Transcription

Evaluation of Selected Dam-BreakFlood-Wave Models by Using Field DataU.S. GEOLOGICAL SURVEYWater-Resources Investigations 80-44

50272-101REPORT DOCUMENTATIONPAGE3. Recipient's Accession No.1. REPORT NO.5. Report Date4. Title and SubtitleEVALUATION OF SELECTED DAM-BREAK FLOOD-WAVE MODELS BYUSING FIELD DATAAugust 19806.8. Performing Organization Rept. No.7. Author(s)USGS/WRI-80-44Larry F. Land10. Project/Task/Work Unit No.9. Performing Organization Name and AddressU.S. Geological Survey, Water Resources DivisionGulf Coast Hydroscience CenterNational Space Technology LaboratoriesNSTL Station, MS 3952911. Contract(C) or Grant(G) No.(C)(G)13. Type of Report & Period Covered12. Sponsoring Organization Name and AddressU.S. Geological Survey, Water Resources DivisionGulf Coast Hydroscience CenterNational Space Technology LaboratoriesNSTL Station, MS 39529Final14.15. Supplementary Notes16. Abstract (Limit: 200 words)Four dam-break flood-wave models have been evaluated by using three field data setsand selected criteria of desirable features. The models include (1) modified Puls (MP),(2) U.S. Army Corps of Engineers' unsteady flow profiles (USTFLO), (3) National WeatherServices' dam-break flood forecast (DBFF), and (4) U.S. Geological Survey's coupled methodof characteristics and a general purpose streamflow simulation (MOC-J879DB). The fielddata sets documented the disasters at Teton Dam, Idaho, Laurel Run, Pa., and ToccoaFalls, Ga.The computed discharges were often within 20 percent of the observed values with theexception of the simulations at Teton Dam and for a short distance below the dams by the *MOC-J879DB. With the same exceptions noted above, the computed flood crests were usuallywithin 2 feet of the observed high-water marks.A modified version of the DBFF model is identified as the most accurate, economical,flexible, numerically stable, easiest to apply, and descriptive of the boundary andflow conditions.17. Document Analysisa. Descriptors*Dam failure, *Floods, *Model studies, Mathematical models, Hydraulic properties,Hydrodynamics, Continuity equations, Momentum equations, Flood routingb. Identifiers/Open-Ended TermsTeton Dam, Idaho, Laurel Run, Pennsylvania, Toccoa Falls, Georgiac. COSATI Field/Group18. Availability Statement19. Security Class (This Report)UNCLASSIFIED20. Security Class (This Page)No restriction on distribution.(See ANSI-Z39.18)21. No. of Pages6022. PriceUNCLASSIFIEDSee Instructions on ReverseOPTIONAL FORM 272 (4-77(Formerly NTIS-35)

EVALUATION OF SELECTED DAM-BREAK FLOOD-WAVE MODELS BY USING FIELD DATABy Larry F. LandU.S. GEOLOGICAL SURVEYWater-Resources Investigations 80-44July 1980I

UNITED STATES DEPARTMENT OF THE INTERIORCECIL D. ANDRUS, SecretaryGEOLOGICAL SURVEYH. William Menard, DirectorFor additional information write to:U.S. Geological Survey, WRDGulf Coast Hydroscience CenterNational Space Technology LaboratoriesNSTL Station, MS 39529II

CONTENTSPageConversion tableAbstractIntroductionMethod of investigationSelection of modelsField data setsTeton Dam, IdahoJohnstown, Pa.Toccoa Falls, Ga.Features of dam-break modelsMathematical modelsMP model- » «USTFLO modelDBFF modelMOC-J879DB modelsVI11223355899111214Characterization of test casesTeton DamMP modelUSTFLO modelDBFF modelMOC-J879DB modelsJohnstownMP modelDBFF modelMOC-J879DB modelsToccoa FallsMP modelDBFF modelMOC-J879DB modelsResultsTeton DamDischargesFlood profileTraveltimeJohnstownDischargesFlood profileTraveltimeToccoa FallsDischargesFlood profileTraveltimeDiscussionIdentification of general-purpose modelSummaryReferences citedIIIT*- 28283333333338393941

ILLUSTRATIONSPageFigures 1-3.4.5.6.7.8.9-11.12.13.14.Maps showing location of:1. Teton Dam, Idaho, study area2. Laurel Run, Pa., study area3. Toccoa Falls, Ga., study areaGraphs showing observed and computed peakdischarges along Teton RiverHydrographs showing computed discharge forthree sites along Teton RiverGraphs showing observed and computed high-watermarks along Teton RiverGraphs showing observed and computed peakdischarges along Laurel RunHydrographs showing computed discharge forthree sites along Laurel RunGraphs showing:9. Observed high-water marks and computedcrests for the upper part ofLaurel Run10. Observed high-water marks and computedcrests for the lower part ofLaurel Run11. Observed and computed peak dischargesalong Toccoa CreekHydrographs showing computed discharge forthree sites along Toccoa CreekGraphs showing the observed high-water marks andcomputed crests for the upper part ofToccoa CreekGraphs showing the observed high-water marks andcomputed crests for the lower part ofToccoa CreekIV4672426272930313234353637

TABLESPageTable 1.2.3.4.5.6.Flow into the three reservoir-stream systems*Relationship of elevation-storage-surface areaoutflow for the three reservoirsChannel characteristic and geometry data coded inthe GEDA format for the Teton Dam reservoirstream systemChannel characteristic and geometry data coded inthe GEDA format for the Laurel Run reservoirstream systemChannel characteristic and geometry data coded inthe GEDA format for the Toccoa Falls reservoirstream systemCode format and variable definitions of the datagiven in tables 3-5434445474954

FACTORS FOR CONVERTING U.S. INCH-POUND UNITSTO SI UNITSByMultiply U.S. inch-pound unitsTo obtain SI unitsfoot (ft)0.3048meter (m)mile (mi)1.609kilometer (km)acre0.4047square mile (mi )2.5902square hectometer (hm )2square kilometer (km )acre-foot (acre-ft)0.001233cubic hectometer (hm )cubic foot per second (ft /s)0.028323cubic meter per second (m /s)VI

EVALUATION OF SELECTED DAM-BREAKFLOOD-WAVE MODELS BY USING FIELD DATABy Larry F. LandABSTRACTFour dam-break flood-wave models have been evaluated by using threefield data sets and selected criteria of desirable features. The modelsinclude (1) modified Puls (MP), (2) U.S. Army Corps of Engineers'unsteady flow profiles (USTFLO), (3) National Weather Services' dambreak flood forecast (DBFF), and (4) U.S. Geological Survey's coupledmethod of characteristics and a general purpose streamflow simulation(MOC-J879DB). The field data sets documented the disasters at TetonDam, Idaho, Laurel Run, Pa., and Toccoa Falls, Ga.The computed discharges were often within 20 percent of the observedvalues with the exception of the simulations at Teton Dam and for ashort distance below the dams by the MOC-J879DB. With the same exceptionsnoted above, the computed flood crests were usually within 2 feet ofthe observed high-water marks.A modified version of the DBFF model is identified as the mostaccurate, economical, flexible, numerically stable, easiest to apply,and descriptive of the boundary and flow conditions.INTRODUCTIONDelineating the extent of flooding caused by a dam failure is oneaspect of flood mapping that has had limited attention. However, thedisasters in the last few years have caused an increased awareness ofthis potential flood hazard. As a result, the federal government isrequiring the inspection of dams that potentially threaten life andproperty. Repairs or alterations must be made to the ones that lackstructural integrity and the ability to handle large storms. For damsthat pose a special threat to the public, engineering studies areconducted to determine the extent of flooding which would result from ahypothetical failure. This latter activity is the major subject of thisreport.Predicting the extent of flooding, including floods caused by damfailures, is one of the activities of the U.S. Geological Survey. Theprimary tool for making the predictions or analyzing a dam-break floodis a mathematical model that can simulate the reservoir-stream system.Several models have been developed and are available. However, the bestmodel for general-use applications has not been identified. As aresult, an investigation was conducted to evaluate several mathematical

models in order to identify the best one. The criteria for modelevaluation were several desirable dam-break model features and themodel's ability to accurately simulate field occurrences. The desirablefeatures included such items as requiring only readily available data,having numerical stability and accuracy, simulating flow on hydraulicallymild or steep slopes, and having simplicity, flexibility and economy.The field data sets available for model testing were the occurrences atTeton Dam, Idaho on June 5, 1976, Johnstown, Pa. on July 19-20, 1977,and at Toccoa Falls, Ga. on November 5-6, 1977.METHOD OF INVESTIGATIONSelection of ModelsNumerous models could have been included in this evaluation; however,the scope of the investigation limited the number to four tested anddocumented models. The selected models ranged in technical sophisticationfrom a straightforward hydrologic routing model to a complex hydraulicrouting model that incorporates shock-wave equations. In the order ofcomplexity, the selected models included a hydrologic model that usesthe modified Puls technique and hydraulic models that use an explicitfinite-difference technique, a nonlinear finite-difference technique,and a coupled method-of-characteristics and linear implicit finitedifference techniques.The simplest and least mathematically rigorous model uses a modifiedPuls (MP) routing technique. Sources of such programs include the U.S.Army Corps of Engineers (1973) and a Survey unpublished report (U.S.Geological Survey, written commun., 1978). The Survey model was selectedbecause it was readily available and, if needed, it could be easilyrevised to handle special dam-break problems. The next model is theCorps of Engineers' (1977) Gradually Varied Unsteady Flow Profiles(USTFLO) model. It is a general purpose streamflow simulation modelthat can be used for a dam-break flood wave simulation by the use ofcertain options and model layouts. The model uses the complete flowequations with an explicit leap-frog finite-difference algorithm. Thethird model was developed by Fread (1977) of the National Weather Servicespecifically for application to dam breaks. This dam-break flood forecast(DBFF) model uses a nonlinear implicit finite-difference algorithm tosolve the complete flow equations. The last dam-break flood wavesimulation model, developed by the Survey (Chen and Druffel, 1977),couples an explicit method-of-characteristics (MOC) model that includesthe shock wave simulation with a general-purpose (J879DB) model using alinear implicit finite-difference alogrithm. The MOC model is useduntil the shock wave nearly dissipates and the simulation is completedwith the J879DB model.

Field Data SetsThe availability of data sets documenting a dam-break flood anddescribing the reservoir-stream system is limited. Three of the bestdata sets are for the occurrences at Teton Dam, Idaho, Johnstown, Pa.,and Toccoa Falls, Ga. The Teton Dam data are mostly in a report by Rayand Kjelstrom (1978). The published Johnstown data are in the reportsby Armbruster (1978) and Chen and Armbruster (1979). The publishedToccoa Falls data are mostly in reports by a Federal Investigative Board(1977), Sanders and Sauer (1979), and Land (1978a). In all cases,additional data were supplied by the local Survey offices.For those readers interested in the details of these three fielddata sets, much of these data are listed in the following tables. Thepresented data include: (1) flow into the reservoir-stream systems(table 1), (2) reservoir elevation-storage-surface area-outflow information(table 2), and (3) reservoir-stream system geometry along with highwater marks (tables 3, 4, and 5). Table 6 is given to show the codeformat and variable definition for the geometry data in tables 3-5. Thedata presented in tables 3-5 have been coded in the GEDA program format;however, only data related to the channel description are given exceptfor the variable HWM which has been added (field 10) to the XI card. Itis the observed high-water mark at that cross section. If the data areestimated, an asterisk (*) precedes the value. If an entire crosssection is estimated, an * is placed in front of the XI card. A completedefinition of the coded data is given in the GEDA documentation by theU.S. Army Corps of Engineers (1976).Teton Dam, IdahoThe failure of Teton Dam on the Teton River, Idaho (fig. 1) onJune 5, 1976 released 251,700 acre-ft of water stored behind the 305-fthigh and 1,200-ft long earth-filled structure. At the time of failure,the reservoir's water level was at an altitude of 5,301.7 ft, about 272ft above the streambed. The altitude of the dam's crest was 5,330 ftwith a spillway crest of 5,305 ft. The first evidence of the dam'simpending failure was noticed at 7:30 a.m. on June 5. The dam wasbreached at 11:57 a.m. The almost fully penetrating breach, approximatelytrapezoidal in shape, included about 40 percent of the dam. The failurewas not associated with any current flooding on the river; in fact, itwas being filled for the first time. A detailed field survey of thebreach was not available. As a result, the breach's dimensions wereestimated from topographic maps and photographs. For purposes of thisreport the breach was assumed to have eroded to an elevation of 5,040ft, and to have a base width of 50 ft and side slopes of 1.52:1.00(horizontal to vertical).

43 50'44 00IDAHO2 30Figure 1. Location of Teton Dam, Idaho, study area.H2 45 iDSite of indirect peakdischarge measurementStntion »n hundreds offeet be ow dam

Teton Reservoir was formed in a canyon generally less than 1,200ft wide. At high stages it was about 15 mi long. Below the dam thecanyon extended another 5 mi before emptying into a valley or floodplain over 2 mi wide. In the flood plain, Teton River widely meandersand divides about 8.5 mi downstream of the dam into two forks which alsomeander considerably. Several stream diversions also exist in thisarea. For purposes of this investigation the flood wave was routed onlyabout 9 mi below the dam. Adaptation of the models to the complexity ofthe stream system below this point is beyond the scope of this study.Johnstown, PennsylvaniaThe failure of the Laurel Run Reservoir Dam near Johnstown, Pa.(fig. 2) released about 450 acre-ft of water in only a few minutes intoa stream that was already flooding from a severe rainstorm. The earthendam had a crest altitude of 1,436.5 ft with a spillway crest of 1,430ft. The streambed altitude was 1,391 ft. The estimated time of failurewas 2:35 a.m. on July 20, with the reservoir's water level at an altitudeof 1,437.2 ft. Chen and Armbruster (1979) suspect that the dam had apartial failure before approximately one-third of the dam failed. Thebreach's shape was approximately triangular. It fully penetrated thedam and had average side slopes of 2.45:1.00.The reservoir was about 0.4 mi long and generally less than 600 ftwide. The stream channel confined the flood to a width of less than 500ft, often less than 200 ft. Laurel Run flows into the Conemaugh River2.5 mi below the dam. Red Run and Wildcat Run Tributaries enter LaurelRun 1,100 and 9,700 ft below the dam, respectively.Toccoa Falls, GeorgiaThe dam-break flood at Toccoa Falls, Ga. (fig. 3) resulted from thefailure of the dam forming Kelly Barnes Lake. The breach completelypenetrated the dam, was 57 ft wide at the base, and had average sideslopes of 0.56:1.00. The time of failure was estimated to be 1:30 a.m.on November 6, 1977. It followed a series of rainstorms that had producedan estimated 7.2 in. of rain in the previous four days.According to a Federal Investigative Report (1977), Kelly BarnesDam had its origin as a rock crib dam in 1899. The dam was made largerin the late 1930's by building an earthen dam over the old rock cribdam. In the late 1940's the dam was made somewhat larger by addingearth to the structure. The base of the dam had an altitude of 1,102.0ft. The normal pool altitude was 1,137 ft, the lower spillway's crestwas 1,136.7 ft, the upper spillway's crest was 1,139.8 ft, and the crestof the dam was approximately 1,147 ft. Surveys show that the maximumwater level for this flood was 1,141.6 ft, but the Federal InvestigativeBoard believes that a series of partial failures lowered the water levelto 1,137.5 ft before the dam completely collapsed. The dam was about400 ft long and 20 ft wide at the crest; the lake was about 0.9 milong.

40 IO40 12 3040 1578 55Figure 2. Location of Laurel Run, Pennsylvania, study area.Site of indirect peakdischarge measurementStation in hundred offeet below dam: .X PL AN AT I ONI78 58PENNSYLVANIALaurel RunReservoir-20(Mile

34 36Site of indirect peakdischarge measurementStation m hundreds ofFigure 3. Location of Toccoa Falls, Georgia, study area.34 37' -4-83 22'83 20'

Toccoa Creek's headwaters are in the steep to very steep mountainousterrain of the Blue Ridge Mountains. After only a few miles, and 2,600ft below Kelly Barnes Dam, the creek passes over the 175-ft Toccoa Fallsand enters the foothills where the streambed is not as steep as in themountains. In the reach above the falls the banks restricted the floodto a width of 300 ft. Below the falls the width of flooding was lessthan 500 ft except in the wider parts of the flood plain where the widthof flooding was as much as 1,000 ft. The flooding was surveyed for adistance of 4.4 mi below the dam.FEATURES OF DAM-BREAK MODELSA researcher or engineer has many choices and approaches that canbe taken in developing a dam-break flood model. The first choice is thebasic type of model, that is, hydrologic or hydraulic. A hydrologicmodel uses the law of continuity, or its equivalent, in conjunction withthe relationship between storage, inflow and outflow from the reach as theyrelate to time. This type of model is generally simpler, numericallymore stable and easier to apply. A hydraulic model is based on thesolution of the two partial differential equations describing unsteadyflow in shallow open channels. This type of model uses equations thatare known as the St. Venant equations, namely the equations of continuityand motion. For the extra effort in achieving a solution, this type isgenerally more accurate and better describes the reservoir-stream system.Following the selection of the basic model type, several solutiontechniques or methods are available for each. For a hydrologic modelthe choice may include Muskingum, modified Puls, or kinematic wavemethods. For the hydraulic model the numerical analysis technique maybe any number of finite-difference, method-of-characteristics, orfinite-element methods.In the design of a general-purpose dam-break model other basicdecisions are also necessary. Alternatives are: (1) to assume the damfailure to be instantaneous or to have a finite duration, (2) to treatthe dam as an internal node that may or may not have special boundaryconditions, or as a downstream boundary, (3) to simulate the shock frontor ignore it, and (4) to route the flood wave in a wet or dry streambed.No matter what choices are made in the model's basic structure, it isdesirable for the model to:(1) Require only data readily available, or easily interpreted, fromfield surveys;(2) Represent the complexity of the reservoir-stream system,including the conveyance and storage areas of channel crosssections;(3) Compute accurate hydrographs at the dam;(4) Have low sensitivity to parameters that are subject tointerpretation or debate;

(5) Allow the dam structure to discharge water other than throughthe breach;(6) Have numerical stability and accuracy;(7) Simulate flow on hydraulically steep or mild slopes;(8) Require no data alteration to get the model operational;(9) Allow for easy redesign and data changes;(10) Print and plot results in an easily readable format; and(11) Have simplicity, economy, flexibility, and a minimum numberof user steps in a complete simulation.MATHEMATICAL MODELSMP ModelThe modified Puls method is a hydrologic routing technique that isbased only on the conservation of mass. The method was originallydesigned for use in routing flood waves through reservoirs. However,it can also be used to route flood waves in riverine systems, which isone of its uses in the dam-break routing problem. According to Chow(1964) the method requires the use of invariable storage-outflowrelationships for each reservoir or stream segment.The modified Puls method arranges the continuity equation,expressed in finite time intervals, in the following form:Sl(1)whereandI0SAtis the inflow rate,is the outflow rate,is storage,is the time interval between time steps 1 and 2 (Soil ConservationService, 1972).f202permits aA so-called working curve of 0 versusLworkingAt. t 2 Jcurve is computedtime-sequence solution of equation 1. Thefrom the time step interval and the storage-outflow table. All termson the left-hand side of equation 1 are known; therefore, the value ofthe P2 *2\ term and the working curve can be used to compute 0 at thenew time step. Then the storage-outflow table is used to determine anew storage value.

The assumptions used in the modified Puls method are:(1) the watersurface in the reservoir is level and in the channel its slope correspondsto steady flow conditions, (2) the water surface responds instantaneouslyto inflows and outflows, (3) outflows are uniquely described as a functionof storage, and (4) a time interval is no larger than 2 x S/0. Themodel's major advantage is simplicity and its major disadvantage is thelack of rigorous mathematical representation of the reservoir- streamsystem and the flow dynamics.A very large part of the application of the modified Puls methodinvolves the stage- storage-outflow tables. For the river below the dam,the tables are computed from several water- surf ace profiles which arefirst computed by a step-backwater subprogram and several discharges.Tables for the dam in the original and breached condition are suppliedby the user as input data.The equation used for the three field sites evaluated in this reportis the dam-break equation for a trapezoidal channel and breach which waspresented by Price and others (1977) . The equation isP*'t * * J « 2)whereandQ-pkgHbTisisisisismaximum discharge,acceleration of gravity,height of initial water level above breach base,width of breach base,top width of breach at initial water level.The model begins its computation by routing a flood wave throughthe reservoir using the tables for the dam in the original condition.When the water level rises to a preselected elevation the table representingthe dam in the breached condition is used, thus simulating an instantaneousfailure. The incoming flood wave is then completely routed through thereservoir with the latter table. Finally, the model begins steppingthrough the river reach, subreach by subreach, until the downstream endis reached. Each time the flood wave is completely routed through asubreach before moving to the next subreach.The design of the step-backwater subprogram used in the MP modelallows a nonprismatic river reach, with or without inactive flow areas,which may have subcritical or supercritical flow in any subreach and atany discharge. The computations are made in the upstream direction forsubcritical flow and in the downstream direction for supercritical flow.Any time the subprogram is unable to solve the equations, it uses thenormal depth for that cross section. The method does not explicitlycalculate the travel of a flood wave moving through a stream system.10

USTFLO ModelA general purpose unsteady open-channel-flow model originallydeveloped by Garrison and others (1969) , and documented by the Corps ofEngineers (1977) , has been used by Price and others (1977) , andGundlach and Thomas (1977) for simulating dam- break floods. The modelis used for dam-break analysis by selecting appropriate designs andoptions.The one -dimensional unsteady flow equations used in the developmentof the USTFLO model follow.-,-0and8xwhereAVxBhtqandSg2xgdtgA V Sf(4)isisisisisisisactive flow area,mean cross-sectional velocity of active flow area,distance along channel,water surface width,water surface elevation,time,lateral discharge per unit length (loss to soil orground -water seepage) ,is the friction slope.The numerical analysis technique is a leap-frog scheme. It isexplicit and has time lines going through alternating odd and evencomputational nodes. The scheme advances the solution along the timeaxis where solutions are alternated between the odd and even nodes inthe computation net. Only the computed values at the odd nodes aresuitable for analysis. The computational nodes must be evenly spaced.For computational stability At must meet certain criteria which are afunction of Ax and channel characteristics.According to Gundlach and Thomas (1977) the characterization of theUSTFLO model to reservoir-stream system for dam-break flood wave analysisdepends on the type of breach. If the breach is 100 percent of thestream cross-section then the dam is treated as an internal node. If apartial breach is simulated, the reservoir part is simulated in one stepand its outflow routed through the channel in another step.The advantages of using the USTFLO model include (1) good documentation,(2) general familiarity owing to its long-term use and availability, and(3) flexibility in boundary conditions. Some of the disadvantages, aspointed out by Gundlach and Thomas (1977) , are difficulty in establishinginitial conditions and tendency for the upstream node to go dry incertain situations, thereby aborting the computer run. The USTFLO model11

documentation states that oscillations are a common problem for verysmall water-surface slopes, such as in reservoirs. For the tests conductedin this investigation these difficulties prevented successful simulationsfor the Johnstown and Toccoa Falls data sets; therefore, no results arepresented for these two sites.DBFF ModelThe DBFF model has been developed and documented by Fread (1977) ofthe National Weather Service. The DBFF model is a hydraulic model thatsolves the complete St. Venant equations of unsteady open-channel flow.It has been developed with the primary purpose of simulating dam-breakflood waves; therefore, it has many useful functions that are usuallymissing in other models. Furthermore, it was designed with an emphasison real-time flood forecasting. However, for this investigation, it hasbeen slightly modified with an emphasis on general applications. Thebasic approach and the equations were not changed.Fread used the conservation form of the two equations in whichdischarge (Q) and water-surface elevation (h) were the dependent variables.One equation is the conservation of mass (continuity)ao(5)atwhereAQ is the inactive cross-sectional area.The other equation is the conservation of momentum (motion)S S ) 0fce(6)whereandSceis evaluated from the Manning equation for steady uniform flow,is the expansion-contraction slope, defined byce2g Ax(7)wherekvaries from -1 to 1 with the negative values for expansionand positive values for contraction,and AVis the difference between the square of the velocities attwo adjacent computational nodes separated by a distance, Ax.The g term is assumed to have no velocity component in the x-direction.12

The numerical solution is obtained by a nonlinear implicit finitedifference technique. Fread selected the "weighted four-point scheme"which allows unequal distance and time steps and exhibits good stabilityconvergence properties. In this scheme the continuous x-t region, inwhich the solutions of h and Q are sought, is represented by arectangular net of discrete points. Fread uses a weighting factor of0.60, meaning that the derivatives are written at a point slightlyforward of and above a grid's center.By use of two boundary conditions, discharge at the upstream endand stage-discharge relation for nonuniform, unsteady flow at the downstreamend, a system of 2N equations and the same number of unknowns result.N denotes the number of nodes in a time line. The resulting system of2N nonlinear equations with 2N unknowns is solved by a quad-diagonalGaussian elimination algorithm (Fread, 1971) and a functional interactiveprocedure, the Newton-Raphson method.Simulation of a reservoir-stream system with the DBFF model beginsby automatically computing the reservoir's outflow hydrograph. Thecomputational method can be either the dynamic formulation describedabove, which is used in the downstream reaches, or a hydrologic (storagerouting) technique. The latter technique uses the continuity equationwhich requires a stage-surface area table. The outflow is the sum ofthe discharges through the outflow structures and the breach. Thebreach is assumed to start forming when the reservoir's water surfacereaches a specified level. The breach initially has its base at thewaterline and has a zero bottom width. During the progressive failure,the base broadens and moves downward at a constant rate, as specified bythe duration of failure. The discharge through the breach (Q- ) isassumed to be a function of head and the breach's shape. Q is computedby the broad-crested weir equation. The equation has two components,one for a triangular shape and the other for a rectangular shape.Together they can represent a trapezoidal-shaped breach. If the breach'soutflow becomes submerged, a submergence correction factor is automaticallycomputed and used to adjust the outflow.After computing the complete outflow hydrograph at the dam, themodel begins routing it through the stream. If needed, the stream canbe divided into segments where the state of flow is always subcriticalor supercritical. In such cases the routing is completed for one segmentbefore moving to the next downstream segment.The advantages of the DBFF model include: (1) reasonable representationof the physical and hydrodynamic aspects of the reservoir-stream system,(2) primarily for dam-break simulations, (3) relatively easy one-stepapplication, and (4) an efficient numerical a

specifically for application to dam breaks. This dam-break flood forecast (DBFF) model uses a nonlinear implicit finite-difference algorithm to solve the complete flow equations. The last dam-break flood wave simulation model, developed by the Survey (Chen and Druffel, 1977), couples an explicit method-of-characteristics (MOC) model that includes