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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 three-dimensional 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 nonpoint-source 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 S-shaped breakthrough curves (concentration versus
time curves), which are characteristic of flow affected by dis-
OCR for page 38
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 contaminant-transport
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 well-known governing equation
referred to as the advection-dispersion 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 advection-dispersion equation is derived by combining
a mass-balance 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 39
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 advection-dispersion 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 advection-dispersion 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 second-rank 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 10-6 cm2/
see, although a range of 1O-5 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 advection-dispersion equation in its most general form
is written
(Do*) -(cri)+ ~ Rk =
dxi dXj Oxi n k= ~ at
dispersion term advection sink/source chemical-reaction (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 chemical-reaction term,
Rk is the rate of production of the solute in reaction k of s
39
different reactions. The problems involved in quantifying the
chemical-reaction term are discussed in Chapter 3 of this re-
port.
If the advection-dispersion 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. 17-27) cites several examples of
laboratory experiments in which the experimental results did
not fit theoretical curves derived from solutions of the advec-
tion-dispersion 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-
centration-time 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 concentration-distance curves will approximate a Gaussian
distribution except for short times f Figure 2.3(a)~. However,
concentration-time 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
concentration-time 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-
tration-time curves in the Lesser Slave River deviated less from
OCR for page 40
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
X10-9
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-
tion-time 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 concentration-time 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
non-Gaussian 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 non-Fickian 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 waste-disposal 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
advection-dispersion 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
non-Fickian 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 advection-dispersion 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 41
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
Lallemand-Barres 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 advection-dispersion equation. Local dispersivi-
ties are on the order of 10-2 to 1 cm for laboratory experiments
and range from 10-i 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 so-called 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
10-2 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 three-dimensional analysis of Gel-
har and Axness (1981), dispersivity is a second-rank 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 two-dimensional 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 three-dimensional 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 three-dimensional 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 advection-dispersion 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 42
42
elapse before the advection-dispersion 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 two-dimensional 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 advection-dispersion 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 time-de-
pendent dispersivity requires evaluation of an integral for which
the small time limit for the case of one-dimensional 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 advection-dispersion 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 advection-dispersion 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 one-dimensional 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 advection-dispersion equation
call for use of time-dependent 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 advection-dispersion 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 43
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 site-specific 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 advection-dispersion equation
can be represented by an expression analogous to Fick's second
law of diffusion Em. (2.2~], and therefore the classical advec-
tion-dispersion 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 advection-dispersion 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 advection-dispersion 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 high-level nuclear waste, and clay, which is also
susceptible to fracturing, is currently used as a disposal medium
for municipal, industrial, and low-level 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 so-called 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 44
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
advection-dispersion equation for one-dimensional 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 finite-element 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 advection-dispersion 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 advection-dispersion equation to handle non-Fickian 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 fractured-rock 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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Representative terms from entire chapter:
water resour
