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Envtronmental
Health
Perspectives
Vol.
83,
pp.
69-75,
1989
Prediction of Contaminant
Retention and
Transport in Soils Using Kinetic
Multireaction
Models
by
H.
M.
Selim*
Mathematical models that describe
the
retention reactions of contaminants in the soil
system are
presented. Single andmultireaction-type
models for
simultaneous retention
and
transport
inthe
soil profile
are
discussed. Single retention models
are
classified
into
two
types:
equilibrium and kinetic
models.
Emphasis
is
given
to the
nonlinearity
and kinetic
behavior
of solute
retention
processes
in
soils.
Two-site
models
that
include the equilibrium-kinetic types
as well as the fully
kinetic
type are
also examined.
A
multireaction-type model
is
also presented, which
includes reversible and irreversible retention processes
of the equilibrium and kinetic types.
Advantages
of the
multireaction approach
over the single or
two-
site models
are
discussed.
The
predictive capability
ofthe two-site model and
the multireaction model
for
their description ofexperimental
results for
phosphorus and
two
heavy
metals (Cd
and
Cr) are
examined.
Introduction
Retention reactions that
occur
in the soil
are
impor-
tant
processes
that
govern
the fate ofchemical
contam-
inants
and hazardous chemicals
in
groundwater. Math-
ematical models that describe the potential mobility of
dissolved
chemicals
must
therefore include the
physical
and chemical,
as
well
as
biological,
processes
that influ-
ence
the
behavior ofthese chemicals
in
the
soil matrix.
Theability
to
predictthemobilityofdissolved chemicals
in
the soil
and
the
potential contamination
of
ground-
water
supplies is important
in
assessment
of hazards
and
is
a
prerequisite
for the
management
of
land
dis-
posal
of
chemicals contaminants.
In
this
paper, a
reviewofwidelyused soluteretention
models is presented. Emphasis is
on
solute retention
mechanisms characterized by
time-dependent (or
ki-
netic) and nonlinear
type
reactions.
Several multireac-
tion
models
for the
transport
and
retention
of
contam-
inants
are
presented.
Equilibrium
Retention
Models
It is well accepted that, under steady
water
flow
con-
ditions,
transport
ofdissolved chemicals
in
soils is
gov-
erned
by the following convection-dispersion
transport
equation (1):
where c is the concentration of the dissolved chemical
in the soil solution (mg/L) and s is the amount ofsolute
retained per unit mass of the soil matrix (mg/kg). In
addition,
D
is the
hydrodynamic
dispersion
coefficient
(cm2/day),
q
is
the
Darcy
water flux
(cm/day),
0 is
the
volumetric soil moisture content
(cm3/cm3),
p
is the soil
bulk
density
(g/cm3/day),
t
is time
(day)
and
x
is soil
depth
(cm). The
two
terms
on the
right-hand
side of
Eq. (1) are
commonly
known as the dispersion and con-
vection terms,
respectively. Theterm
(as/at)
represents
the rate
for
reversible
solute removal from the soil
so-
lution. In
contrast, the
term
Q
is
a source or a sink
representing
irreversible solute
production (Qnegative)
orsolute
removal (Q
positive)
fromthe soilsolution(mg/
cm3
day).
Over the last 2 decades, several analytical models for
the description ofsolute transport in porous mediahave
been
proposed.
One group of models deals with solute
transport
in
well-defined geometrical systems of pores
and/or cracks ofregular shapes orinteraggregate voids
ofknown
geometries.
Examples ofsuch models include
those of
Rao et
al. (2), Rasmuson and Neretnieks (3)
(for
uniform spheres), Tang et al. (4) (for rectangular
voids), van Genuchten et al. (5) (for cylindrical voids),
and Rasmuson (6) (for discrete aggregate or spherical
size
geometries). van Genuchten and Dalton (7) pro-
vided a review of models using
such an approach. So-
lutions of
these models are
analytic,
often
complicated,
and involve severalnumerical
approximating steps. Re-
cent
applications
include transport
in fixed beds con-
sisting of spheres or
aggregates (8,9). Another group
oftransport models that are
widely used are those that
do
not
consider
well-defined geometries of the pore
~~(1)
Paat+eat=eDaa2
qAac Q
at
at
aX
*104
Madison
B. Sturgis Hall, Louisiana
State
University,
Baton
Rouge,
LA 70803.
70
H. M. SELIM
space or
soil
aggregates.
Rather, solute
transport
is
treated
on a macroscopic
basis with
p, 0, q,
and
D of >
Eq. (1) as the associated parameters that
describe the
_
transport processes in the bulk soil. The mobile-im-
z
mobile transport models
are
refinements
of
this mac-
0
roscopic approach. Here,
it is
assumed
that soil-water
<
is divided into two regions. A mobile-water
region
is
I
one
that is considered
to
be
present in
large
pores and
w
through which solute transport occurs
by convection
z
and
mechanical
dispersion.
The other
is
an
immobile- o
water
region
present in
the bulk
matrix and through
U
which
relatively low ornowater
flows.
Mobile-immobile
models have been introduced by Coats
and
Smith (10),
Skopp and Warrick
(11),
van Genuchten and
Wierenga
m
(12),
and Skopp
et
al.
(13).
The mobile-immobile models
have been extensively used to
describe several solutes
[for
a
review
see
Nielsen
et
al. (14)].
Description
ofthe solute retention
mechanisms
as ex-
pressed by the
term
(as/at)
has been the
focus ofinves-
tigators for several
years.
Such
a
description, when
incorporated into Eq. (1), provides a predictive tool
for
'
the
transport
of dissolved chemicals
in
the
soil
profile.
Most
mathematical models that describe the
retention
mechanisms
are
based
on
the
validity
ofthe local
equi-
(
librium assumption
(LEA) in the soil system (15). Here
one assumes
that the reaction of
an
individual solute
species
in
the soil is sufficiently fast
or
instantaneous
and that an apparent
equilibrium condition
may
be ob-
u
served in
a
few minutes or
hours. Such
a
behavior has
been used
as
the basis for soil surface
adsorption
mech-
(
anisms as well
as
ion-exchange reactions. Illustrative
examples of equilibrium-type
solute
retention are
shown
in
Figure 1. The
data
are
from
kinetic batch
experiments of Mg retention by
a
Ca
saturated
Abist
soil and for two aggregate size separates. The
time of
reaction does
not appear to
be significant.
No apparent
change
in
Mg concentration occurred after
4
hr
of
re-
action. Such rapid retention reactions
or
equilibrium
solute behaviorhasbeen observedforother
solutes
(16-
18).
Linear, Freundlich,
and
Langmuir sorption models
are
perhaps
the
most
commonly
used
equilibrium-type
models for describing
the retention
of
a
wide
range of
dissolved chemicals
in
soils.
A
partial listing ofequilib-
rium
type models
are
given in Table
1.
The linear
and
Freundlich models usethe solutedistribution coefficient
(KD),
which partitions the solute between that in the
soil solution and the
amount
sorbed
by
the soil matrix.
A
discussion ofthe
KD
parameter and
its
capability
for
describing
contaminant
migration
is
given by
Reardon
(19). Unlike the Langmuir models, linear and
Freun-
dlich models do
not
include
a
maximum
sorption
term
(Smax)
This is
disadvantageous
since
the capacity ofthe
soil for solute
removal, i.e.,
the total sites,
is
finite
and
should be
an
important
limiting
factor.
Langmuir
models
are
perhaps
the
most
widely
used
equilibrium
models for
describing
the fate
of
solutes
such
as
phosphorus
and
heavy
metals
in
soil
(18,20,21).
The two-site Langmuirmodel
may
be considered
as one
ofthe earliest multireaction
type
models. Here
one as-
0
1 2
24
36
48
60
72
TIME
(HOURS)
0
12
72
24
36
48
60
TIME (HOURS)
FIGURE 1.
Mg
concentration vs. time ofreaction from
batch
kinetic
experiments
for two
aggregate
sizes of
Abist
(Aquic Eutrochrept)
soil (15).
Table
1.
Fast
or
equilibrium
type
models
for
contaminant
retention in
soils.
Model
Formulation
Linear
s
=
KDC
Freundlich
(nonlinear)
s
=
KDcn
Langmuir
s
=
bcsmax/[l
+
bc]
Langmuir
with
sigmoidicity
s
=
bcsmax/[l
+
bc
+
k/c]
sumes complete equilibrium and partitions the reaction
sites
into
two
fractions. Holford
et
al. (22)
were one
of
the earliest
researchers
to
evaluate this
model
for de-
scribing
P retention
by several soils.
Recently,
the two-site
Langmuir
was
modified
to
in-
corporate the
sigmoidal
shape
of Cu, Pb,
and
Cd
sorp-
tion isotherms
observed
at
extremely low
concentra-
tions (23). The
equilibrium
models given
in
Table
1
have
been
used
to
describe adsorption
isotherms
for
a
wide
range
of solutes including major cations (Na, Ca, Mg,
and K), heavymetal species, and organics (16,24).
How-
MODELING CONTAMINANT
RETENTION AND TRANSPORT IN SOILS
71
ever, as
pointed
out
by Veith and Sposito (25) and Spos-
ito
(26),
a
good fit of
a
particular adsorption isotherm
does
not
in itselfconstitute
a
proof
of
any
specific
sorp-
tion
mechanism.
Other
types
of
equilibrium models
are
those based
on
ion-exchange reactions (27,28). Unlike the
previous
models, which
are
empirical
in
nature,
ion-exchange
models
are
based
on
rigorous
thermodynamics
in which
the reaction stochiometry is
explicitly
considered. A
set
ofrecursion formulae has beenformulated by Rubinand
James (27) that describe exchange isotherms
for
mul-
tiple ions in the
soil. Recently,
aqueous
equilibrium
re-
actions, along
with
ion-exchange
reactions, have been
used
to
describe
multiple ion
transport
in soils (29,30).
Ion
exchange has been
used
by several researchers
to
describe the
transport
of
cations present
in the soil
so-
lution (28,31-33).
also
apparent
that
even
after 300 hr,
quasi-equilibrium
conditions
were
not
attained. The results shown in Fig-
ure
3
illustrate the influence ofkinetic reactions
on
the
shape ofsorption isotherms of
P
in
a
Norwood
soil. The
amount
of
P
sorbed
increased
with
time
as
well
as
with
concentration. Itis
apparent
that
the
use
ofequilibrium-
type
models
would
yield inadequate
predictions of the
fate
of such solutes in the
soil
system.
Several
models have
been proposed
to
describe
the
kinetic reactions
of dissolved chemicals
in
the
soil
so-
lution. Most
common
is the first-order
kinetic
reaction,
which
was
incorporated
into the
coivection-dispersion
transport
equation by Lapidus and Amundson (34).
Such
reactions
are
assumed
to
be
fully reversible,
and
the
magnitude
of the
reaction coefficients determines
the time when
apparent
equilibrium
may
be
attained.
The
use
ofsuch linear
models has beenrather
restricted
due
to
the nonlinear behavior of
most
solute retention
reactions, exemplified
by the
cases
shown in Figures 2
and
3.
The first-order kinetic model has been
modified
to account
for the nonlinear-kinetic
behavior of
reten-
tion mechanisms.
Such
a
modified model
was
used
suc-
cessfully for describing the retention of
P
and several
pesticides
in
batch and miscible displacement studies
(16,35). Another fully reversible model is that of the
Langmuir kinetic
type
(Table
2).
Important
features of
this
kinetic
model
are
that it includes
a
maximum
re-
Kinetic Retention
Models
It
has been
observed
that the
amount
of solute
re-
tained (or released) from the soil solution
may
be
strongly time dependent.
Selected
examples of kinetic,
retention for Cd
are
given in Figure 2. Here, the kinetic
dependence
of
Cd reactions, carried
out in
batch
ex-
periments,
is shown for various soils
(24).
The
amount
ofCd retained varied
among
soils with Cecil soil
exhib-
ited the lowest retention, whereas
the
Sharkey
soil
showed maximum Cd retention.
The
sharp
decrease in
Cd concentrations indicates
a
fast-type sorption
reac-
tion, which
was
followed by
slower
type
reactions. It is
80
60
KINETIC EXPERIMENT
01
E
CADMIUM
-
WINDSOR
Co=
1
MG/L
40
O.7
20
0.
*
CECI
L
0
0.5
0
10
20
30
40
50
60
C
(ug/mI)
0.4
C
(MG/L)
0.3
~
~
WIDOR
FIGURE 3. P
sorption
isotherms
after
8, 24,
and
96
hr of reaction
time for
a
Norwood soil. Solid
curves are
predictions using
the
nonlinear Freundlich model.
=
SHARKEY
OLIVIER
NORWOOD
0.2
Table 2.
Insufficiently
fast
or
kinetic-type
models
for
contaminant retention in soils.
I
0.1
Model
Formulation
L
as/at
=
k,
(O/p)
c
-
k2s
First-order
as/at
=
k,
(O/p) c'
-
k2s
Nth order
200
TIME
(HOURS)
300
Irreversible
(sink/source)
as/at
=
k8 (O/p) (c
-
cp)
100
0
Langmuir
kinetic
as/at
=
k,
(O/p)
c
(smax
-
s)
-
k2s
Elovich
as/at
=
A
exp(-Bs)
as/at
=
ki
(O/p) c'sm
Power
Mass transfer
as/at
=
k (O/p)
(c
-
c*)
FIGURE 2.
Cd
concentration
vs.
time of reaction for five soils
(16).
72
H. M.
SELIM
tention capacity
term
and that
it is
nonlinear
in nature
(15). A discussion of the kinetic behavior of the Lang-
muir sorption reaction mechanisms during transport is
presented by
Jennings and
Kirkner
(36).
Ra
a2c
Fp-
ac
(6)
(kl
cn
-k2
at
-2
[8s1)
-
v
-
(7)
Multireaction Kinetic Models
A
widely
used multireaction model is
the two-site
model
proposed
by
Selim
et
al. (37) and
Cameron
and
Klute
(38).
This model
was
developed
for the
purpose
of
describing
observed batch results, which
showed
rapid initial retention reactions followed
by
slower
type
reactions.
The
model
was
also
developed
to
describe the
excessive
tailing
of
breakthrough
results
obtained from
pulse
inputs
in
miscible displacement experiments. Sin-
gle
retention
models of the first and nth-order kinetic
type
failed
consistently
to describe such batch
or
mis-
cible
displacement
results. The two-site model is based
on
several simplifying assumptions.
It is assumed that
a
fraction
ofthe total
sites
(referred
to
as
type I
sites)
are
highly
kinetic
in
nature. As
a
result, type
I sites
were
assumed
to react
slowly
with
the solute
in the soil
solution.
In
contrast,
we
consider
type II sites to
react
rapidly with soil
solution.
The retention reactions for
both
types
of
sites
were
based
on
the
nonlinear
(or
nth
order)
reversible kinetic approach
as
outlined
in Table
2. The
convention-dispersion
transport
equation
with
the two-site
retention mechanism
may
be
expressed
as:
R
=
1
+[P]KDMC
n.1
Ke
s2
=
KD
c
(8)
where Eqs. (7
and
9) describe
equilibrium
reaction of
the Freundlich
type.
The
term
R of Eq.
(7) is
the
re-
tardation factor
which
for
this
nonlinear case is a
func-
tion
of
c.
Selim et
al.
(18) found that
the
use
of
the
equilibrium and kinetic
two-site model provided im-
proved
predictions of breakthrough curves (BTCs)
for
Picloram in
soils. This result was
due primarily to
im-
proved predictions of the excessive tailing of the de-
sorption
or
leaching side
and
the sharp
rise ofthe
sorp-
tion
side
of
the
BTCs
in
comparison to predictions using
single reaction
equilibrium
or
kinetic models. Examples
of
predictions
for two
pesticides using this model
are
shown in
Figure
4.
Here, atrazine
and
(2,4-dichloro-
phenoxyacetic acid)
were applied
as
separate pulses
into
different soil columns. The
equilibrium and kinetic
two-
site
model described in BTCs
adequately for both
pes-
ticides
and for
different input pulse concentrations
(co).
In
order
to
obtain the predictions
shown
in
Figure
4, it
was
necessary that
the
retention reactions
for
the equi-
librium and the kinetic sites were nonlinear with the
values for m and
n
less than unity.
The two-site model has been
used by
several scien-
tists
including
De
Camargo
et
al. (40), Rao et
al.
(41),
Hoffman and Rolston (42), Jardine
et
al. (43), Nkedi-
Kizza
et al. (8),
and Parker and Jardine (44), among
others. The
model
proved successful
in
describing
the
retention
and
transport of several dissolved chemicals,
including
aluminum, 2,4-D, atrazine, phosphorus,
po-
tassium,
cadmium,,
chromium, and methyl
bromide.
Major
disadvantages
of
the
two-site model are
that
itis
restricted
to
reversible mechanisms and
that
is
does
not account
for possible consecutive-type solute inter-
actions in the
soil
system.
Several multireaction
models
have
beenintroduced to
incorporate irreversible
as
well
as
reversible reactions
ofthe concurrent
or the consec-
utive
type.
An example
of
a
multireaction model is
shown by the
schematic diagram of Figure 5. Here
we
consider the solute to
be
present in
the soil
in five
phases:
c,
sl,
S2, S3,
and
se,.
It
is assumed that the rate
eat=c
D-a2
--qac-
(klcn
-k2ps1)
(2)
-(k3
Ocm
-
k
PS2)
as1
kl
[
cn
- k2
s1
(3)
at
as2F
at
k3
[-Jcm
k4
s2
(4)
St
=
Sl
+
s2 (5)
where
s1
and
S2
are
the amounts retained
by
sites I and
sites
II,
respectively,
and
st
is the total amount ofsolute
retained.
The
nonlinear
parameters
m
and
n are
usually
considered
less
than unity
and
n m.
For the
case
n
=
m
=
1,
the
retention reactions
are
ofthe first-order
type, and
the
problem
becomes
a
linear
one.
This
two-
site
approach
was
also
considered for the
case
when
type
II sites
are
assumed to be in
equilibrium
with
the
soil solution. Such conditions
may
be attained when
the
values for the forward
and
backward (or k3
and
k4)
rate
coefficients
are
extremely
large
in
comparison
to
the
water
flow
velocity
(q);
that
is,
the local
equilibrium
assumption
is valid for
type
II
sites
(39).
Under these
conditions,
the
solute
convection-dispersion
transport
equation
for
a
combined modelof
equilibrium
and kinetic
retentions is
(37):
of irreversible reaction
(as./dt),
which
is equivalent to
the
sink
term Q of Eq.
(1),
can be
expressed
as
8
ksc
(9)
Q
=
p
=
a
t
This is a first-order irreversible
kinetic process, and
ks
is the associated rate coefficient
(per day). Mansell
et
al.
(35)
proposed
this approach
for
describingpossible
precipitation of
P in
miscible
displacement
studies. Fis-
kell et
al.
(45) found thatincorporation
of
s,
inthe model
MODELING
CONTAMINANT
RETENTION AND TRANSPORT IN
SOILS
73
was
essential
in
describing
the slow
P
retention
kinetics
for
a
Spodosol
from
deep
and
shallow tilled
treatments.
Recently, Amacher
et
al.
(18) showed thatthe sink
term
was
necessary
to
describe
batchresults for Hg,
Cd,
and
Cr
retention
versus
time for five different soils. This
sink term
is
similar to that for
diffusion-controlled pre-
cipitation
reaction if one assumes that
the
equilibrium
concentration
for
precipitation
is
negligible and
that
k8
is
related
to
the diffusion coefficient. Among
kinetic
models that
are
used
to describe
the
rate
of
irreversible
reactions is the Elovich model given
in
Table
2. For
further discussion of irreversible kinetic models
see
0o Travis and Etnier
(16).
As indicated
by Figure
5, the
s1
and S2
phases
are in
direct contact with c, and
reversible
processes of the
equilibrium
and kinetic types govern their
reactions,
respectively.
The s1 and
S2
phases
may
be regarded
as
the amounts adsorbed on
surfaces
of soil
particles
as
well
as
chemically bound
to Al
and Fe oxides surfaces
or
other
type
surfaces.
Moreover, these
phases
may be
characterized
by
their fast sorption from, as well
as
release
to,
the soil
solution
and, thus susceptibility
to
leaching
in the soil. In contrast,
S3
iS considered here
as
the
amount which is nonlabile, firmly held,
or
fixed
by
the soil matrix.
Furthermore,
this
phase may be
characterized by
its slow
(retention
and
release)
reac-
tions.
Therefore,
itis often assumed thatthefirmlyheld
phase
is
less readily available
to
uptake by plant
roots
or transport in the soil
profile.
The predictive capability of the multireaction model
above was tested for two different solutes [P and
Cr(VI)]
and forvarious soils. Asillustrated bythe BTCs
shown in
Figures
6 and
7,
the model is
capable
of de-
scribing
the behavior
ofthese solutes
adequately.
Such
agreement may be
regarded
as
adding
credence to the
validity
of
the model. For P predictions, we found that
the
presence of a
consecutive
reaction (or
S3)
was
nec-
essaryinordertodescribetheresultsshown.
Moreover,
the use of a simple first-order reaction to describe the
50
,ug/ml
(o
,0
'°°
*9
i
°8,
°
J/°
o
5000
6
140
0
0
to
0
O
2
4
6
8
V/vo
EUSTIS SOIL
1.0
ATRAZINE
.8
.6;
4
5
pg/mI
a
~~~
0P
o
0'
4t2
1~~~~ot
.2
0
4
8
12
16
V/V
0
FIGURE
4.
Breakthrough
curvesfor
2,4-D
amine
and
atrazinefor
two
pulse concentrations.
Dashed
and
solid curves are
predictions
us-
ing the
two-site
model
(21).
36
v/vo
FIGURE
5. A
schematic
diagram
of
a
multireaction model for solute
retention
in the soil
system.
FIGURE
6.
Breakthrough
for
P in
a
Norwood soil. Solid
curve
is the
prediction using the multireaction model.
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