download pdf version of PhD book - Universiteit Utrecht

4. Upscaling Adsorbing Solutes: Pore-Network Modeling
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 4.9: Upscaled distribution coefficient, K c , as a function of
D
local-scale distribution coefficient, k d . The circles are the results of
network simulations. A linear equation fits the data: K c = Sk D d. S is
equal to 5.28 × 10 3 m −1 .
4.5 Discussion
Figures (4.6) and (4.7) show that the core-scale attachment and detachment
rate constants are functions of the local-scale adsorption coefficient and the
average pore-water velocity. Combining these graphs results in a surface plot of
k c det (k d, v), as shown in Figure (4.10) for the network (R throat = 0.17×10 −3 m).
We have fitted an equation to this surface
k c det = D 0.95
0 v 0.05
(0.02 + 0.5 k d )R 0.95 (4.16)
Using a volume averaging method, Raoof and Hassanizadeh [2010a] derived
the following relationship for macro-scale kinetic adsorption coefficient for a
general porous medium
k c det = 2D 0
k d R
(4.17)
If, in Equation (4.16), we neglect the dependency on velocity and approximate
(0.02 + 0.5 k d ) ≈ 0.5 k d , we can confirm that the two equations are in full
agreement.
Clearly, knowing the upscaled detachment rate coefficient and the upscaled
distribution coefficient, we can obtain a relationship for upscaled attachment
coefficient
82