manual on stream gauging volume 11 computation of...

WORLD METEOROLOGICAL ORGANIZATION

Operational Hydrology

Report No. 13

MANUAL ON STREAM GAUGING

VOLUME 11

COMPUTATION OF DISCHARGE

~I WMO-No.519 I

~.~~(\ '-l~~ - tf

....-;:-~ Secretariat of the World Meteorological Organization - Geneva - Switzerland1980

CONTENTS

Page

Foreword .

Summary (English. French, Russian, Spanish)

v

VII

Chapter 1 - Discharge ratings using simple stage-discharge relations

1.1 Introduction............. 11.2 Stage-discharge controls . . . . . . . 21.3 Graphical plotting of rating curves. 21.4 Section controls 81.5 Channel control 141.6 Extrapolation of rating curves . 161.7 Shifts in the discharge rating . . 261.8 Statistical analysis of the stage-discharge relation 281.9 Effect of ice formation on discharge ratings 441.10 Sand channel streams 59-References .. _ . . . . . . . . . . . . . . . . . . . 68

Chapter 2 - Discharge ratings using slope as a parameter 69

2.1 General considerations. . . . . . . . . . . . . 692.2 Theoretical considerations 702.3 Variable slope caused by variable backwater 702.4 Variable slope caused by changing discharge 872.5 Variable slope caused by a combination of variable backwater and changing

discharge 992.6 Shifts in discharge ratings where slope is a factor. . . . . . . . . . . . . . . . 992.7 A suggested new approach to computing discharge records for slope stations 99References 100

Chapter 3 ~ Discharge ratings fOf tidal streams. 101

3.1 General.................. 1013.2 Evaluation of unsteady flow equations 1013.3 Empirical methods. 104References 112

Chapter 4 - Discharge ratings for miscellaneous hydraulic facilities 115

4.1 Introduction....... 1154.2 Dams with movable gates 1154.3 Navigationlocks.. 1384.4 Pressure conduits. . 1444.5 Urban storm drains. 159References _ . 163

IV CONTENTS

Chapter 5 - Computation of discharge records.

5.1 General .5.2 Station analysis . . . . . . . . . . . . . . . . . . .5.3 Computation of discharge records for a non-recording gauging station5.4 Computation of discharge records for a recording station equipped with an

autographic recorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.5 Computation of discharge records for a recording station equipped with a

digital recorderReferences .

Annex - Adjustment of discha:rge in river branches

Acknowledgements

Index .

165

165165177

178

200231

233

239

241

FOREWORD

In view of the broad scope of the subject matter published in WMO guidancematerial and because of the necessary space limitations, the treatment of streamgauging techniques has not previously been presented in sufficient detail to serve as amanual on that topic. This report attempts to fIll the need for such a manual.

WMO Member countries have established procedures for stream gauging but,generally speaking, their basic manuals require updating and their supplementarymanuals that update the description of specific techniques and equipment are frag~mentary. Several international manuals on stream gauging are also available in theform of WMO Technical Notes and reports and Standards of the InternationalOrganization Jar Standardization (ISO), but these deal only with selected aspects ofthe subject. The purpose of this report is to provide a complete, updated, and stan-dardized manual of stream gauging procedures.

At its fourth session (Buenos Aires, April 1972) the WMO Commission forHydrology (CHy) recommended the preparation of guidance material on streamgauging procedures aod requested its Working Group on Hydrological Instrumentsand Methorls of Observation to prepare this material. Since this m.aterial is to providecomprehensive coverage of modern stream gauging procedures, the group recom-mended its preparation and publication in the fonn of a "Manual". At its first session(Geneva, November 1974) the group adopted an outline of the "Manual on StreamGauging" which was subsequently prepared by R. W. Herschy of the Department ofthe Environment, Water Data Unit, u.K., and S. E. Rantz of the Geological Survey,U.S. Department of the Interior. The Maoual also includes material based on thework of many others; those whose work constituted a direct contribution to thisManual are listed in the references appended to each chapter.

At its fIfth session (Ottawa, July 1976), the WMO Commission for Hydrologyagreed that this Manual would be a major contribution ofWMO in meeting the needsof national Hydrological Services and recommended its publication in the WMOOperational Hydrology Reports Series. The Manual is published in two separatevolumes: Volume I - Fieldwork and Volume II - Computation of Discharge.

I am pleased to have this opportunity to express to Messrs. Herschy and Rantzand the other members of the CHy Working Group mentioned above, the sincereappreciation of WMO for the time and effort they have devoted to the preparation ofthis Manual.

Geneva, Angust 1978D. A. Davies

Secretary-General

SUMMARY

This Manual is published in two separate volumes. Volume 1- Fieldwork, andVolume II - Computation of Discharge. Three major topics are discussed in VolumeI, namely selection of gauging-station sites, measurement of stage and measurementof discharge. Volume I, which is aimed primarily at the hydrological technician, con-sists of nine chapters. Chapter 1 - Introduction - briefly discusses streamflow recordsand general stream-gauging procedures. Chapter 2 - SelectIon ofgauging-station sites -discusses the general aspects of gaugingMstation network design and the hydraulicconsiderations which enter into specific site selection. The section on the design ofgaugingstation networks is, of necessity, written for the experienced hydrologistwho plans such networks. Chapter 3 - Gaugingstation controls - reviews the typesof control, the attributes of a satisfactory controi and artificial controls. Chapter 4 -Measurement of stage - discusses the gauge datum, non-recording stream-gaugingstations, recording stream-gauging stations, operation of a recording stream-gaugingstation, factors affecting the accuracy of the stage record and special purpose gauges.Chapter 5 ~ Measurement of discharge by conventional current meter methods ~ dis-cusses -the general description of a conventional current -m~ter measurement of dis-charge, instrwnents and equipment, measurement of velocity, procedure for con-ventional current meter measurement of discharge, special problems in conventionalcurrent meter measurements, summary of factors affecting the accuracy of a diswcharge measurement and accuracy of a discharge measurement made under averageconditions. Chapter 6 - New methods - discusses three new methods of gauging,namely the moving boat method, the ultrasonIc method and the electromagneticmethod. Chapter 7 .~- Measurement of discharge by use of precallbrated measuringstructures ~ discusses ten st:;mdard measuring structures. The methods discussed inChapter 8 - Measurement of discharge by miscellaneous methods - include floatmeasurements, volumetric measurement, ultrasonic and electromagnetic currentmeters, gauging from aircraft, and the use of photographic teclUliques for unstableflow-roll waves or slug flow. Chapter 9 - Indirect determination of peak discharge -provides a general discussion of the procedures used in collecting field data and incomputing peak discharge by the various indirect methods after the passage of a flood.

Volume Il - Computation of discharge - deals primarily with computation ofthe stage-discharge relation and computation of daily mean discharge. It is aimedprimarily at the junior engineer who has a background in basic hydraulics. Volume Ilconsists of five chapters. Chapter 1 - Discharge ratings using simple stage-dischargerelations - is concerned with ratings in which the discharge can be related to stagealone. It discusses stage-discharge controls, graphical plottings of rating curves,section controls, channel control, extrapolation of rating curves, shifts in the ruswcharge rating, statistical analysis of the stage-discharge relation, effects of ice forma~lion on discharge ratings and sand channel streams. Chapter 2 - Discharge ratingsusing slope as a parameter .- deals with variable slope caused by variable backwater,

VIII SUMMARY

by changing discharge and by a combination of both, shifts in discharge ratings whereslope is a factor and presents a new approach to computing discharge records forslope stations. Chapter 3 - Discharge ratings for tidal streams -- reviews two generalapproaches for obtaining a continuous discharge record in tidal streams - the theore-tical approach involving evaluation of the equations of unsteady flow and the empiri-cal approach involving empirical relations whose effectiveness generally varies inverselywith the degree ofimportance of the acceleration head. Chapter 4 - Discharge ratingsfor miscellaneous hydraulic installations - deals with specialized problems in establish-ing discharge ratings for various hydraulic installations such as dams with movablegates, navigation locks, pressure conduits and urban stonn drains. Finally, Chapter 5-- Computation of discharge records _. discusses station analysis and computation ofdischarge records for a non-recording gauging station, as well as for a recordingstation equipped with either an antographic or digital recorder.

The adjustment of discharge in river branches is discussed in the ISO TechnicalNote reproduced in the Annex to this Manual.

RESUME

Le present ManueI se compose de deux volumes distincts: le Volume 1- Tra-vaux sur le terrain - et le Volume 11 - Calcul des debits. Le Volume I traite de troissUjets principaux, a savoir le choix de l'emplacement des stations de jaugeage, lesmesures de niveau et les mesures de debit. Destine principalement au tcchnicien deI'hydrologie, it comprend neuf chapitres. Le chapitre I - Introduction - donne unbref aperc;u des rcleves des debits cl'un coms cl'eau, ainsi que des methodes de jau-geage d'application generale. Le chapitre 2 - Choix de I'emplacement des stations dejaugeage - expose les aspects generaux de la conception des reseaux de stations dejaugeage, ainsi que les facteurs hydrauliques aprendre en consideration dans le choixcl'un site particulier. La section consacree ala conception des reseaux de stations dejaugeage s'adresse naturcllement al'hydrologue experimente qui elabore les plans dereseaux de ce type. Le chapitre 3 - Parametres conditionnant la relation hauteur-debit aune station de jaugeagc - traite des differents types de tron90ns et de sectionsde controle, des caracteristiques auxquelles doivent satisfaire de tels tron90ns ousections, ainsi que des ouvrages de contr6le. Le chapitre 4 -- Mesures de niveau -C!borde les questions suivantes: cote du zero du limnimetre, stations de jaugeageequipees de simples limnimetres, stations de jaugeage equipees de limnigraphes,fonctionnement d'une station de jaugeage equipee de Hmnigraphes, facteurs inter-venant dans la precision des releves de niveau et limnimetres destines a des finsspeciales. Le chapitre 5 - Mesme du debit par la methode c1assique du moulinet -donne une description generale d'une mesure de debit effectuee au moyen d'unmoulinet, presente les instruments et l'equipement utilises a cet effet et parte surd'autres questions telles que la mesure de la vitesse d'ecoulement, la methode c1as-sique de mesure du debit i l'aide d'un moulinet et les problemes particuliers quepose ce genre de mesure; il resume, en outre, les facteurs qui interviennent dans laprecision des mesures de debit et la precision de ces mesures dans des conditionsmoyennes. Le chapitre 6 - Nouvelles methodes - expose trois nouvelles methodesde jaugeage, i savoir la rhethode du bateau en marche, la methode ultrasonique et lamethode electromagnetique. Le chapitre 7 - Mesure du debit au moyen d'ouvragescalibres - analyse dix ouvrages types. Les methodes dont il est question au chapitre 8

- Mesure du debit a l'aide de diverses methodes - englobent les mMhodes cl flot-teurs, les methodes volumetriques, les hydrotachymetres a ultrasons et electro-magnetiques, les jaugeages faits i partir d'aeronefs et l'utilisation de techniquesphotographiques dans le cas de regimes instables avec ondes de translation. Lechapitre 9 - Detennination lndirecte des debits de pointe - donne une descriptiongenerale des m6thodes utilisees pour rassembler les donnces obtenues sur le terrain etcalculer les debits de pointe par differentes methodes indireetes apres une erue.

Le Volume II - Calcul des debits - traite principalement du calcul de la relationhauteur-debit et du debit journalier moyen. nest destine princi.palement aux jeunesingcnieurs ayant re9u une fonnation de base en hydraulique et comprend cinq-

x RESUME

chapitIes. Le chapitre 1 - Etablissement des courbes de tarage sur la base d'unesimple relation hauteur-debit - anaiyse ies cas ou le debit ne depend que de lahauteur. n expose les questions suivantes: parametres qui conditionnent la relationhauteur-debit, pointage des courbes de tarage, sections de contra1e, trongons decontr6le, extrapolation des courbes de tarage, detarage, analyse statistique de larelation hauteur-debit, effets de la formation de glace sur le tarage et califS d'eaudont le lit est sableux. Le ehapitre 2 - Etablissement des eourbes de tarage utilisantla pente comme parametre - parte sur les questions suivantes: pente variable due ades remous variables, a un debit changeant ou cl une combinaison des deux facteurs,detarage lorsque la pente entre en ligne de compte, nouvelle ffiethode de calcul desreleves de debit pour les stations avec une pente marquee. Le chapitre 3 - Etablisse-ment des courbes de tarage pour les rivieres a man!ies - pn!isente deux methodesd'application generale qui pennettent d'obtenir un re1eve de debit continu dans lesrivieres cl marees: la methode theorique, comportant l'evaluation des equations d'unecoulement non permanent, et la methode empirique, faisant intervemr des relationsempiriques dont l'efficacite est en general inversement proportionnelle all degred'importance de l'acce1eration acquise dans le cours superieur. Le chapitrc 4 - Eta-blissement des courbes de tarage pour diverses installations hydrauliques - traite desprob1emes particuliers que pose l'etablissement de ces courbes pour diverses installa-tions hydrauliques telies que barrages cl vannes mobiles, ecluses de navigation, con-duites en charge et canaux de drainage urbains. Enfin, le chapitre 5 - Calcul desreleves de debit - porte sur l'analyse et le calcul en station des releves de debit pourune station de jaugeage non enregistreuse, ainsi que pour une station enregistreuseequipee d'un dispositif d'enregistrement graphique ou numerique.

La Note technique de l'lSO reproduite dans 1'annexe au present Manuel donnedes precisions sur 1'ajustement des debits dans des bras de cours d'eau.

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XII PE3IOME

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RESUMEN

Este Manual se publica en dos volurnenes separados: e1 Volumen I - "Trabajossobre el terreno" - y el Volumen 11- "Calculo de los caudales". El Volumen I tratade tres temas principales, a saber: la selecci6n del emplazamiento de Ias estaciones deaforo, la medida del nivel y la medida del caudal. Esta destinado principalmente a lostecnicos de la hidrologia y consta de nueve capitulos. El Capitulo I - "Introduc-cion" - trata brevemente de 108 registros del flujo de corriente y de 108 metodos deaforo generalmente utilizados. El Capitula 2 ~-- "Selecci6n de 108 emplazamientosde tas estaciones de aforo" -, trata de 108 aspectos generales del disefio de la redes deestaciones de afom y de 108 factores hidniulicos que han de tenerse en cuenta en laselcccion de un emplazamiento especifico. La secci6n dedicada al disefio de las redesde estaciones de afora esta destinada, naturalrnente, a los hidr61ogos experimentadosque eleboran los planes de este tipo de redes. El Capitula 3 - ''Parametros quecandi-cionan la relaci6n altura - caudal en una estacion de afaro" - estudia los distintostipos de tramos y de secciones de control, las caracteristicas que deben reunir talcstramos y seccianes y las obsas de control. El Caprtulo 4 -- "Medida del nivel" - tratadg-la -cotadelcerodelaforo,estaciones no .registradoras delafo1'O de caudales, esta-ciones registradoras del aforo, explotaci6n de una estacion registradora del aforo decaudales, factores que afcctan la precision del registro del nivel, y limnimetros parafines especiales. El Capitulo 5 - "Medida del caudal par el metodo c1asico del moll-nete" - facilita una descripcion general de una medida del caudal efectuada con unmolinete, presenta los instrumentos y el equipo utilizados a estos efectos, y trata deotras cuestiones tales como la medida de la velocidad del flujo de corriente, metodoclasico de medida del caudal con molinete, y problemas especiales que plantea estetipo de medida. Resume ademas los factores que intervienen en la precision de lamedida del caudal y en la precision de la medida del caudal realizada en condicionesmedias. En el Capitula 6 - "Nuevos metodos" - se expanen tres nuevos metodos deafo1'O, a saber: el metado del buque en movimiento, el metado ultrasonico y elmetodo electromagnetico. El Capitula 7 - "Medida del caudal utilizando estructurasde medida precalibradas" -, analiza diez estructuras de medida normalizadas. ElCapitulo 8 - "Medida del caudal par metodas diversos" -, trata de los metodosempleados, incluidos los metodos que utilizan flotadores, los metodos volumetricos,hidrotac6metros ultras6nicos y clectromagneticos, aforos desde aeronaves, y la utili-zacion de tecnicas fotognificas en cl casa de regrmenes inestables con olas dependientes. El Capitulo 9 - "Detcrminacion indirecta de caudales de punta" - con-tiene una discusi6n general de 10s metodas utilizados para la concentraci6n de losdatos obtenidos sobre el teneno y para cl calculo de caudales de punta despues deuna crecida por diferentes metados indirectas.

El Volumen II - "Calculo de caudales" --, trata principalmente del calculo dela rclacion altura - caudal y del caudal media diaro. Est

XIV RESUMEN

cinco capitulos. El Capitula 1 - "Establecimiento de curvas de tarado fundandoseen una simple relaci6n altura-caudal" - analiza 108 casos en 108 que el caudal s610depende de la altma. Tambi

CHAPTER I

DISCHARGE RATINGS USING SIMPLE STAGE-DISCHARGE RELATIONS

1.1 Introduction

Continuous records of discharge at gauging stations are computed by applyingthe discharge rating for the stream to records of stage. Discharge ratings may besimple or complex, depending on the number of variables needed to define the stage-discharge relation. This chapter is concerned with ratings in which the discharge canbe related to stage alone. (The terms "rating", "rating curve", "station rating", and"stage-discharge relation" are synonymous and are used here interchangeably.)

Discharge ratings for gauging stations are usually determined empirically bymeans of discharge measurements made in the field, notable exceptions being theprecalibrated ratings used in several countries for the special weirs and flumesdiscussed in Volume I, Chapter 7. Common practice is to measure the discharge ofthe stream periodically, usually by current meter, and to note the concurrent stage.Measured discharge is then plotted against concurrent stage on graph paper to defmeth.e rating curve. At a new station many discharge measurements are needed to definethe stage discharge relation throughout the entire range of stage. Periodic measure-ments are needed thereafter to either confirm the stability of the rating or to followchanges (shifts) in the rating. A minimum of ten discharge measurements per year isrecommended, unless it has been demonstrated that the stage-discharge relation iscompletely unvarying with time. In that event the frequency of measurements maybe reduced. It is of prime importance that the stage-discharge relation be definedfor flood conditions and for periods when the rating is subject to shifts as a result ofice formation or as a result of the variable channel and control conditions. It isessential that the stream gauging programme provides for the non-routine measure-ment of discharge at those times.

If the discharge measurements cover the entire range of stage experiencedduring a period of time when the stage-discharge relation is stable, there is littleproblem in defining the discharge rating for that period. On the other hand, if, asis usually the case, there are no discharge measurements to define the upper end ofthe rating, the defined lower part of the rating curve must be extrapolated to thehighest stage experienced. Such extrapolations are always subject to error, but theerror may be minimized if the analyst has a knowledge of the principles that governthe shape of rating curves. Much of the material in this chapter is directed toward adiscussion of those principles, so that when the hydrologist is faced with the problemof extending the high water end of a rating curve he can decide whether the extra~polation should be a straight line, or whether it should be concave upward or con-cave downward.

The problem of extrapolation can be circumvented, of course, if the unmeasuredpeak discharge is determined by use of the indirect methods discussed in Volume I,

2 VOLUME II - COMPUTATION OF DISCHARGE

Chapter 9. In the absence of such peak discharge determinations, some of the uncertainty in extrapolating the rating may be reduced by the use of one or more ofseveral methods of estimating the discharge corresponding to high values of stage.Four such methods are discussed in section 1.6.

1.2 Stage-discharge controls

The subject of stage-discharge controls was discussed in detail in Volume I,Chapter 3, but a brief summary at this point is appropriate.

The relation of stage to discharge is usually controlled by a section or reach ofchannel downstream from the gauge that is known as the station control. A sectioncontrol may be natural or manmade; it may be a ledge of rock across the channel,a boulder covered riffle, an overflow dam or any other physical feature capable ofmaintaining a fairly stable relation between stage and discharge. Section controlsaIe often effective only at low discharges, and are completely submerged by channelcontrol at medium and high discharges. Channel control consists of all the physicalfeatures of the channel which determine the stage of the river at a given point for agiven rate of flow. These features include the size, slope, roughness, alignment,constrictions and expansions, and shape of the channel. The reach of channel thatacts as the control may lengthen as the discharge increases, introducing new featuresthat affect the stage-discharge relation.

Knowledge of the channel features that control the stage-discharge relation isimportant. The development of stage~dischargecurves where more than one controlis effective, and where the number of measurements is limited, usually requires judg~ment in interpolating between measurements and in extrapolating beyond the highestmeasurements. That is particularly true where the controls are not permanent andvarious discharge measurements represent different positions of the stage~dischargecurve.

1.3 Graphical plotting of rating curves

Stage- discharge relations are usually developed from a graphical analysis of thedischarge measurements plotted on either rectangular co~ordinate or logarithmicplotting paper. In a preliminary step the discharge measurements available for analysisare tabulated and summarized on a form such as that shown in Figure 1.1. Dischargeis then plotted as the abscissa, coresponding gauge height is plotted as the ordinate,and a curve or line is fitted by eye to the plotted points. The plotted points carry theidentifying measurement numbers given in Figure 1.1; the discharge measurementsare numbered consecutively in chronological order so that time trends can be identi-fied.

At recording gauging stations that use stilling wells, there is often a differencebetween recorded (inside) gauge heights and outside gauge heights during periods ofhigh stage. When that occurs both inside and outside gauge heights for dischargemeasurements are recorded on the form shown in Figure 1.1, and in plotting themeasurements for rating analysis, the outside gauge readings are used first. The stage~discharge relation is extended to the stage of the outside high water marks that areobserved for each flood event. The stage~discharge relation is next transposed to cor-respond with the inside gauge heights obtained from the stage recorder at the timesof discharge measurement and at flood peaks. The rationale behind this procedure isas follows. The outside gauge readings are used for developing the rating because the

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hydraulic principles on which the rating is based require the use of the true stage ofthe stream. The transposition of the rating to inside (recorded) stages is then madebecause the recorded stages will be used with the rating to determine discharge. Therecorded stages are used for discharge determination because if differences existbetween inside and outside gauge readings. those differences will be known only forthose times when the two gauges are read concurrently. If the outside gauge heightswere used with rating to determine discharge, variable corrections. either knownor assumed, would have to be applied to recorded gauge heights to convert them tooutside stages. We have digressed here to discuss differences between inside and out~side gauge heights, because in the discussions that follow no distinction between thetwo gauges will be made.

The use of logarithmic plotting paper is usually preferred for graphical analysisof the rating because in the usual situation of compound controls, changes in theslope of the logarithmically plotted rating identify the range in stage for which theindividual controls are effective. Furthermore, the portion of the rating curve that isapplicable to any particular control may be linearized for rational extrapolation orinterpolation. A discussion of the characteristics of logarithmic plotting follows.

The measured distance between any two ordinates or abscissas on logaritlunicgraph paper, whose values are printed or indiated on the sheet by the manufacturerof the paper, represents the difference between the logarithms of those values. Con~sequently, the measured distance is related to the ratio of the two values. Therefore,the distance between pairs of numbers such as I and 2,2 and 4, 3 and 6, 5 and 10,are all equal because the ratios of the various pairs are identical. Thus the logarithmicscale of either the ordinates or the abscissas is maintained if all printed numbers onthe scale are multiplied or divided by a constaut. This property of the paper haspractical value. For example, assume that the logarithmic plotting paper available has2 cycles (Figure 1.2), and that ordinates ranging from 0.3 to 15.0 are to be plotted.If the printed scale of ordinates is used and the bottom line is called 0.1, the top lineof the paper becomes 10.0, and values between 10.0 and 15.0 cannot be accom-modated. However, the logarithmic scale will not be distorted if all values aremultiplied by a constant. For this particular problem, 2 is the constant used in Figure1.2, and now the desired range of 0.3 to 15.0 can be accommodated. Examinationof Figure 1.2 shows that the change in scale has not changed the distance betweenany given pair or ordinates; the position of the ordinate scale has merely been trans-posed.

We turn now to a theoretical discussion of rating curves plotted on logarithmicgraph paper. A rating curve that plots as a straight line on logarithmic paper has theequation,

where

Q=C(h-a)~ (1.1)

Q(h - a)

ha

C

is discharge;is head or depth of water on the control - this value is indicated by theordinate scale printed by the manufacturer or by the ordinate scale thathas been transposed, as explained in the preceding paragraph;is gauge height of the water surface;is gauge height of zero flow for a control of regular shape, or of effectivezero flow for a control of irregular shape;is the discharge when the head (h - a) equals 1.0 m;

USE OF STAGE-DISCHARGE RELATIONS

z )( O~;J;". /SCQ/~

Figure 1.2 - Example showing how the logarithmic scale of graph paper may be transposed

5


~ is slope of the rating cnrve. (Slope in Equation 1.1 is the ratio of the hori-zontal distance to the vertical distance. This unconventional way of measur-ing slop~ is necessary because the dependent variable Q is always plotted asthe abscissa.)

We assume now that a segment of an established logarithmic rating is linear,and we examine the effect on the rating of changes to the control. If the width of thecontrol increases, C increases and the new rating will be parallel to and to the right ofthe original rating. If the width of the control decreases, the opposite effect occurs;C decreases and the new rating will be parallel to and to the left of the original rating.If the control scours, a decreases and the depth (h - a) for a given gauge height in-creases; the new rating moves to the right and will no longer be a straight line butwill be a curve that is concave downward. If the control becomes built up by deposi-tion, a increases and the depth (h - a) for a given gauge height decreases; the newrating moves to the left and is no longer linear, but is a curve that is concave upward.

When discharge measurements are originally plotted on logarithmic paper, noconsideration is given to values of a. The gauge height of each measurement is plottedusing the ordinate scale provided by the manufacturer or, if necessary, an ordinatescale that has been transposed as illustrated in Figure 1.2. We refer now to Figure 1.3.

2

.3

.--~V~ .~~

--------:;:.V;;.-.;::----- ~ ~a=" 0_- :;.-:.--::.- ./a.o,Je .;;V}.'\.Ge ~

(co~,;:,,,,-:- :'\)'C. /'a ~ 1_-f ~a.\!b ~ ... /

--- -I .' ,se o::f'.Y." ./f I_c~e-~,_ le-/'

----1.;:--'~

// NOTE - All ,curves represent

/the sar.te rating. The truevalue of ua" is 1.5 .

/I

20 30 SO5 7 102 3.1

0.1 .2 .3 .4.7 La

.7

.5

1.03 2.5 2

2.7 2.2

2.5 2.0

2.3

2.2

2.1

~

N ..< rl 0

" " " "" " " "12 11.5 11 10

9 8.5 8 7

6.5 6 5

5 4.5 4 3

4 3.5 3 2

DISCHARGE, IN CUBIC METRES PER SECOND

Figure 1.3 - Rating curve shapes resulting from the use of differing values of effective zero-flow

USE OF STAGE-DISCHARGE RELATIONS 7

The inside scale (a = 0) is the scale printed by the paper manufacturer. Assume thatthe discharge measurements have been plotted to that scale and that they defme thecurvilinear relation between gauge height (h) and discharge (Q) that is shown in thetopmost curve. For the purpose of extrapolating the relation, a value of a is sought,which when applied to h, will result in a linear relation betweeu Ch - a) aud Q. If weare dealing with a section control of regular shape, the value of a will be known; itwill be the gauge height of the lowest point of the control (point of zero flow). Ifweare dealing with a channel control or section control of irregular shape, the value ofa is the gauge height of effective zero flow; it may be determined by successiveapproximations.

In successive trials, the ordinate scale in Figure 1.3 is varied for a values of 1 rn,1.5 m and 2 m, each of which results in a different curve, but each new curve stillrepresents the same rating as the top curve. For example, a discharge of 3 m 3 8-1

corresponds to a gauge height (h) of 4 m on all four curves. The true value of a is1.5 rn, and thus the rating plots as a straight line if the ordinate scale numbers areincreased by that value. In other words, while even on the new scale a discharge of3 m3 s-1 corrcsponds to a gauge height (h) of 4 m, the head or depth on the controlfor a discharge of 3 m3 s-1 is (h -- a), or 2.5 m; the linear rating marked a = 1.5 mcrosses the ordinate for 3 m3 S-1 at 4 ill on the new scale and at 2.5 m on the manu-facturer's, or inside, scale. If values of a smaller than the true value of 1.5 m are used,the rating curve will be concave upward; if values of a greater than 1.5 m are used,the curve will be concave downward. The value of a to be used for a rating curve orfor a segment of a rating curve can thus be determined by adding or subtracting trialvalues of a to the numbered scales on the logarithmic plotting paper until a value isfound that results in a straight line plot of the rating. It is important to note that ifthe logarithmic ordinate scale must be transposed by multiplication or division toaccommodate the range of stage to be plotted, that transposition must be madebefore the ordinate scale is manipulated for values ofa.

A more direct graphical solution for a, as described by Johnson (1952) is il-lustrated in Figurc 1.4. A plot of h versUs Q has resulted in the solid line curve which

(2)2

h, 03 =1 22

(h3-a) ~ Ih ,-allh2-a)h3 2

0 a ~ h1 h2 -h3ah2

h1+ h2-2h3

//

a/

/

//

2 3 1

Figure 1,4 - Schematic representation of the linearization of a curve on logarithmic paper


(I.3)

is to be linearized by subtracting a value of a from each value of h. The part of therating between points I and 2 is chosen, and values of h, h2, Q, and Q2 are pickedfrom the co-ordinate scales. A value of Q3 is next comput~d,such that

Q~ = Q, Q2

From the solid line curve, the value of h 3 that corresponds to Q3 is picked. In ac-cordance with the properties of a straight line on logarithmic plotted paper,

(ft3 - aj2 = (h, - a)(h2 - a) (1.2)

Expansion of terms in Equation 1.2 leads to Equation 1.3. which provides a directsolution for a.

2h,h2 -h 3a :;:

h , +h2 -2h3A logarithmic rating curve is seldom a straight line or a gentle curve for the entire rangein stage. Even where a single crosssection of the channel is the control for all stages,a sharp break in the contour of the cross-section, such as an overflow plain, will causea break in the slope of the rating curve. Commonly, however, a break in slope is dueto the low water control being drowned out by a downstream section control be-coming effective or by channel control becoming effective.

The use of rectangular co-ordinate paper for rating analysis has certain ad-vantages, particularly in the study of the pattern of shifts in the lower part of therating. A change in the low-flow rating at many sites results from a change in theelevation of effective zero flow (a), which means a constant shift in gauge height. Ashift of that kind is more easily visualized on rectangular co-ordinate paper becauseon that paper the shift curve is parallel to the original rating curve, the two curvesbeing separated by a vertical distance equal to the change in the value of a. On loga-rithmic paper the two curves will be separated by a variable distance which decreasesas stage increases. A further advantage of rectangular co-ordinate paper is the factthat the point of zero flow can be plotted directly on rectangular co-ordinate paper,thereby facilitating extrapolation of the low water end of the rating curve. That can-not be done on logarithmic paper because zero values cannot be shown on that typeof paper.

Logarithmic plotting should always be used initially in developing the generalshape of the rating. The final curve may be displayed on either type of graph paperand used as a base curve for the analysis of shifts. A combination of the two typesof graph paper is frequently used with the lower part of the rating plotted on an insetof rectangular co-ordinate paper or on a separate sheet of rectangular co-ordinatepaper.

1.4 Section controls

Artificial controlsAt this point we digress from the subject oflogarithmic rating curves to discuss

the ratings for artificial section controls. A knowledge of the rating characteristics ofcontrols of standard shape is necessary for an understanding of the rating character-istics of natural controls, almost all of which have irregular shape.

Flow measuring structuresThe structures detailed in Volume I, Chapter 7 may be used as controls for


velocity area stations, the low or medium flows being measured by the structureusing the laboraIory rating, the high flows being measured by cableway. For suchdual purpose stations the laboratory rating is used up to the modular limit and therating continued by current meter. For structures such as the Crump weir or Flat-Vweir where a crest tapping is used the range of the structure is increased into thenon-modular region until complete drowning takes place. Flows above this limit willbe dependent on a downstream control or a channel control.

If the weirs described in Volume I, Chapter 7 are compounded without dividewalls being incorporated in the design a loss of accuracy will occur if the laboratoryratings are used. This loss of accuracy may not be serious but it is recommended thatall such non-standard variations be field calibrated. The calibration of a non standardbroad-crested weir follows.

Notched flat-crested rectangular weir

Figure 1.5 shows the notched flat-crested rectangular weir that is the controlfor a gauging station on Great Trough Creek near Marklesburg, Pa, D.S.A.

Because there is a sharp break in the cross-section at gauge height 0.43 m abreak occurs in the slope of the rating curve at that stage. The gauge height of zero

20 30.4 .5 .7 1 ..0 2 4 5 7 10


.,

I .!-G.H... O.431.22m~G.H .. a.Om"'------13.7 m

--c.o1\cll.'le. "" ..........~V~ 'i.a1\'be.;:--V 11-

~ v--- 3./V. ..----'ie.1\~~

1~yV

2.5V V

1/.1'/

f'l,"if

10 VOLUME IT - COMPUTATION OF DISCHARGE

flow for stages between 0.0 and 0.43 m is 0.0 m; for stages above 0.43 m the effec-tive zero flow is at some gauge height between 0.0 and 0.43 m. If the low end of therating is made a tangent, the gauge height of zero flow Ca) is 0.0 m and thc slope ofthis tangent turns out to be 2.5, which, as now expected, is greater than the theoret~kal slope of 1.5. The upper part of this rating curve is concave upward because thevalue of a used (0.0 m) is lower than the effective value of zero flow for high stages.

If the upper end of the rating is made a tangent, it is found that the value ofa, or effective zero flow, must be increased to 0.2 m. Because we have raised thevalue of a, this will make the low water end of the curve concave downward. Thehigh water tangent of the curve, principally because of increased rate of change ofvelocity of approach, will have a slope that is greater than that of the low watertangent of the curve previously described; its slope is found to have a value of3.0.

The low water tangent for the notched control, which is defined by dischargemeasurements, warrants further discussion. Its slope of 2.5 is higher than one wouldnormally expect for a simple flat-crested rectangular notch. Reasons for tIus may be(a) thc velocity of approach factor is included in the rating, (b) a thin-plate weir isfixed to the downstream edge of the notch WitIl its elevation about 0.03 ID above thebase of thc notch and (c) and probably more important, the width of thc notch issmall compared to the total width of the control; this may alter the flow character~istics to the extent that the notch may in fact be operating between rectangular andvnotch conditions. These observations are mentioned here only to warn the readernot to expect a slope as great as 2.5 in the rating for a simple nat~crested rectangularnotch. In fact, the sole purpose here of discussing the low water tangent of the ratingcurve is to demonstrate the effect exerted in the curve by varying the applied valuesof a. The Iow water end of a rating curve is usually well defmed by discharge measure-ments, and if it is necessary to extrapolate the rating downward, it is best done byreplotting the low water end of the curve on rectangular co-ordinate graph paper,and extrapolating the curve down to the point of zero discharge.

Trenton-type controlTIle so-called Trenton-type control is a concrete weir that is popular in the

D.S.A. The dimensions of the cross-section of the crest are shown in Figure 1.6. The

914

FLOW 241127 546

Dimensions in mm

457

Figure 1.6 - Cross-section of Trenton~typecontrol (dimensions in mm)

USE OF STAGE-DISCHARGE RELATIONS I I

crest may be constructed so as to be horizontal for its entire length across the stream,or for increased low flow sensitivity the crest may be given the shape of extremelyflat V. For a horizontal crest, the equation of the stage discharge relation, as ob~tained from a logarithmic plot of the discharge measurements, is commonly of theorder of Q~ 2.31 Yh1.65 The precise values of the constants will vary with theheight of the weir above the stream-bed, because that height affects the velocity ofapproach. The constants of the equation are greater than those for a flat-crestedrectangular weir (see Volume I, Chapter 7) because the cross-sectional shape of theTrenton~type control is more efficient than a rectangle, with regard to the flow of

. water.When the Trenton-type control is built with its crest in the shape of a flat V,

the exponent of h in the discharge equation is usually 2.5 or more, as expected fora triangular notch where velocity of approach is significant. Again, the precise valuesof the constants in the discharge equation are dependent on the geometry of theinstallation.

Columbustype controlOne of the most widely used controls in the U.S.A. is the Columbus-type

control. This control is a concrete weir with a parabolic notch that is designed togive accurate measurements of a wide range of flows (Figure 1 .7). The notch accom-modates low flows; the main section, whose crest has a flat upward slope away from

Profile of weir crest and notch

Coordinates of notcll profile,in melres

Coordinates of cross sectionof weir crest, in metres

F"ol _

7

\}x

Cross section of weir

....".".".00.m.H.H.>0

."." H

.".".m...,16

0.ol8.018.-.00'.-.-.00':~:.O~6

.0/,

.10

.149

.n,

.2U

.l'O

.00

~ '.'.00 '.00

.01l1 ..,.0l22 .".0'1] ....0'" .u.HS .u.l9S .".29l .n

Figure 1.7 - Dimensions of Columbus-type control


the notch, accommodates higher flows. The throat of the notch is convex along theaxis of flow to permit the passage of debris. For stages above a head of 0.2 m, whichis the elevation of the top of the notch, the elevation of effective zero flow is 0.06 rn,and the equation of discharge is approximately,

Q = 12.l4(h - 0.061)3.3

The precise values of the constants in the equation will vary with conditions for eachinstallation. The shape of the crest above a stage of 0.2 m is essentially a flat V, forwhich the theoretical exponent of head is 2.5 in the discharge equation. However,the actual value of the exponent is greater than 2.5, principally because of the in-crease of velocity of approach with stage.

Natural section controlsNatural section controls, listed in order of permanence, are usually a rock ledge

outcrop across the channel, or a riffle composed of loose rock, cobbles, and gravel, ora gravel bar. Less commonly, the section control is a natural constriction in width ofthe channel, or is a sharp break in channel slope, as at the head of a cascade or brinkof a falls.

Where the control is a rock outcrop, riffle, or gravel bar, the stage-dischargerelation, when plotted on logarithmic paper, conforms to the general principles dis-cussed for broad-crested artificial controls. If the natural control is essentially hori~zontal for the entire width of the control, the head on the control is the differencebetween the gauge heights of the water surface and the crest of the control. Theexponent ((J) of the head in the equation of discharge,

Q = C(h - a)~ (1.1)

will be greater than the theoretical value 1.5, primarily because of the increase invelocity of approach with stage. If the crest of the control has a roughly parabolicproftle, as most natural controls have (greater depths on the control near midstream),the exponent {J will be even larger because of the increase in width of the stream withstage, as well as the increase in velocity of approach with stage. The value of {J willalmost always exceed 2.0 and a range of {J from I to 4 has been experienced. If thecontrol is irregularly notched, as is often the case, the gauge height of effective zeroflow (a) for all but the lowest stages, will be somewhat greater than that for thelowest point in the notch.

The above principles are also roughly applicable to the discharge equations foran abrupt width contraction or an abrupt steepening of bed slope. The exponent {Jand the gauge height of effective zero flow are influenced, as described above, by thetransverse profile of the stream~bedat the control cross~section.

Natural compound section controlsWhere the control section is a local rise in the stream-bed, as at a rock outcrop,

riffle, or gravel bar, that cross-section is invariably a control only for low flows. Thegauging station in that circumstance has a compound control, the high flows beingsubject to channel control. Occasionally there is a second outcrop or riffle, down~stream from the Iow water riffle, that acts as a section control for flows of inter-mediate magnitude. When the control for intermediate stages is effective it causessubmergence of the Iow water control. At high flows the section control for inter-mediate stages is in turn submerged when channel control becomes effective. An


example of a compound control involving two section controls follows: an exampleof a compound control involving a section control that is submerged when channelcontrol becomes effective is described in section 1.5.

Figure 1.8 shows the rating for the compound section control at the gaugingstation on Muncy Creek near Sonestowll, Pa, D.S.A. The control consists of two rockledge riffles, effective zero flow (a) for very low stages being at gauge height 0.40 mand for higher stages at gauge height 0.37 m. If the low end of the rating is made atangent, it means that too large a value of a is used for the high end of the rating(0.40 m vs 0.37 m), and the high water end of the curve becomes concave downward.Conversely, if the high end of the curve is made a tangent, the low water end of thecurve becomes concave upward. The high water tangent of the ,one curve has agreater value of ~ than the low water tangent of the other curve. This difference inthe values of ~ reflects the effect of differences in the geometries of the two controlsas well as the effect of increased rate of change of approach velocities at the higherstages. Thc slopes of the two tangents are 2.9 and 2.2, both values being greater thanthe theoretical slope of 1.5.

~-/ OC.H.. 0.40

~."3'6e - .......- G.R. 0.37 _-

\\1.-\,>,," ~~---~--

.ed

2023456810,.2 .3.4,.5.6.81.0

I TT 12.9 ./ill ~,

'ta9Ie .0.

31I

~~o..p ;\.t:~ot1- a.",e.

~'

14 VOLUME II ~ COMPUTATION OF DISCHARGE

1.5 Channel control

Channel control for stable channelsThe term "stable channels", as used in tItis report, is a relative term. Virtually

all natural channels are subject to at least occasional change as a result of scour,deposition, or the growth of vegetation, but some alluvial channels, notably thosewhose bed and banks are composed of sand, have movable boundaries that changealmost continuously, as do their stage-discharge relations. For the purpose of thismanual stable channels include all but sand channels. Sand channels are discussedin section 1.9.

Almost all streams that are unregulated by man have channel control at thehigher stages, and among those with stable channels, all but the largest rivers havesection control at low stages. Because this section of the manual discusses only stablechannels that have channel control for the entire range of stage experienced, thediscussion is limited to the natural channels of extremely large rivers and to artificialchannels constructed without section controls. The artificial channels may be con-crete lined, partly lined or rip-rapped, or unlined_ The Manning discharge equationfor the condition of channel control, as discussed in Volume I, Chapter 9, is

(1.4)

In analysing an artificial channel of regular shape, whose dimensions are ftxed, flowat the gauge is first assumed to be at nniform depth. Consequently for any stage alldimension on the right side of the equations are known except {3. A value of {3 canbe computed for a single discharge measurement, or an average value of (3 can becomputed from a pair of discharge measurements, and thus a preliminary rating curvefor the artificial channel can be computed for the entire range of stage from theresults of a pair of discharge measurements. If subsequent discharge measurementsdepart from the computed rating curve, it is likely that the original assumption offlow at uniform depth was erroneous. That means that the energy slope, S, is notparallel to the bed slope, but varies with stage, and that the value of ~, which wascomputed on the basis of bed slope, is also in error. The rating curve must be revisedto fit the plotted discharge measurements, but the preliminary rating curve may beused as a guide in shaping the required extrapolation of the rating curve. The extra-polation should also be checked by application of the conveyance slope method ofrating extrapolation, which is described in section 1.6.

To understand the principles that underlie the stage-discharge relation forchannel cOl;ttrol in a natural channel of irregular shape we return to the Manningequation and make some simplifying assumptions in that equation. We assume, notunreasonably, that at the higher stages n is a constant and that the energy slope (S)tends to become constant. Furthermore, area (A) is approximately equal to depth(H) tImes width (W). We make the substitution for A in the equation lA, and byexpressing Sl /2 In as a constant, Cl , we obtain

Q ~ C, (H)(W)R 2/ 3 (approx)

If the hydraulic radius (R) is considered eqnal to H, and W is considered a constant,the equation becomes

Q ~ CHl. 6? ~ C(h - a)1.6? (approx) (1.5)

However, unless the stream is exceptionally wide, R is appreciably smaller than H.

USE OF STAGE-DISCHARGE RELATIONS IS

This has the effect of reducing the exponent in the last equation, although this reduc-tion may be offset by an increase of S or W with discharge. Changes in roughnesswith stage will also affect the value of the exponent. The net result of all these fac-tors is a discharge equation of the [ann

Q = C(h - a)~

where {3 will commonly vary between 1.3 and 1.8 and seldom reach a value as highas 2.0.

An example of a discharge rating for channel control in a natural stream isgiven in the following section, where compound controls that involve channelcontrol are discussed.

Compound controls involving channel controlIn the preceding section mention was made of the fact that compound control

of the stage-discharge relation usually exists in natural channels, section controlbeing effective for the lower stages and channel control being effective for the higherstages. An example of that situation is given in Figure 1.9, the rating curve for the

{Section control at low st


out and channel control becomes effective. If the low end of the rating is made atangent, a value of a:::: 0.67 ill must be used. Because the value ofa for the upper endof the rating is something less than 0.67 rn, the high end becomes concave down-ward. tf the high end of the curve is made a tangent, the effective value of a is foundto be 0.0 m. This being too Iowa value of a for the lower end of the curve, the lowend becomes concave upward.

Where the rating for a section control (low end of the curve) is a tangent, thevalue of B is expected to be greater than 2.0. In this example, B= 2.3. Where therating for a channel control (high end of the curve) is a tangent, the value of Bisexpected to be less than 2.0, and probably between 1.3 and 1.8. In this exampleB= 1.3. Should overbank flow occur the rating curve will bend to the right.

It can be demonstrated, non-rigorously> that straight line rating curves forsection control almost always have a slope greater than 2.0 and that those forchannel control have a slope less than 2.0. It has been shown that the equation fora straight line rating on log paper is Q = ChP, where Bis the slope of the line. Thefirst derivative of this equation is a measure of the change in discharge resulting froma corresponding change in stage. The first derivative is:

dQ = C{3H~-ldH

Second differences are obtained by differentiating again. The second derivative is:

~:q = CB(B - I)HP~2

Examination of the second derivative shows that second differences increase withstage when Bis greater than 2.0 and decrease with stage when Bis less than 2.0.

The hypothetical rating for a compound control is shown in Table 1.1. Thisrating represents the condition of section control at the lower stages and channelcontrol at the higher stages. Where two values of discharge are shown for an item inthe rating table, the figure in parenthesis is the exact value and the figure bearing anasterisk is the "rounded" value that normally would be used in the rating table.Experienced hydrologists will recognize this table as being a most typical one. In-spection of the second difference column shows the second differences to be increas-ing at the low water end (section control, B> 2) and decreasing at the high water end(channel control, B< 2). These are the results that one would predict from thediscussion in the preceding paragraph.

1.6 Extrapolation of rating Curves

Rating curves, more often than not, must be extrapolated beyond the range ofmeasured discharges. The preceding material in this chapter explained the principlesgoverning the shape of logarithmic rating curves to guide the hydrologist in shapingthe extrapolated segment of a rating. However, even with a knowledge of those prin~ciples, an element of uncertainty exists in the extrapolation. The purpose of tIussection of the manual is to describe methods of analysis that will reduce the degreeof uncertainty~

Low flow extrapolationLow flow extrapolation is best performed on rectangular coordinate graph

paper because the co-ordinates of zero flow can be plotted on such paper. (Zero


TABLE 1.1Hypothetical stage-discharge rating table for a compound control

17

Gauge Discharge Difference Second Gauge Discharge Difference Secondheight 1m3 ,-1) per hun- difference height Im3 ,-1) per hun- differenceIm) dredth of Im) dredth of

a metre a metre

0.30 2.740.18 0.52 7.82 0.320.31 2.92 0.01 0.53 8.14 0.01

0.32 3.110.19

0 0.54 8.470.33

0.010.33 3.30

0.190 0.55 8.81

0.340.01

0.34 3.490.19

0 0.56 9.160.35

0.010.35 3.68

0.190.01 0.57 9.52

0.360.01

0.36 3.880.20

0 0.58 9.890.37

00.37 4.08

0.200 0.59 10.26

0.370.01

0.38 4.280.20

0.01 0.60 10.640.38

0.010.39 4.49

0.210 0.61 11.03

0.390.01

0.40 4.700.21

0.01 0.62 11.430.40

0.010.41 4.92

0.220 0.63 11.84

0.410

0.42 5.140.22

0.01 0.64 12.25 0.41 00.43 5.37

0.230.01 0.65 12.66

0.410.01

0.44 5.610.24

0.01 0.66 13.080.42

00.45 5.86

0.250 0.67 13.50

0.420.01

0.46 6.110.25

0.01 0.68 13.930.43 0

0.47 6.370.26

0.01 0.69 14.360.43

0.010.48 6.64

0.270.01 0.70 14.80

0.44

0.49 6.920.28

0.01 0.75 17.052.25

0.50 7.210.29 om 0.80 19.35 2.30

0.51 7.510.30

0.01 0.85 21.702.35

0.310.01

18

.13

.12

.11

.10

.09

~

w

" .08...w'"" .07'"'"12 .06w"wD .05 -~D

VOLUME II - COMPUTATION OF DISCHARGE

/ ..03 f_-_ gage height

of zcro flow

.02 -

.01

F.XI'I,ANATION@Discharge measurement

o .1o L~L~I"---_JI_---"-'__IL_J_i__1 I J_---"-_--'-_--'-_--L_:-'

,2 .3 ,4 .S .6 .7 .8 .9 1.0 1.1 1.2 1.3 1.4 1.S


Figure 1.10 - Example of low-flow extrapolation on rectangular-coordinate graph paper

discharge cannot be plotted on logarithmic graph paper.) An example of such extra-polation is shown in Figure 1.10, where the circled points represent dischargemeasurements plotted on the co -ordinate scales of gauge height versus discharge. Therating in the example is defined by the measurements down to a gauge height of0.85 m, but an extrapolatIon to a gauge height of 0.043 m is required. Field observa-tion has shown the low point on the control (point of zero flow) to be at gaugeheight 0.027 m.

The method of extrapolation in Figure 1.10 is self-evident. A curve has beendrawn between the plotted points at gauge heights 0.027 m and 0.085 m, to mergesmoothly with the rating curve above 0.085 m. There is no assurance that the extrapolation is precise -low flow discharge measurements are required for that assurance- but the extrapolation shown is a reasonable one.

High flow extrapolationAs mentioned earlier (section I.!), the problem of high flow extrapolation can

be avoided if the unmeasured peak discharge for the rating is determined by the useof the indirect methods discussed in Volume I, Chapter 9. In the absence of suchpeak discharge determinations, estimates of the discharges corresponding to high


values of stage may be made by using one or more of the following four techniques:

(a) Conveyance slope method;(b) Areal comparison of peak runoff rates;(e) Flood routing;(d) Step backwater method.

The knowledgeable reader of this manual may notice the absence from theabove list of two techniques that were once standard practice - the velocity areamethod and Q vs Ad l/2 method. The Q vs Ad l/2 method was superior to thevelocity area method and largely supplanted it; similarly, the conveyance slopemethod, because of its superiority, has, in the last two decades, largely supplantedthe Qvs Ad l/2 method. Of the three somewhat similar methods only the conveyanceslope method is described here, because a description of the two earlier methods(Corbett, 1943, pp. 91-92) would have only academic, rather than practical, value.

Conveyance slope methodThe conveyance slope method is based on equations of steady flow, such as the

Chezy or Manning equation. In those equations,

Q=KSI/2 (1.6)

In the Chezy equation conveyance, K, equals CAR 1j2, and in the Manning equation

K;::; !AR 2/3, when metric units are used. Values of A and R corresponding to anyn

stage can be obtained from a field survey of the discharge measurement cross-section,and values of the coefficient C or n can be estimated in the field. Thus, the value ofK" embodying all the elements that can be measured or estimated, can be computedfor any given stage. (We shall soon see that errors in estimating C or n are usually notcriticaL) Values of gauge height vs K, covering the complete range of stage up to therequired peak gauge height, are computed and plotted on rectangular graph paper. Asmooth curve is fitted to the plotted points.

Values of slope, S, which is actually the energy gradient, are usually not avail-able even for measured discharges. However for the measured discharges, S1/2 canbe computed by dividing each measured discharge by its corresponding K value; S isthen obtained by squaring the resulting value of S 1/2 , Values of gauge height vs S forthe measured discharges are plotted on rectangular graph paper, a curve is fitted tothe plotted points, and the curve is extrapolated to the required peak gauge height.The extrapolation is guided by the knowledge that S tends to become constant at thehigher stages. That constant slope is the "normal" slope, or slope of the stream-bed.If the upper end of the dermed part of the curve of gauge height vs S indicates that aconstant or near constant value of S has been attained, the extrapolation of the curvecan be made with confidence. The discharge for any particular gauge height will beobtained by multiplying the corresponding value of K from the K curve by the cor-responding value of SI/2 from the S curve. We see that errors in estimating n winhave a minor effect, because the resulting percentage error in computing K is com-pensated by a similar percentage error in the opposite direction in computing S1/2In other words, the constancy of S is unaffected, but if K is, say, 10 per cent high,sI/2 win be 10 per cent low, and the two discrepancies are cancelled when multi-plication is performed. However, if the upper end of the define part of the curve ofgauge height vs S has not reached the stage where S has a near constant value, theextrapolation of the curve will be subject to uncertainty. In that situation the general


slope of the stream-bed, as determined from a topographic map, provides a guide tothe probable constant value of S that should be altained at high stages.

As mentioned in the preceding paragraph, the discharge for any particulargauge height is obtained by the multiplication of appropriate values of K and S'/2,and in that manner the upper end of the stagedischarge relation is constructed.

Figure 1.11 provides an example of the slope conveyance method, as used for

'"

~ 13

~\2 ~Conveyance curve

Slope curve

I -10% J +1%.1

\ I\ f

f\ I\ f\ f

\ /I IIf

2 3 4 5Conveyance (Kt in mitl1on5

6 7 8 9

Slope(S\. in ten-thousandths

Figure 1.11 - High-flow extrapolation by use of conveyance-slope method, Klamath River atSomcs Bar, California, U.S.A.

rating curve extrapolation at the gauging station on Klamath River at Somes Bar,California, U.S.A. The conveyance curve is based on values ofK computed from thegeometry of the measurement cross-section. The slope curve is defined to a gaugeheight of 9 m by discharge measurements (circled points), and extrapolated as thesolid line to the peak gauge height of 20 m. It appears higWy unlikely that the slopecurve at a gauge height of 20 m will fall outside the limiting dashed curves shown inFigure 1.11; in other words, it is highly unlikely that the value of Sat 20 m (0.00095)is in error by more than 10 per cent. If that is true, when the square root of S iscomputed and then used in a computation of peak discharge, the error for both S 1/2and Q reduces to 5 per cent. One can place considerable confidence in the dischargecomputed for a gauge height of 20 m in this example. It should be mentioned herethat the likelihood of a decrease in slope at high stages, as shown by the dashed curveon the left of the slope curve, is greatest when overbank flows occurs.


Areal comparison of peak runoff ratesWhen flood stages are produced over a large area by an intense general storm,

the peak discharges can often be estimated, at gauging stations where they are lacking,from the known peak discharges at surrounding stations. Usually each known peakdischarge is converted to peak discharge per unit of drainage area before makingthe analysis. In other words, peak discharge is expressed in terms of cubic metres persecond per square kilometre.

If there has been relatively little difference in storm intensity over the areaaffected, peak discharge per unit area may be con-elated with drainage area alone.If storm intensity has been variable, as in mountainous terrain, the correlation willrequire the use of some index of storm intensity as a third variable. Figure 1.12illustrates a multiple carrelatiau of that type where tlie iudepeudeut variables usedwere drainage area and maximum 24-hour basin-wide precipitation during the stormof December 1964 iu uarlh coastal Califaruia.

The peak discharges estimated by the above method should be used auly as aguide in extrapolating the rating curve at a gauging station. The basic principlesunderlying the extrapolation of logarithmic rating curves are not to be violated toaccommodate peak discharge values that are relatively gross estimates, but theestimated discharges should be given proper consideration in the extrapolationprocess.

Step backwater methodThe step backwater method is a teet~Hlque in which water surface profiles for

selected discharges are computed by successive approximations. The computationsstart at a cross-section where the stage-discharge relation is known or assumed, andth~y _proceed to the study site, which 1s the gauge site whose rating requires extra-polation. If flow is in the subcritical regime, as it usually is in natural streams, thecomputations must proceed in the upstream direction; computations proceed in thedownstream direction where flow is in the supercritical regime. In the discussion thatfollows, the usual situation of subcritical flow will be assumed.

Under conditions of subcritical flow, water surface profiles converge upstreamto a common profIle. For example, the stage for a given discharge at a gated dammay have a wide range of values depending on the position of the gates. At a studysite far enough upstream to be out of the influence of the dam, the stage for thatdischarge will be unaffected by the gate operations. Consequently, when the watersurface profile is computed for a given discharge in the reach between the dam andthe study site, the segmeut of the computed profile iu the viciuity of the study sitewill be uuaffected by the value of stage that exists at the dam. However, it will benecessary that the computations start at the dam and proceed upstream, subreachby subreach (in "steps"). It follows, therefore, that if an initial cross-section for thecomputation of the water surface profIle is selected far enough downstream from thestudy site, the computed water surface elevation at the study site, corresponding toany given discharge, will have a single value regardless of the stage selected for theinitial site.

A guide for determiuiug the required distance (L) betweeu study site andinitial section is found in the dimensionless graph in Figure 1.13. The graph, whichis nsed iu the U.S.A. (Bailey and Ray, 1966), has for its equation,

LSa SaC2d = 0.86 -" 0.64-

g- (1.7)


~

1""-",~ .,

""""- I

........ 1_.I'-.....

........

"'>f-.'-,

"'-, 1- I.. - I .I i "-I !I i""-

00 50.1 0.2 0.3 0,4 O. 0.6 0.7 0.6 0.9 1.0 1.I 1.2 1.3 1.4se'-""-8

0.7

0.1

0.2

0.8

0.3

0.9

0.6

0.5

~.~0.

Figure 1.13 - Dimensionless relation for determining distance required for backwater profilesto converge

where

L is the distance required for convergence,So is bed slope,d is mean depth for the smallest discharge to be considered,g is the acceleration of gravity,

and

C is the Chezy coefficient.

If a rated cross-section is available downstream from the study site, that cross~sectionwould be used as the initial section, of course, and there would be no need to be concerned with the above computation of L.

After the initial site is selected the next step is to divide the study reach, thatis, the reach between the initial section and the study site, into subreaches. That isdone by selecting cross-sections where major breaks in the high water profIle wouldbe expected to occur because of changes in channel geometry or roughness. Thosecross-sections are the end sections of the subreaches. The cross-sections are surveyedand roughness coefficients are selected for each subreach. That completes the fieldwork for the study.

The first step in the computations is to select a discharge, Q, for study, andobtain a stage at the initial section for use with that value of discharge. If the initialsection is a rated cross-section, that stage will be known. If the initial section is not arated cross-section, an estimated stage there is computed from the estimated meandepth Cd) for discharge Q; d in turn is estimated by cutandtry computations froma variation of the Chezy equation,

24 VOLUME I1- COMPUTATION OF DISCHARGE

(1.8)

where

C is the Chezy coefficient,A is the cross-sectional area corresponding to d, andSo is the bed slope (or water surface slope).

Step backwater computations are then applied to the subreach farthest down-stream. We have a known or estimated stage at the downstream cross-section for thevalue of Q being studied; the object of the computations is to determine the stage atthe upstream and of the subreach that is compatible with that value of Q. Thecomputation for each subreach is based on a steady flow equation, such as the Chezyor Manning equation, after the equation has been modified for non-uniformity in thesubreach by use of the difference in velocity head at the end cross-sections. (VolumeI, Chapter 9.) It will be recalled that the Chezy equation is related to the Manningequation by the formula,

(1.9)

where n is the Manning roughness coefficient and R is hydraulic radius.By shifting terms in the modified Chezy equation, the following equ2tion is

obtained for the difference in water surface elevation (h) between the upstream (sub-script 1) and downstream (subscript 2) cross-sections.

(LWI V2 (0/2 Vi-O/l V2 )!I+k)

h=h I -h2 = 2 1/2 1/2 + (1.10)C RI R 2 2g

where

h is stage;L is the length of the subreach;V is average velocity in the cross-section;g is the acceleration of gravity;k is a constant whose value is zero when a2 V~ al Vi, and whose value is 0.5

when a2 V~ al Vi; andis the velocity head coefficient whose value is dependent on the velocitydistribution in the cross-section.

As for a, in many nations its value is assumed to he 1.1; in the U.S.A. its value isassumed to he 1.0 for cross-sections of simple shape, but its value is computed forcross-sections of complex shape that require subdivision. The equation used for thatpurpose is

'Z(Kt'laJ)0/= (IJI)

K~/A}

where the subscript / refers to the conveyance (K) or area (a) of the individual sub-sections, and the subscript T refers to the conveyance (K) or area (A) of the entirecross-section. With regard to conveyance, K,

Ki ;::: Ci aiRl/2,and

KT = 'ZK,


We return to our computations for the downstream suhreach. A trial value ofstage for discharge Q is selected for the upstream cross-section, and values of A I V,and R are computed for the upstream and downstream cross-sections. Those valuesare substituted in Equations 1.10 aud 1.11 and after solving for h, the computedvalue of h is compared with the difference between the trial value of stage at the up-stream cross-section and the known or assumed stage at the downstream cross-sec-tion. Seldom will the two values agree after a single trial computation; if they do notagree, a second trial value of stage is selected for the upstream cross-section. Thecomputational procedure is repeated and the newly computed value of h is comparedwith its corresponding trial value. The computations are repeated as many times asare necessary to obtain agreement between the computed h and the differencebetween the trial stage at the upstream cross-section and the known or assumed stageat the downstream cross-section.

After a satisfactOly value of stage has been determined for the upstream cross-section, that cross-section becomes the downstream cross~section for the next sub-reach upstream. Computations similar to those described in the preceding paragraphare repeated for that subreach, and for each succeeding subreach, to provide a watersurface profIle extending to the study site that is applicable to the discharge value(Q) being studied.

If the stage corresponding to discharge Q at the initial cross-section was known,the stage computed for the study site, is satisfactory. If the stage at the initial cross-section was estimated from Equation 1.8, it is necessary to repeat the above computa~tions twicc using other values of stage at the initial cross-section for the same dis Mcharge Q. This is done to assure convergence of the water surface profiles at thestudy site. The computations are repeated, first using an initial stage about 0.25 mhigher than that orginally used, and then using an initial stage abQutO .25 ill lowerthan that originally used. All three sets of computations for discharge should resultin almost identical values of stage at the study site for discharge Q. If they do not,the initial cross-section for the step backwater computation should be moved fartherdownstream, and all computations previously described must be repeated. If thethree sets of computations give water surface profiles that converge at a commonstage at the study site, the entire procedure is repeated for other discharges untilenough data are obtained to define the high water rating for the study site.

From the preceding discussion it should be evident that the computations willbe expedited if, in a preliminary step, the three relations of stage versus area (A),hydraulic radius (R), and conveyance (K), are computed for each cross-section. Eventhen, the computations will be laborious and the use of a digital computer is there~fore recommended.

The step backwater method can be used to prepare a preliminary rating for agauging station before a single discharge measurement is made. A smooth curve isfitted to the logarithmic plot of the discharge values that are studied. The prelimiuaryrating can be revised, as necessary, when subsequent discharge measurements indicatethe need for such revision. If the step backwater method is used to define the highwater end of an existing rating curve, the discharge values investigated should includeonc or more of the highest discharges previously measured. By doing so, selectedroughness coefficients can be verified, or can be modified so that step backwatercomputations for the measured discharges provide stages at the study site that are inagreement with those observed. The computations for the high water end of therating can then be made with more confidence, in the knowledge that reasonablevalues of the roughness coefficents are being used. There will also be assurance of


continuity between the defined lower part of the rating and the computed upperpart.

Flood routingFlood-routing techniques may be used to test and improve the overall consis-

tency of records of discharge during major floods in a river basin. The number ofdirect observations of discharge during such flood periods is generally limited by theshort duration of the flood and the inaccessibility of certain stream sites. Throughthe use of flood-routing teclmiques, all observations of discharge and other hydrolog-ical events in a river basin may be combined and used to evaluate the dischargehydrograph at a single site. 11,e resulting discharge hydrograph can then be used withthe stage hydrograph for that gauge site to construct the stage-discharge relation forthe site; or, if only a peak stage is available at the site, the peak stage may be usedwith the peak discharge computed for the hydrograph to provide the end point fora rating curve extrapolation.

Flood-routing techniques, of which there are many, are based on the principleof the conservation of mass - inflow plus or minus change in storage equals outflow.It is beyond the scope of a stream gauging manual to treat the subject of floodrouting; it is discussed in most standard hydrology texts (for example, Linsley,KoWer, and Paulhus, 1949, pp. 485-541).

1.7 Shifts in the discharge rating

Shifts in the discharge rating reflect the fact that stage~discharge relations arenot permanent but vary from time to time, either gradually or abruptly, because ofchanges in the physical features that form the control for the station. If a specificchange in the rating stabilizes to the extent of lasting for more than a month or two,a new rating curve is usually prepared for the period of time during which the newstage discharge relation is effective. If the effective period of a specific rating changeis of shorter duration, the original rating curve is usually kept in effect, but duringthat period shifts or adjustments are applied to the recorded stage, so that the "new"discharge corresponding to a recorded stage is equal to the discharge from theoriginal rating that corresponds to the adjusted stage. For example, assume thatvegetal growth on the control has shifted the rating curve to the left (minus shift), sotllat in a particular range of discharge, stages are 0.015 ID higher than they originallyhad been. To obtain the discharge corresponding to a recorded stage of, say, 0.396 mthe original rating is entered with a stage of 0.381 m (0.396-0.015) and the cor-responding discharge is read. The period of time during which such stage adjustmentsare used is known as a period of shifting control.

Frequent discharge measurements should be made during a period of shiftingcontrol to define the stage-discharge relation, or magnitude(s) of shifts, during thatperiod. However, even with infrequent discharge measurements the stage-dischargerelation can be estimated during the period of shifting control if the few availablemeasurements are supplemented with a knowledge of shifting control behaviour.This section of the report discusses such behaviour. That part of the discussion thatdeals with channel control shifts does not include alluvial channels I such as sandchannels, whose boundaries change almost continuously; sand channels are discussedin section 1.9.

The formation of ice in the stream and on section controls causes shifts in thedischarge rating; ice is discussed separately in section 1.8.


Detection ofshifts in the ratingStage-discharge relations are usually subject to minor random fluctuations

resulting from the dynamic force of moving water, and because it is virtually impos-sible to sort out those minor fluctuations, a rating curve that averages the measureddischarges within close limits is considered adequate. Furthermore, it is recognizedthat discharge measurements are not error free, and consequently an average curvedrawn to fit a group of measurements is probably more accurate than any singlemeasurement that is used to define the average curve. If a group of consecutivemeasurements subsequently plot to the right or left of the average rating curve it isusually clearly evident that a shift in the rating has occurred. (An exception to thatstatement occurs where the rating curve is poorly defined or undefined in the rangeof discharge covered by the subsequent measurements; in that circumstance theindication is that the original rating curve was in error and requires revision.) If, how-ever, only onc or two measurements depart significantly from a defined segment ofthe rating curve, there may be no unanimity of opinion on whether a shift in therating has actually occurred, or whether the departure of the measurement(s) resultsfrom random error that is to be expected occasionally from measurements.

Two schools of thought exist with regard to identifying periods of shiftingcontrol. In some countries, notably the D.S.A., a pragmatic approach is taken thatis based on certain guidelines and on the judgement of the analyst. In other countries,notably the United Kingdom, the approach used is based on statistical theory. (It isreiterated that the discussion that follows excludes the constaudy shifting alluvialchannels that are discussed in section 1.9.)

In the D.S.A., if the random departure of a discharge measurement from adefined segment of the rating curve is within 5 per cent of the discharge valueind-iG-a-ted by -the rating, the measurement is considered to be a verification of therating curve. If several consecutive measurements meet the 5 per cent criterion, butif they all plot on the same side of the defined segment of the rating curve, they maybe considered to define a period of shifting control. It should be mentioned thatwhen a discharge measurement is made, the measurement is computed before thehydrologist leaves the gauging station and the result is plotted on a rating curve thatshows all previous discharge measurements. If the discharge measurement does notcheck a defined segment of the rating curve by 5 per cent or less, or if the dischargemeasurement does not check the trend of departures shown by recent measurements,the hydrologist is expected to repeat the discharge measurement.

In making a check measurement, the possibility of systematic error is eliminatMed by changing the measurement conditions as much as possible. The meter and stop-watch are changed, or the stopwatch is checked against the movement of the secondhand of a standard watch. If the measurements are being made from a bridge, boat,or cableway, the measurement verticals are changed by measuring at verticalsbetween those originally used; if wading measurements are being made, a new meas-urement section is sought, or the measurement verticals in the original section arechanged. If the check measurement checks the original rating curve or current ratingtrend by 5 per cent or less, the original discharge measurement will be given no conMsideration in the rating, although it is still entered in the records. If the check meaSMurement checks the original discharge, or the trend of that measurement if the stagehas changed, by 5 per cent or less, the two measurements are considered to bereliable evidence of a new shift in the stage-discharge relation. If the check measureMment fails to check anything that has gone before, a second check measurement ismade and the most consistent two of the three measurements are used for rating

28 VOLUME 11 ~ COMPUTATION OF DISCHARGE

analysis. TIle need for a second check measurement is a -rarity, but may possiblyoccur.

Thus, in the D.S.A., a single discharge measurement and its check measure-ment, even if unsupported by later measurements, may mark a period of shiftingcontrol. The engineer who analyses the rating does have the responsibility of ex-plaining the reason for the short-lived shift - it can often be explained as havingstarted as a result of fill (or scour) on a preceding stream rise and as having endedas a result of scour (or ftll) on the recession or on a following rise.

In Ihe UniIed Kingdom, the analysis of the rating starts in the usual way; thechronologically numbered discharge measurements are plotted on logarithmic graphpaper and are fitted by eye with a smooth curve and the rating equations establishedby computer. Where compound controls exist there may be one or more points ofinflection in the curve. In the statistical analysis that follows, each segment of therating curve between inflection points is treated separately.

1.8 Statistical analysis of the stage-discharge relation

number of current meter (discharge) observations

standard deviation of the natural logarithms of the discharge observations

~(log,Qi -log,Qif

N-I

current meter observation

average value of ]og,Qi

standard deviation of the stage values (h +a)

~[]og,(h +a) -Iog,(h +a)]2

N -I

The stage discharge relation, being a line of best fit, should be more accuratethan any of the individual gaugings. The equation of the relation may be computedas detailed in Table 1.2, which assumes that the relation plots as a straight line oulogarithmic paper.

The standard error of estimate S,(log, Q) is first calculated for the logarithmicrelation. The uncertainty in Q is subsequently calculated fromS,(log,Q). Uncertain-ties are expressed as percentages at the 95 per cent confidence level.

The standard error of estimate of log,Q may be calculated from one of thefollowing equatious:

S,(log,Q) ~ lE: =~ ~210g'Q_~2S210g,~1/2 (I.! 2). _ + l~(lOg,Qi "]Og,QC)2]1/2

S,(log,Q) - - N _ 2 (I.! 3)

The equation [ [ . ]2] 1/2~ Ql - Qc X 100 (I.!4)

t QcN ~2

S,(Q)

where

N ~

Slog,Q

S2]og,Q ~

Qi

]og,Qi ~

Slog,h

S2]og,h ~


TABLE I.2Typical manual computation of the stage-discharge relation by the method of least squares

29

Obser Q Stage 'h' (h+a) LogQ Log(h+a) XY X2vation Cumecs III where ~Y ~Xeference a=-0.115

number III

I 2.463 0.272 0.157 0.3915 -0.8041 -0.3148 0.64662 2.325 0.273 0.158 0.3664 -0.8013 -0.2936 0.64213 2.923 0.303 0.188 0.4658 -0.7258 -0.3381 0.52684 3.242 0.307 0.192 0.5108 -0.7167 -0.3661 0.51375 3.841 0.334 0.219 0.5844 -0.6596 -0.3855 0.43516 4.995 0.374 0.259 0.6985 -0.5867 -0.4098 0.34427 5.410 0.393 0.278 0.7332 -0.5560 -0.4077 0.30918 5.422 0.394 0.279 0.7342 -0.5544 -0.4070 0.30749 5.883 0.402 0.287 0.7696 -0.5421 -0.4172 0.2939

10 6.154 0.410 0.295 0.7892 -0.5302 -0.4184 0.281111 7.376 0.463 0.348 0.8678 -0.4584 -0.3978 0.210112 9.832 0.520 0.405 0.9926 -0.3925 - 0.3896 0.154113 11.321 0.548 0.433 1.0539 -0.3635 -0.3831 0.132114 12.372 0.576 0.461 1.0924 -0.3363 -0.3674 0.113115 11.825 0.580 0.465 1.0728 -0.3325 -0.3567 0.110616 13.826 0.616 0.501 1.1407 -0.30

manual on stream gauging volume 11 computation of...

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