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II. Transport and Transformation Processes

II. Transport and Transformation Processes

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principal process model descriptions for each element of the complete

balance model, and will discuss the physical significance and realm of

application of each process representation.



The starting point for a model describing the transport and transformations

of a chemical species in soil is the principle of mass conservation. This law

is applied to a chemical mass in an arbitrary volume V of soil, and consists

of a mathematical statement of all of the places where the chemical can

move in the volume, and all of the fates that it can suffer. The law of mass

conservation in words is given in Eq. (1).

Rate of flow of mass into V

- Rate of flow of mass out of V

- Rate of increase of mass stored in V

- Rate of loss of mass in V by reactions = 0


The differential equation corresponding to Eq. (1) is developed by applying the mass conservation principle for a short time interval of duration

At to a small volume element of size V = Ax Ay Az and then letting the size

of the time interval and volume shrink to zero (Jury et al., 1991). In one

dimension the differential equation corresponding to Eq. (1) is written as

+ d J , / d z + r, = 0


where J, is the total chemical mass flux (flow of chemical mass per area per

time), r, is the rate of loss of chemical mass per volume (which will be

negative for source reactions), and Ciis the total resident chemical concentration (mass of chemical/soil volume). The three-dimensional form of

Eq. (2) is similar, except that the chemical flux has x and y components as

well. However, the one-dimensional form given in Eq. (2) is also valid for

a multidimensional flow process, provided that the lateral boundaries

normal to the mean flow direction do not have mass entering or leaving

through them. In this case, the quantities J,, r S , and C: in Eq. (2) are

averaged over the direction normal to flow. Thus, this equation could be

used to describe an area-averaged mass balance for an agricultural field

receiving water and chemical inputs over the inflow surface, provided that

there was no net lateral inflow or outflow from the transport volume.




The total resident chemical concentration C : in Eq. (2) may be divided

up into a series of terms representing the chemical mass storage per volume



in various compartments or chemical phases. The division of chemical mass

into phases is arbitrary, but for practical purposes should obey the following constraints:

1. The mass in each phase should behave differently than the mass in

other phases.

2. Each phase should have transport and reaction laws that are known

or that can be deduced by experiment.

3. The mass in a given phase should if possible be experimentally

distinguishable from the mass in other phases.

The most common phases into which a soil chemical is divided are gaseous,

dissolved, adsorbed, and nonaqueous phase liquid (NAPL). The latter

phase refers to a concentrated mass of pure or formulated liquid that does

not dissolve completely into the water phase.

The principal phase divisions of the total resident concentration are

given in Eq. (3)

c:= ac;+ ec;+ p c ; + qc;


where a is volumetric air content (volume of air per volume of soil), 6

is volumetric water content (volume of water per volume of soil), p is soil

dry bulk density (mass of dry soil per volume of soil), r] is the nonaqueous phase liquid volume fraction (volume of NAPL per volume of soil),

Ci is the dissolved chemical or solute concentration (mass of solute per

volume of water), Ck is the adsorbed chemical concentration (mass of

adsorbed chemical per mass of soil), CL is the gaseous chemical concentration (mass of chemical vapor per volume of air), and CL is the mass density

of the NAPL liquid (mass of NAPL per volume of NAPL).



When the chemical mass flux J, in Eq. (2) is expressed as a function of

the phase concentrations of interest in the problem, the expression is called

a flux law. A separate flux expression is required for each chemical phase

that is mobile in soil. The usual form in which flux laws are expressed is

through differential equations that are functions of the phase concentrations. The dissolved phase flux may also be modeled using a different

formulation that will be discussed after the differential flux laws are


1. Vapor Flux

The mass flux of vapor in soil is assumed to obey Fick’s law of diffusion,

modified to account for the presence of the solid and liquid phases that act



as barriers to vapor movement. This modified expression has the general



Jg =

- t g ( a ) D ;az


where Jg is the gas flux, D: is the binary diffusion coefficient of the vapor in

air, and (,(a) is a tortuosity factor to account for the reduced crosssectional area and increased path length of a vapor molecule in soil. A

variety of gas tortuosity models have been proposed for use in soil (see

reviews by Sallam et al., 1984; Collin and Rasmuson, 1988). The model of

Millington and Quirk (1961) given in Eq. ( 5 )

t g ( a )= a10’3/6,*


where 6, is the porosity, has been shown to represent vapor flow well in soil

over a wide range of water contents (Sallam et al., 1984; Farmer et al.,


2. NAPL Flux

The flux of nonaqueous phase liquid in unsaturated soil can only be

described by using a theory that simultaneously represents the water and

air phases, because the potential energy of the fluid is regulated by interfacial curvature, involving the local geometric configurations reached

by the three phases at their interfaces within the soil pores (Van Dam,

1967). Moreover, the flow depends on the relative permeability of the

medium to NAPL mass, which is regulated by the presence of the water

phase as well as the solid matrix. When these effects are all taken into

account, one writes flux equations of the form shown in Eq. (6) (assuming

one-dimensional flow for simplicity)

where J, is the flux of phase p (water, NAPL, or air), k, is the relative

permeability of the medium to phase p , Ksp is the saturated hydraulic

conductivity of phase p in the medium, p, is the density of phase p , p* is

the density of the phase used as the reference, and h, is the fluid pressure

head of phasep in the soil (Abriola and Pinder, 1985; Parker et al., 1987).

The fluid pressure in each phase depends on the configurations that the

fluid forms at the interfaces within the soil pores.

The three-phase equilibrium pressures can be characterized as a function

of the relative saturation of each phase in controlled laboratory experiments. However, the relative permeability k , must be developed from an



idealized geometric model of the porous medium (Parker et al., 1987), or

else related empirically to the two-phase relative permeability of water

and NAPL (Stone, 1973).

The flux equations for the three phases in soil are combined with mass

conservation equations for each phase to produce a transport model

(Abriola and Pinder, 1985). The theory of NAPL flow in soil is still in its

infancy, and only a few experiments have been performed to test its

hypotheses. A significant barrier to progress in modeling NAPL flow in soil

is the development of fluid instabilities under a wide range of experimental

conditions (Schwille, 1988).

3. Flux of Dissolved Chemical in Soil

Unless a compound is very volatile, the dominant mechanism by which it

is transported through soil is by bulk movement as a dissolved constituent

of the water phase. At the scale of the soil pore, there are two transport

mechanisms that can move solutes through the medium: convection and

diffusion. Convection refers to the transport of a dissolved chemical by

virtue of bulk movement of the host water phase. Thus, the vector of the

local convective solute flux v, may be written as

vsc = V W C I


where vw is the local water flux vector and C, is the corresponding local

fluid concentration. The local diffusion flux vector Vsd within the water

phase is described by Fick's law of diffusion (Bird et al., 1960)


:; 2 )

vsd=-D;hl - i + - j + - k

where 0;"

is the binary diffusion coefficient of the solute in water and i, j,

and k are unit vectors in the x , y , and z directions, respectively.

The total local flux of dissolved chemical vI = v,, + vd describing the

movement of solute within the water phase is exact but impractical, because the local water flux describes three-dimensional flow around the solid

and gaseous portions of the medium and is not measurable. Instead, the

local quantities are volume averaged to produce a larger scale representation of the system properties. The averaging volume must be large enough

so that the statistical distribution of geometric obstacles is the same from

place to place. If the porous medium contains the same material and density throughout, then the mean value produced by this averaging is macroscopically homogeneous over the new transport volume containing the

averaged elements.



For example, a quantity such as the porosity c$ is highly irregular at the

pore scale, with a value of 0 or 1, depending on whether the point of

characterization is within or between the solid particles forming the soil

matrix. However, when many pores lie within the averaging volume, the

value of the porosity represents the fraction of the volume that is not filled

with solid material. This measurement will yield the same value for the

porosity when the averaging volume is centered at different locations

within the porous medium, provided that the material has been packed in a

uniform manner using the same distribution of solid particle sizes and

shapes. The minimum size of the averaging volume that yields the same

value for the porosity is called its representative elementary volume (Hubbert, 1956; Bear, 1972). If the soil matrix does not have the same properties at different locations, then a representative elementary volume

may not exist for that quantity, because the value for its average will then

change continually as the size of the volume is altered (Baveye and

Sposito, 1985).

If we assume that the soil is macroscopically homogeneous, the local

dissolved chemical flux can be volume averaged. The new macroscopic

convective flux may be written as (assuming that the averaged flow is one

dimensional for simplicity).

J,, = -J,c;


where now the volume-averaged water flux J , is regarded as a continuum

quantity that can be described by the Buckingham-Darcy flux law (Bear,

1972). The volume-averaged diffusion flux is modeled with a tortuosity

coefficient similar to the one used in Eq. (4) for the vapor flux

where the dissolved-phase tortuosity factor &(O) may be described with the

Millington-Quirk model, Eq. (9,applied to the water phase

t,(e) = e10’3/c$2


However, unlike the situation at the scale of the soil pore, the volumeaveraged solute flux is not the sum of the convection and diffusion fluxessome of the solute motion has been lost in the averaging process and must

be reinserted into the flux equation as a new term. When the local water

flux is averaged to produce a new continuum quantity, the local threedimensional fluctuations about the average motion due to movement

around the air and solid phases are lost because they do not contribute to

the mean water flow at the new scale. However, dissolved chemicals

convected along these tortuous flow paths do contribute to solute transport



and cannot be neglected. The motion of solute due to small-scale convective fluctuations about the mean motion is called hydrodynamic dispersion

(Bear, 1972).

Development of models to describe the hydrodynamic dispersion flux is

one of the most active research areas in soil physics and hydrology today

(Gelhar, 1986). An appreciation for the physical significance of these

models can only be gained after a detailed discussion of the local mixing

and transport processes that contribute to the bulk flow of solute by

hydrodynamic dispersion. We will illustrate the interplay between lateral

mixing and longtitudinal convection using the classic analysis of Taylor

(1953) on solute transport through a capillary tube.

When a hydrostatic water pressure gradient is placed across a cylindrical

capillary tube of radius R that is saturated with water, the water velocity

distribution that forms under laminar flow is parabolic

v,(r) = vmax(l- r 2 / R 2 )

where v,,, is the maximum water velocity at the center of the tube (Jury

et al., 1991). The area-averaged water velocity (v,) obtained by calculating the water volume per unit area of the entire wetted cylinder is

given by






where CP is the angular coordinate of the circular plane in the cylindrical

coordinate system (Arfken, 1985). Thus, the average water flux does not

change with position along the axis of the cylinder, even though it varies

with radial position within the cross-section. The average water velocity is

therefore one dimensional. When a pulse of solute is added to the inlet end

of the capillary, the solute at the center of the capillary initially moves

ahead of the front. However, random diffusion normal to the direction of

flow causes the solute to migrate into regions of slower velocity, and

eventually the solute molecule samples all of the different flow paths in the

cross-section. The time required for a solute molecule to sample all of

these different flow paths is called the transverse mixing time fm of the

capillary. The mean time required for the solute that enters the capillary

tube at z = 0 to reach a given distance z along the axis of the cylinder is

called the convection time tc of the solute. For this problem the convection

time t, is equal to z/(vw), and the total mean convection time required to

reach the end of the tube is the breakthrough time f b = L/(v,).

The area- or flow-averaged solute concentration at a given distance

along the tube looks very different depending on whether the mixing time

is longer or shorter than the mean convection time required to reach the



point of observation. In the top part of Fig. 1, the solute pulse near the

point of entry has a very short convection time compared to the mixing

time, and the mixing process has just started. Therefore, some of the solute

at the center of the tube has not diffused into slower channels and is still

advancing at nearly twice the average velocity. In the lower part of the

figure the pulse has migrated farther along the tube, the mean convection

time has increased, and solute mixing has allowed each molecule to explore

all of the different flow pathways. As a result, the pulse is spread out along

the direction of flow (because individual molecules have spent different

amounts of time in a given flow region), and the main pulse is now

migrating at the average water velocity (v,) = v,,,/2. Hence, the relative

size of the mixing time compared to the convection time determines the

radial distribution of the solute as it arrives at a given distance from the

point of entry.

Although the capillary tube example is an idealized situation, it contains

several elements that are important to an understanding of solute dispersion in soil. First, the mixing time of the capillary is regulated by the

geometry of the tube. It will be longer for larger capillaries than for smaller

ones. Second, the convection time is defined relative to the distance

traveled or to the time since the pulse was injected. It does not depend on

Casel: t, << ,t



Case2: t c >> ,t


z= L

Figure 1. Schematic illustration of a solute pulse advancing along the axis of a capilIary

tube that has a parabolic water velocity distribution in the radial direction.



the lateral geometry, but rather on the mean convective motion and on the

distance traveled.

In a porous medium, the local water velocity is not distributed in an

ordered parabolic pattern. Rather, it is chaotic because of the complex

geometric configurations formed by the solid and air spaces that act as

barriers to the water flow. However, at any scale of observation, local

velocities are correlated over some distance normal to the flow direction.

Velocities immediately adjacent to each other are more likely to be similar than those farther apart. At any scale of transport, there is a characteristic distance of separation, called the correlation length, within which

the velocities are similar (Lumley and Panofsky, 1964). When solute has

time to explore this zone of similarity, the mixing time is completed,

provided that adjacent zones of this size have the same mean flow characteristics. However, if portions of the soil volume have distinctly different

transport characteristics, such as extremely rapid water flow, then the

area-averaged description of flow will have properties that reflect both the

fast and slow regions of the medium.

4. Asymptotic Dispersion Models

From this discussion it is clear that there are some representative types

of solute flow that apply at different times during the transport event.

These so-called asymptotic extremes of solute movement are useful both

for classifying the kinds of solute behavior and for developing simple

models that are only valid at certain times. For a soil with a unimodal

velocity distribution (i.e., a soil having no local pathways that are extremely fast compared to the majority of the water velocities), the two

extreme flows of interest are the zero-time and the infinite-time models.

a. Zero-Time Model of Solute Dispersion

At zero time, just as the solute enters the medium, solute molecules are

caught in isolated water flow pathways (called stream tubes) moving at

different velocities through the wetted pore space. Because mixing has

not yet begun, diffusion or small-scale transverse convection is neglected,

and the solute convection along the mean direction of flow is described

as parallel flow of solute in the different stream tubes. This is called a

stochastic-convective model, because all of the motion is convective but

encompases a range of velocities along the direction of motion (Simmons,

1982). It applies approximately whenever the convection time is very much

smaller than the mixing time. An average solute flow velocity can be

defined in such a system by averaging over the motion in the different



tubes, and the dispersion or spreading along the flow direction that occurs

in the averaged medium reflects those parts of the fluid body that are

convected at slower or faster than average velocities.

b. Infinite-Time Model of Solute Dispersion

The infinite time model of solute dispersion applies when the solute

molecules have traveled by diffusion into the zones of different water

velocity. Therefore, each molecule has the same mean convective flow rate

through the medium. The spreading of solute molecules about this average

convective motion is due to the varying amounts of time a given molecule

spends in each stream tube; the average motion of the molecules is random

or diffusionlike when reviewed in a frame of reference moving at the mean

convective velocity. The overall motion is described as the sum of the mean

drift motion and the random longitudinal spreading. This transport process

representation is called a convection-dispersion model. It applies approximately whenever the mixing time is very much less than the convection

time (Taylor, 1953). In a recent field-scale test of solute transport in

groundwater, the longitudinal dispersion coefficient reached a constant

asymptotic value after about 26 m of travel distance (Garabedian et al.,

1991). In another aquifer study, however, the asymptotic limit had not

been reached after 90 m of travel (Freyberg, 1986).

c. Preferential Flow of Solute

A field soil is full of local pathways such as structural voids or biological channels that can carry water at velocities much greater than those of

the surrounding matrix, even when the entire field surface is watered

uniformly. Moreover, local obstacles in a heterogeneous soil can cause

water to funnel into narrow plumes under certain circumstances, even

when moving within the soil matrix (Kung, 1990a,b). Also, local variations

in the rate of water and solute input at the soil surface can create preferentially wetted pathways even in homogeneous soils. In any of these circumstances, the mixing and convection times of that portion of the field

that is experiencing much greater than average flow are very different

mixing and convection times than those within the rest of the medium.

Consequently, the asymptotic limits apply to one part of the medium may

not apply to other parts. In this case, a third type of limiting model is

invoked that describes the two parts of the medium with different process

representations. Figure 2 shows schematic illustrations of solute particles

moving through a medium that could be described with these prototype

dispersion models.



Convective-Dispersive Flow


Stochastic-Convective Flow


Preferential Flow




Figure 2. Schematic illustration of three types of solute transport.

5. Stochastic Continuum Model

Modeling solute dispersion with a single continuum model from the time

of solute application onward is possible only if one is able to describe both

the local water flows and the mechanism of lateral mixing in terms of

measurable parameters. At the pore scale, this description is not possible

because it depends on the unobservable local pore geometry and the soil

structural features. At larger scales a local three-dimensional model is used

that has already been volume averaged, so that the local water flux can be

calculated with a transport equation that contains measurable parameters

(such as the saturated hydraulic conductivity if the water flow is modeled

in saturated soil). The volume-averaged local solute transport model is

assumed to be convective-dispersive, which amounts to assuming that the

solute has sufficient time to mix through the regions of different velocity



within the local averaging volume. At the field scale, the local concentration and local transport parameters are assumed to be stationary random

functions oscillating about their mean values. A field-scale description

of solute transport is then derived by statistically averaging the stochastic

local transport model (Gelhar and Axness, 1983), producing a macroscopic

model that depends on the statistical properties of the random transport


A preliminary model of this type has already been developed for saturated soil (Dagan, 1984, 1987). The local water flux that is reaveraged to

create a large-scale model is described by Darcy’s law for saturated flow

in two or three dimensions, and the mixing time is characterized in terms

of the correlation length scale of the saturated hydraulic conductivity,

which is measurable if sufficient numbers of samples can be taken

(Sudicky, 1986). This model has shown promise in characterizing the

migration of solute through aquifers in a transport study for which sufficient numbers of measurements were taken to validate the theory

(Freyberg, 1986; Sposito and Barry, 1987).

The water flux is much more difficult to measure in unsaturated soil than

it is in saturated soil, and the unsaturated hydraulic conductivity is a

function of the saturation state of the soil. Also, flow in groundwater is

usually parallel to the direction of stratification, and therefore the mean

flow properties along the direction of flow may not change appreciably

during the transport event of a practical distance. However, in unsaturated

soil, the normal direction of stratification is perpendicular to the direction

of flow and the mean soil properties can change significantly over short

vertical distances. For these reasons, the stochastic continuum model of

chemical transport is more difficult to formulate in unsaturated soil and

has not been developed or tested at this time. There have been several

stochastic continuum models published for water flow in unsaturated soil

(Yeh et al., 1985; Mantaglou and Gelhar, 1987).

Having completed this brief introduction to the principles of solute

dispersion, we will now return to the discussion of specific models for the

solute flux in soil.



1. Convective-Dispersive Flux

The convective-dispersive model of the chemical flux in the liquid or

dissolved phase J, in one dimension is given by Eq. (14) (Lapidus and

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II. Transport and Transformation Processes

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