Notes on Colloid transport and
filtration in saturated porous
media
Tim Ginn, Patricia Culligan, Kirk Nelson
Purdue Summerschool in Geophysics 2006
But first, we start with
 Brief review of general reactive transport
formalism
Outline
 General reactive transport intro



Multicomponent/two-phase/multireaction
colloid filtration “Miller lite”
Stop and smell the characteristic plane - mcad
 Colloid Filtration “Guiness”




Overview
Processes catwalk
Classical approach
Blocking
 Issues
 Return to macroscale: multisite/population
Gone to mathcad





Some analytical solutions - hope it runs
Just transport
Irreversible filtration no dispersion
Reversible filtration no dispersion
(Dispersion included by superposition.)
Outline
 General reactive transport intro



Multicomponent/two-phase/multireaction
colloid filtration “Miller lite”
Stop and smell the characteristic plane - mcad
 Colloid Filtration “Guiness”




Overview of colloids in hydrogeology
Processes catwalk
Classical approach
Blocking
 Issues
 Return to macroscale: multisite/population
1. Introduction - Background
Particle Sizes
-10
(diameter, m) 10
1Å
10-9
10-8
10-7
10-6
1 nm
10-5
10-4
1 mm
Soils
Clay
10-3
10-2
1 mm
1 cm
Sand
Silt
Gravel
Microorganisms
Viruses
Protozoa
Red blood cell
Blood cells
Atoms,
molecules
Bacteria
White blood cell
Atoms
Molecules Macromolecules
Colloids
Suspended particles
Depth-filtration range
Electron
microscope
Light microscope
Human eye
Problems Involving Particle Transport
through Porous Media in Environmental
and Health Systems
 Water treatment system


Deep Bed Filtration (DBF)
Membrane-based filtration
 Transport of pollutants in aquifers


Colloidal particle transport1
Colloid-facilitated contaminant transport2
 Transport of microorganisms



1.
2.
Pathogen transport in groundwater
Bioremediation of aquifers
…
Ryan, J.N., and M. Elimelech. 1996. Colloids Surf. A, 107:1–56.
de Jonge, Kjaergaard, Moldrup. 2004. Vadose Zone Journal, 3:321–325
…and some more
 In situ bioremediation


transport of bacteria to contaminants1
excessive attachment to aquifer grains – biofouling
 Bacteria-facilitated contaminant transport
(e.g.,DDT2)
 Clinical settings


1.
2.
Blood cell filtration
Bacteria and viruses filtration
Ginn et al., Advances in Water Resources, 2002, 25, 1017-1042.
Lindqvist & Enfield. 1992. Appl. Environ. Microbiol, 58: 2211-2218.
Outline
 General reactive transport intro



Multicomponent/two-phase/multireaction
colloid filtration “Miller lite”
Stop and smell the characteristic plane - mcad
 Colloid Filtration “Guiness”




Overview
Processes catwalk
Classical approach
Blocking
 Issues
 Return to macroscale: multisite/population
Processes in colloid-surface
interaction





Actual colloid,
Inertia in (arbitrary) velocity field
Torque, drag due to nonuniform flow
Diffusion,
hydrodynamic retardation/lubrication

Effective increase in viscosity near surface
 Electrostatic (dynamic) interaction

DLVO (=LvdW + doublelayer model
electrostatics)
 Buoyancy/gravitational force
Overview
 General reactive transport intro



Multicomponent/two-phase/multireaction
colloid filtration “Miller lite”
Stop and smell the characteristic plane - mcad
 Colloid Filtration “Guiness”




Overview
Processes catwalk
Classical approach – “Colloid filtration theory”
and some Details
Blocking
 Issues
 Return to macroscale: multisite/population
Classical take on Processes in
colloid-surface interaction
 Inert, Spherical colloid to Sphere (flat)
 Inertia in (Stokes) velocity field
 Torque, drag due to nonuniform flow

approximated
 Diffusion (superposed)

hydrodynamic retardation/lubrication
 Electrostatic (dynamic) interaction

DLVO (=LvdW + doublelayer model
electrostatics )
 Buoyancy/gravitational force added

So flow must be downward
Forces And Torques – RT model
Trajectory Analysis
Smoluchowski-Levich Solution
(particle has finite
diameter)
(particle diameter = 0)
TD
TD
FG
h
+
=
FB
FI = inertial force due to Stokes flow*
FD = drag force due to Stokes flow*
TD = drag torque due to Stokes flow*
FG = gravitational force
FB = buoyancy force
FvdW = van der Waals force
*with corrections near surface
FI = inertial force due to Stokes flow
FD = drag force due to Stokes flow
TD = drag torque due to Stokes flow
FBR = random Brownian force
Classical CFT :Happel spherein-cell
 Clean-bed “Filtration Theory”
• Single “collector” represents a solid
phase grain. A fraction h of the particles
are brought to surface of the collector
by the mechanisms of Brownian
diffusion, Interception and/or
Gravitational sedimentation.
•A fraction  of the particles that reach
the collector surface attach to the
surface (electrostatic and ionic strength)
• The single collector efficiency is then
“scaled up” to a macroscopic filtration
coefficient, which can be related to
first-order attachment rate of the
particles to the solid phase of the
medium.
h0  h D  h I  hG
Single collector efficiency
h
Filtration coefficient

First-order deposition rate


3(1 n)
h
2dc
katt  u
Bulk “kf” by classical filtration theory
nC
  f c  kC
t
 3 1  n U
k  h
 n
2
d

c 
First-order removal
Rate = filter coefficient * porewater velocity
=> two-step process
n porosity
C aqueous phase concentration of colloid suspension
fc flux of C
U groundwater (Darcy) specific flux
 fraction of colloids encountering solid surface that stick (empirical2,3)
h fraction of aqueous colloids that encounter solid surface (modeled1,3-6)
1. Rajagoplan & Tien. 1976. AIChE J. 22: 523-533.
2. Harvey & Garabedian. 1991. ES&T 25: 178-185.
3. Logan et al. 1995. J. Environ. Eng. 121: 869-873.
3. Nelson & Ginn. 2001 Langmuir 17: 5636-5645
4. Tufenkji & Elimelech. 2004 ES&T 38: 529-536.
5. Nelson & Ginn. 2005 Langmuir 21: 2173-2184
Details1:Happel sphere-in-cell model2
 Happel sphere-in-cell is
porous medium
 Stokes’ flow field
 h calculated via trajectory
analysis1
 Additive decomposition

h=hI+hG+hD
 Initial point of limiting
trajectory
 h = A1/A2 = sin2qs
1.
Rajagoplan & Tien. 1976. AIChE J.
22: 523-533.
2.
Happel. 1958. AIChE J. 4: 197-201.
A1
A2
Detail: Basic solution (analytical)
due to Rajagopalan & Tien (1976)
 Hydrodynamic retardation effect = the increased drag
force a particle experiences as it approaches a surface.
Interception by boundary
 a deviation from Stokes’ law
condition
 Hydrodynamic correction factors
Sedimentation group
 Particle velocity expressions gives:




1 
U


uq r, q  
B s2  D 1  s3  NG sinq
s1
r

1  
ur r, q   t  A 1   
fr 



2
frm
 NG cosq 
London van der
Waals group
 rtd N LO

2

U
2 

2    

where frt, frm, s1, s2, and s3 are the drag correction factors.
Detail: h vs. 
 irreversible adsorption constant, kirr = f(,h)
 h = fraction of colloids contacting solid phase,
calculated a priori from RT model
  = fraction of colloids contacting solid phase
that stick, treated as a calibration parameter
accounting for all forces and mechanisms
not considered incalculation of h
Role of electrostatic forces : aside
Detail: Surface Forces in CFT –
DLVO
 RT model uses DLVO theory for surface
interaction forces:
attractive
repulsive for like charges
potential = van der Waals + double layer
 Theory predicts negligible collection when
repulsive surface interaction exists  RT
model neglects double layer force.
Detail: Surface Forces in CFT –
DLVO
 RT model uses DLVO theory for surface
interaction forces:
attractive
repulsive for like charges
potential = van der Waals + double layer
 Theory predicts negligible collection when
repulsive surface interaction exists  RT
model neglects double layer force.
 Thus, double layer force implicit in .
Highlights of Formulae for h
 Yao (1971)

hydrodynamic retardation and van der Waals force not included
 Rajagopalan and Tien (1976)



deterministic trajectory analysis
torque correction factors
Brownian h added on separately from Eulerian analysis
 Tufenkji and Elimelech (2004)


convective-diffusion equation solution
influence of van der Waals force and hydrodynamic retardation on
diffusion
  fc   UC   D C   

D f 
kT
Diffusion, interception, & sedimentation considered additive
 Nelson and Ginn (2005)

C
Particle tracking in Happel cell – all forces together
Outline
 General reactive transport intro



Multicomponent/two-phase/multireaction
colloid filtration “Miller lite”
Stop and smell the characteristic plane - mcad
 Colloid Filtration “Guiness”




Overview
Processes catwalk
Classical approach – “Colloid filtration theory”
and some Details
Blocking
 Issues
 Return to macroscale: multisite/population
Dynamic surface blocking (ME)
 initial deposition rate (kinetics)
rate  a kc
2
p
 later, when deposition rate drops due to
surface coverage (dynamics)
rate  a kB(s)c
2
p
 retained particles block sites, B is the
dynamic blocking function (misnomer).
B's
 B = fraction of particle-surface collisions that
involve open seats (cake walk).
 Random Sequential Adsorption
 40
176 
3
Bs  1  4ss 
ss    2 s s

 3 3 
6 3

2
Power series in S, for spherical geometry
 Langmuirian Dynamic Blocking
Bs  1   s
  1/ s
Outline
 General reactive transport intro



Multicomponent/two-phase/multireaction
colloid filtration “Miller lite”
Stop and smell the characteristic plane - mcad
 Colloid Filtration “Guiness”




Overview
Processes catwalk
Classical approach – “Colloid filtration theory”
and some Details
Blocking
 Issues
 Return to macroscale: multisite/population
Issues
 CFT coarse idealized model



Chem/env. Engineering, not natural p.m.
Biofilms, organic matter, asperities,
heterogeneity (gsd, psd, surface area,
electrostatic (dynamic), transience, flow
reversal, temperature, etc.
Reversibility ???
 CFT good for trend prediction

Attachment goes up with colloid size, gw
velocity, ionic strength, etc.
 Ultimately need equs for bulk media


Lab
field
Outline
 General reactive transport intro



Multicomponent/two-phase/multireaction
colloid filtration “Miller lite”
Stop and smell the characteristic plane - mcad
 Colloid Filtration “Guiness”




Overview
Processes catwalk
Classical approach – “Colloid filtration theory”
and some Details
Blocking
 Issues
 Return to macroscale: See the data !
Field/Lab observations
 Microbes 1,2,3 and viruses 4,5 first showed
apparent multipopulation rates due to
decreased attachment with scale



Sticky bugs leave early
Readily explained by subpopulations
Some suggest geochemical “heterogeneity”
 Recent surprize is that inert monotype,
monosize and polysize colloids exhibit same6
1.
2.
3.
4.
5.
6.
Albinger et al., FEMS Microbio Ltr., 124:321 (1994)
Ginn et al., Advances in Water Resources, 25:1017 (2002).
DeFlaun et al., FEMS Microbio Ltr., 20:473 (1997)
Redman et al., EST 35:1798 (2001); Schijven et al., WRR 35:1101 (1999)
Bales et al., WRR 33:639 (1997)
Li et al., EST 38:5616 (2004); Tufenkji and Elim. Langmuir 21:841 (2005)Yoon et al., WRR June 2006
Ability-based modeling (because
we can)
 BTCs (first) exhibit long flat tails



Two-site, multisite model1 (google “patchwise”)
Two-population, multipop’n model2 (UAz,
Arnold/Baygents)
Can’t tell the difference
 Profiles (recently) are steeper than expected




1.
2.
3.
Multipopulation works, not multisite (Li et al in 2), 3
This is the location of the front in practice
Upscaling
Alternative explanations
E.g., Sun et al., WRR 37:209 (2001); “patchwise heterogeneity”, CXTFIT ease of use (sorta)
E.g., Redman et al., EST 35:1798 (2001); Li et al. EST 38:5616 (2004)
Johnson and Li, Langmuir 21:10895 (2005); Comment/Reply
Research Needs (at least)
 Formal upscaling



Forces complex but well understood
Approximations tested
Analytical results (Smoluchowski-Levitch1)
 Alternative explanations

C<-> S -> S’ surface transformations 2
Mainly bacteria; need RTD for attachment events



Physical straining of larger sizes (a pop’n model)3
Reentrainment4
Contact (CFT) and surface (multipopn) filtration5
1.
For CFT/Happel cell without interception or sfc forces (LvdW =-hyd. Retardation)
2.
Davros & van de Ven JCIS 93:576 (1983); Meinders et al. JCIS 152:265 (1992); Johnson et al. WRR
31: 2649 (1995); Ginn WRR 36:2895 (2000)
3.
Bradford et al WRR 38:1327 (2002); Bradford et al. EST 37:2242 (2003)
4.
Grolimund et al WRR 37:571 (2001)
5. Yoon et al. WRR June 2006
Appendix: DNS Approach
 Langevin equation of motion


Happel sphere-in-cell
Contemporaneous accounting of all forces
 Solution per colloid
 Calculating h



Monte carlo colloidal release per qs =>
P(qs) frequency of attachment per qs
h as an expectation over P(qs)
Langevin Equation
 Deterministic and Brownian displacements are
combined per time step:
du
mp
 Fh  Fe  Fb
dt
 mp is the particle mass, u is the particle velocity
vector, Fh is the hydrodynamic force vector, Fe is
the external force vector, and Fb is the random
Brownian force vector.
 All three components of random displacement
must be modeled in the axisymmetric (3D  2D)
flow field.
Solution
R  udet Dt  ns R
 R = 3D displacement,
 udet = deterministic velocity vector
 n =3 N(0,1),
 sR = standard deviations of Brownian displacements.
 negligible particle inertia assumed
Dt >> tB (Kanaoka et al., 1983)
tB particle’s momentum relaxation time (=mp/6map).
Thus, tB << Dt < tu
tu is the time increment at which udet is considered con
Highlights of numerical solution
 Stokes’ flow in two-dimensions
 R&T (1976) hydrodynamic drag correction
factors1
 Brownian diffusion algorithm of Kanaoka et
al. (1983)2 for diffusive aerosols
 Coordinate transformation to 2D model
1. Brenner, H., Chem. Eng. Sci. 1961, 16, 242-251; Dahneke, B.E., J. Colloid Interface
Sci., 1974, 48, 520-522.
2. Kanaoka, C.; Emi, H.; Tanthapanichakoon, W., AIChE J., 1983, 29, 895-902.
Coordinates for diffusion
 The Happel model: 3-D -> 2-D polar coordinates
 convert 3-D Brownian Cartesian displacement to spherical, to
polar ˜
˜ y  ny 2DB MDt
˜ z  nz 2 DBM Dt
Rx  nx 2DBM Dt
R
R
y,z, contribute to angular displacements
And thus to r
R˜ y 
˜ q  arcsin 
R
 r 
 
R˜ z 
˜   arcsin 
R
 r 
2 ˜
2 ˜ 
˜R  R
˜  r
 1 sin Rq  1 sin R  2r
r
x


Calculating h
/ 2
h2

Pcollect qS sin qS cosqS dqS
0
 qS starting angle of a colloid
 Pc(qS) frequency of contact with the collector.
 reduces to classical equation when
deterministic (e.g., when Pc(qS) equals one for
all qS < qLT and zero for all qS > qLT).
 task of stochastic trajectory analysis for h is to
find Pc(qS).




Colloid transport and Colloid Filtration Theory
Classical approach
Issues
Direct numerical simulation:


Approach
Examples, Convergence, Testing
 Results
 Blocking - pages from Elimelech's site
 Conclusions
Example Brownian Trajectory
1.6429E-04
1.6428E-04
1.6427E-04
1.6426E-04
r [m]
1.6425E-04
1.6424E-04
1.6423E-04
1.6422E-04
1.6421E-04
1.6420E-04
1.6419E-04
1.1838
1.184
1.1842
1.1844
1.1846
q [rad]
1.1848
1.185
1.1852
1.1854
1.1856
164.44
164.39
164.34
r [ m m]
164.29
164.24
164.19
1.132
1.134
1.136
1.138
1.14
q [rad]
1.142
1.144
1.146
P(qs)
Num ber of bacteria collected w ith (Brow nian m otion included) as function of theta-start
0.025
Bacteria collected
ran1 300 rlzns
ran1 1000 rlzns
MT 12K rlzns
ran1 12K rlzns
0.020
0.015
0.010
0.005
0.000
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
theta-start
0.08
0.09
0.1
0.11
0.12
Convergence of a trajectory - 50K
realizations
Convergence of Collection Freq from ts=.0418 (Case 1, ap = .695 microns, dt = 0.5 msec)
1
0.9
0.8
Frequency of collection
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
10
100
1000
number of realizations
10000
100000
Convergence to deterministic trajectory analysis of
Rajagopalan and Tien (when diffusion is neglected),
Parameters: e = 0.2, as = 50 mm, ap = 0.1 mm, and U = 3.4375 * 10-4 m/s.
The approximate analytical solution is h = 1.5 NR2g2AS (Rajagopalan and Tien, 1976).
Convergence of stochastic simulations for
Smoluchowski-Levich approximation.
Parameters: ap = 0.1 mm, as = 163.5 mm, e = 0.372,
U = 3.4375*10-4 m/sec, m = 8.9*10-4 kg*m/sec, T = 298 K.
8.4E-03
8.3E-03
Dt = 10 ms
8.2E-03
analytical result
D t = 1 ms
8.1E-03
Dt = 100
8.0E-03
h 7.9E-03
7.8E-03
7.7E-03
7.6E-03
7.5E-03
7.4E-03
0
2000
4000
6000
number_of_realizations
8000
10000
12000




Colloid transport and Colloid Filtration Theory
Classical approach
Issues
Direct numerical simulation:


Approach
Convergence
 Results


Smoluchowski-Levitch approximation
General case
 Blocking - pages from Elimelech's site
 Conclusions
Testing comparison to the Smoluchowski-Levich
approximation (external forces, interception neglected).
Parameters: as = 163.5 mm, e = 0.372, U = 3.4375*10-4 m/sec, m = 8.9*10-4 kg*m/sec,
T = 298 K, Dt = 1 ms, N = 6000.
1.E-02
NG04

analytical
h
1.E-03
0
0.2
0.4
0.6
 m)
ap (m
0.8
1
1.2
Comparison of h calculations
R&T (1976) X N&G
- - - T&E (2004)
o N&G Additive
RT_76
NG_04
TE_04
NG_04 additive
RT_76 deterministic
NG_04 deterministic
R&T (1976)
deterministic
N&G deterministic
1E-02
h

1E-03
1E-04
0.2
0.3
0.4
0.5
0.6
0.7
ap (mm)
0.8
0.9
1
1.1
Conclusions
 Lagrangean analysis is viable tool with modern
computers
 Stochastic trajectory analysis suggests diffusion
and sedimentation may not be additive
 More realistic “unit cell” models could be used
 Lagrangean approach allows for arbitrary
interaction potentials



Chemical (mineralogical, patchwise) heterogeneity
Exocellular polymeric substances in bacteria
Polymer bridging, hysteretic force potentials
Parameters
used in
stochastic
trajectory
simulations.
Parameter
Value
Collector radius, as
163.5 mm
Porosity, e
0.372
Approach velocity, U
3.4375 * 10-4 sec
Fluid viscosity, m
8.9 * 10-4 kg·m / sec
Hamaker constant, H
10-20 J
Bacterial density, rp
1070 kg / m3
Fluid density, rf
997 kg / m3
Absolute temperature, T
298 K
Time step, Dt
1 ms
Number of realizations, N
6000
Modification of CFT to Account for EPS
 Distribution of polymer lengths
on the cell surface
 Repulsion modeled by steric
force, Fst(h)1,2
Hypothetical cell (drawn to scale)
O
L
depends on polymer density
and brush length
KT2442
L
E
0.695 mm
 If sufficient polymers contact
collector, cell attaches
depends on polymer density,
length, and adhesion forces
C
h
C
T
O
R
mean polymer length = 160 nm
1. de Gennes. 1987. Adv. Colloid Interface Sci. 27: 189-209.
2. Camesano & Logan. 2000. Environ. Sci. Technol. 34: 3354-3362.
Theoretical Sticking Efficiency
Numerical Calculation of Trajectories
 Steric repulsive force
 Polymer bridging
 Interception
 Sedimentation
 Brownian motion
 London van der Waals
attractive force
 Hydrodynamic retardation
effect
Incorporation of Brownian motion and polymer interactions into
trajectory analysis allows for computation of a theoretical sticking
efficiency.
Theoretical Sticking Coefficient
 Incorporation of polymer interactions and Brownian
motion + assumption that polymers control adhesion 
Trajectory analysis yields the product [h]theo = A1/A2 =sin2 qs
 Then we can define a theoretical value for the sticking
efficiency :
theo = [h]theo / h
where h is the model result without polymer interactions.
 Comparison of theo with experimental  can serve as a
validation tool for the polymer interaction modeling.
Pseudomonas putida KT2442
 Considered for
bioremediation use1,2
 Congo Red stain
image  heavy EPS
coverage on cells
 EPS characteristics
being studied by
Camesano et al.
(WPI)3
KT2442 cells with Congo Red
White areas
indicate EPS
1. Nublein et al. 1992. Appl. Environ. Microbiol. 58: 3380-3386.
2. Dobler et al. 1992. Appl. Environ. Microbiol. 58: 1249-1258.
3. Camesano & Abu-Lail. 2002. Biomacromolecules. 3: 661-667.
Photo credit: Stephanie Smith
Dept. of Land, Air, & Water Resources
Summary
 CFT trajectory analysis modified for explicit
inclusion of Brownian motion and bacterial EPS
interactions
 Brownian trajectory analysis results suggest that
sedimentation and diffusion may not be additive
as previously assumed
 Future work
 comparison of h calculations with
experimental data in the literature
 more realistic modeling of EPS interactions
(e.g., hysteresis)