Branko Bijeljic

Ann Muggeridge

Martin Blunt

Dispersion in Porous Media from Pore-scale Network Simulation

Dept. of Earth Science and Engineering,Imperial College,

London

OVERVIEW

• Dispersion in Porous Media (Motivation)

• Network Model

• Asymptotic Dispersion: Model vs. Experiments

• Pre-asymptotic Dispersion: Model vs. CTRW

• Conclusions

MIXING of FLOWING FLUIDS in POROUS MEDIA

Pore scale mixing processes are COMPLEX:

What is the correct macroscopic description?

MOTIVATION

Describe macroscopic dispersion using a Lagrangian-based pore network model over a wide range of Peclet numbers (0<Pe<105)

• Aquifers

• Contaminant transport

• Oil reservoirs:

• Tracers

• Development of gas/oil miscibility

METHOD

Pore networks from reconstructed Berea sandstone

Stokes equation Random walk

diffadv XX

tdtt

),(),( 0xrxru2 p

structure flow diffusion

Pore network representation

Berea sandstonesample (3mmX3mm)

geologically equivalent network

diamond lattice network (60x60)

LARGE SCALE

Process-based reconstruction

Algorithm

1. Calculate mean velocity in each pore throat by invoking volume balance at each pore

2. Use analytic solution to determine velocity profile in each pore throat

3. In each time step particles move bya. Advectionb. Diffusion

4. Impose rules for mixing at junctions5. Obtain asymptotic dispersion coefficient

MIXING RULES at JUNCTIONS

Diffusio n

Flo w

Pe >>1 Pe<<1

- area weighted rule ~ Ai / Ai ;- assign a new site at random; - forwards and backwards

- flowrate weighted rule ~ Fi / Fi ;- assign a new site at random &

move by udt;- only forwards

Simulation (DL , Pe=0.1)

Random velocity field in heterogeneous network

0

2

4

6

8

10

12

-2 0 2 4 6 8

X(mm)

Y(m

m)

periodic boundary conditions

injection line

mean flow direction

Comparison with experiments asymptotic DL

(0<Pe<105)

- network model, reconstructed Berea sandstone

- Dullien, 1992, various sandstones- Gist and Thompson, 1990, various sandstones- Legatski and Katz, 1967, various sandstones

- Pfannkuch, 1963, unconsolidated bead packs- Seymour and Callaghan, 1997, bead packs- Khrapitchev and Callaghan, 2003, bead packs

21PePePe

FD

D

m

L

- Frosch et al., 2000, various sandstones

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E+03

1.0E+04

1.0E+05

1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05

Pe

DL/D

m

Bijeljic et al. WRR, Nov 2004

Comparison with experiments: DL - Boundary-

layer dispersion

21PePePe

FD

D

m

L

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

0 1 2 3 4 5 6 7 8 9 10 11 12

study number

po

we

r-la

w c

oe

ffic

ien

t

1 - Bijeljic et al. 2004 network model, reconstructed Berea sandstone 2 - Brigham et al., 1961, Berea sandstone3 - Salter and Mohanty, 1982, Berea sandstone4 - Yao et al., 1997, Vosges sandstone5 - Kinzel and Hill, 1989, Berea sandstone6 - Sorbie et al., 1987, Clashach sandstone7 - Gist and Thompson, 1990, various sandstones8 - Gist and Thompson, 1990, Berea sandstone9 - Kwok et al., 1995, Berea sandstone, liquid radial flow10 - Legatski and Katz, 1967, various sandstones, gas flow11 - Legatski and Katz, 1967, Berea sandstone, gas flow12 - Pfannkuch, 1963, unconsolidated bead packs

10<Pe<400; L = 1.19

Comparison with experiments asymptotic DT

(0<Pe<105)

- network model, reconstructed Berea sandstone

- Dullien, 1992, various sandstones- Gist and Thompson, 1990, various sandstones- Legatski and Katz, 1967, various sandstones

- Frosch et al., 2000, various sandstones

- Harleman and Rumer, 1963 (+); (-); - Gunn and Pryce, 1969 (□); - Han et al. 1985 (○)

- Seymour and Callaghan, 1997 () - Khrapitchev and Callaghan, 2003 (∆,◊).

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E+03

1.0E+04

1.0E+05

1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05

Pe

DT/D

m

10<Pe<400; T = 0.94

Pe>400; T = 0.89

Pre-asymptotic regime

Pe =10000

Pe =1000

Pe =100

Pe =10

Pe =1

0.1

1

10

100

1000

10000

100000

1 10 100 1000 10000 100000

Number of pores traversed

DL

/Dm

Probability density distributions

mDLt 222

min2 uLt 2)1(

11~)( ttettt t1 = L/u Scher and Lax, 1973; Berkowitz and Scher, 1995

Comparison with CTRW theory

= 1.80 2)1(11~)( ttettt

mDLt 222 t1 = L/u

Comparison with CTRW theory

DL/Dm ~ Pe 2- = 3- = 1.2

For 2>>1: For > t2/t1 :

DL/Dm ~ Pe for Pe > Pecrit

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E+03

1.0E+04

1.0E+05

1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05

Pe

DL

/Dm

L = 1.19

Dentz et al., 2004

DL/Dm ~ Pe3- for Pecrit > Pe >>1

CONCLUSIONS

-Unique network simulation model able to predict variation of DL,T/D m vs Peclet over the range0<Pe<105.

- A very good agreement with the experimental datain the restricted diffusion, boundary-layer and mechanical dispersion regimes.

- The boundary-layer dispersion regime is related to the CTRW exponent 1.80 where = 3-.

- The cross-over to a linear regime for Pe>400 is due to a transition from a diffusion-controlled late-time cut-off, to one governed by a minimum typical flow speed umin.

THANKS!

Structure-flow relationship

0

100

200

300

400

500

600

0 4 8 12 16 20 24 28 32 36 40 44 48

mean throat radius(m)

freq

uen

cy throat size distribution - uncorrelated

average throat radius, rav = 11.05m

0

20

40

60

80

100

120

140

160

0 4 8 12 16 20 24 28 32 36 40 44 48

mean throat radius(m)

me

an

th

roa

t ve

loc

ity(

mm

/s)

velocity distribution

maximum velocities are in the throats of intermediate radii