1. Significance of Pre-Asymptotic Dispersion in Porous Media
Branko Bijeljic
Martin Blunt
Dept. of Earth Science and Engineering,
Imperial College,
London

2. Solute Dispersion Packed Beds
(Bi=0.3)
Tanks-in-series model: column is idealised
as a number (N=7) of CSTRs in series of
equal volume (V).
Levenspiel; Cramers and Alberd

3. 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?

4. Method

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

6. 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

7. 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

8. Simulation (DL , Pe=0.1)

9. 3D Pe = 100
MEAN FLOW DIRECTION X

10. Pre-asymptotic (non-Fickian) regime - DL
DL/Dm ~ Pe t2-b
b = 1.8

11. 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

12. Pre-asymptotic (non-Fickian) regime -DT

13. Plume Propagation in Simulation of Transverse Dispersion

14. Comparison with experiments asymptotic DT (0<Pe<105)
10<Pe<400; dT = 0.94
Pe>400; dT = 0.89
Harleman and Rumer, 1963 (+)
Hassinger and von Rosenberg, 1968 ()
- Gunn and Pryce, ( )
- Han et al ( )
- Seymour and Callaghan, 1997 ()
- Khrapitchev and Callaghan, 2003 ( , )

15. PDF Comparison Network Model vs. Analytic=
t/t1
Scher and Lax, 1973; Berkowitz and Scher, 1995

16. Pre-asymptotic (non-Fickian) regime - Comparison with CTRW theory
DL/Dm ~ Pe t2-b
b = 1.8

17. Asymptotic regime - Comparison with CTRW theory
d = 1.2 A) t =
t/t1 B)
Dentz et al., 2004

18. Pore Size Distribution vs. “Boundary Layers”

19. 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 experiments
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.

20. CONCLUSIONS - D / D
not 10:1 ! L T
The asymptotic dispersion regime is faster attained
for the transverse dispersion
The pre-asymptotic dispersion regime DL/Dm ~ Pe t2-b
where the CTRW exponent b 1.80!
- Dispersion in porous media is likely to be non-Fickian!!
More heterogeneous porous media will have longer times needed to achieve asymptotic limit

21. THANKS!

22. 3D Simulations