CONCLUSION by MikeJenny

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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

								
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