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

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?

Method

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

Simulation (DL , Pe=0.1)

Pre-asymptotic (non-Fickian) regime

DL (Pe) - Network Model Results vs. Experiments - asymptotic DL II 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

Comparison with experiments: DL - Power-law 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; d = 1.2 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 (∆,◊).

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

PDF Comparison Network Model vs. Analytic b= 1.80

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

Pore Size Distribution vs. “Boundary Layers”

3D Simulations

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

REACTIVE TRANSPORT Cubic network Overlap Update

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)

THANKS!

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