Introduction to Hydrology

Contents PREFACE xiii PART ONE CYCLE THE HYDROLOGIC CHAPTER1 lntroduction 1.1 t.2 r.3 t.4 1.5 1.6 I.7 1.8 3 Hydrolo...

Author: Warren Viessman | Gary L. Lewis | John W. Knapp

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Contents

PREFACE xiii

PART ONE CYCLE THE HYDROLOGIC CHAPTER1 lntroduction 1.1 t.2

r.3 t.4 1.5 1.6 I.7 1.8

3 Hydrology Defined 3 A Brief History 5 The Hydrologic Cycle 5 The Hydrologic Budget 11 HydrologicModels 11 HydrologicData 12 Common Units of Measurement Problems to Environmental Application of Hydrology

CHAPTER2 Precipitation 1 5 2.1 2.2 2.3 2,4 2.5 2.6 2.7

15 Water Vapor 17 Precipitation Distribution of the Precipitation Input 27 Point Precipitation 29 Areal Precipitation 34 Precipitation ProbableMaximum 36 Grossand Net PreciPitation

t2

vi

coNTENTS

CHAPTER3 Interception and Depression Storage 3.1 3.2 3.3

Interception 40 Throughfall 44 DepressionStorage

40

45

CHAPTER4 Infiltration 52 4.I 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10

MeasuringInfiltration 53 Calculation of Infiltration 53 Horton's Infiltration Model 57 Green-AMPT Model 64 Huggins-MonkeModel 67 Holtan Model 68 Recoveryof Infiltration Capacity 69 Temporal and Spatial Variability of Infiltration Capacity SCS Runoff Curve Number Procedure 73 76 @Index

70

CHAPTER5 Evaporation and Transportation 82 " 5.1 5.2 5.3 5.4 5.5 5.6 5.7

Evaporation 86 EstimatingEvaporation 86 EvaporationControl 95 Transpiration 95 TranspirationControl 100 Evapotranspiration 100 EstimatingEvapotranspiration

103

CHAPTER6 Streamflow 111 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8

DrainageBasinEffects 111 The Hydrograph 11,2 Units of Measurementfor Streamflow 113 Measuringand RecordingStreamflow 113 Measurements of Depth and Cross-Sectional Area II4 Measurementof Velocity lI4 RelatingPoint Velocity to Cross-SectionalFlow Velocity The Slope-AreaMethod for DeterminingDischarge II7

115

oONTENTS Vii

PART TWO AND MONITORING 121 fT'IEASUREMENTS HYDROLOGIG CHAPTER7 DataSources Hydrologic 7.1 7.2 7.3 7.4

123

I23 GeneralClimatologicalData 123 Precipitation Data 124 StreamflowData Evaporationand TranspirationData

I24

CHAPTER8 fnstrumentation126 8.1 8.2 8.3 8.4

126 Introduction 127 HYdrologicInstruments 135 TelemetrySYstems 135 RemoteSensing

CHAPTER9 Networks 144 Monitoring The Purposeof Monitoring 9.r 9.2 9.3 9.4

144

I45 SpecialConsiderations I47 Uie of ComPutersin Monitoring 147 Hydrological-MeteorlogicalNetworks

PART THREE SURFACEWATERHYDROLOGY151 CHAPTER1O Runoffandthe Catchnient 10.1 tO.2 10.3 IO.4 10.5

153

153 Catchments,Watersheds,and DrainageBasins 155 Basin CharacteristicsAffecting Runoff RudimentaryPrecipitation-RunoffRelationships 164 166 StreamflowFrequencyAnalysis 168 StreamflQwForecasting

CHAPTER11 Hydrographs 171 11.1 Il.2 11.3

StreamflowHYdrograPhs 171 FactorsAffecting HydrographShape HydrograPhComPonents 174

172

viii

ooNTENTS lI.4 11.5 11.6 Il.7

BaseFlow Separation I77 HydrographTime Relationships Time of Concentration I82 BasinLae Time I82

181

CHAPTER12 UnitHydrographs 188 l2.I 12.2 12.3 12.4 12.5 12.6

Unit HydrographDefinition 188 Derivation of Unit Hydrographsfrom StreamflowData 190 Unit HydrographApplications by Lagging Methods I94 S-HydrographMethod 198 The InstantaneousUnit Hydrograph 201 SyntheticUnit Hydrographs 205

CHAPTER13 Hydrograph Routing 13.1 13.2 13.3

HydrologicRiver Routing 235 HydrologicReservoirRouting 245 Hydraulic River Routing 248

CHAPTER14 SnowHydrology I4.l I4.2 I4.3 I4.4 14.5 14.6

234

265

Introduction 265 Snow Accumulation and Runoff 267 Snow Measurementsand Surveys 268 Point and Areal Snow Characteristics 269 The SnowmeltProcess 271. SnowmeltRunoff Determinations 284

CHAPTER15 Urbanand SmallWatershedHydrology 309 15.1 15.2 i5.3 15.4

Introduction 309 PeakFlow Formulasfor Urban Watersheds 311 PeakFlow Formulasfor Small Rural Watersheds 33I Runoff Effects of Urbanization 344

CHAPTER16 Hydrologic Design 16.l 16.2

359

Hydrologic DesignProcedures 360 Data for HydrologicDesign 363

CONTENTS

16.3 16.4 16.5 t6.6 16.7 16.8

HydrologicDesign-Frequency Criteria DesignStorms 373 Critical EventMethods 391 Airport DrainageDesign 400 Designof Urban Storm Drain Systems FloodplainAnalysis 409

365

402

PART FOUR GROUNDWATER HYDROLOGY 425 CHAPTER 17 Groundwater, Soils,and Geology l7.l I7.2 I7.3 I7.4 I7.5 I7.6

427

Introduction 427 GroundwaterFlow-General Properties 429 SubsurfaceDistribution of Water 429 GeologicConsiderations 430 Fluctuationsin GroundwaterLevel 433 Groundwater-Surface Water Relations 433

CHAPTER 18 Mechanics of Flow

435

18.1 t8.2 18.3 18.4 18.5 18.6

Hydrostatics 435 GroundwaterFlow 436 Darcy's Law 436 Permeability 438 Velocity Potential 440 HydrodynamicEquations 441 r8.7 ' Flowlines and EquipotentialLines 18.8 BoundaryConditions 447 18.9 Flow Nets 449 1 8 . 1 0 VariableHydraulic Conductivity 1 8 . 1 1 Anisotropy 452 18.t2 Dupuit's Theory 453

CHAPTER 19 Wellsand Collection Devices 19.1 19.2 19.3 19.4 19.5

444

451

460

Flow to Wells 460 SteadyUnconfinedRadial Flow Toward a Well 461 SteadyConfined Radial Flow Toward a Well 462 Well in A Uniform Flow Field 463 Well Fields 465

IX

X

CONTENTS

19.6 I9.7 19.8 I9.9 19.10 19.ll 19.12

The Method of Images 466 UnsteadyFlow 467 Leaky Aquifers 4'13 Partially PenetratingWells 473 Flow to an Infiltration Gallery 473 SaltwaterIntrusion 474 GroundwaterBasin Development 475

CHAPTER20 ModelingRegionalGroundwater Systems 20.I 20.2 20.3 20.4 20.5

RegionalGroundwaterModels 481 Finite-DifferenceMethods 484 Finite-ElementMethods 493 Model Applications 494 GroundwaterQuality Models 500

PART FIVE HYDROLOGIC MODELING

505

CHAPTER 21 Introduction to Hydrologic Modeling 2l.I 2t.2 21.3 21.4

481

5O7

HydrologicSimulation 508 GroundwaterSimulation 509 Hydrologic Simulation Protocol 524 Corps of EngineersSimulation Models

526

CHAPTER22 SyntheticStreamflows 535 22.I 22.2

SyntheticHydrology 536 Serially DependentTime SeriesAnalysis

CHAPTER23 Continuous Simulation Models 23.1 23.2

548

ContinuousStreamflowSimulationModels 549 ContinuousSimulation Model Studies 570

CHAPTER24 Single-Event Simulation Models 24.1 24.2 24.3

539

594

StormEventSimulation 594 FederalAgency Single-EventModels Storm SurgeModeling 625

597

CONTENTS Xi

CHAPTER25 UrbanRunotfSimulation Models 25.1 25.2 25.3

Urban StormwaterSystemModels 63I Urban Runoff Models Compared 659 Vendor-DevelopedUrbanStormwaterSoftware 663

PART SIX METHODS STATISTICAL

669

CHAPTER26 Probability and Statistics 26.1 26.2 26.3 26.4 26.5 26.6 26.7 26.8 26.9

2 6 .r0

671

RandomVariablesand StatisticalAnalysis 672 Conceptsof Probability 673 ProbabilityDistributions 676 Moments of Distributions 681 Distribution Characteristics 682 Types of Probability Distribution Functions 685 ContinuousProbabilityDistributionFunctions 685 Bivariate Linear Regressionand Correlation 690 Fitting RegressionEquations 692 Regressionand Correlation Applications 697

CHAPTER27 Frequency Analysis 27.1 27.2 27.3 27.4 27.5 27.6 27.7

630

708

FrequencyAnalysis 708 GraphicalFrequencyAnalysis 709 FrequencyAnalysis Using FrequencyFactors RegionalFrequencyAnalysis 7I9 Reliability of FrequencyStudies 730 FrequencyAnalysis of Partial Duration Series Flow Duration Analysis 737

APPENDICES 751 INDEX 757

7Il

734

Preface

Watermanagementis taking on new dimensions.New federalthrusts,the growprotecing list of global iisues,and strongpublic sentimentregardingenvironmental forces. driving tion havebeen the principal In the early yearsof the 20th century,waterresourcesdevelopmentand management were focuied almostexclusivd on water supplyand flood control' Today,these the environment,ensuring safe drinking issuesare still important, but protecting -and experiencescompete equally for recriatioinal water, and providing aesthetic consciouspublic is pressingfor environmentally an attentionand funds.Furthermore, with fewer structural compopractices, management greaterreliance on improved of continually striving to notion The problems. water ients, to solve this nuiion'. this preciousnatural husbanding of one by replaced provide more water has been resource. There is a growing constituencyfor allocatingwater for the-benefitof fish and wildlife, for protection-of marshesand estuary areas' and for other natural system uses.But estimatingthe quantitiesof water neededfor environmentalprotectionand for maintainingand/or restoringnatural systemsis difficult, and there are still many unknowns.Scilntific data are ,putt", and our understandingof the complexinteracof an scalesis rudimentary.Indeed,this is a critical issue' tions inherentin ecosystems sincethe quantitiesof water involvedin environmentalprotection can be substantial of and competitionfor thesewatersfrom traditional water usersis keen' The nations are that systems-decisions natural regarding decisions the world are facing major ladenwith significantectnomic and social impacts.Thus there is_anurgencyassociatedwith developinga betterunderstandingof ecologicsystemsand of their hydrologic components. Water policies of the future must thereforetake on broader dimensions'More emphasismustbe placedon regionalplanningand management,and regionalinstitutions to accommodatethis muJt be devised.Watermanagementmustbe practicedat, more andbetween,all levelsof government.Land useand wateruseplanningmustbe tightly coordinatedas well'

XIV

PREFACE

Water scientistsand engineersof tomorrow must be equipped to addressa diversity of issuessuch as: the design and operation of data retrieval and storage systems;forecasting;developingalternativewater use futures; estimatingwater requirementsfor natural systems;exploringthe impactsof climate change;developing more efficient systemsfor applyingwaterin all water-usingsectors;and analyzingand designingwater managementsystemsincorporating technical, economic, environmental, social, legal, and political elements.A knowledgeof hydrologicprinciplesis a requisitefor dealingwith suchibsues. This fourth edition hasbeendesignedto meet the contemporaryneedsof water scientistsand engineers.It is organizedto accommodatestudentsand practitioners who are concernedwith the development,management,and protection of water resources.The format of the book follows that of its predecessor, providing material for both an introductory and a more advancedcourse. Parts One through Four provide the basicsfor a beginninglevel course,while Parts Five and Six may be used for a more advancedcourseon hydrologicmodeling. This fourth edition has been updated throughout, and many solved examples havebeen added.In addition, new computer approacheshave been introduced and problem-solvingtechniquesincludethe use of spreadsheets as appropriate.New featuresof eachchapterincludean introductory statementof contentsand,at the conclusion of the chapter,a summaryof key points. Many sourceshave been drawn upon to provide subject matter for this book, and the authorshope that suitable acknowledgmenthas been given to them. Colleaguesand studentsare recognizedfor their helpful commentsand reviews,particularly the following reviewers. Gert Aron, ThePennsylvaniaStateUniversity JohnW. Bird, Universityof Nevada-Reno IstvanBogardi, Universityof Nebraska RonaldA. Chadderton,VillanovaUniversity RichardN. Downer,Universityof Vermont Bruce E. Larock, Universityof Califurnia-Davis Frank D. Masch,Universityof Texas-SanAntonio Philip L. Thompson,FederalHighwayAdministration A specialnote of thanks is due to Dr. John W. Knapp, Presidentof the Virginia Military Institute,coauthorof previouseditionsof this book, for his pastcontributions andvaluableguidance. WarrenViessman.Jr. Gary L. Lewis

PARTONE

CYCLE THE HYDROLOGIC

L.

Chapter1

lntroduction

I Prologue The purPoseof this chaPteris to: . , . .

Define hydrology. . earth -,.1Lscience' Give a brief niJiory of the evolution of this important Statethe fundamentalequationofhydrology' appliedto supplementdecision Demonstratetrow ffiofogic principle, "urib" management' support systemsfor water and environmental

DEFINED 1.1 HYDROLOGY Hydrologyisanearthscience'Itencompassestheoccuffence'distribution,moveA knowledgeof hydrologyis fundamenr, and propertiesof the watersof the earth. mentaltodecisionmutingp,o.",,e,*he,ewaterisu"ompon"nto.f.th.esystemof inextricably linked' and it is important concern.water and environmentalissuesare toclear$understandhowwaterisaffectedbyandhowwateraffectsecosystem maniPulations'

1.2 A BRIEFHISTORY Ancient philosophersfocusedtheir i flows Production of surfacewater oc"ur."n"e of water in variousstag from the seato the atmosPhereto t early speculationwas often faulty'l of large subterraneanreservoirsth is interestingto note, however'tha *suoeriornumbersindicatereferencesattheendofthechapter.

CHAPTER1

INTRODUCTION

Greek aqueductson both conveyancecrosssectionand velocity.This knowledgewas lost to the Romans,and the proper relation betweenarea,velocity, and rate of flow remainedunknownuntil Leonardoda Vinci rediscoveredit duringihe Italian Renaissance. During the first century s.c. Marcus Vitruvius, in Volume 8 of his treatiseDe Architectura Libri Decem (the engineer'schief handbookduring the Middle Ages), setforth a theory generallyconsideredto be the predecessorof modern notionsof the hydrologiccycle. He hypothesizedthat rain und ,no* falling in mountainous areas infiltrated the earth's surface and later appearedin the lowlands as streamsand springs. In spiteof the inaccuratetheoriesproposedin ancienttimes,it is only fair to state that practical applicationof varioustry-orotogic principleswas often carried out with considerablesuccess.For example,about4000 s.c. u du- was constructed acrossthe Nile to permit reclamation of previously barren lands for agricultural production. Severalthousandyears later a canal to convey fresh water from Cairo io Suezwas built. Mesopotamiantowns were protecteduguinrt floodsby high earthen walls. The Greek and Roman aqueductsand early Chineseirrigation and flood control works were also significantprojects. Nearthe endof the fifteenth century the trend towarda more scientific approach to hydrology based on the observationof hydrologic phenomena becameevident. Leonardo da Vinci and Bernard Palissyindepende-ntly reachedan accurateunderstandingof the watercycle.They apparentlybised theii theoriesmore on our"*iion than on purely philosophicalreasoning.Nevertheless, until the seventeenthcentury it seemsevident that little if any effort was directed toward obtaining quantitative measurementsof hydrologicvariables. The adventof what might be calledthe "modern" scienceof hydrology is usually consideredto beginwith the studiesof suchpioneersasPerrault,Mariotte, and Halley in the seventeenthcentury.r'aPerraultobtainedmeasurements of rainfall in the Seine River drainagebasin over a period of 3 years. Using these and measurements of runoff, and knowing,thedrainage areasize,he showeJthat rainfall was adequatein quantity to accountfor river flows. He also mademeasurements of evaporati,onand capillarity. Mariotte gaugedthe velocity of flow of the River Seine. Recordedvelocities were translatedinto termsof dischirgeby introducingmeasurements of the river crosssection'The English astronomerHalley measuredthe rate of evaporation of the MediterraneanSeaand concludedthat the amountof water evaporated was sufficient to accountfor the outflow of rivers tributary to the sea.Measurements suchas these, althoughcrude,permitted reliable conclusionsto be drawn reggrding the hydrologic phenomenabeing studied. brth numerousadvancesin hydraulic theory zometer,the Pitot tube, Bernoulli's theorem, ples.8 perimental hydrology flourished.Significant ydrology and in the measurementof surface water. Such significantcontributionsas Hagen-Poiseuille'scapillary flow equation, Darcy's law of flow in porous media, und th" Dupuit-Thiem well formula were evolved'e-lrThe beginningof systematicstream guoling can also be traced to this period' Although the basis for modern hydrology wui tirrr establishedin the nine,)

BUDGET 1.4 THE HYDROLOGIC

5

of teenth century, much of the effort was empirical in nature. The fundamentals early the In recognized. or widely established yet well been not physicalhydtotogyhad years of tle twJntieth ""niury the inadequaciesof many earlier empirical formulato tions becamewell known. As a result, interestedgovernmentalagenciesbegan rational 1950, to 1930 about From research' of hydrologic programs developtheir own analysis began to ieplace empiricism.3 Sherman's unit hydrograph, Horton's infiltration theory, und Th"it's nonequilibrium-approachto well hydraulicsare outstandingexamplesof the great progressmade'r2-'o Since 1930 a theoreiical approachto hydrologicproblemshas largely replaced permit a less sophisticatedmethods of ttre past. Advancesin scientific knowledge and ' better understandingofthe physicaibasisofhydrologic relations,and the advent continueddevelopnientof high-speeddigital computershavemadepossible,in both would a practical and an economiciense,extensivemathematicalmanipulationsthat past. in the havebeen overwhelming For a more compiehensivi historical treatment, the reader is referred to the works of Meinzer,Jonls, Biswas,and their co-workers'1'2'4'5'15

CYCLE 1 . 3THE HYDROLOGIC the The hydrologiccycle is a continuousprocessby which water is transportedfrom The exist' subcycles Many sea. the to back landind to the oceansto the atmosphere precipitation over land beforereturnevaporationof inlan-dwater and its subsequent for the global watertransportsystem force driving The example. ingio the oceanis one requiredfor evaporation'Note that energy the furnishes is providedby the sun,which the cycle; for example,sea passage through during changes the water quality also evaporation' through water fresh water is convertedto The completewater cycle is global in nature. world water problems require Practical studieson regional,national,internitional, continental,and global scales.16 earth is the to available water fresh of supply total the that significanceof the fact has limited and very small compared with ihe salt water content of the oceans at the receivedlittle attention.Thus watersflowing in one country cannotbe available u's' same time for use in other regions of the world. Raymond L' Nace of the thatoowaterresourcesare a global problem with GeologicalSurvey has aptly sta=ted are obligated to cope with problems requiring hydrologists local roots."tu Mtdern magnitudedifference.In addition, developing of oider of definition in varying scales careful attention, since climatological receive must weather techniquesto contiol hydrology and therefore the water the affect profoundly can changesin one area resourcesof other regions.

BUDGET 1 . 4 THE HYDROLOGIC Becausethe total quantity of water availableto the earth is finite and indestructible, subsysthe global hydrolojic ,yrt"* may be lookedupon as closed.Open hydrologic system' temsare abundantlhowever,and theseare usuallythe type analyzed'For any ' a water budgetcan be developedto accountfor the hydrologiccomponents'

CHAPTER1

,

INTRODUCTION

FiguresI'I,I.2, and 1.3 showa hydrologicbudgetfor the coterminousUnited States,a conceptualizedhydrologiccycle,andthe distributionof a precipitationinput, respectively.Thesefiguresillustrate the componentsof the water cycle with which a hydrologistis concerned.In a practicalsense,somehydrologicregionis dealtwith and a budgetfor that region is established.Suchregionsmay be topographicallydefined (watershedsand river basinsare examples),politically specified(e.g- couniy or city limits), or chosenon some other grounds. Watershedsor drainagi tasins are the easiestto deal with sincethey sharply define surfacewater boundaries.Thesetopographically determinedareasare drainedby a river/streamor systemof connecting rivers/streamssuch that all outflow is dischargedthrough a single outlet. Unfortunately,it is often necessaryto deal with regions that are not well suitedto tracking hydrologiccomponents.For theseareas,the hydrologistwill find hydrologicbudgeting somewhatof a challenge. The primary input in a hydrologicbudgetis precipitation.Figures 1.1-1.3 illustrate this. Someof the precipitation (e.g.,rain, snow,hail) may be interceptedby trees,grass,other vegetation,and structuralobjectsand will eventuallyreturn to the atmosphereby evaporation.Onceprecipitationreachesthe ground,someofit may fill (becomedepressionstorage),part may penetralethe ground (infiltraie) to depressions replenishsoil moisture and groundwaterreservoirs,and some may become surface runoff-that is, flow over the earth's surfaceto a definedchannelsuchas a stream. Figure 1'3 showsthe dispositionofinfiltration, depressionstorage,and surfacerunoff.

and vegetation

*r-d{i;

I

'1.'

Consumptive use 100bgd

bgd = billion gallons per day

Figure 1.1 Hydrologic budgetof cotermiriousunited States.(U.S. Geologicalsurvey.)

BUDGET 7 1.4 THEHYDROLOGIC Clouds and water vaPor

Clouds and water vaPor

ffi

(

)

" 1 ) t t l l ,

--1vtivv

p\--

P

P

P

P

P

P

E E

E

E

t t E' evaporation;P' Figure 1.2 The hydrologic 'surfac-e cycle: ?, transpiration; flow; and I' groundwater runoff; G, R, p.Sqipi*i"tt inflltration.

L

t Precipitation inPut (hyetogaph)

'l,r\ t SSeamflow (hyclrograPh)

Figure 1.3 Distributionof precipitationinput'

CHAPTER1

INTRODUCTION

water enteringthe ground may take severalpaths.Somemay be directly evaporatedif adequatetransferfrom the soil to the surfaceis maintained.This can easily occur where a high groundwater table (free water surface) is within the limits of capillary transportto the ground surface.Vegetationusingsoil moistureor groundwatei directly can also transmit infiltrated water to the atmosphereby a processknown as transpiration.Infiltrated water may likewise replenishsoil moisture deficiencies and enter storageprovided in groundwaterreservoirs,which in turn maintain dry weather streamflow.Important bodies of groundwaterare usually flowing so that inflltrated water reachingthe saturated,on" muy be transportedfor considerable' distancesbefore it is discharged.Groundwatermovement is subject, of course,to physicaland geologicalconstraints. Water storedin depressionswill eventually evaporateor infiltrate the ground I surface.Surfacerunoff otti-ut"ty reachesminor channels(gullies, rivulets, and the : like), flowsto major streamsand rivers,and finally reachesan ocean.Along the course of a stream,evaporationand infiltration can also occur. The foregoing discussion suggeststhat the hydrologic cycle, while simple in concept,is actually exceedinglycomplex.Pathstaken by particlesof water precipitated in any arcaare numerousand varied before the seais reached.The time scale may be on the order of seconds,minutes,days,or years. A generalhydrologicequationcan be developedbasedon theprocessesillustrated in Figs. 1.2 and 1.3. ConsiderFig. 1.4. In it, the hydrologicvariablesP, E, T, R, G, and l are as definedin Fig. 1.2. Subscriptss and g are introduced to denote vectorsoriginatingaboveand belowthe earth's surface,respectivd. For example,R,

[" Earth's surface Surface channels

R2

Level of plastic rock . (no water below this level)

Figure 1.4 Regionalhydrologiccycle.

BUDGET 1.4 THE HYDROLOGIC

9

signifies groundwaterflow that is effluent to a surface streamoand E, represents evaporationfrom surfacewaterbodiesor other surfacestorageareas.Letter S stands for storage.The regionunderconsiderationspecifiedasA hasa lower boundarybelow which water will not be found. The upper boundary is the earth's surface.Vertical boundsare arbitrarily set asprojectionsof the peripheryof the region.Remembering that the water budgetis a balancebetweeninflows, outflows,and changesin,storage, Fig. 1.4canbe translatedinto the following mathematicalstatements,whereall values are given in units of volume per unit time: 1. Hydrologicbudgetabovethe surface P+R1 -RrIRr-E"-7,-

1:AS"

(1.1)

AS,

(1.2)

2. Hydrologicbudgetbelow the surface I + Gt- G2- Rr- E, - 4:

3. Hydrologicbudgetfor the region (sum of Eqs' 1.2 and 1.3) p - (Rr- R,) - (E" + E) - (r" + Tr) - (Gr- G,) : a(S, + ss), (1.3) If the subscriptsare droppedfrom Eq. 1.3 sothat letterswithout subscriptsrefer to total precipitation and net valuesof surfaceflow, undergroundflow, evaporation, transpiration,and storage;the hydrologicbudgetfor a regioncan be written simply as p_R-G_E_T:LS

(1.4)

This is the basicequationof hydrology.For a simplifiedhydrologicsystemwhereterms G, E, and Z do not apply, Eq. 1.4 reducesto p-R:AS

(1.5)

Equation 1.4 is applicableto exercisesof any degreeof complexity and is therefore basic to the solution of all hydrologicproblems. The difficulty in solvingpractical problemslies mainly in the inability to measure or estimateproperly the various hydrologic equation terms. For local studies, reliableestimatesoften are made,but on a global scaleqqantificationis usuallycrude. Precipitationis measuredby rain or snowgaugeslocatedthroughoutan area.Surface flows can be measuredusing various devicessuchas weirs, flumes,velocity meters, and depthgaugeslocatedin the rivers and streamsof the area.Under goodconditions these measurementsare 95 percent or more accufate,but large floods cannot be measureddirectly by current methodsand dataon sucheventsare sorelyneeded.Soil moisturecan be measuredusingneutronprobesand gravimetricmethods;infiltration can be deterrnined locally by infiltrometers or estimated through the use of precipitation-runoff data. Areal estimatesof soil moistureand infiltration are generilly very crude,however.The extentandrate of movementof groundwaterareusually exceedinglydifficult to determine,and adequatedataon quantitiesof groundwaterare not alwaysavailable.Knowledgeof the geologyof aregion is essentialfor groundwater estimatesif they are to be more than just rough guides.The determinationof the

1O

CHAPTER1

INTRODUCTION

quantities of water evaporatedand transpired is also extremely difflcult under the presentstateof developmentiofthe science.Most estimatesof evapotranspirationare obtainedby usingevaporationpans,energybudgets,masstransfermethods,or empirical relations.A predicamentinherent in the analysisof large drainagebasinsis the fact that rates of evaporation,transpiration,and groundwatermovementare often assumedto be highly heterogeneous. The hydrologicequationis a usefultool; the readershouldunderstandthat it can be employed in various ways to estimate the magnitude and time distribution of hydrologicvariables.An introductory exampleis given here,and otherswill be found throushout the book. EXAMPLE 1.1 In a given year,a 10,000-mi2wabrshedreceived20 in. of precipitation.The average rate of flow measuredin the river drainingthe areawasfound to be 700 cfs (cubicfeet per second).Make a rough estimateof the combinedamountsof waterevaporatedand transpiredfrom the region during the year of record. Solution. Beginningwith the basic hydrologicequation P - R - G - E - Z : A S

(r.4)

and sinceevaporationand transpirationcan be combined, ET:P-ft-G-AS

{4 t:

.+ v' a'E-;.

(1.6)

The term EZ is the unknown to be evaluatedand P and R are specifled.The equation thus has flve variables and three unknowns and cannot tre solved without additional information. In order to get a solution, two assumptionsare made. First, since the drainageareais quite large (measuredin hundredsof squaremiles),a presumption that the groundwaterdivide (boundary)follows the surfacedivide is probably reasonable.In this casethe G componentmay be consideredzero. The vector R, existsbut is included in R. The foregoingassumptionis usually not valid forsmall areasand mustthereforebe usedcarefully.It is alsopresupposed that AS : 0, thus implying that the groundwaterreservoir volume has not changedduring the year. For such short periods this assumptioncan be very inaccurate,evenfor well-wateredregionswith balancedwithdrawalsand good rechargepotentials.In arid areaswheregroundwateris beingmined (AS consistently negative),it would be an unreasonablesuppositionin many cases.Nevertheless,the assumptionis made here for illustrative purposesand qualified by sayingthat pastrecordsof waterlevelsin the areahaverevealedan approximate constancyin groundwaterstorage.Hydrology is not an exact science,and reasonablewell-foundedassumptionsare required if practical problemsare to be solved. Using the simplificationsjust outlined, the working relation reducesto ET:P_R

'I1 1.6 HYDROLOGIC DATA which canbe solveddirectly.First, changeR into inchesper yearsothat the units are compatible: _ ft3 r

\

t

-

r

sec

R _

.

,

1

sec

area (m n-l

yt

,

6

^ :R,in.

II

7 0 0 x 8 6 ; 4 0 0 x 3 6 5 x 1 :2 104x (5280)'

0.95in.

Therefore,ET : 20 - 0.95 : 19.05in./yr. The amountof evapotranspirationfor the year in questionis estimatedto be 19.05in. This is admittedlya crudeapproximationbut could serveasa useful guide for water resourcesplanning. ll

1.5 HYDROLOGIC MODELS Hydrologic systemsare generally analyzedby using mathematicalmodels.'These modelsmay be empirical, statistical,or foundedon known physicallaws.They may be usedfor suchsimplepurposesas determiningthe rate of flow that a roadwaygrate mustbe designedto handle,or they may guidedecisionsaboutthe bestway to develop a river basinfor a rnultiplicity of objectives.The choiceof the modelshouldbe tailored to the purposefor which it is to be used.In general,'thesimplestmodel capableof producinginformation adequateto deal with the issueshouldbe chosen. Unfortunately,most waterresourcessystemsof practicalconcernhavephysical, social,political, environmental,andlegaldirnensions;andtheir interactionscannotbe exactly describedin mathematicalterms. Furthermore,the historical data necessary for meaningful hydrologic analysesare often lacking or unreliable. And when one considersthat hydrologic systemsare generallyprobabilistic in nature, it is easyto understandthat the modeler'stask is not a simpleone.In fact, it is often the casethat the best that can be hoped for from a model is an enhancedunderstandingof the systembeing analyzed.But this in itself can be of great value, leading,for example, to the implementationof datacollectionprogramsthat canultimately supportreliable modelingefforts. For the most part, mathematicalmodels are designedto describethe way a system'selementsrespondto sometype of stimulus(input). For example,a model of 'a groundwater systemmight be developedto demonstratethe effectson groundwater storageof various schemesfor pumping. Equations 1.1 and L2 are mathematical modelsof the hydrologicbudget,and Figure 1.3 can be considereda pictorial model of the rainfall-runoff process.In later chapters,a variety of hydrologicmodelswill be presentedand discussed.Thesemodelsprovidethe basisfor informed watermanagement decisions.

DATA 1.6 HYDROLOGIC Hydrologic dataarc neededto describeprecipitation; streamflows;evaporation;soil moisture; snow fields; sedimentation;transpiration;infiltration; water quality; air, s9i!, and water temperatures;and other variablesor componentsof hydrologicsys-

12

CHAPTER 1

INTRODUCTION

tems. Sources of data are numerous,with the U.S. Geological Survey being the primary one for streamflow and groundwaterfacts. The National Weather Service (NOAA or National Oceanicand AtmosphericAdministration)is the major collector of meterologicdata.Many other federal,state,and local agenciesand other organizations also compile hydrologicdata. For a completelisting of theseorganizationssee Refs.3 and 17.

1 . 7 COMMONUNITSOF MEASUREMENT

'

Streamandriver flowsare usuallyrecordedascubic metersper second(m3/sec),cubic feetper second(cfs),or second-feet(sec-ft);groundwaterflowsand watersupplyflo\i/s are commonly measuredin gallons per minute, hour, or day (gpm, gph, gpd), or millions of gallonsper day (mgd); flowsusedin agricultureor relatedto water storage are often expressedas acre-feet (acre-ft), acre-feet per unit time, inches (in.) or centimeters(cm) depth per unit time, or acre-inchesper hour (acre-in./hr). Volumesare often given as gallons,cubic feet, cubic meters,acre-feet,secondfoot-days,and inchesor centimeters.An acre-footis equivalentto a volume of water 1 ft deep over 1 acre of land (43,560 ft3). A second-foot-day(cfs-day,sfd) is the accuinulatedvolumeproducedby a flow of 1 cfs in a24-hr period.A second-foot-hour (cfs-hr) is the accumulatedvolume produced by a flow of 1 cfs in t hr. Inches or centimetersof depthrelate to a volume equivalentto that many inchesor centimeters of water over the areaof concern.In hydrologicmassbalances,it is sometimesuseful to note that 1 cfs-day : 2 acre-feetwith sufficient accuracyfor most calculations. Rainfall depthsare usually recordedin inchesor centimeterswhereasrainfall rates are given in inches or centimetersper hour. Evaporation,transpiration, and infiltration rates are usually given as inchesor centimetersdepthper unit time. Some usefulconstantsand tabulatedvaluesof severalof the physicalpropertiesof water are given in Appendix A at the end of the book.

1.8 APPLICATIONOF HYDROLOGYTO ENVIRONMENTALPROBLEMS It is true that humanscannot exist without water; it is also true that water, mismanaged,or during times of deficiency(droughts),or times of surplus(floods),can be life threatening.Furthermore,there is no aspectof environmentalconcernthat doesnot relate in someway to water. Land, air, and water are all interrelatedas are water and all life forms. Accordingly, the spectrum of issuesrequiring an understandingof hydrologicprocessesis almost unlimited. As waterbecomesmore scarceand as competition for its useexpands,the need for improved water managementwill grow. And to provide water for the world's expandingpopulation, new industrial developments,food production, recreational demands,and for the preservationand protection of natural systemsand other purposes,it will becomeincreasinglyimportant for us to achievea thoroughunderstanding of the underlyinghydrologicprocesseswith which we must contend.This is the challengeto hydrologists,waterresourcesengineers,planners,policymakers,lawyers, economists,and others who must strive to see that future allocationsof water are sufficient to meet the needsof human and natural svstems.

PROBLEMS

13

r summary distribution' moveHydrology is the scienceof water. It embracesthe occurrence, sense,an account-"nt, urii propertiesof the watersof the earth. In a mathematical sothat a history ing may be madeof the inputs,outputs,and waterStofagesof a region of water movementfor the region can be estimated' the hydrologic After reading this chapter you should be able to understand shouldalso You budgetand make a simpleu".ouniing of water transportin a region' to be used facilitate have gained an undersiandingof trow hydrologic analysescan designand managementprocessesfor water resourcessystems'

PROBLEMS of 50 mi2.Convertthis

1.1.. One-half inch of runoff resultsfrom a stormon a drainagearea

amount to acre-feetand cubic meters. surfacearea of t.2. Assume you afe dealing with a vertical walled reservoir having a will it take to hours many How occurs: m3/sec 1.0 of 500,000 m' and that anlnflow raise the reservoirlevel bY 30 cm? time is 15 acre-ft and 1.3. consider that the storageexistingin a river reachat a reference from the reachis outflow the and cfs 500 is reach tie at the sametime the inflow to 650cfs.onehourlater,theinflowis550cfsandtheoutflowis630cfs.Findthe meters' changein storageduring the hour in acre-feetand in cubic walled reservoir was t.4. During a24-hr time period, the inflow to a 500-acre vertical a rise or fall in there in. was 1 was evaporation interval, same the 100 cfs. During centimeters' and in inches surfacewaterelevation?How muchwasit? Give the answer areais 3000 acres' 1.5. The annualevaporationfrom a lake is 50 in. If the lake's surface centimeters? in and acre-feet in rate evaporation daity what would beiire time requiredto raise 1.6. A flow of 10 cfs entersa 1-mi2vertical walledreservoir.Find the the reservoirlevelbY 6 in. Iftheaverageannualrunoffis5l02cfsand areaof4511mi2. t.7. Adrainagebasinhasan lossesfor the areain the averalerainfall is 42.5 in.,estimatethe evaportranspiration is? estimate this you think 1 year. Iiow reliable do Determinethe storage 1.8. The storagein a reachof a river is 16.0acre-ft at a given time. during the hour are outflow and inflow of (u"r"-f""tj t hr later if the averagerates 700 and 650 cfs, resPectivelY. areafor 3 days' (a) L.9. Rain falls atataverage irrtensity of 0.4 in./hr over a 600-acfe the 3-day (b) determine per second; feet Determinethe averagerate ofrainfau in cubic volumeofrainfallinacre-feet;and(c)determinethe3-dayvolumeofrainfallininches of equivalentdepth over the 600-acrearea' 100 acre-ft/day'Deter1.10. The evaporationrate from the surfaceof a 3650-acrelake is year ifthe inflow to the lake a 365-day during lake (feet) in the mine the depthchange is25.2cfs.1s the changein lake depth an increaseor a decrease? acre-feetif the drainage 1.11. One and one-half inchesof runoff areequivalentto how many : ft"') 43,560 areais 25-mi2?lNote: I acte of how many cubic feet L.12. one-half inch of rain per day is equivalentto an averagerate per second? meters many p". ,".ona if the areais 500 acrei?How

14

cHAPTER1 INTRoDUCTIoN

REFERENCES 1. P. B. Jones,G. D. Walker,R. W. Harden,and L. L. McDaniels,"The Developmentof the Scienceof Hydrology," Circ. No. 60-03, TexasWater Commission,Apr. 1963. 2. W. D. Mead, Noteson Hydrology. Chicago:D. W. Mead, 1904. 3. Ven Te Chow (ed)., Handbook of Applied Hydrology. New York: McGraw-Hill,1964. 4. O. E. Meinzer,Hydrology,Vol. 9 of Physicsof the Earth.New York: McGraw-Hlll, 1942. Reprintedby Dover, New York, 1949. 5. P. D. Krynine, "On the Antiquity of Sedimentationand Hydrology," Bull. Geol. Soc. Am. 70. l7 2I - l7 26(1960\. 6. RaphaelG. Kazmann,Modern Hydrology.New York: Harper & Row, 1965. 7. H. Pazwoshand G. Mavrigian, "A Historical Jewelpiece-Discovery of the Millennium Hydrologic Works of Karaji," WaterResourcesBull. 16(6), 1094-1096(Dec. 1980), 8. Hunter Rouseand Simon Ince, History of Hydraulics, Iowa Institute of Hydraulic Re- :. search,State University of Iowa, 1957. 9. G. H. L. Hagen, "Ueber die Bewegungdes Wassersin engen cylindrischenRohren," Poggendorfs Ann. Phys. Chem.16, 423- 442(1839). 10. Henri Darcy, Les fontaines publiques de la ville de Dijon. Paris:.V. Dalmont, 1856. 11. J. Dupuit, Etudesthdoriqueset practiques sur le mouvementdes eaux dans les canauxs dtcouvertset d travers les terrainspermdables,2nded. Paris:Dunod, 1863. 12. L. K. Sherman,"Stream Flow from Rainfall by the Unit-Graph Method," Eng. NewsRec.108(1932). 13. R. E. Horton, "The Role of Infiltration in the Hydrologic Cycle," Trans.Am. Geophys. Union 14, 446- 460(1933). l 14. C. V. Theis,"The RelationBetweenthe Lowering of the PiezometricSurfaceand the Rate and Duration of a Well Using Ground WaterRecharge,"Trans.Am. Geophys.Union 16, 519-524(1935\. 15. Asit K. Biswas,"Hydrologic EngineeringPrior to 600 s.c.," Proc. ASCE J. Hyd. Div., Proc. Paper5431,Vol. 93, No. HY5 (Sept.1967). 16. RaymondL. Nace,"WaterResources:A GlobalProblemwith Local Roots,"Environ. Sci. Technol.1(7) (July i967). 17. D. K. Todd (ed.), The WaterEncylopedia.New York: Water Information Center, 1970.

Chapter2

Precipitation

r Prologue The purposeof this chapteris to: ' Define precipitation, discussits forms, and describeits spatial and temporal attributes. ' Illustrate techniquesfor estimatingareal precipitation amountsfor specific storm eventsand for maximum precipitation-generatingconditions. Precipitation replenishessurfacewater bodies,rbnewssoil moisture for plants, and rechargesaquifers.Its principal forms are rain and snow.The relative importanceof theseforms is determinedby ttre climate of the area under consideration.In many parts of the westernUnited States,the extentof the snowpackis a determiningfactor relative to the amountof waterthat will be availablefor the summergrowing season. In more humid localities, the timing and distribution of rainfall are of principal concern. Precipitatedwaterfollows the pathsshownin Figs. r.2 and,1.3.some of it may be intercepted,evaporated,infiltrated, and becomesurfaceflow. The actual disposition dependson the amountof rainfall, soil moistureconditions,topography,vegetal cover soil type, and other factors Hydrologic modeling and water resourcesassessments dependupon a knowledgeof the form and amountof precipitation occurringin a region of concernover a time period of interest.

2.1 WATERVAPOR The fraction of watervapor in the atmosphereis very small comparedto quantitiesof other gasespresent,but it is exceedinglyimportant to our way of life. Precipitationis derived from this atmosphericwater. The moisture centent of the air is also a significantfactor in local evaporationprocesses.Thus it is necessaryfor a hydrologist to be acquaintedwith waysfor evaluatingthe atmosphericwatervapor contentand to understandthe thermodynamiceffects of atmosphericmoisture.l

CHAPTER2

16

PRECIPITATION

Under most conditions of practical interest (modest ranges of pressureand temperature,provided that the condensationpoint is excluded),water vapor essentialiy obeys the gas laws. Atmospheric moisture is derived from evaporationand transpiration,the principal sourcebeing evaporationfrom the oceans.Precipitation overthe United Statescomeslargelyfrom oceanicevaporation,the watervaporbeing transporatedover the continentby the primary atmosphericcirculation system. Measuresof watervapor or atmospherichumidity are relatedbasicallyto conditions of evaporationand condensationoccurring over a level surfaceof pure water. Considera ilosed systemcontaining approximatelyequal volumesof water and air maintainedat the sametemperature.If the initial condition of the air is dry, evaporation takesplace and the quantity of water vapor in the air increases.A measurement of pressurein the airspacewill reveal that as evaporationproceeds,pressurein the airipaceincreasesbecauseof an increasein partial pressureof the watervapor (vapor preJsure).Evaporationcontinuesuntil vapor pressureof the overlying air equalsthe surfacevapor pressure[a measureof the excessof water moleculesleaving(evaporating from) the water surfaceover thosereturning]. At this point, evaporationceases, and if the temperaturesof the air spaceand water are equal,the airspaceis saidto be saturated.If the containerhad beenopeninsteadof closed,the equilibriumwould not havebeenreached,and all the water would eventuallyhaveevaporated.Somecommonly usedmeasuresof atmosphericmoistureor humidity are vapor pressure,absolute humidity, specifichumidity, mixing ratio, relative humidity, and dew point tem-

,

Perature.

Amount of PrecipitableWater .

Estimatesof the amount of precipitation that might occur over a given region with favorable conditions are often useful. These may be obtained by calculating the amountof water containedin a column of atmosphereextendingup from the earth's surface.This quantity is known as theprecipitable water 14{althoughit cannotall be removed from the atmosphereby natural processes.Precipitable water is usually expressedin centimetersor inches. An equationfor computingthe amountof precipitablewater in the atmosphere can be derived as follows. Considera column of air having a squarebase 1 cm on a side.The total water masscontainedin this column betweenelevationzero and some height z would be

W:

r p*dz

(2.1)

J^

wherep. : the absolutehumidity and IVis the depthof precipitablewaterin centimeters. The integral can be evaluatedgraphically or by dividing the atmosphereinto layersof approximatelyuniform specifichumidities,solvingfortheseindividually, and then summing.Figure 2.1 illustratesthe averageamountof precipitablewater for the continentalUnited Statesup to an elevationof 8 km.2

Geographicand TemporalVariations The quantity of atmosphericwater vapor varieswith location and time. Thesevariations may be attributed mainly to temperatureand sourceof supply considerations. The greatestconcentrationscan be found near the ocean surfacein the tropics, the

2.2 PRECIPITATION

17

Sault Ste.Mtrie Portled

0;7 VCT -NJ 0.8

0.9 1.0

0.8 b.z o.d1'o 1.1 r.2 t.J

Bromsville

Figure 2.L Mean precipitablewater, in inches,to an elevationof 8 km. (U.S. WeatherBureau.)2

concentrationsgenerallydecreasingwith latitude, altitude, and distanceinland from coastalareas. About half the atmosphericmoisturecanbe found within the first mile abovethe earth's surface.This is becausethe vertical transport of vapor is mainly through convectiveaction,which is slight at higher altitudes.It is also of interestthat there is not necessarilyany relation betweenthe amount of atmosphericwater vapor over a regionand the resultingprecipitation.The amountof water vapor containedover dry areasof the Southwest,for example,at times exceedsthat over considerablymore humid northern regions,eventhough the latter areasexperienceprecipitation while the former do not.

2.2 PRECIPITATION Precipitation is the primary input vector of the hydrologiccycle. Its forms are rain, snow,andhail andvariationsof thesesuchasdrizzle and sleet.Precipitationis derived from atmosphericwater, its form and quantity thus being influencedby the action of other climatic factors such as wind, temperature,and atmosphericpressure.Atmosphericmoistureis a necessarybut not sufficient condition for precipitation. Continental air massesare usuallyvery dry sothat mostprecipitationis derivedfrom moist maritime air that originatesoverthe oceans.In North America about50 percentof the evaporatedwater is taken up by continental air and movesback againto the sea.

18

CHAPTER2

PRECIPITATION

Formationof PreciPitation Two processesare consideredto be capableof supportingthe growth of dropletsof sufficient mass(dropletsfrom about 500 to 4000 p'min diameter)to overcomeair resistanceand consequentlyfall to the earth asprecipitation.Theseare known as the process' ice crystalprocessand the coalescence the small cloud dropletsincreasetheir which by process is one The côalescence collision. Water droplets may be through droplets other with size due to contact gravitationaland air resistance to both subjected are that bodies falling consideredas are proportional to the (terminal velocities) equilibrium at effects. Fall velocities descendmore quickly will droplets larger the thus the droplet; of squareof the radius overtakenby larger are often droplets smaller result, As a ones. than the smaller increasingly producing the drops, to unite tend collisions resulting droplets,and the into smallup break in diameter) mm (order 7 of drops large largir particles.Very effect. chain of a produce somewhat and process coalescence dropletsthat repeatitre significant generate to produced be may raindrops large In this *unn"r, sufficiently precipitation. This processis ionsidered to be particularly important in tropical regionsor in warm clouds. An important type of growth is known to occurif ice crystalsand waterdroplets -40'C- Under are found toexist togetherat subfreezingtemperaturesdown to about theseconditions,certain particles t saltsserveasfreezingnucleisothat theseconditionsis higher over the t condensationoccurson the surface unevenparticle sizedistributionsde' with otherparticles.This is considet mechanism. The artificial inducementof precipitation has been studied extensively,and thesestudiesare continuing.It has been demonstratedthat condensationnuclei supplied to cloudscan induceprecipitation.The ability of humansto ensurethe produciion of precipitation or to control its geographiclocation or timing has not yet been attained,however' Many legal as well as technologicalproblemsare associatedwith the prospects ..rain-makiig" processes. Of interesthereis the impacton hydrologicestimatesthat of partially controlled artificial precipitation might have. Many uncontrolled oi onty are consideredas statisticalvariatesthat are variables Lydrologic naturally occurring with a random component.If the distribudistributed or either randomty distrlUuted an inferenceasto the frequencyof modeled, can be variable the tion or time seiiesof given magnitude(suchas precipitaa of events hydrologic occurrenceof significant used and if the effectsof these are controls artificial however, If, tion) can be made. prove to be totally unreliable may analyses frequency predicted, cannot be reliably tools.

PrecipitationTyPes Dynamic or adiabaticcoolingis the primary causeof condensationand is responsible for most rainfall. Thus it can be seen that vertical transport of air massesis a requirementfor precipitation.Precipitationmay be classifiedaccordingto the condi-

'1 2.2 PRECIPITATIOI'|9 tions that generatevertical air motion. In this respect,the three major categoriesof precipitation type are convective,orographic, and cyclonic. Convective Precipitation Convectiveprecipitationis typical of the tropicsand is brought about by heatingof the air at the interfacewith the ground. This heatedair expandswith a resultantreductionin weight.During this period,increasingquantities of water vapor are taken up; the warm moisture-ladenair becomesunstable;and pronouncedvertical currents are developed.Dynamic cooling takes place, causing condensationand precipitation.Convectiveprecipitationmay be in the form of light showersor stormsof extremelyhigh intensity (thunderstormsare a typical example). Orographic Precipitation Orographicprecipitationresultsfrom the mechanical lifting of moist horizontal air currentsover natural barriers suchas mountainranges. This type of precipitation is very common on the West Coast of the United States where moistureladen air from the Pacific Oceanis interceptedby coastalhills and mountains.Factorsthat are important in this processinclude land elevation,local slope,orientation of land slope,and distancefrom the moisture source. In dealingwith orographicprecipitation,it is commonto divide the regionunder study into zonesfor which influencesasidefrom elevationare believedto be reasonably constant.For eachof thesezones,a relation betweenrainfall and elevationis developedfor usein producingisohyetalmaps(seeSection2,5). with the movementof Cyclonic Precipitation Cyclonicprecipitationis associated air massesfrom high-pressureregionsto low-pressureregions.Thesepressuredifferencesare createdby the unequalheatingof the earth's surface. Cyclonicprecipitationmay be classifiedasfrontal or nonfrontal. Any barometric low canproducenonfrontal precipitationasair is lifted throughhorizontalconvergenceof the inflow into a low-pressurearea. Frontal precipitation results from the lifting of warm air over cold air at the contact zone between air masseshaving different characteristics.If the air massesare moving so that warm air replacescolder warm air, the front is known asawarmfront; if , on the otherhand,cold air displaces air, the front is saidto be cold.If the front is not in motion,it is saidto be a stationary front. Figure 2.2 illustratesa vertical sectionthrough a frontal surface.

Figure 2.2 Vertical cross-sectionthrough a frontal surface.

20

CHAPTER 2 PRECIPITATION

Thunderstorms Many areasof the United Statesare subjectedto severeconvectivestorms,which are generallyidentifiedasthunderstormsbecauseof their electricalnature.Thesestorms, although usually very local in nature, are often productive of very intenserainfalls that are highly significantwhen local and urban drainageworks are considered. Thunderstormcells developfrom vertical air movementsassociatedwith intense surfaceheatingor orographiceffects.Thereare threeprimary stagesin the life history of a thunderstorm.Theseare the cumulusstage,the mature stage,andthe dissipating stage.Figure 2.3 illustrateseachof thesestages. All thunderstormsbeginascumulusclouds,althoughfew suchcloudseverreach the stage of developmentneededto produce such a storm. The cumulus stageis characterizedby strongupdrafts that often reachaltitudesof over 25,000ft. Vertical wind speedsat upperlevelsare often as greatas 35 mph. As indicatedinFig.2.3a, there is considerablehorizontal inflow of air (entrainment)during the cumulusstage. This is an important elementin the developmentof the storm, as additional moisture is provided.Air temperaturesinside the cell are greaterthart thoseoutside,as indicatedby the convexity of the isothermsviewed from above.The number and size of The duration ofthe cumulusstage the water dropletsincreaseasthe stageprogresses. is approximatelyi0-15 min. The strong updrafts and entrainmentsupport increasedcondensationand the developmentof waterdropletsand ice crystals.Firrally, whenthe particlesincreasein size and number so that surfaceprecipitation opcurs,the storm is said to be in the mature stage.In this stage strong downdrafts are created as falling rain and ice crystals cool the air below. Updraft velocities at the higher altitudes reach up to 70 mptrin the early periodsof the maturestage.Downdraft speedsof over20 mph are

El

o o F

(a)

(b)

(c)

Figure 2.3 Cumulus,mature,and dissipatingstagesof a thunderstormcell. (Department of the Army.)

2.2 PRECIPITATION 21

usual aboveabout 5000 ft in elevation.At lower levels,frictional resistancetendsto decreasethe downdraft velocity. Gusty surfacewinds move outwardfrom the region of rainfall. Heavyprecipitationis often derivedduring this preiod, which is usuallyon the order of 15-30 min. In the final or dissipatingstage,the downdraftbecomespredominantuntil all the air within the cell is descendingand being dynamically heated.Since the updraft ceases,the mechanismfor condensationends and precipitation tails off and ends.

PrecipitationData Considerabledata on precipitation are available in publications of the National WeatherService.a's Other sourcesincludevariousstateand federal agenciesengaged in water resourceswork. For critical regional studiesit is recommendedthat all possibledata be compiled; often the establishmentof a gauging network will be necessary(seealso Chapters7-9).

Precipitation VariabiIity Precipitationvariesgeographically,temporally, and seasonally.Figure 2.4 indrcates the mean annualprecipitation for the continentalUnited States,while Fig. 2.5 gives an exampleof seasonaldifferences.It should be understoodthat both regional and temporal variationsin precipitation are very important in water resourcesplanning and hydrologicstudies.For example,it may be very important to know that the cycle of minimum precipitationcoincideswith the peakgrowing seasonin a particular atea, or that the periodofheaviestrainfall shouldbe avoidedin schedulingcertainconstruction activities. Precipitation amounts sometimesvary considerablywithin short distances. Recordshaveshowndifferencesof 20 percentor more in the catchof rain gaugesless that2Oft apart.Precipitationis usuallymeasuredwith a rain gaugeplacedin the open so that no obstacleprojects within the inverted conical surfacehavingthe top of the gaugeas its apexand a slopeof45'. The catchofa gaugeis influencedby the wind, which usually causeslow readings.Variousdevicessuchas Nipher and Alter shields havebeendesignedto minimize this error in measurement.Precipitationgaugesmay be of the recording or nonrecordingtype. The former are requiredif the time distribution of precipitation is to be known. Information about the featuresof gaugesis readily available.3 Becauseprecipitationvariesspatially,it is usuallynecessaryto usethe datafrom severalgaugesto estimatethe averageprecipitation for an area and to evaluateits reliability (seeChapter27). This is especiallyimportant in forestedareaswhere the variation tendsto be large. Time variations in rainfall intensity are extremely important in the rainfallrunoff process,particularly in urban areas(seeFi g. 2.6a). The arealdistributionis also significantandhighly correlatedwith the time history of outflow (seeFig. 2.6b).These considerationsare discussedin greaterdetail in following chapters.

g

6

o

n

c)

() H

o U) o E t r 9

.n

EF

C) d

Q

.v) C) k

H

h0 A

C)

!

()

n"i c.)

o

€ a

() 6

s o E o

U F

e
0.04 in., lossesin the rangeof 10-40 percentare realistic.3 Figure 3.1 illustratesthe generaltime distribution pattern of interceptionloss intensity.Most interceptionlossdevelopsduring the initial stormperiod and the rate of interception rapidly approacheszero thereafter.l-6Potentialitorm interception lossescan be estimatedbv usins2'3,6 Zi:,S+KEt

(3.1)

where . L, : the volume of water intercepted(in.) S : the interceptionstoragethat will be retainedon the foliage againstthe forcesof wind and gravity (usuallyvariesbetween0.01 and 0.05 in.) K : the ratio of surfaceareaof interceptingleavesto horizontal projection:, of this area E : the amountof waterevaporatedper hour duringthe precipitationperiod ' (in.) : r time (hr)

i= it+i2+4+i4

Figure 3.1 Disposition of rainfall input in terms of interception, depressionstorage,infiltration, and overlandflow.

3.1 INTERCEPTION 43 Equation3.1 is basedon the assumptionthat rainfall is sufficientto fully satisfy the storageterm S. The following equationwas designedto accountfor the rainfall amountT-e L;:S(1 -e-P/s)+KEt

(3.2)

whereP : rainfall and e is the baseof natural logarithms.Note in Eqs. 3.1 and 3.2 that the storm time duration t is given in hours, while ,L,, S, and E are commonly measuredin in. or mm. It is important to recognizethat forms of vegetationother than trees can aiso interceptlarge quantitiesof water. Grasses,crops,and shrubsoften haveleaf-areato ground-arearatios that are similar to thosefor forests.Table3.2 summarizessome observationsthat havebeen madeon crops during growing seasonsand on a variety of grasses.Interceptedamountsare aboutthe sameasthosefor forests,but sincesome of thesetypes of vegetationexist only until harvest,their annualimpact on interception is generallylessthan that of forestedareas. Precipitationtype, rainfall intensityand duration,wind, and atmosphericconditions affecting evaporationare factors that serve to determineinterception losses. Snow interception,while highly visible,usually is not a major loss sincemuch of the interceptedsnowfall is eventuallytransmittedto the ground by wind action and melt. Interceptionduring rainfall eventsis commonly greaterthan for snowfall events.In both cases,wind velocity is an important factor. The importanceof interceptionin hydrologicmodelingis tied to the purposeof the model. Estimates of loss to gross precipitation through interception can be significantin annual or long-term models,but for heavy rainfalls during individual storm events,accountingfor interceptionmay be unnecessary.It is important for the modeler to assesscarefully both the time frame of the model and the volume of precipitation with which one must deal. TABLE3.2 OBSERVED PERCENTAGES OF INTERCEPTION BY VARIOUSCROPSAND GRASSES'

Vegetation type Crops Alfalfa Corn Soybeans Oats Grassesb Little bluestem Big bluestem Tall panic grass Bindweed Buffalo grass Blue grass Mixed species Natural grasses

Intercepted(%)

Comments

36 16 15 7 50-60-.1

s7 l s 7 f 17 l ) 31 17

Water applied at rate of ] in. in 30 min

Pdor to harvest

)A

t4-t9

"Valuesroundedto nearestpercent.Data for table were obtainedfrom Refs.2,4, and 5, 'Grass heightsvary up to about 36 in.

44

CHAPTER3

INTERCEPTIONAND DEPRESSIONSTORAGE

Equations3.1 and 3.2 canbe usedto estimatetotal interceptionlosses,but for detailedanalysesofindividual storins,it is necessaryto dealwith the areal variability of suchlosses.Generalequationsfor estimatingsuchlossesare not available,however. Most researchhas been related to particular speciesor experimentalplots strongly associatedwith a given locality. In addition, the lossfunction varieswith the storm's character.If adequateexperimentaldata are available,the nature of the varianceof interceptionversustime might be inferred. Otherwise,common priictice is to deduct the estimatedvolumeentirelyfrom the initial period of the storm(initial abstraction). EXAMPLE 3.1 Using the following equationsdevelopedby Horton6for interceptionby ash and oak trees, estimatethe interception loss beneaththesetrees for a storm having a total precipitationof 1.5 in. Solution 1. For ashtrees.

L ; : 0 . 0 1 5+ 0 . 2 3 P : 0.36in. : 0.015+ 0.23(1.5) 2. For oak trees,

L;:0.03+0.22P : 0.36in. rl : 0.03+ 0.22(1.5)

3.2 THROUGHFALL A numberof relationshipsfor estimatingthroughfall for a variety of foresttypeshave been developed.ntt Deiermining factors for throughfall quantities include canopy coverage,total leaf area,numberand type of layersof vegetation,wind velocity, and rainfall intensity.The arealvariability ofthesefactorsresultsin little or no throughfall in somelocationsand considerablethroughfall in others.In general,prediction equations for throughfall mustincludemeasuresof canopysurfaceareaand coverasprime variables.An example of a throughfall relationship for an easternUnited States hardwoodforest follows.l2 For the growing season T n : 0 . 9 0 1 P- 0 . O 3 l n

(3.3)

For the dormant season T n : 0 . 9 I 4 P- 0 . 0 1 5 n where ?1,: throughfall (in.) P : total precipitation (in.) n : number of storms

(3.4)

3.3

DEPRESSIONSTORAGE

45

3.3 DEPRESSION STORAGE Precipitationthat reachesthe ground may infiltrate, flow over the surface,or become trappedin numeroussmall depressionsfrom which the only escapeis evaporationor infiltration. The natureof depressions, aswell astheir size,is largely a funition of the original land form and local land-usepractices.Becauseof extremevariability in the nature of depressionsand the paucity of sufficient measurements,no generalized relation with enoughspecifiedparametersfor all casesis feasible.A rational model can, however,be suggested. Figure 3.1 illustratesthe dispositionof a precipitationinput. A studyof it shows that the tate at which depressionstorageis filled rapidly declinesafter the initiation of a precipitationevent.Ultimately, the amountof precipitation goinginto depression storagewill approachzero,given that thereis alargeenoughvolume of precipitation to exceedother lossesto surfacestoragesuch as inflltration and evaporation.Ultimately, all the water stored in depressionswill either evaporateor seep into the ground.Finally, it shouldbe understoodthat the geometryof a land surfaceis usually complex and thus depressionsvary. widely in size, degreeof interconnection,and contributingdrainagearea.In general,depressionsmay be looked upon as miniature reservoirsand as suchthey are subjectto similar analytical techniques. According to Linsley et a1.13 the volume of water storedby surfacedepressions at any given time can be approximatedusing Y:Sd(l

-e-kP")

r? 5)

where V : the volume actually in storageat sometime of interest S, : the maximum storagecapacityof the depressions P" : the rainfall excess(grossrainfall minus evaporation,interception,and infiltration) k : a constantequivalentto l/So The valueof the constantcan be determinedby consideringthat if P" : 0, essentially all the water will fill depressionsand dv/dp" will equal one. This requires that k : r/Sa. Estimatesof s, may be securedby making samplefleld measurementsof the areaunder study.Combiningsuchdata with estimatesof P" permits a determination of V. The mannerin which Vvarieswith time must still be estimatedif depression storagelossesare to be abstractedfrom the grossrainfall input. One assumptionregarding dVldt is that all depressionsmust be full before overlandflow supply begins.Actually, this would not agreewith reality unlessthe locationsof depressionswere gradedwith the largestonesoccurring downstream.If the depressionstoragewere abstractedin this manner, the total volume would be deductedfrom the initial storm period suchas shownby the shadedareain Fig.3.2. Such postulateshave been used with satisfactory results under special circumstances.ra Depressionstorageintensitycan alsobe estimatedusingEq. 3.5. If the overland flow supplyrale oplus depressionstorageintensityequali - /, wherei is the rainfall intensity reachingthe ground and/is the infiltration rate, then the ratio of overland flow supply to overlandflow plus depressionstoragesupply can be proved equal to

i - f

(3.6)

46

CHAPTER 3 INTERCEPTION ANDDEPRESSION STORAGE

o b0 o

!t o

E

d

4

8

1

2

16

20

Time (min)

Figure3.2 Simpledepression storageabstraction scheme. This expressioncan be derivedby adjudging c : i - f - o (3.7) i - f i - f and noting that o is equal to the derivativeof Eq. 3.5 with respectto time. Then )

o:fiso1t_e-kP")

(3.8)

u : (Soke-kg#

(3.e)

It was shown that k : 1/S, so that

u : ,-o'"d!" dt

(3.10)

The excessprecipitationP, equalsthe grossrainfall minus infiltrated water, and since the derivativewith respectto time canbe replacedby the equivalentintensity(i * f), the intensity of depressionstoragebecomes o:e-or.(i-f)

a

i - f

(i-f)*G-f)e-."" i - f

(3.11)

(3.r2)

o.125

o.25

0.50(turf)

0.315

Mass overlandflow and depressionstoragesupply ( P - F)

0.0625 All depressions filled before overland flow supply begins -= :\

----l

0.0938

0.125(pavements) 1.00

I

on

d

-

9)-

RO

0.80

E

l-

bl I

o "

7

n

Exponential relationship a

tl

i,t

-\

-lP-F)tS, " = -I - e I

F

I

,

i 6 0 6

o b!

OGEE sumrnationof the standardprobability curve

o

g

n5n

a

B o E

a

{ .:

-

o.7o I

-

40

6

B

9 d J U

o

o o0

s2 o

6zo

I o

!)

E

t

o

o

0

50

100

150

200

Mean depth as a percentage of overall depth of depression storage

Figure 3.3 Depth distributioncurve ofdepressionstorage.Enter graph from top, readdown to selectedcurve, and project right or left as desired.(After Tholin and Kiefer.r5) and

o

i * f

-

:(i

:

f)(I

- e-kP")

i - f |

-

g-kP"

(3.13) (3.r4)

Figure 3.3 illustratesa plot of this function versusthe massoverlandflow and depressionstoragesupply(P - F), whereF is the accumulatedmassinfiltrationl5 and

48

CHAPTER3

AND DEPRESSION STORAGE INTERCEPTION

P is the grossprecipitation.In the plot meandepthsof 0.25 in. for turf and 0.0625in. for pavementswere assumed.Maximum depthswere 0.50 and0.125in., respectively. The figure also depictsthe effect on estimatedoverlandflow supplyrate, which is derivedfrom the choiceof the depressionstoragemodel. Three modelsare shown in the figure: the flrst one assumesthat all depressionsare full before overlandflow begins.For a turf area having depressionswith a mean depth of 0.25 in., the figure showsthat for P - F valueslessthan 0.25 in., thereis no overlandflow supply,while for P - F valuesgreater than 0.25 in., the overlandflow supply is equal to i - f . For the exponentialmodel (Model 2), c alwayswill be greaterthanzero. Tholin and Kiefer haverecommendedthat a relation betweenthosepreviouslymentionedis A cumulative normal likely more representativeof fully developedurban areas.15 describedin Fig. 3.3 and is also probability curve was selectedfor this representation (Model 3). Depressionstoragedeductionsare usually madefrom the first part of the storm as illustrated in Fig. 3.2. The amount to be deductedis a function of topography, groundcover,and extentand type of land development.During major storms,this loss is often consideredto be negligible.Someguidelinesfor estimatingdepressionstorage losseshavebeen developedbasedon studiesof experimentaland other watersheds. Values for depressionstoragelossesfrom intense storms reported by Hicks are Tholin andKieferhaveused 0.20 in. for sand,0.15in. for loam,and0.10in. for c1ay.16 Studiesof in. forpavements.ls 0.0625 and valuesof 0.25 in. inpervious urban areas yielded information shownin the by Viessman four small imperviousdrainageareas This is slope. correlated with loss is highly Fig.3.4, where mean depressionstorage

0.15

I Ei

U.IU

0

1

2

3

4

Slope(70)

Figure 3.4 Depression storage loss versus slope for four impervious drainageareas.(From Viessman.ra)

PROBLEMS

49

h

oo

Antecedent rainfall during preceding 30 min o

Time (min)

Figure3.5 Depression storage intensityversustimefor animpervious area.(After Turner.l7)

easilyunderstood,sincea given depressionwill hold its maximum volumeif horizontally oriented. Using very limited data from a small, paved-streetsection, Turner devisedthe curvesshownin Fig. 3.5.17Other sourcesof datarelatedto surfacestorase are availablein the literature.2,18,1e

r Summary Accountingfor the dispositionof precipitation is an important part of the hydrologic modelingprocess.Two abstractionsfrom the precipitation input, intercepiion, and depressionstoragewere coveredin this chapter. Interception lossesduring the courseof a year may be substantial,but during intensestorms,they may be sufficiently small to neglect.Precipitationtype, rainfall intensity and duration, wind, and atmosphericconditions affeiting evaporationare factorsthat serveto determineinterceptionlossesfor a given foresi standor ground coverconfiguration.Interceptionduring rainfall eventsis commonly greaterihan for snowfall events. Depressionstoragedeductionsoccur early in a storm sequenceand they are a function of topography,ground cover, and extent and type of land development. During major storms,this loss is often consideredto be negligible.

PROBI.EMS 3.1. UsingFig. 3.2, estimatethe volume of depressionstoragefor a 3-acrepaveddrainage area' Statethe volume in cubic feet and cubic meters.Convert it to equivalentdep;h over the area in in. and cm.

50

CHAPTER3

INTERCEPTIONAND DEPRESSIONSTORAGE

3.2. Estimatethe percentageof the total volume of rainfall that is indicatedas depression storagein Fig. 3.2. 3.3. Using the averageannual precipitationfor your state,estimatethe annual amountof interceptionloss. 3.4. Refer to Fig. 2.4 and estimate the annual interception lossesin lllinois, Florida, California, and New Mexico. How good do you think theseestimatesare?In which estimatesdo you havethe most confldence?Why? In which of thesestateswould the water budgetbe most affectedby interception? 3.5. Using Fig. 3.4, estimatethe percentageof rainfall that would be lost to depression storagefor a l0-acre parking lot havinga mean slopeof 1 percent.Repeatfor a slope of 3 percent.Using the total rainfall volume determinedin Problem 3.2, estimatethe equivalentdepth over the area of the depressionstorageloss for both slopes.Stater depthsin mm and in. 3.6. Refer to Fig. 3.3 and estimatethe ratio of overlandflow supplyto overlandflow and depressionstoragesupplyif the areais turf, the OGEE summationcurve is the model, and the mean depth of depressionstorageis (a) 75 percent and (b) 125 percent. 3,7. Explain how a relation such as that given in Fig. 3.3 could be used in a simulation model of the rainfall-runoff process' 3.8. Using Eqs. 3.3 and 3.4, estimatethe throughfall in in. for 28 in. of rainfall during the growing season(21 events), and 17 in. of rainfall during the dormant season(13 events). 3.9. Using Horton's equationsgiven in Example 3.1, estimatethe interceptionlossesby ash and oak trees for a storm having a total precipitation of 1.33 in'

REFERENCES "The Effect of Forestand Pastureon the Dispositionof Precipitation," 1. C. E. Schomaker, Maine Farm Res.(July 1966). 2. ven Te chow (ed.), Handbook of Apptied Hydrology. New York: McGraw-Hill,1964. 3 . JosephKittredge, ForestInfluences.New York: McGraw-Hill' 1948. "Interception of Rainfall by HerbaceousVegetation,"Science86(2243), i + . O. n. Ctark,

59r-s92(r937).

"Resultsof the Mountain Home Rainfall Interceptionand Inflltration Project 5. J. S. Beard, on Black Wattle, 1953-1954," J. S. Afr. Foresty Assoc'27,72-85(1956)' "Rainfall Interception," Monthly WeatherRev.47,603-623(L9I9)' 6. R. E. Horton, "A Note on the InterceptionLoss Equation," J. Geoplrys.Res.65' 38507. R. A. Meriam, 1 ( 1 9 6 0 ) . 385 8. D. M. Gray (ed.),Hand.bookon the Principles of Hydrology.National ResearchCouncil, Canada,Port Washington:WaterInformation Center,Inc., 1973. 9. K. N. Brooks, P. F. Folliott, H. M. Gregersen,and J. L. Thames,Hydrology and the Managementof Watersheds.Ames, IA: Iowa StateUniversity Press/Ames,1991. "The InterceptionProcess."In Prediction in CatchmentHydrology,National 10. G. J. Blake, Symposiumon Hydrology,eds.T. G. Chapmanand F. X. Dunin, MelbourneAust. Acad. S c i . , 5 9 - 8 1 1, 9 7 5 . "Throughfall in PlantedStandsof Fourth SouthernPines 11. F. A. Roth, II, and M. Chang, Bulletin 17' 880-885(1981) Speciesin EastTexas,"WaterResources

REFERENCES 51 "canopy and Litter Interceptionby Hardwoodsof Eastern J. D. Helvey and J. H. Patric, Res.l, 193-206(1965)' Resour. United States,"Water New York: R. K. Linsley, Jr., M. A' Kohler, and J. L. H. Paulhus,Apptied Hydrology' McGraw-Hill, 1949. "A Linear Model for synthesizingHydrographsfor Small Drainage warren viessman,Jr., presented at the Forty-eighthAnnual Meetingof the AmericanGeophysical paper Areas," D.C., APr. 1967. Union, Washington, "The Hydrology of Urban Runoff," Trans. ASCE 125, A. L. Tholin una C. J. Kiefer, 1 3 0 8 -1 3 7 91( 9 6 0 ) . ..A Method of Computing Urban Runoff ,', Trans' ASCE |09, I2L,7W. I. Hicks,

Aue. 1966.

Chapter4

lnfiltration

Prologue The purposeof this chapteris to: . Defineinfiltration. ' Indicatethe role infiltration playsin affectingrunoffquantities and in replenishing soil moistureand groundwaterstorages. ' Presentmodelsfor estimatinginflltration and provide examplesof how they can be used. Infiltration is that processby which precipitation movesdownwardthrough the surface of the earth and replenishessoil moisture, rechargesaquifers,and ultimately supportsstreamflowsduring dry periods.Along with interception,depressionstorage, and stormperiodevaporation,it determinesthe availability,if any,of the precipitation input for generatingoverland flows (Fig. 1.3). Furthermore,infiltration rates influence the timing of overland flow inputs to channelizedsystems.Accordingly, infiltration is an important componentof any hydrologicmodel. The ratef at which infiltration occursis influencedby suchfactorsas the type and extent of vegetalcover, the condition of the surfacecrust, temperature,rainfall intensity,physicalpropertiesof the soil, and water quality. The rate at which wateris transmittedthrough the surfacelayeris highly dependent on the condition of the surface.For example,inwashof fine materialsmay seal the surfaceso that infiltration ratesare low evenwhen the underlyingsoils are highly permeable.After water crossesthe surfaceinterface,its rate of downwardmovement is controlled by the transmissioncharacteristicsof the underlying soil profile. The volume of storageavailablebelow ground is also a factor affecting infiltration rates. Considerableresearchon infiltration hastakenplace,but consideringthe infinite combinationsof soil and other factors existing in nature, no perfectly quantified generalrelation exists.

OF INFILTRATION 53 4.2 CALCULATION

4.1 MEASURING INFILTRATION

.

Commonly usedmethodsfor determininginfiltration capacityare hydrographanalysesandinfiltrometer studies.Infiltrometersareusuallyclassifiedasrainfall simulators or flooding devices.In the former, arlificalrainfall is simulatedover a small test plot and the inhltration calculatedfrom observationsofrainfall andrunoff, with consideration given to depressionstorageand surfacedetention.lFlooding infiltrometers are usually rings or iubes insertedin the ground. Water is applied and maintainedat a constantlevel and observationsmade of the rate of replenishmentrequired' Estimatesof infiltration basedon hydrographanalyseshavethe advantageover infiltrometers of relating more directly to prevailing conditions of precipitation and field. However,they areno betterthan the precisionwith which rainfall andrunoff are measured.Of partitular importancein suchstudiesis the areal variability of rainfall. Several meth;ds have been developedand are in use. Reference1 gives a good descriptionof thesemethods.

OF INFILTRATION 4.2 CALCULATION Infiltration calculationsvary in sophisticationfrom the applicationof reported averageratesfor specificsoil types and vegetalcoversto the useof differential equations g6verningthe flow of wateiin unsaturatedporousmedia.For small urban areasthat iespondiapidly to storm input, more precisemethodsare sometimeswarranted'On large waterlhedssubjectto peak flow production from prolongedstorms,averageor representativevaluesmay be adequate. The infiltration pto"".r is -omplicated at best. Even under ideal conditions (uniform soil propertiei andknown fluid properties),conditionsrarely encounteredin practice,the processis difflcult to characterize.Accordingly,therehasbeenconsiderabtestudyof the infiltration process.Most of theseeffortshaverelatedto the development of i1) empirical equationsbasedon field observationsand (2) the solution of equationsbasedon the mechanicsof saturatedflow in porous media.l'2 Later in this chapter,severalcommonly usedinfiltration modelsare discussed. As a prefaceto that discussion,a brief descriptionof the infiltration processfollows' It reviews the principal factors affecting infiltration and points out some of the problemsencounteredby hydrologicmodelers. Webeginour discussionwith an idealcase,onein which the soil is homogeneous throughout the profile and all the pores are directly interconnectedby capillary purru!"r. Furtheimore,it is assumedthat the rainfall is uniformly distributedovef the ur"u of "on"ern. Undertheseconditions,the infiltration processmay be chatactetized as one dimensional and the major influencing factors are therefore soil type and moisturecontent.3 through which The soil type characterizesthe sizeand numberof the passages and relative potential capillary the sets content moisture the the watermustflow while capillary to due head hydraulic the is potential Capillary conductivity of the soil. sign. opposite with potential but capillary as same is the forces. capillary suction

54

CHAPTER4

INFILTRATION

9 aoo

4

Ei o

. 300 -. d

d

t

2oo

0

0.1

0.2

0.3

0.4

0.5

Moisture content, 0 (vol/vol)

Figure 4.1 Typical capillary suction-relativeconductivity-moisture contentrelation.(AfterMein andLarson.e.) Capillary conductivity is the volume rate of flow of water through the soil under a gradient of unity (dependenton soil moisture content). Relative conductivity is the capillary conductivity for a specifiedmoisturecontent divided by the saturatedconductivity. Figure 4.1 illustratesthe relations amongthesevariables.Note that at low moisturecontents,capillary suctionis high while relative conductivityis low. At high moisture contentsthe reverseis true. With this background,an infiltration event can be examined.Consider that rainfall is occurringon an initially dry soil. As shownin Fig. 4.l,the relative conductivity is low at the outsetdue to the low soil moistureconditions.Thus, for the water to move downwardthrough the soil, a higher moisture level is needed.As moisture builds up, a wetting front forms with the moisturecontentbehindthe front beinghigh (essentiallysaturated)and that aheadof the front being low. At the wetting front, the capillary suctionis high due to the low moisture content aheadof the front. At the beginning of a rainfall event, the potential gradient that drives soil moisture movementis high becausethe wetting front is virtually at the soil surface. Initially, the infiltration capacity is higher than the rainfall rate and thus the infiltration rate cannot exceedthe rainfall rate. As time advancesand more water entersthe soil, the wetting zone dimensionincreasesand the potential gradient is reduced.Infiltration capacitydecreases until it equalsthe rainfall rate. This occursat the time the soil at the land surfacebecomessaturated.Figures4.2 and4.3 illustrate

l

4.2

CALCULATIONOF INFILTRATION

55

Moisturecontent.d

d

B

a

I with a constantrainfall Figure4.2 Typicalmoistureproflledevelopment rate. these conditions. Figure 4.2 showshow a moisture profile might developwhen a rainstormofconstant intensityoccurs.In the diagramthe soil moistureat the surface is shownto rangefrom its initial value at the top left to its saturatedvalue at the top right. Thus in moving downwardon the left-handside of the diagram,one can trace the downward progressionof the wetting front for varying levels of soil moisture

92

::

Time,r Figure 4.3 Infiltration rate versustime for a given rainfall intensity.(After Mein and Larson.e)

56

CHAPTER4

INFILTRATION

contentat the land surface.Figure 4.3 indicatesthat until saturationis reachedat the surface,the infiltration rate is constantand equalto the rainfall applicationrate at the surface.At Point 4, apoint that coriespondsto the time at which saturationoccursat the surface,the infiltration rate beginsto proceedat its capacity rate, the maximum rate at which the soil can transmit water acrossits surface.As time goes on, the infiltration capacity continues to decline until it becomesequal to the saturated conductivity of the soil, the capillary conductivity when the soil is saturated.This ultimate infiltration rate is shown by the dashedline to the right of K" in Fig. 4.3. Of particular interest is the determinationof Point 4 on the curve of Fig. 4.3. This is the point at which runoff would beginfor the conditions specifiedabove.It is also the point at which the actual infiltration rate/becomes equal to the infiltration capacityratefo ratherthan the rainfall intensityrate i. The time of occurrenceof this point depends,for a given soil type, on the initial moisture content and the rainfall rate. The shapeof the infiltration curve after this point in time is also influencedby thesefactors. Another factor that must be reckonedwith in the infiltration processis that of hysteresis. In Fig. 4.1 it can be seenthat the plot of capillary suctionversussoil moistureis a loop. The curve is not the samefor wetting and drying of the soil. The curves shown on the figure are the boundary wetting and boundary drying curves, curves applicableunder conditions of continuouswetting or drying. Betweenthese curves, an infinite number of possiblepaths exist that dependon the wetting and drying history of the soil. A numberof approachesto the hysteresisproblemhavebeen reportedin the literature.3 The illustration of the infiltration processpresentedwas basedon an ideal soil. Unfortunately,suchconditionsare not replicatedin natural systems.Natural soils are highly variable in composition within regions and soil cover conditions are also far-ranging. Becauseof this, no simpleinfiltration model can accuratelyportray all the conditionsencounteredin the fleld. The searchhas thus beenfor modelsthat can be called upon to give acceptableestimatesof the rates at which infiltration occurs durine rainfall events. Mein and Larson have describedthree generalcasesof infiltration associated with rainfall.3The first caseis one in which the rainfall rate is lessthan the saturated conductivity of the soil. Under this condition, shownas (4) in Fig. 4.4, runoff never occurs since all the rainfall infiltrates the soil surface.Nevertheless,this condition mustbe recognizedin continuoussimulationprocessessincethe level of soil moisture is affectedeven though runoff doesnot occur. The secondcaseis one in which the rainfall rate exceedsthe saturatedconductivity but is lessthan the infiltration capacity. Curves(I), (2), and (3) of Fig. 4.4 illustratethis condition.It shouldbe observed that the period from the beginningof rainfall to the time of surfacesaturationvaries with the rainfall intensity.The final caseis one in which the rainfall intensityexceeds the infiltration capacity.This condition is illustratedby the infiltration capacitycurve of Fig. 4.5 andthoseportionsof infiltration curves(l), (2), and (3) of Fig. 4.4 that are in their declining stages.Only under this condition can runoff occur. All three caseshaverelevanceto hydrologicmodeling,particularly when it is continuousover time.

4.3

tsal,

tsatt

HORTON'SINFILTRATIONMODEL

57

tsat,

Time, t

Figure 4.4 Inflltration curves for several rainfall intensities.(After Mein and Larson.e)

fo: f"+ Ao-f")"u' Infiltration capacitY curve

f, 0

o

,r*"

Figure 4.5 Horton's infiltration curve and hyetograph.

MODEL INFILTRATION .I3 HORTON'S The inflltration processwas thoroughly studiedby Horton in the early 1930s.oAn outgrowth of his work, shown graphically in Fig. 4.1, was the following relation for determininginfiltration capacity: (+.r; fo: f, + ("fr f")e'n'

58

CHAPTER4

INFILTRATION

where fo : k : f" : ,fo :

the infiltration capacity(depth/time) at sometime / a constantrepresentingthe rate of decreaseinf capacity d final or equilibrium capacity the initial infiltration capacity

It indicatesthat if the rainfall supply exceedsthe infiltration capacity,infiltration tendsto decreasein an exponentialmanner.Although simplein form, difficulties in determininguseful valuesfor/. and fr restrict the useofthis equation.The areaunder the curve for any time interval representsthe depth of water infiltrated during that interval. The infiltration rate is usually given in inches per hour and the time r in minutes,althoughother time incrementsare usedand the coefficientk is determined accordingly. By observingthe variation of inflltration with time and developingplots of / versus/ asshownin Fig. 4.5,we canestimatefsandft. Two setsof/and / are selected from the curve and enteredin Eq. 4.1. Two equationshavingtwo unknownsare thus obtained;they can be solvedby successiveapproximationsforfi and k. Typical infiltration ratesat the end of t hr ( f) areshownin Table4.1. A typical relation betweenf, and the infiltration rate throughout a rainfall period is shown graphically in Fig. 4.6a; Fig.4.6h showsan infiltration capacity curve for normal antecedentconditionson turf. The data given in Table4.1 are for aturf areaand must be multiplied by a suitablecover factor for other types of cover complexes.A range of cover factorsis listed inTable 4.2. Total volumes of inflltration and other abstractionsfrom a given recorded rainfall are obtainablefrom a dischargehydrograph(plot of the streamflowrate versus time) if one is available.Separationof the base flow (dry weather flow) from the dischargehydrographresultsin a direct runoff hydrograph(DRH), which accountsfor the direct surfacerunoff. that is. rainfall less abstractions.Direct surfacerunoff or precipitation excessin inchesuniformly distributedover a watershedcan readily be calculatedby picking valuesof DRH dischargeat equal time incrementsthrough the hydrographand applyingthe formulas P": where P" : 4r : A : r?7:

(0.0371e)() q')

( a) \

Ano

precipitation excess(in.) DRH ordinatesat equal time intervals (cfs) drainagearea(mi2) hurrber of time intervalsin a 24-hr period

For most casesthe differencebetweenthe original rainfall and the direct runoff can be consideredas infiltrated water. Exceptionsmay occur in areasof excessive subsurfacedrainageor tracts of intensiveinterceptionpotential.The calculatedvalue of infiltration can then be assumedasdistributedaccordingto an equationof the form of Eq. 4.1 or it may be uniformly spreadover the stormperiod. Choiceof the method employeddependson the accuracyrequirementsand size of the watershed. To circumvent some of the problems associatedwith the use of Horton's infiltration model,someadjustmentscanbe made.6ConsiderFig. 4.5. Notethat where the infiltration capacitycurve is abovethe hyetograph,the actual rate of infiltration

MODEL 4.3 HORTON'SINFILTRATION

h

o - 1

q

1

1.0 Time (hr) (a)

d

h

U

3.0 2.8

-TT

2.O

-

2.4 2.2 2.0 1.8 1.6 t.4 7.2 1.0 0.8 0.6 0.4 0.2

I

,4 f=0.53+2.4

l

l

l

0691ty

l, = U.UUUUJTIu . ) 9 ( l

l

e fle rOa"

9l

r\$4"

o\

rras!

-iro$dts"

l

l

l

l

l

l

Infiltration capacity curve (/.

0 10

70 80

100

t20

140

160

t80

Time (min) from beginning of infiltration capacity cuve' t/ (b)

Figure 4.6 (a) Typical infiltration curve. (b) Infiltration capacity and mass curvesfor normal antecedentconditionsof turf areas.fAfter A. L. Tholin and "The Hydrology of Urban Runoff," Proc. ASCE J. Sanitary Clint J. Kiefer, Ens. Div. S4(SA2),56 (Mar. 1959).1

59

60

CHAPTER4

INFILTRATION

TABLE 4.1 ryPICAL f, VALUES

Soilgroup

f, (in./hr)

0.50-1.00 0.10-0.50 0.01-0.10

High (sandysoils) Intermediate(loaps, clay, silt) Low (clays,clay loam)

f' (mm/h)

t2.50-25.00 2.so-r2.50 0.25-2.50

Source: After ASCE Manual of Engineering Practice,No.28.

TABLE4,2 COVERFACTORS Cover

Cover fac{or

Permanentforest and grass

Good(1 in. humus) Medium(f-1 in. humus; Poor(< j in. humus)

Close-growingcrops

Good Medium Poor Good Medium Poor

Row crops

3.0-7.5 2.0-3.0 1.2-1.4 2.5-3.0 t.6-2.0 1.11 - .3 1 . 3 -1 . 5 1.1-1 3 1 . 0 -1 . 1

Source: After ASCE Manual of Engineering Practice, No.2t.

is equal to that of the rainfall intensity, adjustedfor interception,evaporation,and other losses.Consequently,the actuafinfiltration is given by

f(t) : minlfof), i(t)l

(4.3)

where/(r) is the actual infiltration into the soil and i(l) is the rainfall intensity.Thus the infiltration rate at any time is equal to the lesserof the infiltration capacity,f,(t) or the rainfall intensity. Commonly,the typical valuesof foandf" are greaterthan the prevailingrainfall intensitiesduring a storm. Thus, when Eq.4.l is solvedforS as a function of time alone, it shpwsa decreasein infiltration capacity even when rainfall intensitiesare much lessthanfo. Accordingly,a reductionin infiltration capacityis maderegardless of the amount of water that entersthe soil. To adjust for this deficiency,the integratedform of Horton's equationmay be used,

F(tp):

lr'' ,

o, : f,tp +

(t - a' *,,1

(4.4)

whereF is the cumulativeinfiltration at time to, as shownin Fig. 4.7.rnthe figure, it is assumedthat the actual infiltration has been equal to$. As previouslynoted, this is not usually the case,and the tnie cumulativeinfiltration must be determined.This

MODEL 4.3 HORTON'SINFILTRATION

0 0

61

tptpt tl Equivalent time

Figure4.7 Cumulativeinfiltration. can be done using F\t) :

l,',u,o,

(4.s)

where/(r) is determinedusingF,q.4.3. Equations4.4 and4.5 may be us6djointly to calculatethe time t, that is, the equivalent time for the actual infiltrated volume to equal the volume under the infiltration capacity curve (Fig. 4.7). The actual accumulatedinfiltration given by Eq. 4.5 is equatedto the area under the Horton curve, F,q. 4.4, and the resulting expressionis solvedfor r' This equation, F:fJrt

(l-e-k'n)

(4.6)

cannotbe solvedexplicitly for to,but an iterativesolutioncan be obtained.It should be understoodthat the time to is lessthan or equal to the actual elapsedtime r. Thus the availableinfiltration capacityas shownin Fig. 4.7 is equalto or exceedsthat given described,fbecomesa functionof the actual by Eq. 4.1. By makingthe adjustments amount of water infiltrated and not just a variable with time as is assumedin the original Horton equation. In selectinga model for use in inflltration calculations,it is important to know its limitations. In somecasesa model can be adjustedto accommodateshortcomings; are not realisticfor the natureof the useproposed, in othercases,if its assumptions the model shouldbe discardedin favor of anotherthat better fits the situation. Part One of this book deals with the principal componentsof the hydrologic cycle. In later chapters,the emphasisis on putting these componentstogetherin

62

CHAPTER4

INFILTRATION

varioushydrolqgicmodelingprocesses.When thesemodelsare designedfor continuous simulation,the approachis to calculatethe appropriatecomponentsof the hydrologic equation,Eq. 1.4, continuouslyovertime. A discussionof how infiltration could be incorporatedinto a simulation model follows. It exemplifiesthe use of Horton's equationin a storm water managementmodel (SWMM)." First, an initial value of /o is determined.Then, consideringthat the value of$ dependson the actualamountof infiltration that hasoccurredup to that time, a value of the averageinfiltration capacity,fo, availableover the next time stepis calculated using

f o : * 1 " = ' *fo' o' ' : l W

(4.7)

-tp

Equation 4.3 is then usedto find the averagerate of infiltration, /.

'v - [f, tt

iti >1, iri I6f k, the Horton curve is approximatelyhorizontal and : f". Once this point hasbeenreached,there is no further needfor iteration since fo and equal to f" and no longerdependenton F. is constant f EXAMPLE 4.I

Givenan initial inflltrationcapacityfiof 3.0 in./hr anda time constantft of 0.29hr-', derive an infiltration capacity vs. time curve if the ultimate infiltration capacity is 0.55 in./hr. For the first ten hours, estimatethe total volume of water infiltrated in inchesover the watershed. Solution. Using Horton's equation(4.1), valuesof infiltration can be computed for various times. The equationis as follows: f : f" + Lfo + f")e-k, Substitutingthe appropriatevaluesinto the equationyields ,f = 0.55 + (3.0 0.55)s-o'zo' Table4.3, valuesof f atecomputedand Then for the times shownin spreadsheet enteredinto the table. Using the spreadsheetgraphics package,the curve of Fig. 4.8 is derived,

4,3 HORTON'SINFILTRATIONMODEL

63

TABLE 4.3 CALCULATIONSFOR EXAMPLE4.1

lnfiltration (in./h0

Time (hr)

Time (hr)

lnfiltration (in./hr)

0.00

3.00

5.00

t.l2

0.10

2.93

6.00

0.98

0.25

2.83

7.00

0.87

0.50

2.67

8.00

0.79

1.00

2.38

9.00

0.73

2.00

t.92

10.00

0.68

3.00

1.58

15.00

0.58

4.00

r.32

20.00

0.56

4.1 can To find the volumeof waterinfiltrated duringthe flrst 10hours,Eq' be integratedover the range of 0-10 v :

I J

[0.5s + (3.0 0.55)e-o2e' ldt

[0.55/ + (2.45I-0.29)e-o2s\o : 12.47in' V

Y:

The volume in inchesover the watershedis thus 12'47 in'

ll

3.0

\ ;

2.0

6 .:

15

l?

-\-\

1.0

"'-0 0.5

2

4

6

L 8

10

Time (hr) Figure 4.8

GraPh for ExamPle 4'1.

12

t4

18

64 4.4

CHAPTER4

INFILTRATION

MODEL GREEN-AMPT The Green-Ampt infiltration model,originallyproposedin 1911,has had a resurgenceof inte1gs1.3'6-1t This approachis basedon Darcy's law (seeChapter18).In its original form, it was intendedfor use where infiltration resultedfrom an excessof water at the ground surfaceat all times. In 1973, Mein and Larson presenteda methodologyfor applying the Green-Ampt model to a steadyrainfall input.eThey alsodevelopeda procedurefor determiningthe valueof the capillary suctionparameter usedin the model. In 1978,Chu demonstratedthe applicability of the model for useunder conditionsof unsteadyrainfall.lo As a result of theseand other efforts, the Green-Ampt model is now employedas an option in such widely used continuous simulation modelsas SWMM.6 The original formulation by Greenand Ampt assumedthat the soil surfacewas coveredby ponded water of negligible depth and that the water infiltrated a deep homogenoussoil with a uniform initial watercontent(seeFig. 4.9). Wateris assumed to enter the soil so as to define sharply a wetting front separatingthe wetted and unwettedregions as shown in the figure. If the conductivity in the wetted zone is definedas K", applicationof Darcy's law yields the equation

"

Io:

K"(r + s) r

(4.10)

where I is the distancefrom the ground surfaceto the wetting front and S is the capillary suction at the wetting front. Referringto Fig. 4.9, it can be seenthat the cumulativeinfiltration F is equivalentto the product of the depth to the wetting front L and the initial moisture deficit, 0, - 0, : IMD. Making these substitutionsin

Ponded depth considerednegligible

I Ho

I

I

I

II

I

Figure 4.9 Definition sketch for GreenAmpt model.

4.4 GREEN-AMPTMODEL

65

Eq. 4.10 and rearranging,we obtain - , \ -, f- : -K.( rp

*

t * jt' ) F

(4.11) /

Considering thatfp : dFldt. we can state

#: *,(t.'#)

(4.r2)

: 0 at t : 0' we obtain Integratingand substitutingthe conditionsthat F

+ I MD X {) : K.t

F - S x I M D X l o g " ({ IMD

X

s

(4.r3)

This form of the Green-Ampt equationis more convenientfor usein watershed than Eq. 4.l}beciuse it relatesthe cumulativeinfiltrationto the modelingprocesses ponded time at which infiltration began.The derivation of this equation assumesa at all capacity infiltration to the equal surfaceso that the actualrate of infiltration is a time, any at infiltration cumulative times.Using Eq. 4.13, we can determinethe equation the in parameters the All featuredesiiablefor continuoussystemsmodeling. are physicalpropertiesof the soil-water systemand are measurable.The determinaparticution of suitablevaluesfor the capillary suctions is often difficult, however, It can be 4.10. Fig. in soil larly for relationssuchas that ihown for a clay-type capillary of variation wide is a there observedfrom the figure that for this curve suctionwith soil moisturecontent.3 The Mein-Larson formulation using the Green-Ampt model incorporatestwo The first stagedealswith prediction of the volume of water that infiltrates stages.3,6 before the surfacebecomessaturated.The secondstageis one in which infiltration water capacityis calculatedusingthe Green-Ampt equation.In the widely usedstorm the of one is infiltration of model ,nunug"*"nt model, the irodif,ed Green-Ampt using made are Computations optiois that can be employedto estimateinfiltration.6

v)

U

Moisture content,0

Figure 4.10 Capillary suction versus moisture contentcurves.

66

CHAPTER4

INFILTRATION

the following equations:for F ( F,(f : i),

F. :7:

IM?

i/K" - |

rori > K"

\4.r4)

and thereis no calculationof F"for i < K"; for F > F,(f : fi):

*\ ' f,: x,(t where

t * j") F

l

(4.11)

f : fo: I : F : { :

acttal infiltration rate (ftlsec) infiltration capacity(ftlsec) rainfall intensity (ftlsec) cumulativeinfiltration volume in the event (ft) cumulative infiltration volume required to causesurfacesaturation (f0 S : averagecapillary suctionat the wetting front (ft of water) IMD : initial moisture deficit for the event (ftlft) K" : saturatedhydraulic conductivity of soil (ftlsec)

Equation 4.10 showsthat the volume of rainfall neededto saturatethe surface is a function of the rainfall intensity.In the modelingprocess,for eachtime stepfor which I ) K", the value of d is computedand comparedwith the volumeof rainfall infiltrated to that time. If F equalsor exceeds{, the surfacesaturatesand calculations for infiltrationthenproceedusingEq.4.I4. Notethat by substituting/fori in Eq.4.l4 and rearranging,the equationtakesthe sameform as Eq. 4.11. For rainfall intensitieslessthan or equal to K", all the rainfall infiltrates and its amount is used olly to update the initial moisture deficit, IMD.6 The cumulative infiltration volume {" is not altered. After saturationis achievedat the surface,Bq.4.l1 showsthat the infiltration capacityis a function of the infiltrated volume,and thus of the infiltration ratesduring previous time steps.To avoid making numerical errors over long time steps,the integratedform of the Green-Ampt equation(Eq. .B) is used.This equationtakes the following form as it is usedin SWMM: (4.15) &Gr- tr): Fz- Cln (F2+ C) - Fr* Cln(F1+ C) where C : IMD X .t (ft of water) / : times (sec) 1,2 : subscriptsindicating the starting and ending of the time steps.

-

Equation4.15 mustbe solvediteratively for F2,the cumulativeinfiltration at the end of the time step.A Newton-Raphsonroutine is used.6 In the SWMM model, infiltration during time step tz - tt is equal to (t, - tr)i if the surfaceis not saturatedand is equal to F, - F, if saturationhas previously occurred and there is a sufficient water supply at the surface.If saturationoccurs during an interval, the infiltrated volumesover each stageof the processwithin the time stepsare computedand summed.When the rainfall endsor becomeslessthan

MODEL 4.5 HUGGINS_MONK

67

theinfiltrationcapacity,anypondedwaterisallowedtoinfiltrateandisaddedtothe cumulative inflltration volume'

MODEL 4.5 HUGGINS-MONKE problemby introducing thetimedependency havecircumvented Severalinvestigators by the followingequationproposed soilmoistureas the depenO"niuutluUfe.2'10-13 HugginsandMonkeis an examPle:2

f : f,*A(

I

(4.16)

coefficients layer (Q ,,orug" potential of a soil overlying the impeding minus antecedentmoisture) F _ the total volume of water that infiltrates impeding stratum T o = the total porosity of soil lying over the sprinkling infiltrometer studies'The The coefficientsare determinedusing data from in the iteration process'At the variableF mustbe catculatedfor eachtime increment

where AandP:

s - itr"

exceedsthe inflltration capacity,the rate zone," which determinesthe soil moistu (1) wherethe moist evaluatedasfollows:10 the field capacity(amountof waterheld in drained),the drainagerate is consideredz to the infiltration rate when the soil is s constan| and (3) if the watercontentis be drainagerate is comPutedas

rate: ,"(t - 2)' drainage

(4.r7)

where P, : the unsaturatedpore volume G:maximumgravitationalwater,thatis,thetotalporosityminusthefield caPacitY Datafromsprinklinginfiltrometerstudiesofvariouswatershedsofinterestare usedto estimatethe coefficientsin Eq' 4'16''

5.6

EVAPOTRANSPIRATION

101

of a masstransferequationthat has often been employedfor this purpose.However, Linsley and co-workers indicate that there is some question as to the adequate The equation is verification of this model to estimate evapotranspirationlosses.27 expressed as 833x2(e1

^ E : @

E : K: €1,€2 : Vr, Vz : Z =

er)(V,

V,)

(s.2s)

evaporation(in./hr) von K6rm6n's constant(0.4) vaporpressures(in. Hg) wind speeds(mph) the mean temperature("F) of the layer betweenthe lower level zt and the upper level z2

It is assumedin Eq. 5.25 that the atmosphereis adiabatic and the wind speedand moistureare distributedlogarithmically in a vertical direction. In view of the sm4ll differencesbetweenwind and vapor pressureto be expectedat two levelsso closely spaced,and sincethesegradientsare directly relatedto the sought-afterevaporation, highly exactinginstrumentationis required to get reliable results.

PotentialEvapotranspiration "the waterlosswhich will occur Thornthwaitedeflnedpotential evapotranspirationas if at no time there is a deficiencyof water in the soil for the use of vegetation."In a practical sense,however,most investigatorshave assumedthat potential evapotranspiration is equal to lake evaporationas determinedfrom National WeatherService ClassA pan records.This is not theoreticallycorrect becausethe albedo(amountof incoming radiation reflectedback to the atmosphere)of vegetatedareas and soils rangesas high as 45 percent.28As a result, potential evapotranspirationshould be somewhatlessthan free water surfaceevaporation.Errors in estimatingfree water evapotranspirationfrom pan recordsare such,however,asto make an adjustmentfor potential evapotranspirationof questionablevalue. An equationfor estimatingpotential evapotranspirationdevelopedby the Agricultural ResearchService (ARS) illustratesefforts to include vegetalcharacteristics and soil moisturein sucha calculation.The evapotranspirationpotentialfor any given day is determinedas follows:2e

E T : G I x k ux , " ( # l where

(s.26)

ET : evapotranspirationpotential (in./day) GI : growth index of crop in percentageof maturity K : ratio of G1 to pan evaporation,usually 1.0-I.2 for short grasses' 1,.2-L6 for cropsup to shoulderheight,and 1'6-2.0 for forest Eo: pan evaporation(in./day) ,S: total porosity SA : availableporosity (unfilled by water) AWC : porosity drainableonly by evapotranspiration x : AWC/G (G : moisture freely drainedby gravity)

102

CHAPTER5

EVAPORATIONAND TRANSPIRATION

0.2 0.1 ,.} 0.0

ti

>

n ,

0.1

J

J

Months (b)

Figure 5.5 Averagedaily consumptionof water: (a) for year 1953 by corn, followed by winter wheat under irrigation; (b) for year 1955, with irrigated first-year freadow of alfalfa, red clover, and timothy. Both measurements taken on lysimeterY 102 C at the Soil and Water ConservationResearchStation. Coshocton.Ohio. (After Holtan et al.2e)

l x F rrllS N I

]l{

tl rH

0

8

1

6

2

r

'

.

3

2

4

0

4

8

Weeks

Figure 5.6 Growth index GI = ETfET^,, from lysimeterrecords,irrigatedcorn,andhayfor 1955,from Coshocton, Ohio.(AfterHoltanet al.2e)

5.7

ESTIMATINGEVAPOTRANSPIRATION 103

TABLE 5.5 HYDROLOGICCAPACITIESOF SOIL TEXTURECLASSES

sa

GD

Textureclass

(v")

(Y")

Coarsesand Coarsesandy loam Sand Loamy sand Loamy fine sand Sandyloam Fine sandy loam Very fine sandy loam Loam Silt loam Sandy clay loam Clay loam Silty clay loam Sandy clay Silty clay Clay

24.4 24.5 32.3 37.0 32.6 30.9 36.6 32.7 30.0

t'7.7 15.8 19.0 26.9 27.2 18.6 23.5 21.0 14.4 11.4 13.4 13.0 8.4 rl.6

JI,J

25.3 25.7 z5-J

19.4 z l.+

o 1

18.8

7.3

AWC" ('/")

x AWC/G

6.7 8.7 13.3 10.1 5.4 t2.3 13.1 11.7 15.6 19.9 tr.9 12.7 t4.9 7.8 12.3 11 . 5

0.38 0.55 0.70 0.38 0.20 0.66 0.56 0.56 1.08 1.74 0.89 0.98 r.77 0.6'l r.34 r.58

aS = total porosity - 15 bar moisture 7o' bG : total porosity - 0.3 bar moisture 7o. "AWC: S - G. "Land Capability: A Hydrologic ResponseUnit in Source: Adaptedfrom C. B. England, U.SlDepartmentof Agriculture,ARS 41-172' Sept' 1970' Asiicultural'Watersheds," Aiter H. N. Holtan et a1.2e

The GI curveshave been develoPed evapotranspirationfor severalcrops (Fig. : daily rate(Fig. 5.6).Equation5.26is used USDAHL-74 model of watershedhydrolq late daily evapotranspiration.Represental Table5.5.

EVAPOTRANSPIRATION 5.7 ESTIMATING Transpirationis an important componentin the hydrologicbudgetof vegetatedareas, but it is a difficult quantity to measurebecauseof its dependenceon phytological variables.It is a function of the number and types of plants, soil moisture and soil type, qeason,temperature,and averageannual precipitation. As noted previously, evaporationand tianspiration are commonly estimatedin their combinedevapotranform. spiration ' If the precipitation and net runoff for an atea ate known, and estimatesof of ET canbe had using groundwateiflow and storagecan be made,rough estimatesihe basic hydrologicequatiJn,Eq. 1.1. A more sophisticatedapproachdevelopedby peaman foilows.t3It iJ representativeof the methodsmost often used'

104

CHAPTER 5 EVAPORATION ANDTRANSPIRATION

The PenmanMethod Both the energybudgetand masstransportmethodsfor estimatingevapotranspiration (ET)have limitations dueto the difficulties encounteredin estimatingparametersand in making other required assumptions.To circumventsomeof theseproblems,penman developeda method to combinethe masstransport and energybudgettheories. This widely usedmethodis one of the more reliableapproachesto estimatingETrates usingclimaticdata.13'rs'23,30 The Penmanequationis of the form of Eq. 5.18; it is theoreticallybasedand showsthat EZ is directly related to the quantity of radiative energy gained by the exposedsurface.In its simplified form, the Penmanequationisls t s t : -

LH + 0.27E L + 0.21

(s.27)

where A : the slopeof the saturatedvapor pressurecurve of air at absolute temperature(mm Hg/'F) H : the daily heatbudgetat the surface(estimateof net radiation) (mm/day) E : daily evapoiation(mm) ET : the evapotranspirationor consumptiveusefor a given period (mm/day) The variablesE and.Fl are calculatedusing the following equations: E : 0.35(e"- e)(l'+

( s .28) 0.0098ar) : where eo the saturationvapor pressureat mean ak temperature(mm Hg) e6 : the saturationvapor pressureat meandew point (actualvapor pressure in the air) (mm Hg) u2 : the mean wind speedat 2 m abovethe ground (mi/day) The equationusedto determinethe daily heat budget at the surface,11,is 11 : R(1 - r)(0.18 + 0.55.t)- 8(0.56 * 0.092e2s)(0.10 (5.29) + 0.905) where R : the mean monthly extraterrestrialradiation (mm HrO evaporatedper dav) : the estimatedpercentageof reflecting surface B = a temperature-dependent coefficient s : the estimatedratio of actual duration of brisht sunshineto maximum possibleduration of bright sunshine. The empirical reflectivecoefficientr is a function of the time of year,the calmnessof the water surface,wind velocity, and water quality. Typical rangesfor r are 0.05 to 0.12.31 valuesof e" andA can be obtainedfrom Figs.5.7 and 5.8, thosefor R and B can be obtainedfrom Tables5.6 and 5.7. The use of Penman'sequationrequiresa knowledgeof vaporpressures,sunshineduration,net radiation,wind speed,and mean temperature.Unfortunately,regular measurements of theseparametersare often unavailableat sites of concern and they must be estimated.Another complication is making a reductionin the valueof EZwhen the calculationsare for vegetatedsurfaces. While results of experimentsto quantify reduction factors have not completelyresolved the problem, there is evidencethat the annual reduction factor is close to

ESTIMATINGEVAPOTRANSPIRATION

5.7 "C 60 50

't22 104

t r 1 n

86

20

68

10

111

40

r04

€ : o

86

!?

o B t ^

o

o F

.F

50

t40

I

€ + o

"C

.F

- 1 0 0

50 10 20

30 40

50 60

70

313 = ts

68

14

s0l. zzV | 0.2

105

|

0.4 0 6

| 0.8

|

I

I

I

303

ct

ro?

!d 6

z6J

6 F.

t273

1.0 1.2 1.4 1.6 1.8

80 90 100

satuated vapor pressure' ea (mm Hg)

Figure 5.7 Relation betweentemperatureand saturatedvapor Pressure.

Valueof A (mmHg/'F) Figure 5.8 TemperatureversusA relation for use *i?rt ,n" PenmanLquation. (After Criddle'23)

that using Thus,unlessthereis evidenceto supportanothervalue'it appears unity.32-34 surfaces for results satisfactory a value of 1 for the reduction coefficient may give evaporation water free of having varied vegetal covers.Accordingly, aty estimate by an appropriatereduction could be used ro estimaieEZ, providin[ it is modified coefficient.

EXAMPLE 5.4

to 5.29, estimateET, giventhe following data: ': using the PenmanMethod, Eqs.5.21 : 30 degreesc' 20 degreesc, temperatureof 1l temperatureat warer r"tru"" : (48 mi/day)' the month is relative humidity : +O p"r""nt, wind-velocity i mph S is found to be 0'75' Juneat latitude 30 degreesnorth, r is given ut 0'07' and Solution l.Giventhedatafortemperature,thevaluesofeoandeacanbedetermined. UsingFig.5.TorAppendixTableA.2,thesaturatedVaporpressuresare 31'83'andfor foundto be l7.53unO-:t'Sl mm Hg respectively' ThI'"-: : : 12'73' X 0'4 a relativehumidity of 40 percent,e,t 31'83 Then,usingEq. 5.28' .83 - 12.73)(1+ 0.0098x 48) E : 0.35(31 E:9.83 mm/daY 2.ThevalueofAisfoundusingFig.5.8;forthegivenlatitudeandmonth,R isobtainedfiomTable5.6;andBisgottenfromTable5.'Tforatemperature : 1'0' R : 16'5' andB : I7 '01' of 30"C.fne vatuesfound are A Then,usingEq.5.29, H: 16'5(I - 0'07)(0'18+ 0's5 x 0'75) - 17.01(0.5 6 - 0.092x 12j30)(0.10 + 0'90 x 0'75) H = 6.04 mm/daY

O \ O h O \ 6 \ t o € O \ \ O € € r ) O o r) F. O N + r) \O F-

t--

l--

F-

O

Ol

t- * co o o\ * o co cn oo oo n tF-O\dcl\fh!nv1 f. In this example,interflow and groundwaterflow are zero,assoil

176

CHAPTER11 HYDROGRAPHS

End ofrainfall

9C

Groundwater flow

Time

of thehydrograph. Figure 11.4 Components

moisture deficiency still exists,although at a reduced level. Channel precipitation likewise constitutesa component. In the final set, Fig. 11.3d, all four componentsexist with rainfall intensity exceedinginflltration rate andthe field capacityofthe soil is reached.This casewould of a large storm event. be typical Figures 1l.je-h illustrate how hydrograph shape can be modified by areal variations in rainfall and rainfall intensity and by watershedconfiguration.8Minor fluctuationsshownin thesehydrographsare linked to variationsin stormintensity.In Fig. 11.3eonly the delayingeffectspertinent to a storm over the upstreamsectionof the areaareindicated.Figure 11.3fshowsthe reverseof this condition.Figures11'3g and h depict the comparativeeffects of basin geometry. In most hydrographanalyses,interflow and channelprecipitation are grouped with surfacerunoff rather than treated independently.Channelprecipitation begins with inceptionof rainfall and endswith the storm.Its distributionwith respectto time is highly iorrelated with the stormpattern.The relative volume contributiontendsto increasesomewhatas the storm proceeds,since stream levels rise and the water surfaceareatendsto increase.The fraction of watershedareaoccupiedby streamsand lakes is generallysmall, usually on the order of 5 percent or less,so the percentage of rrnoff relatedto channelprecipitation is usually minor during important storms. Distribution of interflow is commonly characterizedby a slowly increasingrate up to the end of the storm period, followed by a gradual recessionthat terminatesat the intersectionofthe surfaceflow hydrographand base flow hydrograph.Figure 11.4 illustrates the approximatenature of the componentsof channelprecipitation and interflow. The baseflow componentis composedof the water that percolatesdownward until it reachesthe groundwaterreservoirand then flowsto surfacestreamsasgroundwater discharge.The groundwaterhydrographmay or may not show an increase

11.4 BASE FLOW SEPARATION

'

177

during the actual storm period. Groundwateraccretion resulting from a particular storm is normally releasedover an extendedperiod, measuredin days for small watershedsand often in months or yearsfor large drainageareas. The surfacerunoff component consistsof water that flows overland until a streamchannelis reached.During large stormsit is the most significanthydrograph component.Figure 11.4illustratesthe surfacerunoff and groundwatercomponentsof a hydrograph.As pointed out in Fig. 11.3,the relative magnitudeof eachcomponent for a given storm is determinedby a combinationof many factors.Hydrographsare analyzedto provideknowledgeof the way precipitationand watershedcharacteristics interact to form them. The degreeof hydrographseparationrequireddependson the objective of the study. For most practical work, surfacerunoff and groundwater componentsonly are required.Researchprojectsor more sophisticatedanalysesmay dictate considerationof all components.When multiple storms occur within short periods,it is sometimesnecessaryto separatethe overlappingparts of consecutive surfacerunoff hydrographs.

11.4 BASEFLOWSEPARATION Severaltechniquesare usedto separatea hydrograph'ssurfaceand groundwaterflows. Most are basedon analysesof groundwaterrecessionor depletioncurves.If there is no addedinflow to the groundwaterreservoir,and if all groundwaterdischargefrom the upstreamareais interceptedat the stream-gaugingpoint of interest,then groundwater dischargecan be describedby either e'to er: eoK' or er: eo€-K' where eo : q, : K: e:

( 11 .1 )

a specifiedinitial discharge the dischargeat any time / after flow 46 arecessionconstant baseof natural logarithms

Time units frequently used are days for large watershedsand hours or minutes for small basins.A plot of either yields a straight line on semilogarithmicpaper by plotting / on the linear scale. For most watersheds,groundwaterdepletioncharacteristicsare approximately stable,sincethey closelyfit watershedgeology.Nevertheless, the recessionconstant varieswith seasonaleffectssuchas evaporationand freezingcyclesand other factors. Becauseq, dt is equivalentto -dS, where S is the quantity of water obtainedfrom storage,integrationofEq. 11.1produces t- 4o s - Qlog" K

(rr.2)

This equation determinesthe quantity of water releasedfrom groundwaterstorage betweenthe times of occurrenceof the two dischargesof interest,or it can be usedto calculatethe volume of water still in storageat a time some chosenvalue of flow occurs.To get the latter, 4, is setequalto zero and qsbecomesthe referencedischarge. Figure 11.5ais a plot of Eqs. 11.1and II.2 andprovidesadditionaldefinition.

178

CHAPTER11

HYDROGRAPHS

qo

.. ,?

q

K \r3

/

?r

t (a.)

I

l

I

I

qo discharge at rcginningof arr interval discharge at :nd of an inten,a\ Qt

I

II

Interval = unit period in / whicht is expreissed .

q d

/ qo Qt

Time (b) Figure 11.5 Baseflow model' to

.

t1

Groundwaterdepletioncurvescan be analyzedby variousmethodsto evaluate the recessionconstantK. One of thesewill be described.Data from a stream-gaugi.ng stationare a prerequisiteand shouldreflectrainlessperiodswith no upstreamregulation,'suchas a fesefvoir,to affect flow at the gaugingpoint. Otherwisean adjustment with its own enors is introduced. From the streamflow data, plot a portion of the recession hydrograph (Fig. 11.5b) to find values of dischargeat the beginning and end of selectedtime intervals.Flows at the beginningof eachinterval are analogousto 4e,whereasthose at the end are analogousto q1.Next, selectseveraltime intervalsand plot correspondvalues ing qe'sversusq,'s shownin Fig. 11.6.The time periodbetweenconsecutive of 4 shouldbe identical for eachdatum set. Figurestaken from recessioncurvesof times that still reflect surfacerunoff will usually fall below and to the right of a 45o line drawn on the plot. Thesevalueswill also be associatedwith larger numbersfor 4. Points taken from true groundwater recessionperiods should approximately

11,4 BASEFLOWSEPARATION 179 r.-,x. âre points plotted from discharges for different groundwater discharge periods

,/ o

eo nd qt are taken for equal time

I

,(

I .5r"2'

9/ .,

/(^lI A

/#

- s l o p e o 0f A=

fi:

x

inEq11.1

/ Figure 11.6 Graphical rnethod for determiningrecessionconstant K. (U.S. Departmentof Agriculture, Soil ConservationService.)

describea straightline.Becauseh : eo : 0 when4o: 0, a straightlinecanbefitted graphicallyto the datapoints.The slopeof this line is qrfqo : K. Usingthis value,the depletioncurve plots as a straight line on semilogarithmicpaper (r is the linear scale variable)or as a curveon arithmeticpaper,Fig. 11.5a.

SeparationTechniques r

Severalmethodsfor baseflow separationare used when the actual amount of base flow is unknown. During large storms,the maximum rate of dischargeis only slightly affectedby baseflow, and inaccuraciesin separationmay not be important. The simplestbaseflow separationtechniqueis to draw a horizontalline from the point at which surfacerunoff begins,PointA in Fig. 11.7, to an intersectionwith the hydrographrecessionwherethe baseflow rate is the sameasat the beginningof direct runoff as indicatedby Point B. A secondmethodprojects the initial recessioncurve downwardfromA to C, which lies directly below the peak rate of flow. Then point D on the hydrograph,representingN daysafter the peak, is connectedto point C by a straightline definingthe groundwatercomponent.One estimateof N is basedon the formula3 N - Ao'2

(11.3)

where N : the time in days A = the drainageareain squaremiles A third procedureis to developa baseflow recessioncurve using Eq. 11.1 for data from the segmentFG, and then back-calculateall base flow to the left of Point fl

180

cHAPTER11 HYDRocRApHS

o

9p F

Time

Flgure 11.7 Illustrationof somehydrograph separation techniques. wherethe computedcurve beginsto deviatefrom the actualhydrograph,marking the end of direct runoff. The curve is projected backward arbitrarily to some Point E below the inflection point and its shapefrom C to E is arbitrarily assigned.A fourth widely usedmethod is to draw a line betweenA and F, and a fifth common method is to project the line AC alongthe slopeto the left of A , and then connectPointsC and

o d d

12N 6P

t21|'{{ 6A

12N 6P

tz]0|{ 6,{

I2N

6P

IzM

Figure 11.8 Illustration of base flow separation:hydrograph for the Uharie River near Trinity, North Carolina, February 25, 1939. (U.S. Departmentof Agriculture, Soil ConservationService).

TIME RELATIONSHIPS 181 11.5 HYDROGRAPH

B. All thesemethodsare approximatesincethe separationof hydrographsis partly a subjectiveprocedure. Figure 11.8illustratestwo graphicalseparationtechniquesto determinesurface runoff and groundwaterflow components. Line AD representsthe simpleprocedure of connectingthe point of the beginningof direct runoff with the flrst point on the groundwaterrecessioncurve (an advantageoverthe horizontal line techniquebecause the time baseof direct runoff is much shorter).Ctrve ABCD is constructedfrom the extensionof the baseflow recessioncurve.

TIMERELATIONSHIPS 11.5 HYDROGRAPH Wavetravel time is definedas the time required for direct runoff originating at the most remotepoint in the channelto reachthe outlet. The last drop of direct runoff to passthe outlet conceptuallytravelsoverthe watersurfaceand reachesthe outlet at the speedof a small surfacewave,rather than at a speedequal to the averagevelocity of flow. The wavetravel time is fasterthan the averagevelocity and varieswith channel shapeand other factors.For a rectangularchannel,the ratio is approximately5/3 (see Section 13.1 for other wavevelocities).The time base(Fig. 11.4) of a hydrographis consideredto be the time from which the concentrationcurve beginsuntil the directrunoff componentreacheszero. An equationfor time basemay take the form T6:t"*t,

(rr.4)

where To : the time baseof the direct runoff hydrograph /" : the duration of runoff-producingrain t- : the excessrainfall releasetime Watershedlag time, illustrated in Fig. 11.4, is definedas the time from the penterof tt massof effectiverainfall to the centerof massof direct runoff. Other definitionsand severalequationsrelating lag time to watershedcharacteristicsare providedin S_ecchapters. tion 11.7 and subsequent Becauseofits importancein unit hydrographtheory, the excess-rainfallrelease time is introduced.This is definedasthe time requiredfor the last, most remotedrop ofexcessrain that fell on the watershedto passthe outlet, signallingthe cessationof direct runoff. It is easilydeterminedas the time interval betweenthe end of rain and the end of direct runoff. Only that part of the outflow which classiflesas direct runoff (excessrain) is consideredin dqterminingthe releasetime. Watershedoutflow normally continuesafter cessationof direct runoff, in the form of interflow andbaseflow. Releasetime is very similar by definitionto wavetraveltime and time of concentration (Section11.6). A foundational assumptionof unit hydrographtheoryl2 is that the watershed excessreleasetime is a constant,regardlessof the storm duration, and is related to The excessreleasetime is also basinfactorsratherthan meteorologicalcharacteristics. conceptuallyidenticalwith the time baseof an instantaneousunit hydrograph(IUH). This is the runoff hydrographfrom 1.0 in. of excessrain applieduniformly over the watershedin an instant of time (seeChapter 12). Both wavetravel time and excessrainfall releasetime are often used synonymouslywith time of concentration.

182

CHAPTER11

HYDROGRAPHS

11.6TIMEOF CONCENTRATION The most common definition of time of concentrationoriginatesfrom consideration of overlandflow. If a uniform rain is appliedto atract, the portions nearestthe outlet contribute runoff at the outlet almost immediately.As rain continues,the depth of excesson the surfacegrowsand dischargeratesincreasethroughout.Runoff contributions from variouspoints upstreamarrive at later times, addingthemselvesto continuing runoff from nearerpoints, until flow eventually arrives from all points on the watershed,"concentrating" at the outlet. Thus, concentrationtime is the time required,with uniform rain, for 100percentof a tract of land to contributeto the direct runoff at the outlet.e As a secondpopular definition, the concentrationtime is often equatedwith either the excess-rainfallreleasetime or the wave travel time becausethe time for runoff to arrive at the outlet from the most remotepoint after rain ceasesis assumed to be indicative of the time required for 100 percent contribution from all points during any uniform storm having sufficient duration. The latter definition is often preferred becausefew storm durations exceedthe time of concentration,making determinationof /. possibleonly by examiningexcessrain recession. Becausetime of concentrationis conceptuallythe time requiredfor 100percent of the watershedto contribute, it is also often defined as the time from the end of excessrainfall to the inflection point on the hydrographrecessionlimb (e.g., see Fig. 12.2).The reasoningusedin this definitionis that direct runoff ceasesat the point of inflection. For a small tract of land experiencinguniform rain, the entire areacontributes at approximatelythe sametime that the runoff reachesan equilibrium.This givesrise to yet anotherdefinition of time of concentration.If rain abruptly ceased,the direct runoff would continueonly as long as the excess-rainfallreleasetime t,. On the basis of the seconddeflnition.excessreleasetime and time of concentrationcan be considered equivalent. Numerous equationsrelating time of concentrationto watershedparameters havebeen developed.Table 11.1 summarizesseveralpopular versions.Other variations are presentedin Chapters12, 15,16, and25.

11.7 BASIN LAG TIME The relative timing of rainfall and runoff must be known if drainageareashaving subbasinsare to be modeledor if continuoussimulation is desired.A basic measure of timing is basi'nlag, which locatesthe hydrograph'spositionrelativeto the causative stormpattern.It is most often definedas the differencein time betweenthe centerof massof effectiverainfall and the centerof massof runoff produced.Other definitions are also used.Two of theseare ( 1) the time interval from the maximum rainfall rate to the peakrate of runoff and(2) the time from the centerof massof effectiverainfall to the peak rate of flow. Time lag is characterizedby the ratio of a flow length to a mean velocity of flow and is thus a property that is influencedby the shapeof the drainagearea,the slopeof the main channel,channelroughnessand geometry,and thg'storm pattern.

11.7 BASINLAGTIME

183

TABLE11.1 SUMMARY OF TIMEOF CONCENTRATION FORMULAS Formula for t" (min)

Method and date Kirpich (1940)

tc : L : S :

USBR Design of Small Dams

t" : L :

(r973) Il :

lzzatd (7946)ts

0.00782077S-o38s length of channel/ditch from headwater to outlet, ft average watershed slope, ftlft

Developedfrom SCS data for sevenrural basins in Tennesseewith well-definedchanneland steepslopes(37o to 1O7o);for overlandflow on concreteor asphaltsurfacesmultiply t;by 0.4; for concretechannelsmultiply by 0.2; no adjustments for overland flow on bare soil or flow in roadsideditches.

60(lI.9L31H)o38s length of longest watercourse, ml elevation difference between divide and outlet, ft

Essentiallythe Kirpich formula; developedfrom small mountainousbasinsin California (U.S. Bureauof Reclamation,1973,pp. 67-7I).t4

41.025(0.0007, t

Developedin laboratory experimentsby Bureau of Public Roadsfor overland flow on roadwayand turf surfaces;valuesof the retardance coefficientrange from 0.0070 for very smooth pavementto 0.012 for concretepavementto 0.06 for denseturf; solution requiresiteration; product i times Z shouldbe = 500.

c)Lozz

-c S0.333i0.66?

I : c: Z : ,S: FederalAviation Administration ( 1970)r6

Kinematic WaveFormulas Morgali and Linsley (1965)'? Aron and Erborge (1973)18 SCS Lag Equation (1972)te

rainfall intensity,in/h retardancecoefficient length of flow path, ft slopeof flow path, ftlft

r" = 1.8(1.1- C)Losofso333 C : rational method runoff coefficient l, : length of overland flow, ft S : surface slope, Va O.94Lo6no6 (io 45o 3)

l, n I S

: : : :

length of overland flow, ft Manning roughnesscoefficient rainfall intensity in/h averageoverland slope ftlft t.67 Lo8[(tooo/cN)- 9]oi

,. ._- @

L : hydraulic length of watershed (Iongestflow path), ft CN : SCS runoffcurve number S : averagewatershedslope,Vo

SCS AverageVelocity Charts(1975, 1986),0

Sarrce.' After Ref. 13.

Remarks

',' : l v L 60- v Z : length of flow path, ft V : ureragevelocity in feet per secondfrom Fig. 3- 1 of TR 55 for various surfaces

Developedfrom air field drainagedata assembled by the Corps of Engineers;method is intended for use on airfield drainageproblems,but has been used frequently for overlandflow in urban basins. Overland flow equation developedfrom kinematic wave analysis of surface runoff from developed surfaces;method requiresiteration sinceboth i (rainfall intensity) arrdt, are unknown; superposition of intensity-duration-frequency curve gives direct graphical solution for t". Equation developedby SCS from agricultural watersheddata; it has been adaptedto small urban basinsunder 2000 acres;found generally good where areais completelypaved;for mixod areas it tendsto overestimate;adjustmentfactorsare applied to correct for channelimprovementand imperviousarea; the equationassumesthat t" : 1.67 X basinlag. Overland flow charts in Ref. 20 provide average velocity as function of watercourseslope and surfacecover.

184

CHAPTER11

HYDROGRAPHS

Various studieshave been conductedfor the purpose of developingrelations descriptiveof time lag. Most prominent of thesewas the work by Snyderon large natural watersheds.2lHis original equation has been widely used and modified in variouswaysby other investigators.Eaglesonhas proposedan equationfor lag time on sewereddrainageareashavinga minimum sizeof 147 acres.2z An early investigation (1936) on small drainageareas(2-4 acres)was conductedby Horner in his classicalwork on urbandrainagein St. Louis,Missouri.23 Horner'swork wasinconclusive in that it did not yield a deflned procedure,but he did conclude that the comparativelywide rangein the lag time at eachlocation led to the inferencethat the lag was a variable,its valuebeingdeterminedmore by rainfall characteristicsthan by characteristicsof the drainagearea. Snyder'sstudybasedon data from the AppalachianMountain regionproduced the following equationfor lag time:z1 t,1:

where

c,(L""L)o3

( 11 . s )

/1 : the lag time (hr) betweenthe centerof massof the rainfall excessfor a specifiedtype of storm and the peak rate of flow I'"o : the distancealongthe main streamfrom the baseto a point nearestthe centerof gravity of the basin (mi) I : length of the main stream channel (mi) from the base outlet to the upstreamend of the streamand including the additional distanceto the watersheddivide C, : & coefficientrepresentingvariationsof types and locationsof str'eams

For the areastudied,the constantC, was found to vary from I.8 to 2.2, with somewhatlower valuesfor basinswith steeperslopes.The constantis consideredto includethe effectsof slopeand storage.The value of 4 is assumedto be constantfor a given drainagearea,but allowanceis madefor the useof different valuesof lag for different types of storms. The relation is consideredapplicable to drainage areas rangingin sizefrom 10 to 10,000mi'. In a studyof seweredareasrangingin size from 0.22 to 7.51 mi2,Eagleson22 developedthe equation "t ', where

tr : L :

: -

L ( l . 5 / n ) R 2 / 3s t / 2

(11.6)

lag time, the center of mass of rainfall excess to the peak discharge (sec) the mean travel distance (ft), which is equal to the length of that portion

of the sewerwhich flows full n : the weightedManning's coefficientfor the main sewer : the weightedhydraulic radius of the main sewerflowing full { S : the weightedphysicalslopeof the main sewer Eagleson'sequationdirectly includesthe effects of channelgeometry and slope,as well as basin shape,and thus representsa refinementof the Snyderapproach.It also indirectly includesthe important effect of the storm pattern.

PROBLEMS185 Linsley and Ackerman give examplesof applicationof the following modified form of Snyder'sequation.24

t,:C,+!

(rr.1)

where s is a weighted slope of the channel and the other variablesare as defined previously. Other investigatorshaverepresentedtime lag by equationsof the form

t t =K +

( 11 . 8 )

Vs

Numerous other derivations of relations for watershedlag times can be found in standardhydrologictexts and periodical literature. Others are includedwith someof the syntheticunit hydrographdiscussionsin Chapter 12.

r Summary Understandingthe structureof hydrographSis important to many designand water supply applications.The hydrographrepresentsthe portion of the hydrologiccycle that engineersmostoften needin orderto determineratesof flow in streamsfor setting bridge lengthsand elevations,designingflood protection measures,and establishing areal extent of flooding. Similarly, the volume of drainageinto a reservoiror past a water supply diversionis determinedfrom the areaunder the hydrograph.Accurate estimatesof thesevolumesare important to designof dams,reservoirs,pipelines,and ' numerousother structures. After graspingthe fundamentalsof hydrographcomponents,including the time relationshipspresentedin this chapter,the reader should be well preparedfor the quantitativedevelopmentsof hydrographtheory and applicationspresentedthroughout Chapters12 through 16 and in Part Five.

PROBLEMS 11.1. Referto Fig. 11.1.Replotthis hydrographand usetwo different techniquesto separate the baseflow. 11.2. Obtain streamflowdata for a water courseof interest.Plot the hydrographfor a major runoff event and separatethe baseflow. 11.3. For the event of Problem I1.2, tabtlate the precipitation causingthe surfacerunoff and determinethe duration of runoff-producing.rain. Estimatethe time of concentration and useEq. I 1.4 to estimatethe time baseof the hydrograph.Comparethis with the time basecomputedfrom the hydrograph. 11.4. Tabulatedbelow are total hourly dischargerates at a cross sectionof a stream.The drainagearea abovethe sectionis 1.0 acre. a. Plot the hydrographon rectangular coordinate paper and label the rising limb (concentrationcurve), the crest segment,and the recessionlimb.

186

CHAPTERll HYDROGRAPHS b. Determinethe hour of cessationof the direct runoff usinga semilogplot of Q versus time. c. Use the base flow portion of your semilog plot to determine the groundwater recessionconstantK, d. Carefully constructand label baseflow separationcurveson the graph of Part a, using two different methods.

Time(hr)

Q (cfs)

Time (hr)

@ (cfs)

0 1 2 3

102 100 98 220 512 630 460 330

8 9 l0 11 t2

2lo 150 105 75 60 54 48.5 43.5

I

5 6 7

IJ

t4 15

11.5. On a neatsketchof a typical total runoff hydrograph,showor dimensionthe (a) storm

hyetograph,(b) beginningofdirectrunoff, (c) cessationtimeofdirectrunoff' (d) base fllw separationassumingthat additional contributions to base flow are negligible during ihe period of rise, and (e) crest segmentof the hydrograph' 11.6. For an urban watershedassignedby your instructor,obtain measuresof the watershed area,length, and slope,and compareestimatesof the time of concentrationusing the Kirpich, USBR, FAA, and SCSLag equationsin Table 11.1'

REFERENCES 1. American Society of Civil Engineers,Hydrology Handbook, Manuals of Engineering Practice,No. 28. New York: ASCE, 1957. "A 2. Donn G. DeCoursey, Runoff HydrographEquation," U.S. Departmentof Agricul.ture, AgriculturalResearchService,Feb. 1966,pp.4I-116' 3. R. K' Linsley, M. A. Kohler, and J' L. H' Paulhus,Applied Hydrology' New York: McGraw-Hill, 1949. "Erosional Developmentof Streamsand Their DrainageBasins:HydroA R. E. Horton, physicalApproach to QuantitativeMorphology,"Bull. Geol' Soc'Am' 56(1945)' "An Approach Toward a PhysicalInterpretationof Infiltration Capacity," 5 . if.-n. gorton, Sci. Soc.Am. 5,399-417(1940). Soil Proc. "An Inflltration Equation with Physical Significance,"Soil Sci.77(1954). 6 . J. R. Philip, 7 . R. E: Horion";Surface Runffi Phenomena.Ann Arbor, MI: EdwardsBros., 1935. 8 . R. J. M. DeWiest,Geohydrology.New York: Wiley' 1965' 9 . ,.Hydrology,,'in EngineeringHandbook, Sec. 4, U.s. Department of Agriculture, Soil ConservationService,1972. "Discussionof Analysis of Runoff characteristicsby o. H. Meyet," Trans. 1 0 . B. S. Barnes, ASCEl0s(1940). "Unit HydrographLag and PeakFlow Relatedto Basin 1 1 . A. B. Taylorand H. E. Schwartz, Characteristics," Trans,Am. Geophys. Union 33(1952).

REFERENCES 187 "Streamflow 12. L, K. Sherman, from Rainfall by the Unit-GraphMethod," Eng.News-Rec. 108(1932). 1 3 . D. F. Kilber, "Desk-top methods for urban stormwatercalculation," Ch. 4 in Urban Stormwater Hydrology, Water ResourcesMonograph No. 7, American Geophysical Union, Washington, D. C., 1982. 14. U.S. Bureauof Reclamation,Designof SmallDams,2nd ed., Washington,D.C.,1973. 1 5 . C.F.Izzafi,, "Hydraulicsof RunofffromDevelopedSurfaces,"Proceedings,26th Annual Meetingof the HighwayResearchBoad,26, pp. 129-146, December1946. 16. FederalAviation Administration,"Circular on Airport Drainage,"ReportA/C 050-532058, Washington, D.C., 1970. n. J. R. Morgali, andR. K. Linsley,"ComputerAnalysisof OverlandFlow,"./. Hyd.Div., Am. Soc.Civ.Eng.,9l, no. HY3, May 1965. 1 8 . G. Aron, and C. E. Egborge,"A PracticalFeasibility Study of Flood PeakAbatementin Urban Areas,"U.S. Army Corpsof Engineers,Sacramento, Calif., March 1973. 1 9 . Soil ConservationService,"National EngineeringHandbook, Sec. 4, Hydrology," U.S. Dept. of Agriculture,U.S. GPO, Washington, D.C., 1972. 20. Soil ConservationService,"Urban Hydrology for Small Watersheds,"TechnicalRelease 55, Washington, D.C., 1975(updated,1986). 2 t . F. F. Snyder,"Synthetic Unit Graphs," Trans.Am. Geophys.Union 19, 447-454(1938). 22. Peter S. Eagleson,"CharacteristicsofUnit Hydrographsfor SeweredAreas," paperpresentedbeforethe ASCE, Los Angeles,-CA,1959,unpublished. 23. W. W. Horner, and F. L. Flynt, "RelationBetweenRainfall and Runoff from Small Urban Areas," Trans.ASCE 62(101), 140-205(Oct 1956). aA R. K. Linsley,Jr.,andW. C. Ackerman,"Methodof PredictingtheRunofffromRainfall," Trans.ASCE 107(1942\.

C h a p t e r1 2

Unit Hydrographs

r Prologue The purposeof this chapteris to: . Define unit hydrographsand show their utility in hydrologicstudiesand design. . Developfully the current methodsof obtaining, analyzing,and synthesizing unit hydrographs. . Presentmethodsfor converting unit hydrographsfor one storm duration to other storm durations. Waysto predict flood peak dischargesand dischargehydrographsfrom rainfall events havebeenstudiedintensivelysincethe early 1930s.One approachreceivingconsiderable use is called the unit lrydrographmethod. 12.1 UNIT HYDROGRAPH DEFINITION The concept of a unit hydrographwas first introduced by Shermant'zin 1932. He defineda unit graph as follows:2 givendrainage area, Ifa givenone-day rainfallproduces a 1-in.depthofrunoffoverthe thehydrograph showingtheratesat whichtherunoffoccurredcanbe considered a unit graphfor that watershed. Thus, a unit hydrographis the hydrographof direct runoff (excludesbaseflow) for any stormthat producesexactly 1.0inch of net rain (the total runoff after abstractions).Sucha stormwould not be expectedto occur,but Sherman'sassumptionis that the ordinatesof a unit hydrographare t.O/P times the ordinatesof the direct runoff hydrographfor an equal-duration storm with P inchesof net rain. The term "unit" hasto do with the net rain amountof 1.0inch and doesnot mean to imply that the duration of rain that producedthe hydrographis one unit, whether an hour, day,or any other measureof time. The storm duration,X, that producedthe unithydrographmustbe specifiedbecausea watershedhasa differentunit hydrograph

12.1 UNIT HYDROGRAPHDEFINITION

189

for eachpossiblestorm duration. An X-hour unit lrydrograp,his defined as a direct runoff hydrographhavinga 1.0-in. volumeand resultingfrom anX-hour stormhaving a net rain rate of 1,/Xin.lhr. Az-hr unit hydrographwould havea 1.0-in. volume producedby a 2-hqstorm,and a 1-dayunit hydrographwould be producedby a storm having 1.0 in. of eicessrain uniformly producedduring a 24-hr period. The valueX is often a fraction. Figure 12.1illustratesa2-ltr,l2-hr, and24-hrwthydrograph for a given watershed.

t (b)

24hr

)z nl T (c)

Figure 12.1 Illustration of 2-br, I2-hr, and 24-ht unit hydrographsfor the same watershed(Note: a : b : c : 1' X A).

190

CHAPTER 12 UNITHYDROGRAPHS By Sherman'sassumption,applicationof an X-hourunit graphto designrainfall excessamountsother than 1 in. is accomplishedsimply by multiplying the rainfall excessamount by the unit graph ordinates,since the runoff ordinatesfor a given duration are assumedto be directly proportional to rainfall excess.A 3-hr storm producing2.0 in, of net rain would haverunoff rates2 times the valuesof the 3-hr unit hydrograph.One-half inch in 3 hr would produceflowshalf the magnitudeof the 3-hr unit hydrograph.This principle of proportional flows is expandedin Section 12.3 and appliesonly to equal duration storms. Implicit in deriving the unit hydrographis the assumptionthat rainfall is distributed in the sametemporal and spatialpattern for all storms.This is generallynot true; consequently,variationsin ordinatesfor different stormsof equal duration can be expected. This chapteris organizedto defineunit hydrographsfirst, then presentmethods of deriving unit hydrographsfrom actual rainfall and runoff records (Section 12.2). After familiarizing the readerwith the origin of unit hydrographs,Section I2.3 presentsmethodsof applyingunit hydrographsto generatedirect runoff hydrographsfor any storm with durationsthat are multiple integersof the U.H. duration. The constructionof unit hydrographsfor stormswith other than integermultiples of the derived duration is facilitated by a method known as the S-lrydrograph developedby Morgan and Hulinghorst.3The procedure,as explainedin Section 12.4, employs a unit hydrographto form an S-hydrographresulting from a continuous appliedrainfall. The need to alter duration of a unit hydrographled to studiesof the shortestpossiblestorm duration-the instantaneousunit rainfall. The concept of instantaneousunit hydrograph(IIJH) is tracedto Clark6and can also be used(Section 12.5) is constructingunit hydrographsfor other than the derived duration. The previousdiscussionassumesthat the analysthasrunoff and rainfall datafor deriving a unit hydrographfor the subject watershed.The application of unit hydrographtheory to ungaugedwatershedsreceivedearly attentionby Snyderaand also by Taylor and Schwartz,5who tried to relate aspectsof the unit hydrographto watershed characteristics.As a result, a full set of synthetic unit-hydrographmethods emerged.A numberof theseare presentedin Section12.6.

12.2 DERIVATION FROM OF UNITHYDROGRAPHS DATA STREAMFLOW Data collection preparatoryto deriving a unit hydrographfor a gaugedwatershedcan be extremelytime consuming.Fortunately,many watershedshaveavailablerecordsof streamflowand rainfall, and thesecan be supplementedwith office records of the Water ResourcesDivision of the U.S. GeologicalSurvey.TRainfall records pay be securedfrom ClimatologicalDatas publishedfor eachstatein the United Statesby the National Oceanicand AtmosphericAdministration (NOAA). Hourly rainfall records for recordingrainfall stationsare publishedas a Summaryof Hourly Observationsfor the location. Summariesare listed for approximately300 first-order situationsin the United States. To developa unit hydrograph,it is desirableto acquireas rnanyrainfall records aspossiblewithin the study areato ensurethat the amountand distributionof rainfall

12.2 DERIVATION,OFUNIT HYDROGRAPHSFROM STREAMFLOWDATA

191

over the watershedis accurate$ known. Preliminary selectionof storms to use in deriving a unit hydrographfor a watershedshouldbe restrictedto the following: 1. Stormsoccurring individually, that is, simple storm structure. 2. Storms having uniform distribution of rainfall throughout the period of rainfall excess. 3. Stormshavinguniform spatial distribution over the entire watershed. Theserestrictionsplace both upper and lower limits on size of the watershedto be employed.An upper limit of watershedsize of appro5imately1000 mi2 is overcautious, althoughgeneralstormsover suchareasare not unrealisticand somestudiesof areasup to 2000 mi2 have used the unit-hydrographtechnique.The lower limit of watershedextentdependson numerousother factorsand cannotbe preciselydefined. A generalrule of thumb is to assumeabout 1000acres.Fortunately,other hydrologic techniqueshelp resolveunit hydrographsfor watershedsoutsidethis range. The preliminary screeningof suitable storms for unit-hydrographformation shouldmeet more restrictive criteria before further analysis: 1. Duration of rainfall event should be approximately10-30 percent of the drainagearea lag time. 2. Direct runoff for the selectedstorm shouldrangefrom 0.5 to 1.75 in. 3. A suitablenumberof stormswith the sameduration shouldbe analyzedto obtain an average of the ordinates (approximately five events). Modificationsmay be madeto adjustdifferent unit hydrographsto a single duration by meansof S-hydrographsor IUH procedures. 4. Direct runoff ordinatesfor eachhydrographshouldbe reducedso that each eventrepresents1 in. of direct runoff. 5. The final unit hydrographof a specificdurationfor the watershedis obtained by averagingordinatesof selectedeventsand adjustingthe result to obtain 1 in. of direct runoff. Constructingthe unit hydrographin this way producesthe integratedeffect of runoff resultingfrom a representativeset of equal duration storms.Extremerainfall intensityis not reflectedin the determination.If intensestormsare needed,a study of recordsshouldbe madeto ascertaintheir influenceupon the dischargehydrographby comparingpeaks obtainedutilizing the derived unit hydrographand actual hydrographsfrom intensestorms. Essentialstepsin developinga unit hydrographfor an isolatedstorm are: l. Analyzethe streamflowhydrographto permit separationof surfacerunoff from groundwaterflow, accomplishedby the methodsdevelopedin Section l 1.4. 2. Measurethe total volume of surfacerunoff (direct runoff ) from the storm producingthe original hydrograph.This is the area under the hydrograph after groundwaterbaseflow has been removed. 3. Divide the ordinatesof the direct runoff hydrographby total direct runoff volume in inches,and plot theseresultsversustime as a unit graph for the basin. \-

192

12 UNITHYDROGRAPHS GHAPTER 4. Finally, the effective duration of the runoff-producing rain for this unit graphmustbe found from the hyetograph(time history of rainfall intensity) of the storm eventused. Proceduresother than thoselisted are requiredfor complex stormsor in developing synthetic unit graphs when data are limited. Unit hydrographscan also be transposedfrom one basinto anotherundercertain circumstances.An exampleillustratesthe derivationof a unit hydrograph.

EXAMPLE I2.1 Using the total direct runoff hydrographgiven in Fig. I2.2, derive a unit hydrograph for the l7I5 ac drainagearea. Solution 1. Separatethe base or groundwaterflow to get the total direct runoff hydrograph.A commonmethodis to draw a straightline AC that beginswhen

2-hr rainfall duration

I I

precipitaion = 4.2 in.

*zTotal

)

o

1l

l

t2

Time (hr)

Directrunoff. /T\ ordinate Yl

\

*zTotal directrunotf of l.4l5in.on1715ac

500 o FA

2-hr unit hydrograph of l.u rn. on I / I) ac

400 300 200 .100 0

Basef-tow

/

\,

/

t'-# -j

/^ 3

4

5

6

7

Baseflow separatlon

8

Time(hr) Dfuectrunoffduration

Figure 12.2 Illustration of the derivation of a unit hydrograph from an isolatedstorm.

12,2 DERIVATIONOF UNIT HYDROGRAPHSFROM STREAMFLOWDATA

193

the hydrographstartsan appreciablerise and endswherethe recessioncurve intersectsthe baseflow curve.The importantpoint hereis to be consistentin methodologyfrom storm to storm. 2. The depth of direct runoff over the watershedis calculatedusing > (DR x Ar) _ 2447 cfs-hr -: 1 4" ''* area l7l5 ac

(r2.r)

whereDR is the averageheightof the dfuectrunoff ordinateduring a chosen time period Ar (in this caseA/ : 1.0 hr) . The valuesof DR determinedfrom Fig. I2.2 are listedin Table 12.1. 3. Computeordinatesof the unit hydrographby using

8 "_ Q , 1 v,

(r2.2)

where Q, : the magnitudeof a hydrographordinate of direct runoff having a volume equal to % (in.) at someinstant of time after start of runoff Q, : the ordinateof the unit hydrographhavinga volume of 1 in. at someinstant of time In this examplethe valuesare obtainedby dividing the direct runoff ordinatesby 1.415.Table12.1outlinesthe computationof the unit-hydrograph ordinates. 4. Determinethe duration of effectiverainfall (rainfall that actuallyproduces surfacerunoff). As statedpreviously,the unit hydrographstorm duration

TABLE 12,1 DETERMINATION OFA 2.HRUNITHYDROGRAPH FROM AN ISOLATEDSTORM (1)

(2)

(3)

Time (hr)

Runoff (cfs)

Base flow (cfs)

I 2 3 4 4.7 5 6 7 8 9 10 10.5 lt t2

110 t22 230 578 666 645 434 293 202 160 1t7 105 90 80

t10 t10 t10 110 110 110 110 110 110 110 1r0 105 90 80

(4) Direct runoff, (2)-(3) (cfs)

0 t2 120 468 556 535 324 183 92 50 7 0 0 0

(5) 2-hr unit hydrograph ordinate,(4) + 1.415 (cfs) 0 8.5 84.8 i-l

I

393 379 229 129 65.0 35.3 4.9 0 0 0

194

CHAPTER12

UNIT HYDROGRAPHS

should not exceedabout 25 percent of the drainage atea lag time' but violatesthis rule for the example.From Fig. 12.2, the rain duration is 2 hr. 5. Using the values from Table I2.1, plot the unit hydrograph shown in Fig. 12.2. r I

BY LAGGINGMETHODS APPLICATIONS 12.3 UNITHYDROGRAPH Once an X-hr unit hydrographhasbeenderivedfrom streamflowdata (or synthesized from basin parameters,Section 12.6) it can be used to estimatethe direct runoff hydrographshapeand duration for virtually any rain event.Applications of the X-hr UH to other stormsbeginswithlagging procedures,usedfor stormshavingdurations that are integermultiples of the derived duration. Applications to stormswith fractional multiples of X, known as S-hydrographandIUH procedures,are discussedin Sections12.4 and 12.5. Becauseunit hydrographsare applicableto effective (net) rain, the processof applyrngUH theory to a storm beginsby first abstractingthe watershedlossesfrom the precipitation hyetograph,resulting in an effective rain hyetograph'Any of the proceduresdetailedin Chapter 4 can be applied. The remainder of this discussion assumesthat the analysthas already abstractedwatershedlossesfrom the storm. If the duration of anotherstorm is an integermultiple of X, the storm is'treated as a seriesof end-to-endX-hourstorms.First, the hydrographsfrom eachX increment ofrain are determinedfrom the X-hourunit hydrograph.The ordinatesare then added at correspondingtimes to determinethe total hydrograph. EXAMPLE 12.2 Dischargerates for the 2-hr unit hydrographshownin Fig' I2.3 are'.

Time (hr) O (cfs)

0 0

1 100

2 250

3 200

4 100

5 50

6 0

Develophourly ordinatesof the total hydrographresultingfrom a 4-hr designstorm havingthe following excessamounts:

Hour Excess(in.)

1 Q.5

L

0.5

J

1.0

4 1.0

Solution. The 4-hr duration of the designstorm is an integermultiple of the unit hydrographduration.Thus,the total hydrographcanbe foundby addingthe contributionsof two 2-hr incrementsof end-to-endrain, asshowninFig. l2'3c. The first 2-hr stormsegmenthas 1.0 in. of net rain and thus reproducesa unit 2.0in. of netrain (in 2 hr);thus The second2-hr stormsegmenthas hydrograph. its ordinatesare twice those of a 2-hr unit hydrograph.The total hydrograph,

12.3 F

r>

UNIT HYDROGRAPHAPPLICATIONSBY LAGGINGMETHODS

1.0

h

u.)

c

0 2

3

4 .

5

6

(a) 2-hourunit storn excess

^

300 200

i90 zoo € 1oo '6 0

1.0 ir

0.5 0 2

3

4

5

6

(c)Designstom excess 600 500 '6 400 :i

H 300 i5 200 100 0 r

2

3

4

5

6

7

(d) Contributionof each2-hourstorm 600 500 a

t +oo ; ff:oo '$ zoo -100 0 0

1

2

3

4

5

6

1

8

(e) Total design hydrograph

Figure 12.3 Example 12.2 deivation of total runoff hydrographusing a 2-hr unit hydrograph.

195

196

cHAPTERi2

uNtr HyDRocRApHS

Fig.12.3e, is found by summingthe fwo contributionsat correspondingtimes. Note in Fig, 12.3d that runoff from the secondstorm beginswhen the second rain begins,not at the beginningof the first storm. r I This methodof "lagging" is basedon the assumptionthat linear responseof the watershedis not influencedby previousstorms-that is, one can superimposehydrographs offset in time and the flows will be directly additive. The simplestway to developcompositedirect runoff hydrographsfor multiple-hourstormsis in a spreadsheet.Care must be taken, however,in visually confirming, as in ExampleI2.2, that the start and end points of runoff from eachcontributingX-hr incrementof rain are properly selected.A commonerror is to lag eachadditional contributinghydrograph by Ar, the time interval betweenreadings,rather than X, the associatedduration with the given unit hydrograph.Also, the multiplier for the UH ordinatesmust be the net rain occurring in X hours, not the rain occurring in the time increment A/. Example I2.3 illustratesthesepoints. EXAMPLE 12.3 Using the derived 2-hr lunit hydrographin Table 12.1, determinethe direct runoff hydrographfor a 4-hr. storm havingthe following excessrain amounts:

Hour

I

Excessrain. in.

o.7

2 0.7

t.2

1.2

Solution 1. Tabulatethe unit hydrographat intervalsof the selectedtime interval, A/, as shown inTable 12.2. TABLE 12.2 UNIT HYDROGRAPHAPPLICATIONOF EXAMPLE12.3

Time (h0

Effective rainfall (in.)

0 I

2 J

4 5 6 7 8 9 10 1l t2

0.7 0.7 1 . 2" 1.2

Unit hydrograph (cfs) 0 8.5 84.8 JJI

379 229 129 65 35.3 4.9 0

Contrib. of first 2-hrrain U HX 1 . 4 ' 0 I1.9 l19 463 531 321 181 91 49.4 6.9 0

Contrib. of second 2-hr rain

uH x 2.4'

0 20.4 203 794 910 550 310 156 84.7 11.8 0

Total outflow hydrograph (cfs)

0 11.9 119 483 734 11 1 5 1091 641 359 163 84.7 I 1.8 0

12.gUN|THYDRoGRAPHAPPL|CAT|oNSBYLAGG|NGMETHoDS197

2. Determine the correct UH multiplier for eachX-hr interval. BecauseX is 2 hrs for this example,the first two hours of the storm producea total net rain of 1.4 inches. Similar$, the last two hours of the storm produce 2.4 inchesof net rain. 3. Determinethe correct start and end times for eachof the two hydrographs and tabulatethe contributionof the l.4-inch and 2.8-inch rains at the : 3 hrs, appropriatelag times. Becausethe secondX-hr storm startedat / : I2'2. inTable shown hrs as 3 runoff for this-stormcannotbegin until / hydrographs runoff the total 4. Add the contributionsat eachtime to determine for the 4-hr storm. 5. Checkthe tabular solutionby plotting eachof the two hydrographsand sum rr the ordinatesat each/, as showninFig.I2'4' In addition to using a given X-hr UH for determiningthe runoff hydrographfor a given storm, tagging of ttt" X-hr UH can be used to developother duration unit hyirographs.The pioCedureis the sameas applyingthe X-hr UH to 1.0 in. of net rain in f nouri. As earlier, Y must be an integermultiple of X. For example,if a 1-hr unit hydrographis availablefor a given watershed,a unit hydrographresultingfrom a 2-hr siorm-is tbtained by plotting two L-hr unit hydrographs,with the secondunit hydrographlagged t hr, adding ordinates,and dividing by 2' This is demonstratedin nigl ti.S, riliere the dashedline rept"sentsthe resulting2-ht unit hydrograph'Thus the t in. of rainfall containedin the original 1-hr duration has been distributedover a Z-hr period.

t

2

3

4

5

6

7

8

9

1

0

1

1

1

Time units

Figure 12.4 Synthesizedhydrographfor'Example 12'3 derivedby the unit hydrographmethod'

\_

2

198

CHAPTER 12 UNITHYDRoGRAPHS

l-hr unit hydrogaph 2-hIunithy&ogaph

Time

Figure12.5 Unithydrographlaggingprocedurero developanotherunit hydrograph. Modificationsof the original unit-hydrographduration can be madeso that two 1-hr unit hydrographsare usedto form a 2-hr unit hydrograph;two 2-hr unit hydrographsresult in a 4-fu diagram,and so on. Care must be taken not to mix durations in the lagging procedure, since errors are introduced; a l-hr and a}-hr unit hydrograph do not representa 3-hr unit hydrograph.Lagging procedureis therefoie restrictedto multiples of the original duration accordingto the expression D1 : nD

(I2.3\

where Dl : the possibledurationsof the unit hydrographby lagging methods D : the original duration of any given unit hydrograph fl: I,2,3,..

12.4 S.HYDROGRAPHMETHOD The S-hydrographmethodovercomesrestrictionsimposedby the laggingmethodand allowsconstructionof any durationunit hydrograph.By observingthe lagging system just described,it is apparentthat for a l-hr unit hydrograph,the l-in. rainfall excess hasan intensityof 1 in./hr, whereasthe 2-hr unit hydrographis producedby a rainfall intensity of 0.5 in./hr. Continuouslagging of either one of theseunit hydrographsis comparableto a continuouslyappliedrainfall at either 0.5 in./hr or 1 in./hr intensity, dependingon wfuch unit hydrographis chosen. As an example,usingthe 1-hr unit hydrograph,continouslaggingrepresentsthe direct runoff from a constantrainfall of 1 in./hr as shownin Fig. r2.6a. The cumulative addition of the initial unit hydrographordinatesat time intervalsequalto the unit storm duration resultsin an S-hydrograph(seeFig. r2.7). &aphically, construction of an s-hydrographis readily accomplishedwith a pair of dividers. The maximum

12.4 S-HYDROGRAPHMETHOD

*l

199

Dhr

D-hr S-hydrographlaggedt hr

*l

Time

l-

(a)

Figure 12.6 S-hydrographmethod.

dischargeof the S-hydrographoccursat a time equalto D hourslessthan the time base of the initial unit hydrographas shown inFig. 12.6a. To constructa pictorial 2-hr unit hydrograph,simply lag the first S-hydrograph by a secondS-hydrographa time interval equalto the desiredduration.The difference in S-hydrographordinatesmust then be dividedby 2. Any duration r unit hydrograph may be obtained in the samemanner once another duration D unit hydrqgraphis known. Simply form a D-hr S-hydrograph;lag this S-hydrographf hr, andmultiply the difference in S-hydrographordinatesby D/t. Accuracy of the graphical procbdure dependson the scaleschosento plot the hydrographs.Tabular solution of the Shydrographmethodis also employed,but hydrographtabulationsmustbe at intervals of the original unit.hydrographduration. EXAMPLE 12.4 Given the following 2-hr unit hydrograph,use S-hydrographproceduresto construct a 3-hr unit hydrograph. Time (hr) 0 (cfs)

o 0

| 100

2 250

3 200

4 100

5 50

6 0

Solution. The 2-hr unit hydrographis the runoff from a 2-hr stormof 0.5 in./hr. The S-hydrograph is formedfrom a net rain rate of 0.5 in./hr lasting indefinitelyas shownin Fig. 12.6a.Its ordinatesarefoundby addingthe 2-hr (UH) runoff ratesfrom eachcontributing2-hr block of rain: unit-hydrograph

200

CHAPTER12

UNIT HYDROGRAPHS

I

i

/

i

/,/

I

a

90

-'{.3 u0 Figure 12.7

S-hydrograph.

Time (h0

1st2-hr

0 I z

3 4

5 6 7 8

0 100 250 200 100 5n

0

Time (min.)

2nd Z-hr

0 100 250 200 100 50 0

S-hydrograph

3rd2-hr

' 0 100 250 200 100

0 100 250

0 100 250 300 350 350 350 350 350

To find a 3.-hrhydrograph,the S-curveis laggedby 3 hr and subtractedasshown in Fig. 12.6b.This results in a hydrographfrom a 3-hr storm of 0.5 in./hr, or 1.5 in. total. Thus the ordinatesneed to be divided by 1.5 to producethe 3-hr unit hydrograph:

201

12.5 THE INSTANTANEOUSUNIT HYDROGRAPH

Time (hr)

S-hydrograph

0 1 2 3 4 5 6 7

0 100 250 300 350 350 350 350

Lagged S-hydrograph

0 100 250 300 350

Difference

0 100 250 300 250 r00 '50 0

3-hrUnit hydrograph 0 67 167 200 167 . 6 7 33 0

12.5 THE INSTANTANEOUS UNITHYDROGRAPH The unit-hydrographmethodof estimatinga runoff hydrographcanbe usedfor storms of extremelyshortduration.For example,if the durationof a stormis 1 min and a unit volume of surfacerunoff occurs, the resulting hydrograph is the 1-min unit hydrograph.The hydrographofrunofffor any 1-min storm ofconstant intensitycan be computedfrom the l-min unit hydrographby multiplying the ordinatesof the 1-min unit hydrographby the appropriaterain depth.A storm lasting for many minutescan be describedas a sequenceof 1-min storrns(Fig. 12.8).The runoffhydrographfrom each l-min storm in this sequencecan be obtainedas in the precedingexample.By superimposingthe runoff hydrograph from each of the l-r.nin storms, the runoff hydrographfor the completestorm can be obtained. From the unit hydrographfor any duration ofuniform rain, the unit hydrograph for any other durationcanbe obtained.As the durationbecomesshorter,the resulting unit hydrographapproachesan instantaneousunit hydrograph.The instant4neousunit hydrograph(IUH) is the hydrographof runoff thaf would result if 1 in. of waterwere spreaduniformly over an areain an instant and then allowed to run off.e To develop an IUH, any I in.lhr S-hydrographmust first be obtained. The resulting S-curve is laggedby the interval Ar to developa Ar-hour unit hydrograph. The resulting At-hour unit graph becomesan IUH when Ar is set to 0.0 in the limit. If a continuing 1 in./hr excessstorm produces the original and lagged Shydrographsof Fig. 12.6b,the Ar-hour unit hydrographis the differencebetweenthe two curves,divided by the amount of excessrain depth in A/ hours,or Qo- Q" Q,(Lt-hr UH) : ILt

(r2.4)

The Qo - Q" dtfferencesare dividedby I Lt to convertfrom a stormwith 1Al inches in Al hours to one with 1.0 in. in At hours,which is the definition of a Ar-hour unit graph. As Ar approacheszpro, Eq. 12.4 becomes

: !as 0,(ruH) Idt

(12.s)

202

CHAPTER12 UNIT HYDROGRAPHS

Figure 12.8 Unit-hy&ographdescriptionof the runoff process.(a) Unit hydrograph;(b) a sequenceof l-min storms;(c) superposition for eachof of runoffhydrographs the l-min storms.(After Schaake.e) which showsthat the flow at time I is proportional to the slopeof the S-hydrograph at time r. In applications,the slopeis approximatedbyLQ/A,I, and the IUH ordinates can be estimatedfrom pairs of closelyspacedpoints of the S-hydrograph. Ifan IUH is supplied,the aboveprocesscan be reversed,and any X-hour unit graph can be found by averagingIUH florvsat X-hr intervals, or

Q,(X-tuUH) : 1(IUH,+ IUH,_X)

{r2.6)

Use of this approximate equation is allowed for small X values and permits direct calculationof a unit graph from an IUH, bypassingthe normal S-hydrograph procedure.

UNITHYDROGRAPH 12.5 THEINSTANTANEOUS

203

EXAMPLE 12.5 Given the following 1.0 in./hr S-hydrograph,determinethe IUH, and then use it to estimatea 1-hr UH.

Time (hr) S-curve (cfs)

0 0

0.5 5 0

1.0 200

1.5 450

2,0 500

2.5 650

3.0 700

3.5 750

4.0 800

Solution. The IUH is foundfrom Eq. l2.5.The slopeat time r is approximated bY (Q,*o.,- Q,-o)lLt

IUH = AQlAf

0 0.5 1 1.5 2 2.5 3 3.5 +

5

0 200 400 300 200 200 100 100 50 0 0

0 JU

200 450 500 650 700 750 800 800 800

The 1-hr uH is obtainedfrom F,q.12.6,usingreadingsat 1-hr intervals:

0 1 2 3 A

5 6

luHf

luHr_1

1-hr UH

0 400 200 100 50 0 0

0 0 400 200 100 50 0

0 200 300 150 75 25 n "

l

l

The readershouldverify that the 1-hr UH obtainedthrough use of the IUH is approximatelyt[e sameas that obtainedby lagging the S-hydrographt hr, subtracting, and convertingthe differenceto a 1-hr UH. The ordinatesof the IUH representthe relative effect of antecedentrainfall intensitieson the runoff rate at any instant of time. By plotting the IUH with time increasingto the left rather than to the right (seeFig. I2.9), andthen superimposing this plot over the rainfall hyetograph(plotted with time increasingto the right as in pig.iZ.g),the'relative weight given to antecedentrainfall intensities(asa function of time into the past) is easily observed.In other words, the runoff rate at any time is

/

204

12 UNITHYDROGRAPHS CHAPTER

Time-reversed image of the instantaneous unit hydrograph

Time into the past Antecedent iainfall intensities

oo=[;f@xi(t-r)dr

unit hyFigure 12.9 Calculationof runoff rates with the instantaneous antecedent the of average is a weighted drograph.The runoff rateat arrytime unit hyimageof the instantaneous rainfall intensities.The time-reversed the weightingfurtction.(After Schaake'v) drographrepresents computedas a weighted averageof the previousrainfall intensities.Therefore,the computedrunoff hydrographis the weighted,moving averageof.the rainfall pattern and itre weighting irtt"iiott is the time-reversedimage of the unit hydrograph'e Statedmathematically,the runoff rate at any time is given by the convolution integral

Q(A:

l,', cA,t Z*io ft

9,000 7,500 15,000

mV = wave celerity l*ln LR

mV ft/sec

0.5 0.7

LR ft 18,000

5,000

12,000 10,000

3,500

7,000

r 10

\ .=

rt \

M,= M^

Mrt

.

-4 (r 'F))- 3.30 =3.821F+ .7s7

\M,

-o.5 0.2

0.0

0.4 .'

1.0

0.6

Forestcanopycover,F (a)

(l) ,*^=l,r*

0.5

H o.o Mf

-0.5

'--1

Mr, + Mr1

4

4

1-

Mrt

? 'qTF r

I

-n - 3.30 7s7Q.

-1.0 0.0

0.6

0.4

0.8

Forestcanopycover,F (b)

- Figure 14.4 Daily radiationmelt in the forestwith clear skies:(a) duringspring,May 20; and (b) duringwinter, February15.(AfterU.S.Army Corpsof Engineers.2) Becausesnow radiates as a blackbody, the amount of radiation is related to its temperature(Planck's law), and total energyradiated is accordingto Stefan'slaw. Long-waveradiation by a snowpackis determinedin a complexfashionthrough the interactionsof temperature,forest cover, and cloud conditions'

PROCESS 281' 14.5 THESNOWMELT Direct solar short-waveradiation receivedat the snow surfaceis not all transferredto sensibleheat.Part of the radiationis reflectedand thus lost for melt purposes. Short-wavereflection is known as albedo and ranges from about 40 percent for melting snow late in the seasonto approximately80 percent for newly fallen snow. This property Valuesas high as 90 percenthavealsobeenrepofted in severalcases.22 of the snowpackto reflect large fractions of the insolation explainswhy the covers persistand air temperaturesremain low during clear, sunny,winter periods. That portion of short-waveradiation not reflectedand availablefor snowmelt may becomelong-waveradiation or be conductedwithin the snowpack.Some heat may also be absorbedby the ground with no resultantmelt if the ground is frozen. An expressionfor hourly short-waveradiation snowmeltis given as2 M-

H^

203.2Q,

(r4.33)

H^ : the net absorbedradiation (langleys) where ' 203.2 : a conversionfactor for changinglangleysto inchesof water When the snowquality is 1, long-waveradiation is exchangedbetweenthe snowcover and its surroundings.Snowmelt from net positive long-wave radiation follows Eq. 14.33.If the net long-waveradiation is negative (back radiation), there is art equivalentheat loss from the snowpack. An approximatemethodof estimating12-hr snowmeltDn (periodsmidnight to noon, noon to midnight) from direct solarradiation has been given by Wilson.2oThe relation is ofthe form (14.34) Dp = Do(l - 0.75m) where Do : the snowmeltoccurring in a half-day in clear weather m : thedegreeof cloudiness(0 for clearweather,1.0 for completeovercast) Suggested valuesfor Do are 0.35 in. (March), 0.42 in. (April), 0.48 in. (May), and 0.53 in. (June)within latitudes 40-48".2o

Rainfall Heat derivedfrom rainfall is generallysmall, sinceduring thoseperiodswhenrainfall occurson a snowpack,the temperatureof the rain is probablyquite low. Nevertheless, at highertemperatures,rainfall may constitutea significantheat source;it affectsthe aging processof the snow and ffequently is very important in this respect. An equationfor hourly snowmeltfrom rainfall isrs

(14.3s) P : the rainfall (in.) T- : the web-bulb temperatureassumedto be that of the rain This equationis basedon the relation betweenheatrequiredto melt ice (I44 Btu per pound of ice) and the amountof heatgiven up by a pound of water when its temperature is decreasedby one degree. where

282

14 SNOWHYDROLOGY CHAPTER

Daily snowmeltby rainfall estimatesare given by Ma: 0.007Pd(T"- 32)

(r4.36)

where Md : the daily snowmelt(in.) , P d : the daily rainfall (in.) To : the meandaily air temperature('F) of saturatedair taken at the 10-ft

level23

Conduction Major sourcesof heatenergyto the snowpackare radiation, convection,and condensation. Under usual conditions, the reliable determinationof hourly or daily melt quantitiescanbe foundedon theseheatsourcesplus rainfall ifit occurs.An additional sourceof heat,negligiblein daily melt computationsbut perhapssignificantover an entire melt season,is ground conduction. Ground conductionmelt is the result of upward transferof heatfrom ground to snowpackdue to thermal energythat was storedin the ground during the preceding summerand earlyfall. This heatsourcecanproducemeltwaterduringwinter and eady springperiods when snowmeltat the surfacedoesnot normally occur. Heat transfer by ground conductioncan be expressedby the relation2 dT Hn: K-where

(14.37)

K : the thermal conductivity "r'J"n dTlda : the temperaturegradient perpendicularto soil surface

The snowmeltfrom ground conductionis generallyexceedinglysmall. Wilson notes that after about 30 days of continuoussnowcover,heat transferredfrom the The amountof snowmeltfrom ground conducground to the snow is insignificant.20 tlon during a snowmelt seasonhas been estimatedat approxirnately0.02 in.lday'23 Groundconductiondoesact to providemoistureto the soil; thus,whenotherfavorable conditionsfor snowmeltoccur, a more rapid developmentof runoff can be expected. This sectionhasemphasizedthephysicsof snowmelt.The mannerin which heat Equations14.27-I4.3I and canbe providedto initiatethemelt processwasdiscussed. L4.33-I4.37 inclusivecan be usedto estimatethe melt at a given point. The task of computingrunoff from snowmeltin a basin cannot be approachedin sucha simple fashion,sincethere are many complexfactorsoperative.The remainderof this chapter is devotedto the general subject of runoff from snowmelt investigations.Figure 14.5 illustrateshourly variation in the principal heat fluxes to a snowpackfor a cloudy day. EXAMPLE 14.2 During a completelycloudy April period of l2hr, the following averagesexistedfor a ripe snowpacklocated at 10,000 ft above sea lel'el at a latitude of 44" N: air temperature50' F; mean wind velocity, 10 mph; relative humidity, 65Vo;avenge rainfall intensity,0.03 in./hr for I2hr; wet bulb psychrometerreading,48oF. Estimate the snowmeltin in. of water for convection,condensation,radiation, and warm rain for the 12 hr period.

April 23 100

--- Short-waveradiation Long-wave radiation

80 9 6 0

ffi

Incident short-wave radiation

nm

Reflected shon-wave raoanon

G *

Absorbedshort-wave radiation

Eo

F

7

E

0

1

0

2 0 MaY1955

31

20 10 June1955

Figurel4.l2Hydrometeorologicdataandcomputationofwatergenerated. (AfterU.S.Army Corpsof Engineers.') the contributblanket is highly important. where only snowmeltflows are developed, If rainfall ing areaneel not U"itr" entiredrainage-only that portion with-snowcover' while bare areas from o"-"o., during the snowcoverperiod, contributionscan come may cases in such losses other expansesmay producecombinedrunoff' The natureof differ greatly for nonsnowoverlayedand coveredlocations' subThe altitude is an exceedinglypertinent factor in the hydrology of tracts reducgeneral a to due jected to snowfall.Ratesof .no*tnitid"crease with elevation tion in temperaturewith height. orographic effects and the temperature-elevation snowcover relations tend to raise the amount of precipitation with altitude' Greater a result' As rates' melt depth occursbecauseof increasedprecipitation and reduced is apsnowline the as the basin-wide melt and cover-area increase with height

14.6 SNOWMELTRUNOFFDETERMINATIONS

299

proached,then diminish with elevationover the higher placesnormally completely snowcovereduntil late in the season.A snowpackexhibitsanotherimportant trait in relation to rainstorms.In the spring, relatively little runoff occurs from snow-free regionscomparedwith that from a snowfieldfor moderaterainfalls. During very cold weather,the situationduring heavyrains is often reversed,sincea dry snowpackcan retain significantamountsof water. Two basic approachesintroduce elevation effects into procedures for hydrographsynthesis.2 The first dividesthe basin into a seriesof elevationzoneswheie the snowdepth,precipitationlosses,and melt are assumeduniform. A secondmethod considersthe watershedas a unit, so adjustmentsare made to accountfor the areal extentof the snowcover,varying melt rates,precipitation, and other factors. To synthesizea snowmelthydrograph,information on the precipitation losses, snowmelt, and time distribution of the runoff are needed.Snowmelt is generally estimatedby index methodsfor forecasting,but in design flood synthesisthe heat budget approach,is the most used. Precipitation is determinedfrom gaugingsand historicor generateddata.Lossesare definid in two wayswheresnowmeltis involved. For rain-on-snowhydrographsall the water is considereda lossif delayedvery long in reaching a stream. This is basically the concept of direct runoff employed in rainstormhydrographanalysis.For hydrographsderivedprincipally from snowmelt, only that part of the waterwhich becomesevapotranspiration,or deeppercolation,or permanentlyretained in the snowpackis consideredto be lost. Assessingthe time distribution of runoff from snow-coveredareasis commonly done with unit hydrographsor storagerouting techniques.For rain-on-snow events,normal rainfall-type unit hydrographsare applied;for the distribution of strictly snowmeltexcess,special long-tailedunit graphsare employed.Storagerouting techniquesare widely exeicised to synthesizespring snowmelthydrographs,perhapsdividing them into severalcomponentsand different representativestoragetimes. The time distribution of snowmehrunoff differs from that of rainstormsdue mainly to large contrastsin the ratesof runoff generation.For flood flows associated with rainfall only, direct runoff is the prime concern,and time distribution of base flow is only approximated.Big errorsin estimatesof baseflow arenot generallyof any practical significance where major rainstorm floods occur. In rainstorm flows, infiltrated water is treated as part of the base flow component and little effort is directedtoward determiningits time distribution when it appearsas runoff. In using the unirhydrograph approachto estimatesnowmelthydrographs,it is customaryto separatethe surfaceand subsurfacecomponentsand route them independently. Storagerouting has beenusedextensivelyfor routing floodsthrough reservoirs or river reaches.It is also applicablein preparingrunoff hydrographs.In snowmelt runoff estimates,the rainfall and meltwaterare treatedasinputs to be routedthrough the basin, using storagetimes selectedfrom the hydrologic characteristicsof the watershed.Two basichydrologicrouting approachesare relatedto the assumptionof ( 1) reservoir-typestorageor (2) storagethat is a function of inflow and outflow.These methodswere treatedin depthin Chapter13. Storagerouting techniquesthat separaterunoff into surfaceand groundwater components,assigndifferent empirically derivedstoragetimesto each,and thenroute them separatelyhavebeen employed.26 An additional systemusesa multiple storage,

300

CHAPTER14

SNOW HYDROLOGY 120

,:

Rn

o 6 U il

Three 6-hr stages

One 18-hr stage

Time (hr)

storagerouting' Figure 14.13 Example of multiple-stagereservoir-type storageroutreservoir-type iti. ngur" illustratesih" ut" of multiple-storage to unit analogous ttrunn". u in runoff ing for"evaluatingtime distribution of Engineers'2) of Corps tiOrog.uptr. (Af1erU.S. Army

is routed throrrgh two or reservoir-type storagescheduling.2In this method inflow suchan approach'Any more stagesof storageru"""rtiu!fy. Figure 14.13 illustrates the storagetime and.the desiredtravel time can be obtainedby properly selecting varied to reflectvarious tf st4ges'Retentiontimes betweenstepsmay alsobe the. use of single-stage drologic characteristics.clark has suggestedthat to be^^simplified.27 rfter translatirrgirp", in time permits Jo=mputations hydrographshas runoff e most practicJd method for synthesizingsnowmelt primarily in differs graphs ,,nir hrrrrrnoranhThe characterof snowmeltunit 12' chapter in plots' As disclssed time base length fiom that of rainstorm unit In events' storm single isolated rainstorm unit hydrograph, ort"n are derived from snowmeltrunoff,ratesofwaterexcessaresmallandapproximate$continuous'Asa result, the use of S-hydrographsis indicated'2 since it allows (1) adiustThe S-hydrogrupfrri"titoA has considerableutility, generationrates,(2) adjusting ments to the derivedoni, hydrographfor nonuniform of the areaunder iirn" p"rtoo to; d;ired interval, (3) ready adjustments il;;;t; veragingseveralhydrographsto get a unit

;?ilt1,"1*T,::t1il1""t"1':*"ff rydiographmethodin adjustingfor nonuniform generationrates of water excess' Onceap"r""nrug"S-hyatogtupftisderived'aunithydrographofanydesired are of the S-hydrograph periodcanbe obtaineias indlcatedm nig. 14.15.ordinates

312

HYDROLOGY ANDSMALLWATERSHED 15 URBAN CHAPTER Both categoriesof peakflow determinationhavehad wide application;however, two relatively major difficulties are normally encounteredin applyingthe techniques. First, the rainfall-runoff formulas,suchas the rational formula, aie difficult to apply unless the return periods for rainfall and runoff are assumedto be equal. Also, estimatesof coefficientsrequiredby theseformulas are subjectiveand havereceived considerablecriticism. The empiric and correlativemethodsare limited in application becausethey are derivedfrom localizeddata and are not valid when extrapolatedto otherregions. The most fundamentalpeak flow formulas and empiric-correlativQmethods, ilue to their simplicity,persistin dominatingthe urban designscene,and severalof the most popular forms are briefly describedto acquaintthe reader with methodsand assu-ption*. Urban runoff simulationtechniquesare describedin Chapter25.

RationalFormula The rational formula for estimatingpeak runoff rates was introducedin the United Sincethen it has becomethe most widely used Statesby Emil Kuichlingin 1889.18 method for designingdrainagefacilities for small urban and rural watersheds.Peak flow is found from QO: CIA

(1s.1)

where Qo: the peak runoff rate (cfs) C _ the runoff coefficient(assumedto be dimensionless) I _ theaveragerainfall intensity(in./hr), for a stormwith a durationequal to a critical period of time /" t" : the time of concentration(seeChapter Ii) A : the size of the drainagearea (acres) cI : the averagenet rain intensity (in./hr) for a storm with duratiofl: t, The runoff coefficientcan be assumedto be dimensionlessbecause1.0 acre-in./hr is equivalentto 1.008 ft3lsec.Typical C valuesfor stormsof 5-10-year return periods are providedin Table 15.1. The rationale for the method lies in the conceptthat application of a steady, uniform rainfall intensity will causerunoff to reachits maximum rate when all parts of the watershedare contributingto the outflow at the point of design.That condition is met after the elapsedtime t", the time of concentration,which usually is taken as the time for a waveto flow from the most remotepart of the watershed.At this time, the runoff rate matchesthe net rain rate. Figure 15.1 graphically illustrates the relation. The IDF curve is the rainfall intensity-duration-frequencyrelation for the areaandthe peakintensityofthe runoff is Q/A: 4, which is proportional to the value of 1 defined at t". The constantof profottionatity is thus the runoff coefficient,C : (QIA)lL Note that QIA is a point value and that the relation, as it stands,yields nothing of the nature of the rest of the hydrograph. The definition chosenfor /" can adverselyaffect a designusing the rational formula. If the averagechannelvelocity is usedto estimatethe travel time from the most remote part of the watershed(a common assumption),the resulting design

FORURBANWATERSHEDS 313 15.2 PEAKFLOWFOHMULAS TABLE15,1 ryPICALC COEFFICIENTS FOR5TO 1O-YEAR FREQUENCY DESIGN Descriptionof area Business Downtown areas ' Neighborhoodareas Residential Single-family areas Multiunits, detached Multiunits, attached Residential(suburban) Apartment dwelling areas Industrial Light areas Heavy areas Parks,cemeteries Playgrounds Railroad yard areas Unimproved areas Streets Asphaltic Concrete Brick Drives and walks Roofs Lawns; sandy soil: Flat,2Vo Avenge,2-7Vo Steep,TVo Lawns; heavy soil: Flat,2Vo Average,2-7Vo Steep,TVo

Q

Flunoffcoefficients

0.70-0.95 0.50-0.70 0.30-0.50 0.40-0.60 0.60-0.75 0.25-0.40 0.50-0.70 0.50-0.80 0.60-0.90 0.10-0.25 0.20-0.35 0.20-0.40 0.10-0.30 0.70-0.95 0.80-0.95 0.70-0.85 0.75-0.85 0.75-0.9s 0.05-0.10 0.10-0.15 0.15-0.20 0.13-0.17 o.r8-0.22 0.25-0.35

H

o

Time (min)

Figure L5.1 Rainfall-runoff relation for the rational method.

314

CHAPTER15

URBANAND SMALLWATERSHEDHYDROLOGY

dischargecould be less than that which might actually occur during the life of the project. The reason is that wave travel time through the watershedis faster than averagedischargevelocity (seeSection 13.1).As a result of using the slowervelocity I{ the peak time (/.) is overestimated,the resultingintensityl from IDF curvesis too small, and the rational flow rate p is underestimated. Rational Method Applications Most applications of the rationalformulain determining peak flow rates utilize the following steps:, 1. Estimatethe time of concentrationof the drainagearea. 2. Estimatethe runoff coefficient,Table 15.1. 3. Selecta return period T, and find the intensity of rain that will be equaled or exceeded,on the average,once every I years.To produceequilibrium flows, this design storm must have a locally derived IDF curve such as Fig. 27.I3 or Fig. 15.2usinga rainfall durationequalto thetime of concentration. . 4. Determinethe desiredpeak flow Q,from Eq. 15.1. 5. Somedesignsituationsproducelargerpeak flowsif designstormintensities for durationslessthan /" are used.Substitutingintensitiesfor durationsless than t" is justified only if the contributingarea term in Eq. 15.1 is also reducedto accommodatethe shortenedstorm duration. One of the principal assumptionsof the rational method is that the predicted peak dischargehas the same'returnperiod as the rainfall IDF relation used in the IDF curves for storms in vicinitv of example site

-

a

4

E

"

0 5 10 15 20

30

40

50

60

70

80

90

100 110 r20

130

Time (min)

Figure 15.2 Intensity-duration-frequency curvesusedin Example 15.1.

15.2 PEAKFLOWFORMULASFORURBANWATERSHEDS 315

prediction.Another assumption,and onethat hasreceivedclosescrutinyby investigais the constancyof the runoff coefficient during the progressof individual tors,re'2o stormsand also from storm to storm. The"coefficientis usually selectedfrom a list basedon the degreeof imperviousnessand infiltration capacityof the drainagesurface. BecauseC : I,.rf I,the coefficientmust vary if it is to accountfor antecedent moisture,nonuniform rainfall, and the numerousconditions that causeabstractions and attenuationof flood-producingrainfalls. In practice,a composite,weightedaveragerunoff coefficientis computedfor the various surfaceconditions.Times of concentration are determinedfrom the hydraulic characteristicsof the principal flow path, which typically is divided into two parts, overland flow and flow in defined channels;the times of flow in eachsegmentare addedto obtain /". Another assumptionwith the rational method is that the equation is most applicableto antecedentmoisture conditions that exist for frequent storms,in the rangeof the 2- to 10-yrrecurrenceinterval,representativeof stormstraditionally used for design of residential storm drain systems.Becausemore severe,less frequent stormsoften have wetter antecedentmoisture conditions,the rational coefficient is increasedby multiplying it by a frequencyfactor.The commonly usedmultipliers for lessfrequent stormsare: Returnperiod(yrs)

2-to 25 )U

100

Multiplier

1.0 1.1 t.2 1.25

EXAMPLE 15.I Usethe rational.methodto find the 10-yearand 5O-yeardesignrunoff ratesfor the area shownin Fig. 15.3.The IDF rainfall curvesshownin Fig. 15.2arc applicable. Solution 1. Time of concentration: t,:tt*tz:15+5:20min

At = 3 acres

cr = o'3

tr = Az= Cz = tz =

15min 4acres o'7 5min

Figure 15.3 Hypotheticaldrainagesystem for Example15.1.

316

HYDROLOGY CHAPTER15 URBANAND SMALLWATERSHED

2. Runoff coefflcient: c : [(3 x 0.3) + (4 x 0.7)]lQ + 4) :0.53 for 10-yrevent C : 1.2(0.53): 0.64 for 50-yr event 3. Rainfall intensity-from Fig. 15.2: Irc : 4'2 in'/ht 1so: 5'3 in'/hr 4. Designpeak runoff: Q r c : C I A : 0 ' 5 3x 4 ' 2 x 7 Q s o : C I A : 0 ' 6 4x 5 ' 3 x 7

16 cfs 24 cfs rl

Rational Method Discussion The runoff coefficientin the rational formula is dependenton the soil type, antecedentmoisturecondition, recurrenceinterval, land use, slope, amount of urban development,rainfall intensity, surface and channel roughness,and durationof storm.Tablesand graphsgenerallyallow determinationof C from only two or three of thesefactors.Nomographsand regressionequationscan providerelationsamongmore factors.One suchrelation, applicableonly in the region for which it was derived,is2r 2(0.001CN1 48)0 ts-o't{(P+ I)/zfo j e1-s0 C : j .Z(t1-i)CN3To05[(0.01CN)o

n5.2) where CN : SCScurvenumber(Chapter4) T : recurrenceinterval ( years)

s:

averageland slope (7o) I : rain intensity(in./hr) P : percentimperviousness The rational formula is a simplemodelto expressa complexhydrologicsystem.Yet the methodcontinuesto be usedin practicewith resultsimplying acceptanceby designers, officials, and the public. The methodis easyto apply and givesconsistentresults. From the standpointof planning,for example,the methoddemonstratesin clearterms the effectsof development:runoff from developedsurfacesincreasesbecausetimes of concentrationdecreaseand runoff coefficientsincrease. For storm drainagesystems,the designeris normally askedto estimatethe peak flow rate that might be equalledor exceededat leastonce in a given numberof years (describedas the frequency- see Section 10.4). For designsusing the rational formula, the frequencyof the peak runoff eventis assumedequal to the frequencyof the rain event(an eventbeing deflnedas somerain depthin a given duration).Studies Figure 15.4 showscumulativelog-normal probabilhaveexploredthis assumption.z2 ity functions (Chapter 26) fitted to observationsof rainfall and runoff on a 47-acre area in Baltimore, Maryland, with an averageSurfaceimperviousnessof 0.44. The data are partial seriesfitted independentlyto the observedrainfall sequenceand the observedrunoff sequence.Thus the largestrunoff doesnot necessarilycorrespondto the largestranked rainfall, and a similar lack of correspondencebetweenany runoff

FORURBANWATERSHEDS 317 15,2 PEAKFLOWFORMULAS Percentage of sample values equal to or greater than indicated value

99. 5 9 9 9 8

95 90

80 7060504030

20

10

Rainfall frequency curve (rr = 7.5 min)

o

l"

!2 k

..t

y" o

tt

't'2'

5

2

10.s

./

o

" i/

o-' oo

2

o

-

q

a

1 0.9 0.8 0.7 0.6

a

a o

\

Peak runoff ftequency curve

0.5 cd

0.4 0.3

5 1020 Recurrenceinterval (Years)

Figure 15.4 Distributions of recorded rainfall and runoff' (After Schaake.22)

and the rainfall that producedit holds for the ranked position of the observationsin In Fig. 15.4, the 5-yearrainfall frequencyof the arraysof the two i"putut" sequences. io a runofffrequencyof4'0 cfs/acre;theratio indicatesa runoff 6.5in./hr corresponds coefficientof approximately0.6. Although the two sequencesare eachcloselylognormal, they tend to converge,which suggeststhat the runoff coefficient increases slightly with more intense,iess frequent storms.In the designrange,however,the t"*ttr tend to support the assumptionof the rational method that the recurrence interval of the runoff equalsthe reiurrence interval of the rainfall. It shouldbe noted that the rainfall distributionsin Figs. 15.1and 15.4havesimilarproperties.All IDF curvesare drawnthrough the averagerainfall intensitiesderivedfrom many different stormsof record; any single IDF curve dses not representthe progressof a single storm.For lack of historicalrunoff data, the designerturns to the rational methodto construct from the rainfall history what amounts to a runoff intensity-durationrelation. frequency The most critical (highestpeak) runoff eventis often assumedto be causedby a storm havinga duration -qual to the time of concentrationof the watershed'If the rainfall IDF curve is steepin the designrange,severaldurationsshouldbe testedfor the given frequency to assurethat no other storm of equal probability producesa higlier peak runoff iate. Most applicationsof the rational method do not includethis testbecausethe assumptionthai ihe peak occursat /" is commensuratewith the other inherent assumptions.

31 8

CHAPTER15


The rational method is used in the designof urban storm drainagesystems servingareasup to six hundredacresin size.For areaslarger than I mir, liydrograph or other techniquesare generally warranted.Considerabl" 3udg-"nt is riquir-eAin selectingboth the runoff coefficientsand times of concentratloi. a common procedure is to selectcoefficientsand assumethat they remain constantthroughout the storm.As the designproceedsfrom point to point downstream,a compositeweighted C factoris computedfor the drainageareaaboveeachpoint. The time of concentration is composedof an inlet time (the overlandand any channelflow timesto the first inlet) plus the accumulatedtime of flow in the systemto the point of design. Figure 15.5 is an exampleof a designaid for prJdicting overland flow times. calculation of flow time in stormdrainscan readily be estimatid knowing the type of pipe, slope,size,and discharge.23 Generally,the pipe is assumedto flow full foi this calculation.(see Fig. 15.6.) Nomographsalso are availableto solvethe Manning equationfor flow in ditchesand gutters.The estimationof inlet time is frequentl| basedsolelyon judgment; reportedvaluesvary from 5 to 30 min. Denselyaevitopei areaswith impervioustractsimmediatelyadjacentto the inlet might be assignedinlet periods of 5 min, but a minimum value of 10-20 min is more uiual.

d o

10n "-"

4U

Fo F

Figure 15.5 Surfaceflow time curves.(After FederalAviation Agency.23)

15.2 PEAKFLOWFORMUI.AS FORURBANWATERSHEDS 3.I9 1,500

2,400 2,000

1,000 800

1,500

600 500 400

1,000 800

0.7

s00

0.9

400

1.0

200

300 200

0.02

10 8 6 5 i

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0.8

600

100 80

60 50 40 )n

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300

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100 80 60 50 40 30

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: 8 o

0.00002 o h o

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0.00001 0.000008 0.000006 0.000005 0.000004

Figure 15.6 Flow in pipes (Manning's formula); (After Ref. 24.)

Most designersapplyingthis methoddo not usethe time of concentrationin its strictestsense;rather, the largestsum of inlet time plus travel time in the storm drain systemis taken as the time of concentration.Caution is required in app$ing the method' Peak dischargeis not the summationof the individual dischapges, b"Jarr* peaks from subareasoccur at different times. The runoff from subareasshould be recheckedfor eacharea under consideration.The averageintensity / is that for the time of concentration of the total area drained. While I decreaiesas the design ploceeds downstream,the size of the contributing area increasesand normally e

320

CHAPTER15


increasescontinuously.It shouldbe noted that the designat eachpoint downstream is a new solutionof the rational method.The only direct relation from point to point derivesfrom the meansfor determiningan incrementof time to be addedfor a new time of concentration.The effect is to provide an equal level of protection (i.e., an equal frequencyof surcharging)at all points in the system.Example 15.2 is reproduced from standarddesignreferencesto illustrate the application of the rational method to an urban sYstem.2a EXAMPLE 15.2 Basedon the storm sewerarrangementof Fig. 15.7a,determinethe outfall discharge. Assume that C : 0.3 for residentialareasand C = 0.6 for businesstracts. Use a 5-yearfrequencyrainfall from Fig. 15.7b andassumea minimum 20-min inlet time. Solution. The principal factors in the designare listed in Table 15.2' Additional columns can be provided to list elevationsof manhole inverts, sewer inverts, and ground elevations.This information is helpful in checkingdesigns use in drawingfinal designplans.(see Table 15.3.) lI and for subsequent 'orational"in that thepeak Modified Rational Method Therationalmethodis truly flow rate is simply set equal to the net rain rate after sufflcient time occurs for the entire watershedto contributerunoff. This resultsfor any storm equallingor exceed-

design,requiring volume of runoff as well as peak flow rates. OF COLUMNHEADINGSIN TABLE15.3 TABLE 15.2 DEFINITION Column

1 4 5 6 7 8 9 10 t1 T2 t3 14 15 t6 t7

Comment Line being investigated Inlet or manholebeing investigated Length of the line Subareaof the inlet Accumulatedsubareas Value of the concentrationtime for the area draining into the inlet Travel time in the pipe line WeightedC for the areabeing drained Rainfall intensitybasedon time of concentrationand a 5-yearfrequencyculve Uqitrunoff q: CI Accumulatedrunoff that must be carried by line Slopeof line Size of pipe Pipe capacity Velocity in full pipe Actual velocity in pipe

15.2 PEAK FLOW FORMULASFOR URBANWATERSHEDS

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l

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MH1-1 r ,\e / --

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contourline - areaoutline Drainage

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10-yr averagefrequencY

30

r

!

l

Ij-'ff:::i::::?

K:---t.-

40

50 60 70 80 Duration(min)

90

100 110 120

Figure 15.7 Sample storm drainage problem: (a) typical storm rainfall seier design plan anO (U) intensity-duration-frequency (After 24.) Ref' Iowa. for Davenport, curves

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WATERSHEDS323 FORURBAN 15,2 PEAKFLOWFORMULAS In the modified rational method.,a full hydrographis developqdrather than simply estimating the peak flow rate, using the following reasoning.If the storm duration exceedslhetime of concentration,the runoff rate would dse to the rational formula peakvalue,then stayconstantuntil net rain ceases.At that point, runoffrates would decreaseto zero as excessrain is releasedfrom the basin.Ifthe rainfall-excess releasetime (seeChapter 11) is equal to the time of concentration,the hydrograph : t",femaining flat would havean approximatetrapezoldshaperising to the peak al t D I t". until/: therainduration,D,andthenfallingalongastraightlineuntilt: rational modified the incorporate hydrology urban packages for Many software method for hydrographanalysis.the method is approximateand shouldnot be applied to watershedsover 50 acresin size.

SCSTR-55Method The U.S. Soil ConservationServicedevelopedproceduresfor estimatingrunoff volThey are known collectively as ume and peak ratei of dischargefrom urban areas.zs TR-55und indiuidually as thegraphical method,chart method, andtabular method. The threemethodsadjustrural proceduresin NEH-426to urbanconditionsby increasing the curve number CN foi impervious areas and reducing the lag time /1 for and channel improvements.Allowances are also made for various imlperviousness watershedshapes,slopes,and times of concentration.The SCS designedthe first two methodsto be usedfor estimatingpeak flows,and the third for synthesizingcomplete hydrographs.The tabular method and chart method (usedfor small watershedsup to aredescribedhereto help explainthe evolution 2000 acres)wererevisedin 1986,21but of the methods.All three were developedfor use with 24-hr storms'Use with other storm durationsis not advised. watersheds,up to 20 mr2 The graphicalmethodwasdevelopedfor homogeneous by the runoff curve represented in size, on which the land use and soil type may be a third variable is simply number. As shownin Chapter4, the runoff curve number in a graph of rainfall versusrunoff. Tie SCS peak dischargegraph shown in Fig. 15,8 is limited to applications (see where only the peak flow rate ii Oisired for 24 hr, Type-II storm distributions expethunderstorm the 24-hr of Chapter f 6l. A Type-II storm distributionis typical rienced in all staiei exceptthe Pacific Coaststates.Figure 15.8 was developedfrom numerousapplicationsofine SCSTR-20 eventsimulationmodeldescribedin Chapter 24. To apply Fig. 15.8, the watershedtime of concentrationin hours is enteredinto net ttre grapir'to prJdu." the peak dischargerate in cfs/mi2of watershedper inch of gross 24-hr the from rain during tk Z+-nr period. The 24-hr net rain is estimated amountusingthe scS curve number approachdescribedin chapter 4. the effectof urbanizationcanbe estimatedusingFig. 15.9.Oncethe composite curvenumber(CN) hasbeenestimatedfor the previousarea,a modifiedcurvenumber is determinedby enteringFig. 15.9 with the value of the percentimperviousareaon the modified watershed,r"ading vertically to the curve correspondingto the CN for the pervious watershed,and then reading horizontally to determinethe modified compositerunoff curve numberthat would be usedin determiningthe net rain depth for the urbanizedwatershed' Useof the 1975graphicalmethodis restrictedby the assumptionsof the tabular This method.The methodii a -ompositeof resultsfor one caseof the tabularmethod.

324

CHAPTER15

URBANNNO SUNU-WATERSHEDHYDROLOGY

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100 0.1

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3.0 4.0 5.0

(cfs/mi2lin.) Figure 15.8 Peakdischarge of runoff versustime of con(AfterU.S.SoilConsercentration /. for 24-hr,Type-IIstormdistribution. vationService.25) restricts its applicationsto runoff volumes greaterthan about 1.5 in. (if the curve numberis lessthan 60). Time of concentrationshouldrangebetween0.1 and 2.0hr, and the initial abstractionshould not exceedabout 25 percent of the precipitation. The chart method allows determinationof peak flows for 24-hr Type-II storms over watershedshaving a fixed length/width relation and no ponding areas.Three chartsareusedfor flat, moderate,or steepslopesof approximatelyI,4, or 16percent. Tablesof adjustmentsfor intermediateslopesare provided in the technicalrelease. Severalmicrocomputersoftwarepackagesfor urbanhydrologyhavebeendeveloped.28 Over two-thirdsarebasedon SCSprocedures, but cautionshouldbe applied

80

z

L) e .E

7n

u 6 0

0

l0

20

40 60 70 50 Connected imperviousarea(7o)

30

80

90

100

Figure 15.9 Percentage of impervious areas versus composite CNs for given pervious area CNs. (After U.S. Soil Conservation Service.2s)

WATERSHEDS325 FORURBAN 15,2 PEAKFLOWFORMULAS in assumingthat the commercialprogramsfully imitate TR-55 or other SCS handbook methods.An ideal TR-55 packagewould includeall three methods,would carry SCS endorsement,would stateall assumptionsand limitations, and would incorpopercentageof rate all SiS adjustmentsfor peak coefflcient,percentimperviousness, channelimproved,pondingor swampyareas,length/widthratio variations,and slope. Its use shoutAako be cautionedfor other than 24-hr stormshaving a Type-II SCS distribution. Packagesnot adheringto these limitations would not be qualified as TR-55 procedures. A significantproblem in someof the commercial softwarepackagesis the use of a trianlular-shaped unit hydrographfor convolutionto producehydrographsfor stormsof-various durations.The SCS used a triangular shapeto conceptualizethe peak flow rate of a curvilinear unit hydrograph,but has never endorseduse of other than either the curvilinear shapediscussedin Section I2.5 ot the tabulatedhydrg; graphsgiven in the TR-55 manual.For further reading,the SCS publisheda guide2e for the useof the 1975TR-55 intendedto clarify proceduresin the original technical release.

PrevailingSCSTR-55Method for use. ratherthan the 1975version,is recommended The 1986editionof TR-55,27 . It incorporatesseveralyearsof resultsof researchand experienceswith the original edition. The revisionsinclude the following:

1. Three additional rain distributions(seeFig. 16'17). ,, Expansionof the chapteron urban runoff curve numbers' 3. A procedurefor calculatingtravel times of sheetflow' 4. Deletion of the chart method.

Modifications to the graphical peak dischargemethod and tabular hydrographmethod. 6. TR-55 computerProgram.

Ratherthanrelyingtotally on Fig 15.9,the newTR-55usesTablet5.4 andFig.15'10 to provide urban runoff curve numbersfor certain instancesindicated in the table' For the new graphicalmethod,an urban curvenumberand the 24-hr designrain depth are estimated,ih"n un initial abstraction1, is determinedfrom the SCSrunoff equation(Chapter4) or from Table 15.5.The peak flow is found from linear interpolation of the curvesin Figs. 15.ll,15.I2, !5.13, or 15.I4, dependingon the rainfall distributiontype (Fig. rcn).If the computedI"f P ratio falls outsidethe curves,the nearestcurvi should be used. If the watershedcontains a pefcentageof ponds or swampyareas,the peak flow is multiplied by a reductioncoefficientfrom Table 15'6' EXAMPLE 15.3 concentration,CN: 75 A 1280-acreurbanTennesseewatershedhasa6.0-hrtimeof from Table15.4,and5 percentof the areais ponded.The25-year,24-httainis 6'0 in' Find the 25-yearpeak discharge.

TABLE 15.4 RUNOFFCURVE NUMBERSFOR URBANAREAS (see Sec. 4.9 foT other values)

Curvenumbers for hydrologic soilgroup'

Cover descriotion

Cover type and hydrologiccondition

Average percent imperviousareaD

A

Fully developed urban areas (vegetationestablished) Open space(lawns, parks, golf courses,cemeteries,etc.)" Poor condition (grasscover 757o) 39 Impervious areas Pavedparking lots, roofs, driveways,etc. (excluding right- of-way) 98 Streetsand roads Paved;curbs and storm sewers(excluding right-of-way) 98 Paved;open ditches(including right of-way) 83 Gravel (including righrof-way) 76 Dirt (including right-of-way) 72 Westerndeserturban areas Natural desertlandscaping(pervious areas only)' 63 Artificial desertlandscaping(impervious weed barrier, desertshrub with 1-2-in. sandor gravel mulch and basin borders) 96 Urban districts Commercial and business 89 85 Industrial 81 72 Residentialdistricts by averagelot size 77 65 f acre or less(town houses) j acre 61 38 57 30 I acre 25 I acre I acre 20 5l 2 aqes l2 46 Developingurban areas Newly gradedareas(pervious areasonly, no vegetation)" 77 Idle lands (CNs are determinedusing cover types similar to thosein Table 4.7).

B

c

79 69 6l

86 79 74

98

98

98 89 85 82

98 92 89 87

77

85

96

96

96

92 88

94 91

95 93

85 75 72 70 68 65

90 83 81 80 79 77

92 87 86 85 84 82

86

91

89 84 80

98 93 9r 89

'Average runoff condition, and 1" : 9.25. 'The averagepercent impervious area shown was used to developthe composite CNs. Other assumptionsare as follows: impervious areas are directly connected to the drainage system,impervious areas have a CN of 98, and pervious areas are consideredequivalent to open spacein good hydrologic condition. CNs for other combinations of conditionsmay-becomputedusingFig. 15.9 or 15.10 " CNs shown are equivalent to those of pasture. Composite CNs may be computed for other combinations of open spacecovol type. dComposite CNs for natural desertlandscapingshould be comppted using Fig. 15.9 or 15.10 basedon the impervious areapercentage(CN : 98) and the pervious area CN. The pervious area CNs are assumedequivalent to desert shrub in poor hydrologic condition. eComposite CNs to use for the designof temporary measuresduring grading and construction should be computed using Fig, 15.9 or 15.10basedon the degreeofdevelopment (impervious areapercentage)and the CNs for the newly graded pervious areas. Source: U.S. Soil ConservationService,2T

0.0

n5

> o

e

r.o I I

70

60

50

CompositeCN

Total impervious arca (Vo)

Figure 15.10 Graph of 1986 TR-55 composite CN with unconnected imperviousarea,or total imperviousarea,lessthan 30 percent.(After U.S. Soil . ConservationService.2T) TABLEls.s /aVALUESFORRUNOFFCURVENUMBERS Curvenumber

L fin.)

Curvenumber

L (in.)

40 4l 42 43 44 45 46 47 48 49 50

3.000 2.878 2.762 2.65r 2.545 2.444 2.348 2.255 2.167 2.082 2.000 1.922 r.846 1.774 1.704 1.636 1.571 r.509 1.448 1.390 1.333 1.279 1.226 I.t75 t.t25 1.077 1.030 0.985 0.941 0,899

70 71 72

0.857 0.817 0.'778 0.740 0.703 4.667 0.632 0.597 0.564 0.532 0.500 0.469 0.439 0.410 0.381 0.353 0.326 0.299 0.273 0.247 0.222 0.198 0.174 0.151 0.128 0.105 0.083 0.062 0.041

)l

52 53 54 55 56 57 58 59 60 6l 62 63 o+

65 66 67 68 69

Source:U.S. Soil ConservationService.

t5

74 IJ

76 77 78 79 80 8l 82 83 84 85 86 87 88 89 90 9l 92 93 94 95 96 o1 98

328

HYDROLOGY CHAPTER15 URBANAND SMALLWATERSHED

300

u

tnn

o P0 R I

3 S

100 R

o 60

40 01

0.6

0.8 l

Time of concentration, Z; (hr)

Figure 15.11 Unit peak discharge(q*) for SCS Type-I rainfall distribution. (After U.S. Soil ConservationService.)

9

100

Po

Ro

ci

60

30 0.1

0.2

0.4

0.6 0.8 1

2

4

6

810

Time of concentration,Zr (hr)

Figure 15.12 Unit peakdischarge(q,) for SCSType-IA rainfall distribution.(After U.S. Soil ConservationService.)

15.2 PEAK FLOW FORMULASFOR URBAN WATERSHEDS

329

1000 800

^

600 500

E

400

{, s 300 o 90 d

€ 2oo € 6

o

= P 1oo 80 60 50 0.1

0.2

0.4

0.6 0.8 1

2

4

6

810

Time of concentration,Z" (hr)

Figure 15.13 Unit peak discharge(q,) for SCSType-II rainfall distribution. (After U.S. Soil ConservationService.)

700 600 500 400 €< 300

-* d

,nn

E E

= 5 roo 80 60 40 0.1

0.2

0.4

0.6 0.8 1

2

4

6

810

Time of concentration, Z, (tn)

Figure 15.14 Unit peak discharge(q,) for SCSType-III rainfall distribution.(After U.S. Soil ConservationService.)

L

330

CHAPTER15


TABLE 15.6 ADJUSTMENTFACTOR(Fp)FOR POND AND SWAMP AREAS THAT ARE SPREADTHROUGHOUTTHE WATERSHED Percentageof pond and swamp areas

0 0.2 1.0 3.0 5.0

1.00 0.97 0.87 0.75 0.72

Source:U.S. Soil ConservationService.

From Solution. From Fig. 16.17, the Type-II storm appliesto Tennessee. q,: csm/in. 96 Table15.5,I":0.667.Thus I,fP: 0.11.FromFig. 15.13, From Chapter4, tberunoff from 6.0 in. is 3.28 in. Since5 percentof the area is ponded,the peak flow is adjustedusingTable 15.6,giving 4 : 0.72. Thus g : (96 csm/in.)(3.28in.)(2.0 mr')(0.72): 453 cfs rr The graphicalmethodprovidespeak dischargesonly. If a hydrographis needed or watershedsubdivisionis required, the tabular method2Tshouldbe used.The event simulationmodel TR-20 shouldbe usedif the watershedis very'complexor a higher degreeof accuracyis required (seeChapter24). Assumptionsof the graphicalmethod include: The method shouldbe used only if the weighted CN is greaterthan 40. The ?i valueswith the method may range from 0.1 to 10 hr. The watershedmust be hydrologicallyhomogeneous,that is, describable by one CN. Land use, soils, and cover must be distributed uniformly throughoutthe watershed. The watershedmay haveonly one main streamor, if more than one, the branchesmust havenearly equal times of concentration' The method cannotperform channelor reservoirrouting. The Fofactorcan be appliedonly for pondsor swampsthat are not on the flow path. Accuracy of peak dischargeestimatedby this method will be reducedif I"fP va\uesare usedthat are outsidethe range given. When -this method is used to develop estimatesof peak dischargefor presentand developedconditionsof a watershed,usethe sameprogedure for estimating[. Both the graphicalandtabular methodsare derivedfrom TR-20 output.The use of I permits them to be usedfor any size watershedwithin the scopeof the curves or tables. The tabular method can be used for a heterogeneouswatershedthat is Hydrographsfor the subwatersubwatersheds. dividedinto a numberof homogeneous shedscan be routed and added.

15,3

PEAK FLOW FORMULASFOR SMALL RURAL WATERSHEDS

331

The tabularmethodis describedin the technicalreleaseand is not detailedhere. In using the method, the following stepsare employed:

'

1. Subdividedthe watershedinto areasthat are relatively homogeneousand haveconvenientrouting reaches' 2. Determinedrainageareaof eachsubareain squaremiles' 3. Estimate T"for eachsubareain hours. The procedurefor estimatingI is outlinedinTR-55. 4. Find the travel time for eachrouting reachin hours' 5. Developa weightedCN for eachsubarea. 6. Selectan appropriaterainfall distribution accordingto Fig. 16.17. 7. Determine the 24-hr rainfall for the selectedfrequency(Chapter 16). 8. Calculatetotal runoff in inchescomputedfrom CN andrainfall (Chapter4)' 9. Find I,fot eachsubareafrom Table 15.5. 10. Usingthe ratio of I,f P andT,for eachsubarea,selectone of the hydrographs tabulatedin TR55. 11. Multiply the hydrographordinates(csm/in.) by the area (mi2) and runoff (in.) of eachrespectivesubarea. 12. Route and combinethe hydrographs. The SCS recommendsthat TR-20, rather than the tabular method,be used if any of the following conditions apply: Travel time is greaterthan 3 hr. f is greaterthan2hr. Drainageareasof individual subareasdiffer by a factor of 5 or more. The TR-55procedureshavebeenincorporatedby SCSin a computerprogram.Copies are availablefrom the U.S. National TechnicalInformation Service.

FORSMALLRURALWATERSHEDS 15.3 PEAKFLOWFORMULAS SCSTP-149Method TR-55 is the SCS procedure for urban watersheds,TR-20 is the unit-hydrograph procedurefor larger agriculturalwatersheds(seeChapter24), andTP-149wasdevelbped to allow esiimation of peak flow rates from small (5-2000 acres)agricultural It consistsof a seriesof 42 charts from which the peak dischargeof a watersheds.3o 24-hr ruinfall can be determined. Input to the procedure is the drainage area, averagewatershedslope, storm distributiontype (I or II), watershedcompositecurve number, and depth of rainfall. Figures15.15ind 15.16illustratethe numerouschartsin the TP. Shownare type-I with CN : 70 for both. Similar and type-Il curvesfor moderatelyslopedwatersheds, 15.7.Applicationsof TP 149 Table given in chartsare availablefor the combinations incrementsof Table 15.7, 5-unit the than other to watershedshavingcurve numbers by arithmetic or be accomplished percent, can 16 4, or or for slopesother ihan I, values' chart adjacent logarithmic interpolationbetween

332

CHAPTER15


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338

CHAPTER15


.--Duratio4

of Overbank Flow-->l InstantaneousPeak

o o

Drainage Ditch CaPacitY "Removal" Rate Based on

Iime

Figure 15.18 Illustration of relation befweenCyplys Creek stJntaneousflow. (After Soil ConservationService'35)

"removal" rate and peak in-

time for The selectedduration was} hours,consideredto be the maximum allowable to developed was Creekformula, Cyprus the called inundationof crops.An equation, equation, The rate. removal 24-hr the called rate, flow determinethe canal design curve based on rainfall depth,-contributingdrainage area,and the SCS composite number is Q:

CA5/6

( 1s .10)

where Q : reqrriredchannelcapacity fot 24-ht removal (cfs) C : drainagecoefflcient A: drainageareaGq mi) bY The drainagecoefficient,C,fot Eq. 15.10is found from an equationdeveloPed Stephensand Mi1ls36 (15.1) C : 16.39+ (14.75Q,",) where designeventfrom Fig' 4'I4' Q"", : the scS direct runoff (in.) for the 24-hr peak flow rate is Once Eq. 15.10is solvedfor the given frequency,the instantaneous from 1 to about areas drainage obtainedfrom Fig. 15.19.The procedureis limited to to the 24-hr rate peak instantaneous that ratios of the 200 squaremiles.It is suggested areas flatland For 1.0. to equal canaliemoval rateUetirniLO b values greaterthan or peak flows the that that havepart of the areain storm ,"wJts, the SCS recornmends 15.20.TheSCSfurther fromFig. ts.tqu.increasedbytheamountsindicatedinFig. that are recomniendsrestrictingur" of thit procedureto watershedsthat haveslopes TR-55' TR-20, as such methods lessthan 0.002. For stJeperslopewatersheds,other TP l4g, or regressionequationsare recommended'

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340

CHAPTER15


d

s40 o !l

o 9 ^ ^ i 4rl

0 Percentageof Area Servedby Storm Sewers

Figure 15.20 Effect of urban storm sewerson peak dischargefor urban areas. (After U.S. GeologicalSurvey.aT)

EXAMPLE 15.5 Usethe Cypro, Creek methodto determinethe peak 50-yr flow rate fron a 1.0 sq mi drainageareathat hasa CN : 80,is 50 percentstormsewered,andhasa50-yr,24-hr rainfall depthof 12.0inches. Solution. From Fig. 4.I4, thedirectrunoff for 12 inchesof rain is 9.45in. The drainagecoefficient,C, is found from Eq. 15.11,

: 155.t C : 16.39+ (14.75X9.45)

15.3 PEAK FLOW FORMULASFOR SMALLRURALWATERSHEDS 341

The 24-hr removalrate is found from Eq. 15.10' : 0 : 155'8(1'0)5/6 155'8cfs rate to removalrate is 2.0, giving a From Fig. 15.19the ratio of instantaneous designflow rate of 311.6cfs if no stormsewersexisted.From Fig. 15.20,it is found that the unseweredareadischargeshouldbe increasedby 35 percentfor a watershedwith 50 percent storm sewers.The final designflow is 1.35 x 31'1.6: 420.7 cfs. rr

U.S.GeologicalSurveyRegressionEquations for UrbanAreas

.

The U.S. GeologicalSurvey,in cooperationwith the FederalHighway Administration, conducteda nationwidestudy of flood magnitudeand frequencyin urban waterat 56 citiesin 31 states,includinvolved26ggaugedbasins The investigation sheds.37 Basin sizesrangedfrom 0.2 to 15.21. Fig. in are shown locations The Hawaii. ing 100 mi2. Multiple linear regression(see,Chapter27) of a variety of independentparameters was conductedto developpeak flow equationsthat could be applied to small, ungaugedurban watershedsthroughoutthe United States.Similar USGS regression equationsfor large rural basinsare describedin Chaptet27' The simplestform of the developedregressionequationsinvolvesthe three most significantvariablesidentified.Thesewere contributing areaA (mi2),ba-sin^developandthe correspondingpeak flow RQ,(cfs)for the lth ment factorBDF (dimensionless), in the sameregionasthe urbanwatershed.The basin rural identical an from frequency and estimatescan be developedfrom variations, regional for accounts latter vaiiable (seeSection27.4). The threereports frequency flood USGS the applicable any of for the 2-,5-,IO-,25-,50-, 100-,and500-yearflowsaregiven parameter equations as37

ez: l3.2Ao.zt(13 BDFl-o.azpnotz es : 10.6Ao.rz(13 BDF)-o.3eReo18 t0(13- BDnl-otuRQ?dn Qto : 9.5rAo ts(13- BDF)-o'z+P9o'to Qt5 : 8.68Ao ts(13- BDFl-o'zzR03o" Qso: 8.o4Ao ts(13- BOrT-ot'RQ?r!& Qno: 7.70Ao - BDF)-'*RQ1i& : 7.47A0'16(13 Qsoo

(Ls.r2) ( 1s . 1 3) (15.14)

(1s.1s) (1s.16) (15.17) ( 1s . 18 )

Thesewere developedfrom data at 199of the 269 original sites.The other siteswere deletedbecauseof the presence.ofdetentionstorageor missingdata. All theseequations havecoefficientsof determinationabove0.90. of estimatedand observedvaluesused Figure 15.22showsthe correspondence fall within one standarddeviation values the percent of Forty 15.15. Eq. in devedping are similar to the 1O-year intervals recurrence for other line. Graphs regression of the graphshownin Fig. 15.22.

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PROBLEMS 353

15.5,

rainfall duration? The 4-hr unit hydrographfor a 5600-acrewatershedis n

Time (hr) 0 (cf9

0

1

400

8 4 6 1000 800 400

10 200

t2 0

: discussthe Rework Example 15.2 basedon a C : 0.2 and C 0'4' Compare and effect of C on the dischargeat the outfall' AwatershedhasareaA.Startingwithatriangular.shapedunithydrographwithabase : 484A/to' lengthof 2.67t, and a heightof [0, deriveEq. 15'9 (seealsoEq' t2'25)' Qo derivation' the State and carr units of eachterm usedin

UsingtheSCSdimensionlessunithydrographdescribedinChapter12,determinethe peak for a net storm of 101n.in 2 hr on a 400-acrebasinwith a time to peaidischarge ôf + nt and i lag time of 3 hr. Comparewith Eq' 12'17' rainfall at a A 10.00-mi2watershedwith a 100-min time of concentrationreceives min' 200 of period rate of 2.75 in.lhr for a (cfs)from the watershed1f C : 0'4' a. Determinethe peak d-ischarge rainfall. b. Estimatethe dischargerate lcfs) 150 min after the beginningof the beginning of after min 40 watershed c. Estimate the dischar"gerate from the rainfall. F

E * n

{ *

F 6 0

F

/

9 4 0 o o

E ) i o - " s? o

o

2

-=

-Time{min)-after-be€ianing-of rainfall

0

100

354

CHAPTER15

URBANANDSMALLWATERSHED HYDROLOGY

1s.13. A storm gutter receivesdrainagefrom both sides.On the left it drains a rectangular 600-acreareaof t" : 60 min. On the right it drains a relatively steep300-acrearea of t" : 10 min. The f index on both sidesis 0.5 in.ihr. Use the intensity-durationfrequency curves in Fig. 15.7 to determinethe peak discharge(cf$ with a25-year recurrenceinterval for (a) the 600-acreareaalone, (b) the 300-acrearea alone,and (c) the combinedareaassumingthat the proportion of the 600-acreareacontributing to runoff at any time r after rain beginsis l/60. t5.14. A drainagebasin has a time of concentrationof 8 hr and producesa peak Q of 4032 cfs for a 10-hr storm with a net intensity of 2 in./hr. Determinethe peak flow rate and the time base(duration)of the direct surfacerunoff for a net rain of 4 in./hr lasting (a) 12 hr, (b) 8 hr, and (c) 4 hr. State any assumptionsused.

15.15. A 1.0-mi2parking lot has a runoff coefficientof 0.8 and a time of concentrationof 40 min. For the following three rainstorms,determinethe'peakdischarge(cf$ by the rationalmethod:(a) 4.0 in./hr for 10 min, (b) 1.0in./hr for 40 min, and (c) 0.5 in./hr for 60 min. State any assumptionregarding area contributing after various rainfall durations. 15.16. The concentration time varies with dischargebut is relatively constant for large discharges.From this statement,why do engineersfeel confidentin usingthe rational formula? 15.17. Determinethe 50-yearflood for a20-mi2 basin at the northwestcorner of Nebraska. Use the index-flood method and assumethat Fig. 26.4 appbes. 1s.18. Determinethe entire frequencycurve for the basin in Problem 15.17 and plot it on probability paper.

15.19. Use the index-flood method to determinethe 10- and 50-yearpeaksfor a 6400-acre drainagebasin near Lincoln, Nebraska.Assumethat Fig. 26.4 applies. 15.20. For the drainagebasin in Problem 15,19 determinethe probability that the 20-year peak will be equaledor exceededat leastonce (a) next year and (b) in a 4-yr. period. Referto Section26.1.

15.2L. For a 100-mi2drainagebasinnearLincoln, Nebraska,usethe index-flood methodto determinethe probability that next year's flood will equal or exceed3000 cfs. 15.22. UseFig. 26.4 to determinethe return period (years)of the meanannualflood for that region.How doesthis comparewith the theoreticalvalue for a Gumbel distribution? How doesit comparewith a normal distribution?Refer to Section26.6. ts.23. Usethe Cyprus Creekmethodto determinethe 25-yr peak dischargefor the watershed describedin Example 15.3. Assumethat the watershedis nearly flat.

15.24. You are asked to determinethe magnitudeof the S0-yearflood for a small, rural drainagebasin (nearyour town) that hasno streamflowrecords.Statethe namesof at leasttwo techniquesthat would provide estimatesof the desiredvalue. 't5.25. The drainageareas,channellengths,and relevantelevations(underlined)for several subbasinsof the Oak Creek Watershedat Lincoln, Nebraska,are shownin Fig. 24.8. The watershedhas a SCScurve numberof CN : 75 which may be usedto determine the direct runoff for any storm.Assumethat IDF curvesin Fig. 27.13 applyat Lincoln. Treat the entire watershedasa singlebasinand determinethe 50-yearflood magnitude at Point 8 using: a. The rational method. b. The SCSpeak flow graph,Fig. 15.8. c. Snyder'smethod of syntheticunit hydrographs,Eq. 15.5. d. The USGS index-flood method. Figure 26.4 applies.

REFERENCES 355

15.26. RepeatProblem 15.25 with SubareaI excluded.compare the results with Prob-

lem 15.25 and comment on the effectivenessat Point 8 for the 5O-yearevent of the BranchedOak Reservoirat Point 9. (This reservoirwill easilystorethe 100-yearflood from Area I.) 1s.27. RepeatProblem 15.25 for SubareaA' 15.28. RepeatProblem 15.25 for SubareaI' 15.29. Describecompletelyhow the magnitudeof the 30-yearflood for a watershedis determined by the USGS index-flood method. 15.30. A rural watershedwith a composite cN of 70 is being urbanized. Eventually' 36 percentof the areawill be impervious.Determinethe increasein runoff that can for a 6.2-in.rain. be expected peak flow for the SCS dimensionlessunit hydrographin Ch. l2,_determine 15.31 Using the the piak dischargefor a net storm of 10 in. in Zhr on a 400-actebasinwith a time to peak of 4 hr and a lag time of 3 hr. L5.32. A timber railroad bridgein Nebraskaat Milepost 27I.32 ontherailroad systemshown in the sketchis to be replacedwith a new concretestructure.The 50- and 100-year flood magnitudesare neededto establishthe low chord and embankmentelevations, respectively.Determine the designflow rates using the scs TP-149 method. The bridge drainsthe zone marked,about45 acres.The moderatelyslopedbasinlies in a rype-n stormregion,the curve numberis 70, and the 24-hr 50- and 100-yearrainfall depthsare 8.6" and g.4" respectively'

Bidge27l.32

The 15.33. Repeatproblem 15.32usingthe FHWA HEC-19 peakflow SCSdesignmethod. relationships the from be determined 1o can of Values hrs. is 0.2 time of concentration , in Fis. 414. Provide the answersin both metric and English units.

REFERENCES "A Critique of Current Methods in Hydrologic Systems 1 . J. Amorocho and W. E. Hart, Investigations ," Trans.Am. Geophys.Union 45(2),301-321(Jwe 1964)' ..NonlinearInstantaneousUnit-HydrographTheory," ASCE J. Hyd. Div. jingh, p. 2. f. 90(HY2), Par I, 313-347(Mar' 1964).

356

HYDROLOGY 15 URBANANDSMALLWATERSHED CHAPTER "ContinuousHydrographSynthesiswith an 3. W T. Sittner, C. E. Schauss,and J. C. Monro, API:Iype HydrologicModel," WaterResourcesRes.5(5), 1007- 1022(1969). 4. J.E.Nash,"TheFormoftheInstantaneousUnitHydrograph,"Int.Assoc.Sci.Hyd.3@5),

r14-L2r(r9s7).

"Mathematical Models of CatchmentBehavior," Proc. 5. D. R. Dawdy and T. O'Donnel, ASCEJ. Hyd. Div.91(HY4), 124-127(Iuly 1965). 6. S. L. S. Jacoby,"A MathematicalModel for Nonlinear Hydrologic Systems,"J. Geophy. Res. 7l(20), 48t | - 4824(0ct. 1966). 7. R. Prasad, "A Nonlinear Hydrologic System ResponseModel," Proc. ASCE J. Hyd. Div. 93(HY4)(1967). "Hydrology of Urban Runoff," J. ASCE 85, 418. A. L. Tholin and C. T. Keifer, 1959). 106(Mar. "Digital Simulationin Hydrology:StanfordWater9. N. H. Crawfordand R. K. Linsley,Jr., shedModel IV," Department of Civil Engineering,Stanford University, Stanford, CA, Tech.Rep.No. 39, July 1966. 10. JohnC. Schaake,Jr., "synthesisof the Inlet Hydrograph,"Tech.Rep. 3, Storm Drainage ResearchProject, JohnsHopkins University,Baltimore, MD, June 1965. "WaterPollution Aspectsof UrbanRunoff," Federal 11. AmericanPublic WorksAssociation, Water Pollution Control Administration, 1969. "Urban Water ResourcesRe12. Arnerican Society of Civil Engineers,First Year Report, search,"Sept. 1968. 13. W. Viessman,Jr., "Modeling of Water Quality Inputs from Urbanized Areas," Urban Water ResourcesResearch, Study by ASCE Urban Hydrology Research Council, Sept.1968,pp. A79-A103. "Characterization, 14. S. R. Weible,R. B. Weidner,A. G. Christianson,and R. J. Anderson, Treatment,and Disposal of Urban Storm Water," in Proceedingsof the Third International Conference,International AssociationonWater Pollution Researcft(S. H. Jenkins, ed.). Elmsford,NY PergamonPress,1969. "Pesticidesand Other 15. S. R. Weible,R. B. Weidner,J. M. Cohan,and A. G. Christianson, Contaminants in Rainfall and Runoff," '/. Am. Water Works Assoc. 58(8), 1675(Aug.1966). 16. Division of WaterResources,Departmentof Civil Engineering,University of Cincinnati, Cincinnati. OH. "Urban Runoff Characteristics,"Water Pollution Control ResearchSeries. EPA. 1970. 17. Metcalf and Eddy, Inc., University of Florida, Gainewille, Water ResourcesEngineers, Inc., "Storm Water ManagementModel," Environmental Protection Agency, Vol. 1,

r971.

18. E. Kuichling, "The Relation Betweenthe Rainfall and the Dischargeof Sewersin Populous Districts,"Tians.ASCE,20(1889). 19. W. W. Horner, "Modern Procedurein District SewerDesign," Eng. News 64,326(1910). "Relation BetweenRainfall and Runoff from Small Urban 20. W. W Horner and F. L. Flynt, Areas," Trans.ASCE 20(140),( 1936). 21. R. L. Rossnriller,"The Runoff Coefficient in the Rational Formula," EngineeringResearchInstitute, Iowa State University,Feb. 1981. "Experimental Examination of the 22. J. C. Schaake,Jr., J. C. Geye1,and J. W. Knapp, RationalMethod," Proc.ASCEJ. Hyd. Div.93(HY6) (Nov. 1967). "Airport Drainage," Advisory 23. FederalAviation Agency, Departmentof Transportation, D.C.: U.S. GovernmentPrintingOfflce, 1970. Circular, AIC 150-5320-58.Washington,

Chapter16

HydrologicDesign

Prologue The PurPoseof this chaPteris to:

r

. Introducethe hydrologistto proceduresusedin the United Statesfor designing structuresfor safe and effectivepassageof flood flows' , Give sufficient information for the designerto selectthe applicablecriteria for designinghYdrauhcstructures' , provide a discussionof designstorm hyetographsand provide methodsfor design. selectingthe duration, depth, and distribution of plecipitation for precipitation . DemonJtratehow designiloods can be developedwithout using data' ' Discussparticular designmethods including airport drainage'urban storm sewerdeiign, and flood control reservoirdesign' . Describethe U.S. Federal EmergencyManagementAgency (FEMA) flood of flood plain managementsystemand piesent the hydrologicfundamentals Plain analYsis. piotto studying Readersare encouragedto reviewthe materialin Chapters26 and27 designprocedurespresentedin this PredictingPeakdischargerater for use in designingminor and ma aspects of engineering hydrology' I small crodsroadculverts, levees,dt akPort drainage structuresto the lumped together'with major structr design information. GenerallY,a h dischargefor a designfrequencY,a dischargehYdrograPhfor a design rates,low-flow frequencYanalysis, are often conductedas part of a designproiect'

360

DESIGN 16 HYDROLOGIC CHAPTER Mostdesignsinvolvinghydrologicanalysesuseadesignfloodthatsimulates recordsare someseverefutirre eventorlmitates ime historicalevent.If streamflow records storm unavailable,designflood hydrographsare synthesizedfrom available are cases in rare Only usingthe rainfall--runoff proceduresof Chapters2, 12,and 15. watersheds' in small streamflowrecords adequatefor complex designs,particularly in Chapter 15 are Regionalanalysesand the empiric-coirelativemethodsdiscussed in chappresented Methods usefulfor determiningpeak flow ratesat ungaugedsites' for necessary hydrographs ter 12 andin this chapterare used for developingentire manYengineeringdesigns. are describedin uyirologic;ethJds for designingminor and major structures rnethodsfor levels, this chapter.included are discussions;f data needs,frequency dams' and for floodplains synthesizingdesignstorms,and hazard assessments

DESIGNPROCEDURES 16.1 HYDROLOGIC in either the peak flow Proceduresfor estimatingdesignflood flows (interestcan be historical or projected rate or the entire hydrogiaph)includemethodsthat examine and methodsthat flood flowsto arrive at i sultauteestimate(flow-basedmethods), to flood flow rates storms evaluatethe stormsthat producefloods,andthen convertthe on selectinga based be can In eachcase,the analysis (precipitation-basedmethods). meth' (callfrequency-based flood i"rign rr"qo"ncy and determiningthe ass,ociated final the narrowing and_ ods), developingdesignsfor a ringe of flood frequencies or methods)' (called risk-based choice on the basisof long-term c6sts and benefits designingonthebasisofanestimateoftheprobablemaximumstormormanmum RooAtnit could occur at the site (calledcritical-event methods). basedon frequencyMinor Structure Design Minor structuredesignis largely approachto hydrologic the in steps Several basedor sometimesrisk]basedmethods. techadopted and handbooks design most to minor structtJredesign are comrnon are: subsequently) (each is illustrated niques.The generalsteps to the time 1. Determinethe duration of the critical storm,usually equated concentrationof the watershed. t Choosethe designfrequencY. 3. Obtain the storri OeptltUaseaon the selectedfrequencyand duration' in 4. Qomputethe net direct runoff (severalmethodswere presented ter 4). 5. Selectthe time distribution of the rainfall excess' 6.Synthesizetheunithydrographforthewatershed(seeChapterl2). T.Applythederivedrainfall_excesspatterntothesyntheticunithydrograph get the runoff hYdrograPh. the frequen-cyiftn" calculatedflood (usuallyassumedequalto t Establish 8. designstorm frequencY).

DESIGNPROCEDURES 16.1 HYDROLOGIC

361

Major Structure Design Hydrologicdesignaspectsof maior structuresare considerably more complexthan thoseof a small dam, crossroadculvert, or urban drainage system.A designstorm hydrographfor a'large dam still is required but it is put to greater use. The designstorm hydrographis routed to determinethe adequacyof spillwaysand outlets operatedin conjunction with reservoir storage.The economic selectionofthe spillwaysizefrom the variouspossibilitiesdictatesthe final designand is a function of the degreeof protection providedfor downstreamlife and property, project economy,agencypolicy andconstructionstandards,andreservoiroperational requirements.Major structuredesignis largely basedon critical eventmethodspresentedin Section16.5. Water Resource System Design Most information and techniquespresentedin this chapterare directedtoward the flood protection aspectof small and large structures.Needlessto say,a major structureis designedfor more than just flood protection; it is multipurposeand may provide storagefor irrigation, power, water supply, navigation,and low-flow augmentation.The proper allocationof storageto theseuses requiresan understandingof the entire streamflowhistory in terms of the frequency of occurrenceof low flows and averagemonthly, seasonal,and yearly flows, as well as the historical and designfloods. Material is presentedin Part Five to provide a hydrologistwith the tools to developcomplete streamflowhistories for a complex multipurposesysteminvolving various combinationsof minor and major structures, water developmentprojects, and managementpractlces.

Flow-BasedMethods For designlocationswhererecordsof streamflows are available,or whereflows from anotherbasin can be transposedto the designlocation, a designflood magnitudecan be estimateddirectly from the streamflows by any of the following methods: '

1. Frequencyanalysisof flood flows at the designlocation or from a similar basin in the region. 2. Useof regionalflood frequencyequations,normally developedfrom regression analysis(seeChapter 26) of gaugedflood data. 3. Examination of the stream and floodplain for signs of highest historical floods and estimationof the flow rates using measurementsof the crosssectionand slopeof the stream.

Methods Precipitation-Based Where stream-gaugingrecordsare unavailableor inadequatefor streamflow.estimation, designfloods can be estimatedby evaluatingthe precipitation that would producethe flood, and then convertingthe frecipitation into runoffby any ofthe rainfallrunoff methodsdescribedin Chapters10-15 or 2l-27. Typical methodsinclude: 1. Design using the greateststorm of record at the site, by converting the precipitation to runoff. 2. Transpositionof a severehistoricalstormfrom anothersimilar watershedin the region.

362

CHAPTER16

HYDROLOGIC DESIGN

3. Frequency analysis of precipitation and conversion of design storm to runoff. 4. Useof a theoreticalprobablemaximumprecipitation (PMP), or fraction of PMP, basedon meteorologicalanalyses. Becausethe flood flow rate is desiredin all cases,the flow-basedmethodsare preferred over conversionof precipitation to runoff. Due to the relatively longer period of time and greaternumberof locationsat which precipitation amountshave been recorded, precipitation-basedmethods are used in the majority of designs, especiallywith small and very large basins.Flow-basedmethodsare typically used in the midrangeof basin sizes.

Frequency-BasedMethods Regardlessof whetherflow or precipitation dataareused,designsmost often proceed by selectinga minimum acceptablerecurrenceinterval and using proceduresfrom Chapter27 to determinethe correspondingworst condition storm or flood that could be equalledor exceededduring the selectedrecurrenceinterval. Criteria for selecting designrecurrenceintervals are summarizedin Section 16.3. Resultsfrom frequency analysisof flood flow data normally provide reliable estimatesof 2-, 5-, 10-, and 25-yearflows.Extrapolationbeyondthe rangeof the period of flow recordsis allowed, but is lessreliable.

Risk-BasedMethods Recenttrends in designof minor (and major) structuresare toward the use of economic risk analysesrather than frequgncy-based designs.The risk methodselectsthe structuresizeas that which minimizestotal expectedcosts.Tfreseare madeup of the structurecostsplus the potential flood lossesassociatedwith the particular structure. The procedureis illustrated in Fig. 16.1.The total expectedcost curve is the sum of

q

o b0 q

Optimal structure size, S* (least total expected cost) Sîn

Structure size, S

Figure 16.1 Principlesof economicriskanalysisforstructure size selection. (U.S. Federal Highway Administration, Hydraulic EngineeringCircular No. 17).

16.2 DATAFORHYDROLOGIC DESIGN 363 the other two curves.Risk costs(flood damages,structuredamages,road and bridge losses,traffic interruptions)and structurecostsare estimatedfor eachof severalsizes. The optimal sizeis that with the smallestsum.Structuresselectedby risk analysisare normally constrainedto sizesequal to or larger than thoseresultingfrom traditional frequency-basedmethods.

CriticalEventMethods Becauseof the high risk to lives or property below major structures,their design generallyincludesprovisionsfor a flood causedby a combinationof the most severe meteorologicand hydrologic conditions that are possible.Instead of designingfor somefrequencyor leastexpectedtotal cost,flood handlingfacilities for the structures are sizedto safelystoreor passthe most critical storm or flood possible.Methodsfor designingby critical eventtechniquesinclude: Estimating the probable maximum precipitation (PMP) and determining the associatedflood flow rates and volumesby transformingthe precipitation to runoff. ) Determiningthe probablemaximum flood (PMF) by determiningthe PMP and convertingit to a flood by applicationof a rainfall-runoff model,including snowmeltrunoff if pertinent. 3. Examining the flood plain and stream to identify palaeo-floodevidences such as high-water marks, boulder marks on trees or banks, debris lines, historical accountsby local residents,or geologicor geomorphologic evidences. 4. In somecases,the critical eventmethodinvolvesestimatingthe magnitude of the 500-yr eventby various frequencyor approximatemethods.Often, suchas in mappingfloodplains,the 500-yr flood is estimatedas a multiple of the 100-yr event, ranging from 1.5 to 2.5. Due to lack of longer-term records, frequency-basedestimatesare seldom attemptedfor recurrence intervals exceeding500 years.

16.2 DATAFORHYDROLOGIC DESIGN The designof any structurerequires a certain amount of data, even if only a field estimateof the drainageareaand a descriptionof terraintype and cover.The following material identiflessomegeneraldata types and sources.

PhysiographicData The hydrologic study for any structurerequires a reliable topographicmap. United StatesGeologicalSurveytopographicmapsusually are available.The mappingof the United Statesis almost completewith 15-minute quadrangles,and many of these areasaremappedby 7.5-minutequadrangles.County mapsand aerialphotoscan also be usedto advantagein making preliminary studiesof the watershed.

364

DESIGN CHAPTER16 HYDROLOGIC

drainage Based on an area map, a careful investigation of the watershed's maps USGS from obtained be can information behaviormust be made. Additional erosive and inflltration the and types Soil that depict predominantrock formations. or univercharacteristicsof soilscanbe securedfrom U.S. Soil Conservationdistricts sity extensiondivisions. of an The drainageareascontributing to large dams require stricter analysis of a possibility The structures. minor designing in area,shydrotogyitranis necessary large for uniformly intenserainfall over the entire basin is an unrealisticassumption thus should the rainfall of variations spatial and The influenceof temporal watersheds. "worst possible" rainfall values are estimated the be considered.For major dams, in reservoir generallyconvertedto a designdischargehydrograph,which is then used and storage, surcharge size, spillway and reservoii routing calculationsto propoition downsustained or power requirements maintain any additional outlets neededto in hydrologic streamflow for navigation,irrigation, or watersupply.The basicconcern estimatefor realistic a using interests downstream designof a large Aamis to protect the designstorm hYdrograPh. purpose the Topographic'rnuf o"tuit necessarilyshifts with the type and .of of understanding the pi"ta increases always reconnaisiance structurebeing design"a. be. might structule the insignificant an area,shydrology*nomatter how

,

HydrologicData the regionunder one difficulty in hydrologicdesignis that of gettingadequatedatafor issuedby published-reports data canbe-acquiredfrom pr-eviously study.ConsiOerabie agencies federal of list is a governmentalagenciesand/or universities.The following that PublishhYdrologicdata: '

egricultural ResearchService Soil ConservationService Forest Service U.S. ArmY CorPsof Engineers National Oceanicand AtmosphericAdministration Bureauof Reclamation DePartmentof TransPortation U.S. GeologicalSurvey' TopographicDivision Division WaterResources U.S. Geological.Survey, governments,interAdditional dlta often canbe procuredfrom departmentsof state statecommissions,and regional and local agencies'

MeteorologicData and Atmospheric The National WeatherService, couchedin the National Oceanic in a variety published data meteorologic of source Administration, is the primary 16.2 Figure (HMR) series. Report of forms, including their Hyirometeorologic showstheapplicablereportsforvariousgeographicandtopographicregionsofthe

DESIGN-FREQUENCYCRITERIA 16.3 HYDROLOGIC

365

Figure16.2HydrometeorologicalreportseriescoverageofconterminousUnitedStates. (U.S.Bureauof Reclamation') state,and local agenciescollect and analyze United States.lNumerousother federal who design,inspect, or regulatelarge precipitation information-"tp""iuriv-tr'rore

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require:.5::S:1g:,:f stormhyetographs design forestimating or precrp-

in the region,maximum amount ttre meteoroioii" .huru"t"ristics of Jtot-t storm of precipitation,frequenciesof total itable moisturein the atmospheJo;;r;r the over storms for snowmelt and influence.of durationsbf U;t, depthsf*;;;i"", topography chains, mountain of major region.tn ,o1n" areassuch", f;;;iii'*gion, precipitation' on has a very distinct impact

CRITERIA DESIGN-FREQUENCY 16.3 HYDROLOGIC S e l e c t i o n o f f r e q u e i s m o s t o f t e n b a s e d o n p o t e n t i a l d a m a g e t o p r o p e r r : l o s s e s s u c h a s i n t e r r u p t i o n o f commefce'A stanc the worst conditio involved,a greal a an A11projects involve somerisks to property thror.t proceed can design human tife li absent'the tn quencylevel and designof the leastcost structure alternativetoleastStructurecost,economicriskana] rather than the final designfrequencyis optimized trequencret several for would accommodatestorms includenot only the actualconstfuction costs itr"r" u*"J. is leasttotal expectedcost "o** du9 to interruption of servicesand costsbut also the flood dama!;;irk una economicanalysescan be used' commerce.Either annual or p"resentworth

366

CHAPTER16

HYDROLOGICDESIGN

MinorStructures The designfrequenciesshown in Table 16.1 are typical of levels generallyencountered in minor structuredesign.An exampleof variationsthat do occur is the design backwatercould effectivelyhalt frequencyof a culvert,which undercasesof excessive trafflc. The Soil ConservationServicerecommendsthe use of a2l-year frequencyfor minor urban drainagedesignif there is no potential loss of life or risk of extensive damagesuch as first-floor elevationsof homes.A 100-yearfrequencyis commonly recommendedwhen extensiveproperty damagemay occur.t TABLE 16.1 MINORSTRUCTUREDESIGNFREQUENCIES

Typeof minorstructure Highwaycrossroad drainage" ADT' 0-400 400-1700 ADT 1700-5000 ADT ADT 5000Airfields Railroads Stormdrainage Levees Drainage ditches

Returnperiod,4 10yr 10-25yr 25 yr 50 yr 5yr 25-50 yr 2-10 yr 2-50 yr 5-50 yr

= 1/7, Frequency 0.10 0.10-0.04 0.04 o.o2 0.20 0.04-0.02 0.50-0.10 0.50-0.02 0.20-0.02

'ADT : averagedaily traffic. (After Ref. 3).

Large Dams Damsrequirehydrologicanalysisduringthe designof the original structureandduring periodic safetyevaluations.Significanteconomicand humanlossesarepossiblewhen large quantitiesof water are rapidly releasedfrom storage. Initial heightsof retardedwater behind the dam, disregardingthe total volume of stored water, can produce destructiveflood wavesfor a considerabledistance downstream.Basedon two criteria, the TaskForceon SpillwayDesignFloodsrecommeridedthe classificationof large damsas li,stedin Table 16.2.The type of construction has not been included in this grouping, althoughit affects the extentof failure resulting from overtopping. Many of the federalagencieshaveadopteddefinitionsfor hydraulic elementsof dams.The following list is usedby the Soil ConservationService: A spillwuy is an open or closedchannel,or both, used to convey excess water from a reservoir.It may contain gates,either manually or automatically controlled, to regulatethe dischargeof excesswater' Theprincipal spillwayis the ungatedspillwaydesignedto conveythe water from the retarding pool at releaserates establishedfor the structure. The emergencyspillway of a dam is the spillwaydesignedto conveywater in excessof that impoundedfor flood control or other beneficialpurposes.

16.3 HYDROLOGICDESIGN-FREQUENCYCRITERIA

367

FORI-ARGEDAMS TABLE16,2 DESIGNCRITERIA danger lmpoundment Potential Category (1) Major; failure cannot be tolerated

Storage (acre-ft)o (2)

Height (ft) (3)

Failure damage Potential' Loss of life (4) Considerable

>50,000

Damage (5) Excessiveor as matter of PolicY

lntermediate

1000-50.000 40-100

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FOR CRIPPENAND BUE PEAK DISCHARGE TABLE 16,11 COEFFICIENTS CURVES ENVELOPE Fig.16.29 Region I

2 3 4 5

6 '7 8 9 10 11 12 IJ

I4 15 16 L I

Nationwide

Coefficients

Upperlimit(sqkm) 26000 7800 26000 26000 26000 26000 26000 26000 26000 2600 26000 18100 26000 26000 50 2600 26000 2600

Source: AfIer Crippen,J. R., and C. D. Bue, WaterSupPlYPaPer1887, 1977.

c1 3.203865 3.4'10923 3.330746 3.258400 3.126412 3.500489 3.326333 3.236183 3.503734 3.314692 3.231389 3.596209 3.461373 3.07349'l 3,451746 3.s65536 3.389030 3.743026

c2 .8049163 .74'72908 .8443r24 .8906783 .796472r .9123848 .8503960 .9193289 .8054884 1.0386350 .8867450 .8806263 .8519276 .64'727rO .9718339 .9699340 .9445212 .7918884

c3

c4

-.002975'7 -.0394382 - . 0 5 5 1 7 8 0 -.0000965 -.0642062. -.0021362 -.0870959 .0022803 -.0899000 .0022'744 -.1013380 .00496r4 -.0998'74'7 .0042129 -.0947436 .0029486 -.0890172 ,0018961 -.059'7463 -.0042542 -.102053s .0045531 -.0747598 .0000138 -.1094456 .0058948 -.0038285 -.0252243 -.00s7110 -.0617496 -.0034'776 -.0649503 - . 0 6 7 8 1 3 1 -.002'7647 .0244991 -.0192899

..Maximum Flood flows in The ConterminousUnited States,''U.S.G.S

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i:::\

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including storm Figure 16.36 Children's artist rendering of urban undergroundsysteT, with perReprinted Macaulay. David by 1976 A driins. From UNDERGROUND,Copyrighl missionof HoughtonMifflin Co' Al1 rights reserved'

and detention in gutters, house drains, catchbasins,and the storm sewer systems, interceptionin extensivelylandscapedlocations' Two items normally accountedfor in urban storm drain designare: many 1, Infil,tration. The ability of the soil to infiltrate water dependson given characteristicsof the soil as noted in chapter 3. The rangeof values continuous in the following table is typical of variousbare soils after t hr of rainfall' RATEs rvircnr- |NFILTRATIoN Soilgroup

(in./hr) Infiltration

High (sandy,oPen-structured) Intermediate(loam) Low (clay,close-structured)

0.50-1.00 0.10-0.50 0.01-0.10

The influenceof grasscover increasesthesevalues3 to 7'5 times'

16.8 FLOODPLAINANALYSIS

409

2. Retention This is usually assumedto be 0.10 in. for pervioussurfacessuch as lawns and normal urban pervious surfaces' Developmentof hydrologicparametersfor designof storm sewerpipes, street gurrers,or detentionbasinsis by the rational method(or modified rational rnethodiee Chapter25) when peak flow ratesand approximatehydrographsare adequate,or unit hydrographand kinematic wave hydrographsynthesismethodswhen greater detail is needed.The latter usually involveuseof public domainor vendor-developed stormwaterdesignsoftware.The hydrologicaspectsof computerizedhydrologicdetext' signtools are deiailedin Chapter25. In addition to the material presentedin this ILLUDAS' method, rational modified descriptionsof usesof the rational method, TR-sj, SWMM, DR3M, and other tools in designingurban storrrtdrainagefacilities are addressedin numerousurban drainagedesigntextsand handbooks.Additionally, many statedepartmentsof transportationor city and county engineer'soffices have developedtocattyapplicabledrainagedesignmanuals.As well, the American Society i'model" drainagedesignmanual for local adaptaof Civil Engineeishas developeda tion, availableby contacting ASCE in New York. Finally, the discussionof urhan "shopper'sguide" to urban drainageanalysis modelsin Chaptlr 25 includesa useful and designsoftware.

16.8 FLOODPLAIN ANALYSIS

.

Due to heavymonetary and other floodplainlossesover the years,the federalgovernment hasbeenconductingstudiesof floodplainsof the nation's waterwaysand methods of protecting life und prop"tty and pieventing overdevelopmentthat causesinin creasedwater levelsand more widespreadflooding. Hydrology is a key ingredient open dams, of effects studying rates, flow potential these studies for identifying channels,and.otherwater control structureson hydrographsand determiningvolumes of floodwatersthat will needto be safelystoredand conveyedby the waterwaysand floodplains.

U.S.NationalFlood lnsuranceProgram(NFIP) In 1968, the U.S. Department of Housing and Urban Development(HUD), later to called the FederalEmlrgency ManagementAgency (FEMA), initiated the NFIP of mapping with floodplains of identify flood hazard areasand to provide occupants the flood-proneareasand accesstolorv-costflood insurance'The NFIP requireslocal gou"rn-"nt5, to adoptand implementflood managementprogramsthat preventdevelopmentsin excessof national standards' Since the inception of the National Flood InsuranceProgram, flood hazatd The areas have been mapped in over 18,000 communities in the United States. maintenance a to converted since has and programcost over $t.O Uittion to complete effort of updating and expandingthe maps as developmentsoccur'-Each of these for studieshas required eithei approximateor detailedevaluationof peak flow rates has base the called discharge, flood, a range of recurrenceintervals. The 100-year beendeterminedin all cases.The portion ofthe floodplainoccupiedby the baseflood

4'10

oHAPTER 16 HYDRoLoGIc DESIGN

has been mapped,allowing communitiesto determinewhether a property is in the 100-yr floodplain,and in many cases,what water surfaceelevationwould be experiencedat the property during the baseflood. Figure 16.37illustratesthe typical NFIP mappingand floodplainmanagement procedure.Surveyedvalley and channelcross-sectionsare used in determiningthe 100-yr flow depth, allowing the hydrologistto delineatethe lateral extentof flooding duringthe 100-yrflood. Then afloodwaywidth is generallydeterminedasthat portion of the floodplain that is reservedin order to dischargethe 100-yearflood without cumulativelyincreasingthe water surfacemore than 1.0 ft. This procedureis illustrated in Fig, 16.38.The floodwayis most often centeredover the main stream channel,but can be offset or even split into severalzones. Developmentwithin the floodwayis allowed only if compensatedby relocating the floodway or mitigating the water surfaceincreasedue to the development.The flood fringe is that portion of the floodplain outsidethe floodwayin which development is allowed,up to a point of full encroachmentby buildings,roadbeds,berms,and so forth. As much as sevento ten percentof the total land area of the United States lies within the 100-yearfloodplain.The largestareasof floodplainare in the southern parts of the country, and the most populatedfloodplainsare alongthe north Atlantic coast,the GreatLakesregion,and in California. The floodplain mapping effort produced a large amount of data and analyses useful to designhydrologists.The productsof the program include: 1. The 10-, 50-, 100-, and 500-yearfrequencydischargefor streams. 2. The 10-, 50-, 100-,and 500-yearflood elevationsfor riverine,coastal,and lacustrinefloodplains.

Floodway Flood Fringe

i

_

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Flood Fringe

',100_year"Floodplain

I

Channel

Figure 16.37 Definition sketchof floodplain delineations.

16,8 FLOODPLAINANALYSIS

411

100Year FloodPlain Floodway

Flood Fringe

Encroachment

the floodwaywidth' for determining Figure 16.38 Procedure 3.The100-and500-yearmappedfloodplaindelineationsatscalesranging from 1:4800to 1:24,000' 4. The 100-yearfloodwaydata and mapping' wave haz5. Coastalhigh hazard irea mapping 1*"u* subjectto significant ards). 6. FloodwaYflow velocities' 7. Insurancerisk zones' This information is provided in the form of three products: |.FloodlnsurancestudyReportsprovidegeneralprogramandcommunity floodway backgroundinformation,tauutateaflood dischatgedata,tabulated. tabuinformation' surcharge datalncluding velocity, floodwaywidth, and and 100-' 50-' 10-' the of lated flood insurance'zonedati, and profi'les flooding' riverine for 500-yearflood elevationversusstreamdistances of the 1002. Flood InsuranceRateMaps (FIRM maps)provide delineations and500-yearfloodplains,basefloodelevations'coastalhighhazardareas' andinsuranceriskzonesonaplanimetricbaseatascalebetweenl:4800 and 1:24,000. 3.FloodBoundaryHazardMapsprovidedelineationsofthel00-and500crosssections yearfloodplains,locationsof surveyedfloodplainand channel on a floodway 100-year the of usedin hydraulic analyses,and delineations 1:24'000' and 1:4800 planimetricor topographicbaseat a scalebetween

HydrologYfor FloodPlainStudies

NFIP studiesare basedon Flood flow frequency estimatesfor gaugedlocationsin records.Annual peak log-pearsonType III (seechapter 27; analysisof streamflow recommendedby FEMA' flows and historical data arcfitted accordingto procedures

412

CHAPTER16

HYDROLOGIGDESIGN

For ungaugedlocations,flood flow frequency'estimatesare developedthrough regionalfrequencyanalysisor throughrainfall-runoff modeling.Equationspublished by the U.S. GeologicalSurveyrelatepeak dfschargesof variousfrequenciesto various drainagebasincharacteristicssuchassize,slope,elevation,shape,andland use.These equationsare developedusing multiple regressiontechniques(see Chapter 26) at gaugedsitesthroughoutthe region. Rainfall-runoff modelingtechniques(Chapter24) use syntheticrainfall hyetographs.Storm-eventmodels, such as the Corps HEC-I and SCS TR-20 packages, employdesignstormsof particular frequenciesand then mathematicallysimulatethe physical runoff process.The resulting peak dischargeis assumedto have the same frequencyas the rainfall.

U.S.Flood Hazards Despiteconsiderableeffort and expenditurein identificationof floodplainsand flood hazardareas,dam failuresand other catastrophiescontinueto resultin severedamage to life, property, and the environment.Floods from hurricanes,intenserainstorms, and rapid snowmelt or structure failure have all contributed to the loss of life. A tabulationof eventscausingmore than 100 deathsin the United Statesis providedin Table 16.15. As indicated, the majority are hurricane related, principally concentrated in the east-coastand Gulf of Mexico regionsas sfown in Fig. 16.39. Monetary lossesfrom floodsare also large.Table 16.16showsa numberof past producingover $50 million in flood damageseach,given in 1966dollars. floods U.S. Collectively, these floods have produced flood dam4gesin billions of dollars, disI tributed through the yearsas shown in Fig. 16.40. The Federal InsuranceAdministration evaluatedthe floodplain areas in the communitiesmappedby FEMA. By using demographicand economicinformation, projectionsof future property at risk of flooding could be made.Results.suggest that property in have occurred of investments in flood damageable floodplains. billions Table 16.17lists the breakdown,by state,of estimated1990 developmentvalue that will be in harm's way.

Dam Break Hazards '

.-"-fabir- 16.18lists the outflow rates,peak depth, and storageat the time of failure for 18 significantdam failures in the United States.The death rate for dam failures is relatedto the polpulation at risk (PAR). This term describesthosepeoplewho would need to take someaction to avoid the rising water. Figures 16.41 and 16.42showthe lossesas functions of PAR for low (lessthan 1.5 hr) and high (greater than 1.5 hr) advancewarning times, respectively.The high-warning-timelossesare significantlyless.This strongly supportsthe incorporation of early warningand flood delayfeaturesin the designof any structure.Data used in plotting Figs. 16.41 and 16.42 are given in Table 16.19. Table 16.20providesa typical time line requiredfor alerting downstreamresidentsof a severestorm and potential dam failure. The valuesgiven are hypothetical, and apply to an assumed15-mi reach betweenthe storm center and the populated atea.

SUMMARY 413 TABLE 16.15 FLOODSCAUSING1OOOR MOREDEATHSIN THE UNITEDSTATES

Year

Streamor place

Lives lost

1831 1856 1874 1875 1886 1889

BaratariaIsle, LA Isle Derniere,LA ConnecticutRiver tributarY Indianola,TX

1893 1899 1900

Vic. GrandIsle,'LA PuertoRico Galveston,TX

150 320 t43 176 150 2t00 2000 3000 6000+ 100+ 247 151 700 46'7 177 550 284 t20 350 100+ t20 300 2400 350 ,))\

1903 1903 1906 1909 1913 1913 1915 l9l9 l92l 1926 1927 1927 1928 1928 1928 1932 1935 1935 1936 1937 1938 1955 195'l 1960 1972 ' 1972 t1976

Sabine,TX Johnstown,PA

Central States HePPner,OR Gulf coast Gulf coast-New Orleans Miami, Muskingham,and Ohio Rivers Brazos River, TX Louisianaand TexasGulf coast Louisianaand TexasGulf coast Upper ArkansasRiver Miami and Clewiston,FL Lower MississiPPiRiver Vermont Puerto Rico Lake Okeechobee,FL San Francisco,CA PuertoRico Florida KeYs RePublicanRiver, KS, NE NortheasternUnited States Ohio River New Englandcoast NortheasternUnited States Westcoast,LA PuertoRico Buffalo Creek, WV RaPid.Creek,SD Big Thompson,Co

400 110

ro7 137 200 115 556 r07 t25 245 r39

Cause Hurricane tidal flood Hurricane tidal flosd Dam failure Hurricane tidal flood Hurricane tidal flood Dam failure Hurricane tidal flood Hurricane tide and waves Hurricane tidal flood Rainfall-river floods Rainfall-river floods Hurricane tidal flood Hurricane tidal flood Rainfall-river floods Rainfall-river floods Hurricane tidal flood Hurricane tidal floo 0. K : 0.5 m/day and S : 0.02. Use the explicit method to determineheadsat future times. 20.2. Referto Fig. 20.8.Assumethe elementlengthis 10 ft andthe thicknessof the confined aquiferis 8 ft. The headat the left and right is 21 ft at t : 0, and it drops on the right sideto 8 ft for all t > 0. K: 1.5 ftlday-ands : 0.02. use the explicitmethodto calculatefuture heads. 20.3. Referto Fig. 20.12.Asidefrom the trend, what elsecanyou deducefrom studyingthis figure? 20.4. Discusshow you would go about designinga grid for a regional groundwaterstudy. What types of boun{ary conditions might you specify?Why?

REFERENCES 1. J. W. Mercer and C. R. Faust,Ground-WaterModeling.Worthington,OH: National Water Well Association,1981. 2. C.A.AppelandJ.D.Bredehoeft,"statusofGroundwaterModelingintheU.S.Geological Survey," U.S, Geol. Survey Circular 737(1976). "Utilization of Numerical Groundwa3. Y. Bachmat,B. Andres,D. Holta, and S. Sebastian, ter Models for Water ResourceManagement,"U.S. Environmental Protection Agency ReportEPA-600/8-78-012. "Contribution of Ground-waterModeling to Planning," J. Hydrol' 43(Oct. 4. J.E. Moore,

r979). 5. T. A. Prickett,

"Ground-water Computer Models-State of the Att,'l Ground Water

r7(2),t2r-r28(r979). 6. R. A. Freezeand J. A. Cherry, Groundwater.EnglewoodCliffs, NJ: Prentice-Hall,1979. 7. G. D. Bennett, Introduction to Ground Water Hydraulics, book 3, Applications of Hydraulics. Washington,D.C.: U.S. GeologicalSurvey,U.S. GovernmentPrinting Offlce, 1976. 8. D. B. McWhorter and D. K. Sunada, Ground Water Hydrology and Hydraulics. Fott Publications,1977' Collins,CO: WaterResources 9. D. K. Todd,GroundwaterHydrology,2d ed.New York: Wiley' 1980' "PredictedWater-LevelDeclinesfor Alternative GroundwaterDevelop10. P. W. Huntoon, Big Blue River Basin,Nebraska,"ResourceRep.No. 6, Conservation in the Upper ments and Survey Div., University of Nebraska,Lincoln, 1974. "The Numerical Solution of Parabolic and 11. D. W Peacemenand H. H. Rachford, Jr., Afpl. Math. J.3,28-4I(I955)' Indust. Elliptic DiffetentialEquations,"Soc. "Application of th€ Digital Computer for Aquifer D. Bredehoeft, and J. F. Pinder 12. G. Evaluation,"WaterResourcesRes.4(4), 1069- 1093(1968). 13. I. Remson,G. M. Hornberger,and F. J. Molz, NumericalMethodsin SubsurfaceHydrology. New York: Witey, 1971. for Groundwa14. T. A. Prickett and C. G. Lonnquist,SelectedDigital ComputerTechniques ter ResourceEvaluation,Illinois StateWater Survey Bull. No. 55, 1971.

PARTFIVE

MODELING HYDROLOGIC

Chapter21

lntroductionto Hydrologic Modeling

r Prologue The purposeof this chapteris to: . Introduce the types and classesof hydrologicmodels. . Illustrate the limitations, alternatives,steps,general components,and data needsof hydrologicsimulation models. . Presenta philosophicalprotocol for performing successfulmodelingstudies. . Give an overview of groundwatermodel types. . Distinguishthe need for separate,specificproceduresdetailedi4 subsequent Chapters22, 23, 24, and25. Information regardingratesandvolumesof flow at any point of interestalonga stream is necessaryin the analysisand designof many types of water projects. Although many streamshavebeen gaugedto provide continuousrecords of streamflow,plannersand engineersare sometimesfacedwith little or no availablestreamflowinformation and must rely on synthesisand simulatlon as tools to generateartificial flow sequencesfor use in rationalizing decisionsregardingstructuresizes,the effects of land use, flood control measures,water supplies,water quality, and the effects of natural or inducedwatershedor climatic changes. The problemsof decisionmaking in both the designand operationof large-scale systemsof flood control reservoirs,canals,aqueducts,and water supplysystemshave resultedin a need for mathematicalapproachessuchas simulation and synthesisto investigatethe total project. Simulationis definedas the mathematicaldescriptionof the responseof a'hydrologic water resourcesystemto a seriesof eventsduring a selectedtime period.For example,simulationcanmeancalculatingdaily, monthly, or seasonalstreamflowbasedon rainfall; or computingthe dischargehydrographresulting from a known or hypotheticalstorm; or simply fllling in the missingvaluesin a streamflowrecord. Simulation is commonly used in generating streamflow hydrographsfrom rainfall and drainagebasin data. The philosophiesand overall concepts used in simulation are introduced in this chapter. Chapter 22 summarizesconcepts of

508

To HyDRoLocrcMoDELING cHAprER21 rNTRoDUcroN streamdowsynthesisby stochasticmethods.Chapters23-25provide detailsregarding determinislic continuousmodels, single-eventmodels,urban runoff and storm sewerdesignmodels,and water quality models. Stochastictechniquesusedto extendrecords,either rainfall or streamflow,are classifiedas synthesismethods.This procedurerelies on the statisticalpropertiesof an existingrecord or regionalestimatesof theseparameters.An overviewof synthesis techniquesis presentedin Chapter22.

21.1 HYDROLOGIC SIMULATION In this chapter, simulation of all or parts of a surface,groundwater,or combined systemimplies the use of computersto imitate historical eventsor predict the future responseof the physicalsystemto a specificplan or action. Physical,analog,hybrid, or other modelsfor simulatingthe behaviorof hydraulic and hydrologicsystemsand systemcomponentshave had, and will continue to have, application in imitating prototype behSviorbut are not discussedhere. A few of the numerousevent,continuous,andurbanruneff computermodelsfor simulatingthe hydrologiccycle are comparedin Table 21.1. As shown in the tabfe, mostof the modelsweredevelopedfor, or by, universitiesor federalagencies.All have moderate-to-extensive input data requirements,and all have from 1 to 10 percentof TABLE 21.1 DIGITALSIMULATIONMODELSOF HYDROLOGICPROCESSES

Code name

Model name

Percentage of inputsby Date of original Agencyor organization judgmenf development

Continuous simulationmodels-Chapter23 streamflow API USDAHL SWM-IV HSPF NWSRFS SSARR PRMS SWRRB

Antecedent Frecipitation Index Model 1970,1973,1974 RevisedWatershedHydrology Stanford WatershedModel IV Hydrocomp Simulation Program-FORTRAN National Weather Service Runoff Forecast System Streamflow Synthesisand Reservoir Regulation Precipitation-Runoff Modeling System Simulator for Water Resourcesin Rural Basins

HEC-1 TR-20 USGS HYMO SWMM

Rainfall-runoff event-simulation models-Chapter HEC-I Flood HydrographPackage Corps Computer Program for Project Hydrology scs USGS Rainfall-Runoff Model USGS Hydrologic Model Computer Language ARS Storm Water ManasementModel EPA

UCUR STORM MITCAT SWMM ILLUDAS DR3M PSURM

University of Cincinnati Urban Runoff Model Quantity and Quality ofUrban Runoff MIT Catchment Model Storm Water ManagementModel Illinois Urban Drainage Area Simulator Distributed Routing Rainfall-Runoff Model PennsylvaniaState Urban RunoffModel

I

Private ARS Stanford University EPA

I 10 10 10 3 5 10

Corps USGS USDA

1969 t970 1959 196l t972 1958 1982

r990

24

10 1 5

1973 I 965 t972 r972 t971.

z 3 5 5 I 5 5

t972 r974 t970 I9'71 1972 1978 1979

I 5

models-Chapter 25 Urbanrunoffsimulation University of Cinci4natr Corps MIT EPA Illinois State Survey USGS PennsylvaniaState University

"Judgment percentagesare from U.S. Army WaterwaysExperiment Station.r

SIMULATION509 21.1 HYDROLOGIC by repeatedtrials inputs that arejudginent pafameters.Theseare normally validated modelsbut simulation event primarily are with the models.The urban runoff models deferred are models urban of descriptions havebeenisolatedin Table2 1. 1 becausethe to ChaPter25. modelsshown Severalof the major eventand continuousstreamflowsimulation models HEC-1 and Stanford and24.The 23 in Table2!.1aredescribedin chapters briefly are 2I't Table in listed models most are emphasized.For further referince, "Models and Methpublication in the models, described,alongwith about 100 other Fleming's text plesentscomplete ods Applicable to Corps of EngineersStudi-es."1 and other models'2 USDAHL' descriptionsof the SSARR, S'[iM, HSP,

Classificationof SimulationModels In recent decadesthe scienceof ct water resourcesystemshas Passed engineeringprocedure.The variedni has causeda proliferation of catego classiflcationsare Presented. physical vs. Mathematical Models Physicalmodelsinclude analogtechnologies In contrast' mathematical and principlesof similitude appliedto smali-scalemodels' A laboratory flume system' the represent to models,"iy on mathematicafJtut"-.nt, theory of hydrograph unit the while rhay be a 1:10 physical model of a stream, effective various to watershed a of response Chapter12 is a mathematicalmodelof the rain hYetograPhs.

'

by consideris achieved Continuous vs. Discrete Models A secondclassification processes the because continuous as models ing physical,analog,and some digital occur and are modeledcontinuous necessityand advantagesof slicing qualifY as discrete models' A we indication method for routing a flo instantaneousreservoirdischargerz time. over time and timeDynamic vs. static Models Processesthat involve changes modelsthat contrast' In models' dynamic varying interactionscan be simrrlatedby hydrologic Few static' called irequently ui" examine time-independentprocer*"* simulation modelsfall into the latter category' havehad the greatest Descriptive vs. conceptual Models Descriptivemodels becausethey are appticaiionand are of particular interestto practicinghydrologists and the useof basic designedto accountfor observedphenomenathrough empiricisrn concepassumptions' conservation momentum fundamentalssuchas continuity or rather interpletphenomena to theory on heavily tual models,on the other hand, rely on based models include latter the of Examples than torepresentthe physicalpto""tt.

510

CHAPTER21

INTRODUCTIONTO HYDROLOGICMODELING

probabiliiy theory.Recenttrendsin the useof artificial intelligenceand expertsystems in water systemmodelingwould classifyas conceptualmethods. Lumped vs. Distributed Parameter Models Modelsthat ignore spatialvariations in parametersthroughoutan entire systemare classifiedas lumpedparameter models.An exampleis the use of a unit hydrographfor predicting time di;tributions of surfacerunoff for different stormsover a homogeneous drainagearea.The "lumped parameter" is the X-hour unit hydrographusedfor convolutionwith rain to givelhe storm hydrograph.The time from end of rain to end of runoff is also a lumped parameteras it is held constantfor all storms.Distributedparametermodelsaccount for behavior variations from point to point throughout the *yst"*. Most modern groundwatersimulationmodelsare distributedin that they allow variationsin storage andtransmissivityparametersovera grid or lattice systemsuperimposedoverthe plan of an aquifer.More recently,surfacewater systemsare being analyzedthroughuseof distributedparameterGeographicalInformation System(GIS) technologies. Black-Box vs. Structure-lmitating Models Both of thesemodelsacceptinput and transform it into output. In the former case,the transformationis accomplished by techniquesthat havelittle or no physicalbasis.The alchemist'spurported ability to transform lead into gold or plants into medicinewas accomplishedin a black-box fashion- In hydrology, black-box models may sometimestransform "plants" into "medicine" even though the reasonsfor successare not clearly understood.For example,a modelthat acceptsa sequenceofnumbers,reduceseachby 20 percent,and outputsthe resultsmight be entirely adequatefor predictingthe attenuationof a flood waveas it travelsthrough a reachof a given stream.In contrast,a structure-imitating modelwould be designedto useacceptedprinciplesof fluid mechanicsand hydraulics to facilitate the transformation.

'

Stochastic vs. Deterministic Models Many stochasticprocessesare approximated by deterministic approachesif they exclude all considerationof random parametersor inputs. For example,the simulation of a reservoir systemoperating policy for water supply would properly include considerationsof unceitainties in natural inflows, yet many water supply systemsare designedon a deterministic basisby masscurve analyses,which assumethat sequencesof historical inflows are repetitive. Deterministic methods of modeling hydrologic behavior of a watershedhave becomepopular. Deterministic simulation describesthe behaviorof the hydrologic cycle in terms of ma[hematicalrelations outlining the interactionsof variousphases of the hydrologicclcle. Frequently,the modelsare structuredto simulatea streamflow value,hourly or daily, from given rainfall amountswithin the watershedboundaries. The model is "verified" or "calibrated" by comparingresults of the simulation with existingrecords.Oncethe modelis adjustedto fit the known period of data,additional periods of streamflowcan be generated. Event'Based vs. Contlnuous Models Hydrologicsystemscan be investigatedin greaterdetail if the time frame of simulationis shortened.Many short-termhydrologic modelscould be classifiedas event-simulationmodelsas contrastedwith seauential

' 21.1 HYDROLOGIC SIMULATION

.

511

or coniinuousmodels.An exampleof the former is the Corps of Engineers-singlemodel eventmodel, HEC-1,3 and an exampleof the latter is the Stanfordwatershed three' simulate to operated normally is which developedby Crawford and Linsley,a use might model simulation event A typical four, five, oi *or" yearsof streamflow. min' I perhaps even a time incrementof t hr or have arisen water Budget vs. Predictive Models sevelal model classiflcations comparison important One that distingtish betweenthe purposesof the model types. precipiis whetherthe model proposesto predict future conditionsusing synthesized tation and watershedconditions or model is definedas a model or set of of inflows, outflows,and changesin advisedthat simulationmodel studier use the Parametersthat affirm the b shed.For example,meteorologicdat watershed'A water application amounts might be known for a given agricultural (ET) formula budgetmodelwould be uied to determinethe correctevapotranspiration equation continuity in the parametersby testing a range of valuesuntil a balance month-byor day-by-day occursfor all time increments.This is often performedon a budget model' month basis. once the ET parametersare derived from the water or farming conditions' predictive simulationsof diffirent crop patterns,meteorologic the model in relationships practicescould be performedwith the satisfactionthat the corroboratehistorical water budget outputsare measured(precipitation studiesrequire the simultaneousdt ondarY Processessuch as ET, infil spatial distribution of water applications'

Limitationsof Simulation

t-

systems'some Becausesimulation entails a mathematicalabstractionof real-world to which the extent The canôccur. behavior system degreeof *i,,"p,",entation of "depends a developed of test The factors' many on model and systemoutputs vary is consisbehavior the that demonstrating by simulationmodelconsistsof u"iifi"uiion system' physical the tent with the known behaviorof resources Even verified simulation models have limitations in usesfor water of assessments performance allow will models planhing and analysis.Simulation particularly optispecificschemesbut cannotbe usedefficiently to generateoptions, some formulated^by plan is near-optimal a once mal plans, for stated objectives. testing for effective normally are runs simulation of othertechnique,a limited number variablesusing ranand improving the plan by modifying combinationsof decision optimal plans are generating for Techniques dom or systeriaticsamplingtechniques. describedin Section21'3' proceAnother limitation of simulationmodelsinvolveschangingthe operating Programming modeled' being system the of duresfor potentialor existingcomponents for example,requires u "o*pot", to handle reservoir storageand releaseprocesses' is rereprogramming considerable and rules, large portions to define the operatin! qoir"a if other operatingproceduresare to be investigated.

512

oHAPTER 21 INTRoDUoTIoN To HYDRoLoGIc MoDELING A'fourth limitation of simulationmodelsis the potential overrelianceon sophisticated output when hydrologicand economicinputs are inadequate.The techniques of operational hydrology can be used to obviate data inadequacies,but these also require input. Controversyover the use of syntheticdata centerson the questionof whetheroperationalhydrologyprovidesbetter information than that containedin the input.

Utilityof Simulation Computersimulation of hydrologicprocesseshas severalimportant advantagesthat shouldbe recognizedwheneverconsideringthe merits of a simulation approachto a problem that has other possiblesolutions.One alternativeto digital simulationis to build and operateeither the prototype systemor a physically scaledversion.Simulation by physical modeling has been applied successfullyto the analysisof many componentsof systemssuchas the designof hydraulic structuresor the investigation of streambank stability.However,for the analysisof complexwaterresourcesystems comprisedof many interactingcomponents,computersimulation often provesto be the only feasibletool. Another alternative to digital simulation is a hand solution of the governing equations.Simulation models,once formulated, can accomplishidentical results in lesstime. Also, solutionsthat would be impossibleto achieveby hand are frequently achievedby simulation.In addition, the systemcan be nondestructivelytested;prdposedmodificationsof the designsof systemelementscan be testedfor feasibility or comparedwith alternatives;andmanyproposalscanbe studiedin a shorttime period. An often overlookedadvantageof simulation includes the insight gained by gathering,organizing,and processingthe data, and by mentally and mathematically formulating the model algorithmsthat reproducebehaviorpatternsin the prototype.

Stepsin DigitalSimulation A simulationmodel is a set of equationsand algorithms(e.g.,operatingpolicies for reservoirs)that describethe real systemand imitate the behaviorof the system.A fundamentalfirst stepin organizinga simulationmodelinvolvesa detailedanalysisof all existingand proposedcomponentsof the systemand the collection of pertinent data. This stepis called the systemidentification or inventory phase.Includeditems of interestare site locations,reservoircharacteristics.rainfall and streamflowhistories, water and power demands,and so forth. Typical inventory items requiredfor a simulatiol study and data needsthat are specificto someof the modelsare detailed in subsequentparagraphs. The second-phase is model conceptualiTation,which often providesfeedbackto the first phasebf defining actual data requirementsfor the planner and identifying systemcomponentsthat areimportant to the behaviorof the system.This stepinvolves (1) selectinga techniqueor techniquesthat are to be used to representthe system elements,(2) formulating the comprehensivemathematicsof the techniques,and (3) translatingthe proposedformulation into a working computerprogramthat interconnects all the subsystems and algorithms. Following the systemidentification and conceptualizationphasesare several stepsof the implementationphase. Theseinclude(1) validatingthe model,preferably

SIMULATION 513 21.1 HYDROLOGIC for the by demonstrating that the model reproduces any available observed behavior the improve to necessary as algorithms (2) the modifying actual or a similir system; simulathe out carrying by work to model (3) putting the accuracy of the model; and tion exPeriments.

Model Protocoi Five axiomsfor performingsuccessfulmodelstudies,adaptedfrom recommendations by Friedrich,5are: 1. Evaluatethe data beforebeginning. t Document assumPtions. 3. Plan and control the sequenceof computerruns' of output. 4. Insist on reasonableness 5 . Document,document,document. Examining and evaluatingthe basic data are essential.An annotated,bibliographic program \ record of the data ,orrr."* shouldbe maintained.It is alwaysgood adviceto numerical of the modelsthat output (echo)datavaluesasthey arereadin. Verification valuesand proper entry of the data can be establishedfrom the echo. Statisticssuchasih" 1n"un,mode,median,range,standarddeviation,skewness, for kurtosis, and rank order are often helpful in locating entry errors' Checking Are the rainfall? of ahead inconsistenciescan identify errors. Didthe runoff occur characters waterlevelsgraduallyvaried,or are there discontinuities?Do alphabetical or missing as be interpreted appearin th! data?Will blank valuesin the data sets hyFor zeto)? (division by zeros?Will zeros in the data sets result in overflows drographrouting, doesthe time interval selectedfall betweenthe limits recommended for staUltityand convergenceof the numerical method usedto solvethe differential equations? ^ Assumptionsare also important to the successof a simulation' Assumptions assumpwere madeby the p.og.u--"iwhen developingthe mod91,and additional deviation standard the tions aremadeby userJ.Fo, "*u-ple, a programthat calculates must be from an unbiased estimating equation assumesthat the sample size = considered often is 30 N sufficiently lafge to validate ihe estimate.A value of used' as minimal. For a TP-149 (Chapter 15) application, is a 24-hr storm being and reading prior to used be assumedby the method?No computerprogram should of the aware becoming and undershndingthe assumptionsmadeUy ttre programmer programmed' assumptionsimplicit in the hydrologicprocessthat was 'ih" lor cpst of simulationcan reiult in unnecessaryruns and may enticeusers bstantively to the information originally purposeofruns) and working the plan can

asmarr isonrY dtli|l,""i:#THfftime monetary limits to

r alproximate time and in a single useasa guideduringa simulationproject. combining severalinvestigations available models of the Some run is anotherway to conductarrefficient simulation. allowthis.Forexample,TR-20(seeChapter24)allowsthegenerationofflood

CHAPTER21

514

'

MODELING TO HYDROLOGIC INTRODUCTION

hydrographsfrom severalstormsat once.It is often desirableto generatethe2-,5-, 10-, 25-, 50-,100-, and 500-yearflood dischargeat a singlewatershedlocation. The computer is able to generatefar more output than the hydrologistcan analyze.Most modelsincorporateoptionsallowingthe userto specifyoutputquantity. In additionto controlling outpul, a predeterminationshouldbe madeof which specific analyseswill be performed. A tabulation of key output data can be developedto compile and evaluatetrends (and make coursgcorrections)after eachrun. Because deterministichydrologyis about 80 percent acbounting,many opportunitiesexist in water budgelbalances.If the total rechargeto an aquifer is simulation for assessing less than the total outflow and withdrawls, but simulatedwater tables are rising, a checkof input and model parametersshouldbe made.Writing important conclusions on the printed output of simulationruns helpsdocumentthe study and guiderevisions in future runs. Documentation of simulation studies is generally deficient in practice. The record should communicatethe findings in a way that provides a later reviewer generalunderstandingof the work plan followed,decisionsmade,andreasonsfor each run. The documentationshouldstateassumptionsmade,provide samplesof the input and output,explaininput preparationrequirements,statehow sensitivethe resultsare to parameterchangesand assumptions,and documentreasonsout-of-rangeparameters were accepted. Documentationis an ongoingand continual task. It is especiallycrucial if the model will be employedin regulatory proceduresor litigation. A comprehensive processwould6: documentation Include an outline descriptionofthe problem being studied. Identify the equations,techniques,and methodsused. Demonstratethe model's validity to this problem. Discussthe code. Include all assumptionsusedin the code and in preparingthe input. List publishedor known limitations or rangesof the applicability of the model. 7. Characterizethe uncertaintiesin the model; describesensitivitytests. 8. Describeparametersand data setsused. 9. Statethe regulatory or legal criteria incorporatedin the model. 10. Describe the verification, whether with test data or analytical solutions. 11. Include a narrative descriptionof the results,indicating any unexpectedor unusualoutcomes. 12.Presentany other details deemedrelevant. 13.Discusithe modelused. 14.Documentchansesmade in the model code. 1. 2. 3. 4. 5. 6.

Componentsof HydrologicSimulationModels Numerousmathematicalmodelshavebeen developedfor the purposeof simulating various hydrologicphenomenaand systems.A generalconceptualmodel including mostof the importantcomponentsis shownin Fig. 21.1;severalothersare described subsequeht$.Irnportedwater in the lorygl leJtcould be input to reservoiror ground-

)

SIMULATION 21,1 HYDROLOGIC

515

System outflow

Snow accumulation and melt Overiand flow direct runoff

Depression storage

System outflow

SYsteminflow

System rntlow

Figure2l..l.Componentsofasurfaceandsubsurfacewaterresourcesystem. allocationson water storageor channelflow, or it might be guideddirectly to water The routing of the far rlgtriif either storageor distribuiion were deemedunnecessary. parameter channelflo* o, overlandflow could be accompdbhedby simple lumped of segments for discrete flow eq-uations techniques,or solutionsof the unsteady-state algorithms and the channelcould be used.In other words,the selectionof techniques as output desired refinement of d9ere9 the on depends to fepresenteachcomponent justifled is and also on knowledgl of the system.A distributedparameterapproach in described only when availableinformation is adequate.Componentsof modelsare ChaPters22-25.

DataNeedsfor HydrologicSimulation

,

aspart of the The simulation6f all or part of a water systemrequiresa data inventory or mofe) are (90 percent initial planningproc"ts. Most modelinput datarequirements engineering from obtained or map or tield ariaitable,or canbe empiricallydetermined encompasses that topics inventory handbooksand equations.A generallist of data most hydrologic- economicmodelingneedsfollows' A. Basin and SubbasinCharacteristics l, Lagtimes, travel times in reaches,times of concentration' 2.Contributingareas,depressions,meanoverlandflowdistancesandslopes'

\--

516

CHAPTER21

INTRODUCTION TO HYDROLOGIC MODELING

3. Designstorm abstractions:evapotranspiration, infiltration, depression,and interception losses.composite curve numbers,infiltr_ationcapacitiesand parameters,@indexes. 4. Land-usepractices,soil types, surfaceand subsurfacedivides. 5. Water-usesites for recreation,irrigation, flood damagereduction, diversions,flow augmentation,and pumping. 6. Numbering systemfor junctions, subareas,gaugingand precipitation statrons. 7. Imprevious areas, forested areas, areas between isochrones, irrigable acreages. B. ChannelCharacteristics 1". Channelbed and valley floor profiles and slopes. 2. Manning or Ch6zycoefficientsfor variousreaches,or hydrhulicor field data from which thesecoefficientscould be estimated. 3. Channeland valley cross-sectionaldata for eachriver reach. 4. Seepageinformation; channellossesand baseflows. 5. channel and overbank storagecharacteristics,existingor proposedchannelization and leveedata. 6. Sedimentloads,bank stability, and vegetativegrowth. c. MeteorologicData 1. Hourly and daily precipitation for gaugesin or near the watershed. 2. Temperature,relative humidity, and solar radiation data. 3. Data on wind speedand direction. 4. Evaporationpan data. D. WaterUse Data 1. Flows returned to streamsfrom treatmentplants or industries. 2. Diversionsfrom streamsand reservoirs. 3. Transbasindiversionsfrom and to the basin. 4. Stream and ditch geometricpropertiesand seepagecharacteristics. 5. Irrigated acreagesand irrigation practices,including water useefficiencies. 6. Crop types and water consumptionrequirements. 7. Pastconservationpracticessuchas terracing,insfallation of irrigation return pits, and conservationtillage. 8. Stock wateringpractices. 9. Presenceand types of phreatophytesin stream valleys and along ditch banks. E. StreamflowData 1. Hourly, daily, monthly, annual streamflow data at all gauging stations, includin$ statisticalanalyses. 2. Flood frequencydata ani curvesat gaugingstations,or regionalcurvesfor ungaugedsites,preferablyon an annual and seasonalbasis. _3. Flow duration data and curves at gauging stations (also any synthesized data for ungaugedareas). 4. Rating curves; stage-discharge, velocity-discharge, depth-discharge curvesfor certain reaches.

SIMULATION517 21.1 HYDROLOGIC $, Flooded area curves. 6. Stageversusarea flooded' 7. StageversusfrequencYcurves' 8. Stageversusflooi damagecurves,preferablyon a seasonalbasis. 9. Hydraulic radius versusdischargecurves' 10. Sireamflowsat ungaugedsitesas fractions of gaugedvalues. ll. Returnflows as fractionsof water-useallocationsdivertedfor consumptive use. 12. Seasonaldistributionsof allocationsto users' 13. Minimal streamflowto be maintainedat eachsite' 14.Masscurvesandstorage_yieldanalysesatgaugedsites' F. Design Floods and Flood Routing l. Designstormand flood determination;temporaland spatialdistributionand mtenslty. 2. Maximum regional stormsand floods' 3. Selection.andverificationof flood routing techniquesto be usedand necessary routrng parameters. 4. Baseflow estimatesduring designfloods' 5. Availablerecords of historic floods' Information Reservoir G. L. List of potential sitesand location data' 2. Elevation-storagecurves. 3. Elevation-area curves. 4. Normal, minimal, other pool levels' 5. Evaporationand seepageloss data or estimates' 6. Sediment,dead storagerequirements' 7. Reservoireconomiclife. 8. Flood control operatingpolicies and rule curves' 9. Outflow characteristics,weir and outlet equations,controls. 10. Reservoir-basedrecreationbenefit functions' L1. Costsversusreservoirstoragecapacities' 12. Purposesof eachreservoirand beneficiariesand benefits.

NonmodelingAssessments can be After researchingthe availabledata and information, the needfor simulation can be model If a decisionis made to proceed,the appropriatesimulation assessed. selected,a sequenceplanned,and data prepared' Transformationof raw data into usabieform doesnot alwaysrequire a simulacan tion model.Much of the usualinformationneededfor waterresourcesassessments microcomputer in be preparedby hand or by using analytical proceduresavailable format.Typicalnonmodelinganalysesincludethefollowing: 1. Identify water-user groups and all basin sites for hydropower producirrigation, flood damagereduction from tion, reservoir-based-recreation,

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INTRODUCTION TO HYDRoLOGICMODELING

reservoircapacity,industrial and municipal water supply,diversions,and flow augmentation. t Compile annualand seasonalstreamflowsand flood recordsat eachgauged site for the period of record at eachsite. 3. Determine the fraction of the allocation to each consumptiveuse that is assumedto return to the streamat eachuser site in the basin. 4. Perform frequencyanalysesof annual streamflowand flood valuesat each gaugesite in the basin. Determinefor eachreservoirsite the eVaporationand seepagelosses. 6. Selectmeanprobabilitiesto be usedin the firm and secondaryyield analySES.

7. Developflood peak probability distributionsat eachpotential flood damage center or reachin the basin. 8. Determinethe fraction of water to be allocatedduring eachperiod to each water-usesite in the basin. 9. Determine existing and proposed hydropower plant capacitiesand load factors. 10. Identify any minimal allowablestreamflowsto be maintainedfor flow augmentation at eachflow augmentationreachin the basin. 11. Specify any maximal or minimal constraintson any of the annual or seasonal water allocations,storagecapacities,or target yields. 12. Specify any constraintson maximal or minimal dead storage,active storage, flood control storage,or total storagecapacitiesat any or all of the reservoirsitesin the basin. 13. Determine annual capital, operation, maintenance, and replacement (OMR) costsat eachreservoirsite as functions of a rangeof total reservoir 'capacities or scalesof development. 14. Determinebenefitsas functions of energyproduced. L5. Determineannual capital and OMR costsat eachhydropowerproduction site as functions of variousplant capacities. 16. Determine benefit-loss functions for a variety of allocationsto domestic, commercial,industrial, and diversionuses. 17. Determine short-run lossesas functions of deviations (both deficit and surplus)in plannedor target allocationsto user sites. 18. Developbenefit functions at eachirrigation site in the basin.This analysis requiresinformation on the area of land that can be irrigated per unit of water allocated,the quantitiesof eachcrop that can be producedper unit areaof land, the total fixed and variablecostsof producingeachcrop, and the unit"pricesthat will clear the market of any quantity of eachcrop. 19. Developflood-damage-reductionbeneflt functions at eachpotential flood damagesite. This analysisrequiresrecords of historical and/or simulated floods, channel storagecapacities,and flood control reservoir operating policies. 20. Developreservoir-basedrecreationbenefitfunctions at eachrecreationsite in the basin.

SIMULATION 21.2 GROUNDWATER

519

21.2 GROUNDWATER SIMULATION Digital simulation models are used in a different manner to study the storageand movementof water in a porous medium. Distributedrather than lumped parameter models are used to imitate observedevents and to evaluatefuture trends in the developmentand managementof groundwatersystems.The equationsdescribingthe flow of waterin a poroui mediumwerederivedin Chapter18 andmodelingof regional systemswas discirssedin Chapter 20. This section deals primarily with techniques used in solving the hydrodynamicequationsof motion and continuity, followed by brief discussionsof (i) typical input requirements,(2) techniquesof calibrating and verifying the models, and (3) the sensitivity of groundwatermodels to parameter changeslAnexampleof the calibrationand applicationof a groundwatermodelis also provided.

ModelTypes Groundwaterstudiesinvolve the adaptationof a particular code to the problem at hand. Severalpopular public domain computer codesfor solving various types of groundwaterflow problemsare listedin Table2L2.The codesbecomemodelswhen the systembeingsiudiedis describedto the codeby inputting the systemgeometryand known internai operandi (aquifer and flow field parameters,initial and boundary conditions, and water use and flow stressesapplied in time to all or parts of the system).Codes have emerged in four general categories:groundwaterflow codes,solutetransport codes,particle tracking codes,andaquifer testdata analysis programs.to Groundwaterflow codesprovide the user with the distribution of headsin an aquifer that would result from a simulated set of distributed recharge-discharge stressesat cells or line segments.From Darcy's law, the flow passingany two points can be calculated from the head differential. The codes are used to model both confined and unconfinedaquifers.Eachcan be structuredto model regional flow, or flow in proximity of a singG well or wellfield. Steady-stateand transientconditions canbe evaluated.Boundaiiescanbe barriers,full or partially penetratingstreamsand lakes,leaky zones,or constantheador constantgradientperimeters.By application of Darcy's 1aw,the seepagevelocitiesof groundwatercan be determinedafter solving for the head differentials. When groundwaterseepagevelocitiesareknown, the advection,dispersion,and changesin concentrationof iolutes can be modeled.Solutetransportmodelsbuild on groundwaterflow modelsby the addition of advection,dispersion,and/or chemical reactionequations.If the chemical,dispersion,or dilution concentrationchangesdue m groundwaterflow are not important, particle tracking codesmodel transport by advectionand providean easiermethodthan solutetransportmodelsto track the path and traveltimes of solutesthat moveunderthe influenceof headdifferentials.Aquifer test data programsprovide userswith computersolutionsto many of the hand calculations (Ctrapter li) neededto graph and interpret aquifer test data for determining aquifer and well Parameters' I

\.

520

CHAPTER21

INTRODUCTION TO HYDROLOGIC MODELING

TABLE 21.2 GROUNDWATER MODELINGCODES

Acronymfor code PLASM MODFLOW AQUIFEM-1 GWFLOW GWSIM-II GWFL3D MODRET SUTRA RANDOMWALK MT3D AT123D MOC HST3D FLOWPATH PATH3D MODPATH WHPA TECTYPE PUMPTEST THCVFIT TGUESS

Description Groundwaterflow models Two-dimensionalfi nite difference Three-dimensionalflnite difference T!wo-and three-dimensionalfinite element Packageof1 analytical solutions Storageand movementmodel' Three-dimensionalfinite difference Seepagefrom retention ponds Solutetransooftmodels Dissolvedsubstancetransport model Two-dimensionaltransientmodel Three-dimensionalsolutetransport Analytical solution package Two-dimensionalsolutetransport 3-D heat and solutetransport model Particletrackingmodels Two-dimensionalsteadystate Three-dimensionaltransient solutions Three-dimensionaltransient solutions Analyticai solution package Aquifertest analyses Pump and slug test by curve matching Pumping and slug test Pumping and slug test Specific capacitydetermination

Source

ru.sws USGS MIT IGWMC TDWR TDWR USGS

1971 1988 r979 1975 1 9 81 r991 1992

USGS ill. SWS EPA DOE USGS USGS

1980 1981 1990 1981 t978 r992

SSG Wisc GS USGS EPA

1990 1989 t991 1990

ssG

1988 1980 1989 1990

IGWMC IGWMC IGWMC

Note: IGWMC : International Groundwater Modeling Center; Ili. SWS : Illinois State Water Survey; SSG : Scientific Software Group; EPA : Environmental Protection Agency; USGS : U.S. Geological Survey; Wisc. GS = Wisconsin Geological Survey; MIT : MassachusettsInstitute of Technology; TDWR : Texas Department of Water Resources;DOE : Department of Energy.

SolutionTechniques With few exceptions,the hydrodynamic equations for groundwaterflow have no analyticalsolutions,and groundwatermodelingreliesonfinite-dffirence and,finiteelementmethodsto provide approximatesolutionsto a wide variety of groundwater problems.The choice of method is normally driven by the systemto be modeled. Other numericalmethodsincludeboundary integral methods,integratedfinitedffirence methods,and analytic elementmethods. Thesesolutions,as with streamflowsimulation models,are facilitated by first subdividingthe region to be modeledinto subareas.Groundwatersystemsubdivision dependsmore on geometriccriteria and lesson topographiccriteria in the sensethat the region is overlaidby a regular or semiregularpattem of node points at which (or betweenwhich) specificmeasuresof aquifer and water systemparametersare input and other parametersare calculated.Approximate solutionsof simultaneouslinear and nonlinearequationsare found by making initial estimatesof the solution values, testingthe estimatesin the equationsof motion and continuity, adjustingthe values, andfinally acceptingminor violationsin the basicprinciplesor making further adjustments of-the parametersin an orderly and convergingfashion.

SIMULATION 21.2 GROUNDWATEFT

r '

521

Theorderly solutionof finite differenceanalogsof the steady-stateor unsteadypartial difierential equation of motion for flow of groundwaterin a confined stare aquifer or an unconfinedaquifer is obtainedby rela_xationmethods'An eady relaxation solutionof the equationis discussedby Jacob.TFor two-dimensionalproblems, the iterativealternating-direction-implicit(ADI) methoddevelopedby Peacemanand Rachfordsis often adoPted. prickett and Lonnquisteused the ADI techniqueto calculate fluctuations in watertable elevationsat all nodesin an aquifermodelby proceedingthrough time in small incrementsfrom a known initial state.Their modelis computationallyefficient and readily appliedand is particularly attractivefor usewith problemsinvolvingtime variablesind nu-"tous nodes.The primary aquiferparametersare the permeability and storagecoefficient,which, if assumedconstantover the aquifer plan, result in a homogenJousandisotropigcondition.For thosefamiliar with relaxationmethods'the over-relaxation(SOR)methodshavehad application Gauss-seidelandthe successive equations. in solvingdifference

DataRequirements Input to groundwatersystemmodelsmay be classifiedas spatialand temporal' Spatial input inciudesinitial oi projectedwater table maps,saturatedthicknessdata over the ."gion, land surfacecontourmaps,transmissivitymaps,regionalvariationsin storage coefficients,locationsand typei of wells and canals,locationsand types of aquifer boundariesboth lateral and vertical, a nodecoordinatesystem,actualor net pumpage rates,percolationand rechargerates for precipitation and other appliedwaters,logs of drilled wells, geologicstratigraphy,and soil types and cropping patterns. Time-dependenidatarequirementsfor aquifer models involve principally the formulation of ti-" schedules,using a rangeof time incrementsfor suchvariablesas pumpingrates,precipitationhyetographs,canaland streamflowhydrographs,groundrates,and developmentvariablessuchasthe timing of added waterevapotranspiration Becauseeachtemporal schedulecan apply only system components. wells or other the time-dependentrequirementsare also positions, node of particular iubset to a spatial. In addition to the listed input parameters,aquifer modelsrequire reliable estimatesof the percentagesof waters in the land phasethat actually percolateto the aquifer being modeled.Theseestimatescan be basedon knowledgeof the physical processesinvolved in unsaturatedflow through porous medium but are most often obtainedasjudgmentparametersthat are modifiedduringthe calibrationphaseof the simulation. Simply stated,the lateral movementand the changesin piezometeror watertable leneli are easilymodeledif the node-by-nodestresses(withdrawalratesor rechargerates)aie known. The latter parametersare governedby the complexmovement of water in the unsaturatedsoil zone and by the random precipitation and consumptiveusepatternsof the region.The art of modelinggroundwatersystemslies in the ability to evaluatetheseparameters'

Calibration

I L,

Groundwatermodelcalibrationremovessomeof the guessworkinvolvedin parameter determination.Severalcombinationsof parameters,basedon availableknowledgeof the physical syqtem,are testedin the model during a period for which records are

522

21 INTRODUCTIoN CHAPTER To HYDRoLoGIc MoDELING availabi-e. Simulatedresultsare then comparedwith historical events.After structuring the model,calibrationis achievedby operatingthe model during the studyperiod by imposinghistoricalprecipitationamounts,canaldiversions,evaporationand evapotranspirationrates, streamflowsand stream levels, pumping rates during known periods, and other stresseson the aquifer. Calibration is achievedafter the flow, storage,and other parametershavebeenadjustedwithin reasonablelirnits to produce the best imitation of recordedevents.

CaseExample A typical finite-differencestudyinvolvingsurfacewaterand groundwatermodelingin central Nebraskawas performed by Marlette and Lewis.tt The region involved is shownin Fig. 2I.2.In additionto the surfaceirrigation systemrepresented by the severalcanalsand laterals,over 1200wells withdraw waterfrom the aquiferbetween the PlatteRiver and the Gothenburgand DawsonCounty canals.The aquiferrecharge and withdrawal amountsas percentagesof precipitation, snowfall, pumped water, deliveredcanal water, evaporation,and evapotranspirationwere estimatedusing a mix of judgment andphysicalprocessevaluations.The resultingsetthat producedthe bestcomparisonwith recordedeventsat the six observationwells showninFig.2l.2 is summarizedin Table 21.3. Samplesof the comparisonbetweenrecordedand simulatedwaterlevelsin the DawsonCounty study during aZ-yearcalibrationperiod are shownin Figs. 21.3 and2l.4. The Prickettand Lonnquistmodelwasappliedin the DawsonCounty study.The storagecoefflcientfor this unconfinedaquiferwas establishedby calibrationtrials as 0.25 and the adopted permeability was 61 mlday. Other trials were made using variouscombinations of S andK, with S rangingfrom 0.10to 0.30andwith Kranging between4I and I02 mlday. As with most unconfined aquifer models,water table elevationswere most sensitiveto fluctuationsin the storasecoefflcient.Fisure 21.5 is

tt'iorX \F

'q;

Figure 21.2. Grid coordinatesfor Dawson County, Nebraska,aquifer model. o : observationwell.

SIMULATION 21.2 GROUNDWATER

523

FORWATERALLOCATIONS CRITERIA RECHARGE TABLE..21.3 ADOPTED AQUIFER COUNTY OVERTHEDAWSON

Systemcomponent Rainfall Snowfall Pumpedwater Delivered canal water

Evaporation Evapotranspiration

Aquiferrecharge/withdrawal ot as a percentage appliedamount

Allocationand appliedamounts Recordeddepth if daily amount exceeded 0.25 cm at all nodes 25Voofrecorded depthsat all nodes Averagerate of 50 l/sec at-all well nodes during irrigation seasons Recordeddaily rates,applied to land surfaceone node laterally uphill and two nodesdownhill from canal Observeddaily lake evaporationdepth at all marsh and water surfacenodes I25Voof daily lake evaporation,applied at all alfalfa nodes

30 30 50 30 100 15

d

o o 6

B

0.999

d

0.998 A M J J A S O N D J F M A M J J A S O N

1970

1971

Figure2L.3. Simulatedandrecordedwaterlevelsat observation w e l l Di n F i g . 2 1 . 2 .:I8 2 ; i : 3 7 . a typical summary of the calibration results at a single observationwell located at PositionF in Fig. 21.2. After vefification, the Dawson County model was applied to investigatethe short-terminfluenceof severalmanagementschemes.Included among the schemes were investigationsinvolving the completeremoval or shutdownof the surfacewater canals,and other testsin which isolatedcanal contributionsto rechargewere determined by operatingthe model with singlecanals and comparingresults with water table fluctuationsfor identicalconditionswith all canalsremoved.Many other applicationsof the model are possible.This particular study revealedthat rechargefrom

524

CHAPTER21


o a d

F

0.999

0.998 A M J J A S O N D J F M A M J J A S O N l97l

1970

Figure 21.4. Simulated and recorded water levels at observation well F in Fig. 21.2. I : 97; i : 42.

a

lu3

P (J

o q)

(.1 'F1

H 0)

i3 -

702

A

M

J

J

A

S

O

N

D

t9'70

J

F

M

A

M

J

J

A

S

O

r97|

with constantpermeabilityandvarying changes Figure 21.5. Water-level Well4 Fig. 21.2;K : 6Im/day;I : 97;J : 42. storagecoefficients. the existingcanal systemcontributesto the waterbalancqof the aquiferbut is not the dominantfactor\n the shortrun. The naturalrechargefrom precipitationandfrom the Platte River accountfor the long-term water table stability in the region.

PROTOCOL SIMULATION 21.3 HYDROLOGIC The useof hydrologicsimulationas a tool in the decision-makingprocessis not new but is of a different, more sophisticatedand more encompassingform. A model is a rgplesentationof an actualor proposedsystemthat permitsthe evaluationand-rmanip-

21.g HYDROLOGICSIMULATIONPROTOCOL

525

useof ulation of-manyyearsof prototypebehavior.This is the featurethat makesthe largest, the of even thesetooii so attractiveand hoids suchpotential for the analysis so well mostcomplexsystems.It is alsothe prin-ipal featurethat makesthis approach suitedto water resourcessystemplanning and analysis' Apart from the useof cot1entionalhandmethodsand someelementarymodels, planninghastraditionally beena practiceof judgment.This is changing,however,as quantititive tools are developedthat permit the analysisof large numbersof alternaout but is tives and plans.Judgment,a-nessentialelementof the process,is not ruled planning the in to those strengthenedthrough new insights that were not available professiona few years ago. "What if ?" Plannersare conti;ually required to anticipatethe future and ask and "What's best?" questions.Quantitativeplanning techniques,suchas simulation cost than can provide detailedinformation aboutmore planning alternativesfor less at principally occurred has uny oth", approachavailable.Developmentof thesetools universitiesand federal agencies.

CombinedUse of Simutationand OptimizationModels at this An important secondtype of quantitativeplanningtool shouldbe mentioned a select to information point.Screeningmodelsare designedto utilize llmiPd system Hence objectives. of set or bestplan u*oni many alternativesfor a specifiedobjective toward screeningmodJs, or optimizationmodelsasthey are often called,are oriented plan formulation rn contrast to th€ Simulation models are suited to de reliable information on which to bi "If modelsaddressthe question, out tion models,on the otherhand,ask, will the systemlook like after we arr special merits of each, these two Completeder planningtechnologies. ing, oPtimization,and simulation r Final designvaluesshouldbe and operatinga detailedsimulation model over time using a to the systemel-ements and/or streamflows,while at sequenceof known or synthesizedprecipitation-amounts reservoirand streamcontrol, the sametime accumulatingbenefiisovertime for flood siderecreation,wateryields,strean and anY other factorsnot consider model. Severalsimulationruns wi resultin a plan that bestmeetsthe c generatedby coilventionalmethods' inforResultsof optimization modelswill provide readily obtainedand useful promismost the test to in order mation for initiating more refinedsimulationanalyses of ing measuresand arrive at final plans for the design,construction,and operation a water resourcesystem.Even thor for decisionsregardingboth the de postoptimizationsimulation is rec assumptionsoften requiredin prel

L

526

CHAPTER21


for preliminary screeningof developmentalternativesowing to time and cost limitations.Unlessa new generationof computersevolves,currenttime and sizelimitations do not allow screeningby simulating all iilternativesunlesssubstantialsacrificesin realismare made.For the present,preliminary screeningfollowedby detailedsimulation appearsto be the most effectivemeansfor arriving at optimal waterdevelopment and managementPlans.

MODELS SIMULATION 21.4 CORPSOF ENGINEERS In 1964,the U.S. Army Corps of Engineersdevelopeda specialtybranch locatedat the HydrologicEngineeringCenter(HEC) in Davis,California. The facility provides a centerforipplying academicresearchresultsto practical needsof the Corps fleld offices.In addition, the centerprovidestraining and technical assistanceto governmenr agenciesin advancedhydrology,hydraulics,and reservoiroperations. Over the years, a large number of analytical tools were developedat HEC. Table 2I.4 summarizesthe computerprogramsin categoriesof hydrology,river/reservoir hydraulics, reservoir operations, stochastichydrology, river/reservoir water TABLE 21.4 HEC WATER RESOURCECOMPUTERPROGRAMS Name

HEC-I, Flood HydrograPhPackage

Basin Rainfall and SnowmeltComputation

Date of latest version HydrologyModels September1980

Purpose Simulatesthe precipitation runoff processin any comPlexriver

July 1966 many subbasilsof a river basin using gaugedata and weightings (includedin HEC-1).

Unit Graph and HydrographComputation

Iuly 1966

Unit Graph Loss Rate OPtimization

August 1966

HydrographCombining and Routing

August 1966

Computessubbasin interception/infiltration, unit hydrographsbaseflow, and runoff hydrograph(included in HEC-1). Estimatesbest-fitvaluesfor unit graph and lossrate Parameters from given precipitation and subbasinrunoff (included in HEC-1). Combinesrunoff from subbasinsat confluencesand routes througha river hydrographs network using hydrologicrouting methods(includedin HEC-1).

21.4 CORPSOF ENGINEERSSIMULATIONMODELS TABLE21.4

527

(Continued)*

Name

Dateof latestversion

Purpose

Streamflow Routing OptiffIization

July 1966

Estimatesbest-fit valuesfor hydrologic streamflow routing parameterswith given upstream' downstream,and local inflow hydrographs(includedin HEC-l).

Interior Drainage Flood Routing

November1978

Computes seepage,gravitY and pressue flow, pumping and overtopping dischargesfor Pond areasbehind leveesor other flow obstructions.Main river elevation and ponding area elevation-area-capacitydata are usedin computingdischarges.

Storage,Treatment, Overflow, Runoff Model (..STORM")

JuJy1976

Simulatesthe precipitation runoff processfor a single, usuallY urban, basin for manY Yearsof hourly precipitation data. Simulates qualitY of urban runoff and dry weather sewageflow. EvaluatesquantitY and qualilY of overflow for combinations of sewagetreatmenl plant storage and treatment rate.

hYdraulics River/reservoir HEC-2, Water SurfaceProfiles

August 1979

super-critical. AnalYzes allowableencroachmentfor a given rise in water surface. Gradually Varied UnsteadyFlow Profiles

Iarnary L976

Simulatesone-dimensional, unsteady,free surfaceflows in a branching river network. Natural and artificial cross sectionsmaY be used. Uses an exPlicit centered difference computational scheme.

Geometric Elements from Cross Section Coordinates("GEDA')

June 1976

Computestables of hYdraulic elementsfor use bY the GraduallY Varied UnsteadYFlow Profiles or other programs.InterPolates values for area, top width, n value, and hydraulic radius at evenly spacedlocations along a reach.

534

CHAPTER21


PROBLEMS 21.1. Simulation and synthesisare treatedseparatelyin Chapters'22and 23. List the most distinguishingcharacteristicsof eachmethod and give an exampleof each. 21.2. Listatleastthreereasonsmanyofthedevelopedmodelsoftherainfall-runoffprocess might not be usedby hydrologists. 21.3. You are askedto determinea designinflow hydrographto a reservoirat a site where no recordsof streamfloware'available.I,!st generalstepsyou would take as a hydrol' ogist in developingthe entire designinflow hydrograph.

REFERENCES "Models and MethodsApplicableto Corps of 1. U.S. Army WaterwaysExperimentStation, EngineersUrban Studies,"MiscellaneousPaperH-74-8, National TechnicalInformation Service,Aug.1974. 2. GeorgeFleming, ComputerSimulation Techniquesin Hydrology. New York: American Elsevier,1975. "HEC-I Flood HydrographPackage,"Users and ProU.S. Army Corps of Engineers, grammersManuals,HEC Program723-X6-L20I0, Jan.1973. "Digital Simulationin Hydrology:StanfordWater+ . N. H. Crawford and R. K. Linsley,Jr., shedModel IV," Department of Civil Engineering,Stanford University, Stanford, CA, Tech.Rep.No. 39, July 1966. 5 . A. J. Friedrich, "Managementof ComputerUse in SolvingEngineeringProblems,"U.S. Army Corps of Engineers,Hydrologic EngineeringCenter,Davis,CA, 1979. 6 . National ResearchCouncil, Ground WaterModels-Scientific and RegulatoryApplications. Water Scienceand TechnologyBoard, Commissionon PhysicalSciences,MatheD.C., 1990. National AcademyPress,Washington, matics,and Resources, 7 . C. E. Jacob,"Flow of Groundwater,"in EngineeringHydraulics (HunterRouse,ed.)' New York: Wiley, 1950. "The Numerical Solution of Parabolicand Elliptic 8 . D. W. Peacemanand N. H. Rachford, Indust. Appl. Math.3' (1955). DifferentialEquations,"J. Soc. "selectedDigital ComputerTechniquesfor Ground9 . T. A. Prickett and C. G. Lonnquist, waterResourceEvaluation,"Ilinois StateWaterSurveyBull. No. 55,197I. 1 0 . D. R. Maidment, (ed.), Handbookof Hydrology.New York: McGraw-Hill, 1993. "Digital Simulationof Conjunctive-Useof Groundwater 1 1 . R. R. Marlette and G. L. Lewis, in DawsonCounty, Nebraska,"Civil EngineeringReport, University of Nebraska,Lincoln, 1973. 12. W. K. Johnson,"Use of SystemsAnalysis in Water ResourcePlanning," Proc- ASCE J. Hyd. Div. (1974). 1 3 .R. deNeufvi[e and D. H. Marks, SystemsPlanning and Design CaseStudiesin Modeling . EnglewoodCliffs, NJ: Prentice-Hall,I914. Optimizationand EvaluatioiT 14. D. P. Loucks, "stochasticMethodsfor Analyzing River Basin Systems."Cornell Univer', sity Water Resourcesand Marine SciencesCenter,Ithaca, NY, Aug. 1969. 1 5 . A. Maass,(ed.),Designof WaterResourcesSystem*Cambridge,MA: HarvardUniversity Press,1962. t6. A. F. Pabst, "Next Generation HEC Catchment Modeling," Proceedings,ASCE HydraulicsDivision Symposiumon EngineeringHydrology,SanFrancisco,CA, July 25-30' 1993.

Chapter22

Time":?J5: Hydrologic

r Prologue The purposeof this chaPteris to: . Show how time seriesanalysisis used for generatingsynthetic hydrologic records. . Give definitionsof termsusedto describethe stochasticaspectsof hydrologic series. . Introduce fundamentalsof streamflowsynthesisincluding masscurve analysis, random generationof sequences,serial-dependentsequences,and sequenceshavingprescribedfrequencydistributions' Time-seriesanalysisof hydrologicvariableshas becomea practical methodology for generatingsyntheticsequencesof precipitation or steamflowvaluesthat can ui-usedlor u tung" of applicationsfrom filling in missingdata in a gaugedrecord to, extendingmonthiy streamflowrecordsl, and from analyzinglong-term reliability of or reservoirst'oto forecastingfloodsor snowmeltrunoff quantiyields of-watersheds2 Thesesynthetic hydrologytechniques iies from syntheticprecipitation sequences.5 wideaugmentttre simulationtools describedin Chapter21. Both have,experienced generation of a spiead use by hydrologistsand engineers.6Synthesisinvolves the or annual). ,"qu"n"" of valuesfor somehydrologicvariable(daily, monthly, seasonal, The techniquesare most often applied to produce streamflowsequencesfor use in reservoir designor operation studiesbut can also be used to generaterainfall sequencesthat can subsequentlybe input to simulation models' If historioalflows iould be consideredto be representativeof all possiblefuture variationsthat someproject will experienceduring its lifetime, there would be little needfor synthetichyOroiogy.The hiitorical record is seldomadequatefor predicting future eventswith certainty,however.The exacthistoricalpatternis unlikely to recur, sequencesof dry years (or wet years) may not have been as severeas they may beiome, and the singlehistorical record gives the planner limited knowledgeof the magnitudeof risks involved.

536

CHAPTER22

HYDROLOGICTIME SERIESANALYSIS

particularly if Syhthesisenableshydrologiststo deal with data inadequaci.es, of hydrologic records historical Short record lengths are not sufficiently extensive. synhydrologic sequenceslrsing longer variables*"fu u, streamfloware extendedto hydrology'1 as operational known science thesisand other techniquesof the broad either preservethe statisticalcharacterof the historThesenew, syntheticsequences ical records or follow a prescribedprobability distribution, or both. When coupled with computer simulation techniques,the techniquesprovide hydrologistswith improved designand analysiscapabilities. The rnethodsdescribedin this chapterare basedon probability and statistics. The material presentedin Chapter 26 should be reviewed prior to studying this chapter.

HYDROLOGY 22.1 SYNTHETIC

,

Hydrologic synthesistechniquesare classifiedas (I) historical repetition methods, suchas masscurve analyses,which assumethat historical recordswill repeatthemsevlesin as many end-to-endrepetitionsas required to bracketthe planningperiod; (2) random generationtechniques,such as Monte Carlo techniques,which assume that the historicalrecordsare a numberof random,independentevents,any of which could occur within a definedprobability distribution; and (3) persistencemethods, such as Markov generationtechniques,which assumethat flows in sequenceare dependentand thit the next flow in.sequenceis influencedby some subsetof the previousflows. Historical repetition or random generationtechniquesare normally applied only to annual or seasonalflows. Successiveflows for shortertime intervals are usually correlated,necessitatinganalysisby the Markov generationmethod. As with most subfieldsof hydrology,a number of computerprogramsfor timeseriesanalysisandhydrologicdatasynthesishavebeendeveloped.One ofthe first, and one of the most widely app1i"d,wasthe U.S. Army Corps of Engineersmodel HEC-4 (seeSection21.a) pubfishedin I97IJ Its use is limited, though, to synthesizing of seriaitydependentmonthly streamflowsin a river reach.Other codess'e sequences are avarlableto thi hy-drologist,however. Additional models and descriptionsof theory and applicationsof time-seriesanalysisof precipitation and streamfloware detailedin a number of availabletexts and publications'10-l3

MassCurveAnalysis One of the earliestand simplestsynthesistechniqueswas devisedby Ripplla to investigate reservoirptoragecapacityrequirements.His analysisassumesthat the future inflowsto a reseivoirwill be a duplicateof the historicalrecord repeatedin its entirety as many times end fo end as is necessaryto span the useful life of the reservoir. Sufficient storageis then selectedto hold surpluswaters for releaseduring critical periods when inflows fall short of demands.Reservoirsize selectionis easily accompmfr"a from an analysisof peaks and troughs in the mass curve of accumulated Future flows can be similar, but are unlikely to be syntheticinflow versustime.15-17 identicalto pastflows.Randomgenerationand Markov modelingtechniquesproduce t-hatare difJerentfrom, although still representativeof, historical flows. seqUences

HYDROLOGY 537 22.1 SYNTHETIC

-

EXAMPLE 22.1 Streamflowspast a proposedreservoir site during a 5-year period of record were, in eachyear14,000,10,000,6000,8000,and 12,000acre-ft.use Rippl's respectively, to determinethe size of reservoir neededto provide a yield of method ,nur. Crrru" 9000 acre-ft in eachof the next 10 years. solution. A lo-year sequenceof syntheticflows,usingRippl's assumptions,is shown in Table Z2.l.Inflows are set equal to the historical record repeated twice. rl FOREXAMPLE22'1 TABLE22.1 STREAMFLOWS Flows (thousandsof acre-ft) Year Inflow Cumulative inflow

r 1 14

2 4

3 4 0 6 8 38 30

1 24

5 6 1 2 1 64 50

7 8 9 1 0 4 1 0 6 8 1 2 100 88 80 74

draft of 9000 acre-ft per year for 10 years.

90,000 80,000

^ I

70,000

B

60,000

Storagerequfued for 9000-acre-ft/Yr draft is maximum

Cumulative draft, slopeof9000 acre-fl/Yr

,' ,

50,000 4000 acre ft storagerequired

40,000 U

30,000 . 20,000 10.000 1

2

3

4

5

6

Year

Eigarc 22.1 Mass curve for ExamPle 22.1: inflow; --- cumulativedtaft.

cumulative

538

CHAPTER 22 HYDRoLoGIc TIMESERIES ANALYSIS

RandomGeneratiOn one method of generatingsequencesof future flows is a simplerandom rearrangement of past records.If the streamis ungaugedand recordsare not available,a probability distributioncan be selectedand a sequenceof future flows that follow the distribution and haveprescribedstatisticalmomentsis generated. Wheneverhistorical flows are available,a reasonablesequenceof future flows can be synthesizedby first consultinga table of randomnumbers,selectinga number, matchingthis with the rank-in-file numberof a pastflow, and listing thecorresponding flow as the first value in the new sequence.The next random numberwould be used in a similar fasion to generatethe next flow, and so on. Randomnumbershavingno correspondingflows are neglectedand the next randomnumberis selected.Table-B.3 in Appendix B is a table of uniformly distributedrandom numbers(eachsuccessive numberhas an equalprobability of taking on any of the possiblevalues).To illustrate the use of Table B.3 in the random generationprocess,the first three yearsof a syntheticflow sequencecould be generatedby selectingthe 53rd, 74th, and23rdfrom the list of past flows. Alternatively, the flows in 1953, 1974, and 1923 could also be selectedas the new randomsequence. Most computershave random number generationcapabilitiesin their system libraries. Rather than storing large tables of numberssuchas Table B.3, successive random integersare usually generatedby the computer. EXA]I/IPLE22.2 Annualflowsin CrookedCreekwere 19,000,14,000,21,000,8000,11,000,23,000, 1 0 , 0 0 0 , a n d 9 0 0 0 a c r e - f t , r e s p e c t i v e l y , f ol ,r2y,e3a, 4 r s, 5 , 6 , 7 , a n d 8 . G e n e r a t e a 5-year sequenceof annual flows, O,, by matching five random numberswith year numbers. Solution. Randomintegersbetween0 and 9 are generatedfrom the computer. The Q, valuesin Table22.2 areselectedfrom the eight given flowsby matching the respectiveyear numberwith the random number.The digit t has no correspondingflow in the 8-year sequence,so the next random number, 2, places the 14,000-cfs flow in Year 2 in the first position of the synthetic 5-year sequence.rI TABLE 22.2 DEVELOPMENTOF s.YEAR SYNTHETIC SEQUENCE Randomdigit I

o

2 3

2 5 8 I

I

5 6

I

Q(acre-ft)

Skip 14,000 11,000 9,000 19,000 8.000

544

CHAPTER22

HYDROLOGICTIME SERIESANALYSIS

3.- a regressioncoefficient of the standarddeviation for the daily flow logarithms within each month of record to the logarithm of the monthly total flow.,

'

Given the calculatedstatistics,the simulationof daily flows could be structured in the following manner: variatesfrom Eq. 22'1,0. 1. Generatestandardized 2. Use the logarithm of the monthly mean flow as an initial estimateof the mean of the logarithmic daily flows for the month' 3. Calculatethe standarddeviationof flow logarithmsby the previouslydetermined regressionequation. 4. Apply the inverseof Eq. 22.1I, k--

?lz(' -:)

.

'l-; 13

1

(22.r2)

to transformthe standardizedvariatesto flows,multiplying by the appropriate standarddeviationand addingthe mean. 5. Add the differencebetweenthe total monthly flow generatedand the given monthly flow to the given monthly flow, and repeatthe simulation. 6. Multiply daily results of the secondsimulation by the ratio of the given monthly total to the generatedmonthly total. Each simulation techniquecommonly requiresmodificiationswhen applied to individual problems.Methodsoutlined thus far can be utilized as guidesin establishing a procedureto follow in synthesizingflows.Simulationof flows for a given station hasbeenpresentedwith serial correlation,skewness,means,and standarddeviations for an entire system,the preserrnaintained.When generatingstreamflowsequences vation of cross-correlationbetweenstationsbecomesa significantfactor. are employedto determinethe capacSyntheticallygeneratedrunoff sequences ities of reservoirsto satisfy specifieddemands.Individually generatedflow magnitudesare uncertain, as are the syntheticallygeneratedsequencesof flows. Hydrologistscanestimateprobabilitiesof flowsby generatingseveralequally likd sequences of flowsand then evaluatingrecurrencesof certain values.Herein lies one of the most useful applicationsof Markov generatingtechniques.

i Summary Hydrologicmodelingis often presentedas comprisingonly the deterministicmodels of the rainfall-runoff processdescribedin Chapters2I,23,24, and 25. The fully equippedhydrologistincorporatesthe synthetichydrology models describedin this chapterin the analysisand designof water resotrces systems.A growing numberof projects are constructedor operatedon the basis of synthetichydrology and timesedesanalysiseachyear.

PROBLEMS

545

PROBLEMS 22'2 and detet22.1. plot cumulativeinflows versustime for the S-yearrecord in Example yield of

neededto provide a mine by mass curve analysisthe size of the r_eservoir maximumyield possible? is,the What 12,000acre-ft in eachoithe next 24years. sequenceof synthetic 'r, t use the annual rainfall trom Table 26.2to generatea lO-year assumption' annual rain depthsfor Richmond using Rippl's masscurve curve methods'Use 22.3. RepeatProblem 22.2 wingrandom generationrather than mass with the last two digits of these match and taUl'n.: numberJfrom i*o-Olglt.unaom the yearnumbersin Problem26.32.

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23.2 CONTINUOUSSIMUI-ATIONMODEL STUDTES

-

591

6000 5000

^

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Figure 23.21 Sensitivityof modelresponseto the length-of-overland-flow parameter.(After Clarke.16)

KSF = 0.85

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Figwe 23.22 Sensitivityof modelresponse to thechannelroutingparameter. (After Clarke.16) usingcontinuousmodelingincludewatershedyield studies,reservoirdesignand operation studies,sedimentyield estimatingfor reservoirdesignor analysisof impactsof erosion controls on water quality, water supply studiesfor municipal, industrial or agricultural demands,litigation over impactsof wellfieldsor direct diversions,determination of hydropower production potential, evaluationsof flows that will pass through critical in*stream or riparian habitat reachesof a stream,identification of flows that will be availablefor recreationalusesof a stream,and, amongnumerous other applications,analysisof water quantity and quality impacts of removingdams or rnaking other major upstreamchanges. The modelsdescribedin this chapter,and other similar continuoussimulation codes,are availablefrom the federal or state agenciesthat originatedthe code, or from numerouspublic outlets or vendorswho havebeen authorizedto distributethe software.

592

MODELS SIMULATION CHAPTER23 CONTINUOUS

Problems 23.r. Assumethat a 30-mi2rural watershedin your localereceivesh 3-in. rain in a 10-day

period. Reconstructthe block diagram ofFig. 23.2 and plot approximatepercentages io show,for averageconditions,how the rain would be distributed(a) initially and (b) after 30 days. 23.2. A sloping,concreteparking lot experiencesrain at afate of 3.0 in.i hr for 60 min' The tot is SOOft deepand has a slopeof 0.000-1ftlft. If the water detentionon the lot is zero at the start of the storm,calculatethe completeoverlandflow hydrographfor 1 ft of width using the SWM-IV equations.Use a 5-min routing interval and continue computationsuntil all the detainedwater is discharged. 23.3. Calculatethe SWM-IV overlandflow time-to-equilibrium for the lot of Problem23'2 and compareit with the Kirpich time of concentrationfor the lot. Should thesebe equal? 23.4. Cornpare,by listing traits and capabilitesof each,the SWM-IV with its more sophisticated offspring HSP and HSPF. 23.5. Discussthe primary differencesamong the four versionsof the Stanfordwatershed model describedin this chaPter. 23.6. Verify Eqs. 23.23 and 23.24by starting from Eqs. 13.4 and 13 '33' "typical" 23.7. Discuss the watershedbehavior that is depicted in Fig. 23.7. Is this a watershed? 23.8. Comparethe differencesbetweenthe two U.S. Departmentof Agriculture continuous simuiationmodels,USDAHL and SWRRB,and discussthe applicationsthat would be best suitedto each. 23.9. Review the differencesbetweenwater budget and simulation models discussedin Chapter2l anddeterminewhich of the continuoussimulationmodelsdescribedhere could be usedto perform water budgetcalculations. 23.10. For what applicationsmight the following be best suited? API model USDAHL HSPF PRMS SWRRB 23.1r. For the continuoussimulationmodel selectedby your instructor,describefour different types of problemsthat could be analyzedif you were given the ful1, calibrated model.

REFERENCES "ContinuousHydrographSynthesiswith an 1 . W. T. Sittner, C. E. Schauss,and J. C. Monro, API-Type Hydrologic Model," WaterResourcesRes'5(5), 1007'1022(1969)' "Digital Simulationin Hydrology:StanfordWaterL. N. H. diawford and R. K. Linsley,Jr., shedModel IV," Departmentof Civil Engineering,Stanford University, Tech. Rep. No. J.

3 9 .J u l y 1 9 6 6 . "An Evaluationof RelationshipBetweenStreamflowPatternsand Watershed L. D. James, CharacteristicsThrough the Use of OPSET," ResearchRep. No' 36, Water Resources Institute, University of Kentucky,Lexington, 1970'

Chapter24

Simulation Single-Event Models

r Prologue The purposeof this chaPteris to: . Describehow storm event models are structuredand how they are used to simulatedirect runoff hydrographsfor singlestorms. . Describethe five most widely usedfederal agencysingle-eventmodels(note that popular single-eventurban runoff simulation models are describedin Chapter25). ' Provide a detailedcasestudy using one of the models,HEC- 1' . Introducethe emergingtechnologyof storm surgemodeling-the simulation of hydraulic surgesresulting from wind energy acting on the ocean surface. Many severefloodsare causedby short-duration,high-intensityrainfall events. A single-eventwatershedmodel simulatesrunoff during and shortly following these discreterain events.Usersof single-eventmodelsare normally interestedin the peak flow rate, or the entire direct runoff hydrographif timing or volume of runoff is needed.Single-eventmodelssimulatethe rainfall-runoff processand makeno special effort to account for the rest of the hydrologic cycle. Few, if any, simulate soil moisture,evapotranspiration,interflow, baseflow, or other processesoccurring between discreterainfall events. Modelsdescribedin this chapterare applicableto studiesof watershedsthat are primarily rural in makeup.Urbanizedsubareasare allowed,but for watershedsthat ire principally urbanized, the single-event and continuous models described in Chapter2i aremoreapplicable.Coastalfloodingthat is inducedby surgescreatedby wind action on the ocean surfaceis modeledby a different class of single-event models,describedin Section24'3.

24.1 STORMEVENTSIMULATION Event simulation model structurescloselyimitate the rainfall and runoff processes suchasunit-hydrograph developedin earlierchapters.Lumpedparameterapproaches, methods,are generally incorporated even though some use distributed parameter

SIMULATION595 EVENT 24,1 STORM Preparationfor implementingmost single-eventsimulationstudiesbegins techniques.. with a watershedsubdivisioninto homogeneoussubbasinsas illustrated inFig' 24'1' Computationsproceed from the most remote upstream subbaslnin a downstream direction, In any single-eventmodelfor a typical basin(Fig. 24.I),the runoff hydrographs for eachof subbasinsA, B, . . . , E are computedindependently,and then routed and cornbinedat appropriatepoints (callednodes)to obtain designhydrographsthroughout the basin.The modelieads input parametersfor the storm; then appliesthe storm to the first upstreamsubbasin,Bf computesthe hydrographresultingfrom the storm event; repeatsthe hydrographcomputationfor subbasinA; combinesthe two computed'hydrographsinto a singlehydrograph;routesthe hydrographby conventional iechniquest[rough reachC to the upstreamend of reservoirR, whereit is combined with the compuied hydrographfor subbasinC; and so on through the procedure detailedinFig.24.l. Hydrographcomputationsfor subbasinsare most often determinedusing unitis hydrogiaph proceduresas illustrated in Fig. 24.2. The precipitation hyetograph leaving abstracted, are losses input iniiormly overthe subbasinarea,andptecipitation un "*..r, prelipitation hyetographthat is convoluted (see Chapter 12) with the

Outlet (point of concentration)

1. Subdividebasinto accommodatereservoirsites,damagecenters' diversionpoints, surfaceand subsurfacedivides,gaugingstations' precipitationstations,land uses,soil types,geomorphologicfeatures' 2. Lomputation sequencein eventsimulationmodels: a. Computehydrographfor subbasinB' b. Computehydrographfor subbasinA. c. Add hydrograPhsforA and B. d. Routecomtined hydrographto upstreamend of reservoirR' b. Compute hydrograph for subbasinC. f. Computehydrographfor subbasinD. at R. g. CombinethreehYdrograPhs h. Route combinedhydrographthroughreservoirR' i. Route reservoir outflow hydrograph to outlet' j. Computehydrographfor subbasinE' k. Combinetwo hydrographsat outlet. Figure 24.1 Typical watershed subdivision and computation sequence for event-simulation models.

Precioitation

I

- ?lg HJ'VhYetograPh a Time

s / Outlet

/

Gross precipitation

Excess (net

Time

I c"-bt". I - -

\Z

Total streamflowhydrograph

Figure24.2 Typical lumpedparameterevent-simulationmodelof the rainfallrunoff process.

MODELS 597 AGENCY SINGLE.EVENT 24.2 FEDERAL

'

prescribedunit hydrographto producea surfacerunoff hydrographfor the subbasin. The abstractedlossesare divided among the loss componentson the basis of prescribedparameters.Subsurfaceflows and waters derived from groundwaterstorage are transformedinto a subsurfacerunoff hydrograph,which when combinedwith the surfacerunoff hydrographforms the total streamflow hydrographat the subbasin outlet. This hydrographcan then be routed downstream,combined with another contributing hydrograph,or simply output if this subbasinis the only, or the final, subbasinbeine considered.

MODELS 24.2 FEDERALAGENCYSINGLE-EVENT depictedin Figs.24.1and24.2 arerecognizedby most The rainfall-runoff processes of the eventsimulationmodelsnamedin Table21.1. Specificcomputationtechniques for losses,unit hydrographs,river routing, reservoirrouting, and baseflow are comparedin Table24.1for five of the major federalagencyrainfall-runoff eventsimulation models.All the modelsallow selectionamongavailabletechniques.Brief descriptions of eachof thesemodelsare followedby an illustrative exampleof an application of the HEC-I model to a single storm occurring over a 250-mf watershednear Lincoln, Nebraska.

U.S.GeologicalSurveyRainfall-RunoffModel The USGSmodel can be usedin evaluatingshort streamflowrecordsand calculating peak flow ratesfor natural drainagebasins.lThe programmonitorsthe daily moisture content of the subbasinsoil and can be used as a continuousstreamflowsimulation model.The model is classifiedas an eventsimulationmodelbecauseits calibrationis based on short-termrecords of rainfall, evaporation,and dischargesduring a few documentedfloods.It hasbeenmodified severaltimes and hasevolvedinto the USGS urban continuoussimulation model, DR3M, describedin detail in Chapter25. Input to the model consistsof initial estimatesof 10 parameters,which are modified by the model through an optimization fitting procedurethat matchessimulated and recordedflow rates. Other input includesdaily rainfall and evaporation, close-interval(5-60 min) rainfall and dischargedata, drainageareas,impervious areas,and baseflow rates for eachflood. Phillip's2infiltration equationis usedto determinea rainfall excesshyetograph, which is translatedto the subbasinoutlet and then routed through a linear reservoir, using the time-.areawatershedrouting techniquedescribedin Chapter 13. The USGSrainfall-runoff model can be usedto simulatestreamflowsfor relatively short periodsfor small basinswith approximatelylinear storage-outflowcharacteristicsin regionswheresnowmeltor frozen groundis not significant.Output from the model includes a table showingpeak discharges,storm runoff volumes, storm rainfall amounts,and an iteration-by-iterationprintout of magnitudesof parameters and residualsin fitting volumesand peak flow rates.

598

CHAPTER24

SINGLE-EVENTSIMULATIONMODELS

TABLE 2II"1 HYDROLOGYPROCESSESAND OPTIONSUSED BY SEVERAL AGENCY RAINFALL-RUNOFFEVENTSIMULATIONSMODELS Model code names (sleeTable 21.1)

Modeledcomoonents Infiltration and losses Holtan's equation Horton's equation Green-Ampt Phillip's equation SCS curve number method Exponentiallossrate Standardcapacitycurves Unit hydrograph Input Clark's Snyder's Two-parametergamma response SCS dimensionlessunit hydrograph River routing Kinematic wave Full dynamic wave Muskingum Muskingum-Cunge Modified Puls Normal depth Variable storagecoefficient Att-kin method Translation only Reservoirrouting Storage-indication(Puls) Baseflow Input Constantvalue , Recessionequation Snowmeltroutine

HEC-1 (Corps)

TR-20

(scs)

USGS (USGS)

HYMO (ARS)

SWMM (EPA)

x X

x

X X X X X

X X

X X X Yes

No

No

Yes

ComputerProgramfor ProjectFormulation Hydrology(TR-20) A particular$ powerful hydrologicprocessand water surfaceprofile computerprogram was developedby CEIR, Inc.3and is known by the codename TR-20, which is an acronymfor the U.S. Soil ConservationServiceTechnicalReleaseNuntber20. The model is a computerprogram of methodsusedby the Soil ConservationServiceas presentedin lhe National Engineering Handbook.a The program 'is recognized as an engineer-orientedrather than computeroriented package,having been developedwith easeof use as a purpose.Input data sheetsand output data are designedfor easein use and interpretationby field engi-

24.2 FEDERALAGENCY SINGLE-EVENTMODELS

599

neers,and the programcontainsa liberal numberof operationsthat are user-accommodating,even at the expenseof machinetime. The TR-20 was designedto use soil and land-use information to determine runoff hydrographsfor known stormsand to perform reservoirand channelrouting of It is a single-eventmodel,with no provisionfor additional hyOrograpns. the generateO lossesor inflltration beiweendiscretestorm events.The programhasbeenusedin all 50 statesby engineersfor flood insuranceand flood hazardstudies,for the designof reservoirand channelprojects, and for urban and rural watershedplanning. Surfacerunoff is computedfrom an historicalor syntheticstormusingthe SCS curvenumberapproachdescribedin Chapter4 to abstractlosses'The standarddimensionlesshydrogiaphshownin Fig. 12.13is usedto developunit hydrographsfor each subareain the watershed.The ixcess rainfall hyetographis constructedusing the effectiverain and a given rainfall distributionand is then appliedincrementallyto the unit hydrographto obtain the subarearunoff hydrographfor th9 storm' As shown in Table 24.I, TR-20 usesthe storage-indicationmethod to route hydrographsthroughreservoirs(seeSection13.2).The baseflow is addedto the direct .onoffhyatographs at any time to producethe total flow rates.The programusesthe logic depictel in fig. Z+.iby computingthe total flow hydrographs,routing the flows thiough streamchannelsandreservoirs,combiningthe routedhydrographswith those from6ther tributaries,and routing the combinedhydrographsto the watershedoutlet' Prior to 1983, the model routed stream inflow hydrographsby the convex method (Sec(Section13.1),which has sincebeenreplacedby the modifiedatt-kinmethod tion 13.3).Asmanyas200channelreachesandggreservoirsorfloodwater-retarding structurescanbe accommodatedin any singleapplicationof the model.To add to this the capability,the programallowsthe concurrgntinput of up to 9 different stormsover watershedarea. Subdivisionof the watershedis facilitatedby determiningthe locationsof control points. Control points are defined as stream locations correspondingto crossor sectionaldata, reservoirsites,damagecenters,diversionpoints, gaugingstations, requiredata Subarea desired. be may tributary confluenceswhere hydrographdata ments include the drainagearea,the time of concentration,the reachlengths,strucreach ture data as describedin-Section 21.1, andeither routing coefflcientsfor each proare dala cross-sectional Whenever or cfoss-sectionaldata along the channels. peak flow the to addition in elevations vided, the model calculatesihe water surface by the dictated are sizes Subarea rates and time of occurrence at each section. it is information, hazatd flood and routing locations of control points. To provide of the characteristics hydraulic the so that necessaryto defineenoughcontiol points instreamaie definedbetwJencontrof sections.Applications with TR-20 normally The apart' more or 2 mi to feet hundred few corporatecont*rolpoints spacedbetween-a mi2' ."rolting subarLasthat contributerunoff to a control point are usually lessthan 5 is no there though even mi2 25 than less Common subareasizesfor structuresare program' the within limitation on reachlength or subareasize Minimal input daia requirementsto TR-20 includethe watershedcharacteristics, at leastone actual or syntheiicstorm including the depth, duration, and distribution; routing the discharge,capaciiy, and elevation data for each structure; and the

600

CHAPTER24


coefficientsor cross-sectionaldata for eachreach.Input can be describedaccording to the following outline: 1. Watershedcharacteristics. a. The area(in mi2)contributingrunoff to eachreservoirand crosssection. b. Runoff curve number CN for eachsubarea.(SeeChapter4.) c. The antecedentmoisturecondition associatedwith eachsubarea,coded as dry, normal, or wet. d. The time of concentrationfor eacfi subarea(hr). e. The length of eachchannelrouting reach and subareamainstream. 2. Velocity-routingcoefficienttable. a. A table containingrouting coefficientsfor a rangeof velocities(ftlsec). This table is containedwithin the program and need only be enteredif the user desiresdifferent velocities. 3. Dimensionlesshydrograph. a. This table is containedwithin the program and need only be enteredif the user desiresa different hydrograph. 4. Actual hydrograph. a. Actual hydrographscan be introduced at any point in the watershed. Hydrographordinates are read as dischargerates (cfs) spacedat equal time incrementsapart, up to a maximum of 300 entries. b. Baseflow rates (cfs)can also be specifiefl' 5. Baseflow. a. In additionto the option of specifyingthe baseflow ratesassociatedwith a hydrographthat was input, the baseflow can be specifiedor modified at any other control Point. 6. Stormdata. a. Stormsare numberedfrom 1 to 9 and are input as cumulativedepthsat equally spacedtime increments. b. As an alternativeto specifyingcumulativedepthsat varioustime increments, a dimensionlessstorm can be input, and up to 9 stormscan be synthesizedby specifyingeachstorm depth and duration' data. 7. Streamcross-sectional a. Up to 200 crosssectionsmay be input for a singlerun. Cross-sectional data consistof up to 20 pairs of valuesof the dischargeversusflow area. b. If cross-sectionaldata are provided,the routing coefficientsare determined from them. In the absenceof suchdata, the user must specify a routing coefficientfor eachreach. 8. Structure.data. a. The rbservoirdataconsistof up to 20 pairsof outflow dischargerates(cfs) versusstorage(acre-ft). b. A maximum of 99 structuresare allowed in a run. The desiredoutput from TR-20 must be specifiedby a set of input file control variables.Hydrographsat eachcontrol point for eachstormcanbe printedby specifying the control point identification in the control cards. Any combination gf the following items can be producedat eachcontrol point:

MODELS 601' AGENCY SINGLE-EVENT 24,2 FEDERAL The peak dischargerate, time of peak, and peak watet'surfaceelevations. The dischargerates in tabular form for the entire hydrograph' Water surfaceelevationsfor t'heentire duration ofJunoff' The volume of direct runoff, determinedfrom the area under the hydrograph. 5. Hydrographordinatesin any specifiedformat. 6. Summary tablescontainingwater balanceinformation'

* . 2. 3. 4.

Basicdata neededby the computerprogramare determinedfrom field surveys' Rainfall frequency data are input from daia in the U.S. Weather Bureau TP-40 report.s Peak-dischargeand area-floodedinformation for presentand future conditions for severalreturn periodsare output by TR-20 in a form suitablefor direct use in an economicevaluationmodel.

ComputerLanguagefor Hydrologic Problem-Oriented Modeling(HYMO) A unique computerlanguagedesignedfor use by hydrologistswho haveno conventional computir programming experiencewas developedby Williams and Hann.6 Once the progru- has been compiled,the user forms a sequenceoJ commandsthat synthesiz{ route, stoie, plot, or add hydrographsfor subareasof any watershed. Seventeencommandsare availableto usein any sequenceto transformrainfall data into runoff hydrographsand to route thesehydrographsthrough streamsand.reservoirs. The HYMO model also computesthe sedimentyield irt tons for individual stormson the watershed. Watershedrunoff hydrographsare cornputed by HYMO using unit-hydrograph techniques.Unit hydrographscan either be input or synthesizedaccordingto the dimensionlessunit hydrographshown in Fig. 24.3. Tetms in the equationsare 1.0 .

-

i l - I

q = q' " ( - \ \'p

ttr-'trtrtr-tt I

t6 inflection point

s ls'o

q = qos(to-t)tx

tp

Figure 24.3 Dimensionlessunit hydrographused in HYMO. (After Williams and Hann.6)

602

CHAPTER24


4 : flow rate (ft3lsec)at time t q, : peak flow rate (ft3lsec) te = time to peak (hr) n : dimensionlessshapeparameter q6 : flow rate at the inflection point (cfs) /o : time at the inflection point (hr) K : recessionconstant(hr) Once K and to and 4oare known, the entire hydrographcan be computedfrom the three segmentequationsshownin Fig. 24.3. The peak flow rate is computedby the equation u ^ - -

BAO

(24.r)

where B : a watershedparameter,related to n as shown inFig.24.4 A : watershedarea (mi2) O: volume of runoff (in.), determined by HYMO from the SCS rainfall-runoff equationdescribedin Chapter4 t The duration of the unit hydrographis equatedwith the selectedtime increment.The runoff Q for the unit hydrographwould of course be 1.0 in. The parametern in Fig. 24.4 is obtainedfrom Fig. 24.5. ParametersK and to for ungaugedwatersheds

500

50

10 0

2

4

6

8

1

0

1

2

n

Figure 24.4 Relation between dimensionless shape parametern and watershedParameterB. (After Williams and Hann.6)

MODELS 24.2 FEDEMLAGENCYSINGLE-EVENT

603

L2

10

8

s

6

4

2

0

0

.

5

1

1

.

5

2

2

.

5

3

K tp

shape' Figure 24.5 Relation betweendimensionless parametern and ratio of recessionconstantto time to peak.(AfterWilliamsandHann.6.1 are determinedfrom regional regressionequationsbasedon 34 watershedslocated in Texas,Oklahoma, Arkansas,Louisiana, Mississippi,and Tennessee,ranging in size from 0.5 to 25 mi2, or

and

sLp-oi.'t(#,)"K : 27.oAo 231

(24.2)

! tp : 4.63Ao+"5rr-o''u(

(24.3)

)t'"

where SLP : the differencein elevation(ft), divided by floodplain distance(mi), betweenthe basin outlet and the most distant point on the divide LfW : the,basinlength/width ratio River routing is accomplishedin HYMO by a revised variable storqge cofficient (VfC) method.TThe continuity equation,I - O : dS/dt, and the storage equation,S : KO, are combinedand discretizedaccordingto the methodsoutlined in Chapter 13. The VSC methodrecognizesthe variability in K as the flow leavesthe confinesofthe streamchannelandinundatesthe floodplainand valley area.Relations betweenK and O are determinedby HYMO from the input cross-sectionaldata, or HYMO will calculatethe relation using Manning's equation if the floodplain and channelroughnesscoefficientsare specified.The bed slopeand reachlength are also part of the required input.

604

CHAPTER24


The Widelyadoptedstorage-indicationmethod(seeChapter 13) is usedto route inflow hydrographsthroughreservoirs.The storage-outflowcurvemustbe determined externallyby the userand is input to the prolram asa table containingpairedstorages and outflows,using storagedefinedas zero wheneverthe outflow is zero. The user-orientedcomrnandsand the datarequirementsfor eachcommandare as follows: 1. Sta;t: the time rainfall beginson the watershed. 2. Storehydrograph:the time incremenfto,be used,the lowestflow rate to be stored, the watershed atea, and the successiveflow rates spacedat the specifledtime increment. 3. Develophydrograph:the desiredtime increment, the watershedatea, the SCSrunoff curve numberCN, the watershedchannellength and maximum differencein elevation,and the cumulativerainfall beginningwith zero and accumulatedat the end of eachtime incrementuntil the end of the storm. 4. Computethe rating curve:cross-sectionalidentificationnumber,numberof points in the crosssection,the maximum elevationof the crosssection,the main channel and left and right floodplain slopes,Manning's n fot each segment,and finally pairs of horizontal and vertical cpordinatesof the points describingthe crosssection. 5. Reachcomputations:the numberof crosssectionsin the routing reach,the time increment to be used in routing, the reach length, and the discharge rates for which the variable storagecoefficientis to be computed' 6. Print hydrograph:the idpntification nurnberof the cross sectionat which hydrographsare to be printed. 7. Plot hydrograph:the identificationnumberof the crosssectionsat which the hydrographsare to be plotted. 8. Add hydrographs:the identification numbers of the hydrographsto be added. 9. Route reservoir:the identificationriumbersof the locationsof .the outflow relation for the reservoir. andinflow hydrographs,andthe discharge-storage L0. Compute travel time: the reach identification number, reach length, and reach slope. 11. Sedimentyield: severalfactors describingsoil erodibility, cropping management,erosioncontrol practices,slopelength, and slopegradient. Output from HYMO includes the synthesizedor user-providedunit hydrographs,the storm runoff hydrographs,the river- bnd reservoir-routedhydrographs, and the sedimentyield for individual storms on each subwatershed.Hydrographs computedby HYMO comparedcloselywith measuredhydrographsfrom the 34 test watersheds.

Storm Water ManagementModel (SWMM) The EnvironmentalProtectionAgency model,SWMM,s is Hstedin Table21.1 in two locationscorrespondingto rainfall-runoff eventsimulationand urban runoff simulation. The model is primarily an urban runoff simulation model and is describedin detail-in Chapter25.

MODELS 605 SINGLE-EVENT AGENCY 24.2 FEDERAL and Lilie mostothers,the SWMM modelhasundergonenumefousmodiflcations a single-event improvementssinceits first releasein 1972.The initial version8was urban storm of modeling in continuous ri'se its allow model, and newer versionse,lo a new snowmelt waterflows and waterquality parameters.The latestreleaseincludes scour and routine, a new storm watei tto.ugt and treatment package'a sediment depositionroutine, and a revisedinfiltration simulation' hydrologic' swMM's hydrographandrouting routinesarehydraulicratherthan of single A distributed parameier approachis used for subcatchmentsconsisiting as routed parking lots, city lots, and so on. Accumulatedrainfall on theseplots is first closed or overlandflow to gutter or storm drain inlets,whereit is then routed asopen Of the five channelflow to the receivingwatefsor to sometype of treatmentfacility' detail greatest event-simulationmodelscoripared in Table 24.l,the SWMM givesthe in simulation,but cannotbe usedin large rural watershedsimulations. step using overland flow depths and flow rates are computedfor each time depth over a Manning's equation along with the continuity equation. The water depth reachesa subcatchmentwill increale without inducing an outflow until the over the subspecifieddetentionrequirement.If and wheneverthe resulting depth rate D., is largei than the specifieddetentionrequirement,Da, an outflow câtchment, is computedusing a modified Manning's equation

r .49.-

V:-J1-(D,-

and

Q*:

D,)2/3St/2

vw(D, - D)

(24.4)

(24.s)

where V:thevelocity n : Manning's coefflcient S : the ground slope W : the width of the overlandflow Q. : the outflow dischargerate they are After flow depthsand rates from all subcatchmentshave been computed, the total to form guttef combinedalong with the flow from the immediateupstream flow in eachsuccessivegutter. The gutter and PiPeflows are routed to any points of interestin the network wh ordinatesfor each time step in the routir incrementsuntil the runoff from the storm ters of the gutter shape,slope,and length and are availroughnessJoef{cienti ior ttre pipes or channelsmust also be supplied able in most hydraulicstextbooks' includethe Other iniut requiredfor a typical simulationwith the SWMM model following: percent im1. Watershedcharacteristicssuchas the infiltration parameters, pervious area, slope, area, detention storage depth' and Manning's coefficientsfor overlandflow' 2. The rainfall hyetographfor the storm to be simulated'

606


CHAPTER24

3. The land-use data, averagemarket values of dwellings in subareas,and populationsof subareas. 4. Characteristicsof gutters such as the gutter geornetry,slope, roughness coefficients,maximum allowabledepths,and linkageswith other connecting inlets or gutters. 5 . Street cleaningfrequency. 6. Treatmentdevicesselectedand their sizes. 7. Indexesfor costsof facilities. 8. Boundary conditionsin the receiving waters. 9. Storagefacilities, location, and volume. 1.0. Inlet characteristicssuchas surfaceelevationsand invert elevations. 11. Characteristicsof pipes such as type, geometry,slope,Manning's n, and downstreamand upstreamjunction data.

HEC-I Flood HydrographPackage(HEC-1) The U.S. Army Corpsof EngineersHydrologicEngineeringCenterdevelopeda series of comprehensive computerprogramsascomputationalaidsfor consultants,universities, and federal, state, and local agencies(see Section 21.4). Programsfor flood hydrographcomputations,watersurfaceprofile computations,reservoirsystemanalyses,monthly streamflowsynthesis,and reservoir systemoperation for flood control comprisethe series.The single-eventflood hydrographpackage,HEC-1, is described here.tt

The HEC-1 model consistsof a calling program and six subroutines.Two of these subroutinesdeterminethe optimal unit hydrograph,loss rare, or streamflow routing parametersby matchingrecordedand simulatedhydrographvalues.The other subroutinesperform snowmelt computations, unit-hydrographcomputations,hydrographrouting and combiningcomputations,and hydrographbalancingcomputations. In addition to being capable of simulating the usual rainfall-runoff event processes,HEC-1 will also simulatemultiple floods for multiple basin development plansandperform the economicanalysisof flood damagesby numericallyintegrating areasunder damage-frequencycurvesfor existingand postdevelopmentconditions. HEC- 1 underwentrevisionsin the early 1970sand againin the 1980s.Several features were added (e.g., SCS curve number method, hydraulic routing), and a microcomputerversion was developedin 1984. The 1985 releaseexpandedearlier versionsto include kinematic hydrographrouting, simulation of urban runoff, hydrographanalysisfor flow over a dam or spillway,analysisof downstreamimpactsof dam failures,multistagepumpingplants for interior drainage,and flood control sysfem economics.The 1990versionof HEC- 1, availablefor PCs or Harris minicomputers, incorporatBsyet other improvements.It adds report-quality graphic and table capability, storageand retrieval of data from other programs,and new hydrologic proceduresincluding the popular Green and Ampt infiltration equation(Chapter4) and the Muskingum-Cungeflood routing method (Chapter 13). In addition to the unit-hydrographtechniquesof the earlier versions, the modified HEC-I allows hydrographsynthesesby kinematic-waveoverland runoff techniques,similar to those developedfor use in SWMM. The runoff can either be concentratedat the outlet of the subareaor uniformly addedalong the watercourse leogth through the subarea,distributingthe inflow to the channelor gutter in linearly


607

increasingamountsin the downstreamdirection. The 1990 versionallows the use of the Muskingum-Cungerouting method in a land surfacerunoff calculation mode. Precipitation can be directly input, or one of three synthetic storms (refer to Chapter 16) can be selected.A standardproject storm (SPS) is availablefor large basins (over 10 mi2) located east of 105' longitude, using proceduresdescribedin Corpsof Engineersmanuals.A 96-hr durationis synthesized,but the stormhasa 6-hr peak during eachday. A secondtype of storm is the probablemaximum precipitation (PMP), using estimatesfrom National Weather Service hydrometeorologicreports availablefor different locations(Chapter 16 describesthese).A minimum duration is 24ht, and stormsup to 96 hr long may be analyzed.The third method allows synthesisof any duration from 5 min to 10 days.The userneedonly specify the desiredduration and depth,and the programbalancesthe depth aroundthe central portion of the duration usingthe blockedIDF methodof Section16.4. The later versionsof HEC-I include all the precipitation loss, syntheticunit hydrograph,and routing functions developedfor earlier versions.Additional loss methodsincludeboth the SCS curve numbermethodand Holtan's lossrate equation (an exponentialdecayfunction). Becauseof the popularity of SCS techniques,the HEC-1 now includesTR-20 proceduresfor lossesand hydrographsynthesis.The duration ofthe SCS dimensionlessunit hydrographis interpretedby HEC-1 as approximately0.2 times the time to g 0.25 times the time to peak (this convertsto 0.29 times the peak, but not exceedin lag time). For routing through streamsand reservoirs,the newestversion of HEC-1 includesall previousmethods,and additionally performskinematic-wavechannelrouting for severalstandardgeometriccross-sectionshapes. In comparisonto other event-simulationmodels,HEC-1 is relatively compact and still able to executea variety of computationalproceduresin a singlecomputer run. The model is applicableonly to single-stormanalysisbecausethere is no provision for precipitation loss rate recovery during periods of little or no precipitation. After dividing the watershedinto subareasand routing reachesas shown in Fig. 24.6,the precipitation for a subareacan be determinedby one of four methods: (1) nonrecordingand/or recordingprecipitation station data,(2) basinmeanprecipitation, (3) standardproject or probablemaximum hypotheticalprecipitation distributions,or (4) syntheticbalancedstormmethodusingIDF data (Section16.4).Either actual depthsor net rain amountsmay be input, dependingon the user's choice of techniquesfor abstractinglosses.The HEC-1 lossrate function is easily bypassedif the net rain is availablefor direct input. The programlogic for HEC-1 is shownin Fig. 24.6.Hydrologic processessuch as the subarearunoff computation, routing computation, hydrographcombining, subtractingdivertedflow, balancing,comparing,or summarizingare specifiedin the illustratedinFig.24.L input usingthe sequence One lossrate in the HEC-1 modelis an exponentialdecayfunction that depends on the rainfall intensity and the antecedentlosses.The instantaneousloss rate, in in./hr, is

L, = K'Pf

(24.6)

608

CHAPTER24


READINPUTDATA; REFORMATDATAAND WRITE TO WORKINGFILE

REA,DANDPRINT JOB SPECIFICATION;

READ ANDPRINT DATA

COMPUTE RL]NOFF HYDROGRAPH

1 Figure 24.6 HEC-1 Program OperationsOverview.l

where

L, : P, : E : K' :

the instantaneousloss rate (in./hr) intensity of the rain (in./hr) the exponentof recession(rangeof 0'5-0'9) acoefficient, decreasingwith time as lossesaccumulate K, : KoC-cuMlllo+ AK

where

Q4.7)

Ko : the loss coefficient at the beginning of the storm (when CUML = 0), an averagevalue of 0'6 CUML : the accumulatedloss(in.) from the beginningof the stormto time t C : a coefficient,an averageof 3.0

If AK is zero, the loss rate coefficientK' becomesa parabolicfunction of the accumulatedloss,CUML, and would thus plot asa straight-linefunction of CUML on semilogarithmicgraphpaperif K were plotted on the logarithmic scale.The straightln Fig.24.7, showingthe decreasein the lossrate coefficient fine reLtion t aepiCteO asthe lossesaccumulateduring any storm.Becauselossratestypically decreasemuch more rapidly during the initial minutesof a storm,the loss tate K' is increasedabove the straightJineamountby an amountequal to AK, which in turn is madea function of the amountof lossesth;t will accumulatebefore theK' value is againequalto the


0.2 cuMLl

609

A,K=Oif CUML> CUMLy

Ko

0

(in.) loss Accumulated ' CUML Figure 24.7 Variationof the lossrate coefficientK' with the lossamountCUML. accumulated

straight-linevalue,K. This initial accumulatedloss,CUML,, is user-specifieda-nd-rs rela6d to AK in sucha fashionthat the initial lossrateK' is 20 percenttimes CUMLI greaterthan Ko (seeFig. 24.7).Initialloss coefficientsKo are difficult to estimate,and itandard purvesin Chapters4 and 23 are availableto determineinitial infiltration rateS,26. For gaugedbasins,HEC- 1 allows the userto input rainfall and runoff data from which the lossrate parametersare optimizedto give a bestfit to the information provided. Estimatesof parametersfor ungaugedbasinsfall in the judgment realm noted in Table 21.1. An alternativeto the describedlossrate function is availablein HEC-1, which is an initial abstractionfollowed by a constantloss rate, similar to a @index. The HEC-1 model provides separatecomputations of snowmelt in up to 10 elevationzones.The precipitationin any zoneis consideredto be snowif the zone temperatureis lessthan abase temperature,usually 32'F, plus 2".Thp snowmeltis computedby the degree-dayor energybudgetmethodswheneverthe temperatureis "qou1to or greaterthan the basetemperature.The elevationzonesareusually considered in incrementsof 1000 ft althoughany equal incrementscan be used. unit hydrographsfor eaph subareacan be provided by the user, or clark's methodl3 o{ synthesizingan instantaneousunit hydrograph (IUH) can be used. Clark's method is more commonly recognizedas the time-area curve method of hydrographsynthesisdescribedin Section12.6.The time-area histogram,determined from an isochronal map of the watershed,is convolutedwith a unit design-storm hyetographusingEq. 12.38,as illustratedin Fig. t2.l9.The methodsdescribedin Section 1.3.2arc then usedto route the resultinghydrographthrough linear reservoir storageusingEq. 12.35with a watershqdstoragecoefficientK. Input data for Clark's

610

24 SINGLE-EVENT MoDELS cHAPTER SIMULATIoN

method eonsistsof the time-area curve ordinates,the time of concentrationfor the Clark unit graph,and the watershedstoragecoefficientK. If the time-area curve for the watershedunder considerationis not'available,the model-providesa synthetic time-arca curve at,the user's request. Becausethe Corps of EngineerscommonlyusesSnyder'smethod(Chapter 12) in unit-hydrographsynthesisfor large basins,the Snydertime lag from Eq. 11.5 and Snyder'speakingcoefficientCofrom Eq.12.17 can be input, and Clark's parameters will be determinedby HEC-1 from the Snydercoefficients.The actual or synthetic time-area curve is still required. Baseflow is treatedby HEC-1 as an exponentialrecessionusingan exponentof 0.1 in the following equation:

O r :#

(24.8)

where Q, : the flow rate at the beginningof the time increment Q2 : the flow rate at the end of the time increment R : the ratio of the base flow to the base flow 10 time incrementslater The base flow determinedfrom this equation is added to the direct runoff hydrographordinatesdeterminedfrom unit-hydrographtechniques.The startingpoint for the entire computationis the user-prescribedbaseflow rateat the beginningof the simulation, which is normally the flow severaltime incrementsprior to any direct runoff. Ifthe initial baseflow rate is specifiedas zero, the computerprogram output containsonly direct runoff rates. "storageThe HEC-I packageallows the user a choiceof severalhydrologicor routing" techniquesfor routing floodsthroughriver reachesand reservoirs.All usethe continuity equation and some form of the storage-outflow relation; some are described in more detail in Chapter 13. The five routing proceduresincluded in the programare the following: 1. Modified Puls-this methodis also calledthe storageor storage-indication methodand is a level-pool-routingtechniquenormally reservedfor usewith reservoirsor flat streams.The techniquewas describedin detail in Section I3.2. . 2. Muskingum-describedin detail in Section13.1. 3. Muskingum-Cunge-a blendedhydrologicand hydraulic routing method detailedin Section13.1. 4. Kinematicwave-described in Section13.3. 5. Straddle-stagger-also known as the progressiveaveragelag method.The techniqpesimply averagesa subsetof consecutiveinflow rates, and the averagedinflow value is lagged a specifiednumber of time incrementsto form the outflow rate. Input to HEC- 1 is facilitatedby arrangingthree categoriesof datain a sequence compatiblewith the desiredcomputationsequence,summarizedin Table 24.2. The individual input records are precededby a two-charactercode. The first character

MODELS 611 AGENCYSINGLE.EVENT 24.2 FEDERAL OF INPUTDATAFORHEC-111 TABLE24.2 SUBDIVISIONS Jobcontrol I -, Job initialization V-, Variableoutput summary O , Optimization J -" Job type

andhydraulics Hydrology K-, Job step control H-, Hydrographtransformation Q-, Hydrographdata B_, Basin data P _, Precipitation data L-, Loss (infiltration) data U_, Unit graph da-ta M-, MeIt data R_, Routing data S-, Storagedata D_, Diversion data W-, Pump withdrawal data

andendof job Economics E_, etc., Economics, data ZZ,End of Job

in Example24.1.

EXAMPLE 24.1 In June 1963 the Oak Creek watershedshown in Fig. 24.8 experienceda severe

over Subarea1. For the remainingelongatedwatershedareaA-H within the boldface border,usethe Junestormto simulatehydrographsat eachof Points1 8 usinga single 3 and 8' Points at run of HEC-I and comparepeak flows with recordedvalues

SCS runoff equationaand a basin-widecompositecurve number of 73 (see Chapter4).

612

CHAPTER 24 SINGLE.EVENT SIMULATION MODELS

-o+fr; rsog

paralso

KA\ \:

\%

, 1236

0

1

2

3

4

Scalein miles

;"'F4 d"fs;

r--l l

Rafmond Nebraska I 190 \o

{

\ @ \8.

34 ,,,G 'I-t

I

B C D E F G H f

TOTAL

.

Miles 2

Net rain (in)

33.4 26.9

7.8

27.3 9.2 28.3 17.0 5.0 28,0 82.9 258.0

dt\ -

4

t).

65

+.-)

4.1 2.8 A A

t.7 t;7 1.0 N.A.

Figure 24.8 Oak Creek watershedsubareamap.anddata sheet,June 1963.

5

MODELS 613 SINGLE-EVENT AGENCY 24.2 FEDERAL *

The computation logic to simulate runoff for this storm consistsof the following22 steps: L. Computethe hydrographfor AteaA at Point 1. 2. Routethe A hydrographfrom Point 1 to Point 2. 3. Computethe hydrographfor Area B atPoint 2. 4. Combine the two hydrographsatPoint 2. 5. Route the combinedhydrographto Point 3. 6. Computethe hydrographfor Area C-at Point 3. 7. Combine the two hydrographsat Point 3. 8. Route the combinedhydrographto Point 4. 9. Computethe hydrographfor Atea D at Point 4. 10. Combinethe two hydrographsat Point 4. 1L. Routethe combinedhydrographto Point 5. 12. Computethe hydrographfor AreaE at Point 5. 13. Combinethe two hydrographsat Point 5. 14. Routethe combinedhydrographto Point 6. 1.5. Computethe hydrographfor Atea H at Point 6. 16. Combine the two hydrographsat Point 6' 17. Route the combinedhydrographto Point 7. 18. Computethe hydrographfor Area F at Point 7' 19. Combine the two hydrographsat Point 7. 20. Route the combineAhyJrographto Point 8. 21. Computethe hydrographfor Area G at Point 8' 22. Combinethe two hydrographsat Point 8. Runoff hydrographsfor subareasare simulatedby convoluting the net storm hyetographwith unit hydrographssynthesizedby Clark's method using Snyder'i coiffiiients (seeChapter I2). A Co vabteof 0.8 is applied for Oak Crlek becauseof the moderately high retention capacity of the watershed. Subareatime lag valuesfor eachsubareaare found from Eq. (11.5) usinga C' valueof 2.0. Hydrographstreamrouting is performedusing the Muskingumtechnique (chapter 13) with x : 0.15 andK: the approximatereachtraveltime, using length dividedby the avefagevelocity. A Ch6zy averagevelocity determinedas 100 times the squareroot of the averagereachslopeis used.If K exceedsthree routing increments, the reach is further subdividedby HEC-1 into shorter lengthsto ensurecomputationalresolution. A sampleof the input and output for this job is shownasTable24.3.Each of the 22 computationalstepsare separatedin sequence.Only Steps 1-5 are includedin the sample.Note that the HEC-1 lossrate function was not usedso that the end-of-perlod excessand rain depths are equal. Note also that hydrographrouting of the A hydrographfrom Point 1 to Point 2 was facilitated (AMSKK) of l.2hr. using - three equal reachlengthseachwith a K-value ratesfor eachof steps flow peak time-averaged and 1 HECof summary A peak at Point 3 is 27,539 simulated the that Note given Table 24.4. in is l-22 The correspond27 of val:ue recorded the ,500' well with very agrees cfs, which -ing simulatedand observedpeak flows at Point 8 arc 22,453and 21,600cfs, -' - - - -respectively. Il

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E 8 i l f rs s H h E a E E E S S A A E S S R F S R F F 2

= EBagBBaata=R===B=sBB33EEE S E E E E = T E E = 5 !Fi = H N N F K = 3 =S= S E N

o U

= z. z. o

z. o

N F

E

z. L

V

z. z. z. z. z. z. z. z. z z. z. z. z. z. z z. z. z z. z. z. = = z z

{ O

+ + < + + + + + + + + + + + + { $ 9 e ! Q o o h o h d N N d N N d N N N N N N N N N N N N N N N N N N

= < 9 d

ot + Gt ci 6i - N - m N 6 6 F o + O + O @ O @ O = - t s h + o N N F

-

@ t s € o o F N O + h @ F @ O Q - N O S 9 @ r € o +o { + + s + + N N N o o o 6 o d s N - o r i

= =

= = =

= = = = = = = = = = = = = = = = = = = = = f - - - - - - - - - - - - - - - - - - - - - - - a -

F E

u J

6

u

E

E

r; r; Kj + ci + o o o @ O @ o O O @ + N - + O F h + N - F F

N

qt

a

gi

ot

ct

di

>

q q q = 9 q g g = g s s s s s < s < s < s d N N d N N N N N N N N N N N N N N N N N N N N N

-

a ?

- - ' t

-

- - - -

- - -

-

-

a -

-

- -

ifE

N

a z. E V

-

u

a -

{ c

+!Pct hH!2

=x

@ F o

x > =

'

@

=

c t d d d F +

N

=

o o r

O

J

r

o'

3 $ B S g E R S N J H E F E E J = a s 3 f f i $ K 9 A A S 5 = 3 6 D N

l

E

-

N

m

+

b

@

F

6

O

N

N

N

N

N

F

F

a . ar '+ t r r

F

a = Z =

=

= - - -

o - N o + 6 @ F @ o Q - N A + E r

>+

z. E

z + v i Y

8 E A o - 9

e

u ^ E 5 o -

n -E ra o Gi!P+

€r

- z z.\ tr-g?Yx = * = 6 EZ>O> - = '
= = = = = 1 = = = =- 5- == = - - -

3 x x x x * N t N * x * * F * R * N N R R K R R K K

lo F

u J

*

-

dl

a 3 $ p F $ HEE 5 E F H= F E P

u

E F

@ r

E a

U =

x

*

*

*

*

*

*

s 3 n j6 '6 id jq ' @+

e R N R R E E S E S B H h S g S S S S $ s 9 + s g B z

6 ? Y x

E

s E E

z. z. z. z z z z. z z. z z z. z z z z z 1 1 1 z < = z z z' = = = = = 5 = = 5 = = 5 - 5 -= i =- = =r ?- 5? - ? = ?- -r = - ? - - - -

z>> - 3 5

E;\u M F 9

=gFFEEEF-9 I EEEEgEEEEEEEEEEEE J L

-

,,,=

= - - = = - =

O

i B ot

==

U

E == > q< n < F

---

---

---

- - -

*N X * * N * * X * X tN t* tfi * R x * fi N tx

r!6=

giH

U

;+ sHm =

N

=

E= EHii

;

- z z +

E FTVX = - t o

-

"o-oNEH

FEHffEPE$nHHFEE=EEE q

o
@ > N

a = G V

\ o

- N

o + 4 @ r @ o o F N 9 S P 9

= = E R F N , R X

-

U ^

z.

z

Y Y

=

R

o @ E 5 o -

a l

ES F

-

EgEEEEEEEEEEFF=FFFFREE=EE = -^ z .z z z z z 1 1 1 = 1 z = 4 = 4 = = < = = = = =

E

6

E

u

o

F

= @ -

E

z u

F

s

r

-

' :

U

\

(5 o U F F

6 9 u z.

< = , 6 @

u

@

Y

2

o

5 E ts

u

a Y

@

x

O

C

E @

F

F

C

d

d

d

d

ci

d

ct ct d

d

d

d

d

ci

ct

d

ci

d

d

d

ct d

d

d

d

ct d

d

d

d

ci

c; ci

c;

= q @ U

ceeeEaeeeEeEEeecca€EEeeeEeEEeeeEE

d

EEEEEEEEEEEEEEEEgEEEEEEEECqEEqqEE

6 q

za = seECqeeeeeeEeeeeeeEeaeEE - - =- -=- = - -=- = - -F = E

F N o + 4 @ F @ q o F N o + h @ r € - o F S R S F R F R R B E S g 6 @ @ F l 6 @ @ @ @ @ @ @ ; e h h 6 h 6 6 h h 4 o

z.

EEEEEFFEEEEFFFFRRRFEEEgFEEEEgEEEEE 4, z z z z. z. z z z z z. z. z. z z z z. z. 1 1 1 z z z = 1 1 1 1 1 2 z z z >

=

=

-

-

-

-

r

-

-

5

-

-

-

-

-

-

-

-

-

-

?

-

-t

1

a

-t

-.

-

-

-

-

-

-

z

a

E x x * * R f i N S t t N N S S N F t F K K K R K R K R R F R K K R K

q

*

E

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

+

*

*

*

*

*

*

x

? - s e f i s s i i * B $ = R i E E s B 6 R $D E EB s D R s

L

*

J

F

L

*

*

F

O

F

6

+

O

N

N

F

-

g g a s a E gs s s $ s s s s g B E K NRR3 6 s s

d r d

-

=

N

==

z.

E EEEEgEEEEEEEEEEEEEFFFEEEE €s L E

-

o
o +

q!ni!!

= o

EEESgEEEEEEEEEEEE=EFFEEEg ? s --Ki F a

z z 2 z. z z z. z z 1 z z z- = = = = = = = z z z z z: = = = = = = = = = = = J -

- a i

- - - = - - - - i -

-

- -

-

-

- -

- -

- -

* t N * t s * * N * S F S t * N S S * S N t S N S *

*

*

*

*

*

*

*

*

*

*

*

*

*

*

+

*

*

*

*

*

*

x

z

v v

-

U < E

5u?6 o 6 r

-*RNEFBSEus'pRxf HexnsE neEfr F o o

- \ o +

h @ F @ o o

= S p =

= g =

P P R

i s i

= F x

d 5 i i6 ;N \ NE

> =

*

@ I

@ts u +

24

F N R t R

z.

-

E

nRRFEE$$EHR;$€sFEil

F a

@

J

F 6 0 @ O F € O

F

< 6 @ F A < < q 9

=

t

*

O

u

o

u

F

U ^

E=

=

EFEEEEEFEEEEEFF=FRRFEEE=EE = -^ 4 z,zz.z.z=1=zz==z.=z==4z zz v \r

q \ m s

R(U

6

q n

F

Eo

Y - Y - i O \ F - * i cnr)'am\o

Y * N

- v l O O ne{

Vcn

cn v

\ori 6\O i

OCIOl €S re.lcnt N ca

F-O

\o

N6 hr 61 6l

e..l\o N 0, sincenegativeprobabilitieshaveno meaning. Also, the function has the property that ()61\\

1

f(x) dx:

which again is the requirement that the probabilities of all outcomessum to 1. Furthermore,the probability that x will fall betweenthe limits a and b is written

p(a-x-b):lu,r*,o*

(26.16)

Note that the probability that x takeson a particular value,saya, is zero;that is, fo

J

ft-l dx : o

(26.17)

which emphasizesthat nnite proUalilities are definedonly as areasunder the PDF betweenflnite limits. The CDF can now be deflnedin terms of the PDF as P(-*

< X < x) : P(X= x) : F(x) :

t'

I f(u) du

(26.18)

J_*

whereu is usedas a dummy variableto avoidconfusionwith the limit of integration. The area under the CDF has no meaning,only the ordinates,or the difference in ordinates.For example,P(*r3 X = x2),which is equivalentto Eq. 26.16, can be evaluated as.F(x) - F(xt). that cannotbe summarizedin integral form, there are For discrete.distributions analogousarithmetic statementscorrespondingto the propertiesgiven in Eqs. 26.15, 26.16, and26.18.In particular, the distributionof sampleddatetaken from a continuousdistributionis a specialcaseof discretedistributionsand canbe givenin the form Thus of arithmeticsummations.s

),f(',):

t

(26.re)

x1= b

P(a-X o

a

l

,

'\-..7

d

+

.i l \ F t.x

x -:

Nl'\a

t v

bl tl H

T

b

, tl h

x el

r-

a f f tx o o

s

1

+

d

\,

Ir o

o (!

tr

s \/" x rir

: "

" vr x

'

v

I

o \/t :'

x

\/r "'

8 vr '' R \,r vt

-r

IL

tr l dl

o

fr

o

z

r

;

Vl

.,

/\r

o o

a o z o

E U' 0)

o o lt

o tu J o

s

o E E

(!

I

IL

F

a? (o 6|

UJ @

E€E E ;F = .E

d

E E6 a. f

'.

-E

r

E

E "

= i ' 5E

E 3 E !

E6iE

l

s

F[ E S EZ B i ' oO eF E

l

l

-{.

a

-\? .

8

+

+ -

^r

8 vl ^

z o ut a U)

B l

+ x J CO-

N I N t i - l xrnl -il

f

sr

{ o J o

dlJa , +'

o ,? ,9, k

vl x

"'

'

'

\/l

M

u ;

d

6

I

l\t

^

26.7 CONTINUOUSPROBABILIW DISTRIBUTIONFUNCTIONS

687

randomnumberssoimportant in simulationstudies.The wholebody of materialin the area of reliability and estimatingdependson derived distributionslike Student'sl, chi-squared,and the F distribution. The explanationsthat follow concernthe more The readeris referred common distributionsappliedin fitting hydrologicsequences. to standardtexts for more detailedtreatment.3-6

NormalDistribution frequencyfunction, alsoknown The normal distributionis a symmetrical,be,ll-shaped as the Gaussiandistributionor the natural law of errors. It describesmany processes that are subjectto random and independentvariations.The whole basisfor a large body of statisticsinvolving testing and quality control is the normal distribution. Although it often does not perfectly fit sequencesof hydrologic data, it has wide application, for example, in dealing with transformed data that do follow the normal distributionand in estimatingsamplereliability by virture of the central limit theorem. The normal distribution has two parameters,the mean p, and the standard deviationa, for which 7 and s, derivedfrom sampledata, aresubstituted.By a simple transformation,the distribution can be written as a single-parameterfunction only. Definingz : \x - t-i/o, dx : o dz,the PDF becomes

(26.3s) and the CDF becomes F\z) :

-r*l-

e-"2/2du

(26.36)

The variablez is called the standardnormal variate; it is normally distributed with zero mean and unit standard deviation. Tables of areas under the standard normal curve, as given in Appendix B, TableB.1, serveall normal distributionsafter standardizationof the variables.Given a cumulativeprobability,the deviatez is found in the table of areasand x is found from the inversetransform: x:p+za

or x:7lzs

(26.37)

EXAMPLE 26.5 Assumethat the Richmond,Virginia, annualrainfall in Table 26.2 follows a noimal distribution.Usethe standardnormal transformationto find the rain depththat would havea recurrqnceinterval of 100 years. Solution. From example26.4,the meanis 41.5 in. and the standarddeviation is 6.7 in. This gives x:

4L5 + z(6.7)

Equation 26.18 showsthat the areaunder the PDF to the right of z is the exceedence probabilprobability of the event.For the 100-yrevent,F;q.26.7 givesthe exceedence Table 8.1 in From the figure accompanying tlI00:0.01. ity Pk): llT,:

688

CHAPTER26

PROBABILIry AND STATISTICS

Appendix B, F(z) : 0.5 - p(z) : 0.49, and z : 2.326 by interpolating the table' The expected100-yr rain depth is therefore x : 4t.5 + O.32O X 6.7 :57.1 in: The 100-yr event for a normal distribution is 2.326 standarddeviationsabove the mean. rt

Log-NormalDistribution partly dueto the influence Many hydrologicvariablesexhibit a markedright skewness, other lower limit, and or some zero, greater than values phenomena having of natural frequencieswill cases, In such range. the upper in theoreiically, beingunconstrained, distribunormal a follow may logarithms but their distribution, normal not follow the y : substituting from comes log-normal for the 26.3 in Table PDF shown tion.7The ln x in the normal. With p, andcy as the mean and standarddeviation,respectively, the following relations have been found to hold betweenthe characteristicsof the untransformid variatex and the transformedvariatey:r'7

p:exp(p'y+412) oz:p,2lexp(d)_11 a : lexp(3fi) - 3 exp(fi) + 2lC3 C,:lexp(dr) - r1t'' C,:3C" * Cl

(26.38) (26.3e) (26.40) (26.4r) (26.42)

Also p, : lfl M, whereM is the median value and the geometricmean of the x's. The log-normal is especiallyusefulbecausethe,transformationopensthe extensive body oI theoreticaland applied usesof the normal distribution. Since both the normal and log-normal are two-parameterdistributions,it is necessaryonly to compute the mean-andvarianceof the untransformedvariatex and solveEqs. 26.38 and 26.39 simultaneously.Information on three-parameteror truncatedlog-normal distributionscan be found in the literature.r'7

Gamma(and PearsonTYPelll)

-

The gammadistributionhaswide applicationin mathematicalstatisticsandhas been usedincreasinglyin hydrologicstudiesnow that computingfacilities make_iteasyto evaluate the gimma functioi insteadof relying on the painstakingmethod of using tablesof the incompletegammafunction that lead to the CDF, P(X < x). In greater useis a specialcaseof gamma: tbePearsonType/1L This distributionhasbeenwidely adoptedas the standardmethodfor flood frequencyanalysisin a form known as the log-pearson /11 in which the transform y :1og x is used to reduce skewness.8-r0 Aithough all three momentsare requiredto fit the distribution,it is extremelyflexible in that a zeroskewwill reducethe log-PearsonIII distribution to a log-normal and the pearsonType III to a normal. Tablesof the cumulativefunction are availableand A very important property of gamm-avariates will be explainedin a later section.lo'11 aswell asnormal variates(includingtransformednormals)is that the sumof two such variablesretains the samedistribution. This feature is important in generatingsyn-thefie hy-drologic sequences.l''''

26J

CONTINUOUSPROBABILITYDISTRIBUTIONFUNCTIONS

689

Gumbel'sExtremalDistribution The theory of extreme valuesconsidersthe distribution of -the largest(qr smallest) observationsoccurringin eachgroup of repeatedsamples.The distribution of the nt extreme values taken from n, samples,with each samplehaving n2 observations, dependson the distribution of the nrn, total observations.Gumbel was the first to employ extremevalue theory for analysisof flood frequencies.laChow has demonstratedthat the Gumbel distribution is essentiallya log-normal with constantskewness.tsThe CDF of the densityfunction given in Table 26.3 is

P(X - x) : F(x) : exp{-expl-o(, - u)l}

(26.43)

a convenientform to evaluatethe function. Parametersa andu are given asfunctions of the meanand standarddeviationin Table26.3.Tablesof the doubleexponentialare usually in terms of the reducedvariate,y - a(x - u).tuGumbel also has proposed anotherextremevaluedistributionthat appearsto fit instantaneous(minimumannual) droughtflows.17'18

CDFsin Hydrology Normal and Pearsondistributionscan often be usedto describehydrologicvariables if the variableis the sum or mean of severalother random variables.The sum of a numberof independentrandom variablesis approximatelynormally distributed.For example,the annualrainfall is the sumof the daily rain totals,eachof which is viewed as a random variable. Other examplesinclude annual lake evaporation, annual pumpagefrom a well, annual flow in a stream,and mean monthly temperature. The log-normal CDF hasbeen successfullyusedin approximatingthe distribution of variablesthat are the product of powersof many other randomvariables.The logarithm of the variableis approximatelynormally distributedbecausethe logarithm of productsis a sum of transformedvariables. Examplesof variablesthat havebeenknown to follow a log-normal distribution include:

1. Annual seriesof peak flow rates. 2. Daily precipitationdepthsand stremflowvolumes(alsomonthly, seasonal, and annual).

3. Daily peak dischargerates. 4. Annual precipitation and runoff (primarily in the westernUnited States). Earthquakemagnitudgs, 6. Intervalsbetweenearthquakes. 7. Yield stressin steel. 8. Sediment sizes in streamswhere fracturing and breakageof larger into smaller sizesis involved. 5.

The PearsonType III (a form of gamma) has been applied to a number of variablessuch as precipitation depths in the easternUnited Statesand cumulative watershedrunoff at any point in time during a given storm event.The transformed log-PearsonType III is most usedto approximatethe CDF for annualflood peaks.If the skew coefficient C" of the variable is zero, the CDF reverts to a log-normal.

690

ANDSTATISTICS 26 PROBABILIry CHAPTER It hasalso.beenusedwith monthly precipitationdepthandyield strengthsof concrete members. A useful CDF for values of annual extremeis the Gurnbel or extreme value probability of distribution.The meanof the distributionhas a theoreticalexceedance streamshave peaks in natural years. Flood 2.33 T of interval 0.43 and a recurrence 2.33-year with means including disffibution, to this conformance exhibited strong variptraight-line fit for Gumbel a produces paper that Graph recurrenceintervals. is A sample extremes. annual of graphical tests for useful and ables is a available peak rates, discharge annual peak to applie! has been 27 The CDF shownin Fig. .2. wind velocities,drought magnitudesand intervals, maximum rainfall intensitiesof given durations,and other hydrologicextremesthat are independentevents.

AND CORRELATION LINEARREGRESSION 26.8 BIVARIATE Correlation and regressionproceduresare widely used in hydrology and other sciences.teThe premiseof the methodsis that one variableis often conditionedby the value of another,or of severalothers,or the distribution of one may be conditioned by the value of another. Just as there are probability density functions (PDFs) for evaluatingthemarginal probability of a variable(seeSection26.2), so also are there PDFsforlhe conditional probabilities (also describedin Section 26.2) of variables. The conceptis illustrated in Fig. 26.7. For two variables,the bivariatedensityfunction,/(y li,), ptottedin the vertical on the frgure,changesfor eachvalueof x' The one shownappliesonly to variationsin y whenx : xr. Different distributionsmight occur for other valuesof x. A measureof the degreeof linear correlationbetweentwo variablesx and y is thelinear correlation cofficient, P*,y. Avalue of p',, : 0.0 indicatesa lack of linear

Pylr regressionline

Figure 26.7 Bivariate regressionwith conditional probability function.

ANDCORRELATION691 REGRESSION LINEAR 26.8 BIVARIATE correlation andp,,, : + 1.0 meansperfect correlation.The correlationcoefficientis found from cr.v

cov(x, y)

u - , , :

(26.44)

-

(l*ay

aroy

where o, ando, are the variancesof eachvariable,respectively,(seeEq. 26.30), and cov(x, y) is the covariancesharedby the two variables,definedas

y) : c,.,: cov(.r,

f _f _Q'-

p)(y - p,)f(x,v)dvdx

(26'4s)

The samplecorrelationcoefficient,r : s,.rfs*s* is usedto estimatep',r. The sample covarianceis found from the squareroot of

s?,,:

(26.46)

The regressionline shown on Fig. 26.7 is derived to passthrough the mean valuesof the distributions,so that for any given value of x, the mean value of y I x (read"y givenx ") can be estimatedby the regressionline. The standarderror of the estimateof y I x is depictedby the line drawnthrough the conditional distributionsat a distanceof one standarddeviationfrom the mean.If the conditionaldistributionsat all x-valuesare normal,it canbe shownthat the meanvalue,&y1,,of the conditional distribution is related to the meansof .r and H or a & , 1 , : l t ', * ( p 2,( * and the variance is

4t.

: #['" -

-

tr,)

, @- p)'1

d

l

t26.47)

(26.48)

where

a?: 40 - p')

(26.4e)

which is the variance of the residualsof the regression.Just as the mean of the distribution requiressubstitutionof the given value of x into Eq. 26.47, so also does thevariance,Eq.26.48. Whenthe valueof x in Fig. 26.7is setiequalto A the standard error of the meanis O-t=:

ae -----7

(26.s0)

VN

Equation 25.47 is linear and expressesthe linear dependencebetweeny and .r as slrownin Fig. 26J.The meanvalue of y can be computedfor fixed valuesof x' Also, if the correlationbetweenthem is significant,one can predict the valuesof y with less error than the marginal distribution of y alone.In fact, from Eq. 26.49, the fraction of the original varianceexplainedor accountedby the regressionis

o":t-*

(26.5r)

692

CHAPTER26


It can be seenalso from Eq. (26.47) that the slopeof the regressionline is cy

PA:

tl,yl,

-

lLy

(26.s2)

r - rr^

or, ifx andy are standardized,then p itself is the slope,where - p,r)/a, }rnt. (26.s3) p = (x - t*)/o, The bivariatecasecan be expandedto coverhigher-order,multivariatedistributions.

EQUATIONS 26.9 FITTINGREGRESSION

ing value of x. The line to be fitted is

(26.s4)

-l Bx; Y,: a

N 6 I

Year

, ? ; xbo

A1

J . F

/1 !

Q

43 44

3[i e

o

46 47 48 49 50 51 52

F r ;

h

d

B U r40 120 100 80 Lowest annualflow for 1 day (cfs) JacksonRiver at FallingSprings,Virginia, 1941-1952 60

Mean = Standard deviation =

Jackson River

Cowpasture River

61 92 65 72 82 67 74 t 18 t24 r08 65 88-

58 81 70 63 68 58 74 105 134 108 93 85

84.7

83.1

21.7

23.2

: 0.86. : Figure 26.8 Cross-correlationof low floWs.Regressionline: Iz 4.94 + 0.923X; r

EQUATIONS 693 26.9 FITTINGREGRESSION The best estimates of o and B are sought. Thus to minimize

) 0, - f), : ) ly, (o + px,)1,-

(26.ss)

wherey, are the observedvalues and!,ate the estimatedvaluesfrom Eq. 26.54,take partial derivativesas follows:

- (q+ n')r} *{> b, - (o* B';l'} tv, #{>

(26.s6) (26.s7)

After carrying out the differentiationsand summations,two equationsresultin a and B, callednormal equations.

o

(26.s8)

2*,y,-")xt-F2*?:o

(26.se)

)y,

- n d- B ) " , :

SolvingEqs. 26,58 and 26.59 simultaneouslyyields

2 v , F 2 ^' i : y - B T

e:--

n

(26.60)

n

P: =7 - (>;y[

(26.6r)

Recall the slopeis p(arf o), or as estimatedfrom sampledata B:

rt;

(26.62)

Also, the unexplainedvariancein the regressionequationis

4:4Q-p')

(26.63)

and is the squareroot of which is the standarddeviationof residuals(seeFig' 26'8) cailed the standard error of estimate.Thesecan be estimatedfrom

*-: 4 ns -l (z r - r , )

(26.64)

s2":

(26.6s)

2(y,-il'

wherey, and i, are as definedpreviously(seeEq' 26'55)' ivtanynyarotogicvariablesare linearly related,and after estimatingthe regresrangeof sion coeffici"ntr, p."di"tion of y can be madefor any value of x within the be should but performed often is observedx values.Extrapolationoutsidethe range

7

694

CHAPTER26


done wfth caution.Equation 26.48 showsthat the variancein the estimateof y for a givenx valuebecomeslargewhenx is severalstandarddeviationsaboveor belowthe mean. EXAMPLE 26.6 The lowestannualflows for a l2-yr period on the Jacksonand CowpastureRivers are tabulatedin Fig. 26.8. The stationsare upstr€amof the confluenceof the two rivers that form the JamesRiver. Find the regressionequationand the correlationbetween low flows. Solution

t

: The basicstatisticsare2 x : 1016;) y : 997;2 x2 : 9 1 , 2 1 6 ; 2 l " 88,777;and2 xy : 89,209. For the two-variableregressiona and B are found from Eqs.26;60and 26.61,. _ [ ( 8 9 , 2 0 e ) ( l 0 t 6 x e e 7 ) / ( 1 2 ) ] :o Q o ? (9r,216)- (tor6)' 102)

o:

ee7 (0.e23x1016) : 4'9r i

The regressionis y : 4.91 + 0.923x. 3. The correlation coefficientfrom Eq.26.62 is

(0.e23)(2r.7) : 23.2

0:86

4. From F;q.26.64the standarderror of estimate,s,, is 11.7,which is plotted line in Fig. 26.8. rl as limits aroundthe regression

Coefficientof Determinationfor the Regression A regressionequationreplaces(and extends)the data used in its development.Becauseit cannot reproduceall the basedata, the processresults in the loss of some information. This not only includeslossof information aboutparticular pairs of data, but alsoaboutthe variability of the data.The variancesf is a statisticalmeasureof the variability of the measuredvaluesof y. The greaterthe valueof sl, the wider the spread of points aroundthe mean.The percentageof information aboutthe variancein y that is retained,or explainedby, the regrdssionequationis called the cofficient of determination, Cr. To determineits value,the residualsor departures(differencesbetween actual and estimatedy values)haveknown variance4, which representsthe unaccountedvariance in the regressionequation.The explainedvariance would be the difference, 4 - o2, and the percentageretained (coefficient of determination)is

(26.66)

695

26.9 FITTINGREGRESSIONEQUATIONS

Comparisonwith Eq. 26.49 revealsthat Co= Pz

,

(26.67)

Thus the squareofthe correlationcoefficientp is the percentageof d, explained by the regression.For any sample of data the coefficient of determinationr2 is estimatedas sl,rlslsl . A large r2 indicatesa goodfit of the regressionequationto the databecausethe equationaccountsfor or is able to explaina large percentageof the variation in the data. EXAMPLE 26.7

Determinethe coefficientof determinationfor the regressionin Example26.6. Solution, From Eq. 26.67,the coefficientof determination,r2, is 0.7396. "accountsfor" about 74 Thus, the regressionequationadequatelyexplainsor pefcent of the original information abouty containedin the raw data. Twentysix percentof the information is lost. I r For examThe bivariateexamplecan be extendedto multiple linear r'bgressions. ple, the linear model in three variables,with y the dependentvariable andx1 andx2 the independentvariables,has the form y:a*F$t*Fzxz

(26.68)

y : an-r Br) t' * FrZ *,

(26.6e)

The normal equationsare )

xtt FrZ*?+ Fr2*,*,

(26.70)

2 yr r : * ) xzI Fr 2 *r *, + F"2 *7

(26.7r)

)y"':

")

The squareof the standarderror of estimateis

s7:

2(y,-r,)'

(26.72)

where y, are the observedvaluesand y-,are predictedby Eq. (26.68). The multiple correlation coefficient is

+) ^2\ | /2

n : ( t -

s;/

(26.73)

in Hydrology LinearTransformations Strongnonlinearbivariateand multivariatecorrelationsare also commonin hydrology, ind various mathematical models have been used to describethe relations. Piiabolic, exponential, hyperbolic, power, and other forms have provided better

696

CHAPTER26

PROBABILITYAND STATISTICS

graphicalfits than straight lines. Becauseof difficulties in the derivation of normal equationsusing least squaresfor thesemodels,many can be transformedto linear forms. The most familiar transformationis a linearizationof mtiltiplicative nonlinear relationsby using logarithms.For example,the equation Y :

(26.74)

g,xft1$z

becomes linear when logarithms are taken, or

log y : log * + Bllo$ x1 -f B"Iog x2

(26.7s)

The log transformationprocedureresultsin a linear form when the logarithms are substitutedin Eqs. 26.60 and 26.61.For of one or both setsof measurements example,if a bivariateparabolicform I : qXb is suggestedby the data, logarithms allow use of the linear form log Y : log a -t b log X. The normal equationscan be usedby redefiningy : log Y,x : logX, e : log a, andF : b,thereby transforming the equationto y : a * Bx. The regressioncannow be performedon the logarithms, valuesof a and B determined,and the estimateof a is found by taking the antilog of a. This transformationis possiblefor severalother nonlinearmodels,someof which are shown in Table 26.4. The variablesx and y must be nonnegative,with values preferably greaterthan 1.0 to avoid problemswith the log transformation. OF NONLINEARFORMS TABLE 26,4 LINEARTRANSFORMATIONS

Equation Y=A+BX Y = BeAx Y:AXB Y:ABx

Abscissa X log X X

Ordinate I

log Y log Y log Y

Eouationin linearform

lY): A + B[x] tbc rl : locB + A(1oge)[x] tbc rl : bc A + B[tocX] lloCrl : loCA + (loCB)[X]

Note.'Variables in brackets are the regressionvariates.

EXAMPLE 26.8 In the following exhibit (Table 26.5) preparedby Beard,20the regionalcorrelationis soughtof the standarddeviationof flow logarithmswith the logarithmsof the drainage areasize'andthe numberof rainy daysper year;X, is setequalto ( 1 t log s) to avoid negativevalues.Find the regressionequationand the multiple correlationcoefflcient. Solution 1. From Eqs.26.69,26.70, and26.7l, the parametersare -0.49 a : 1.34; Fr : -0.013; Fz: and the regressionequationis Xt : 1.34- 0.0I3X2- 0.49X3 log s : 0.34 - 0.013log(DA) - 0.49 log(days) or 2. The multiple correlation coefficient from Eqs' 26.72 and 26.73 is R : 0.56. ll

697

APPLICATIONS AND CORRELATION 26.10 REGRESSION TABLE 26.5 LOGARITHMICDATA FOR 50 GAUGINGSTATIONS Xr:1 + logs Station number (1) I

2 3 4 5 o

7 8 q

t0 11 t2 I J

t4 15 16 17 18 19 20 21 22 LJ

24 25 26 27 28 29 30 3l 32

Xz: logDA

X2 (2)

x3 (3)

x1 (4)

1.61 2.89 4.38 3.20 3.92 1.61 3.2r 3.65

2.11 2.12 2.ll 2.04 2.07 2.04 2.09 1.99

0.29 0.18 o.l7 0.44 0.38 0.3'7 0.30 0.35 0.16 0.11 0.32 0.34 0.25 0.43 0.2'l 0.25 0.52 0.18 0.39 0.40 0.25 0.23 0.54 0.51 0.45 0.63 0.45 0.59 0.46 0.32 0.96 0.12

3.23 z.rs

2.08 4.33 1.60 2.09 2.00 2.82 2.00 2.40 2.09 3.69 2.18 2.19 2.09 2.17 |.91 4.48 1.9s 4.95 2.21. r.97 2.08 3.4r 4.82 l 88 r.93 r.78 L.74 4.39 3.23 2.01 3.58 2.04 1.64 1.78 1 . 76 4.58 3.26 1.93 1.81 4.29 1.23 1.89 1.48 3.44 |.97 2.lt

Per Year & : log numberof rainY-daYs

Station numDer (5) 33 34 35 36 37 38 39 40 4I 42 43 44 46 4'7 48 49 50

>X x 2 XX, 2 X2 X2/n

x2 (6)

xs

1.94

1.87 t.36 1.81 1.58 1.48 1.89

z.tJ

3.63 1.91 2.26 2.97 0.70 0.30 3.38 2.87 2.42 4.53 3.04 4.13 |.49 5.37 1.36 2.31

r.32 1.54 1.62 2.03 2.26 1.93 1.78 2.00 2.Or 1.95 2.tl 2.23

x1 (8) 0.20 0.58 0.64 0.37 o.27 0.54 0.63 0.78 0.46 0.44 0.24 -0.03 0.30 0.17 0.14 0.10 0.27 0.18

r4'7.55 2.951 503.7779 435.4200 68.3579

96.24 1.925 285.5627 284.0042 1.5585

17.89 0.358 51.1527 52.7934 -r.640' 7

1.5585

r8'7.59r2 185.2428 2.3484

33.2598 34.4347 -r.r749

2 XX, 2 X2 X,ln

2X2X'ln 2 xx1

(7\

-1.6407

Note: x = X - X. (AfterBeard.2o)

2 6 ' I o R E GR E S S |o N A N D C o RRELAT|oNAPPL|CAT|oNS

8.1635 6.4010 L7625

698

26 PROBABILIry ANDSTATISTICS CHAPTER desiredJtatisticalparameteras dependentvariable,and the appropriatephysicaland The proceclimatic variableswithin the basinor regionas the independenlvariables. dures are signiflcantly better than using relatively short historical sequencesand point-frequencyanalysis.Not only doesthe methodreducethe inherently large sampling errors but it furnishesa meansto estimateparametersat ungaugedlocations. There are limitations to the techniquesof Section 26.9. First, the analyst assumesthe form of the model that can expressonly linear, or logarithmically linear, dependence.Second,the independentvariablesto be includedin the regressionanalysis are selected.And, third, the theory assumesthat the independentvariablesare indeedindependentand are observedor determinedwithout error. Advancedstatistical methodsthat are beyondthe scopeof this text offer meansto overcomesomeof theselimitations but in practiceit may be impossibleto satisfy them. Therefore,care must be exercisedin selectingthe model and in interpretingresults. Accidental or casualcorrelationmay existbetweenvariablesthat are not functionally correlated.For this reason,correlationshouldbe determinedbetweenhydrologic variablesonly when a physicalrelation can be presumed.Becauseof the natural dependencebetween many factors treated as independentvariables in hydrologic studies,the correlationbetweenthe dependentvariableand eachof the independent variablesis different from the relative effect of the sameindependentvariableswhen analyzedtogetherin a multivariatemodel. One way to guard againstthis effect is by screeningthe variablesinitially by graphical methods.Another is to examine the results of the final regressionequationto determinephysicalrelevance. Alternatively,regressiontechniquesthemselvesaid in screeningsignificantvariables.When electroniccomputationis available,a procedurecanbe followedin which successive independentvariableSare addedto the multiple regressionmodel, and the relative effect of eachis judged by the increasein the multiple correlationcoefficient. Although statisticaltestscan be employedto judge significance,it is useful otherwise to specify that any variable remain in the regressionequation if it contributesor explains,say,1 or 5 percentofthe total variance,or ofR2. A frequentlyusedrqethod is to computethe partial correlation cofficients for each variable.This statistic representsthe relative decreasein the varianceremaining( 1 - R') by the addition of the variablein question.If the varianceremainingwith the variableincludedin the regression is (I - Rz) : pz and the variance remaining after removal is (l - R'') : D'', then the partial regression correlation coefficient is

\D'' - D')lD''.

Most PC spreadsheetsoftware packageshave statistical routines for all the analysesdescribedhere and many more. Most are extremelyflexible,requiringminimal instructions-andinput data other than raw data. Specialmanipulationscan effect an interchangeol dependentand independentvariables,bring one variableat a time into the regression equation, rearrange the independent variables in order of significance,and perform various statisticaltests.

ExtendingHydrologicRecords Regressiontechniquesfrequently can be used to extend short records if significant correlation existsbetweenthe station of short record and a nearby station with a - lorigerreeord.Iq Example26.6,if the JacksonRiver recordswerecompletefromI94l

PROBLEMS 699 to datebut the Cowpasturerecordswere incompleteafter 1952,the cross-correlation could be usedto estimatethe missingyearsby solvingthe regressionequationfor I from 1953 on usingthe X flows as observed.The reliability of suchmethodsdepends on the correlationcoefficientand the length ofthe concurrentrecords.Ifthe concurrent record is too short or the correlation weak, the standarderror of the parameter to be estimatedcan be increasedand nothing is gained.The limiting value of crosscorrelation for estimatingmeansis approximatelyp : l/\/ n, where n is the length of the concurrentrecord.21Thus any correlation above0.3 would improve the Cowpasturerecords.Estirnatesof other parameterswith larger standarderrors require highercross-correlationfor significantimprovement.Extendingor filling in deficient recordsoften is necessaryfor regional studiesin which every record usedshould be adjustedto the samelength.

HydrologicVariables Regionalized Predicting.

'

Cruff and Rantzz2studiedvariousmethodsof regional flood analysisand found the multiple regressiontechniquea better predictor than either the index-floodmethod (Chapter27) or the fitting of theoreticalfrequencydistributionsto individual historical records.They flrst usedregressiontechniquesto extendall recordsto a common base length. Next they extrapolatedby various methods to estimate the 50- and 100-yearflood events and with multiple correlation examined several dependent variables including the drainage areaA, the basin-shapefactor (the ratio of the diameterof a circle of sizeA to the length of the basin measuredparallel to the main channel)Sa,channelslopeS, the annualprecipitationP, and others.They found only A and S, to be significant, which resulted in prediction equations of the form Q, : cAS!,. These equationswere superior to those of the other techniques.The multiple correlation coefficientwas as high as 0.954.It is interestingthat regression techniqueswereemployedin still a third way,that is, to estimateregionalvaluesof the mean and standarddeviation after adjustingthe record length. Example26.8 illustrated the applicationof regressionanalysisto regionalizethe standarddeviationof annual maximum flow logarithms as a function of the drainage area size and the number of rainy dayseachyear.

r summary Statisticsis a diversesubject,and the treatmentin this chapterhasbeennothing more than an introduction. Seriousstudentsand practitionersmust return againand again to the theory in standardworks.23They will find that evaluatingnew developments of statisand techniquesmust claim a large shareof their time. Only certain aspects, tical hydrologyhave been presented,principally the common distributionsand the methodsfor analyzingfrequency of eventsobservedat a single point. In the next chapterthis information is extendedto common applicationsin hydrology.

PROBLEMS 26.1. The probabilitiesof eventsE1 andE2 arc each.3. What is the probability that E1 or E2 will occur when (a) the eventsare independentbut not mutually exclusive,and - (b) whenthe probabilityof Et, given E2is .l?

700

CHAPTER26

ANDSTATISTICS PROBABILITY

26.2. EventsA and B are independenteventshaving marginal probabilities of.4 and .5, respectively.Determine for a single trial (a) the probability that both A and B will occur simultaneously,and (b) the probability that neither occurs. 26.3. The conditional probability, P(E, I E,r),of a power failure (given that a flood occurs) is .9, and the conditionalprobability,P(Ez I E), of a flood (given that a powerfailure occurs)is .2. If the joint probability, P (\ andE), of a power failure and a flood is .1, determinethe marginal probabilities,P(E) and P(E). 26.4. Describetwo random eventsthat are (a) mutually exclusive,(b) dependent,(c) both mutually exclusiveand dependent,and (d)"neithermutually exclusivenor dependent. 26.5. A temporar;1cofferdamis to be built to protect the 5-yearconstructionactivity for a major crossvalley dam. If the cofferdam is designedto withstand the 20-yearflood, what is the probability that the structurewill be overtopped(a) in the flrst year, (b) in the third year exactly,(c) at leastonce in the 5-yearconstrucfionperiod, and (d) not at all during the 5-yearperiod? 26.6. A 33-yearrecord of peak annualflow rateswas subjectedto a frequencyanalysis.The median value is defined as the midvalue in the table of rank-ordered magnitudes. Estimatethe following probabilities. , a. The probability that the annualpeak will equalor exceedthe medianvaluein any singleyear. b. The averageretlrrn period of the median value. c. The probability that the annual peak in 1993 will equal or exceedthe median value. d. The probability that the peak flow rate next year will be less than the median value. yearswill e. The probability that the peak flow rate in all of the next 10 successive value. be lessthan the median f. The probability that the peak flow rate will equal or exceedthe median value at years. leastoncein l0 successive g. The probability that the peak flow rates in both of two consecutiveyears will equal or exceedthe median value. h. The probability that, for a2-yearperiod, the peak flow rate will equal or exceed the median value in the secondyear but not in the first' 26.7. What return period must an engineeruse in his or her designof a bridge openingif there is to be only a 50 percent risk that flooding will occur at least once in two successiveyears?Repeatfor a risk of 100 percent. 26.8. A temporary flood wall has been constructedto protect several homes in the floodplain. The wall was built to withstand any dischargeup to the 20-year flood magnitude.The,wall will be removed at the end of the 3-year period after all the homeshavebeen relocated.Determinethe probabilities of the following events: a. The wall will be overtoppedin any year. b. The wall will not be overtoppedduring the relocation operation. c. The wall will be overtoppedat leastonce before all the homesare relocated. d. The wall will be overtoppedexactly once before all the homesare relocated. e. The wall will be adequatefor the flrst 2 years and then overtoppedin the third year. 26.9. Waveheightsand their respectivereturn periods(shownon the next page)are known for a 40-mi long reservoir.Ownersof a downstreamcampsitewill accepta 25 percent risk that a proiective wall will be overtoppedby wavesat least once in a 2}-yeat period. Determinethe minimum height of the protective wall.

PROBLEMS 701 Waveheight (ft)

Returnperiod (years)

10.0 8.5 7.4 5.0 3.5

100 50 30 10 5

26.10. Assumethat the channel capacityof 12,000cfs near a private home was equaledor exceededin 3 of the past 60 years.Find the following: a. The frequencyof the 12,000-cfsvalue. b. The probability that the home will be floodednext year. c. The return period of the 12,000-cfsvalue. d. The probability that the home will not be floodednext year. e. The probability of two consecutive,safeyears. f. The probability of a flood at leastonce in the next 20 years. g. The probability of a flood in the second,but not the first, of two consecutiveyears. h. The 20-yearflood risk. 26,11, The distribution of mean annual rainfall at 35 stations in the JamesRiver Basin, Virginia, is given in the following summary:

Interval(2-in. groupings) Numberof observations

36 or 37 in.

38 or 39 in. 4

40 or 41 in. j

42 or 43 in.

Z

Interval (2-in. groupings) Numberofobservations

44 or 45 in. 5

46 or 47 in. 4

48 or 49 in. 2

50 or 51 in. 2

Computethe relative frequencies(seeChapter27) andplot the frequencydistribution andthe cumulativedistribution.Estimatethe probability that the meanannualrainfall (a) will exceed40 in., (b) will exceed50 in., and (c) will be betweenthesevalues. 26.12. Write a simpleprogram to READin N data points and compute the mean, standard deviation,and skewnesscoefficient. 26.13. A normally distributedrandom variablehas a mean of 4.0 and a standarddeviation of 2.0. Determinethe value of f@

I

-

dx I f(x)

"8

26.14. For a standardnormal densitv ' function. use Table B.1 to determinethe value of fr+'o

I

fG) dx

J*-ro 26.15. A normal variableX has a meanof 5.0 and a standarddeviationof 1.0.Determinethe value of X that has a cumulativeprobability of 0.330. 26.16. If the mode of a PDF is considerablylarger than the median, would the skew most likely be positive or negative?

l

702

ANDSTATISTICS 26 PROBABILIW CHAPTER 26.17. tomplete the following mathematicalstatementsabout the properties of a PDF by insertingin the boxeson the left the correct item numberfrom the right. Assumethat X is a seriesof annual occuffencesfrom a normal distributibn. I r.Zerc a. I f(x) dx: J t " b.

d.

: f'_ro,dx r

2.Unity

dx: '34 l-.o 'o'

3. Valuewith 5 percentchanceofexceedanceeachyear

f,o"

dx:r

dx: .5 f-to, f(x) dx : Z

I rt, dx: .02

4. 0.68 5. Valueexpectedevery 50 r"urc on the average 6. P(X < mr) 'X'*r) 7.P(m1 8 . P ( m 1- X = m z )

l.

9. Median

10. Standarddeviation 26.18, The mean monthly temperaturefor Septemberat a weather station is found to be normally distributed.The mean is 65.5" F, the varianceis 39.3'F2, and the record is completefor 63 years.With the aid of TableB.1, find (a) the midrangewithin which two thirds of all future mean monthly valuesare expectedto fall, (b) the midrange within which 95 percentof all future valuesare expected,(c) the limit below which 80 percentof all future valuesare expected,and (d) the valuesthat are expectedto be exceededwith a frequencyof oncein l0 yearsand oncein 100years.Verify the results by plotting the cumulativedistribution on normal probability paper. 26.L9. The total annualrunoff from a small drainagebasinis determinedto be approximately normal with a mean of 14.0 in. and a varianceof 9.0 in.2.Determinethe probability that the annualrunoff from the basinwill be lessthan I 1.0 in. in all three of the next three consecutiveyears. 26.20. In the past60 years,a dischargeof 30,000cfs at a streamgaugingstationwasequaled or exceededonly three times. Determine the averagereturn period (years) of this value. 26.21. EventsA and B are independentand havemarginal probabilitiesof .4 and .5, respectively. Determine the following for a single trial: a. The probability that both A and B occur. b. The probability that neither occurs. c. The probability that B, but notA, occurs. Existingrecordsrevealthe following information aboutEventsA and 4 whereA = a ?6il,

IongMerch'warmspellandB 1q

:!94!$99!.

PROBLEMS

703 1n

Year A : warm March? B : April flood?

No Yes

No No

Yes No

No Yes

Yes Yes

No Yes

Yes No

No Yes

Yes Yes

No No

On the basisof the 10-yearrecord, answerthe following: a. Are variablesA and B independent?Prove. b. Are variablesA andB mutually exolusive?Prove. c. Determinethe marginal probability of an April flood. d. Determinethe probability of having a cold March next year. e. Determinethe probability (onevalue)of havingboth a cold March and a floodfree April next year. f. If a long March warm spell hasjust endedtoday,what is the best estimateof the probability of a flood in April? 26.23. Two dependenteventsarc A : a flood will occur in Omaha next year and B : an ice-jam will form near Omahain the Missouri River next year. Useyour judgment to rank from largestto smallestthe following probabilities:P(A), P(A andB), P (A or B), P(A I B). 26.24. The probability of having a specifiedreturn period, [, is definedas: I P(annualvaluewill be equaledor exceeded : /, \'-' exactlyoncein a periodof r : I years) T,f \' Also, p (annualvalue will be equaledor exceeded _ exactly r times in a period of n years)

pn_r(l _ p)r

the secondprobability shouldequal a. According to the descriptionsin parentheses, the first when n and r are equal to what values? b. Showthat both equationsresultin the sameprobability for an annualvaluewhose years.Discusss. frequencyis 33{ percentand the return period is Z.: /:3 26.25. For the function describedbelow, find (a) the number b that will make the function a probability densityfunction, and (b) the probability that a singlemeasurementof x will be lessthanl.

.l \^)

-

forx ( 0 for0<x=b forxlb

{i",,'

26.26. The random variabler representsdepth of precipitation in July. Betweenvaluesof -r : 0 and x : 30, the probability densityfunction has the equation/(-r) : x/40p',. In the past, the averageJuly precipitation p,, was 30 in. a. Determine the probability that next July's precipitation will not exceed20 in. b. Determinethe singleprobability that the July precipitatjon will equal or exceed 30 in. in all of five consecutiveyears. 26.27. The random variable-r representsdepth of precipitation in July. Betweenvaluesof x : 0 and .r : 30, the probability densityfunction has the equation/(.x) : x/1200. a. Determine the probability that next July's precipitation will not exceed20 in. b. Determinethe probability that next July's precipitationwill equalor exceed30 in.

l---

704

ANDSTATISTICS 26 PROBABILITY CHAPTER 26.28.

26.29.

26.30. 26.31.. 26.32.

26.33. 26.34.

discharge Measured (1000acre-ft)

recharge Estimated (1000acre-ft)

12.2 10.4 10.6 1.2.6 14.2 13.0 14.0 t2.0 10.4 tl.4

t2.o 9.8 I 1.0 t3.z

14.6 14.0 14.0 I z,+

10.4 11.6

26.35. F i t a r e g r e s s i o n e q u a t i o n t o t h e d a t a i n P r o b l e m 2 6 ' 3 4 , t r e a t i l g d i s c h a r g e a s t h e dependentvariable.computethestandarderrorofestimate.Estimatetheexpeoted be the estimateof dischargeif no dischargewhen recharg"ir 13 Kti what would relative improvementprovided the is information were availatie on recharge?what by the regressionestimate? linear regression'The program 26.36. prepare a computerprogramfor simple,two-variable, (b) computethe means'variances' X' should(a) read in N pain oioUt"tuuiions, Y and the regressionconstants'the (c) find and andX, Y and standarddeviationsoiUott' coefficient. verify with the data in standarderror of estimal-, u"J,ir" correlation Problem26.34. of the mean annual rainfall with the 26.37. From the following observations of variation How iinea, predi.tion equationfor the catchment. altitude of the gauge,d";;;;; well correlatedare rainfall and altitude?

)

i

PROBLEMS Gauge number

Mean annual rainfall(in.)

1 2 3

22 28 25 31

4

5 6 7 8 9 10 1l 12

705

Altitudeof gauge (1000ft) A A

4.4 1 < < A

5.6 5.6 5.8 6.0 6.6 6.6 6.8 7.0

JZ J I

36 35 36 46 4l 4I

26.38. Estimatethe expectedrainfall in Problem 26.31 for a gaugeto be installed at an altitude of 5500 ft. estimatesof A and B in the bivariateregressionequationY : A + 26.39. The least-squares definedaslogro)andXis BXarcA: 2.0 andB : 3.0,whereYis atransformation : the valuesof a and b. determine y axb, by related Ify and r are as 1o916;r. defined for 26.40. The time of rise of flood hydrographs(Z), deflnedasthe time a streamto rise from low waterto maximum depthfollowing a storm,is relatedto the streamlength (L) and the averageslope(S). From the information given below for 11 watershedsin Texas, New Mexico, and Oklahoma, derive a functional relation of the form T,: aLbS'.

Watershed number I

2 3 A

5 6 7 8 9 l0 11

Ir

L

D

(min)

(1000ft)

(fv10oo ft)

150 90 60 60 100 75 90 30 30 45 50

18.5 14.2 25.3 tt.7 9.7 8.1 21.'7 3.9 1.2

7.93 19.0 t2.a 13.3 11.0 15.0 16.7 146.0 20.0 64.0 33.0

J.J J.)

26.4r, Repeat the exercise in Problem 26'40 by fitting the relation T,: dF", whete

with t in mi and S in ftimi. Plot the results on log-log paper. 26.42. The squareof the linear correlationcoefficientis calledthe proportion of the variance that is "explainedby the regression."Describethe meaningof this phraseby evaluating the equationsgivenin the text.What varianceis explained,and what doesthe term "explained" mean? 26.43. Twenty measuredpairs of valuesof normally distributedvariablesX and Y are analyzed, yielding valuesof X : 3O,V : 20, s' = 20, ands, : 0. Determinethe values F : L/{S

706

CHAPTER26

PROBABILITY ANDSTATISTICS

d a andb and the standarddeviationof residualsfor a least-squares fit usingthe linear equationY:a+bX. 26.44. The least-squares estimatesof A and B in the bivariateregres*sion equationy : A + BXareA:2.OandB : I.0, whereyis atransformation definedaslog1eLIf Iand X are relatedby Y : a(.b)',determinethe valuesof a and b.

26.45. Given a table of ten valuesof mean annual floods and correspondingdrainageareas for a numberofdrainagebasins,statehow linear regressiontechniqueswould be used to determinethe coefficientand exponent(p and 4) in the equation Qzzz : pAq.

26.46. What choiceof transformedvariablesI and X would provide a linear transformation for y : a/(x3 + b)? Also, if a regressionon these transformed variables yields I : 100 + 10X determinethe correspondingvaluesof a and b. Would the linear transformationbe applicableto all possiblepairs and valuesofx and y? 26.47. Which measureof variation in a regressionY : a * bX is generallylargerin magnitude, the standarddeviation of I or the standarddeviation of residuals?For what condition would the two valuesbe equal?

REFERENCES 1. Ven T. Chow, "Statisticaland ProbabilityAnalysisof HydrologicData," Sec. 8-1, in Handbookof Applied Hydrology (V. T. Chow, ed.). New York: McGraw-Hill, 1964. 2. M. B. Fiering, "Information Analysis," in Water Supply and Waste Water Removal (G. M. Fair, J. C. Geyer,and D. A. Okun, eds.).New York: Wiley, 1966,Chap.4. 3. J. R. Benjamin and C. Cornell, Probability, Statisticsand Decisionfor Civil Engineers. New York: McGraw-Hill. 1969. 4. A. M. Mood and F. A. Graybill, Introduction to the Theory of Statistics, 2nd ed. New York: McGraw-Hill. 1963. 5. A. J. Duncan,Quality Control and Statistics.Homewood,IL: RichardD. Irwin, Inc., 1959. 6. P. G. HoeI, Introduction to MathematicalStatistics,3rd ed. New York: Wiley, 1962. 7. J. Aitchison and J. A. C. Brown, The Log-Normal Distributlon. New York: Cambridge UniversityPress,1957. 8. H. A. Foster,"TheoreticalFrequencyCurves,"Trans.ASCE 87,142-203(1924). 9. L. R. Beatd, Statistical Methods in Hydrology, Civil Works Investigations,U.S. Army Corps of Engineers,SacramentoDistrict, 1962. 10. "A Uniform Techniquefor DeterminingFlood Flow Frequencies,"Bull. No. 178, U.S. GeologicalSurvey,1989. 11. "New Tablesof PercentagePoints of the PearsonType III Distribution," Tech. Release No. 38, CentralTechnicalUnit, U.S. Departmentof Agriculture,1968. 12. M. B. Fiering, StreamflowSynthesis.Cambridge,MA: Harvard University Press,1967. 13. F. E. Perkins,SimulationLecture Notes,SummerInstitute, "Applied MathematicalProgramming in WaterResources,"University of Nebraska,1970. 14. E. J. Gumbel, "The Return Period of Flood Flows," Ann. Math. Statist. l2(2), L63190(June1941). 15. Ven T. Chow, "The Log-Probability and Its EngineeringApplication," Proc. ASCE 80, 1-25(Nov. 1954). 16. "Probability Tables and Other Analysis of Extreme Value Data," Series22, National Bureauof StandardsApplied Mathematics,1953. 17. E. J. Gumbel, Statisticsof Extremes.New York: ColumbiaUniversity Press,1958.

Chapter27

FrequencyAnalysis

r Prologue The purposeof this chapteris to: . Presentmethodsused in hydrology to evaluatethe recurrenceof particular magnitudesand durationsof random hydrologicvariables. . Elaborateon the definitionsof freqUency,reiurrence interval, return period, and risk analysisintroducedin Chapter10, Section10.4. . Illustrate the diverseapplicationsof frequencyanalysisin hydrology. . Teachseveralmethodsfor conductingfrequencyanalyses,includingthe useof frequencyfactors that allow estimationof recurrenceintervals for variables that follow conventionalprobability distribution functions. . Introduce methods of point and regional frequency analysis and describe regional USGSregressionequationsthat have been 4doptedthroughoutthe U.S. for estimating flood flows for use in structure design and floodplain analysis. . Establishhow to estimatethe reliability of estimatesderived from point or regional frequencyanalyses. Explainthe widelyusedBulletin No. 17BLog-PearsonType III proceduresfor performing uniform flood flow frequencyanalyses. Describehow variousfederal agenciesapply frequencymethodsin designor analysisof water resourcessystems. In Chapter 26, probability and statistical characteristicsof random variables were introduced,alongwith common distributionfunctions and principlesof regression and correlatiqn.The presentchapterprovidesapplicationsof theseprinciplesto common hydrologicvariables.

27.1 FREQUENCY ANALYSIS The statisticalmethodspresentedin Chapter26 areusedmostfrequentlyin describing hydrologicdata suchas rainfall depthsand intensities,peak annual discharge,flood flows, low-flow durations,and the like. Frequencyanalysiswas introduced in Sec-

ANALYSIS 709 FREQUENCY 27.2 GRAPHICAL tion 10.4 and is defined as the inve recurrencAor probabilitiesof magnit wise, the frequencYof a hYdrologic discretevariablewill occur or some' exceededin anY given Year'The lat freq-uenc probability or exceedance ir"qo"n"Y is a ProbabilitYand has ceedancsfrequencY,as shownbYEc Two methods of frequencYa plotting techniqueto obtain the cum iu"to.t. The cumulativedistribution the probability of an eventequal to is used to obtain recurrencerntervz tioned whenworking with recordssl ofexpectedhydrologiceventsgreaterthantwicetherecordlength.

ANALYSIS FREQUENCY 27.2 GRAPHICAL The frequencYof an event can be When annualmaximum valuesare imated as the meantime in Years'\ exceededonceon the average'The t be shownto be m

tuf,lj

x:

(21.r)

the mean number of exceedances the number of future trials -oi of values the number to I descendingvalues,with largestequal therunt t : 1' N : T' and If the mean number of exceedances

where

7 = N = n : m:

4 I

-

_

n ) l

(21.2)

m

indicatingthattherecurfenceintervalisequaltothenumberofyearsofrecordplus 1, divided bY the rank of the event' They give different results as Severalpictting p*liion formulas are available'l for 10 yearsof record .1,.The rangein recurrencerntervalsoutained notedin Table2'7' plotting position formulasdo not account is illustratedin the right-hani column.Most formula ihat doesaccountfor samplesize for the samplesizeor length of record. One generalform wa, giuen by Gringorten' and has the .r1 -

r -

n * l - 2 a m - a

(27.3)

710

CHAPTER2TFREQUENCYANALYSIS TABLE 27.1 PLOTTINGPOSITIONFORMULAS

Form:1 andn=10 Method

Solve for P (X > x\

P

m

.10

l0

.05

20

.067

14.9

.091

11

m-0.3 n+0.4

.06'l

14.9

Blom

,m-i n+i

.061

t6.4

Tukey

3m-l 3n*l

.065

15.5

California

n 2m-l

Hazen

1 - (0.5)'/'

Beard

fn

Weibull

n * l

Chegadayev

where n : the number of yearsof record the rank a parameterdependingon n as follows:

n a

10 0.448

20 0.443

30 0.442

40 0.441

50 0.440

n a

60 0.440

70 0.440

80 0.440

90 0.439

100 0.439

In general,a : 0.4 is recommendedin the Gringortenequation.If the distribution is approximatelynormal, , : fi is used.A value of a : 0.44 is usedif the data follows a Gumbel distribution. The techniquein all casesis to arrangethe datain increasingor decreasingorder of magnitudeand to assignorder number m to the ranked values.The most efficient formula for computingplotting positionsfor unspecifieddistributions,l and the one now commonly usedfor most sampledata, is the Weibull equation P _

m n-fI

(27.4)

Whenm is rankedfrom lowestto highest,P is an estimateof the probability of values being equalto or lessthan the rankedvalue,that is, P(X < x); whenthe rank is from the value highestto lowest,P is P(X > x). For probabilitiesexpressedin percentages, is IA\ml@ + 1). The probability that X: .x is zero for any continuousvariable.

27.4 REGIONALFREQUENCYANALYSIS

721

regiond'lstudies.Methodsof "smoothing" and averagingregionalvaluesof skewness havealsobeenproposed.ro'15 Many techniquesused in the past for generalizingregi6nal characteristicsdid not rely on statisticalconsiderations.The so-calledstation-yearmethodof extending rainfall recordshasprovedhelpful but has questionablestatisticalvalidity, especially areas.The method if applied to dependentseriesor to stationsin nonhomogeneous has been used to combine, say,two 25-yearrecords to obtain a single 50-year sequence. In practice, the analyst may have to use imagination and ingenuity to summarizeregional characteristics,while remaining awareof actual and theoretical considerations, lndex Flood Method The index-floodmethodusedin the pastby the U.S. GeologicalSurveyis an example of summarizingregional characteristicssuccessfully.t''tuThe method usesstatistical andgenerally databut combinesthem in graphicalsummaries.It canbe supplemented improved by using statisticalmethods,employing,for example,the regressiontechniquesexplainedin Chapter26.The index method,as illustrated in Fig. 27.4, canbe outlined as follows. 1. Preparesingle-stationflood-frequencycurves for each station within the homogeneous region (Fig. 27.4a). 2. Compute the ratio of flood dischargestaken from the curves at various frequenciesto the mean annual flood for the samestation. 3. Compileratios for all stationsand find the medianratio for eachfrequency (Fi5.27.4b). 4. Plot the median ratios againstrecurrenceinterval to produce a regional frequencycurve (Fig. 27.4c). Two statistical considerationsinvolved are (1) a homogeneitytest to justify definitionof a region, and(2) a methodfor extendingshortrecordsto placeall stations on the samebaseperiod. A somewhatsimilar techniquewas developedby Potterfor the Bureau of Public Roads.rTIt relies on the graphical correlation of floods with physical and climatic variables and is thus a techniquethat refers in part to the in Chapter26 (seealso Chapter16). discussion

U.S.G.S.RegionalRegressionEquations Early in the 1950s,the U.S. Geologicalsurvey institutedaprocessof correlatingflood flow magnitudqsand frequencieswith drainagebasin characteristics.Setsof regressionequationsfor the 2-,5-, l0-,25-,50-, and 100-yearfloodshavebeendeveloped regionin every state.The work was for practically everyhydrologicallyhomogeneous largely inauguratedto developmethodsfor estimatingpeak flow rates for designof highway structuresat ungaugedbasins.Data from gaugedsites was evaluatedby regional analysisto provide the best fit of regressionmodelsto the data. Continuouswater stagerecordersand crest-stagegaugedata were consultedto developfrequencycurvesfor all gaugedwatersheds.Given the frequencycurves,a

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