'/
 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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