Dispersion in Porous Media from Pore-scale Network Simulation Branko Bijeljic Ann Muggeridge Martin Blunt 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 network representation Process-based reconstruction LARGE SCALE

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

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

Simulation (DL , Pe=0.1)

Comparison with experiments asymptotic DL (0<Pe<105) Bijeljic et al. WRR, Nov 2004 - 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  - Pfannkuch, 1963, unconsolidated bead packs - Seymour and Callaghan, 1997, bead packs - Khrapitchev and Callaghan, 2003, bead packs

Comparison with experiments: DL - 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 10<Pe<400; dL = 1.19 11 - Legatski and Katz, 1967, Berea sandstone, gas flow 12 - Pfannkuch, 1963, unconsolidated bead packs

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. 1985 (○) - Seymour and Callaghan, 1997 () - Khrapitchev and Callaghan, 2003 (∆,◊).

Pre-asymptotic regime

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

Comparison with CTRW theory b= 1.80

Comparison with CTRW theory B) Dentz et al., 2004

CONCLUSIONS - Unique network simulation model able to predict variation of D ,T/ D vs Peclet over the range 0< Pe <10 5 . L m - The boundary-layer dispersion regime is related to the CTRW exponent b 1.80 where d = 3-b. - 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 maximum velocities are in the throats of intermediate radii