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Chapter 6. Modeling the Transport and Retention of Inorganics in Soils

Chapter 6. Modeling the Transport and Retention of Inorganics in Soils

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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 well-defined 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 well-defined 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 “mobile-immobile”

transport models, wherein local nonequilibrium conditions are due to

diffusion or mass transfer of solutes between the mobile and immobile

regions. The mobile-immobile 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 above-mentioned more

exact approach for systems of well-defined 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 inner-sphere solute-surface

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 solute-surface site complexes.



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 time-dependent 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 two-site 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 first-order kinetic reversible reaction for

describing time-dependent adsorption/desorption in soils. Others include

linear irreversible and nonlinear reversible kinetic models. Recently, a

combination of equilibrium and kinetic-type (two-site) models and consecutive and concurrent multireaction-type models have been introduced.

In this article, major features of retention models that govern retention

reactions of solutes in the soil are presented. Single-reaction models of

the equilibrium type are first discussed, followed by models of the kinetic

type. Retention models of the multiple-reaction type, including the twosite equilibrium-kinetic models, the concurrent- and consecutive-multireaction models, and the second-order approach will be derived. This is

followed by multicomponent or competitive-type 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 water-saturated and water-unsaturated

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



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




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




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 one-dimensional

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 well-known 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



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




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 convective-dispersive

equation for solute transport, which is valid for soils under transient and

unsaturated soil-water 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 convection-dispersion equation as

where v (cm/hr) is known as the pore water velocity and is given by (4/0).

Solutions of the above convection-dispersive 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 first-order type of boundary condition such that a solute pulse input is










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


among others. The more precise third-type 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 flux-type pulse input as

O = - D d2





Advantages of using the third-type 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),


t r o


which is used to deal with solute effluent from soils having finite lengths.

However, it is often convenient to solve the dispersion-convection equation wherein a semi-infinite rather than a finite length ( L ) of the soil is

assumed. Under such circumstances, the appropriate condition for a semiinfinite medium is





Analytical solutions to the convection-dispersion 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 closed-form solution is

obtainable. A number of closed-form 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).





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

convection-dispersion 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 two-site 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 kinetic-type reactions and their implication for single and multireaction retention and transport models.


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


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



Table I

Selected Equilibrium and Kinetic-Type Models for Solute

Retention in Soils"





Equilibrium type


Freundlich (nonlinear)


Langmuir with sigmoidicity

Kinetic type

First order

n-th order

Irreversible (sinklsource)

Langmuir kinetic



Mass transfer

S = KdC

S = KdCb

s = WCS,,/(l+

S = oCS,,,/(l


+ 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


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.



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 best-fit curve provides the nonlinear parameter b and





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.












1 %










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















C (mg/liter)

Figure 3. Retention isotherms Co, Cd, Cu, and Pb on Alligator soil. From Buchter et al.

(1989), with permission.



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




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 L-curve 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 two-surface isotherm was proposed (Holford



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 two-site Langmuir

model. From Selim and Amacher (1988), with permission.


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 two-surface Langmuir equation is the

incorporation of a sigmoidicity term, where





1+ W l C + (a1/C)


(1 - f b 2 C

1 + w,c + (a,/C)


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



5.5o= 1

n = 2

A = 3




0.020 0.025

Equilibrium Concentration (mg/ml)




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 two-site Langmuir

model with sigmoidicity. From Schmidt and Sticher (1986), with permission.

the two-surface 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 two-surface Langmuir with sigmoidicity has been used to describe

solute retention during transport in soils.


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

fast-type sorption reaction that was followed by slower type reactions. It is

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Chapter 6. Modeling the Transport and Retention of Inorganics in Soils

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