Transport with ion-exchange

[maxim.yutkin]

Good day!

I am trying to match ion exchange with transport for a coreflood experiment and its analytical solution from SPE 6771 (POPE, LAKE, HELFFERICH), fig 2 (see attached).

The authors report 0.0345 eq/L (of pore volume) of CEC.
Initial Na: 30 mM
Initial Ca: 5 mM
Initial Cl: 40 mM

Injected Na: 50 mM
Injected Ca: 5 mM
Injected Cl: 60 mM

Core parameters:
dia = 2in
L = 24in
porosity ~ 0.22
PV = 272 mL


They also report exchange constant (in their another manuscript SPE 6770) as 0.0053 (dimentionless) or 1/0.0053 for the following association reaction:

2NaX + Ca+2 = CaX2 + 2Na+

My input file looks the following:

Code: [Select]

EXCHANGE_MASTER_SPECIES
Z Z-
EXCHANGE_SPECIES
Z- = Z-
-log_k 0.0

Na+ + Z- = NaZ
-log_k 10

-gamma 1e7 0

Ca+2 + 2Z- = CaZ2
-log_k 22
-gamma 1e7 0

SOLUTION 0 Injected solution
    units     mol/L

Na 0.05
Ca 0.005
Cl 0.06

SOLUTION 1-100 Initial solution for column
        units            mol/L
       

Na 0.03
Ca 0.005
Cl 0.04


EXCHANGE 1-100
-equilibrate 1
Z 0.0345



TRANSPORT
    -cells                 100
    -shifts                200
-length 100*0.0061
-boundary_conditions flux flux
-dispersivities 100*0
-correct_disp true
-diffusion_coefficient 0

 
    -punch_cells           100
    -punch_frequency    1
    -print_cells 100


USER_GRAPH 1
        -headings Ca
        -chart_title "1D advection history"
        -axis_titles "Pore volumes" "Ca concentration"
        -axis_scale x_axis 0.9 2
        -axis_scale y_axis auto auto a a

  -start
 10 REM plot_xy (step_no+0.5)/100, MOL("Na+"), color = Red, line_width = 2, \
symbol = Square, y-axis = 1
 20 plot_xy (step_no+0.5)/100, MOL("Ca+2")/0.005, color = Green, line_width = 2, \
symbol = Diamond, symbol_size = 7, y-axis = 1
 50 REM end
-end


In general, the analytical solution can be matched by adjusting exchange constant and by refining the grid (to tackle a little bit of numerical dispersion). However, I would like to match it by specifying correct parameters.

I notice that in the first manuscript authors use Gapon expression, and I am using mass-action. Nevertheless, they report that good fits were obtained for both approaches. Also, they seem to not include chloride and CEC dependence into their fit of constant the real constant that they fit should be Cl/(0.0053*CEC). Yet even using this new value, the model does not match (the result deviates even more).

I tried to adjust the volume of the initial and injected solutions as well as CEC per slice, but realized that it is their ratio that affects exchange, so I use them per 1 L or KGW.

Is there anything I am missing in PHREEQC formulation?