the cumulative effect of random Brown-
ian motion cannot overtake the persistent effect of the
DLVO force. A small time step is used to advance col-
loid trajectory when a colloid is very close to a surface,
in order to accurately track colloidal retention and reen-
trainment.
The magnitudes of Brownian force, drag force and
colloidal forces near SEM are also displayed in Table 1.
Brownian force and colloidal forces are similar in mag-
nitude, while the drag force is relatively small. Com-
paring the magnitude of Brownian force and colloidal
force near SEM helps to explain why a colloid retained
in SEM could escape to re-enter the bulk flow. Although
the drag force is smaller comparing to the other forces, it
could still dominate the tangential movement of a colloid
along collector surface when it is captured. This then
drives a captured colloid moving towards a flow stagna-
tion point, as seen in Fig. 5a where a colloid is purposely
released close to the collector surface. When a colloid is
retained in SEM, it stays in the region around SEM since
in our simulations the primary energy barrier is large and
it is impossible for a colloid overcome the primary bar-
rier. Similar simulation results were reported in Johnson
et al. (2007). Colloid retention at the rear stagnation
zone was also observed in some experiments (Kuznaret
al. 2007).


Results: The overall colloid retention

We shall first examine the total number of colloids re-
tained on grain surface and channel wall by the SEM
well as a function of the total number of colloids re-
leased at the inlet (or equivalently time). We must first
note that a threshold gap distance is used to count the
number of retained colloids. Namely, a colloid is viewed
as having been retained by the SEM if it reaches a gap
distance below the threshold distance, although in reality


(a)


80 -


60-


40


20


19.78


19.76


19.74


19.72


19.70


19.68


I I I I


I I I I I
60 80 100 120 140
X


99.90 99.93 99.96
99.90 99.93 99.96


99.99 100.02


100.05


x

Figure 5: The trajectory of a colloid released near front
stagnation point of a collector surface at ionic strength of
0.1 M and flow velocity 8 m d 1. (a) Colloid retained
in SEM moves along collector surface till rear stagna-
tion point. (b) Zoom in of (a) when the colloid first
approaches the collector surface by attractive colloidal
forces and moves into SEM. Blue curve indicates the
SEM location and Brown curve marks the primary en-
ergy barrier location, collector surface is almost overlap
with Brown line as the distance between collector sur-
face and the primary energy barrier is very small com-
paring to the gap distance to the secondary energy mini-
mum.



this colloid could still moves within the SEM region or it
could re-enter the bulk flow later when a strong thermal-
fluctuation disturbance occurs. This distance is set to
50 nm for the solution ionic strength that is 0.01 M or


particle trajectory


particle trajctory


/, /!"4~-




ICMF 2010, Tampa, FL, May 30- June 4, 2010


Table 1: The magnitudes of Brownian force, drag force and colloidal forces at various distances and ionic
strengths,Brownian force and drag force are calculated at separation distance of 10nm, all the forces are normalized
by xlO9N
Brownian Drag Force Colloidal forces at the following distances(nm)
(M) Force Normal* Parallel* 0.20 0.78 1.00 2.00 5.00 10.00 20.00 50.00 100.00
0.001 -9.55 1.64 1.60 1.25 1.04 0.73 0.34 1.60E-2 3.07 E-
0.010 -7.15 4.97 4.85 3.87 1.84 0.36 0.17 -2.13E-4 -5.35E-
0.050 5E-3 5E-6 2.5E-5 0.58 8.51 7.62 4.23 0.52 5.5E-3 -1.4E-3 -2.14E-4 -5.35E-
0.100 1.19 7.98 6.78 2.79 0.11 -4.4E-3 -1.4E-3 -2.14E-4 -5.35E-
0.300 0.02 5.55 4.08 7.64 -0.017 -4.9E-3 -1.4E-3 -2.14E-4 -5.35E-


300



200



100


I I I I II


| -r I I I I I + '
0 300 600 900 1200 1500 1800 2100
No. of injected colloids


Figure 6: Number of colloids retained on collector sur-
face at different ionic strengths and fluid velocity of
8 m d 1. The results for the favorable surface condi-
tion are plotted for comparison. For the favorable condi-
tion, colloids are deposited at the PEM. For unfavorable
conditions, colloids are retained at the SEM.


higher. This distance corresponds to roughly a DLVO
energy at -3 kT (see Fig. 2b) and it is larger than the
gap distance corresponding to the SEM location. How-
ever, for the lowest ionic strength (0.001 M) considered,
the SEM well is shallow and is located outside 50 nm.
The threshold location is adjusted to 150 nm. While the
quantitative value of colloid retention could depend on
this threshold distance especially at low solution ionic
strength, the overall conclusion to be drawn later is not
changed.
For the favorable surface condition, all the colloids
are deposited at the primary energy minimum and we
simply use a threshold distance of 0.157 nm. Increasing
the threshold distance to 50 nm in this case does not alter
the results on the number of retained colloids, as there
very few colloids found between 0.157 nm and 50 nm
gap distances from the collector surfaces.


."EihII"e


,/
//


Fig. 8 shows the number of colloids retained at dif-
ferent time colloidss are randomly released at inlet at a
rate of Ippm, so the number of injected colloids is pro-
portional to the colloidal transport time). Several obser-
vations can be made. First, at a given ionic strength,
the rate of retention as measured by the slope of the
curves increases with time. This is due to the fact that
as colloids move downstream, they cover a larger region
and more grain and wall surface area becomes available.
Second, at any given time, the number of retained col-
loids increases with increasing ionic strength. This is
expected as the SEM well depth increases with the ionic
strength (see Fig. 2b). The number of retained colloids
increases rapidly with the ionic strength in the range of
0.01 M to 0.05 M. Third, the number of retained col-
loids no longer increases with the ionic strength when
the ionic strength is above 0.05 M. This can be explained
by the fact that the SEM well has a DLVO energy of
-10kT or deeper. No colloids will be able to return to
the bulk flow (see below) when entering the SEM re-
gion. This combined with the fact that DLVO profiles
overlap outside the SEM region make the dynamics of
colloids very similar for higher IS cases. In a way, the
SEM at high IS behaves like a favorable surface when
the solution IS is maintained. This is demonstrated by
the results for a favorable surface condition also shown
in the figure. The main difference is that the retention is
at SEM for the former, rather than at the primary energy
minimum for the latter. Another well-known difference
is that the retained colloids at SEM could return to the
bulk flow if the SEM well is removed by altering the
solution IS.
Another measure of overall retention is the ratio of ac-
tual number of colloids retained at the SEM at the end
of each simulation to the total number of colloids in-
jected. This ratio was found to be 0.9 %, 2.7%, 11.9%,
18.9%, 19.7%, and 19.7% for ionic strengths of 0.001,
0.01, 0.03, 0.05, 0.1, and 0.3 M, respectively. For the
favorable case, this ratio was 18.4%, and all the colloids
are deposited at primary energy minimum. While for
unfavorable conditions, all colloids are retained in the
SEM.
















0.6 r

0.3 -

0.0 -i
0 20 40 60 80
Pore Volume

Figure 7: Colloid breakthrough curves at the ionic
strength of 0.1M. Breakthrough curves are expressed
by the ratio of colloid concentration at the 4th periodic
length to the concentration at inlet of Ippm.


Fig. 6 shows that the colloid retention number for the
favorable condition is slightly lower than those at ionic
strengths of 0.3M and 0.1M. This is not realistic and
could be a result of statistic fluctuations causing some-
what less colloids to move close enough to collector sur-
faces than in the higher ionic strength cases. This issue
should be investigated further.
In Fig. 7, we show colloid breakthrough curves as the
number of pore volume injected. Here the averaged con-
centration in the 4th period length normalized by the in-
let concentration is shown. After colloids are initially
released at the inlet, it takes some time for them to ar-
rive the 4th periodic length. During this time, the av-
erage concentration in the 4th periodic length increases
quickly with the pore volume (or time). It then becomes
saturated. Given the small number of colloids used (or
the small volume of the periodic domain size), there are
large fluctuations. If the retained colloids are also in-
cluded, the concentration increases with time as more
and more colloids retained in the SEM regions and stay
there. The normalized concentration in the bulk flow
will likely reach a steady-state value and it should be
smaller than one, although the large statistical fluctua-
tions make it difficult to see the steady-state level.

Results: Reversible retention around the SEM

As indicated in Fig. 4, the magnitude of Brownian force
is comparable to (or slightly larger than) the DLVO force
when a colloid is located in the region near the SEM. As
a result, a colloid that has entered the SEM could escape
the SEM region and return back to the bulk flow (i.e.,
re-entrainment). This makes the retention of a colloid by


S ..---concentration including colloids in SEM
-.--concentration in effluent


ICMF 2010, Tampa, FL, May 30- June 4, 2010


favorable
5 -0.300M
-5 0.100M
0.050M
S-0.030M
1.2 ---0.010M
---0.001M

o 0.9


0.6


0.3


0.0 I
0 300 600 900 1200 1500 1800 2100
No. of injected colloids

Figure 8: Retention efficiency at the SEM as a function
of time. It is defined as the actual number of colloids re-
tained in SEM at a given time divided by the total num-
ber of colloids once entered SEM (each colloid entered
in SEM is counted as one no matter if it is still in SEM
at the considered time).



the SEM dynamically reversible if the depth of the SEM
is shallow. Because the attractive DLVO force near SEM
is smaller at lower ionic strength, we expect that the per-
centage of retained colloids that could return to the bulk
flow increases with decreasing solution ionic strength.
We stress that this dynamically reversible transport is
different in nature from the re-entrainment observed in
experiments when the solution ionic strength is signifi-
cantly altered. Here the solution ionic strength is fixed
for each simulation case. In fact, it is difficult, if not
possible, to observe the dynamically reversible transport
experimentally.
Fig. 8 shows the retention or attachment efficiency as
a function of time. The attachment efficiency is defined
as the ratio of the actual number of retained colloids at a
given time over the total number of colloids that have
once entered the SEM region. If the attachment effi-
ciency is one, then all colloids entering the SEM region
will stay there and never return to the bulk flow, which is
the case when the surface is favorable or when the ionic
strength is equal to 0.1 M or higher. A close to one at-
tachment efficiency is the necessary condition for the re-
tention rate to become independent of the solution ionic
strength. A lower attachment efficiency implies a higher
reversibility. Indeed, the retention efficiency decreases
with decreasing ionic strength. The efficiency decreases




ICMF 2010, Tampa, FL, May 30- June 4, 2010


with time initially due to the initial transient effect, but
reaches a steady state value for each case at long times.
The steady state value is established when, on average,
the rate of escape is balanced by the rate of entry.
Analytically the reversible retention of colloids may
be understood by comparing the magnitude of colloidal
thermal energy (Brownian motion) and the depth of the
secondary energy well. At the mesoscopic level, the
thermal fluctuation could be modeled by the Maxwell-
Boltzmann distribution for the microscopic velocity of
colloids as done in Kubo (1966) and Shen et al. (2007)
Following O'Melia et al. (2 11 4) and Shen et al. (2007),
we assume that colloids retained in SEM can move back
to the bulk flow if they have sufficient thermal energy.
Our simulation results shown in Fig. 8 suggest that col-
loids can not escape the SEM region when the SEM
depth is deeper than roughly -10kT.
In Fig. 9 we compare the simulated colloidal steady-
state attachment efficiency with the predicted attachment
efficiency based on the Maxwell model from Shen et
al. (2007). The attachment efficiency of large colloids
(1156 nm) from the experiment of Shen et al. (2007)
and our numerical simulation (500 nm) were both less
than the Maxwell model prediction when the secondary
energy well is shallow. One possible reason is that the
Maxwell model does not account for hydrodynamic ef-
fect on the Brownian force near collector as seen in
Fig. 4. Brownian force is larger when the colloid is
closer to collector surface, so it is easier to escape the
SEM with the modified Brownian motion, making the
attachment efficiency smaller.

Conclusions

In this paper, we have developed a Lagrangian colloid
tracking model to simulate colloid transport and reten-
tion in a 2D model porous medium. An accurate numer-
ical integration with small time step sizes ensures that
the colloid trajectories near a surface are properly sim-
ulated. Hydrodynamic corrections near a grain surface
are considered in treating the drag force and Brownian
fluctuations. Since the assumed primary energy barrier
was very larger (above 1000 kT), no retention at the pri-
mary energy minimum was found to be possible. In-
stead, colloids could be retained at the secondary energy
minimum and the rate of retention depended on the so-
lution ionic strength. When a colloid was captured by
the SEM, it was found to migrate towards the rear flow
stagnation region due to tangential drag force, similar to
what was observed in Johnson et al. (2007). The rate of
retention reaches the maximum when the ionic strength
increases to 0.05 M ~ 0.1M, namely, further increase in
the ionic strength does not alter the retention rate. These
observations are consistent with previous observations


1.5


1.2


0.9


0.6


0.3


0.0


I I . I . I I I


+ numerical simulation
----Maxwell model
,-- ---- -------------
/
i









0 5 10 15 20 25 30 -
interaction energy (kT)


Figure 9: Colloidal collision efficiency from Maxwell
model (Shen et al. 2007) and our numerical simulations.
a is collision efficiency and interaction energy repre-
sents secondary enegry minimum in our numerical sim-
ulations here.


in Kuznaret al. (2007); O'Melia et al. (2 11 4); Shen et
al. (2007).
The most interesting feature of this paper is the direct
demonstration of dynamic reversibility of retention at
the SEM. When the depth of SEM well is below 15 kT,
it was found that a portion of retained colloids could es-
cape the SEM region and return to the bulk flow. The
degree of reversibility increases as the solution ionic
strength is reduced (or the depth of the SEM well is
reduced). The simulations showed that a steady-state
attachment efficiency can be realized. Increasing the
depth of secondary energy minimum increases colloid
attachment efficiency until it reaches the maximum of
100%. The minimum SEM well depth for this to occur
was found to be around -15 kT in our numerical simu-
lations. The simulated attachment efficiency was found
to be smaller than the Maxwell model prediction of Shen
et al. (2007) when the depth of the SEM well is below
10kT. This could be due to the hydrodynamic modifica-
tion of the Brownian force which is not considered in the
model.


Acknowledgements

This study is supported by the US Department of Agri-
culture (NRI-2006-02551, NRI-2008-02803), US Na-
tional Science Foundation (NSF CBET-0932686), and
National Natural Science Foundation of China (Project






No. 10628206). We are grateful to Dr. Chongyang Shen
for his comments on the paper.


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