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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)
(t ) ~ 1 t t1
(1 ) t t 2
e
What is the correct macroscopic description?
Method
structure flow diffusion
Pore networks Stokes equation Random walk
from reconstructed
Berea sandstone p 2 u r (x, t dt ) r (x 0 , t )
X adv X diff
Pore network representation
Process-based
reconstruction
Berea sandstone geologically equivalent
sample (3mmX3mmX3mm) network
(Øren & Bakke, 2003)
diamond lattice
network (60x60)
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
1 d L
2
5. Obtain dispersion coefficient DL
2 dt
Simulation (DL , Pe=0.1)
Random velocity field in
yticolev
heterogeneous network
periodic boundary conditions
12
injection line
10
8
Y(mm)
6
4
2
mean flow direction
0
-2 0 2 4 6 8
X(mm)
Pre-asymptotic (non-Fickian)
regime
100000
10000 Pe =10000
1000 Pe =1000
DL /Dm
100 Pe =100
10
Pe =10
1
1.0= eP Pe =1
0.1
1 10 100 1000 10000 100000
Number of pores traversed
DL (Pe) - Network Model Results
vs. Experiments - asymptotic DL
I II III IV
1/(F) DL ~ Pe
= 1.2 DL ~ Pe
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 2 - Brigham et al., 1961, Berea sandstone
3 - Salter and Mohanty, 1982, Berea sandstone
DL ~ Pe
1.9 4 - Yao et al., 1997, Vosges sandstone
5 - Kinzel and Hill, 1989, Berea sandstone
1.8 6 - Sorbie et al., 1987, Clashach sandstone
power-law coefficient
7 - Gist and Thompson, 1990, various sandstones
1.7 8 - Gist and Thompson, 1990, Berea sandstone
9 - Kwok et al., 1995, Berea sandstone, liquid radial flow
10<Pe<400; = 1.2
1.6 10 - Legatski and Katz, 1967, various sandstones, gas flow
11 - Legatski and Katz, 1967, Berea sandstone, gas flow
1.5 12 - Pfannkuch, 1963, unconsolidated bead packs
1.4
1.3
1.2
1.1
1
0 1 2 3 4 5 6 7 8 9 10 11 12
study number
Comparison with experiments
asymptotic DT (0<Pe<105)
1.0E+05
10<Pe<400; T = 0.94
1.0E+04
Pe>400; T = 0.89
1.0E+03
- network model, reconstructed Berea sandstone
1.0E+02
DT/Dm
- Dullien, 1992, various sandstones
- Gist and Thompson, 1990, various sandstones
1.0E+01 - Legatski and Katz, 1967, various sandstones
- Frosch et al., 2000, various sandstones
- Harleman and Rumer, 1963 (+); (-);
1.0E+00 - Gunn and Pryce, 1969 (□);
- Han et al. 1985 (○)
- Seymour and Callaghan, 1997 ()
1.0E-01
- Khrapitchev and Callaghan, 2003 (∆,◊).
1.0E-02
1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05
Pe
Probability density functions
(t ) ~ 1 t t1 (1 ) t t 2
e t1 = L/u t2 L2 2 Dm
Scher and Lax, 1973; Berkowitz and Scher, 1995 t = t/t1 t2 L umin
PDF Comparison
Network Model vs. Analytic
(t ) ~ 1 t t1
(1 ) t t 2
e = 1.80
t1 = L/u t2 L 2 Dm
2
Comparison: Experiment,
Network Model and CTRW
1.0E+05
1.0E+04
1.0E+03
1.0E+02
DL /Dm
1.0E+01
1.0E+00
= 1.2
1.0E-01
1.0E-02
1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05
Pe
For t > t2/t1 :
For 2>>1: A DL/Dm ~ Pe3- for Pecrit > Pe >>1
DL/Dm ~ Pe t2- t = t/t1 = 3- = 1.2
Dentz et al., 2004 B DL/Dm ~ Pe for Pe > Pecrit
Pore Size Distribution vs.
“Boundary Layers”
600
500
400
frequency
300
200
100
0
0 4 8 12 16 20 24 28 32 36 40 44 48
mean throat radius(m)
160
mean throat velocity(mm/s)
140
120
100
80
60
40
20
0
0 4 8 12 16 20 24 28 32 36 40 44 48
mean throat radius(m)
3D Simulations
600
500
400
frequency
300
200
100
0
0 4 8 12 16 20 24 28 32 36 40 44 48
mean throat radius(m)
160
mean throat velocity(mm/s)
140
120
100
80
60
40
20
0
0 4 8 12 16 20 24 28 32 36 40 44 48
mean throat radius(m)
3D Pe = 100
t = 0s
t ~ 0.1s
t ~ 1s
REACTIVE TRANSPORT
Cubic
Overlap network
Update
CONCLUSIONS
- Unique network simulation model able to predict
variation of DL,T/Dm vs Peclet over the range 0< Pe <105.
- 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 1.80 where = 3- =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
Diffusion
Flow
- flowrate weighted rule ~ Fi / Fi ; - area weighted rule ~ Ai / Ai ;
- assign a new site at random & - assign a new site at random;
move by udt; - forwards and backwards
- only forwards
