United States
Environmental Protection
Agency
Robert S. Kerr Environmental
Research Laboratory
Ada OK 74820
Research and Development
EPA/600/S2-86/114 Sept. 1987
&EPA Project Summary
Stochastic Prediction of
Dispersive Contaminant Transport
Efstratios G. Vomvoris and Lynn W. Gelhar
The objective of this research agree-
ment was to develop mathematical
models to quantify the concentration
variability observed in field measure-
ments of concentration plumes.
The concentration variability is attri-
buted mainly to spatial heterogeneity
of the hydraulic conductivity field. Since
limited information exists about actual
distributions of hydraulic conductivity
in a given site, the log-hydraulic con-
ductivity is modeled as a three-dimen-
sional anisotropic stationary random
process. It is characterized by its mean
and its covariance function or spectrum.
Implementing Darcy's equation and the
convective-dispersive equation at the
local scale, relationships among con-
centration variations and characteristic
parameters of the porous medium are
found. Approximate analytical expres-
sions are also developed.
It is shown that the concentration
variance, which measures the intensity
of the variations, is proportional to the
mean concentration gradient and to the
variance and correlation scales of the
log-hydraulic conductivity; it is inversely
proportional to the local dispersivity
values. The main contribution to the
concentration variability results from
the coefficients corresponding to the
longitudinal mean concentration
gradients.
Analysis of a portion of the data from
the Canadian Forces Base, Borden,
Ontario, shows the applicability of the
developed results in field situations.
Simple analytical examples demonstrate
the way to use the results in a predictive
context.
This Pro/ecf Summary was developed
by Massachusetts Institute of Technology
for EPA's Robert S. Kerr Environmental
Research Laboratory, Ada, OK, to an-
nounce key findings of the research
project that Is fully documented In a
separate report of the same title (see
Protect Report ordering Information at
back).
Introduction
Field observations of solute transport
in ground water indicate that dispersivities
are scale dependent and can range over
many orders of magnitude. This discrep-
ancy between the laboratory and field
experience has been attributed mainly to
the spatial variation of the flow properties
of the natural porous earth materials.
Therefore, over the last decade a sub-
stantial portion of research in the area of
subsurface hydrofogy has focused on
understanding the effects of the spatial
heterogeneity of natural formations on
flow and solute transport.
Continuously expanding computer
capabilities have facilitated the use of
deterministic numerical models with pa-
rameters that may be spatially variable
over a grid and have, in principle, made it
possible to obtain solutions that previously
could be described only qualitatively.
However, such solutions are limited to
specific applications and are unable to
generate valid relationships over a broad
range of problems. More importantly, in
order for a numerical model to reproduce
actual details of a given field situation, it
must have sufficient grid resolution to
represent the velocity field producing the
mass transport and information about
the controlling parameters, such as hy-
draulic conductivity and porosity, at each
grid point. While the former may be
practical for small idealized flow systems
(tens of meters), the number of nodes
required to resolve the actual three-
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dimensional heterogeneity at a more
practical field scale (1 km) can easily
exceed the capacity of even the largest
super-computers. Furthermore, actual
measurement of the required parameters
with that degree of resolution is
impractical.
It is the lack of information about the
spatial distribution of controlled param-
eters that has led to modeling them as
realizations of random fields. Before
measuring the value of a random variable,
the only informtion known is a range for
its possible values as defined by its mean
and variance. In a random field it is
further assumed that there is a correlation
in space among the different values, so
that knowledge of the value at one loca-
tion carries information about the possible
values for the hydraulic conductivity at
different locations in such a way that
they fall into the prespecified ranges and
have the appropriate correlation in space
as defined by the correlation function.
This approach is coined as the stochastic
approach. The fundamental assumption
is that for each one realization the classi-
cal laws of flow and mass transport hold
at some representative scale.
Procedure
In the stochastic approach, the behavior
of the mean or expected value of the
concentration field is found to be equiva-
lent to the classical dispersive transport
equation. The theory also predicts the
resulting field-scale dispersion coeffici-
ents in terms of the variability of the
hydraulic conductivity. However, the
mean concentration represents the aver-
age behavior of the ensemble rather than
the actual realization of the aquifer.
Therefore, a classical transport model
will simulate a smoothed approximation
of an actual field simulation; the actual
concentration will vary around the smooth
mean. In order to realistically present the
results of model simulations, it is neces-
sary to understand and quantify this error
in classical models.
The stochastic approach can be used to
quantify the variation around the mean
concentration. The main focus of this
research is the development of the theory
to describe such variability. The degree of
variability will be characterized by the
concentration variance, and it will be
shown that this depends on measurable
quantities characterizing the aquifer
material such as correlation scales and
the variance of the log-hydraulic con-
ductivity, the inclination of the stratifica-
tion, and local dispersivity. It also depends
on the mean concentration gradient
which, through the macrodispersivities,
represents the overall effect of the aquifer
heterogeneity on mass transport.
The stochastic approach is comple-
mentary to classical deterministic model-
ing. The predicted concentration variance
can be used as a realistic and physically
based calibration target for the large-
scale numerical models. Furthermore, one
can use any classical numerical and
analytical model that reproduces the
general characteristics of the contamina-
tion event and, in addition, obtains a mea-
sure of the expected variability around
the model results. Finally, knowledge of
the possible deviations of the actual plume
from the predicted one can improve sub-
stantially the design of monitoring
schemes.
The present study is focusing on quan-
tifying the variation of the concentration
measurements or model predictions, in
heterogeneous porous formations. Ex-
amples demonstrate the practical applica-
tion of these results.
Results
The analysis of the field data demon-
strates the validity of stochastic theory
results for concentration variance. The
predicted concentration intervals are
consistent with observed fluctuations
around a smooth mean and capture most
of the measured values. The effect of the
longitudinal concentration gradient in the
calculation of variance is dominant except
when focusing on vertical sections where
the gradient is higher by 2 or 3 orders of
magnitude.
The estimated effective hydraulic con-
ductivities are quite close to the observed
geometric mean of the measured hydrau-
lic conductivities. The estimated longi-
tudinal macrodispersivity is also compar-
able to that estimated through the method
of moments.
The analysis reveals the need for the
development of a formal procedure for
the estimation of the mean. Analysis of
data from subsequent sampling dates is
needed to show the robustness of the
results.
Discussion
A simple theoretical result was devel-
oped to quantify the concentration vari-
ability. When measurements of the
concentration exist, it can be thought of
as giving a measure of the range of
expected variation of the concentration
around a smooth mean concentration
distribution. The theory is also useful as a
predictive tool when the solute distribution
is extrapolated for distances far down-
stream using a classical dispersive trans-
port model.
The functional form of the concentration
variance is easy to implement. It is a local
relationship that involves the mean con-
centration gradient, the variance and cor-
relation scales of the log-hydraulic
conductivity field, and the local dis-
persivity values. The mean concentration
gradient can be estimated either in a
predictive context with numerical or
analytical models, or through a simple
curve fitting with existing plume mea-
surements.
The variance of the log-hydraulic con-
ductivity can be measured in situ, or as
more information for the behavior or dif-
ferent types of aquifer materials is ac-
cumulated, it can be estimated from
information about geologically similar
sites.
The local dispersivities can be also
found from laboratory tests of soils from
the site or estimated based on the general
properties of the soil. It should be noted
that the effect of local dispersivity on the
variance is significant and cannot be
neglected. This behavior is in contrast to
that of the macrodispersivity which is not
sensitive to the local dispersivity.
An important factor needed for the
variance estimation is the correlation
scales.
The predicted concentration variance
is an appropriate calibration target for the
numerical models. Other sources of error
such as measurement errors or dis-
cretization errors associated with the
numerical scheme should also be con-
sidered, but typically those effects are not
expected to be predominant. Therefore, if
the root mean squared difference between
the observed and simulated concentration
fields is used as an index of the model
error, the calibration of the numerical
model could be considered adequate
when the model error and the error pre-
dicted from the stochastic theory are of
the same magnitude.
Although the results of the comparison
with field data are encouraging and in-
dicate the workability of stochastic theory
to predict concentration variance, there
are a number of approximations and/or
refinements which need to be evaluated
before the full range of applicability of the
theory can be established.
Conclusions
The concentration variability resulting
from the heterogeneity of natural aquifers
can be quantified using stochastic analy-
sis in terms of measurable or deter-
minable properties of the hydraulic
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conductivity. It can be expressed mathe-
matically as the sum of the products of
the concentration gradients weighted by
appropriate coefficients, which depend
on the variance and the correlation scales
of the log-hydraulic conductivity, the local
dispersivity values and the orientation of
the stratification. The dominant coefficient
corresponds to the gradient along with
mean flow and is inversely proportional
to the local dispersivity and proportional
to the correlation scales.
The effects of the orientation of the
stratification in a preliminary analysis
appear to be less important, but further
investigation is required for definitive
conclusions. The structure of the con-
centration covariance for the particular
spectrum used results in a singular effect,
namely, high correlations along the mean
flow direction.
Approximate analytical results have
been developed that establish the role of
key parameters.
The preliminary analysis of field data
shows that the results of the proposed
theory can be applied in practical situa-
tions and produce consistent confidence
intervals for the anticipated concentration
variability.
Recommendations
A local relationship that quantifies the
concentration variability anticipated in
mass transport in heterogeneous aquifers
has been found. The relationship involves
the gradient of the mean concentration,
the correlation scales of the log-hydraulic
conductivity field and the local dispersivity
values. It can be readily used and produce
adequate confidence intervals for the
concentration values, even with crude
estimation of the mean concentration
field.
The obtained concentration standard
deviation ean be implemented for the
calibration of numerical models. The
discretization inherent in numerical
models induces a smoothing or averag-
ing of the true concentration field and
makes it impossible to reproduce the
measured values. The estimated stand-
ard deviation of the concentration can be
used as a measure of the acceptable
deviation of numerical model results
from measured values.
The preliminary application of the re-
sults to a field site was encouraging, but
further analysis is needed to explore the
full three-dimensional nature of the con-
trolled plume and to follow its movement
and its response to the orientation of
stratification. Important issues such as
spatial averaging of the concentration
field introduced by the sampling technique
and its associated filtering properties need
to be addressed.
The developed theory established the
basic relationships among the aquifer
characteristics, the flow parameters and
the concentration variance. However, the
effects of unsteady concentration input,
initial conditions, flow reversal, and un-
known history of injection need to be
investigated. The singular behavior of the
concentration covariance along the mean
flow direction requires additional in-
vestigation before being accepted as a
valid descriptor of the physical system.
Finally, the issue of the mean con-
centration field, central to the theme of
all stochastic theories, should be investi-
gated. Methods to enhance the ability to
identify or estimate the mean concentra-
tion need to be developed.
Efstratios G. Vomvoris and Lynn W. Gelhar are with Massachusetts Institute
of Technology. Cambridge, MA 02139.
Joseph F. Keely is the EPA Project Officer (see below)
The complete report, entitled "Stochastic Prediction of Dispersive Contaminant
Transport." (Order No. PB 87-141 479/AS; Cost: $18.95, subject to change)
will be available only from:
National Technical Information Service
5285 Port Royal Road
Springfield. VA22161
Telephone: 703-487-4650
The EPA Project Officer can be contacted at:
Robert S. Kerr Environmental Research Laboratory
U.S. Environmental Protection Agency
P.O. Box 1198
Ada, OK 74820
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