1. Dispersion and Reactive Transport in Porous Networks
Branko Bijeljic
Martin Blunt
Imperial College, London

2. Mixing of Flowing Fluids in Porous Media
Pore scale mixing processes are COMPLEX:
t1 = L /u
t2 = L2 /2Dm
Pe = t2 / t1
Pfannkuch (1963)
What is the correct macroscopic description?

3. Method

4. Pore network representation
Process-based reconstruction
Berea sandstone
sample (3mmX3mmX3mm)
(Øren & Bakke, 2003)
LARGE SCALE

5. 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 by a.
Advection b.
Diffusion 4.
Impose rules for mixing at junctions

6. Simulation (DL , Pe=0.1)

7. Pre-asymptotic (non-Fickian) regime

8. DL (Pe) - Network Model Results vs. Experiments - asymptotic DLII
III IV
1/(F)
d = 1.2
Bijeljic et al., 2004
Pfannkuch,1963
Seymour&Callaghan, 1997
Khrapitchev&Callaghan 2003
Kandhai et al., 2002
Stöhr, 2003
Legatski & Katz, 1967
Dullien, 1992
Gist et al., 1990
Frosch et al., 2000

9. Comparison with experiments: DL - Power-law dispersion
1 - Bijeljic et al network model, reconstructed Berea sandstone
2 - Brigham et al., 1961, Berea sandstone
3 - Salter and Mohanty, 1982, Berea sandstone
4 - Yao et al., 1997, Vosges sandstone
5 - Kinzel and Hill, 1989, Berea sandstone
6 - Sorbie et al., 1987, Clashach sandstone
7 - Gist and Thompson, 1990, various sandstones
8 - Gist and Thompson, 1990, Berea sandstone
9 - Kwok et al., 1995, Berea sandstone, liquid radial flow
10 - Legatski and Katz, 1967, various sandstones, gas flow
10<Pe<400; d = 1.2
11 - Legatski and Katz, 1967, Berea sandstone, gas flow
12 - Pfannkuch, 1963, unconsolidated bead packs

10. Comparison with experiments asymptotic DT (0<Pe<105)
10<Pe<400; dT = 0.94
Pe>400; dT = 0.89
- 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 (○)
- Seymour and Callaghan, 1997 ()
- Khrapitchev and Callaghan, 2003 (∆,◊).

11. Probability density functions=
t/t1
Scher and Lax, 1973; Berkowitz and Scher, 1995

12. PDF Comparison Network Model vs. Analytic
b= 1.80

13. Comparison: Experiment, Network Model and CTRW=
t/t1 B)
Dentz et al., 2004

14. Pore Size Distribution vs. “Boundary Layers”

15. 3D Simulations

16. 3D Pe = 100
MEAN FLOW DIRECTION X
t = 0s
t ~ 0.1s
t ~ 1s

17. REACTIVE TRANSPORT
Cubic network
Overlap
Update

18. CONCLUSIONS - Unique network simulation model able to predict
variation of D
,T/ D
vs Peclet
over the range
0< Pe
<10 5 . L m -
A very good agreement with the experimental data
in the restricted diffusion, power-law -
and
mechanical dispersion regimes.
- The power-law dispersion regime is related to
the CTRW exponent b 1.80 where d = 3-b =1.2!!
- 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.
- Dispersion at large scale (non-Fickian vs. Fickian)

19. THANKS!

20. Mixing Rules at Junctions
Pe >>1
Pe<<1
- flowrate weighted rule ~ Fi /  Fi ;
- assign a new site at random &
move by udt;
- only forwards
- area weighted rule ~ Ai /  Ai ;
- assign a new site at random;
- forwards and backwards