Dr. Branko Bijeljic

Dr. Ann Muggeridge

Prof. Martin Blunt

Diffusion and Dispersion in Networks

Dept. of Earth Science and Engineering,Imperial College,

London

OVERVIEW

• Introduction to Diffusion and Dispersion in Single Ducts and Porous Media

• Motivation

• Model

• Results and Discussion

• Conclusions and Further Work

MIXING of FLOWING FLUIDSin SINGLE DUCTS

- BY DIFFUSION: DUE TO RANDOM MOLECULAR MOTION

HYDRODYNAMIC DISPERSION: SPREAD OF CONC. DISTRIBUTION WIDER AT OUTPUT

- BY ADVECTION: DUE TO NON-UNIFORM VELOCITY FIELD

MOLECULAR DIFFUSION

A)

B)

t = 0 t = tfinal adv.onlyadv.+diff.

MIXING of FLOWING FLUIDS in POROUS MEDIA

Pore scale diffusion processes are COMPLEX:

What is the correct macroscopic description?

MOTIVATION

Describe macroscopic dispersion using particle tracking pore network model.

• Oil reservoirs:

• Tracers

• Development of gas/oil miscibility

• Aquifers

• Contaminant transfer

Network Modelling of Diffusion and Dispersion

Model tracer flow initially:1. Calculate mean velocity in each pore throat

using existing network simulator2. Use analytic solution to determine velocity

profile in each pore throat3. In each time step particles move by

a. Advectionb. Diffusion (random walk)

4. Impose rules for mixing at junctions5. Obtain DL and DT

Unit Cell in Network Modelling of Diffusion and Dispersion

?

?

?Advection Diffusion

?

B. C. in throats- advection: no-slip velocity- diffusion: angle of incidence = angle of reflection

RULES FOR MIXING at JUNCTIONS-

Mixing ~ geometry, discharge pattern, conc. distributions

Previous work:(e.g. Sahimi et al., Chem. Eng. Sci. , 1986; Berkowitz et al.,

Water Resour. Res., 1994; Park and Lee, Water Resour. Res., 1999) a) stream tube routing b) complete mixing

Diffusio n

Flow

Pe >>1 Pe<<1

RULES FOR MIXING at JUNCTIONS-OUR STUDY

a) Particle leaves the junction in diffusive step:- area weighted rule ~ Ai / Ai ;- assign a new site at random; - forwards (outgoing) and backwards allowed

b) Particle leaves the junction in advective step:

- flowrate weighted rule ~ Fi / Fi ;- assign a new site at random and move by udt;- only forwards (outgoing) allowed

CURRENT WORK

• Verify the particle-tracking advection/random walk in single ducts

• cf. Taylor-Aris analytical solutions

• Implement and test junction rules

• cf. experimental data from literature

ANALYSIS of DISPERSION by the PTRW MODEL

Pe = 10N = 5024 r = 50 mDm= 1.0 ·10-10 m2/sDL= 3.1 ·10-10 m2/s

-60

-50

-40

-30

-20

-10

0

-60 -50 -40 -30 -20 -10 0

Y-DISTANCE from CENTRE (micron)

Z-D

IST

AN

CE

fro

m C

EN

TR

E (

mic

ron

)X

0

500

1000

1500

2000

2500

0 1000 2000 3000 4000 5000 6000

particle label

X d

isp

lace

men

t(m

icro

n)

t = 0s t = 60s

0

200

400

600

800

1000

1200

-1500 -1000 -500 0 500 1000 1500

displacement(micron)

Nu

mb

er o

f p

arti

cles

model

analytical

RESULTS (1) Longitudinal Dispersion in a duct with

circular cross section: TA vs. model

TA: DL/Dm=1+(Pe2); -shape factor

0

10

20

30

40

50

60

0 10 20 30 40 50 60

Pe=u*h/Dm

DL

/Dm

model analytical

y = 0.0208x + 1.0015

R2 = 1

0

10

20

30

40

50

60

0 500 1000 1500 2000 2500 3000

Pe^2

DL

/Dm

model Linear (model)

= 0.0208 = 1/48

RESULTS (2) Longitudinal Dispersion between two

infinitely long parallel plates: TA vs. model

TA: DL/Dm=1+(Pe2); = 0.0190=2/105

0

10

20

30

40

50

60

0 10 20 30 40 50 60

Pe=u*h/D

DL

/Dm

analytical model

y = 0.019x + 1.0038

R2 = 1

0

5

10

15

20

25

30

35

0 500 1000 1500 2000

Pe^2

DL

/Dm

model Linear (model)

RESULTS (3) Longitudinal Dispersion in a duct with square

cross section: model

DL/Dm=1+(Pe2); = 0.0342

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60

Pe=u*h/Dm

DL

/Dm

y = 0.0342x + 1.0032

R2 = 0.9999

0

10

20

30

40

50

60

70

80

90

100

0 500 1000 1500 2000 2500 3000

Pe^2=(u*h/Dm)^2

DL

/Dm

square duct Linear (square duct)

RESULTS(1-3) – CONCLUSIONS

DL - longitudinal dispersion coefficient (asymptotic)

Cross section

DL/Dm=1+(Pe2)

Parallel plates 0.190

Circular 0.208

Square 0.342

Dm - molecular diffusion coefficient

Pe – Peclet number ; - shape factor

Conclusion: longitudinal dispersion increases with greater wall friction

• New pore-scale model to describe dispersion in 2D networks at pore scale

• Compare results with 2D square networks of Bernabe and Bruderer (Water Resour. Res., 2001) – flow orientation important

• Compare results with experimental data for dispersion in beadpacks, unconsolidated sandpacks and sandstones

FUTURE WORK – short term

EXPERIMENTAL DISPERSION IN SANDPACKS

(Fried&Combarnous, Adv. Hydrosc.,1971)

PeFD

D

D

Dregimediffusion

m

T

m

L 3.0;1

:)1

53.0;:)2 PePeD

Dregimetransition

m

L

510300;:)4 PePeD

Ddispersionmechanical

m

L

510;,,:)5 PeRePeD

D

D

Ddispersionturbulent

m

T

m

L

9.0,05.001.0;1

2.1,5.0;1

3005:)3

TTTm

T

LLLm

L

T

L

PeFD

D

PeFD

D

Pedispersionlayerboundary

DL/Dm

DT/Dm

I)

II)

III)

IV)

V)

22,: PePeD

D

D

Ddispersionholdup

m

T

m

L

• Investigate influence of network properties on longitudinal and transverse dispersion coefficients

• Extension of the model to

• two-phase flow

• M > 1 / miscible gas displacements

FUTURE WORK – long term