# CONCLUSION by MikeJenny

VIEWS: 19 PAGES: 20

• pg 1
```									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 ) 
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
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

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
3D Simulations
600

500

400
frequency

300

200

100

0
0   4   8   12 16 20 24 28 32 36 40 44 48

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
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

```
To top