(abstract from the Users' Manual)

The Full EQuations (FEQ) model is a computer program for solution of the full, dynamic equations of motion for one-dimensional unsteady flow in open channels and through control structures. A stream system that is simulated by application of FEQ is subdivided into stream reaches (branches), parts of the stream system for which complete information on flow and depth are not required (dummy branches), and level-pool reservoirs. These components are connected by special features; that is, hydraulic ontrol structures, including junctions, bridges, culverts, dams, waterfalls, spillways, weirs, side weirs, and pumps. 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 by means of an implicit finite-difference approximation at fixed points (computational nodes). The hydraulic characterstics of (1) branches including top width, area, first moment of area with respect to the water surface, conveyance, and flux coefficients and (2) special features (relations between flow and headwater and (or) tail water elevations, including the operation of variable-geometry structures) are stored in function tables calculated in the companion program, Full EQuations UTiLities (FEQUTL). Function tables containing other information used in unsteady-flow simulation (boundary conditions, tributary inflows or outflows, gate settings, correction factors, characteristics of dummy branches and level-pool reservoirs, and wind speed and direction) are prepared by the user as detailed in this report. In the iterative solution scheme for flow and depth throughout the stream system, an interpolation of the function tables corresponding to the computational nodes throughout the stream system is done in the model. FEQ can be applied in the simulation of a wide range of stream configurations (including loops), lateral-inflow conditions, and special features. The accuracy and convergence of the numerical routines in the model are demonstrated for the case of laboratory measurements of unsteady flow in a sewer pipe. Verification of the routines in the model for field data on the Fox River in northeastern Illinois also is briefly discussed.

The basic principles of unsteady-flow modeling and the relation between steady flow and unsteady flow are presented. Assumptions and the limitations of the model also are presented. The schematization of the stream system and the conversion of the physical characteristics of the stream reaches and a wide range of special features into function tables for model applications are described. The modified dynamic-wave equation used in FEQ for unsteady flow in curvilinear channels with drag on minor hydraulic structures and channel constrictions determined from an equivalent energy slope is developed. The matrix equation relating flows and depths at computational nodes throughout the stream system by the continuity (conservation of mass) and modified dynamic-wave equations is illustrated for four sequential examples. The solution of the matrix equation by Newton's method is discussed. Finally, the input for FEQ and the error messages and warnings issued are presented.

(Abstract from Proceedings of the First Federal Interagency Hydrologic Modeling Conference, held in Las Vegas, NV, April 19-23, 1998, prepared by the Subcommittee on Hydrology of the Interagency Advisory Committee on Water Data)

A near real-time flood-simulation system is being developed by the U.S. Geological Survey (USGS) in cooperation with Du Page County Department of Environmental Concerns for a 15-mile reach of Salt Creek in Du Page County, Ill. The Hydrologic Simulation Program-FORTRAN (HSPF) is being utilized to simulate rainfall-runoff for input to the Full EQations (FEQ) model for dynamic-wave routing. The meteorological inputs for the rainfall-runoff simulation are obtained by Internet access and radio-telemetered precipitation gages. Boundary conditions for the dynamic-wave routing model are obtained from telemetered stream-elevation gages and rating curves. The interface for data-base management, developing and processing simulation files, and analysis of simulation results is the program GENeration and analysis of model simulation SCeNarios (GENSCN).

The flood-simulation system is being developed to estimate the downstream effects of diverting streamflow into or out of the Elmhurst Quarry Flood Control Facility, located about 10 miles from the downstream boundary, under various real-time or forecasted rainfall distribution scenarios. The flood-wave characteristics of the stream system are highly dependent upon the distribution of rainfall in taime and space and, thus, the most effective management of diversions is dependent on the ability of facility managers to quickly simulate rainfall and snowmelt effects on the stream system.

The full benefit of understanding the complex model output from dynamic-wave routing can be realized only when the results can be quickly visualized and analyzed. Time series and model input files must be effeciently managed and the database made user friendly. Current operational procedures for the flood control facility are partly based upon operator intuition, which does not provide a systematic means to evaluate alternative operational schemes. The interfaced system makes it possible to test and compare various potential rainfall/diversion scenarios. This includes clickable maps, animated water-surface profiles, and a variety of graphical and analytical tools for evaluating the output scenarios.

¹US Geological Survey, Urbana, Ill.

²Du Page County Dept. of Environmental Concerns, Wheaton, Ill.

(abstract from US Geological Survey Water-Supply Paper 2455, 1996)

A one-dimensional, unsteady-flow model, Full EQations (FEQ) model, based on the Saint-Venant equations for dynamic flow in open channels, was calibrated and verified for a 0.75-mile reach of Spring Brook, a tributary to the West Branch Du Page River, near Warrenville in northeastern Illinois. The model was used to simulate streamflow in a small urban stream reach with two short culverts, one with overbank flow around the culvert during high flows. Streamflow data were collected on the reach during three high-flow periods. Data from one period were used to calibrate the model, and data from the other two periods were used to verify the model. Stages and discharges over the periods were simulated, and the results were compared graphically with stage and discharge data collected at 10 sites in the study reach. Errors in simulated stage and discharge were small except when debris, not represented in the model, clogged the culvert.

The effects of changes in physical and computational model parameters also were studied. The model was insensitive to replacement of measured cross sections with interpolated cross sections, especially if the measured thalweg elevations was preserved. Variation of the roughness, slope, and length of the culvert overbank section, caused only slight changes in the simulated peak stage and discharge. Changes in the modeled culvert area caused large differences in the simulated high flows in the vicinity of the culvert, whereas simulated low flows were unaffected. At all flows, the misrepresentation of the culvert area caused the simulated water-surface elevations to deviate from the measured elevations, especially on the falling limb of the stage hydrograph. The FEQ model, including the routines for modeling culvert and overbank flows, was evaluated as accurate and effective for this application.

¹US Geological Survey, Urbana, IL 61801

(abstract from US Geological Survey Water-Supply Paper 2477, 1996)

A one-dimensional, unstead-flow model utilizing the Full de Saint-Venant EQations (FEQ) for one-dimensional, unsteady flow in open channels was verified for a 30.6-mile reach of the Fox River in northeastern Illinois. The model, which was calibrated prior to the verification study by the Illinois Department of Natural Resources and the Illinois State Water Survey, was used to simulate a period of unsteady, within-bank flow induced by dam operations at the upstream end of the river reach, Stratton Dam near McHenry, Illinois, During November 1990. The river reach included three low-head dams that resulted in backwater effects where the channel slope was small. The river stages and streamflows imulated by the model, together with dye-injection rate and concentration data measured at Stratton Dam, were used as input for a transport model, the Branched Lagrangian Transport Model (BLTM). The simulation results from both models were compared graphically with stage, streamflow, and (or) dye-concentration data collected during the unsteady-flow period at a total of 31 downstream sites. The celerity of the induced low-flow wave was simulated accurately, with no significant error at any location. Differences during low-flow conditions between measured and simulated stage were less than about 0.2 foot at most of the sites, although differences up to 0.8 foot resulted at four sites where depths were shallow or head losses were inadequately represented through bridges. The differences may have resulted from the increase in effective roughness in the channel at very low depths that was not effectively modeled. Furthermore, accurate and representative measurements were difficult under some conditions of very low velocities or water-head buildup on the upstream side of bridges. The traveltime and concentration attenuation of the dye cloud were accurately simulated.

The effects of the physical and computational model parameters also were examined. The converged model was insensitive to distance-step and time-step size. The initial conditions were varied by 50 percent, and the simulated stage and discharge still converged to a common solution within twelve 1-hour time steps. The sensitivity of the model to geometric data was studied by replacing measured cross sections with interpolated cross sections within branches. The changes in distance-step size and geometric information had no effect on flood-wave celerity or discharge, but simulated stage was affected by how well the remaining cross sections represented local channel geometry. Deletion of bridge representations from the model caused no significant effects on the overall hydraulic routing, and only local effect on stage probably because the period simulated did not include high flow. Because of low-head controlling dams throughout the study reach, sensitivity to error in gage datum depended on the type of boundary condition used and whether the datum error was in the upstream or downstream boundary. The FEQ model was evaluated as accurate and robust for this application.

¹US Geological Survey, Urbana, IL 61801

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.

Snohomish County Public Works, 1989, Snohomish River unsteady flow model (FEQ)- Results of model calibration and reevaluation of the five-year + one foot level of protection: Surface Water Management Division, 52 p.

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