'/
United States
Environmental Protection
Agency
Robert S. Kerr Environmental
Research Laboratory
Ada OK 74820
Research and Development
EPA/600/S2-85/067 Aug. 1985
Project Summary
Plume 3D: Three-Dimensional
Plumes in Uniform Ground
Water Flow
Jan Wagner, S. A. Watts, and D. C. Kent
A closed-form analytical solution for
three-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.
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
The full document describes a mathe-
matical model and the associated com-
puter program that can be used to esti-
mate concentration and distributions in
a leachate plume which emanates from
one or more point sources. The model
includes both linear adsorption and
first-order reactions.
The use of the computer program is
fairly simple but represents only one
tool which can aid in the analysis and
understanding of ground-water con-
tamination problems. The user must
select the appropriate tools for the prob-
lem at hand, based on a sound under-
standing of the principles of ground-
water hydrology, the physical problem,
and the assumptions and limitations of
the mathematical model.
Model Formulation
The differential equation describing
the conservation of mass of a compo-
nent in a saturated, homogeneous
aquifer with uniform, steady flow in the
x-direction can be written as
D f*C .,* r)C _ n* H2C
Rd-^f + V — - Dx —
-RdAC (1)
where
C = component mass per
unit of fluid phase M/L3
D* = dispersion coefficient
in x-direction L2/t
D* = dispersion coefficient
in y-direction L2/t
D* = dispersion coefficient
in z-direction L2/t
Rd = retardation coefficient
V* = average interstitial
velocity in x-direction L/t
x,y,z = rectangular coordinates L
X = first-order decay
constant 1/t
The retardation coefficient accounts for
partitioning of the component between
the fluid and solid phases using a linear
adsorption isotherm and is defined as
Rd = 1 + ~ K
where
PB = bulk density of the
aquifer
0 = effective porosity
Kd = distribution constant for
a linear adsorption
isotherm
(2)
M/L3
M/M
M/L3
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A closed-form analytical solution to
Equation 1 for an infinite aquifer with a
continuous point source of strength M0
at the origin can be written as (Hunt,
1978; Turner, 1972)
Cc =
V Dy D2
Lp(15U)erfc(;^±5
[ "2Dx; \2VRdD*
(2)
vRdDxt
where
and
= X2+
2 + Rx
DVY D:
1/2
1/2
(3)
(4)
The steady-state solution for a contin-
uous point source is (Hunt, 1978)
C0Q
™-^r <5)
Equations 2 and 5 can be used to cal-
culate the concentrations in a leachate
plume under the following assumptions
and limitations:
1. The ground-water flow regime is
completely saturated.
2. All aquifer properties are constant
and uniform throughut the aquifer.
3. The ground-water flow is horizon-
tal, continuous, and uniform
throughout the aquifer.
4. The aquifer is infinite in extent.
5. The leachate source is a point lo-
cated at the origin of the coordi-
nate system.
6. The mass flow rate of the source is
constant.
7. At zero time the concentration of
leachate in the aquifer is zero.
The assumptions of an infinite aquifer
depth and a uniform source mass rate
can be overcome by using the principles
of superposition in space and time, re-
spectively (Walton, 1962). Both of these
provisions have been incorporated in
the computer program developed in
this project. Superposition is also used
to include multiple 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 that may be required to
implement the code on a given system,
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 listed in Table 1 are required to ini-
tiate a new problem. The user is
prompted for the required data through
a series of input commands.
Table 1. Input Data Required for the An-
alytical Three-Dimensional
Plume Model
Title - Units for length, time, and con-
centration
Saturated thickness {for aquifer of fi-
nite depth)
Effective porosity
Ground water interstitial velocity
Retardation coefficient
Longitudinal dispersion coefficient
Transverse dispersion coefficient
Vertical dispersion coefficient
First-order decay constant
Type of solution (transient or steady-
state)
Number of sources
Location and rate schedules for each
source
Coordinates of observation points
Observation times (for transient solu-
tion)
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 2 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
mathematical model(s). Unfortunately,
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.
Table 2.
Edit Commands
Command
ST
PO
VX
RD
DE
DX
DY
DZ
RT
OB
XC
ZC
YC
TC
AS
CS
MU
LI
RN
NP
DN
Variable changed/Execution
Saturated Thickness
Porosity
New Seepage Velocity
Retardation Coefficient
Decay Constant
X-Dispersion Coefficient
Y-Dispersion Coefficient
Z-Dispersion Coefficient
Source Rate Schedule
Observation Points
X-Coordinates
Z-Coordinates
Y-Coordinates
Observation Times
Aquifer Sectioning
Change Solution/Sources
Menu of Edit Commands
List input data
Run
New Problem
Done
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References
Hunt, B., 1978, "Dispersive Sources in
Uniform Ground-Water Flow," Jour-
nal of The Hydraulics Division, ASCE,
Vol. 104, No. HY1, pp. 75-85.
Turner, G. A., 1972, Heat and Concentra-
tion Waves, Academic Press, New
York, New York, 233 pp.
Walton, W. C., 1962, "Selected Analyti-
cal Methods for Well and Aquifer
Evaluation," Bulletin 49, Illinois State
Water Survey, Urbana, Illinois, 81 pp.
Jan Wagner. S. A. Watts, and Douglas C. Kent are with Oklahoma State
University, Stillwater, OK 74078.
Carl G. Enfield is the EPA Project Officer (see below).
The complete report, entitled "PLUME 3D: Three-Dimensional Plumes in Uniform
Ground Water How, "(Order No. PB 85-214 443/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
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Agency
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