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