``` '/ 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 ``````------- 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 ``````------- 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 ``````------- 4 United States Environmental Protection Agency Center for Environmental Research Information Cincinnati OH 45268 Official Business Penalty for Private Use \$300 EPA/600/S2-85/067 QCQC329 PS U S ENVIR PROTECTION AGENCY REGION 5 LIBRARY 230 S DEARBORN STREET CHICAGO IL 60604 ````------- `