The Hydraulic Geometry * of Stream Channels and Some Physiographic Implications By LUNA B. LEOPOLD and THOMAS MADDOCK, JR. GEOLOGICAL SURVEY PROFESSIONAL PAPER 252 Quantitative measurement of some of the hydraulic 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 simple power functions. Their interrelations are described by the term "hydraulic geometry" UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1953 UNITED STATES DEPARTMENT OF THE INTERIOR Oscar L. Chapman, Secretary GEOLOGICAL SURVEY W. E. Wrather, Director For sale by the Superintendent of Documents, U. S. Government Printing Office Washington 25, D. C. - Price 35 cents (paper cover) 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 benefited 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. HI GLOSSARY Antidune. A transient form of ripple on the stream bed analogous to a sand dune; an antidune progressively moves upstream. Control. The "control" of a stream-gaging station refers to the 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. i) Froude number. A measure of tranquility of flow defined as ~j== Gage height. Th'e' water-surface elevation referred to some arbitrary datum. Hydraulic gradient: The slope of the energy grade line; or slope of the line representing 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 gjiven cross sectior 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 mechanical 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 the station. These two types of data are used to furnish a continuous record of wate1* flow. Suspended sediment-discharge curve. A graph of water discharge plotted against concomitant m easured sus­ pended-sediment load. SYMBOLS a coefficient representing width at unit discharge k coefficient representing velocity at unit discharge 6 exponent in width-discharge relation L suspended-sediment load, weight per unit of time c coefficient representing depth at unit discharge m exponent in velocity-discharge relation C Chezy coefficient in v=C-^Rs n Manning roughness factor d mean depth p coefficient representing load at unit discharge D pipe diameter Q water discharge in cubic feet per second (cfs) / exponent in depth-discharge relation R hydraulic radius, approximately equal to mean F a function of depth in wide channels <7 sediment-grain diameter s slope of water surface in open channels g acceleration due to gravity v velocity j exponent in suspended load-discharge relation w width CONTENTS Page Page A bstract _ __________________________________________ 1 The hydraulic geometry of stream channels in relation to General statement- _________________________________ 1 sediment load—Continued The hydraulic geometry of stream channels ____________ 2 Interrelations of width, depth, velocity, and sus­ Concept of frequency of discharge._______________ 2 pended load to a variable discharge _____________ 24 Variation of hydraulic characteristics in a particular Interrelations of width, depth, velocity, and bed load cross section of a river____---_-________________ at a given discharge---________________________ 28 Variation of hydraulic characteristics in a down­ Channel-shape adjustment during individual floods-- 30 stream direction ______________________________ The role of channel roughness and slope in the adjust­ Relation of channel shape to frequency of discharge _ 16 ment of channel shape to sediment load_________ 35 The hydraulic geometry of stream channels in relation to Some physiographic implications-_____________________ 43 sediment load____________________________________ 19 The stable irrigation canal—an analogy to a graded Relations of suspended sediment to discharge in a river _ _______'_________----_----_------____--- 43 particular cross section of a river.______________ 19 Sediment and the longitudinal profile._____________ 48 Relations of suspended sediment to discharge in the Summary and interpretation_________________________ 51 downstream direction _________________________ 21 References.________________----_-___-__----------_- 53 Interrelations of width, depth, velocity, and sus­ Appendix___ _-_____----------_---------__-_--------. ' 54 pended load at a given discharge________________ 23 Index _____________________________________________ 57 ILLUSTRATIONS FIGURE 1. Cumulative frequency, or flow-duration, curve for Powder River at Arvada, WyO-____-_-___---_-_---------- 3 2. Comparison of different rates of discharge at a station and downstream._______________._______---_-_------ 4 3. Typical relations among width, depth, velocity, discharge, and gage height, Cheyenne River near Eagle Butte, S. Dak_ ______________________________________________________________________________.._ 5 4. Relation of width, depth, and velocity to discharge, Powder River at Arvada, Wyo., and at Locate, Mont_____ 6 5. Width, depth, and velocity in relation to mean annual discharge as discharge increases downstream, Powder River and tributaries, Wyoming and Montana_______ ___________________________-___-____-.----.--------. 10 6. Width, depth, and velocity in relation to mean annual discharge as discharge increases downstream, Bighorn River and tributaries, Wyoming and Montana______-___-_________-_-_-_---_----_--_------------------ H 7. Width, depth, and velocity in relation to mean annual discharge as discharge increases downstream, Arikaree, Republican, Smoky Hill, and Kansas Rivers in the Kansas River system, Kansas and Nebraska------------ 12 8. Width, depth, and velocity in relation to mean annual discharge as discharge increases downstream, main trunk of Missouri and lower Mississippi Rivers.____________-_______________-_--__-------___-___------------ 13 9. Width, depth, and velocity in relation to mean annual discharge as discharge increases downstream in various river systems.___________________________________-______-___-____----_-_-_-_---___---------_------ 15 10. Width, depth, and velocity in relation to discharges representing two frequencies, Maumee and Seioto River basins, Ohio.-__________________________________________-_______--_-______-----_--___-._----------- 17 11. Width, depth, and velocity in relation to discharge of various frequencies, Maumee and Seioto River basins, Ohio_ 18 12. Diagrammatic relations of width, depth, and velocity to discharge at a station and downstream._____________ 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-sediment load to channel-shape factors at a constant discharge of 500 cfs (from river data in appendix)___________________________________________________________-__--__'•__________ 23 16. Average relation of suspended-sediment load to channel-shape factors at four discharge rates (from river data in appendix)______________!________________________________--_____-_________---_-_-_-------__----- 24 17. Example of width, depth, velocity in relation to discharge (constructed from fig. 16)________________________ 25 18. Relation of &,-p and j (constructed from fig. 16)_______________________________________________________ 25 19. Average hydraulic geometry of river channels___________________________________-__-__----_____------- 27 20. Relation of bed load to channel-shape factors at constant discharge, from Gilbert data_______________________ 29 21. Changes in