1. Branko Bijeljic Martin Blunt Dispersion in Porous Media from Pore-scale Network Simulation Dept. of Earth Science and Engineering, Imperial College, London

2. OVERVIEW Dispersion in Porous Media (Motivation) Network Model Asymptotic Dispersion: Model vs. Experiments Pre-asymptotic Dispersion: Model vs. CTRW Conclusions

3. MIXING of FLOWING FLUIDS in POROUS MEDIA Pore scale mixing processes are COMPLEX: What is the correct macroscopic description?

4. MOTIVATION Describe macroscopic dispersion using a Lagrangian-based pore network model over a wide range of Peclet numbers (0<Pe<10 5 ) Aquifers Contaminant transport Oil reservoirs: Tracers Development of gas/oil miscibility

5. METHOD

6. Pore network representation LARGE SCALE Process-based reconstruction

7. 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 5.Obtain asymptotic dispersion coefficient

8. MIXING RULES at JUNCTIONS Pe >>1 Pe<<1 - area weighted rule ~ A i /  A i ; - assign a new site at random; - forwards and backwards - flowrate weighted rule ~ F i /  F i ; - assign a new site at random & move by udt; - only forwards

9. Simulation (D L, Pe=0.1)

10. Comparison with experiments asymptotic D L (0<Pe<10 5 ) - 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  - Frosch et al., 2000, various sandstones Bijeljic et al. WRR, Nov 2004

11. Comparison with experiments: D L - Boundary-layer dispersion 1 - Bijeljic et al. 2004 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 11 - Legatski and Katz, 1967, Berea sandstone, gas flow 12 - Pfannkuch, 1963, unconsolidated bead packs 10<Pe<400;  L = 1.19

12. Comparison with experiments asymptotic D T (0<Pe<10 5 ) - 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 (∆,◊). 10<Pe<400;  T = 0.94 Pe>400;  T = 0.89

13. Pre-asymptotic regime

14. Probability density distributions Scher and Lax, 1973; Berkowitz and Scher, 1995

15. Comparison with CTRW theory  = 1.80

16. Comparison with CTRW theory   Dentz et al., 2004

17. PORE SIZE DISTRIBUTION vs. “BOUNDARY LAYERS”

18. CONCLUSIONS -Unique network simulation model able to predict variation ofD L,T/,T/ D m vs Peclet over the range 0<Pe<10 5. - 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 u min.

19. 3D NETWORKS – INJECTION

20. TRACKING in 3D

21. Pe = 100 MEAN FLOW DIRECTION X

22. Pe = 0.1 MEAN FLOW DIRECTION X

23. THANKS!