1. Dr. Branko Bijeljic Dr. Ann Muggeridge Prof. Martin Blunt Diffusion and Dispersion in Networks Dept. of Earth Science and Engineering, Imperial College, London

2. OVERVIEW Introduction to Diffusion and Dispersion in Single Ducts and Porous Media Motivation Model Results and Discussion Conclusions and Further Work

3. MIXING of FLOWING FLUIDS in 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 = 0t = t final adv.only adv.+diff.

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

5. MOTIVATION Describe macroscopic dispersion using particle tracking pore network model. Oil reservoirs: Tracers Development of gas/oil miscibility Aquifers Contaminant transfer

6. Network Modelling of Diffusion and Dispersion Model tracer flow initially: 1.Calculate mean velocity in each pore throat using existing network simulator 2.Use analytic solution to determine velocity profile in each pore throat 3.In each time step particles move by a.Advection b.Diffusion (random walk) 4.Impose rules for mixing at junctions 5.Obtain D L and D T

7. Unit Cell in Network Modelling of Diffusion and Dispersion ? ? ? AdvectionDiffusion ? B. C. in throats - advection: no-slip velocity - diffusion: angle of incidence = angle of reflection

8. 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 Pe >>1Pe<<1

9. RULES FOR MIXING at JUNCTIONS- OUR STUDY a) Particle leaves the junction in diffusive step : - area weighted rule ~ A i /  A i ; - assign a new site at random; - forwards (outgoing) and backwards allowed b) Particle leaves the junction in advective step : - flowrate weighted rule ~ F i /  F i ; - assign a new site at random and move by udt; - only forwards (outgoing) allowed

10. 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

11. ANALYSIS of DISPERSION by the PTRW MODEL Pe = 10 N = 5024 r = 50  m D m = 1.0 ·10 -10 m 2 /s D L = 3.1 ·10 -10 m 2 /s X t = 0st = 60s

12. RESULTS (1) Longitudinal Dispersion in a duct with circular cross section: TA vs. model TA: D L /D m =1+  (Pe 2 );  -shape factor  = 0.0208 = 1/48

13. RESULTS (2) Longitudinal Dispersion between two infinitely long parallel plates: TA vs. model TA: D L /D m =1+  (Pe 2 );  = 0.0190=2/105

14. RESULTS (3) Longitudinal Dispersion in a duct with square cross section: model D L /D m =1+  (Pe 2 );  = 0.0342

15. RESULTS(1-3) – CONCLUSIONS D L - longitudinal dispersion coefficient (asymptotic) Cross section  D L /D m =1+  (Pe 2 ) Parallel plates 0.190 Circular 0.208 Square 0.342 D m - molecular diffusion coefficient Pe – Peclet number ;  - shape factor Conclusion: longitudinal dispersion increases with greater wall friction

16. 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

17. EXPERIMENTAL DISPERSION IN SANDPACKS (Fried&Combarnous, Adv. Hydrosc.,1971) D L /D m D T /D m I) II) III) IV) V)

18. 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