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OCR for page 37
Movement of Contaminants
in Groundwater:
Groundwater r ~ ~
~ ransport
Advection and Dispersion
2
.
INTROD UCTION
MARY P. ANDERSON
University of Wisconsin, Madison
The relative success of attempts to model a process is a measure
of how well it is understood. A first level of understanding
produces a conceptual model and a higher level of understand
ing results in a quantitative model. Failure to understand dis
persion in porous media at a level necessary for constructing
reliable mathematical models has impeded progress in studying
contaminant transport in groundwater. However, within the
past 5 years there have been many attempts at improving our
understanding of the nature of dispersion in porous media. As
a result there has been significant progress in refining contam
inant transport models. Some of this progress as well as back
ground information on dispersion in porous media is reviewed
in this chapter.
Dispersion in porous material refers to the spreading of a
stream or discrete volume of contaminants as it flows through
the subsurface. For example, if a spot of dye is injected into
porous material through which groundwater is flowing, the spot
will enlarge in size as it moves downgradient. More specifically,
in a threedimensional Cartesian coordinate system where the
average groundwater velocity is parallel to the x axis, a sphere
of dye moving horizontally along the x axis will undergo lon
gitudinal spreading or dispersion parallel to the x axis and
transverse dispersion parallel to the y and ~ axes.
Dispersion causes mixing with uncontaminated groundwa
ter, and hence dispersion is a mechanism for dilution. More
over, dispersion causes the contaminant to spread over a greater
volume of aquifer than would be predicted solely from an analy
sis of groundwater velocity vectors. This spreading effect will
be of particular concern when toxic or hazardous wastes are
involved. Dispersion is chiefly of importance in predicting
37
transport away from point sources of contamination but is also
influential in the spread of nonpointsource contaminants, al
though of lesser importance. Contaminants introduced into the
subsurface from nonpoint sources will be spread over a rela
tively large area because of the nature of the loading pattern.
In this case, dispersion merely causes a relatively large zone
of contaminated water to acquire some rough fringes. Disper
sion is of interest because it causes contaminants to arrive at
a discharge point (e.g., a stream or a water well) prior to the
arrival time calculated from the average groundwater velocity.
The accelerated arrival of contaminants at a discharge point is
a characteristic feature of dispersion that is due to the fact that
some parts of the contaminant plume move faster than the
average groundwater velocity.
Dispersion is caused by both microscopic and macroscopic
effects. Mechanical dispersion on a microscopic scale (Figure
2.1) is a result of deviations of velocity on a microscale from
the average groundwater velocity. These velocity variations
arise because water in the center of a pore space travels faster
than the water near the wall and because diversion of flow
paths around individual grains of porous material causes vari
ations in average velocity among different pore spaces. These
two factors create mechanical dispersion on a microscopic scale.
In addition, it is customary to include molecular diffusion as a
component of microscopic dispersion. Molecular diffusion oc
curs as species move from higher to lower concentrations. Thus,
microscopic dispersion includes the effects of mechanical dis
persion and molecular diffusion.
One of the first field and laboratory investigations of dis
persion in porous media was performed by Slichter (1905~. He
obtained Sshaped breakthrough curves (concentration versus
time curves), which are characteristic of flow affected by dis
OCR for page 37
38
FIGURE 2.1 Microscopic dispersion (adapted from Freeze
and Cherry, 1979).
persion, during field tracer tests that he performed for the
purpose of estimating groundwater velocity. He correctly at
tributed the effect to dispersion:
The writer formerly supposed that the gradual appearance of the elec
trolyte at the downstream well was largely due to the diffusion of the
dissolved salt, but it is now evident that diffusion plays but a small
part in the result. The principal cause of the phenomena [sic] is now
known to be due to the fact that the central thread of water in each
capillary pore of the soil moves faster than the water at the walls of
the capillary pore, just as the water near the central line of a river
channel usually flows faster than the water near the banks.... Owing
to the repeated branching and subdivision of the capillary pores around
the grains of the sand and gravel, the stream of electrolyte issuing from
the well will gradually broaden as it passes downstream. The actual
width of this charged water varies somewhat with the velocity of the
ground water.... (Slichter, 1905, p. 23).
In subsequent laboratory experiments Slichter (1905, p. 41)
studied the phenomenon more carefully and concluded that
"the spread of the electrolyte, as shown by these experiments,
is not to be explained by the diffusion of the salt, but must be
explained by the continued branching and subdivision of the
capillary pores around the individual grains of the sand.,'
In fact, some of the dispersion observed by Slichter in field
studies was probably a result of macroscopic dispersion. On a
macroscopic scale, dispersion is caused by the presence of large
scale heterogeneities within the subsurface. For example, Ski
bitzkie and Robinson (1963) demonstrated that lenses of high
permeability material within a matrix of lower permeability
caused the spreading of streams of dye as water and dye moved
through a tank filled with sand (see Figure 2.21. It is now
generally recognized that the presence of heterogeneities in
the subsurface, rather than microscopic dispersion alone, is
responsible for the appreciable spreading of contaminants doc
umented in a number of field studies (e.g., Anderson, 1979~.
FIGURE 2.2 Macroscopic dispersion
(adapted from Skibitzkie and Robinson,
1963).
MARY P. ANDERSON
Pioneering theoretical work on dispersion was done by Tay
lor (1921, 1953), and serious efforts at applying modified forms
of this theory to field studies involving the transport of con
taminants in groundwater have been under way since the early
1970s. However, there is still considerable uncertainty con
cerning methods for quantifying dispersion and for measuring
dispersion in the field. To some extent this uncertainty has
impeded progress in developing reliable contaminanttransport
models. However, within the past 5 yr there has been much
effort and some progress in quantifying macroscopic dispersion.
These efforts are reviewed below.
ADVECTIOI~DISPERSION EQUATION
Most attempts at quantifying contaminant transport have relied
on a solution of some form of a wellknown governing equation
referred to as the advectiondispersion equation. Advection
refers to the transport of contaminants at the same speed as
the average linear velocity of groundwater (v), where
v = Klln (2.1)
and K is hydraulic conductivity, I is the head gradient, and n
is effective porosity. The velocity defined by Eq. (2.1) has also
been called the average pore velocity. The nomenclature used
here (i.e., average linear velocity) was introduced by Freeze
and Cherry (19791. Moreover, the term affection is used here
in preference to the term convection because convection often
carries the connotation of transport in response to temperature
induced density gradients.
The advectiondispersion equation is derived by combining
a massbalance equation with an expression for the gradient of
the mass flux (see Bear, 1972; Wang and Anderson, 19821. The
in AVE RAG E FLOW  _
~ A.' ..~.
A I \COARSE LENS
_
OCR for page 37
Contaminants in Groundwater: Transport
difficulties in quantifying dispersion are related to the fact that
field studies of flow through porous media are by necessity
conducted at a macroscopic rather than a microscopic level.
For example, Darcy's law, the fundamental equation for de
scribing flow through porous media, is a macroscopic equation.
That is, K, I, and n in Eq. (2.1) are measured in some rep
resentative elementary volume (REV), and these values are
assumed to represent an average of K, I, and n within the REV.
Likewise, spatial averaging is routinely done when deriving
the advectiondispersion equation. Fried (1975) noted that "The
basis of dispersion theory is a measurement problem.... The
oretical macroscopic concentration, for instance, which appears
in mathematical models, should correspond to the experimen
tal concentration; this is not simple...."
The standard approach in analyzing dispersion, which is em
bodied in the advectiondispersion equation, is to use an av
erage linear velocity. A number of investigators maintain that
this approach is reasonable because it will never be practical
to define the velocity field in detail.
The key assumption in deriving a term to represent disper
sion is that dispersion can be represented by an expression
analogous to Fick's second law of diffusion:
mass flux due to dispersion = ~ (D,* ~ ), (2.2)
where c is concentration and D`*. is the coefficient of dispersion
(the i, j indices refer to Cartesian coordinates). The coefficient
of dispersion can be shown to be a secondrank tensor, where
D,*. = Dij + D,`. (2.3)
Then, Dij is the coefficient of mechanical dispersion and Dot is
the coefficient of molecular diffusion (a scalar>. An effective
diffusion coefficient is generally taken to be equal to the dif
fusion coefficient of the ion in water (D,') times a tortuosity
factor. The tortuosity factor has a value less than 1 and is needed
to correct for the obstructing effect of the porous medium.
Effective diffusion coefficients are generally around 106 cm2/
see, although a range of 1O5 to 10 7 cm2/sec is not incon
ceivable (Grisak and Pickens, 1981~. Except for systems in
which groundwater velocities are very low, the coefficient of
mechanical dispersion generally will be one or more orders of
magnitude larger than D,'. Therefore, in many practical appli
cations the effects of molecular diffusion may be neglected (D,~
= 0~. The coefficient of mechanical dispersion is routinely taken
to be the product of the magnitude of the velocity vector times
a parameter known as dispersivity, which is commonly and
somewhat vaguely referred to as a characteristic mixing length.
The advectiondispersion equation in its most general form
is written
(Do*) (cri)+ ~ Rk =
dxi dXj Oxi n k= ~ at
dispersion term advection sink/source chemicalreaction (2.4)
term term term
where c' is the concentration of solute in a source or sink fluid,
and W is the volume flow rate of the sink or source fluid per
unit volume of porous material. In the chemicalreaction term,
Rk is the rate of production of the solute in reaction k of s
39
different reactions. The problems involved in quantifying the
chemicalreaction term are discussed in Chapter 3 of this re
port.
If the advectiondispersion equation is to be used to evaluate
dispersion in groundwater systems, two questions must be ad
dressed:
1. Is it valid to assume that the dispersion component of
Eq. (2.4) can be represented by a form of Fick's law tEq. (2.2~?
2. Can dispersivity be defined in terms of physically mea
surable parameters?
IS DISPERSION A FICKIAN PROCESS ?
Much ofthe theory on which the analysis of dispersion in porous
material and in rivers is based stems from the pioneering work
of Taylor (1921, 1953), who suggested that dispersion could be
represented as a Fickian process. This assumption as applied
to porous media has since been questioned by several re
searchers. Fried (1975, pp. 1727) cites several examples of
laboratory experiments in which the experimental results did
not fit theoretical curves derived from solutions of the advec
tiondispersion equation, and Dagan (1982) noted that"there
is no a priori reason to believe that the diffusion type equation
is valid at all" for contaminant transport through porous media.
Taylor and others (e.g., Fischer, 1973) who continued the
development of the theory of dispersion clearly recognized that
the Fickian assumption is valid only after a certain length of
time has elapsed in which the dispersion process develops. A
procedure for predicting the length of this initial development
period has not yet been perfected. Indeed, until recently it
was not recognized that for the heterogeneous systems en
countered in the field this development process requires sub
stantial transport from the source. For example, some re
searchers believe that dispersion becomes Fickian only at
distances on the order of lOs to lOOs of meters from the source
in porous media (Matheron and DeMarsily, 1980; Gelhar and
Axness, 1981; Dagan, 1982) and on the order of kilometers in
rivers (Beltaos and Day, 19781. Furthermore, in certain hy
drogeologic settings dispersion may never become Fickian
(Matheron and DeMarsily, 1980; Smith and Schwartz, 19801.
After the initial development period, that is when dispersion
has become Fickian, the concentration distribution and con
centrationtime profiles should behave according to particular
solutions of Eq. (2.41. Specifically, the solution for instanta
neous release of a contaminant from a point source predicts
that concentrationdistance curves will approximate a Gaussian
distribution except for short times f Figure 2.3(a)~. However,
concentrationtime curves are typically skewed on the right
except for long times or large distances from the source tFigure
2.3(b)1
The problems in applying Taylor dispersion theory to rivers
are discussed in some detail by Beltaos and Day (1978~. Most
concentrationtime curves obtained from tracer studies in riv
ers are skewed on the right, and Gaussian curves are rare (Day
and Wood, 1976~. Beltaos and Day (1978) noted that concen
trationtime curves in the Lesser Slave River deviated less from
OCR for page 37
40
SPRT I RL D I STR I BUT I ON
.10
.08
.06
.04
.02 ~
.°Ot 1 2 4 6
IT= 1
,IT= 1.5
it_
_ _ ~ ~ ,_ ~ ~ ~
8 10 12 14 16 18 20 22
D I STRNCE
{a)
TEMPOROL D I STR I BUT I ON
X109
20 .
o
4. 6. 8. 10. 12. 14.
~10
/
1~.,.1.,.1.,.1.,.
b ~
TIME
(b)
FIGURE 2.3 (a) Concentration distribution in space; (b) concentra
tiontime curves. C/CO is a dimensionless concentration. The dimen
sions of distance and time and other variables and parameters are any
consistent set of units (e.g., distance in meters, time in days, and
velocity in meters/day). In this example, velocity and longitudinal dis
persivity were set equal to 1.0 and transverse dispersivities were
1/20.
the Gaussian than did several mountain streams studied by
Day (1915) and concluded that the Lesser Slave River is less
"irregular" in that it deviates less from prismatic laboratory
flumes where Gaussian curves are normally obtained relatively
quickly. They also demonstrated that concentrationtime curves
for this river approximated Gaussian curves at a distance of
about 14 km from the tracer release point. However, the growth
of the variance of the concentration distribution during the
nonGaussian period did not follow a Fickian model. If dis
persion is a Fickian process, then the variance of the concen
tration distribution should increase linearly with time or dis
tance, and the longitudinal dispersion coefficient and the
dispersivity will be a constant for constant velocity. However,
the variance of the concentration distribution in the rivers con
sidered by Beltaos and Day (1978) increased with the square
of the distance and the dispersion coefficient, and the disper
sivity also increased with distance for constant velocity. Based
MARY P. ANDERSON
on these analyses it appears that the initial development period
of nonFickian behavior may be long for many rivers.
Similar behavior has been predicted for porous media on the
basis of theoretical studies (Gelhar et al., 1979; Matheron and
DeMarsily, 1980; Dagan, 1982~. Specifically, for long times or
large travel distances, dispersion in some geologic settings can
be accurately represented as a Fickian process. However, under
certain conditions (e.g., flow parallel to bedding where lateral
dispersion across bedding planes is negligible) theoretical stud
ies suggest that the Fickian limit (also known as the Taylor
limit) may never be reached and dispersion may never become
Fickian (Mercado, 1967; Gelhar et al., 1979; Matheron and
DeMarsily, 1980~. Similarly, Smith and Schwartz (1980>, who
conducted many numerical experiments of hypothetical aqui
fers, found that for most of the situations that they modeled
the variance of the concentration distribution was highly ir
regular and did not approximate a Gaussian distribution within
the lengths of the simulated flow paths.
Field evidence to corroborate these theoretical findings for
porous media is sparse. Results from a field tracer test de
scribed by Sudicky and Cherry (1979) suggested that the con
centration patterns were somewhat irregular near the source
of the tracer but became approximately Gaussian farther down
gradient. An analysis of the variance of the concentration dis
tribution by Sudicky and Cherry (1979) indicated that the growth
of the variance was nonlinear and the dispersivity increased
with distance. Several other investigators (Molinari et al., 1977;
Lee et al., 1980; Pickens and Grisak, 1981a, 1981b) also noted
an increase in dispersivity with distance from the tracer source,
which indicates that dispersion was not Fickian within the dis
tances in which the measurements were taken (see Figure 2.41.
DEFINITION OF DISPERSIVITY
Contaminant transport models usually have been applied to
existing wastedisposal sites where a contaminant plume had
been identified during a field monitoring program. The stan
dard modeling procedure has been to adjust values of disper
sivity until the model correctly reproduces the observed con
centration distribution. Anderson (1979), among others, noted
that dispersivity values used in most applications of contami
nant transport models through 1979 were calibration (or fittings
parameters. Values in the range of 3 to 200 m have been re
ported (see Anderson, 1979), but these values may not be
physically meaningful. In order to obtain meaningful predic
tions of contaminant movement at existing sites and to apply
advectiondispersion models to new or proposed sites, an ac
curate way of quantifying dispersivity needs to be developed.
Results from theoretical studies suggest that dispersion is
nonFickian near the source of the contaminant but that dis
persion generally becomes Fickian at large times or travel dis
tances, when a constant dispersivity value is achieved. There
fore, for large times, it should be possible to use the standard
form of the advectiondispersion equation fEq. (2.4~] if the
dispersivity parameter in the coefficient of mechanical disper
sion can be evaluated in terms of parameters that can be mea
sured in the field.
OCR for page 37
41

00 _
>
in
llJ
~ 10
J
Cl
Z
1
z
o
J
Contaminants in Groundwater: Transport
1 ~ ~
SAND,GRaVEL,
SaN DSTONE
LIMESTONE, BaSALT, /
GRANITE a SCHIST /
.
~ _/ '1 ~

16. /11
· ~
· e/
1.
· 
_ /~,
/
y 
. I I I
1 10 100 1000
DISTANCE (m)
FIGURE 2.4 Variation of dispersivity with distance (adapted from
LallemandBarres and Peaudecerf, 1978).
Gelhar et al. (1979) showed that near the source, dispersivity
steadily increases with distance and at the Taylor limit ap
proaches an asymptotic value. Gelhar et al. (1979) and Gelhar
and Axness (1981, 1983) derived a variety of expressions for
evaluating the asymptotic longitudinal and transverse disper
sivities. In their analyses, the asymptotic dispersivity is ex
pressed in terms of various statistical properties of the hydraulic
conductivity distribution. For example, for the perfectly strat
ified aquifer considered by Gelhar et al. (1979>, the asymptotic
value for the longitudinal dispersivity is cx, + A=, where
Ad =Irk I IK j2, (2. 5)
30`T
where ELK iS the standard deviation of the log normal hydraulic
conductivity distribution; l is a correlation length, and K is the
mean hydraulic conductivity. The parameters cx, and a.,. are,
respectively, the local longitudinal and transverse dispersivities
defined at the level of the representative elementary volume
(REV) used in the spatial averaging that is routinely done when
deriving the advectiondispersion equation. Local dispersivi
ties are on the order of 102 to 1 cm for laboratory experiments
and range from 10i to 10 m for tracer tests in the more
heterogeneous porous material typically encountered in the
field (see Klotz et al., 1980~. However, it is likely that some
of the dispersivities calculated on the basis of field tracer tests
are biased by the socalled scale effect (see Figure 2.4~. That
is, because dispersivity increases with distance from the injec
tion point, some of the values reported from tracer tests are
too high to be representative of local dispersivities and in fact
are equivalent dispersivities that represent dispersion between
the measuring point and the injection point. Typical values for
the local longitudinal dispersivity are probably on the order of
102 to 1 m (Gelhar et al., 1979; Matheron and DeMarsily,
1980; Gelhar and Axness, 1981~. Transverse dispersivity is smaller
than longitudinal dispersivity; ratios of longitudinal to trans
verse dispersivity on the order of 10 to 100 have been suggested
(see Anderson, 1979~.
According to the general threedimensional analysis of Gel
har and Axness (1981), dispersivity is a secondrank symmetric
tensor (Aij). Their result is in contrast to the somewhat alarming
finding of de Josselin de Jong (1968), who concluded that dis
persivity is a tensor of infinite rank in the anisotropic case.
Gelhar and Axness (1981) showed that in the general three
dimensional case with random orientation of stratification, the
longitudinal dispersivity is ax, + All, where
Ail = ~` Alleys (2.6)
and A* is a correlation scale; fly is a mean flow parameter that
can be approximated by 1 + /6 for small values of ~K or
exp ~ cr~2/6) for large ~K. Gelhar and Axness (1981) also present
more involved expressions for the other components of Aij.
Matheron and DeMarsily (1980) analyzed dispersion for the
case of twodimensional flow through a stratified aquifer. Ac
cording to their analysis the magnitude of the asymptotic lon
gitudinal dispersivity is dependent on the magnitude of the
ratio of vertical to horizontal velocities. As this ratio increases,
the asymptotic dispersivity becomes small. They were pessi
mistic about predicting the asymptotic dispersivity a priori
because the value will change with changes in the flow field.
Gelhar and Axness (1981) analyzed threedimensional disper
sion in a stratified aquifer, and their expression for longitudinal
dispersivity was also dependent on a mean flow parameter as
is Eq. (2.6~. However, they suggest that the mean flow param
eter (~y) can be expressed in terms of the variance of the hy
draulic conductivity distribution. Dagan (1982) showed that for
a porous media having a threedimensional isotropic but het
erogeneous structure, the asymptotic value for the dispersivity
is equal to land, where l is the length characterizing the ex
ponential autocorrelation of the natural logarithm of the hy
draulic conductivity and (~,;2 is the variance of the natural log
arithm of the hydraulic conductivity.
For large times or large distances it may be possible to use
expressions like Eqs. (2.5) and (2.6) to calculate values for
dispersivity for use in the advectiondispersion equation. If
dispersion does in fact become Fickian at large times or dis
tances, a key question is: What is the length of time that must
OCR for page 37
42
elapse before the advectiondispersion equation is applicable
and expressions such as Eqs. (2.5) and (2.6) are valid? Matheron
and DeMarsily (1980) found that for a typical set of aquifer
parameters, the Taylor limit would be reached after 140 days
and 600 m of travel. Gelhar and Axness (1981) and Smith and
Schwartz (1980) also suggested that the Taylor limit will not
be reached until the contaminant has traveled on the order of
tens or hundreds of meters from the source. According to a
theoretical analysis by Dagan (1982), dispersion should be Fick
ian at a distance from the source approximately equal to SOL,
where L is the integral scale of the natural logarithm of the
hydraulic conductivity distribution. For twodimensional flow
fields, L may be hundreds to thousands of meters, but for three
dimensional analyses L is expected to be on the order of meters.
While there has been much progress in defining dispersivity
at large distances from the source, no clear consensus has emerged
regarding the appropriate way to quantify dispersion near the
source before the Taylor limit is reached. Gelhar et al. (1979)
derived a revised form of the advectiondispersion equation,
which includes several additional terms of higher order that
are required to represent dispersion close to the source. They
then analyzed the development of the dispersion process for
early times and showed that the definition of the timede
pendent dispersivity requires evaluation of an integral for which
the small time limit for the case of onedimensional flow through
a stratified aquifer is a~ + A*, where
A* = (~KIKj2x (2.7)
and "x is the distance traveled (x = fit). The value of dispersivity
given in Eq. (2.7) is equal to the value of dispersivity implied
in an analysis of dispersion by Mercado (1967), where
and then
(rx= (V~K/K)t (2.8)
A* = (d(J2/dt)/2v, (2.9)
from which Eq. (2. 7) follows. Here, OK is the standard deviation
of the hydraulic conductivity distribution and of is the standard
deviation of the concentration distribution.
According to Eq. (2.7), dispersivity should increase linearly
with distance traveled. According to Mercado's result tEq.
(2.8~], the standard deviation of the concentration distribution
increases linearly with time or distance. If dispersion is Fickian,
the standard deviation increases with the square root of the
distance traveled. Thus, the dispersed zone grows more rapidly
before the Taylor limit is reached.
Pickens and Grisak (1981a) suggest using Eq. (2.7) or some
similar equation in the standard advectiondispersion equation
to describe the early portion of the dispersion process. Math
eron and DeMarsily (1980) consider a similar approach based
on the concept of an equivalent dispersivity:
(X' j = crxi2/2vt, (2. 10)
where crX,2 is the variance of the concentration distribution at
time i. Eq. (2.10) follows directly from Eq. (2.9) for a discrete
time interval. The values of longitudinal dispersivity (~ ~) are
defined such that Ax represents dispersion for the time from
MARY P. ANDERSON
t = 0 to t = t1 and CXL1 represents dispersion for the time from
t = t1 to t = to, for example. However, Matheron and DeMarsily
(1980) caution that because the use of Eq. (2.10) implies that
dispersivity is really a constant and because the advection
dispersion equation is actually inapplicable for early times, the
use of an equivalent dispersivity is not completely satisfactory.
They conclude that, "A better mathematical formulation of the
transport process in porous and fractured media, valid for all
time, seems necessary." More specifically, according to Gill
ham and Cherry (1982~: "The present challenge is to develop
a physically based transport model that incorporates spatially
and/or temporally variable dispersion parameters that can be
determined in a practical manner and with an acceptable de
gree of certainty."
SUMMARY AND DISCUSSION
The discussions of dispersion in the preceding sections focused
on applications to continuous porous media; dispersion in frac
tured porous media will be considered in a later section of this
chapter. The state of the art for quantifying dispersion in con
tinuous porous media is summarized below.
Results from several studies suggest that dispersion is non
Fickian near the source of the contaminant and therefore the
standard form of the advectiondispersion equation does not
apply. A modified form could be derived for analyzing non
Fickian transport for small times. Gelhar et al. (1979) presented
such an equation for onedimensional flow in a perfectly strat
ified aquifer. However, they caution that the approach that
they used to derive the equation requires certain restrictions
that may not be valid at small times. Other approaches that
rely on the standard form of the advectiondispersion equation
call for use of timedependent dispersivities (Gill and Sankar
asubramanian, 1972; Matheron and DeMarsily, 1980; Pickens
and Grisak, 1981b).
The standard approach in analyzing dispersion involves the
use of an average linear velocity Em. (2.1~] and the use of a
dispersion term tEqs. (2.2) and (2.3~] to represent deviations
of velocity from the average. The rationale for this approach is
based on the premise that a dispersion term is necessary be
cause it will never be practical to define the velocity field in
detail. Moreover, according to Dagan (1982) it is inappropriate
to assume that local dispersion does not influence advection.
Hence, several investigators (Gelhar et al., 1979; Gelhar and
Axness 1981, Dagan, 1982) maintain that the key to a rational
application of the advectiondispersion equation is to define
the dispersivity in terms of various statistical properties of hy
draulic conductivity.
Another approach, used by Schwartz (1977) and Smith and
Schwartz (1980, 1981a, 1981b) among others, is based on at
tempts to represent the velocity field in detail by defining the
hydraulic conductivity in a stochastic manner and in this way
to simulate the effects of macroscopic dispersion directly. Smith
and Schwartz (1980) concluded that for their simulations local
dispersion was negligible because most dispersion observed in
their modeled systems occurred at a macroscopic scale. There
fore, in subsequent studies (Smith and Schwartz, 1981a, 1981b)
OCR for page 37
Contaminants in Groundwater: Transport
they used an advective model to simulate contaminant trans
port in a number of hypothetical systems characterized by dif
ferent arrangements and shapes of heterogeneities. Results of
their numerical experiments demonstrated that the solution of
the contaminant transport model is quite sensitive to the struc
ture of the heterogeneities within the porous material. Con
sequently, detailed information on the arrangement and shapes
of the heterogeneities, knowledge of the values of hydraulic
conductivity for the various units, and information on head
gradients (or direct measurements of velocities) are essential
for accurate prediction. Although they demonstrated that it is
theoretically possible to predict contaminant transport given
sufficient hydrogeologic data, they remained pessimistic re
garding the practical limitations of obtaining the detailed in
formation needed (Smith and Schwartz, 1981b). In addition
they concluded that ". . . large uncertainties can be associated
with transport predictions in heterogeneous media. These un
certainties must be dealt with in order to develop confidence
in the application of mathematical models of sitespecific prob
lems. Unfortunately, the variety of possible sources of uncer
tainty and the difficulty in controlling their size suggest that
progress will be slow." Similarly, Dagan (1982) concluded that
"longitudinal dispersion can be represented asymptotically by
a Fickian equation, with dispersivity much larger than pore
scale dispersivity," but given the uncertainties involved in de
fining the hydrogeologic system a stochastic approach is nec
essary and "the traditional approach of predicting solute con
centrations by solving deterministic partial differential equations
is highly questionable in the case of heterogeneous formations."
The conclusion drawn from these studies is that although
there has been considerable progress within the past 5 years
in understanding the nature of macroscopic dispersion in po
rous media, to date, a credible, practical, and reliable model
for analyzing contaminant transport near the source has not
been identified. However, theoretical studies by several re
searchers suggest that for large times (or large travel distances)
the dispersion component of the advectiondispersion equation
can be represented by an expression analogous to Fick's second
law of diffusion Em. (2.2~], and therefore the classical advec
tiondispersion equation will accurately simulate contaminant
transport in porous media, provided that dispersivity can be
quantified in a meaningful way.
Gelhar et al. (1979), Gelhar and Axness (1981), and Dagan
(1982) present formulas for defining dispersivity at large times
in terms of statistical parameters. However, several investi
gators suggest that the length of time that must elapse before
the advectiondispersion equation is applicable may be appre
ciable. Specifically, the travel distance of the contaminant must
be on the order of 10s or 100s of meters from the source before
the advectiondispersion equation is valid. Simulations of con
taminant transport by Pickens and Grisak (1981b) suggest that
the scale dependence of dispersivity at early times may have
little effect on results for large times or travel distances. Hence,
the use of an asymptotic dispersivity may be adequate for these
conditions provided that sufficient geologic data are available
to characterize the statistical parameters of the hydraulic con
ductivity distribution (the variance and the correlation scales),
as well as the orientation of the geologic units. More attention
43
is needed in characterizing geologic systems in terms of these
statistical parameters (e.g., see Smith, 1981; Newman, 1982~.
Moreover, field data needed to corroborate the theory itself
are lacking. There is also a need for carefully designed field
tracer tests to determine whether the theory is generally ap
plicable and to serve as a basis for estimating the length of time
before the Taylor limit is reached and Eq. (2.4) is valid. Data
from such tracer tests would allow an assessment of the reli
ability of recently derived expressions for dispersivity.
DISPERSION IN FRACTURED ROCK
The discussion in previous sections focused on dispersion in
continuous porous media. Application of dispersion analysis to
contaminant transport in fractured rock is in its infancy, yet it
is of considerable importance. Certain types of fractured rock,
such as shale, granite, and salt, are likely candidates for re
positories of highlevel nuclear waste, and clay, which is also
susceptible to fracturing, is currently used as a disposal medium
for municipal, industrial, and lowlevel radioactive wastes.
Recent work on flow through fractured rock has emphasized
laboratory investigations and development and testing of ~nath
ematical models, most of which are based on the concept of a
dual porosity medium (e.g., Grisak and Pickens, 1980; Ner
etnieks, 1980) as shown in Figure 2.5. In a few cases, models
have been used in conjunction with results frown laboratory
experiments (Grisak et al., 1980; Tang et al., 1981; Neretnieks
et al., 1982) and field data (Bibby, 1981~. These results support
the following conclusions:
1. Diffusion of contaminants from fractures to the rock ma
trix can serve as a significant retardation mechanism (see Grisak
and Pickens, 1980; Neretnieks, 1980; Abelin et al., 1982; Bir
gersson and Neretnieks, 1982~. This phenomenon is illustrated
schematically in Figure 2.5.
2. Dispersion in the fracture can significantly accelerate the
arrival of contaminants at a discharge point when velocities in
the fractures are relatively low.
Experimental results suggest that velocity in fractures can
be represented using the socalled cubic law (see Witherspoon
_ _
FRACTURE
FLOW
I DIFFo5
POROU S
ROCK MATRIX
_
FIGURE 2.5 Schematic diagram representing flow through fractures
and diffusion of contaminants from fractures into the rock matrix of a
dual porosity medium.
OCR for page 37
44
et al., 1980), which is a form of Darcy's law in which the
equivalent hydraulic conductivity of the fracture (Kf) is
Kf = (pg/12~)b, (2.11)
where b is the aperture of the fracture, g is the constant of
acceleration of gravity, p is the density of water, and ~ is the
dynamic viscosity. The cubic law for the volumetric flow rate
through the fracture is
Q= (pg/121l)1Wb3, (~2.12)
where I is the head gradient across the length of the fracture
segment and W is the width of the fracture segment.
A number of models (Grisak and Pickens, 1980; Tang et al.,
1981; Neretnieks et al., 1982) consider dispersion in the frac
ture. Tang et al. (1981) derived an analytical solution of the
advectiondispersion equation for onedimensional contami
nant transport with longitudinal dispersion in the fracture, cou
pled to a solution of a model representing diffusion of solute
frown the fracture into the rock matrix. Grisak and Pickens
(1980) solved a similar problem using a finiteelement model.
Both concluded that while longitudinal dispersion in the frac
ture can be important, little is known regarding the magnitude
of dispersivity in fractures except that it is a function of the
roughness of the fracture. Experiments under way (see With
erspoon, 1981) may help in quantifying fracture roughness,
which is a first step toward quantifying dispersivity in fractures.
Analysis of flow of water through networks of fractures has
been attempted (Bibby, 1981; Neuzil and Tracy, 1981; Sudicky
and Frind, 1982~. But much additional work is needed to test
and to modify these types of models. A major impediment to
testing of fracture flow models is the difficulty of obtaining field
data that describe the hydrogeologic characteristics of fractured
rock.
CONCLUSIONS
1. There has been significant progress in quantifying dis
persion in porous material since 1978.
2. The consensus is that dispersion in porous material is non
Fickian near the source of the contaminant, and therefore for
small times or small distances from the source, the standard
form of the advectiondispersion equation does not apply.
3. There is no consensus regarding the best way to quantify
dispersion near the source of the contaminant. Some research
ers prefer to aim at attempting to describe the velocity field
in sufficient detail so that contaminant transport can be sim
ulated by advection alone. Others prefer to rely on refining
the advectiondispersion equation to handle nonFickian dis
persion.
4. Theoretical studies suggest that for many hydrogeologic
settings, dispersion should become Fickian for large times or
large distances (on the order of 10s to 100s of meters). In this
case it is possible to express the dispersion parameter known
as dispersivity in terms of statistical properties of the hydraulic
conductivity distribution.
5. Carefully designed field tracer tests are needed to eval
MARY P. ANDERSON
uate the applicability of the theory for large times and to refine
the theory for small tines.
6. Detailed hydrogeologic information is needed to predict
dispersion for both small and large times.
7. The concept of a dual porosity media (Figure 2.5) is cur
rently used for modeling groundwater and contaminant flow
through fractured rock. However, the difficulties involved in
characterizing the geometry of a fracturedrock system in the
field complicates the testing of the theory. As a result, none
of the models developed to simulate dispersion and diffusion
of contaminants in fractured rock have been verified by field
experiment.
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