Geoscience Reference
In-Depth Information
where q
b
denotes bulk mass density of the porous medium and S is the adsorbed
phase concentration. The sorbed phase S subsequently is quantified in terms of an
isotherm, as discussed in
Chap. 5
. Usually, the simple linear Freundlich isotherm
is assumed to be valid, so that S = K
d
c, where K
d
is the distribution coefficient (or
sorption coefficient). More complicated adsorption-desorption terms can be
incorporated, for example, for the case of different forward and reverse sorption
rates. Then, Eq. (
11.1
) can be written in terms of a retardation coefficient, R, in the
form
;
R
o
c
ot
¼
o
ox
ð
vc
Þþ
o
D
h
o
ox
ð
11
:
2
Þ
ox
where
R
1
þ
q
b
K
d
h
:
ð
11
:
3
Þ
Thus, the travel time for an adsorbed chemical, t
a
, can be related to the travel
time for a mobile, conservative (nonsorbing) chemical, t
a
,by
t
a
¼
ð
1
þ
q
b
K
d
=
h
Þ
t
m
Rt
m
:
ð
11
:
4
Þ
Equation (
11.4
) also can be written as v
a
= vR, where v
a
denotes the average
velocity of a sorbing contaminant. Thus, the concentration profile of a sorbing
contaminant is retarded, relative to a nonsorbing contaminant, as shown sche-
matically in Fig. 10.1c.
Sorption can be included in the more general CTRW transport equation discussed
in
Sect. 10.3
.
Margolin et al. (
2003
) show the relation between macroscopic
transport behaviors of passive and sorbing (reactive) tracers in heterogeneous media.
In the framework of CTRW, they formulate the sorption process using a ''sticking''
rate and a ''sticking'' time distribution and derive a relation between the distributions
of the sorbing and nonsorbing tracer in terms of these quantities. Alternatively,
sorption can be included directly in the definition of the transition time distribution,
w(s, t). This approach is valid when chemical sorption acts only as a relatively gentle
''modifying'' mechanism on the overall transport, in which the long-time tail in a
breakthrough curve is further delayed. In other physical situations, however,
transport may be governed by two highly distinct rate spectra for transport, such as
reactive tracers undergoing (slow) sorption (adsorption-desorption) during migra-
tion in a heterogeneous advective flow field (e.g., Starr et al.
1985
). Berkowitz et al.
(
2008
) show that this ''two-scale'' behavior can be quantified by appropriate mod-
ification of the governing transport equation.
Regardless of the transport equation considered, the major effect of sorption on
contaminant breakthrough curves is to delay the entire curve on the time axis,
relative to a passive (nonsorbing) contaminant (Rubin et al.
2012
). Because of the
longer residence time in the porous medium, advective-diffusive-dispersive
interactions also are affected, so that longer (non-Fickian) tailing in the break-
through curves is often observed.
