1

Transport of suspensions in porous mediaAlexander A. Shapiro*

Pavel G. Bedrikovetsky**

* IVC-SEP, KT, Technical Univ. of Denmark (DTU)**Australian School of Petroleum, Univ. of Adelaide

2

Applications - petroleum

INJECTIVITY INDEX vs TOTAL WATER INJECTED

0

10

20

30

40

50

60

70

80

90

100

0 500 1000 1500 2000 2500 3000Wi

II fi

nal

/ II

inic

ial

(%)

POÇO A POÇO B POÇO C

POÇO D POÇO E POÇO F

POÇO G POÇO H PÇO H_ PDG

Potência (PÇO H_ PDG)

Injectivity decline

(e.g. under sea water injection)

Formation damageby drilling mud

(creation of the filter cakes)

Deep bed filtration

gravel

fines/sand

oil

screen

Productivity decline for the gravel pack with screen

Migration of reservoir fines

in unconsolidated rocks

Thermal reservoirs?

3

Applications within EOR

• Erosion of the rock – e.g. under injection of carbon dioxide

• Filtration of large molecules– e.g. under polymer flooding

• Propagation of bacteria in porous media– e.g. under microbial recovery (to some extent)

• Behavior of drops/emulsions in porous media• Similar models describe flow of tracers in porous media

4

Micro-Physics

Straining

Gravity

Bridging

Electric

forces

Van Der Waals

forces

Sorption

Gravity

-+

+ ++

++- -

---

• Transfer of particles in the flow• Complex interaction with the flow• Multiple mechanisms of capturing• Variation of particle sizes

5

Micro-Physics (2)

- Motion of particles in porous medium is to some extent similar to ”motion in a labirynth”;

- It is characterized by the different residence times and steps in the different capillaries/pores

- Dispersion of the times and steps requires stochastic modeling

6

Traditional model

2

2

( ), ( )

c c cU D

t x x

cUt

k k

Iwasaki, 1937;

Herzig, 1970, Payatakes, Tien, 1970-1990; OMelia, Tufenkij, Elimelech, 1992-2004

porosity

suspended concentration

filtration coefficient

flow velocity

dispersion coefficient

- Advection-dispersion particle transfer- ”First-order chemical reaction” type particle capture mechanism- Empirical equations for porosity/permeability variation- No particle size or pore size distributions- No residence time dispersion

( , ); ( , ); ( , )....c c x t U U x t x t

Experimental observations

(After Berkowitz and Sher, 2001)

The observed profiles do not correspond to predictions of the traditional model

Breakthrough times

0.6, 1.8Johnson,W. et al, 1995,Water Resources Research

0.5Camesano, T. et al, 1999,Colloids and Surfaces A:Physicochemical andEngineering Aspects

1.6Harvey, R. et al., 1995,Applied and EnvironmentalMicrobiology

0.4, 0.8, 0.5Bolster, C. et al., 2001, J. ofContaminant Hydrology

105Roque, C., et.al., 1995,SPE 30110

15, 72, 65, 83Chauveteau, G., et. al, 1998,SPE 39463

0.75, 0.8, 0.85, 0.85Kretzschmar, R. et al, 1997,Water Resources Research

Breakthrough time, p.v.i.Paper

The traditional model predicts breakthrough after 1 porous volume injected (p.v.i.)

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Problems with the traditional model

Contrary to predictions of the model:- Particles may move (usually) slower and (sometimes) faster than the flow;- There may be massive ”tails” of the particles ahead of the flow;- The distributions of retained particles are ”hyperexponential”

0 0.2 0.4 0.6 0.8 1

1

2

3

4

X Xlab,

Tufenkji and Elimelech, 2005 Bradford et al., 2002

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Goals

• Creation of a complete stochastic model of deep bed filtration, accounting for:– Particle size distribution– Dispersion of residence steps and times

• Averaging of the model, reduction to ”mechanistic” equations

• Clarification of the roles of the different stochastic factors in the unusual experimental behavior

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Previous work(all > 2000)

Approach Authors Particlesizedistrib.

Step dispers.

Empirical distributed model for the capture coefficient

Tufenkji and Elimelech

√

Boltzmann-like population balance models

Bedrikovetsky, Shapiro, Santos, Medvedev

√

Continuous-time random walk (CTRW) models

Cortis, Berkowitz, Scher et al.

√

Elliptic transport CTRW-based theory

Shapiro, Bedrikovetsky

√

Essence of the approachRandom walk in a lattice Random walk in one dimension

-The particles jump between the different points of the

”network” (ordered or disordered)

-They spend a random time at each point

-The step may also be random (at least, its direction)

Einstein, Wiener, Polia, Kolmogorov, Feller, Montrol…

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Direct numerical experiment:Random walks with distributed time of jump τ

•1D walk•2-point time distribution•Expectation equal to 1

1

12

:

11 :

1

p T

pTp T

p

-20 -15 -10 -5 0 5 10 15 20

0

500

1000

1500

2000

2500

p=0.5, T1...

X

c

-15 -10 -5 0 5 10 150

500

1000

1500

2000

p=0.5, T1=0.5

x

c

Original

Smoothed

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Direct numerical experiment:Random walks with distributed time of jump τ

-15 -10 -5 0 5 10 150

500

1000

1500

2000

p=0.5, T1=0.5

p=0.9, T1=0.1

x

c

2 2

22x t

c cD

t x

cD

t

Classical behavior with low temporal dispersion

Anomalous behavior with high temporal dispersion:

More particles run far away, but also more stay close to the origin

”Einstein-like” derivation( , ) ( , ) ( , , )c x t pc x l t f l p dld dp

Probability to do not be captured Joint distribution of jumpsand probabilities

Expansion results in the elliptic equation:

2 2 2

2 2x t xt

c c c c cv D D D c

t x x t x t

New terms compared tothe standard model

A stricter derivation of the equation may be obtained on the basis of the theory of stochastic Markovian semigroups (Feller, 1974)

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Monodisperse dilute suspensions

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

x

c

Maximum moves slower than the flow The ”tail” is much larger

- Qualitative agreement with the experimental observations

2 2 2

2 2x t xt

c c c c cv D D D c

t x x t x t

Pu

lse

inje

ctio

n p

rob

lem

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Generalization onto multiple particle sizes

2 2 2

, , ,2 2i i i i

x i t i xt i i i

c c c c cU D D D c

t x x t x t

1( , ,..., ) ( 1,..., )i ii n

h Ucx h h i n

t

2 2 2

2 2x t xt

c c c c cv D D D c

t x x t x t

Monodisperse:

For the particles of the different sizes 1,..., :nR R

For concentrated suspensions coefficients depend on the pore size distributions. The different particles ”compete” for the space in the different pores. Let be numbers of pores where particles of the size will be deposited. Then it may be shown that

,i iD

ih iR

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General theory2 2 2

2 2x t xt

c c c c cv D D D c

t x x t x t

Monodisperse:

2 2 2

, , ,2 2x s t s xt s s

C C C C Cv D D D C

t x x t x t

Polydisperse:

( , )c x t ( , , )sC r x t

The transport equation becomes:

Distribution of the particles by sizes sr

All the coefficients are functions of , ,sr x t

They depend also on the varying pore size distribution ( , , , ) :pH l r x t

( , , , )( , , , ) ( , , , , ) ( , , )p

p s p s s

H l r x tH l r x t r l r x t C r x t dr

t

Length of one step andradius of a pore

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Initial and boundary conditions2 2 2

2 2x t xt

c c c c cv D D D c

t x x t x t

0c c

0c

0c

0c

0c

x

0x

t

0T

1TConsider injection of a finite portion of suspension pushed by pure liquid.

Presence of the second time derivative in the equation requires ”final condition”.

After injection of a large amount of pure liquid, all the free particles are washed out and their concentration becomes (efficiently) zero. This gives the final condition at

1 0 , 1T AT A liquid

suspension

liquid

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Numerical solution (SciLab)

2 2 2

, , ,2 2i i i i

x i t i xt i i i

c c c c cU D D D c

t x x t x t

1( , ,..., )i ii n

h Ucx h h

t

1 1( ,..., ); ( ,..., )i n i nD h h h h

Solve the system of the transport equations under known coefficients:

Solve equations for pore size evolution under known concentrations:

Determine the coefficients:

Normally, convergence is achieved after 3-4 (in complex cases, 5-6) iterations.

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Results of calculationsx

tx

t

Concentration

Porosity

(The values are related to the initial value)

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Retention profiles2 2

, ,2 2i i i i

x i t i i i

c c c cU D D c

t x x t

Temporal dispersion coefficient

ln x ln xLow temporal dispersion High temporal dispersion

1=2

totaltotal

1=2

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Different capturing mechanisms vs temporal dispersion

1=2

total

total

1 2

High temporal dispersion Different capturing mechanisms

ln x

ln x

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Conclusions• A stochastic theory of deep bed filtration of suspensions has been

developed, accounting for:– Particle and pore size distributions– Temporal dispersion of the particle steps

• Temporal dispersion leads to an elleptic transport equation• Pore size distribution results in a system of coupled elliptic

equations for the particles of the different sizes• Coupling appears in the coefficients: particles ”compete” for the

different pores• The temporal dispersion seems to play a dominating role in

formation of the non-exponential retention profiles• Difference in the capture mechanisms may also result in the

hyperexponential retential profiles, but the effect is weaker• No ”hypoexponential” retention profiles has been observed

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Future work

• A new Ph.D. student starts from October (Supported for Danish Council for Technology and Production)

• Collaboration with P. Bedrikovetsky (Univ. of Adelaide)• More experimental verification (not only qualitative)• More numerics

– Scheme adjustement and refinement– Softwareing

• Incomplete capturing• Errosion