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

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

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

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)

3D Pe = 100 MEAN FLOW DIRECTION X

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

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

Pre-asymptotic (non-Fickian) regime -DT

Plume Propagation in Simulation of Transverse Dispersion

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, 1969 ( ) - Han et al. 1985 ( ) - Seymour and Callaghan, 1997 () - Khrapitchev and Callaghan, 2003 ( , )

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

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

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

Pore Size Distribution vs. “Boundary Layers”

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.

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

THANKS!

3D Simulations