The Hydraulic Geometrv J J of Stream Channels and Some Physiographice8 Implications GEOLOGICAL SURVEY PROFESSIONAL PAPER 252 b UNITED STATES DEPARTMENT OF THE INTERIOR FRED A. SEATON, Secretary GEOLOGICAL SURVEY Thomas B. Nolan, Director Reprinted 1954, 1959 For sale by the Superintendent of Documents, U.S. Government Printing Office Washington 25, D. nts (paper cover) Price 40 cents The Hvdraulic Geometrv J J of Stream Channels and Some Physiographic Irn plica tions By LUNA B. LEOPOLD and THOMAS MADDOCK, JR. GEOLOGICAL SURVEY PROFESSIONAL PAPER 25 2 Quantitative measurement of some of the hydrauZic factors that help to determine the shape of natural stream channels: depth, width, velocity, and sus- pended load, and how they vary with discharge as simp Ze power functions. Their interrelations are described by the term “hydradicgeometry.” UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1953 PREFACE This report was prepared by the Technical Coordination Branch of the Water Resources Division of the Geological Survey as a part of the investigation of water-utilization problems. The junior author, Mr. Maddock, is assistant chief of the Hydrology Division, Bureau of Reclamation. The authors have benefit,ed greatly from constructive suggestions made by many friends and colleagues. Particular acknowledgment is due J. Hoover Mackin and Walter B. Langbein whose counsel and criticism contributed greatly to whatever of value is contained in this report. Among the persons who read an early draft and whose suggestions are highly valued are A. N. Strahler, John P. Miller, E. W. Lane, L. C. Gottschalk, K. B. Schroeder, C. C. Nikiforoff, Charles E. Stearns, and W. M. Borland. The continuous encouragement of W. W. Rubey is appreciated. L. B. L. m GLOSSARY Antidune. A transient form of ripple on thc stream bed analogous to a sand clunc.; an antidune progressively moves upstream. Control. The “control” of a stream-gaging station refers to tho nature of the channel downstream from the meas- uring section which determines the stage-discharge relation. Current meter. An instrument for measuring water speed; consists of cups which are rotated by the flowing water; rate of rotation is a function of water speed. Cumulative frequency, or flow-duration, curve. Graph of rate of daily discharge in relation to cumulative fre- quency, indicating the percent of time a given discharge is equalled or exceeded. Discharge. The amount of water flowing in a channel expressed as volume per unit of time. 2, Froude number. A measure of tranquility of flow defined as 48__ Gage height. The water-surface elevation referred to some arbitrary datum. Hydraulic gradient. The slope of the energy grade line, or slope of the line reprcsenting the sum of kinetic and potential energy along the channel length. It is equal to the slope of the water surface in steady, uniform flow. Hydraulic radius. A measure of water depth in a channel defined as cross-sectional area of flowing water divided ~ by the length of wetted perimeter; approximates mean depth in a wide channel. Manning formula. A formula including empiric constants relating open-channel water velocity to hydraulic gradient, hydraulic radius, and channel roughness. Rating curve. The graph of discharge plotted against water-surface elevation for a given cross section of a stream. Sediment station. A river section where samples of suspended load are taken each day, or periodically. Stage, or river stage. Synonymous with gage height. Stream-gaging station. An installation including a mcchanical continuous recorder of water-surface elevation; at the same cross section periodic velocity measurements are made, and are used to construct the rating curve for thc station. These two types of data are used to furnish a continuous record of water flow. Suspended sediment-discharge curve. A graph of water discharge plotted against concomitant measured sus- pcnd ed-setlimen t load. SYMBOLS a coefficient representing width at unit discharge k coefficient representing velocity at unit discharge I, exponent in width-discharge relation L suspended-sediment load, weight per unit of time c coefficient representing depth at unit discharge m cxponent in velocity-discharge relation C Chezy coefficient in v=CJRS n Manning roughness factor d mean depth P coefficient representing load at unit discharge D pipe diameter 0 water discharge in cubic feet per second (cfs) .f exponent in depth-discharge relation R hydraulic radius, approximately equal to mean F a function of depth in wide channels 7j sediment-grain diameter S slope of water surface in open channels g acceleration due to gravity 2) velocity j exponent in suspended load-discharge relation W width Iv CONTENTS Page Page Abstract,._. .-. ~. ~ .. .~ ~ ~.~.~ ~ ~ - ~ . -. .~.~~. .. 1 The hydraulic geometry of si ream cliannels in rrlation to .. ~ ~ ~ ~ . .~ ~.~.~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ General st,atemerit,. , . 1 sediment load-Continued The hydraulic geometry of stream channels ~ . .~ 2 Interrelations of width, depth, vrlocitx, and suq- Concept, of frequency of dischargc: ~ ~ ~ ~ ~ 2 pended load to a variable discharge 24 Variation of hydraulic characteristics iii a pari icrilar Interrelation5 of nidth, depth, velocily, aiid bt~lload ect.ion of a river--. .~ ~ ~ ~ - . ~ .~~ ~ ~. 1 at a given discharge - 28 Variat,ion of hydraulic characteristics in a dowii- (’hannel-shape adjustnient during individual floods 30 stream direction ~... ~ ~ ~.~ ~ ~ ~ ~ ~ ~ - - ~ ~ ~ . 9 The role of channel roughness and qlopc 111 thr adjrl.;t- Relation of channel shape t,o frrquency of discharge- 10 merit of channel shape to srdlm(~ritload 35 The hydraulic geometry of stream chaririels in relat.ion t,o Sonic physiographic implications- 43 sediment load ~ ~ ~ .. - ~ .. -.- .. -. ~ . ~ ~.. 19 The stable irrigation canal--an analoa\ to a gt adrd Relations of suspendcd seditrient io tlisrliarge in a river. - __. ._ 13 particular cross sectioii of a rivrr ~ . 19 Sediment and 1 Iir longitudlnal piofilc .- 48 Relations of suspended scdimeiit, to discharge in 11ie Srirriniary and interpretation 51 downstream direct,ioii.. ~ .- -.. ~ .~ ~ 21 References. -. 53 Interrelations of width, dcpt,li, velocity, and 811s- App~iidiu 54 pciided load at a given dischargr-~ ~ . ~~..23 Iiidev 57 ILLUSTRATIONS FIGURE1. Cumulative frequency, or flow-duration, curve for Powder River at Arvada, Wyo--- .-._~___. ---.-.---~ ---- - 3 2. Cornparison of different rates of discharge at a station and downstream-- ____.__ ___- -.-. -. -. ~ ~ - - -- - ~ -- - 4 3. Typical relat.ions among width, depth, velocity, discharge, and gage height, (:lieyennc River iic’ar lhgle But,t>c, S. nak__..__.__________-.-----.-..--.-----------------------------------.----.--.---.--~~---.-.-5 4. Relation of widtali, depth, and velocity to discharge, Powder River at Arvada, \i-yo., and at locate, >lotit. ~ ~ - - 6 5. Width, depth, and velocity in relation to mean annual discharge as discharge increases dowiistrearii, Powder River and tributaries, Wyoming and Montana- - - - - - - - - - - - - - - - - - - - - - - -.. - - - - - - - - - - - - - .- - - .-. .- - - - - .- - 10 6. Width, depth, and velocit,y in relation to mean annual discharge as discharge incrcascs downstrearii, 13igIiorn River and tributaries, Wyoming and Montana--. -.. - - - __ - - - - - - - - - - - - - - - .- - - - - ~... - ~ - - .- - - - .. - .. ~ ~ - - 11 7. Width, depth, and velocity in relation to mean annual discharge as discharge incrrases downstrcarn, Arikaree, Republican, Smoky Hill, and Kansas Rivers in t,lie I<aiisas River system, Kansas aiid Kebraska. - -.- __ .- .- 12 8. Width, depth, and velocity in relation to mean annual discharge as discharge increases clowislream, inaiii triink of Missouri and lower Mississippi Rivers______ __ - -.- ____ ____ _____._.__ __.- __ -. __ - -.- -.~ ~... - -.- . 1 3 9. WVidt,h, depth, and velocity in relation to mean aiinual discharge as discharge increases dowiistreaiii in various river systems 15 10. Width, depth, and velocity in relat,ion to discharges representing two frrqueilcics, R/Iaumcc and Scioto River basins, Oliio._____________-_-------------.-------------------------.--------~.~-------. _...-~-- 17 11. Width, depth, and velocity iii relation to discharge of various frequencies, Mauniee and Scioto Ijivcr basins, Ohio. 18 12. Diagraniiriatic relations of widt,h, dept.h, and velocity t,o discharge at a station and dowiislreairi-. .. ~ - - ~.~ -. 19 13. Relation of suspended-sediment load to discharge, Powder River at Arvada, Wyo .__._.__._-..~. -.--. .. ..--- 20 14. Typical relations in the hydraulic geometry of natural stream channel systems ._____.__ _.___-_--. .~ ~ -. .--- 22 15. Average relation of suspended-scdiment load t,o ciiannel-shape factors at a colistant, discharge of 500 cfs (froin river data in appendix) - - - ~.- - ~ - - - __ - - - - - - - - - - - - - - - - - ___ - - - - - - - - .. - - - - - - - .. ~. ~ ~ ~.~ .~. 23 16. Average relation of suspended-scdimelit, load to channel-shape factors at four discharge rates (frolii river data in appendix) _____ __ __ __ __.-.__ __- ____ __-. ___ ___ ___ ___ __ __ -.__ -______ - -. .. .. _.- ~ ~..-. ... - -.-. - 24 17. 8;xample of width, depth, vt:locit,y iii relation to discharge (coiist,ructed froin fig. 10) .... ~__..- - . ._._. .. _~_. 25 m 18. ltelation of b, 7’and j (construct,cd froin fig. 16) .______._____._.___.._____.______ .___.. ~ ~._ . ~. ~. 25 19. Average hydraulic geometry of river chaniiels ...____.._.______._________________ ._..~-. .__~ - __... ._.. - 27 20. Relation