This report describes a particle tracking post-processing package, MODPATH/ MODPATH-PLOT, that computes and displays three-dimensional pathlines based on output from MODFLOW, the U. S. Geological Survey finite-difference groundwater flow model. MODPATH is intended for general use and may have to be modified by the user for specific applications. The methods used by MODPATH are based on specific assumptions that result in limitations that must be thoroughly understood to obtain meaningful results. Users are strongly encouraged to carefully read Chapter 2 (Particle Tracking Methodology) before undertaking analyses with MODPATH.

A semianalytical particle tracking method was developed for use with velocities generated from block centered finite‐difference ground‐water flow models. The method is based on the assumption that each directional velocity component varies linearly within a grid cell in its own coordinate directions. This assumption allows an analytical expression to be obtained describing the flow path within an individual grid cell. Given the initial position of a particle anywhere in a cell, the coordinates of any other point along its path line within the cell, and the time of travel between them, can be computed directly. For steady‐state systems, the exit point for a particle entering a cell at any arbitrary location can be computed in a single step. By following the particle as it moves from cell to cell, this method can be used to trace the path of a particle through any multidimensional flow field generated from a block‐centered finite‐difference flow model.

The illustration on the cover of this report shows the local, intermediate, and regional scale flow systems that develop in reponse to sinusoidal variations in water table elevation about an average linear regional gradient In this figure, a regional groundwater divide exists at the right boundary and the regional discharge area is in the upper left corner of the system. The analysis on which this illustration is based was originally done by Toth (1963) using analytical solutions for the groundwater flow equation. The cover illustration was generated by a particle tracking analysis based on a finite difference solution of case 2f in Toth's paper. The finite-difference solution used 100 cells in the vertical direction and 200 cells in the horizontal direction.

A complete understanding of multicomponent gas transport in porous media (unsaturated zones) requires a knowledge of Knudsen transport, the molecular and nonequimolar components of diffusive flux, and viscous (pressure‐driven) flux. The constitutive equations relating these flux components are available from the “dusty gas” model of Mason et al. (1967). This paper presents a brief discussion of the principles underlying each of the above flux mechanisms, illustrated with binary systems, and then casts the constitutive equations in forms thought to be most useful for the study of natural unsaturated zones. A derivation is presented showing that the constitutive equations maintain the same form when expressed in terms of the potentiometric head of a gas column in a gravitational field, a conclusion of considerable practical importance for the study of natural systems. Very small pressure gradients (1 Pa/m or less) can produce viscous fluxes greater than or equal to diffusive fluxes; conversely, pressure gradients of this magnitude can be generated by diffusive processes. The viscous and diffusive fluxes are coupled in the constitutive equations through the Knudsen diffusivities; a knowledge of Knudsen diffusivities is necessary to calculate the viscous component of flux and pressure gradients. Equations are derived allowing calculation of Knudsen diffusivities from measurements of the Klinkenberg effect. The accuracy of Fick's first law (and by inference, Fick's second law) is shown to depend primarily on the relative magnitudes of the viscous and diffusive flux components. Methods are presented for approximating these flux components, in order to determine whether the multicomponent equations are needed for a given problem. These estimates depend on a knowledge of concentration profiles of stagnant (zero flux) gases. To a first approximation, Ar and N2 are considered to be stagnant gases in subsoil environments. Fick's law(s) are, essentially by definition, inadequate to deal with stagnant gases. In the examples presented, the error associated with estimating total fluxes of nonstagnant gases by Fick's law ranges from a few percent to orders of magnitude. Other conclusions are as follows: (1) major gas concentration gradients can be produced solely by transport phenomena; (2) any study of natural systems that requires a knowledge of N2 (or Ar) fluxes, as opposed to assuming them to be stagnant, must be based on a multicomponent analysis; (3) any study of systems in which a nonatmospheric gas constitutes a significant fraction of the gases present must be based on multicomponent analysis; (4) concentration profiles of stagnant gases can be used to determine the direction, and semiquantitatively the magnitude, of the net gas flux into or out of unsaturated zones; (5) with permeability equal to 10−12 m2, for pressure gradients greater than 10 N m−3 viscous fluxes predominate, the Stefan‐Maxwell equations become inapplicable, and the general equations must be used; (6) the net gas flux may be dominated by viscous effe...

Techniques and Methods 6-A41Cover. The cover image shows the local, intermediate, and regional-scale flow systems that develop in response to sinusoidal variations in water-table elevation about an average linear regional gradient. A regional groundwater divide exists at the right boundary and the regional discharge area is in the upper left corner of the system. The analysis upon which this illustration is based was originally presented by Toth (1963) using an analytical solution of the groundwater flow equation. The cover illustration was generated using MODFLOW and MODPATH for case 2f in Toth (1963). Although the system represents flow in a vertical cross section, it is simulated here by MODFLOW using a single horizontal model layer with 200 columns and 100 rows. The head boundary condition at the water table was applied to row 1 using the general head boundary package. The color shading represents the residence time of water. The youngest water is located in the tan regions at the top of the flow system. Residence time increases with depth and from right to left in the direction of groundwater flow. For more information on the USGS-the Federal source for science about the Earth, its natural and living resources, natural hazards, and the environment, visit http://www.usgs.gov or call 1-888-ASK-USGS. User Guide for MODPATHFor an overview of USGS information products, including maps, imagery, and publications, visit http://www.usgs.gov/pubprodTo order this and other USGS information products, visit http://store.usgs.gov Any use of trade, product, or firm names is for descriptive purposes only and does not imply endorsement by the U.S. Government.Although this report is in the public domain, permission must be secured from the individual copyright owners to reproduce any copyrighted materials contained within this report. AbstractMODPATH is a particle-tracking post-processing model that computes three-dimensional flow paths using output from groundwater flow simulations based on MODFLOW, the U.S. Geological Survey (USGS) finite-difference groundwater flow model. This report documents MODPATH version 6. Previous versions were documented in USGS Open-File Reports 89-381 and 94-464.The program uses a semianalytical particle-tracking scheme that allows an analytical expression of a particle's flow path to be obtained within each finite-difference grid cell. A particle's path is computed by tracking the particle from one cell to the next until it reaches a boundary, an internal sink/source, or satisfies another termination criterion.Data input to MODPATH consists of a combination of MODFLOW input data files, MODFLOW head and flow output files, and other input files specific to MODPATH. Output from MODPATH consists of several output files, including a number of particle coordinate output files intended to serve as input data for other programs that process, analyze, and display the results in various ways.MODPATH is written in FORTRAN and can be compiled by any FORTRAN compiler that fully supports FORTRAN-2003 or...