Chapter 6. Modeling the Transport and Retention of Inorganics in Soils
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332
H. M. SELIM
chemicals in the soil is of considerable value in managing land disposal of
wastes and in fertilizer applications. Such predictive capability requires
knowledge of the physical, chemical, and biological processes influencing
solute behavior in the soil environment.
Over the last three decades, a number of theoretical models for solute
transport in porous media have been proposed. One group of models deals
with solute transport in welldefined geometrical systems wherein it is
assumed that the bulk of solute moves in pores and/or cracks of regular
shapes or through interaggregate voids of known geometries. Examples of
such models include those dealing with a soil matrix of uniform spheres,
rectangular or cylindrical voids, and discrete aggregate or spherical size
geometries. Solutions of these models are analytic, often complicated, and
involve several numerical approximating steps. In contrast, the second
group of transport models consists of empirical models that do not consider welldefined geometries of the pore space or soil aggregates. Rather,
solute transport is treated on a macroscopic basis, with the water flow
velocity, hydrodynamic dispersion, soil moisture content, and bulk density
being treated as the associated parameters that describe the soil system.
Refinements of this macroscopic approach are the “mobileimmobile”
transport models, wherein local nonequilibrium conditions are due to
diffusion or mass transfer of solutes between the mobile and immobile
regions. The mobileimmobile models have been used to describe transport of several solutes in soils. These empirical or macroscopic models are
widely used and are far less complicated than the abovementioned more
exact approach for systems of welldefined porous media geometries.
To predict the transport of reactive solutes in the soil, models that
include retention and release reactions of solutes with the soil matrix are
needed. Retention and release reactions in soils include ion exchange,
adsorption/desorption, precipitation/dissolution, and other mechanisms
such as chemical or biological transformations. Retention and release
reactions are influenced by several soil properties, including bulk density,
soil texture, water flux, pH, organic matter, and type and amount of
dominant clay minerals. Adsorption is the process whereby solutes bind or
adhere to soil matrix surfaces to form outer or innersphere solutesurface
site complexes. In contrast, ion exchange reactions represent processes
whereby charged solutes replace ions on soil particle surfaces. Adsorption
and ion exchange reactions are related in that an ionic solute species may
form a surface complex and may replace another ionic solute species
already on surface sites. The term “retention,” or the commonly used term
“sorption,” should be used when the mechanisms of solute removal from
soil solution are not known, and the term “adsorption” should be used
only to describe the formation of solutesurface site complexes.
MODELS OF INORGANICS IN SOILS
333
Solute retention proceses in soils have been quantified by several scientists along two different lines. One represents equilibrium reactions and the
second represents kinetic or timedependent types of reactions. Equilibrium models are those for which solute reaction is assumed to be fast or
instantaneous in nature and “apparent equilibrium” may be observed in a
relatively short reaction time. Langmuir and Freundlich models are
perhaps the most commonly used equilibrium models for the description of
fertilizer chemicals, such as phosphorus, and for several heavy metals.
These models include the linear and Freundlich (nonlinear) and the oneand twosite Langmuir type. Kinetic models represent slow reactions
whereby the amount of solute sorption or transformation is a function of
contact time. Most common is the firstorder kinetic reversible reaction for
describing timedependent adsorption/desorption in soils. Others include
linear irreversible and nonlinear reversible kinetic models. Recently, a
combination of equilibrium and kinetictype (twosite) models and consecutive and concurrent multireactiontype models have been introduced.
In this article, major features of retention models that govern retention
reactions of solutes in the soil are presented. Singlereaction models of
the equilibrium type are first discussed, followed by models of the kinetic
type. Retention models of the multiplereaction type, including the twosite equilibriumkinetic models, the concurrent and consecutivemultireaction models, and the secondorder approach will be derived. This is
followed by multicomponent or competitivetype models wherein ion exchange is considered the dominant retention mechanism. Selected experimental data sets will be described for the purpose of model evaluation
and validation, and necessary (input) parameters are discussed. For convenience, we first present formulations of the transport equations that
govern the transport of solutes in a watersaturated and waterunsaturated
porous medium. Retention reactions of the reversible and irreversible
types are incorporated into the transport formulation. Boundary and initial conditions commonly encountered under field conditions are also
presented.
II. TRANSPORT EQUATIONS
Dissolved chemicals present in the soil solution are susceptible to transport through the soil subject to the water flow constraints in the soil
system. At any given point within the soil, the total amount of solute y,
(pg/cm3) for a species i may be represented by
XI = oci
+ psi
(1)
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H. M. SELIM
where S is the amount of solute retained by the soil (pg/g soil), C is the
solute concentration in solution (pg/ml), 0 is the soil moisture content
(cm3/cm3), and p is the soil bulk density (g/cm3). The rate of change of x
for the ith species with time is subject to the law of mass conservation such
that (omitting the subscript i)
d(0C + pS)
= div J  Q
dt
(2)
or
where t is time (hr) and J,, J y , and J, represent the flux or rate of
movement of solute species i in the x , y , and z directions (pg/cm2.hr),
respecitvely. The term Q represents a sink (Q positive) or source that
accounts for the rate of solute removal (or addition) irreversibly from the
bulk solution (pg/cm3 * hr). If we restrict our analysis to onedimensional
flow in the z direction, the flux J, , or simply J, in the soil may be given by
where Dm is the molecular diffusion coefficient (cm2/hr), DLis the longitudinal dispersion coefficient (cm2/hr), and q is Darcy’s flux (cm/hr). Therefore, the primary mechanisms for solute movement are due to diffusion
plus dispersion and by mass flow or convection with water as the water
moves through the soil. The molecular diffusion mechanism is due to the
random thermal motion of molecules in solution and is an active process
regardless of whether there is net water flow in the soil. The result of the
diffusion process is the wellknown Fick’s law of diffusion wherein the flux
is proportional to the concentration gradient.
The longitudinal dispersion term of Eq. (4)is due to the mechanical or
hydrodynamic dispersion phenomena, which are due to the nonuniform
flow velocity distribution during fluid flow in porous media. According to
Fried and Combarnous (1971), nonuniform velocity distribution through
the soil pores is a result of variations in pore diameters along the flow path,
fluctuation of the flow path due to tortuosity effect, and variation in
velocity from the center of a pore (maximum value) to zero at the solid
surface interface (Poiseuille’slaw). The effect of dispersion is that of solute
spreading, which is a tendency opposite to that of piston flow. Dispersion
is effective only during fluid flow; i.e., for a static water condition or when
water flow is near zero, molecular diffusion is the dominant process for
MODELS OF INORGANICS IN SOILS
335
solute transport in soils. For multidimensional flow, longitudinal dispersion
coefficients ( D L )and transverse dispersion coefficients (DT)are needed to
describe the dispersion mechanism. Longitudinal dispersion refers to that
in the direction of water flow and that for the transverse directions for
dispersion perpendicular to the direction of flow.
Apparent dispersion D is often introduced to simplify the flux Eq. (4)
such that
I3C
J=OD+qC
d.2
where D now refers to the combined influence of diffusion and hydrodynamic dispersion for dissolved chemicals in porous media. Incorporation of
flux Eq. (5) into the conservation of mass Eq. (3) yields the following
generalized form for solute transport in soils in one dimension,
The above equation is commonly known as the convectivedispersive
equation for solute transport, which is valid for soils under transient and
unsaturated soilwater flow conditions. For conditions wherein steady
water flow is dominant, D and 0 are constants; i.e., for uniform 0 in the
soil, we have the simplified form of the convectiondispersion equation as
where v (cm/hr) is known as the pore water velocity and is given by (4/0).
Solutions of the above convectiondispersive Eqs. (6) and (7) yield the
concentration distribution of the amount of solute in soil solution C and
that retained by the soil matrix S with time and depth in the soil profile. In
order to arrive at such a solution, the appropriate initial and boundary
conditions must be specified. Several boundary conditions are identified
with the problem of solute transport in porous media. The simplest is that
of a firstorder type of boundary condition such that a solute pulse input is
described:
C=C,,
z=O,
t
(8)
C=O,
z=O,
trT
(9)
where C, (pg/cm3) is the concentration of the solute species in the input
pulse. The input pulse application is for a duration T , which is then
followed by a pulse input that is free of such a solute. Such a boundary
condition was used by Lapidus and Amundson (1952) and Cho (1971),
3 36
H. M. SELlM
among others. The more precise thirdtype boundary condition at the soil
surface was considered by Brenner (1962) in his classical work, wherein
advection plus dispersion across the interface was considered. A continuous solute flux at the surface can be expressed as,
and a fluxtype pulse input as
O =  D d2
~+vC,
z=O,
trT
(12)
Advantages of using the thirdtype boundary conditions have been discussed by Selim and Mansell (1976) and Kreft and Zuber (1978). The
boundary conditions at some depth L in the soil profile are often expressed
as (Danckwerts, 1953; Brenner, 1962; Lindstrom et al., 1967; Kreft and
Zuber, 1978),
z=L,
t r o
(13)
which is used to deal with solute effluent from soils having finite lengths.
However, it is often convenient to solve the dispersionconvection equation wherein a semiinfinite rather than a finite length ( L ) of the soil is
assumed. Under such circumstances, the appropriate condition for a semiinfinite medium is
dCldz=O,
ac/dz=o,
tzo
(14)
Analytical solutions to the convectiondispersion equation subject to
the appropriate boundary and initial conditions are available for a number
of situations whereas the majority of the solute transport problems must be
solved using numerical approximation methods. In general, whenever the
form of the retention reaction is a linear one, a closedform solution is
obtainable. A number of closedform solutions are available from Crank
(1956), Ozisik (1968), Kreft and Zuber (1978), and van Genuchten and
Alves (1982). However, most retention mechanisms are nonlinear and
time dependent in nature and analytical solutions are not available. As a
result, a number of numerical models using a finite difference or finite
element have been utilized to solve nonlinear retention problems of
multireaction and multicomponent solute transport for one and twodimensional geometries (Rubin and James, 1973; Valocchi et al., 1981;
Miller and Benson, 1983; Cederberg et al., 1985; Selim et at., 1987; Mansell
et al., 1988).
z+a,
MODELS OF INORGANICS IN SOILS
337
III. EQUILIBRIUM RETENTION MODELS
The form of solute retention reaction in the soil system must be identified if prediction of the fate of reactive solutes in the soil using the
convectiondispersion Eq. (7) is sought. The reversible term (&/at) is
often used to describe the rate of sorption or exchange reactions with the
solid matrix. Sorption or exchange has been described by either instantaneous equilibrium or a kinetic reaction whereby concentrations in solution an sorbed phases vary with time. Reviews of various forms of
equilibrium and kinetic models are given by Murali and Aylmore (1983),
Selim (1989), Selim et al. (1990a). Recently, Nielsen et al. (1986) presented a comprehensive discussion of significant features of sorption exchange reactions of the equilibrium and kinetic type. Linear, Freundlich,
and one and twosite Langmuir equations are perhaps most commonly
used to describe equilibrium reactions. Freundlich and Langmuir reactions
and their use in describing equilibrium retention are discussed in subsequent sections. This is followed by kinetictype reactions and their implication for single and multireaction retention and transport models.
A. FREUNDLICH
The Freundlich equation is perhaps the simplest approach for quantifying the behavior of retention of reactive solute with the soil matrix. It
is certainly one of the oldest of the nonlinear sorption equations and has
been used widely to describe solute retention by soils (Helfferich, 1962;
Travis and Etnier, 1981; Murali and Aylmore, 1983; Sposito, 1984). The
Freundlich equation is
S = KdCb
(15)
where S is the amount of solute retained by the soil in pg/g, C is the solute
concentration in solution in pg/ml, Kd is the distribution coefficient in
cm3/g, and the parameter b is dimensionless and typically has a value of
b < 1. The distribution coefficient describes the partitioning of a solute
species between solid and liquid phases over the concentration range of
interest and is analogous to the equilibrium constant for a chemical reaction. For b equals unity, the Freundlich equation is often referred to as the
linear retention equation (see Table I).
There are numerous examples for solute retention, which was described
successfully by use of the Freundlich equation (see Sposito, 1984; Travis
and Etnier, 1981; Murali and Aylmore, 1983; Sparks, 1989). Selected
examples of linear and Freundlich (or nonlinear) retention are shown in
Fig. 1 for phosphate (P) sorption from batch studies for Al and A2
H. M. SELIM
338
Table I
Selected Equilibrium and KineticType Models for Solute
Retention in Soils"
Model
Formulationb
~
~~
Equilibrium type
Linear
Freundlich (nonlinear)
Langmuir
Langmuir with sigmoidicity
Kinetic type
First order
nth order
Irreversible (sinklsource)
Langmuir kinetic
Elovich
Power
Mass transfer
S = KdC
S = KdCb
s = WCS,,/(l+
S = oCS,,,/(l
OC)
+ OC+ u/C)
d S / d t = k f ( O / p ) C  kbS
a s / d t = k f ( @ / p ) C " kbs
dS/Jt = k,(O/p)(C  C,)
JS/Jt= kf(O/p)C(S,,,  S )  kbs
dS/dt=Aexp(BS)
as/at = K(O/p)C"Sm
d S / d t = K(O/p)(C C * )
"Adapted from Selim er al. (1990a), with permission.
'A, B , b, C', C,, K, Kd, kb, k f , k,, n , m, Lax,
0, and
adjustable model parameters.
ff
are
horizons of an Oldsmar fine sand (Mansell et al., 1977). For both
isotherms, it appears that the P isotherms can be adequately described by
the Freundlich equation. Logarithmic representation of the Freundlich
equation is frequently used to represent the data as illustrated in Fig. 2.
Here the slope of the bestfit curve provides the nonlinear parameter b and
1
0
0
,
Figure 1. Equilibrium phosphate adsorption isotherms in surface A, and subsurface A2
soils of an Oldsmar fine sand. From Mansell et al. (1977). with permission.
MODELS OF INORGANICS IN SOILS

1000
I
I
I
339
Chromium
loot
a
1
1 %
1.0
.
.
0.01
A
I.o
0.I
100
10
C (rng/liter)
Figure 2. Retention isotherms for chromium on three surface soils,Alligator (Al), Kula
(Ku), and Windsor (Wi). From Buchter et al. (1989), with permission.
the intercept as & according to log(S) = & + b log(C) as long as a linear
representation of the data in the log form is achieved. In Fig. 2, we
illustrate the use of the Freundlich equation for Cd retention for three
different soils, whereas Fig. 3 shows Pb, Cu, Cd, and Co isotherms for one
(Alligator) soil (Buchter et al., 1989).
Although the Freundlich equation has been rigorously derived (Sposito,
1980), the goodness of fit of the Freundlich equation to solute retention
data does not provide definitive information about the actual processes
involved, because the equation is capable of describing data irrespective of
the actual retention mechanisms. Often complex retention processes can at
1000
100

f
10
v1
1.0
0.I
0.001
0.01
0.I
1.0
10
100
C (mg/liter)
Figure 3. Retention isotherms Co, Cd, Cu, and Pb on Alligator soil. From Buchter et al.
(1989), with permission.
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H. M. SELIM
least in part be described by relatively simple models such as the Freundlich equation. Therefore, the Freundlich parameters Kd and b are best
regarded as descriptive parameters in the absence of independent evidence
concerning the actual retention mechanism.
The Langmuir isotherm is the oldest and most commonly encountered
isotherm in soils. It was developed to describe the adsorption of gases by
solids when a finite number of adsorption sites in the surface is assumed
(Langmuir, 1918). As a result, a major advantage of the Langmuir equation over linear and Freundlich types is that a maximum sorption capacity
is incorporated into the formulation of the model, which may be regarded
as a measure of the amount of available retention sites on the solid phase.
The standard form of the Langmuir equation is
S

, ,s

wc
1+wc
where w and S,
are adjustable parameters. Here w (cm3/kg) is a measure of the bond strength of molecules on the matrix surface and S
,,
(pg/g soil) is the maximum sorption capacity or total amount of available sites per unit soil mass. In an attempt to classify the various shapes
of sorption isotherms, it was recognized that the Langmuir isotherm
is the most commonly used and is referred to as the Lcurve isotherm
(Sposito, 1984).
The Langmuir sorption isotherm has been used extensively by scientists
for several decades. Travis and Etnier (1981) provided a review of studies
in which the Langmuir isotherm to describe P retention for a wide range of
soils was used. Moreover, Langmuir isotherms were used successfully to
describe Cd, Cu, Pb, and Zn retention in soils. Figure 4 shows experimental and fitted isotherm examples of use of the Langmuir equation
to describe Cr(V1) retention for three soils (Selim and Amacher, 1988).
Based on several retention data sets, the presence of two types of surface
sites responsible for the sorption of P in several soils was postulated. As a
consequence, the Langmuir twosurface isotherm was proposed (Holford
MODELS OF INORGANICS IN SOILS
341
c, rng liter 1
Figure 4. Chromium sorption isotherms for Cecil, Windsor, and Olivier soils after 14
days of reaction. Solid curves are calculated isotherms using equilibrium twosite Langmuir
model. From Selim and Amacher (1988), with permission.
el
al., 1974) such that
where f (dimensionless) is considered as a fraction of type 1 sites to the
total sites and w1 and 0, are the Langmuir coefficients associated with sites
1 and 2, respectively. The above equation is an adaptation of the original
equation proposed by Holford et al. (1974) and was used to describe P
isotherms by Holford and Mattingly (1975) for a wide range of soils.
A more recent adaptation of the twosurface Langmuir equation is the
incorporation of a sigmoidicity term, where
=
S
s,
fOlC
1+ W l C + (a1/C)
+
(1  f b 2 C
1 + w,c + (a,/C)
(18)
The terms al and a, are the sigmoidicity coefficients (pg/cm3) for type 1
and 2 sites, respectively. Schmidt and Sticher (1986) found that the introduction of this sigmoidicity term was desirable in order to adequately
describe sorption isotherms at extremely low concentrations. Examples of
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H. M. SELIM
5.5o= 1
n = 2
A = 3
0.000
0.005
0.015
0.020 0.025
Equilibrium Concentration (mg/ml)
0.010
0.030
O.(
Figure 5. Sorption isotherms for lead (l), cadmium (2), and copper (3) in the Ah horizon of a Luvisol. Solid curves are calculated isotherms using equilibrium twosite Langmuir
model with sigmoidicity. From Schmidt and Sticher (1986), with permission.
the twosurface Langmuir with sigmoidicity are shown in Fig. 5 for Pb, Cd,
and Cu in the Ah horizon of a Luvisol (Schmidt and Sticher, 1986).
Although the Langmuir approach has been used to model P retention and
transport from renovated wastewater, we are not aware of studies wherein
the twosurface Langmuir with sigmoidicity has been used to describe
solute retention during transport in soils.
IV.KINETIC RETENTION MODELS
For several solutes, retention reactions in the soil solution have been
observed to be strongly time dependent (e.g., phosphorus, several heavy
metals, and organics). Selected examples of kinetics retention for Cd are
given in Fig. 6. Here, the kinetic dependence of Cd retention, carried out
in batch experiments, is shown for various soils (Selim, 1989). The amount
of cadmium retained varied among soils, with Cecil soil exhibiting the
lowest retention, whereas Sharkey soil showed maximum Cd sorption from
soil solution. The fast decrease in Cd concentration (with time) indicates a
fasttype sorption reaction that was followed by slower type reactions. It is