Water Resources Applications Software

Summary of FEQ


       feq - Full EQuations Model
       fequtl - Full EQuations UTiLity

       FEQ simulates flow in a stream system by solving the full, dynamic
       equations of motion for one-dimensional unsteady flow in open
       channels and through control structures.  The structure of the
       program is designed to follow the structure of a stream system while
       providing maximum generality and flexibility of description.  A
       stream system that is simulated with FEQ is subdivided into three
       broad classes of flow paths: (1) stream reaches (branches), (2)
       parts of the stream system for which complete information on flow
       and depth are not required (dummy branches), and (3) level-pool
       reservoirs.  These components are connected by special features or
       hydraulic control structures, such as junctions, bridges, culverts,
       dams, waterfalls, spillways, weirs, side weirs, pumps, and others.
       The hydraulic characteristics of channel cross sections and special
       features are stored in function tables calculated by the companion
       program FEQUTL.  FEQ can interpolate hydraulic properties of cross
       sections between measured sections.  FEQ can be applied in the
       simulation of a wide range of stream configurations (including
       loops), lateral-inflow conditions, and special features.  Boundary
       conditions can be water-surface stage, discharge, or the stage-
       discharge relationship at a node.  Wind stress terms are supported.
       The effects of lateral inflows can also be simulated in FEQ when
       given local runoff intensity data.

       In FEQ, the principles of conservation of mass and conservation of
       momentum are used to calculate the flow and depth throughout the
       stream system resulting from known initial and boundary conditions
       with an implicit finite-difference approximation.  FEQUTL is used to
       compute the hydraulic properties of various structures, each with
       its own computational theory.  Thorough discussion of computational
       theory is given in the sources listed under DOCUMENTATION.

       The Full Equations (FEQ) model for the simulation of one-dimensional
       unsteady flow in open channels and through control structures was
       first developed in 1976 to model the Sanitary and Ship Canal,
       Chicago, Ill., as part of the 208 Water Quality Management Studies
       conducted by the Northeastern Illinois Planning Commission with
       principal support from the U.S. Environmental Protection Agency.
       This original version was then expanded over the next several years.
       The first numbered version, 2.0, appeared in March 1986 when a
       version for personal computers was prepared.  In subsequent years,
       the capabilities of the software were expanded at frequent intervals
       to meet the needs of the expanding user base.  The USGS published
       documentation for version 8.10 in 1997.  The later features of FEQ 
       are documented below.  Refer to the file RELEASE.TXT distributed 
       with the software for additional information on these enhancements 
       and code corrections.
       Version 9.98, posted 2005/05/31 - (A) Generalized effect of detention 
          of flow within reservoirs and delay of flow from tributary areas
          enabled.  (B) Control of gate operations as a function of 3 
          exterior nodes.  (C) Irrigation runoff and withdrawal effects 
          from near-surface and imported water.  (D) Table numbers changed
          to labels resulting in heading-dependent formats for many blocks.
          Convertfeq utility available for conversion.  (E)Rainfall and 
          evaporation from open surfaces.  (F) Adjustment factor for 
          conveyance in a branch.  (G) Crest and flow adjustment for side 
          weirs.  (H) Master files and scenario-selector variable option for 
          complex project management.  (I) Automatic counting of nodes and 
          branches, disabling of weirs with datum errors.  (J) Formats to 
          facilitate conversion between linux and dos operating systems 

       Version 8.92, posted 1999/04/07 - (A) The negative constant-flow boundary
          was corrected from an applied constant value of 0.  Positive
          constant flow boundaries into and out of the system were
          unaffected.  (B) An undefined variable was defined to end
          potential error in the interpolation between weir and orifice
          flow for underflow gates.  (C) An undefined variable that caused
          potential error in checking for overflow in side-weir 2-D tables
          was defined.

       Version 8.92, posted 1997/06/18 - (A) Blank lines are now allowed 
          anywhere in the input except in the Branch Description Block where 
          they indicate repeated cross sections.  The blank lines are echoed to
          the output.  (B) FORTRAN I/O numbers are no longer needed.  They
          can be left in old inputs however.  (C) Space-delimited format is
          allowed for the Network Matrix Control Input Block, when the
          block is specified as NEW Network Matrix Control Input Block.  A
          asterisk is used as a place holder for optional items.  (D) The
          Special Output Block has been modified to allow user-customized
          specification of output variables or single-line output.  The new
          values are given in an OPTIONS line following the UNIT line.  The
          variables that may be output include mean velocity (V), cross-
          sectional area (A), main channel area (MCA), flow (MCQ), and
          velocity (MCV), flood plain flow area (FPA), flow (FPQ), and
          velocity (FPV).  In order to have single-line special output, the
          Special Output Block is specified as "Special" rather than
          "SPECIAL."  Only flow and velocity will be output.  (E) Access to
          HECDSS time series is supported for the input of flow or
          elevation at a boundary node, for output of flow or elevation
          from any node, and for the unit-area runoff intensity used to
          compute lateral inflow from a tributary area.  (F) Explicit
          specification of side nodes (Code 13) is allowed to provide for
          conservation of momentum for inflows.  The angle of entry is also
          specified.  (G) The difference in elevation between two exterior
          nodes can be used as the argument for control structures in the
          Operations Control Block.  The difference from the NODE elevation
          is specified by identifying the second node in the KEY variable
          field.  (H) An optional Define Macros and Instructions Block has
          been added to streamline the definition of the Network Control
          Matrix.  (I) Two-way pumps have been added.  (J) An argument
          scale factor has been added for function tables type 2,3, and 4
          to allow for variable unit systems.  (K) A linear reservoir delay
          factor for tributary runoff areas can be specified by the
          variable KLR added at the end of the branch specification line.
          KLR has the units of minutes delay time for a linear reservoir
          interposed between the branch and the tributary area.

       The Full Equations Utilities (FEQUTL) model for the approximation of
       hydraulic characteristics of open channels and control structures
       during unsteady flow was first developed in early 1984-85 using some
       ideas from earlier piecemeal utility programs.  The first numbered
       versions appeared in early 1988 as the user base expanded.  The
       software has been expanded at frequent intervals to meet the needs
       of the user base.  The U.S. Geological Survey published the
       documentation for FEQUTL Version 4.15 in 1997.  Many additional
       capabilities have been added to FEQ and FEQUTL during the time the
       documentation was prepared.  The capabilities added to FEQUTL are
       described below.  Refer to the file RELEASE.TXT distributed with the
       software for additional information on these enhancements and code

       Version 5.46, posted 2005/05/31 - (A) Gate control table development 
          from underflow-gate rating partly supported in routine--INV_GATE.
          (B) Sluice gate on upstream face of box culvert routine--UFGCULV.
          (C) Support for function-table labels rather than function-table
          numbers, requires conversion to heading-dependent formats in many
          routines, see release.txt for details.  Conversion program,
          convertutl, available.  (D) Exact specification of conversion 
          factors for Manning's n and the acceleration due to gravity 
          available.  (E) Cross-section conversion from FEQUTL to WSPRO
          format.  (F) Estimate of surface area for level-pool reservoirs
          where only elevation-storage capacity relation is know.  (G) 
          Formats to facilitate conversion between linux and Dos operating
          systems incorporated.
       Version 4.68, posted 1997/05/29 - (A) SI units are now supported. The
          subdirectory METRIC under the test directory in FEQUTL contains
          the SI version of the standard example file and the SI version of
          the weir coefficients for embankment-shaped weirs.  (B) Added
          another global convergence tolerance to the header block for
          FEQUTL. The new convergence tolerance, EPSABS, is to be used in
          those cases in which the residual function returns a length
          value.  Thus when switching to using meters for the length unit,
          EPSF will remain unchanged but EPSABS must be changed to reflect
          the larger length unit.  In the US standard unit system, EPSABS
          has the same numeric value as EPSF but has a different meaning.
          EPSABS is located in the next field from EPSF.  If omitted, the
          value of EPSF is used.  (C) Modified the Preissman slot for
          closed conduits to reflect the unit system. The maximum slot
          level is 150 meters, which is not exactly equal to 500 feet used
          in the US unit system.  However, the round number is indicative
          of an arbitrary selected value. The slot detection code was
          modified also to find the vertical diameter of closed conduits.
          The slot width used for detection of closed conduits remains at
          0.07 feet or 0.021336 meters.  A slot width larger than this will
          not be detected and FEQUTL will treat the cross-section function
          table as being a normal open channel and not a closed conduit in
          any context in which a closed conduit must be detected.  (D)
          Changed the means for eliminating close values of depth in
          computing cross section tables.  Previous versions had used an
          absolute tolerance for the minimum difference between adjacent
          depths.  This has been changed to a relative tolerance to be
          scale independent.  (E) HEC2X command has been modified to
          convert units from SI to English or from English to SI under user
          control.  The default action is no conversion of the elevations
          and offsets on the cross section.  Adding the word CONVERT after
          the MODE response will cause conversion of units.  The conversion
          of station values is governed by SFAC only and is set by the
          user.  (F) Provided additional options following the unit system
          selection in the standard header to force FEQUTL to use the more
          exact value for the factor in Manning's equation.  The factor is
          technically the cubic root of the number of feet in a meter,
          which to single precision in a 32-bit IEEE floating point
          representation is about 1.485919. For nearly all practical
          purposes this can be taken as 1.49.  (G) Provided the option to
          use an equation to compute the value of g given a latitude and an
          elevation.  This happens whenever the exact value for the factor
          in Manning's equation is requested.  The usual value of g is 32.2
          f/s^2 or  9.815 m/s^2 with an error less than 0.2 percent across
          the United States.  Using more than one value of GRAV may result
          in slight differences between SI and US standard units.  (H)
          Added vertical scale factor (VSCALE), and horizontal shift amount
          (HSHIFT), to FEQX cross sections.  (I) Added vertical scale
          factor (VSCALE), vertical shift (CSHIFT), horizontal scale factor
          (HSCALE) to EMBANKQ.  (J) Added an argument scale factor to the
          input of function tables of type 2, 3,and 4.  The argument scale
          factor is placed after the function scale factor.  This moves the
          SHIFT input item to the right.  The SHIFT item is little used and
          may be discontinued.  In any case if you are using it you will
          have to move the SHIFT item to the right by 16 columns to leave
          space for the reading of the argument scale factor.  (K) Added
          two new commands to create a bottom slot in a cross section.  The
          commands are SETSLOT and CLRSLOT.   The first command defines a
          bottom slot and this slot is add to all cross sections that
          FEQUTL encounters until the command CLRSLOT is found.  Thus the
          addition of a bottom slot to a cross section is like a switch:
          either on or off.  When it is on it will appear in all cross
          sections processed.  (L) Flapgate losses from submerged culvert
          flow are supported.  Culverts with risers and flow through
          orifices are also supported.  (M) CHANRAT and EMBANKQ have
          optional variables LIPREC and MINPFD to request optimization of
          interpolation of two-dimensional tables of type 6 or 13.  LIPREC
          is the Linear Interpolation Precision specification in terms of
          relative error.  MIPFD is the minimum partial free drop to be
          computed.  (N) Computation of pump rating curves and pump loss
          tables supported.

       FEQ reads an input file that contains specifications of run control
       parameters, an encoding of the stream schematic, and initial
       conditions.  This file can contain boundary-condition tables and
       function tables for special features, or it can identify additional
       files that contain the information.

       FEQUTL computes function tables from specifications and data
       provided in input files.  FEQUTL can read HEC-2 and WSPRO cross-
       section input data and calculate cross-section function tables for
       use in FEQ simulation.  Function tables for bridges are computed
       using the program WSPRO to compute a suite of upstream and
       downstream water-surface elevations.  FEQUTL can create input files
       for WSPRO and convert tables output by WSPRO into a format suitable
       for FEQ.

       The FEQ simulation process is documented in an output file.  In
       addition, the results of simulation at nodes selected by the modeler
       can be sent to data output files, special output files, or files
       readable by the USGS program GenScn.

       FEQUTL creates two output files, one which documents the computation
       processes in detail, and the other which contains only function
       tables.  The latter file can be identified in an FEQ input file to
       be read directly by FEQ.

       The program was written in Fortran 77 for PC.  The code is easily
       ported to Unix and linux systems.  Both PC and UNIX versions are 

       Snohomish County Department of Public Works, Surface Water
          Management Division, 1989, Snohomish River Unsteady Flow Model
          (FEQ):  Report submitted to the U.S. Army Corps of Engineers,
          Seattle District.  The model developed for the Snohomish River has
          been extended to include the lower portions of the Skykomish and
          Snoqualmie Rivers.  This model will eventually become part of a
          flood-forecasting system.

       Northwest Hydraulic Consultants Inc., 1993, Mill Creek (Auburn)
          Hydraulic Modeling, Report to King County, Division of Surface
          Water Management, Seattle, Washington.

       Model of Mississippi River from Keokuk, Iowa, to Thebes, Illinois,
          including its major tributaries as well as most of its minor
          tributaries.  About 600 miles of stream are represented.  Seven
          dams and six sets of operable gates, under automatic program
          control, are included.  A report is in progress.

       Johnstown Flood of 1977.  FEQ was applied to this flood as part of a
          legal action stemming from this flood.  The lower reaches of the
          Little Conemaugh and Stony Brook as well as the upper reaches of
          the Conemaugh River are simulated.

       DuPage County, Illinois Stormwater Management Plan.  As part of
          their overall planning and regulation effort, DuPage County has
          applied FEQ/FEQUTL to a wide variety of streams in DuPage County.
          These streams include Winfield Creek, Waubaunsee Creek, Salt
          Creek, East Branch DuPage River, Klein Creek, Black Partridge
          Creek, Willoway Brook, and numerous other streams.

       Illinois Department of Transportation, Division of Water Resources.
          The Bureau of Planning has developed models used for planning and
          regulation for the Fox River, Farmer-Prairie Creek, Midlothian
          Creek, and the Skokie Lagoons.  The Fox River model includes
          dynamic modeling of alternative dam-gate control operations.

       Franz, D.D., and Melching, C.S., 1997, Full Equations (FEQ) model
          for the solution of the full, dynamic equations of motion for
          one-dimensional unsteady flow in open channels and through
          control structures: U.S. Geological Survey Water-Resources
          Investigations Report 96-4240, 258 p.

       Franz, D.D., and Melching, C.S., 1997,  Full Equations Utilities
          (FEQUTL) model for the approximation of hydraulic characteristics
          of open channels and control structures during unsteady flow:
          U.S. Geological Survey Water-Resources Investigations Report
          97-4037, 205 p.

       Franz, D.D., 1982, Tabular representation of cross-sectional
          elements: Journal of the Hydraulics Division, American Society of
          Civil Engineers, v. 108, no. 10, p. 1070-1081.

       Ishii, A.L., and Turner, M.J., 1997, Verification of a one-
          dimensional, unsteady flow model for the Fox River in Illinois:
          U.S. Geological Survey Water-Supply Paper 2477, 65 p.

       Ishii, A.L., and Wilder, J.E., 1993, Effect of boundary condition
          selection on unsteady-flow model calibration, in Proceedings of
          the XXV Congress of International Association for Hydraulic
          Research, Tokyo, p. 193-200.

       Knapp, H.V., and Ortel, T.W., 1992, Effect of Stratton Dam operation
          on flood control along the Fox River and Fox Chain of Lakes:
          Illinois State Water Survey Contract Report 533, 79 p.

       Turner, M.J., 1994, Data-collection methods and data summary for
          verification of a one-dimensional, unsteady-flow model of the Fox
          River in Illinois: U.S. Geological Survey Open-File Report
          93-483, 40 p.

       Turner, M.J., Pulokas, A.P., and Ishii, A.L., 1996, Implementation
          and verification of a one-dimensional, unsteady-flow model for
          Spring Brook near Warrenville, Illinois:  U.S. Geological Survey
          Water-Supply Paper 2455, 35 p.

       One-week courses in FEQ modeling are offered about once a year in
       DuPage County as demand dictates.  Contact the local American
       Society of Civil Engineers section or:
          Department of Environmental Concerns
          DuPage County Center
          421 N. County Farm Road
          Wheaton, IL 60187

       Additional courses at other locations are offered but only as demand

          U.S. Geological Survey
          Illinois Water Science Center
          Watershed and River Modeling Support 
          1201 W. University Avenue
          Urbana, IL  61801

       Official versions of U.S. Geological Survey water-resources analysis
       software are available for electronic retrieval via the World Wide
       Web (WWW) at:


       and via anonymous File Transfer Protocol (FTP) from:

         (path: /pub/feq).

       The WWW page and anonymous FTP directory from which the FEQ software
       can be retrieved are, respectively:


       bltm(1) - Branched Lagrangian Transport Model
       branch(1) - One-dimensional Branch-network flow model
       daflow(1) - Streamflow routine in upland channels of
                   channel networks
       wspro88(1) - A computer model for Water-Surface PROfile
       GenScn(1) - GENeration and analysis of model simulation SCeNarios

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