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VI. Transport and Ion Exchange
H. M. SELIM
the governing reactions. In addition, due to their complexity, several of
these models have not been fully validated. CHEMTRAN and TRANQL
models were used successfully to describe the Ca and Mg concentration
data presented earlier by Valocchi et al. (1981). Kirkner et al. (1985)
utilized the FIESTA model to describe Ni and Cd breakthrough results on
a sandy soil. Model predictions provided higher retardation of Cd and
lower retardation for Ni. However, improved predictions were obtained
when a kinetic approach was used with parameters obtained from batch
An overview of ion exchanged and transport models is presented below.
The intent is to emphasize the importance of ion exchange as the retention
mechanism during water flow in soil. Models for which nonequilibrium ion
exchange behavior is described using physical (two-region) or chemical
kinetic approaches using specific sorption are discussed.
We consider ion exchange as the governing mechanism for the retention
of cations on soil matrix surfaces. In a standard mass-action formulation,
the exchange reaction for two competing ions i and j may be written as
where TKii denotes the thermodynamic equilibrium constant and a and a*
(omitting the subscripts) are the ion activity in soil solution and on the
exchanger surfaces, respectively. Based on Eq. (54), one can denote the
parameter vKi, as
where “K is the Vanselow selectivity coefficient and 6 is the activity
coefficient on the soil surfaces. For binary homovalent exchange, i.e.,
vi= vj = v, and assuming similar ion activities in the solution phase, rearranging Eq. (55) yields
where Kij is a generic selectivity coefficient of ions i over j (Rubin and
James, 1973) or a separation factor for the affinity of ions on exchange
MODELS OF INORGANICS IN SOILS
sites (Helfferich, 1962). In addition, ciand cj are the relative ion concentrations (dimensionless) such that ci= Ci/CT and c, = Ci/CT, where Ci, Cj,
and CT [mmol( +)/ml] are the concentrations in the soil solution of ions i
and j the total concentration. Also, si and sj are amounts retained on the
solid matrix surfaces (dimensionless) and are expressed as equivalent fractions where si= SJR and sj = S,/R. Here, Si and S, are the amounts
adsorbed [mmol(+)/g soil] and R is the cation exchange (or adsorption)
capacity of the soil [mmol( +)/g soil]. Therefore, for homovalent exchange
we have (Valocchi ef al., 1981)
rearrangement of Eq. (57) yields the following isotherm relation (for ion 1)
Because C, and R are assumed constant, the respective values for ion 2
(i.e., c2 and s2) can be easily obtained. This isotherm relation Eq. (58)
indicates that for K12# 1 we have a nonlinear-type sorption isotherm.
Upon incorporation of the above ion exchange sorption isotherm into the
convective-dispersive transport Eq. (7), we have (for ion 1)
where the term R may be referred to as the retardation factor, expressed as
and R is a function of relative concentration when K12# 1.
Breakthrough results for Ca and Mg leaching from Abist soil columns,
packed uniformly with two aggregate sizes, are shown in Fig. 21. These
results are from Selim et al. (1987), wherein a Mg pulse was introduced into
Ca-saturated soil columns and the total concentration (CT) was maintained
constant. The solid and dashed lines shown are model calculations of Ca
and Mg results based on the classical convective-dispersive transport equation with ion exchange as the retention mechanism. In general, adequate
model predictions for both ions were achieved for the two aggregate sizes
for early times or pore volumes. Less than adequate predictions were
obtained, however, for large times, i.e., during leaching of the Mg pulse by
Ca. Selim et al. (1987) suggested that such model deviations may be due to
lack of complete local equilibrium between the ions in solution and those
H. M. SELIM
1-2 rnm (aggregates)
V N ,
/ - -
Figure 21. Calcium and Mg breakthrough results from soil columns for two aggregate
sizes of Abist soil. Predictions obtained using the classical ion exchange model are shown by
the smooth curves. From Selim et al. (1987),with permission.
on the ion exchange surfaces. Similar findings were reported by Gaston
and Selim (1990) for binary (Ca and Mg) and ternary (Ca, Mg, and Na)
systems in a well-aggregated Sharkey clay soil.
B. M O B I L E - ~ O BAPPROACH
In an attempt to describe the transport and exchange reactions of cations
in aggregated porous media, the physical nonequilibrium approach of the
mobile-immobile (or two-region) concept was utilized. Van Eijkeren and
Loch (1984) and Schulin et al. (1986) considered the exchange of cations to
be governed by a general form of the exchange equation, which was then
incorporated into the mobile-immobile, convective-dispersive transport
equation. The capability of this approach was examined by Selim et al.
(1987) for describing the mobility of Ca and Mg ions in soil columns,
MODELS OF INORGANICS IN SOILS
packed with 1- to 2-mm and 2- to 4-mm aggregates, under conditions of
constant and variable ionic strength of the soil solution. Mansell et al.
(1988) examined this mobile-immobile approach for the transport and
exchange reactions for ternary systems (Na-Ca-Mg) in soil. In their
analysis, Mansell et al. (1988) allowed cation exchange selectivities to vary
with fractional coverage of the exchange sites, which provided improved
predictions of cation breakthrough results.
To describe ion retention in the dynamic and less accessible regions of
the mobile-immobile concept, the sorption process is governed by the
equilibrium ion exchange relationship of Eq. (57). We also assume that
such a relationship is valid for the exchange sites of both regions, i.e., ion
affinity for the dynamic region is the same as that for the less accessible
Such an assumption was used by van Eijkeren and Loch (1984), Selim et al.
(1987), and Gaston and Selim (1990).
To test the capability of the mobile-immobile concept of describing the
transport of cations, the breakthrough results shown previously (see
Fig. 21) were utilized. Model predictions of Ca and Mg concentrations in
the leachate for the two aggregate sizes are given by the solid and dashed
lines shown in Fig. 22. Obvious improvements in model predictions, in
comparison to the classical approach (Fig 21), were achieved and may be
considered as evidence of lack of local equilibrium between the ions in the
soil solution and those on the exchange surfaces. Improved predictions
using this approach were achieved by Schulin et al. (1989) for Ca and Mg
transport under conditions of variable ionic strength for two wellaggregated forest soils and by Mansell et al. (1988) for a ternary (Na-CaMg) system in a Yo10 loam soil.
(K12Irn= (K12)'" = K12
In the preceding classical and mobile-immobile concepts, the distribution of each pair of cations, i and j , can be described by a constant
selectivity coefficient K i j . Such assumptions make it possible to arrive at
recursion formulas for multiple ions, as were introduced by Rubin and
James (1973). Generalized isotherms for multiple ions were based on
binary exchange coefficients for all combinations of ions present in the soil
system. Such generalized isotherms were used by Valocchi et al. (1981) and
Mansell et al. (1988). However, such an assumption of constant exchange
selectivity is often unfounded as evidenced by several isotherm data in the
H. M. SELIM
I -2 mm (aggregates)
Figure 22. Calcium and Mg breakthrough results from soil columns for two aggregate
sizes of Abist soil. Predictions obtained using the two-region ion exchange model are shown
by the smooth curves. From Selim ef nl. (1987),with permission.
literature (Sposito, 1981; Jardine and Sparks, 1984; Parker and Jardine,
1986; Mansell et al., 1988). As a result, Kij coefficients are no longer
constant but vary with the relative fraction of cations on the exchange
Mansell et al. (1988) utilized the sorption isotherm results of Lai et al.
(1978) for Mg-Ca and Na-Ca in order to calculate binary selectivity
coefficients Kij versus relative ion concentration (C/C,), shown in Fig. 23.
The dashed curves represent least-squares best fit of the data to the
empirical relation log(Kij) = a + bC. The results clearly show a strong Kij
dependency over the concentration range with high affinity of Mg over Ca
for C/CT less than 0.4. As expected, adsorption of Ca or Mg was preferred to adsorption of Na. However, the results show an increased Kij for
Na --* Ca at low (C/CT < 0.4) and high (C/CT > 0.85) relative concentrations. The influence of variable selectivity coefficients on the predictions of
cation transport was investigated by Mansell et al. (1988). The transport
data sets used were those from miscible displacement experiments of Lai
et af. (1978). Mansell et al. (1988) incorporated additional terms into the
MODELS OF INORGANICS IN SOILS
Mg --+ Ca
- - --
classical (convective-dispersive) equation in order to account for variable
Kji values for the ternary systems (Na-Ca-Mg) of Lai et al. (1978). They
also utilized the mobile-immobile approach in conjunction with variable
Kji values in order to describe the same data set. Mansell et al. (1988)
found that good breakthrough predictions were obtained for the relatively
noncompetitive Na when either constant or variable Kii values were used
with the classical model (see Fig. 24). However, the use of constant Kij
values underestimated the tailing of the Mg breakthrough data. Description of Mg tailing was improved when variable Kii values were used, but
the extent of Mg retardation (peak location) was somewhat overestimated.
The combined use of the mobile-immobile approach and variable K j j
values provided the best overall description of Na and Mg breakthrough
In this approach, two mechanisms were considered as the dominant
retention processes in the soil, namely, ion exchange and specific sorption.
We consider ion exchange as a nonspecific sorption/desorption process.
Ion exchange is a fully reversible mechanism and is assumed here to be
H. M. SELIM
Figure 24. Sodium and Mg breakthrough results from soil columns of Yolo soil (Lai
etal., 1978). Solid and dashed curves are predictions using the classical and two-region models,
respectively. Exchange selectivity coefficients used in model predictions were considered constant (top figure) or variable (bottom figure). From Mansell et al. (1988), with
either rapid (i.e., instantaneous) or may be considered as a kinetic process.
Specific sorption is considered as a kinetic process wherein ions have high
affinity for specific sites on matrix surfaces. Furthermore, retention of ions
via specific sorption is regarded as an irreversible or weakly reversible
The kinetic ion exchange reaction was analogous to mass transfer or
between the solid and solution phase such that (for ion i)
ds,/dt = a(si*- S i )
where si is the amount sorbed (at time t ) on matrix surfaces, sT is the
equilibrium sorbed amount, and a is an apparent rate coefficient (hr-').
Here s* (for ion 1) was calculated using the equilibrium isotherm Eq. (58).
Obviously, as t + and/or large values of a,s* and s become equal and
equilibrium conditions prevail, Expressions similar to the above equation
have been used to describe mass transfer between mobile and immobile
water and chemical kinetics (Parker and Jardine, 1986; Selim and
Amacher, 1988). Studies that illustrate kinetic ion exchange behavior
include those of Jardine and Sparks (1984) and Ogwada and Sparks (1986).
MODELS OF INORGANICS IN SOILS
It was postulated that in 2 :1 types of minerals, intraparticle diffusion as a
possible rate-controlling mechanism governs the kinetics of adsorption of
cations (Sparks, 1989).
The specific sorption process was considered as a kinetic reaction whereby the rate of sorption is governed by a second-order mechanism such that
where k f and kb are the forward and backward rate coefficients (hr-'),
the amount of vacant specific sites, and I)is the amount specifically sorbed
[mmol( +) g-'1, respectively. Vacant specific sites are not strictly vacant.
They are assumed occupied by hydrogen, by hydroxyl, or by other specifically sorbed species.
The role of specific sorption and its influence on the behavior of metal
ions has been recognized by several investigators. Sorption/desorption
studies showed that highly specific sorption mechanisms are responsible for
metal ion retention for low concentrations (Tiller et al., 1979, 1984). The
general view was that metal ions have a high affinity for sorption sites of
oxide minerals surfaces in soils. In addition, these specific sites react slowly
with heavy metals and are weakly reversible. In the absence of competing
metal ions for specific sites (e.g., Ni, Co, and Cu), as is the case in this
study, it is reasonable to consider specific sorption as an irreversible
process. Therefore, the above second-order reaction was modified to describe irreversible or weakly reversible retention by setting the backward
rate coefficient k b as zero,
p(a+/at) = k@+C
For several metal ions (e.g., Cd, Ni, Co, and Zn), specific sorption was
shown to be dependent on time of reaction. Therefore, the use of kinetic
rather than an equilibrium sorption mechanism is recommended.
Although our model formulation is based on direct reaction between
metal ions in soil solution and specific sorption sites, others have considered a consecutive-type approach for Cd sorption. According to Thesis
et al. (1988), a set of two second-order reactions was considered; one fully
reversible step was followed by an irreversible reaction. Ion exchange, as
discussed above, was not considered. Theis et al. (1988) argued that the
amount adsorbed on geothite surfaces was susceptible to migration (via
surface diffusion) from primary to secondary surface sites. Other possible
mechanisms may include formation of surface complexes, hydrolysis of
sorbate at the surface, and surface complexation. Tiller el al. (1979) quantified specific sorption as the amount of sites that retain metal ions following several washings of the soil with high concentrations of a nonspecifically sorbed cation (0.01 M Ca2NO3). As a result, metal ions on specific
H. M. SELIM
sites are not easily replaceable by Ca ions, but can be replaced (exchanged)
by competing (specifically sorbed) ions such as Ni, Cd, Co, and Zn.
Figure 25 shows miscible displacement results of Cd in a Windsor sandy
loam soil. A solution of 0.005 M Ca(N03)2 was applied to a packed soil
column at a Darcy flow velocity of 0.271 m/day. A pulse of Cd(N03)2
dissolved in 0.005 M Ca(NO& solution was applied to the soil column and
the effluent solution was collected and analyzed for Cd. The concentration
of Cd in the input pulse was 100 mg/liter, therefore a small change in the
total cationic concentration occurred from 10 mmol,/liter for the background solution to 11.786mmol,/liter for the Cd pulse. The use of the
competitive transport model, assuming equilibrium conditions and ignoring specific sorption (kf= 0), resulted in an overestimation of the peak
concentration and the BTC was more retarded than that experimentally
measured (curve A, Fig. 25). For this model prediction, a selectivity
coefficient (KCdCa)of 2, indicating a slight preference of Cd over Ca to the
exchange surfaces, was used. This value was based on experimental
measurements for a Cd-Ca sorption isotherm. All other parameters were
experimentally measured or estimated using other experimental measurements, e.g., the dispersion coefficient was estimated using 3H20 BTCs.
The use of the kinetic ion exchange mechanism along with specific sorption appears to provide an improved prediction of miscible displacements
for Cd as shown by curves B-D in Fig. 25. Although the use of smaller
values for a provided improved prediction of the observed tailing, BTCs
were less retarded in comparison to the measured BTC. It is obvious that
additional evaluation of this approach is needed for the prediction of the
retention of other ions during transport in the soil profile.
Cd Breakthrough Curve, 100 mglliter - Windsor Soil
PORE VOLUME (V/V,)
Figure 25. Measured (closed circles) and predicted BTCs for Cd in Windsor soil. Prediction using equilibrium ion exchange model is shown by curve A. Curves B, C, and D are
calculations using the kinetic ion exchange model with a of 2 day-' and specific sorption (kf)
values of 0, 0.5, and 1.0 day-', respectively.
MODELS OF INORGANICS IN SOILS
VII. TRANSPORT IN LAYERED SOIL
None of the mathematical models presented thus far deals with the
problem of the fate of reactive or noreactive solutes in nonhomogeneous or
layered soils. Because soil profiles are seldom uniform, it is essential to
consider the fate of dissolved chemicals in stratified soil systems. The study
of Shamir and Harleman (1967) is one of the earliest papers dealing with
nonreactive solute transport through layered porous media having great
depths ( z --* w). Others, including Rubin and James (1973), Selim et al.
(1977), Selim (1978), and Barry and Parker (1987), considered the fate of
reactive solutes in layered soils under various flow and boundary conditions. A schematic diagram of a soil of length L with three distinct layers I,
11, and I11 is shown in Fig. 26. Each layer has specific but not necessarily
the same 0,p, and solute retention characteristics. First, we consider the
case of solute transport in layered soils wherein fully saturated conditions
under constant Darcy’s flux (steady flow) prevail. This is followed by cases
for water-unsaturated layered soils under steady and transient water flow
Simulations were carried out using equilibrium linear and nonlinear
(Freundlich) [see Eq. (15)] as well as first-order kinetic (reversible and
Figure 26. Schematic diagram of a three-layered soil.
H. M. SELIM
irreversible) retention [see Eqs. (19) and (23)] to describe solute behavior
in each layer of a multilayered soil profile. Different simulations were
conducted to evaluate the importance of soil layer stratification and adsorption characteristics on the shape and position of effluent concentration distributions
~ soil profiles.
For the linear case, a retardation factor R [ R = 1 + p K d / O ;see Eq. (29)]
represents the magnitude of retention for each layer and is represented by
R 1 , R 2 , etc. (Selim et al., 1977). Solid lines in Fig. 27 are simulated BTC
results from columns in which the solute passed first through L1 and then
L 2 ; open circles are calculated results of solute flow in the opposite
direction. The dashed lines are for the homogeneous cases wherein L = L1
or L = L2 and these cases have the appropriate retardation factors, R1 or
R 2 , respectively. As expected, the BTCs for the two-layered cases lie
between the homogeneous cases R1 and R2 (dashed lines). The retardation
factor R1 equals one and represents a nonreactive solute whereas R2 equals
10. Increasing L2 (or decreasing L,) causes the breakthrough curves to
move to the right toward the homogeneous R2 case. The most striking
result in Fig. 27 is the failure of the order of soil layers to influence the
shape or position of the effluent concentration distribution. Based on these
results, a layered soil profile could be regarded as homogeneous with an
average retardation factor used to calculate emuent concentration distributions. An average retardation factor R for N-layered soil can simply be
Figure 27. Simulated effluent concentration distribution for two-layered soils with retardation factors R, and RZ (linear sorption) and varying lengths L , and Lz. Solid lines are
simulations wherein layer 1 with R, is first encountered, whereas open circles are for flow in
the reverse direction. Dashed curves are for homogeneous soils. From Selim et al. (1977),