Application of particle methods to reliable identification of groundwater pollution sources

Abstract

An alternative strategy for identifying sources of contamination in groundwater systems is presented. Under the assumption that the remediation cost is affected by the level of contamination, the proposed scheme provides probabilistic estimates of source locations and spill-time histories. Moreover, the method successfully assesses the relative importance of each potential source.

Full text

Water Resources Management
6:
15-23, 1992
© 1992 Kluwer Academic Publishers. Printed in the Netherlands.
Application
of
Particle Methods to Reliable
Identification
of
Groundwater Pollution Sources
AMVROSSIOS
C.
BAGTZOGLOU*
Department
of
Civil Engineering, University
of
California, Irvine,
CA
92717, U.S.A.
DAVID
E.
DOUGHERTY
Department
of
Civil Engineering and Mechanical Engineering, University
of
Vermont, Burlington,
VT
05405, U.S.A.
and
ANDREW
F. B.
TOMPSON
Earth Sciences Department, Lawrence Livermore National Laboratory, Livermore,
CA
94550, U.S.A.
(Received: 28 May 1991; accepted:
15
November 1991)
15
Abstract. An alternative strategy for identifying sources
of
contamination in groundwater systems
is
presented. Under the assumption
that
the remediation cost
is
affected by the level
of
contamination,
the proposed scheme provides probabilistic estimates
of
source locations and spill-time histories. Moreover,
the method successfully assesses the relative importance
of
each potential source.
Key
words. Particle methods, random walk, reversed time simulation, conditional simulation, kriging,
Monte Carlo simulations, correlation length, turning bands method, inter-sampling distance.
1.
Introduction
Groundwater
(GW)
contamination by chemicals
and
hazardous waste materials
which are either
not
properly disposed
of
or
applied for agricultural purposes,
has reached alarming levels over the past decade. Remediation
of
GW
contaminated
sites
is
a complex
and
lengthy process, which
is
usually initiated by attempting
to
identify the contamination source.
The
construction, testing,
and
sampling
of
numerous observation wells reveals, most
of
the times, the
nature
of
contamination
and
identifies the extent
of
the migrating plume.
Thousands
of
GW
contamination
incidents are known
and
well documented in the United States
and
clean-up costs
are staggering.
For
example, clean-up costs
of
between $50
and
$100 billion are
estimated in the U.S.
Department
of
Energy
(DOE)
properties alone. Methods
that
provide a rational tool for assessing responsibility for
GW
pollution
can
lead to
reduced taxpayer
burdens
by assigning remediation costs directly
to
identified
responsible parties.
Typically, each
and
every one
of
the probable locations
is
considered as
an
a
* Now
at
Center for Nuclear Waste Regulatory Analyses, Southwest Research Institute, San Antonio,
TX 78228-0510, U.S.A.

16
AMVROSSIOS
C.
BAGTZOGLOU ET AL.
priori
known
feasible solution. Extensive simulations running forward in time,
provide a good understanding
of
how close
to
the present
contaminant
spatial
configuration this potential source leads. This process, repeated as
many
time as
the
number
of
potential sources, ultimately pinpoints the most probable solute
source. Gorelick et al. (1983), Wagner
and
Gorelick (1987, 1989), Lindstrom
and
Boersma (1989),
among
others, have addressed this issue within a forward-time
simulation-optimization framework.
2. Particle Methods
and
the Irreversible Nature
of
Diffusion
The
balance
of
a reactive solute in a
porous
medium
of
constant porosity may
be expressed as
ac
- +
'V
·
(vc)-
'V
· (D ·
'Vc)
= r(c,
t),
at
(1)
where
cis
the solute concentration,
vis
the fluid velocity, D
is
a velocity dependent
dispersion tensor,
and
r(c,t)
is
a net species
production
rate.
Particle methods have been widely used to solve problems in fluid dynamics,
plasma physics, as well as
heat
transfer (Hackney
and
Eastwood, 1988). In the
context
of
solute
transport,
particles represent elements
of
solute mass distributed
in space.
The
particle representation for the concentration
at
some
point
and
time
is
achieved
through
the use
of
projection functions
and
is
of
importance when
accurate/smooth
concentration fields are sought (Bagtzoglou, 1990; Bagtzoglou et
al.,
1991
).
In
the first
part
of
the
operator
splitting algorithm employed in the
present work, a
random
walk particle
method
(RWPM)
is
used
to
compute
concentration changes due
to
advection
or
dispersion processes. Over a time step
I:J.t,
the position
of
each particle will be changed according to
(2)
In
this expression,
xn
is
the position
of
the particle
at
time level
tn,
and
zn
is
a normalized
random
vector
of
zero mean
and
unit variance. In the limit as the
number
of
particles
Np
-oo,
it
can
be shown (Kinzel bach, 1988; Tompson
and
Gelhar,
1990)
that
the mass density
of
the particle distribution
that
evolves
through
repeated
use
of
(2)
is
a solution
of
the nonreactive form
of
(1) if
A = v +
'V
· D
and
B · BT = 2D. (3)
For
reactive
transport,
we
were able
to
develop consistent projections from
concentration changes
to
particle mass changes
through
an
iterative process, although
the computational costs were overwhelmingly large.
In
some cases, however, the
application
of
inconsistent backward interpolation was proven
to
be
an
approximate,
cost effective alternative (Bagtzoglou
and
Dougherty, 1990).
In
the usual
RWPM,
diffusion
is
represented by the
random
step znv2D!J.t. There

IDENTIFICATION
OF
GROUNDWATER POLLUTION SOURCES
17
is
no way to simply
'adjust'
this algorithm, which acts
on
individual particles
undergoing independent flights, to induce variance reduction as time proceeds.
Instead,
we
must look to more global (i.e. correlated) flights. This led to a reversed
time random particle method which, in some cases (uniform coefficients) gave
satisfactory results (Bagtzoglou, 1990). The notion
of
reversing the time in
random
walk (RW) simulations
is
challenging
and
counter-intuitive, since it
is
opposing
a well-established perception
of
a natural phenomenon; the irreversible diffusion
process.
3.
A Particle Method for Pollutant Source Identification
Based
on
the operator-splitting approach, it
is
relatively straightforward to reverse
the advective
part
of
the method. A particle location,
at
a time b.t in the past,
can be calculated by simply advecting it from the present position under a reversed
velocity field (Cady
and
Neuman, 1988).
In
the present work,
we
'reverse' time
by reversing the velocity field
and
leaving the diffusion/dispersion unchanged.
In
order to determine a
GW
pollution incident,
we
need to have concentration
measurements. Thus, these measurements indicate
not
only
that
contamination does
(or does nqt) exist,
but
also where solute mass
is
now located. Therefore,
we
know
from the 'current' measured state
of
the plume where solute mass
is
now.
If
we
can determine the velocity fields
that
have existed in the
GW
region for a period
of
time,
at
least as long as the contamination history suggests, then
we
can use
the 'reversed time' idea
to
determine the probability density function (pdf) indicating
the probability
that
a particle
of
mass now at X originated at
an
arbitrary point
x, T units
of
time ago. Typically,
we
do
not
know
T,
the actual
amount
of
time
the pollutant has been in the
GW
system.
We
assume, however,
that
there
is
some,
at least limited, knowledge
of
the site history
and
pollutant processing practices.
One approach
is
to make use
of
some time constraints
and
present spatial dis-
tributions
of
pdfs as snapshots
at
different times within the time period
that
satisfies
the time constraints. The most likely sources are identified by the loci
of
maxima
of
the
pdf
spatial distribution. The second approach
is
to present the temporal
variation
of
the pdf,
at
specific points
of
interest (i.e. probable sources). When
information concerning
both
the probable location
and
the time history
of
the
contamination incident
is
available, a combined approach
is
utilized
that
attempts
to further restrict the solution.
If
the velocity field
is
not
known, then the
pdf
resulting from
our
methodology
is
subject to uncertainty. Given a probabilistic
statement
that
describes the velocity field
we
can construct probabilistic statements
about
the
pdf
In so doing,
we
can quantify the reliability
of
our
predictions for
the location
of
the source
and
the time
of
occurrence
of
the contamination incident.
Kriging, a stochastic interpolation technique,
is
used to embed field measurements
within a Monte Carlo framework to perform conditional simulations.

18
AMVROSSIOS
C.
BAGTZOGLOU ET AL.
4. Heterogeneous Porous Medium with Perfectly
Known
Conductivity Fields
Flow
and
transport
in a square domain
50
X
50
m in area, discretized into 2500
square elements
of
~x
=
~y
= 1 m,
is
considered. The effective hydraulic conductivity
is
Kef!=
Ka
= 5
m/d
and
the
standard
deviation
of
the logconductivity field (ay)
ranges from 0 (perfectly uniform media) to 4 (extremely heterogeneous media).
An exponentially decreasing isotropic spatial correlation length
Ax
=
Ay
=
2~x,
and
the turning bands method (Tompson et al., 1987, 1989)
is
used for the hydraulic
conductivity field generation. The computational resolution
~x
and
the correlation
length A are significantly less
than
L (the characteristic length
of
the domain),
ensuring a resolution
that
will allow local behavior to be observed. The flow domain
is
subjected to the following Dirichlet
and
no-flux
boundary
conditions
ah ah
h(O,
y)
=
1,
h(L,
y)
= 0,
-(x,
0) = 0,
-(x,
L)
=
0.
ay
ay
(4)
The mixed finite element method, in conjunction with a preconditioned conjugate
gradient algorithm,
is
employed for the velocity field determination. Figure 1 shows
the aquifer system
and
three potential sources
of
contamination. The disposed
amounts
and
the locations
of
the potential contamination sources were selected
so as to enhance the pollutant plume interference.
For
all simulations, presented in this work, the spill incidents are assumed to
be occurring simultaneously. Twenty numerical experiments were conducted, with
50~--------------------------~
45
40
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Iii 0
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25
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0 0 ' 0
B o
A o o
1!1
o o o
....
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Iii o 0 o
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0 o o
',,
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00
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o
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',
o
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___________
,
0
',
__
,
15
5
0+--+--~~--r-~--r--r--r-~~
0 5 10 15
20
25
30 35
40
45
50
Distance (m)
----
Boundary
of
existing
plume
00
Potential
solute
source
o
Observation
well
Fig.
1.
Hypothetical aquifer system for solute source identification with conditioned hydraulic con-
ductivity fields.

IDENTIFICATION OF GROUNDWATER POLLUTION SOURCES
19
the following distinguishing parameters:
five
values
of
heterogeneity level (ay = 0,
0.5,
1,
2,
and
4),
and
four types
of
contamination scenarios depending
on
the location
of
the active source
and
the mass
of
the pollutant spilled.
For
all simulations,
the time step Llt = 0.25 d, a two-dimensional particle projection function was used
with a support
of
H =
1.5
m, Llx = 1 m,
Lly
= 1 m,
500
particles were released
at
each spill point, the total forward simulation time
tp
=
50
d, the dispersivities
aL
=
0.5
m
and
ar
= 0.05
m.
The effect
of
the stochastic nature
of
the RWPM
was eliminated by conducting repeated solute transport realizations
and
evaluating
statistics
of
the concentration
pdf
It
was found that, for the dispersivities used,
10
Monte Carlo runs were sufficient for the RW transport realizations. The numerical
Fig. 2(a).
Fig. 2(b).

20
0.9
0.8
.5
0.7
~
~
~
0.6
Q.
"'
0.5
0
...
:E
0.4
:.:;
.l!
£ 0.3
0.2
0.1
0
0
A
··-----------------
-----------------------
Reversed
Time
AMVROSSIOS C. BAGTZOGLOU
ET
AL.
8
Fig. 2(c)
Fig.
2.
Pdf
values(%)
for hypothetical solute source identification scenario with a
ay
=
1.
(a) Forward-
time simulation
at
time
50
days. (b) Backward-time simulation at reversed time
50
days.
(c)
Temporal
variation
of
pdf
experiments can be outlined as follows:
For
every run, a forward-time simulation
was conducted. The concentration result was then assumed to be the present
contamination configuration,
and
a particle representation
of
the concentration
field was evaluated. Then, the stochastically generated hydraulic conductivity field
was assumed to be know a priori and, thus, the velocity field was reversed. Twenty
backward simulations, for the same conductivity field realization, yielded the average
and
standard
deviation
of
concentration
at
each point within the flow domain.
Finally, the concentration field was transformed to a
pdf
field. Figure 2 corresponds
to identification scenario No. 1 with
ay
=
1,
and presents the average,
of
10
Monte
Carlo runs,
pdf
field
at
the end
of
a time reversal
that
lasted
50
days. Within
the resolution
of
one computational cell, the method identifies source A as the
responsible one (Figure 2b
).
In
Figure 2c the temporal variation
of
the
pdf
at the
three points
of
interest (A,
B,
C)
is
depicted. Again,
point
A
is
identified as the
most probable source
of
contamination which occurred
at
time
50
days in the
past.
An
erroneous contribution
of
liability to
point
B
is
evaluated. This
is
due
to the fact
that
point
B
is
located exactly downstream
of
point
A, thus responding
to all solute mass being released from A. Nevertheless, within the probabilistic
framework
of
this method, a relative liability
of
950% between points A
and
B
is
considered quite satisfactory.
5. Heterogeneous Porous Medium with Conditioned Conductivity Fields
The proposed methodology
is
now applied to a flow field where the only a priori
knowledge
of
the conductivity field
is
the
trend
(or mean), the variance and the
covariance structure
of
the logconductivity field. First, the effect
of
20
repeated
unconditional simulations
is
assessed. A significant decrease in identification power

IDENTIFICATION
OF
GROUNDWATER
POLLUTION
SOURCES
21
is
introduced.
For
the example presented before, three (instead
of
one) potential
sources were observed (Figure 3a). Moreover, the values
of
standard deviation for
this
pdf
distribution are extremely high relative to the actual
pdf
values. This implies
that
the reliability
of
prediction
is
of
very low value. Next, conditional simulations
are performed as
part
of
the Monte Carlo analysis. Each Monte Carlo sample
employs a
random
field conditioned
upon
observations. The observation points
(Figure
1)
are locations at which actual values
of
the hydraulic conductivity,
and
possibly concentration, are observed. Kriging
and
sample field generation are
accomplished using exactly the same covariance structure as was used to generate
0.35
0.3
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0.25
0
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..
..
0.2
·c.
"'
"0
~
0.15
:g
J:>
£
0.1
0.05
0
0
0.9
0.8
0.7
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·;;
0.6
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..
..
·c.
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"0
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0.3
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c..
0.2
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///
~-----·
/
Reversed
Time
Reversed
Time
A
c
50
Fig. 3(a).
A
B
50
Fig. 3(b).
Fig.
3.
Temporal variation
of
pdf
values(%)
for hypothetical solute source identification scenario with
a
ay
= I. (a) Unconditioned repeated realizations. (b) Conditioned repeated realizations with
60
conditioning points.

22
AMVROSSIOS C. BAGTZOGLOU
ET
AL.
the
'known'
conductivity fields. Twenty runs were used in each Monte Carlo
computation with 20, 40,
and
60
observation points.
The reliability
of
the source identification process clearly improves (Figure 3b ),
compared to the unconditional simulation results, when the number
of
observation
points increases. The temporal variation
of
the
pdf
becomes almost identical to
the one corresponding to a perfectly known conductivity field when
60
wells are
used. The inter-well distance, relative to the correlation length,
is
the controlling
parameter
in these simulations.
In
our
work, the inter-well distance
is
defined as
the minimal distance from one well to every other well. When
60
sampling points
are used, the average inter-well distance
is
about
1 to
1.5
correlation lengths. Only
at
this level oflocalization does the kriging process and, consequently, the conditional
simulation, becomes effective enough to yield results comparable to those coming
from use
of
the perfectly known conductivity field.
6. Conclusions
The proposed methodology provides crucial information for the design
of
monitoring
and
data-collection networks in support
ofGW
pollution source identification efforts.
The network must be designed with minimum inter-sampling distances
of
about
one
to
two correlation lengths
A.
When this
parameter
is
not
known, preliminary
sampling will be necessary for its estimation.
The
importance
of
data
in making
reliable pollutant source identifications
is
crucial. Methods to reduce the number
of
necessary measurements,
and
simultaneously increase the
amount
of
information
being used, are needed. One possible approach to bypass the problem
of
having to
deal with the cost
of
well construction, development,
and
maintenance
is
to make
use
of
multiple sources
of
information. The application
of
tomography
or
the use
of
seismic/remote sensing
data
in conjunction with observation wells
is
promising.
Acknowledgements
We
acknowledge computer time granted by the University
of
California, Irvine
that
contributed to this work. Portions
of
the work were conducted under the
auspices
of
the U.S.
Department
of
Energy by Lawrence Livermore, National Labo-
ratory
under contract W-7405-Eng.-48
and
supported
by
the Subsurface Science
Program
of
the Office
of
Health
and
Environmental Research
of
the U.S.
DOE.
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IDENTIFICATION
OF
GROUNDWATER
POLLUTION
SOURCES
23
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