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3. Upscaling of Adsorbing Solutes; Pore Scale
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∂s ∗
∂t ∗ = k∗ attc ∗ − k ∗ dets ∗ (3.36)
where, P e = vR0
D L
, which involves the average flow velocity v and the longitudinal
dispersion coefficient, D L . Note that P e is the tube-scale Peclet number;
it is different from the pore-scale P e p , which is based on a diffusion coefficient
(see Equation (3.31)).
Further, katt ∗ and kdet ∗ are dimensionless adsorption and desorption rate coefficients,
respectively. These are related to the dimensional coefficients, k att and
k det , through the following relationships:
katt ∗ = k attR 0
, kdet ∗ = k detR 0
v
v
(3.37)
These equations contain three parameters: P e, katt ∗ and kdet ∗ . We have evaluated
these parameters by fitting the solution of this set of equations to the
average breakthrough concentration, ¯c ∗ (z ∗ , t ∗ ), obtained from the Single-Tube
Model for various values of pore-scale Peclet number (P e p ) and κ. This procedure
results in relationships between the three upscaled parameters (P e, katt
∗
and kdet ∗ ) and their corresponding pore-scale parameters (P e p and κ). We have
used the cross-sectional averaged concentrations from the Single-Tube Model
to find corresponding upscaled adsorption parameters. A similar approach was
used by Li et al. [2008] for upscaling of dissolution processes under steady-state
flow conditions. They studied upscaling of mineral dissolution within a single
pore. Reactive flow experiments were performed in a cylindrical pore, 500 µm
in diameter and 4000 µm long, drilled in a single crystal of calcite. They employed
the Single-Tube Model to simulate the concentration of Ca 2+ resulting
from dissolution of calcite. A kinetic calcite dissolution formula (from Chou
et al. [1989]) was assumed to hold at the wall of tube. Then, the flux-average
concentration of Ca 2+ , calculated from the Single-Tube Model, was compared
to the measured Ca 2+ concentration from the experiment for different pH and
flow conditions. They found excellent agreement between modeling results and
results of experiment. This, we believe, is an indication of the applicability of
our procedure in upscaling from pore to tube scale.
3.3.3 Upscaled Peclet number (Pe)
Here we assume that the dispersion is not affected by the adsorption process.
Therefore, we evaluate the upscaled Peclet number (P e) for the case of a nonadsorbing
solute (i.e., κ = 0). This is done by fitting the breakthrough curve
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