United States
Environmental Protection
Agency
Robert S. Kerr Environmental
Research Laboratory
Ada OK 74820
''
Research and Development
EPA/600/S2-85/065 Aug. 1985
rn. a? . TI
^Fi*
&EPA Project Summary
Plume 2D: Two-Dimensional
Plumes in Uniform Ground
Water Flow
Jan Wagner, S. A. Watts, and D. C. Kent
A closed-form analytical solution for
two dimensional plumes was incorpo-
rated in an interactive computer pro-
gram. The assumption of an infinite
aquifer depth and uniform source mass
rate and source location was overcome
by using the principal of superposition
in space and time. The source code was
written in a subset of FORTRAN 77 and
can be compiled with FORTRAN IV,
FORTRAN 66 as well as FORTRAN 77.
As a result, the code is nearly indepen-
dent of hardware and operating sys-
tem. The model can be solved for either
vertically or horizontally averaged con-
ditions.
This Project Summary was devel-
oped by EPA's Robert S. Kerr Environ-
mental Research Laboratory, Ada, OK,
to announce key findings of the re-
search project that is fully documented
in a separate report of the same title
(see Project Report ordering informa-
tion at back).
Introduction
Relatively simple analytical methods
can often be used to evaluate ground-
water contamination problems, de-
pending upon the complexity of the sys-
tem and the availability of field data.
Analytical models can also serve as
valuable tools in developing parame-
ters for more sophisticated numerical
models. Although the numerical evalua-
tion of an analytical solution to a
ground-water problem may be mathe-
matically complex, analytical models
are well suited for interactive use on
digital computers. Many analytical solu-
tions to ground-water contamination
problems can be coded on program-
mable hand-held calculators. In general,
very few input parameters are required
to define a given problem, and numeri-
cal results can be calculated in a few
seconds.
This report presents analytical solu-
tions to two ground-water pollution
problems—two-dimensional plumes in
uniform ground-water flow. An interac-
tive computer code has been developed
which enables the user to modify the
definition of a given problem, and thus
gain some insight into the effects of var-
ious parameters on the extent of a con-
taminant plume.
Model Formulation
The differential equation describing
the conservation of mass of a compo-
nent in a saturated, homogenous aqui-
fer with uniform, steady flow in the
x-direction can be written as
rlC
f)X
= D
i>2C
D;
(1)
where
C = component mass per unit vol-
ume of fluid phase
D* =dispersion coefficient in x-
direction
D* =dispersion coefficient in y-
direction
Rd = retardation coefficient
V* = average interstitial velocity in x-
direction
x,y = rectangle coordinates
\ = first-order decay constant.
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The retardation coefficient accounts for
partitioning of the component between
the fluid and solid phases using a linear
adsorption isotherm and is defined as
= 1 + y Kd
(2)
where
PB = bulk density of the rock
0 = effective porosity
Kd = distribution coefficient for a lin-
ear adsorption isotherm.
Closed-form analytical solutions for
the two types of ground-water contami-
nation problems shown in Figure 1 are
included in this report. The first is a
vertically-averaged solution which de-
scribes a contaminant plume in the x-y,
or horizontal, plane (Figure 1a). The sec-
ond is a horizontally-averaged solution
describing a contaminant plume in a
vertical plane (Figure 1b).
The vertically-averaged solution ap-
plies to an aquifer of infinite areal extent
and finite depth. The contaminant is as-
sumed to be well mixed over the satu-
rated thickness. The source of contami-
nation is a vertical line source located at
the origin of a coordinate system in the
x-y plane. The conceptual model is sim-
ilarto an injection well which fully pene-
trates the saturated zone or a finite verti-
cal segment of the aquifier. Wilson and
Miller (1978) have also applied this solu-
tion down-gradient from a contaminant
source at the water table. For a rela-
tively thin saturated zone, vertical dis-
persion will tend to mix the contami-
nant vertically. The concentration
distribution can be considered as being
two-dimensional in a horizontal plane at
distances downstream of the source for
the concentration distribution to be-
come uniform with depth. For a continu-
ous source of strength MQ at the origin,
the vertically-averaged solution is
(Hunt, 1978; Wilson and Miller, 1978)
EXP
C =
V
2D
4ir8(DxDv)05
ir-W(U,B)
(3)
where
U =
V*\2
vv\
4V2t
(4)
Water O
Table O
Figure 1 .
and
B = i
Coordinate systems for (A) vertically averaged solution and (B) horizontally averaged
solution.
which might fit this conceptual model is
seepage from a trench.
The closed-form analytical solution
follows directly from the vertically-
averaged solution. Since the water table
represents a no-flow boundary passing
through the origin, the horizontally-
averaged solution can be written di-
rectly as
Dy \D
(5)
The function W(U,B) is defined as
Mi EXP
-€-d? (6)
C =
V*
(D*D*)05
W(U,B)
(8)
where 4 is a dummy integration vari-
able. This function is often referred to as
the "well function for leaky artesian
aquifers" (Hantush, 1956). The corre-
sponding steady-state-solution of Equa-
tion 1 is
The steady-state solution is
C =
M0 EXP,2D,
-ire (D*D*
Ko(B)
(9)
i EXP
2TT6
Ko(B)
(7)
where K0(B) is the modified Bessel func-
tion of the second kind of order zero.
The horizontally-averaged solution is
based on the conceptual model shown
in Figure 1b. A line source is located at
the water table and normal to the direc-
tion of ground-water flow. A problem
Equations 5 and 7 and 8 and 9 can be
used to calculate concentrations in con-
taminant plumes under the following
assumptions and limitations:
1. The ground-water regime is com-
pletely saturated.
2. All aquifer properties are constant
and uniform throughout the prob-
lem domain.
3. The ground-water flow is horizon-
tal, continuous, and uniform
throughout the aquifer.
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4. The aquifer is infinite in extent.
5. The contaminant source is a line
located at the origin of the coordi-
nate system.
6. The mass flow rate of the source is
constant.
7. At zero time the concentration in
the aquifer is zero.
The assumptions of an infinite aqui-
fier and uniform source rates can be
overcome by using the principles of su-
perposition in space and time. Superpo-
sition can also be used to include multi-
ple sources.
Computer Program
The closed-form analytical solutions
for the two-dimensional plumes as pre-
sented above have been incorporated in
an interactive computer program. The
source code has been written in a sub-
set of FORTRAN 77 and can be compiled
with FORTRAN IV, FORTRAN 66, as well
as FORTRAN 77 compilers. As a result,
the code is almost entirely independent
of hardware and operating systems.
Those changes which may be required
to implement the code on a given sys-
tem, such as assigning logical devices
are clearly identified.
The program has been developed for
interactive use and requires input data
under two modes of operation—"Basic
Input Data" and "Edit." The basic input
data are required to initiate a new prob-
lem. The user is prompted for the re-
quired data through a series of input
commands.
Once the basic input data have been
entered, the problem as currently de-
fined is listed and the program enters
the "edit" mode. The two character edit
commands listed in Table 1 can be used
to redefine the problem, run the calcula-
tions, and terminate the program.
The program has been written to re-
quire a minimum of machine resources
and will run on both 8 and 16 bit micro-
computers under CP/M, MS-DOS, and
PC-DOS as well as larger minicomput-
ers and mainframe machines.
Summary
The models and computer codes de-
veloped in this project are intended to
serve as additional tools in the analysis
of ground-water contamination prob-
lems. The user must select the best tool
for the problem at hand based on a
sound understanding of the principles
of ground-water hydrology, the physi-
cal problem, and the limitations of the
nathematical model(s). Unfortunately,
Table 1. Edit Commands
Command Variable changed/Execution
ST
PO
VX
RD
DE
DX
DY
DZ
RT
OB
XC
YC
ZC
TC
CS
MU
LI
RN
NP
DN
Saturated Thickness
Porosity
New Seepage Velocity
Retardation Coefficient
Decay Constant
X-Dispersion Coefficient
V'-Dispersion Coefficient
Z-Dispersion Coefficient
Source Rate Schedule
Observation Points
X-Coordmates
Y-Coordinates
Z-Coordinates
Observation Times
Change Solution/Sources
Menu of Edit Commands
List input data
Run
New Problem
Done
these computer programs cannot sub-
stitute for an understanding of the pro-
cesses and mechanisms of solute trans-
port in ground-water systems or sound
judgement based on training and expe-
rience.
References
Hantush, M. S., 1956, "Analysis of Data
from Pumping Tests in Leaky
Aquifers," Transitions, American
Geophysical Union, Vol. 37, No. 6, pp.
702-714.
Hunt, B., 1978, "Dispersive Sources in
Uniform Ground-Water Flow," Jour-
nal of The Hydraulics Division, ASCE,
Vol. 104, No. HY 1, pp. 75-85.
Wilson, J. L. and P. J. Miller, 1978,
"Two-Dimensional Plume in Uniform
Ground-Water Flow," Journal of the
Hydraulics Division, ASCE, Vol. 104,
No. HY4, pp. 503-514.
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Jan Wagner, S. A. Wans, and Douglas C. Kent are with Oklahoma State
University, S til/water, OK 74078.
Carl G. Enfield is the EPA Project Officer (see below).
The complete report, entitled "PLUME 2D: Two-Dimensional Plumes in Uniform
Ground Water Flow."(Order No. P8 85-214 450/AS; Cost: $11.50. subject to
change) will be available only from:
National Technical Information Service
5285 Port Royal Road
Springfield, VA 22161
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
United States
Environmental Protection
Agency
Center for Environmental Research
Information
Cincinnati OH 45268
Official Business
Penalty for Private Use $300
EPA/600/S2-85/065
ECTS SGENCY
CHICAGO
U.S. GOVERNMENT PRINTING OFFICE:1985—559-Qlb/27117
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