II. Transport and Transformation Processes
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WILLIAM A. JURY AND HANNES FLmLER
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.
A. MASSCONSERVATION
LAW
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
(1)
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
(2)
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.
B. CHEMICAL
PHASE
CONCENTRATIONS
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
TRANSPORT OF CHEMICALS THROUGH SOIL
145
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;
(3)
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).
C. CHEMICAL
MASSFLUXLAWS
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
presented.
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
146
WILLIAM A. JURY AND HANNES F L m L E R
as barriers to vapor movement. This modified expression has the general
form
acy,
Jg =
- t g ( a ) D ;az
(4)
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,*
(5)
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.,
1980).
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
TRANSPORT OF CHEMICALS THROUGH SOIL
147
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
(7)
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
:; 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.
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WILLIAM A. JURY AND HANNES FLUHLER
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;
(9)
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
(11)
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
TRANSPORT OF CHEMICALS THROUGH SOIL
149
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
(vw>=
p
v,(r)rdrd@
=
2
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
lS0
WILLIAM A. JURY AND HANNES F L m L E R
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
z=L
z=o
Case2: t c >> ,t
z=o
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.
TRANSPORT OF CHEMICALS THROUGH SOIL
151
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
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WILLIAM A. JURYAND HANNES FLmLER
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.
TRANSPORT OF CHEMICALS THROUGH SOIL
153
Convective-Dispersive Flow
INLET SURFACE
Stochastic-Convective Flow
INLET SURFACE
Preferential Flow
.
INLET SURFACE
0
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
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WILLIAM A. JURY AND HANNES F L m L E R
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
properties.
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.
D. CONVECTION-DISPERSION
REPRESENTATION
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