NAMEmodfe - Modular finite-element model for areal and axisymmetric ground-water flow problemsABSTRACTThis MODular, Finite-Element digital-computer program (MODFE) was developed to provide solutions to ground-water-flow problems based on the governing equations that describe two-dimensional and axisymmetric-radial flow in porous media. The documentation is divided into three parts. Part 1 (Torak, L.J., 1993a) is the user's manual that describes hydrologic features and simulation capabilities of MODFE. Descriptions are given for preparing hydrologic data to characterize aquifer properties and boundary conditions by zone. Examples of data input and model output are provided to demonstrate the different types of ground-water problems that are solved by using the simulation capabilities of MODFE. Guidelines for designing the finite-element mesh and for node numbering and determining bandwidths are given to instruct users in the appropriate application of MODFE to ground-water problems of their choosing. Part 2 (Cooley, R.L., 1992) derives the finite-element equations by minimizing a function of the difference between the true and approximate hydraulic head, which produces equations that are equivalent to those obtained by either classical variational or Galerkin techniques. Spatial finite elements are triangular with linear basis functions, and temporal finite elements are one dimensional with linear basis functions. Comparison of finite- element solutions with analytical solutions is given for five example problems. Part 3 (Torak, L.J., 1993b) contains descriptions of subroutines, programming details, and program structure diagrams. Descriptions of subroutines that execute the computational steps of the modular- program structure are given in tables that cross reference the subroutines with particular versions of MODFE. Programming details of linear and nonlinear hydrologic terms are provided. Structure diagrams for the main programs show the order in which subroutines are executed for each version and illustrate some of the linear and nonlinear versions of MODFE that are possible. Computational aspects of changing stresses and boundary conditions with time and of mass balance and error terms are given for each hydrologic feature. Program variables are listed and defined according to their occurrence in the main programs and in subroutines. Listings of the main programs and subroutines are given.METHODAquifer geometry, flow boundaries, and variations in hydraulic properties are represented by triangular elements or element sides in a finite-element mesh. Time variations in hydraulic properties are represented by one-dimensional elements. Linear coordinate functions are used to approximate variations in hydraulic properties within elements. The finite-element matrix equations are solved by using either a direct symmetric-Doolittle method of triangular decomposition or an iterative method that uses the modified, incomplete-Cholesky, conjugate-gradient method. The direct method can be efficient for small- to medium-sized problems (less than about 500 nodes), and the iterative method is generally more efficient for larger-sized problems. Simulation capabilities and uses of MODFE are: transient or steady- state conditions; nonhomogeneous and anisotropic flow where directions of anisotropy change within the model region; vertical leakage from a semiconfining layer that contains laterally nonhomogeneous properties and elastic storage effects; point and areally distributed sources and sinks, specified head (Dirichlet); specified flow (Neumann); and head-dependent (Cauchy-type) boundary conditions; vertical cross-section and axisymmetric cylindrical flow; confined and unconfined (water-table) conditions; partial drying and resaturation of a water-table aquifer; conversion between confined- and unconfined-aquifer conditions; and nonlinear head- dependent fluxes (for simulating line, point, or areally distributed sources and sinks). Aquifer stresses and boundary conditions can be changed on a time-step basis, a stress-period basis, or both. Hydraulic properties and boundary conditions can be input by zone.HISTORYDocumentation for MODFE was developed on the basis of material presented within the period 1982-92 by the authors in the course held at the USGS National Training Center (see TRAINING).DATA REQUIREMENTSAquifer and confining-bed characteristics, two-dimensional space coordinates in Cartesian or radial (r-z) system, and time-step sizes (see DOCUMENTATION).OUTPUT OPTIONSHydraulic head and water-balance summary information of each time step (not an option), suppression of input data, and output of nodal flow rates from vertical leakage and boundary condition zones.SYSTEM REQUIREMENTSMODFE is written in Fortran 77. Generally, the program is easily adapted to most computer systems. The code has been used on UNIX- based computers.APPLICATIONSBuxton, H.T., and Modica, E., 1992, Patterns and rates of ground- water flow on Long Island, New York: Ground Water, v. 30, no. 6, p. 857-866. (Solutions of stream and potential functions, cross- section simulations, flow-net analysis) Czarnecki, J.B., and Waddell, R.K., 1984, Finite-element simulation of ground-water flow in the vicinity of Yucca Mountain, Nevada- California: U.S. Geological Survey Water-Resources Investigations Report 84-4349, 38 p. (Spring flow, water-table conditions, recharge) Iverson, R.M., and Reid, M.E., 1992, Gravity-driven groundwater flow and slope failure potential, 1. Elastic effective-stress model: Water Resources Research, v. 28, no. 3, p. 925-938. (Ground- water-flow field, total-body-force field, and effective-stress field generated in cross section showing that ground-water flow can influence shear stresses as well as effective-normal stress on hill slopes) Lowther, R.A., and Kuniansky, E.L., 1992, Documentation of finite- element mesh generation programs using a geographic information system: U.S. Geological Survey Water-Resources Investigations Report 92-4155, 187 p. (GIS applications programs written in ARC5) Reid, M.E., and Iverson, R.M., 1992, Gravity-driven groundwater flow and slope failure potential, 2. Effects of slope morphology, material properties, and hydraulic heterogeneity: Water Resources Research, v. 28, no. 3, p. 939-950. (Sensitivity analysis of hydraulic conductivity contrasts and their effects on ground-water seepage forces, effective stresses, and slope- failure potentials) Torak, L.J., Davis, G.S., Herndon, J.G., and Strain, G.A., 1992, Geohydrology and evaluation of water-resource potential of the upper Floridan aquifer in the Albany area, southwestern Georgia: U.S. Geological Survey Water-Supply Paper 2391, 59 p. (Model application to well-field development, calibration, sensitivity analysis, flow-vector analysis) Torak, L.J., Davis, G.S., Strain, G.A., and Herndon, J.G., 1996, Geohydrology and Evaluation of Stream-Aquifer Relations in the Apalachicola-Chattahoochee-Flint River Basin, Southeastern Alabama, Northwestern Florida, and Southwestern Georgia: U.S. Geological Survey Water-Supply Paper 2460, 95 p. Torak, L.J., and McDowell, R.J., 1996, Ground-Water Resources of the Lower Apalachicola-Chattahoochee-Flint River Basin in Parts of Alabama, Florida, and Georgia--Subarea 4 of the Apalachicola- Chattahoochee-Flint and Alabama-Coosa-Tallapoosa River Basins: U.S. Geological Survey Open-File Report 95-321, 145 p. Reports in review: Maslia, M.L., Prowell, D.C., and Jones, L.E., Effect of faults on fluid flow and chloride contamination in the Floridan aquifer system, Brunswick, Glynn County area, Georgia: interpretation of field data, conceptual model development, and numerical simulation: U.S. Geological Survey Water-Supply Paper in colleague review. Torak, L.J., Computational Extensions of a MODular Finite-Element Model (MODFE) for Confined Multilayer Ground-Water-Flow Problems: U.S. Geological Survey Open-File Report in colleague review. Torak, L.J., Davis, G.S., Herndon, J.G., and Strain, G.A., 1996, Geohydrology and evaluation of stream-aquifer relations in the lower Apalachicola-Chattahoochee-Flint River Basin, southeastern Alabama, northwestern Florida, and southwestern Georgia: U.S. Geological Survey Water-Supply Paper 2460, 94 p.DOCUMENTATIONCooley, R.L., 1992, A MODular Finite-Element model (MODFE) for areal and axisymmetric ground-water-flow problems, part 2--derivation of finite-element equations and comparisons with analytical solutions: U.S. Geological Survey Techniques of Water-Resources Investigations, book 6, chap. A4. Torak, L.J., 1993a, A MODular Finite-Element model (MODFE) for areal and axisymmetric ground-water-flow problems, part 1--model description and user's manual: U.S. Geological Survey Techniques of Water-Resources Investigations, book 6, chap. A3. Torak, L.J., 1993b, A MODular Finite-Element model (MODFE) for areal and axisymmetric ground-water-flow problems, part 3--design philosophy and programming details: U.S. Geological Survey Techniques of Water-Resources Investigations, book 6, chap. A5.RELATED DOCUMENTATIONTorak, L.J., 1992a, A MODular Finite-Element model (MODFE) for areal and axisymmetric ground-water-flow problems, part 1--model description and user's manual: U.S. Geological Survey Open-File Report 90-194, 153 p. Torak, L.J., 1992b, A MODular Finite-Element model (MODFE) for areal and axisymmetric ground-water-flow problems, part 3--design philosophy and programming details: U.S. Geological Survey Open- File Report 91-471, 261 p.REFERENCESTorak, L.J., Davis, G.S., Strain, G.A., and Herndon, J.G., 1991, Geohydrology and evaluation of water-resource potential of the upper Floridan aquifer in the Albany area, southwestern Georgia: U.S. Geological Survey Open-File Report 91-52, 86 p. AND U.S. Geological Survey Water-Supply Paper 2391.TRAININGFinite-Element Modeling of Ground-Water Flow (GW2097TC), offered annually at the USGS National Training Center.CONTACTSOperation and Distribution: U.S. Geological Survey Hydrologic Analysis Software Support Program 437 National Center Reston, VA 20192 h2osoft@usgs.gov Official versions of U.S. Geological Survey water-resources analysis software are available for electronic retrieval via the World Wide Web (WWW) at: http://water.usgs.gov/software/ and via anonymous File Transfer Protocol (FTP) from: water.usgs.gov (path: /pub/software). The WWW page and anonymous FTP directory from which the MODFE software can be retrieved are, respectively: http://water.usgs.gov/software/modfe.html --and-- /pub/software/ground_water/modfe See http://water.usgs.gov/software/ordering_documentation.html for information on ordering printed copies of USGS publications.SEE ALSOmoc(1) - Two-dimensional method-of-characteristics ground-water flow and transport model modflow(1) - Modular three-dimensional ground-water flow model Modflowp(1) - Parameter-estimation version of the modular model sutra(1) - Saturated and (or) unsaturated, constant or variable-density fluid flow, and solute or energy transport (2-dimensional finite-element code) vs2di(1) - A graphical software package for simulating fluid flow and solute or energy transport in variably saturated porous media

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