Transport and Sorption - Environmental Modeling - page 114

In groundwater or soil systems, advective or diffusive fluxes in the solid phase can
be neglected. But there are exceptions: imagine, the upper soil horizon is turned
over due to agricultural practice. That could be described by a diffusion term. In
aquatic sediments even more processes contribute to diffusive and advective
processes.
As both ( 6.11 ) denote a total mass balance, the exchange terms are necessarily
equal. In a two-phase environment the sinks of one phase are the sources of the
other. Therefore, it is sufficient to introduce one exchange term only and omit the
other one ( e fs ), i.e. e fs ¼e sf . What is gained in one phase from the sorption
process must be lost in the other phase. The describing set of equations then
becomes:

@
@t y ðÞ ¼r y

ðÞ ylc e fs

j

(6.13)

@
@t r b c s

ð Þ¼ r r b j s ð Þr b l s c s þ e fs

6.2 Retardation

In case of fast sorption it turns out that the exchange terms in ( 6.13 ) can hardly be
quantified. They surely change with time and space. Also the sign changes: in front
of an advancing concentration front there is a net gain of the solid phase and losses
of the fluid phases. The situation is contrary after a front has passed: there are net
gains of the fluid phase and losses of the solid phase. For a quantitative analysis
of transport problems it is therefore convenient to find a mathematical formulation
in which the exchange term disappears. This is achieved here easily by adding both
equations of ( 6.13 ). If one neglects decay or degradation, one obtains:

@
@t yc þ r b c s

ð Þ ¼ r y ðÞrr b j s j ð Þ (6.14)

In order to take advantage of the summation the unknown variable c s is
eliminated. In the case of fast sorption it is possible to reduce the system by utilizing
the isotherm relationship ( 6.1 ). Equation 6.14 can be re-written as:

@
@t Ryc

þ r b
y

c s
c

ð Þ¼ r y ðÞrr b j s j ð Þ R ¼

with 1 (6.15)

where R is the so called retardation factor . The formulation ( 6.15 ) is frequently
used by geochemists (Postma and Appelo 2000 ). In groundwater studies an alter-
native formulation often can be found that is valid for the constant porosity
situation. Using the chain rule @c s =@t ¼ @c s =@c ð Þ @c=@t ð Þ on the left side, the

retardation factor appears outside of the time derivative: