Colloidal Transport: Modeling the “Irreversible” attachment of colloids
to porous media
Summer School in Geophysical Porous Media, Purdue University
July 28, 2006
Son-Young Yi, Natalie Kleinfelter, Feng Yue, Gaurav Saini, Guoping Tang, Jean E Elkhoury, Murat Hamderi, Rishi Parashar
Advisors: Patricia Culligan, Timothy Ginn, Daniel Tartakovsky
Jean E. Elkhoury
Grad. Student
Geophysics, UCLA
Son-Young Yi
PhD
Mathematics, Purdue
Guoping Tang
PhD
Civil Engg., Northeastern
Gaurav Saini
Grad. Student
Env. Engg., OSU
Murat Hamderi
Grad. Student
Civil Engg., Drexel
Rishi Parashar
Grad. Student
Civil Engg., Purdue
Feng Yue
Grad. Student
Civil Engg., UIUC
Natalie Kleinfelter
PhD
Mathematics, Purdue
3
• Motivation• Classical Filtration Theory• Experimental Setup• Limitation of Classical Theory
• Single Population Model
• Multi Population Model • 5-population model• 2-population model
• Conclusions• Future Work
Outline
4
Motivation
• Decrease in attachment rate with transport distance indicates deficiencies in classical clean-bed filtration theory.
• Modeling unique experimental dataset to explore alternative approaches to describe irreversible attachment of colloids.
• Distance dependent attachment rate constant• “Multi-population” based modeling approach.
5
Classical Clean-Bed Filtration Theory
ηD: Brownian diffusion
ηI: Interception
ηG: Gravitational Settling
6
Variation of η with velocity & dp
0
0.05
0.1
0.15
0.2
0.25
0.3
0 5 10 15 20 25
particle size (micron)
eta
fast flow
Medium flow
slow flow
7
Macro-scale Approach
First-order deposition rate
kirr = λu
8
Classical Transport Equations
2
2h
C S C CD u
t t x x
Rate of Change of Fore Fluid Concentration
Rate of Change of Adsorbed Concentration
Hydrodynamic dispersion term
Advection term
kirr=attachment rate constant
Ckt
Sirr
9
Classical Theory Inferences
Kirr
Kirr
KirrC
10
Experimental Set-up (Yoon et al. 2006)
• Porous Media: Glass beads (dc= 4mm), surface roughness = 2 μm (rough beads only!)
• Colloids: dp=1-25 μm (d50=7 μm), fluorescent
• C0 = 50 ppm
• Injection period ≈ 11.5 PV (fast/medium) & 9.5 (slow)• Flushing period ≈ 11 PV (fast/medium) & 7 (slow)• Fast (0.0522 cm/s), Medium (0.0295 cm/s)& slow
(0.0124 cm/s) flow• Laser induced fluorescence and digital image processing
11
Schematics
(Yoon et al., 2006)
12
Rough Beads (Contact + Surface Filtration)
13
Smooth Beads (Contact Filtration)
14
Breakthrough Curve
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25
Pore volumes
Fast Flow Rough
Medium Flow Rough
Slow Flow Rough
C/C0
15
Concentration Profile (d~13 cm)
0
0.5
1
1.5
2
2.5
3
3.5
0 5 10 15 20 25
PV
(C+S
)/C
0
Fast Flow Rough
Medium Flow Rough
Slow Flow Rough
Sirr
16
Collector Efficiency
0
2
4
6
8
10
12
14
16
18
0.0000 0.0020 0.0040 0.0060 0.0080 0.0100
collector efficiency (alpha*eta)
Dep
th
Fast flow rough
medium flow rough
slow flow rough
17
Irreversible Adsorption
0
2
4
6
8
10
12
14
16
18
0 0.5 1 1.5 2 2.5 3 3.5
Sirr/C0
Dept
h
Fast flow rough
Medium flow rough
Slow flow rough
18
CXTFIT Analysis
• Fast flow, rough beads
• Deterministic Equilibrium Model
• Predictions:• D = 0.2834 cm2/s
• kirr = 0.165/s 0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25
PV
C/C0
Experimental values
D = 0.042 cm2/s
Kirr = 1.48 x10-4/s
Classic Transport Model can not explain the observed behavior
19
Variation in irreversible attachment with depth (constant kirr)
0
3
6
9
12
15
18
0 1 2 3 4
Sirr/C0
Depth
(cm
)
Slow Flow Rough-Observed
Slow Flow Rough-Simulated
Medium Flow Rough-Observed
Medium Flow Rough-Observed
Fast Flow Rough-Observed
Fast Flow Rough-Simulated
20
Approaches Explored
• Distance dependent attachment rate constant• Multi-Population approach
• 5-population• 2-population
21
Distance dependent kirr
0
2
4
6
8
10
12
14
16
0 1 2 3 4Sirr/C0
Dep
th (cm
)
Slow Flow Rough-Observed
Slow Flow Rough-Simulated
Slow Flow Rough-StepFunction like kirr Assumption
Step Function like kirr
kirr
8 cm
Dep
th
3.5 x10-4
1.2 x10-4
22
Summary
• Constant or step function distribution of rate constant (kirr) does not explain the observed behavior.
• Particle sorption can be predicted given the distribution of kirr with depth (which is unlikely!)
23
Multi-Population Modeling
Governing equations
t
S
z
CD
z
Cu
t
C massimassimassimassi
,2,
2,,
massimassirrimassi Ckt
S,,,
,
i : population
24
Analytical Solution
Dt
utzerfc
D
uz
Dt
utzerfc
D
uz
CktzfCktzC massoimassirrimassoimassirrimassi
22
)1(exp
22
)1(exp
2
1
),;,(),;,( ,,,,,,,,,
Runkel (1996)
H21 2
,,2
u
DkH massirriwhere
,
(t<τ)
),;,(),;,(),;,( ,,,,,,,,,,,,, massoimassirrimassoimassirrimassoimassirrimassi CkTtzfCktzfCktzC
(t>τ)
25
5-Population Model
26
Predictions with constant kirr
0
2
4
6
8
10
12
14
16
0 5 10 15 20
Sirr/Co
Depth
(cm
)
Slow Flow Rough 1.5 micron
Slow Flow Rough 3 micron
Slow Flow Rough 7 micron
Slow Flow Rough 12 micron
Slow Flow Rough 18 micron
27
Simulation Inputs
Average particle size (μm) kirr (10-4)
Mass fraction (input) weighted Mass fraction
1.5 0 0.04 0.16
3 0.17 0.16 0.32
7 2.5 0.37 0.31
12 8.6 0.33 0.17
18 15.5 0.10 0.04
28
Slow Flow: Absolute mass fraction as input
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6 7
(C+S)final/C0 or Sirr/C0
De
pth
, c
m
Observed
Predicted
Medium flow: Absolute mass fraction as input
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6 7
(C+S)final/C0 or Sirr/C0
De
pth
, c
m
Observed
Predicted
Fast flow: Absolute mass fraction as input
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6 7
(C+S)final/C0 or Sirr/C0
De
pth
, c
m
Observed
Predicted
29
Slow Flow: Weighted mass fraction as input
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6 7
(C+S)final/C0 or Sirr/C0
De
pth
, c
m
Observed
Predicted
Medium flow: Weighted mass fraction as input
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6 7
(C+S)final/C0 or Sirr/C0
Dep
th, c
m
Observed
Predicted
Fast flow: Weighted mass fraction as input
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6 7
(C+S)final/C0 or Sirr/C0
De
pth
, c
m
Observed
Predicted
30
Slow Flow: Weighted mass fraction and weighted kirr
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6 7
(C+S)final/C0 or Sirr/C0
De
pth
, c
m
Observed
Predicted
Medium flow: Weighted mass fraction and weighted kirr
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6 7
(C+S)final/C0 or Sirr/C0
De
pth
, c
m
Observed
Predicted
Fast flow: Weighted mass fraction and weighted Kirr
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6 7
(C+S)final/C0 or Sirr/C0
De
pth
, c
m
Observed
Predicted
31
Summary of 5-Population model results• 5-population model provides good fit to the
observed data.• Mass fraction weighting alone does not explain
the observations.• Mass fraction weighted with specific surface
provides good fit, using kirr values from slow flow test.
• Very good fits were obtained when kirr were weighted according to breakthrough data.
32
Motivation for 2-population model
33
2-Population Model
• Population distribution not available.• Assumptions:
• kirr(pop1) > kirr(pop2)
• Approach• Least square fit to observed data (Sirr/C0)
• Predicted rate constants and concentrations (C0 (1) & C0 (2))
34
2-Population model fit
0
3
6
9
12
15
18
0 1 2 3 4
Sirr/C0
Dep
th
Medium flow-observed
Medium flow-fitted
Fast flow-observed
Fast flow-fitted
Slow flow-observed
Slow flow-fitted
35
2-Population model predictions
Slow Medium Fast
Kirr 1(Experimental Kirr
values)
5.34E-03(2.36E-04)
4.44E-03(2.25E-04)
1.75E-03(1.48E-04)
Kirr 21.69E-04 1.70E-04 4.75E-05
C0 (1)/ C0(1+2)3.75E-02(3.75%)
3.34E-02(3.34%)
7.60E-02(7.60%)
36
Summary Results
• 2-population model approach is a potent tool when particle distribution is unknown.
• Optimized k-values found using 2-population model are of the same order as the observed values.
• A small fraction (~5%) of population having high kirr values can explain variations in kirr with transport distance.
37
Conclusions
• Multi-population models capture the trend of decreasing kirr with transport distance.
• Multi-population models can be used to obtain reasonable predictions if particle population distribution is known.
• If particle population distribution is unknown, a 2-population model with optimization can be used to obtain parameters for predictions.
• In homogenized, clean bed-filters, decreases in kirr with transport distance are best explained by distributions in particle populations and not medium properties.
38
Suggestions for future work
• Analysis of Reversible attachment (Srev)
• k = f (particle size, media roughness,fluid velocity)?
• Quantitative measurements of C & S (use of fluorescent bacteria…)
• Explore 1-site model with long-tail distribution functions for residence time.
Questions
Supplementary Slides….
41
Properties of particles
42
43
44