Publication No. FHWA NHI 01-004

December 2001

 Hydraulic Design Series Number 6 

 

 RIVER ENGINEERING FOR HIGHWAY ENCROACHMENTS 

 Highways in the River Environment 

Technical Report Documentation Page

1. Report No. FHWA NHI 01-004 

HDS 6 

2. Government Accession No. 3. Recipient's Catalog No. 

4. Title and Subtitle 5. Report Date December 2001 

RIVER ENGINEERING FOR HIGHWAY ENCROACHMENTS 

Highways in the River Environment 6. Performing Organization Code 

7. Author(s) 

E.V. Richardson, D.B. Simons, P.F. Lagasse 

8. Performing Organization Report No. 

9. Performing Organization Name and Address 10. Work Unit No. (TRAIS) 

Ayres Associates 

3665 JFK Parkway 

Building 2, Suite 200 

Fort Collins, Colorado 80525 11. Contract or Grant No. 

DTFH61-99-T-25051 

12. Sponsoring Agency Name and Address 13. Type of Report and Period Covered 

Office of Bridge Technology National Highway Institute 

FHWA, Room 3203 4600 North Fairfax Dr., Suite 800 

400 Seventh Street, SW Arlington, Virginia 22203 

Washington, D.C. 20590 

14. Sponsoring Agency Code 

15. Supplementary Notes 

COTR: Larry Jones, NHI 

Project Manager: Philip L. Thompson, FHWA 

Technical Assistants: Larry Arneson, J. Sterling Jones, Jorge E. Pagán-Ortiz, FHWA; P.E. Clopper, J.D. Schall, S.A. Schumm, 

L.W. Zevenbergen, Ayres Associates 

The Federal Highway Administration document "Highways in the River Environment - Hydraulic and Environmental Design 

Considerations" was first published in 1975, was revised in 1990, and is now issued as Hydraulic Design Series 6, "River Engineering for 

Highway Encroachments." This document has proven to be a singularly authoritative document for the design of highway associated 

hydraulic structures in moveable boundary waterways. This revised document incorporates many technical advances that have been

made in this discipline since 1990. In addition, Hydraulic Engineering Circulars (HEC) 18, 20, and 23, have been published since 1990. 

This document and the HECs provide detailed guidance on stream instability, scour, and appropriate countermeasures. In HDS-6, 

hydraulic problems at stream crossings are described in detail and the hydraulic principles of rigid and moveable boundary channels are 

discussed. 

In the United States, the average annual damage related to hydraulic problems at highway facilities on the Federal-aid system is $40 

million. Damages by streams can be reduced significantly by considering channel stability. The types of river changes to be carefully 

considered relate to: (1) lateral bank erosion; (2) degradation and aggradation of the streambed that continues over a period of years, 

and (3) natural short-term fluctuations of streambed elevation that are usually associated with the passage of floods. The major topics 

are: sediment transport, natural and human induced causes of waterway response, stream stabilization (bed and banks), hydraulic 

modeling and computer applications, and countermeasures. Case histories of typical human and natural impacts on waterways are 

analyzed. 

17. Key Words 

aggradation, degradation, alluvial channel, alluvial fan, 

river training, geomorphology, headcutting, lateral 

migration, riprap, sediment transport, scour, stable 

channel design, countermeasures 

18. Distribution Statement 

This document is available to the public through the National 

Technical Information Service, 

Springfield, VA 22161 (703) 487-4650 

and isddc.dot.gov 

19. Security Classif. (of this report) 

Unclassified 

20. Security Classif. (of this page) 

Unclassified 

21. No. of Pages 

646 

22. Price 

Form DOT F 1700.7 (8-72) Reproduction of completed page authorized 

 i 

TABLE OF CONTENTS 

LIST OF FIGURES............................................................................................................ xiii

LIST OF TABLES ............................................................................................................. xxv 

ACKNOWLEDGMENTS .................................................................................................. xxix 

LIST OF SYMBOLS......................................................................................................... xxxi 

GLOSSARY...................................................................................................................xxxvii 

1. INTRODUCTION...................................................................................................... 1.1 

 1.1 CLASSIFICATION OF RIVER CROSSINGS AND ENCROACHMENTS ......... 1.1 

 1.1.1 Types of Encroachment...................................................................... 1.2 

 1.1.2 Geometry of Bridge Crossings............................................................ 1.2 

 1.2 DYNAMICS OF NATURAL RIVERS AND THEIR TRIBUTARIES.................... 1.2 

 1.2.1 Historical Evidence of the Natural Instability of Fluvial Systems.......... 1.3 

 1.2.2 Introduction to River Hydraulics and River Response ......................... 1.7 

 1.3 EFFECTS OF HIGHWAY CONSTRUCTION ON RIVER SYSTEMS............... 1.9 

 1.3.1 Immediate Response of Rivers to Encroachment ............................... 1.9 

 1.3.2 Delayed Response of Rivers to Encroachment..................................1.10 

 1.4 EFFECTS OF RIVER DEVELOPMENT ON HIGHWAY 

 ENCROACHMENTS......................................................................................1.11 

 1.5 TECHNICAL ASPECTS..................................................................................1.13 

 1.5.1 Variables Affecting River Behavior.....................................................1.14 

 1.5.2 Basic Knowledge Required................................................................1.14 

 1.5.3 Data Requirements............................................................................1.14 

 1.6 FUTURE TECHNICAL TRENDS ....................................................................1.15 

 1.6.1 Adequacy of Current Knowledge .......................................................1.15 

 1.6.2 Research Needs................................................................................1.16 

 1.6.3 Training .............................................................................................1.17 

 1.7 OVERVIEW OF MANUAL CONTENTS..........................................................1.17 

 1.7.1 Chapter 2 - Open Channel Flow ........................................................1.17 

 1.7.2 Chapter 3 - Fundamentals of Alluvial Channel Flow...........................1.18 

 1.7.3 Chapter 4 - Sediment Transport ........................................................1.18 

 1.7.4 Chapter 5 - River Morphology and River Response...........................1.19 

 1.7.5 Chapter 6 - River Stabilization and Bank Protection .........................1.19 

 1.7.6 Chapter 7 - Scour at Bridges .............................................................1.19 

 1.7.7 Chapter 8 - Data Need and Data Sources .........................................1.20 
 ii 

 1.7.8 Chapter 9 - Design Considerations for Highway Encroachment 

 and River Crossings .......................................................1.20

 1.7.9 Chapter 10 - Overview Examples of Design for Highways in the 

 River Environment ........................................................1.21 

 1.7.10 Chapter 11 - References ...................................................................1.21 

 1.7.11 Appendices........................................................................................1.21 

 1.8 DUAL SYSTEM OF UNITS.............................................................................1.21 

2. OPEN CHANNEL FLOW ........................................................................................... 2.1 

 2.1 INTRODUCTION ............................................................................................. 2.1 

 2.1.1 Definitions........................................................................................... 2.1 

 2.2 THREE BASIC EQUATIONS........................................................................... 2.3 

 2.2.1 Conservation of Mass ......................................................................... 2.3 

 2.2.2 Conservation of Linear Momentum ..................................................... 2.6 

 2.2.3 Conservation of Energy ...................................................................... 2.8 

 2.3 HYDROSTATICS............................................................................................2.12 

 2.4 STEADY UNIFORM FLOW ............................................................................2.14 

 2.4.1 Introduction........................................................................................2.14 

 2.4.2 Shear Stress, Velocity Distribution, and Average Velocity .................2.16 

 2.4.3 Other Velocity Equations ...................................................................2.21 

 2.4.4 Average Boundary Shear Stress........................................................2.24 

 2.4.5 Relation Between Shear Stress and Velocity.....................................2.27 

 2.4.6 Energy and Momentum Coefficients for Rivers..................................2.28 

 2.5 UNSTEADY FLOW.........................................................................................2.32 

 2.5.1 Gravity Waves ...................................................................................2.32

 2.5.2 Surges ...............................................................................................2.34 

 2.5.3 Hydraulic Jump..................................................................................2.35

 2.5.4 Roll Waves ........................................................................................2.35 

 2.6 STEADY RAPIDLY VARYING FLOW.............................................................2.38 

 2.6.1 Flow Through Transitions ..................................................................2.38 

 2.6.2 Specific Energy Diagram ...................................................................2.39 

 2.6.3 Specific Discharge Diagram...............................................................2.40 

 2.6.4 Transitions With Super Critical Flows ................................................2.43 

 2.6.5 Flow Over Drop Structures ................................................................2.44 

 2.7 FLOW IN BENDS ...........................................................................................2.45 

 2.7.1 Types of Bends..................................................................................2.45

 2.7.2 Transverse Velocity Distribution in Bends..........................................2.46 
 iii 

 2.7.3 Subcritical Flow in Bends...................................................................2.47 

 2.7.4 Supercritical Flow in Bends................................................................2.49 

 2.8 GRADUALLY VARIED FLOW ........................................................................2.50 

 2.8.1 Introduction........................................................................................2.50 

 2.8.2 Classification of Flow Profiles ............................................................2.50 

 2.8.3 Standard Step Method for the Computation of Water Surface 

 Profiles ..............................................................................................2.55 

 2.9 STREAM GAGING .........................................................................................2.56 

 2.9.1 Introduction........................................................................................2.56 

 2.9.2 Gaging Station...................................................................................2.56 

 2.9.3 Measuring Water Discharge ..............................................................2.60 

 2.9.4 Stage-Discharge Relation..................................................................2.61 

 2.10 HYDRAULICS OF BRIDGE WATERWAYS....................................................2.62 

 2.10.1 Introduction........................................................................................2.62 

 2.10.2 Backwater Effects on Waterway Openings........................................2.63 

 2.10.3 Effects of a Submerged Superstructure.............................................2.63 

 2.10.4 Effects of Supercritical Flow...............................................................2.63 

 2.10.5 Types of Flow in Bridge Openings .....................................................2.63 

 2.11 COMPUTER MODELS FOR HIGHWAY BRIDGES........................................2.65 

 2.11.1 One-Dimensional Computer Models ..................................................2.65 

 2.11.2 Two-Dimensional Computer Models ..................................................2.67 

 2.12 HYDRAULICS OF CULVERT FLOW..............................................................2.67 

 2.12.1 Introduction........................................................................................2.67 

 2.13 ROADWAY OVERTOPPING..........................................................................2.68 

 2.14 SOLVED PROBLEMS OPEN CHANNEL FLOW (SI) .....................................2.68 

 2.14.1 PROBLEM 1 Evaluation of Correction Factors α and β .....................2.68 

 2.14.2 PROBLEM 2 Velocity Profiles and Shear Stress................................2.71 

 2.14.3 PROBLEM 3 Superelevation in Bends...............................................2.74 

 2.14.4 PROBLEM 4 Maximum Stream Constriction Without Causing 

 Backwater..........................................................................................2.75 

 2.14.5 PROBLEM 5 Maximum Water Surface Elevation Upstream of a 

 Grade Control Structure Without Backwater......................................2.76 

 2.15 SOLVED PROBLEMS OPEN CHANNEL FLOW (ENGLISH) .........................2.79 

 2.15.1 PROBLEM 1 Evaluation of Correction Factors α and β .....................2.79 

 2.15.2 PROBLEM 2 Velocity Profiles and Shear Stress................................2.81 

 2.15.3 PROBLEM 3 Superelevation in Bends...............................................2.84 
 iv 

 2.15.4 PROBLEM 4 Maximum Stream Constriction Without Causing 

 Backwater..........................................................................................2.85 

 2.15.5 PROBLEM 5 Maximum Elevation of a Grade Control 

 Structure Without Backwater .............................................................2.86 

3. FUNDAMENTALS OF ALLUVIAL CHANNEL FLOW ................................................. 3.1 

 3.1 INTRODUCTION ............................................................................................. 3.1 

 3.2 SEDIMENT PROPERTIES AND MEASUREMENT TECHNIQUES ................. 3.1 

 3.2.1 Particle Size........................................................................................ 3.1 

 3.2.2 Particle Shape .................................................................................... 3.2 

 3.2.3 Fall Velocity ........................................................................................ 3.4 

 3.2.4 Sediment Size Distribution.................................................................. 3.6 

 3.2.5 Specific Weight..................................................................................3.11 

 3.2.6 Porosity..............................................................................................3.11 

 3.2.7 Cohesion ...........................................................................................3.12 

 3.2.8 Angle of Repose ................................................................................3.12 

 3.3 FLOW IN SANDBED CHANNELS ..................................................................3.12 

 3.3.1 Regimes of Flow in Alluvial Channels ................................................3.13 

 3.3.2 Bed Configuration..............................................................................3.15 

 3.3.3 Plane Bed Without Sediment Movement ...........................................3.17 

 3.3.4 Ripples...............................................................................................3.17 

 3.3.5 Dunes ................................................................................................3.17 

 3.3.6 Plane Bed With Movement ................................................................3.18 

 3.3.7 Antidunes...........................................................................................3.18 

 3.3.8 Chutes and Pools ..............................................................................3.19 

 3.3.9 Regime of Flow, Bed Configuration, and Froude Number..................3.19 

 3.3.10 Bars ...................................................................................................3.19 

 3.4 RESISTANCE TO FLOW IN ALLUVIAL CHANNELS .....................................3.21 

 3.4.1 Depth.................................................................................................3.22 

 3.4.2 Slope .................................................................................................3.23 

 3.4.3 Apparent Viscosity and Density .........................................................3.23 

 3.4.4 Size of Bed Material...........................................................................3.24 

 3.4.5 Size Gradation ..................................................................................3.25 

 3.4.6 Fall Velocity .......................................................................................3.25 

 3.4.7 Shape Factor for the Reach and Cross-Section.................................3.26 

 3.4.8 Seepage Force ..................................................................................3.26 

 3.4.9 Bed Material Concentration ...............................................................3.26 

 3.4.10 Fine-Material Concentration...............................................................3.26 

 3.4.11 Bedform Predictor and Manning's n Values for Sand-Bed Streams ...3.27 

 3.4.12 How Bedform Changes Affect Highways in the River Environment....3.28 

 3.4.13 Alluvial Processes and Resistance to Flow in Coarse Material 

 Streams.............................................................................................3.29 
 v 

 3.5 BEGINNING OF MOTION ..............................................................................3.32 

 3.5.1 Introduction........................................................................................3.32 

 3.5.2 Theoretical Considerations ................................................................3.32 

 3.5.3 Theory of Beginning of Motion...........................................................3.33 

 3.5.4 Shields Diagram ................................................................................3.36 

 3.5.5 Equations for Flow and Sediment Variables for Beginning of Motion.3.39 

 3.5.6 Tables for Determining Critical Velocities...........................................3.43 

 3.5.7 Figures for Determining Critical Shear Stress or Velocity ..................3.44 

 3.5.8 Summary ...........................................................................................3.45 

 3.6 SEDIMENT DISCHARGE MEASUREMENT...................................................3.46 

 3.6.1 Introduction........................................................................................3.46 

 3.6.2 Terminology.......................................................................................3.46 

 3.6.3 Suspended Sediment Discharge Measurement.................................3.47 

 3.6.4 Velocity Weighted Mean Suspended-Sediment Concentration ..........3.47 

 3.6.5 Suspended Sediment Hydrograph .....................................................3.49 

 3.6.6 Determination of Daily Suspended-Sediment Discharge....................3.49 

 3.6.7 Total Sediment Discharge..................................................................3.51 

 3.7 SOLVED PROBLEMS FOR ALLUVIAL CHANNEL FLOW (SI).......................3.52 

 3.7.1 PROBLEM 1 Sediment Properties and Fall Velocities .......................3.52 

 3.7.2 PROBLEM 2 Angle of Repose...........................................................3.56 

 3.7.3 PROBLEM 3 Resistance to Flow in Alluvial Channels........................3.57 

 3.7.4 PROBLEM 4 Beginning of Motion......................................................3.59 

 3.8 SOLVED PROBLEMS FOR ALLUVIAL CHANNEL FLOW (ENGLISH) ..........3.61 

 3.8.1 PROBLEM 1 Sediment Properties and Fall Velocities .......................3.61 

 3.8.2 PROBLEM 2 Angle of Repose...........................................................3.62 

 3.8.3 PROBLEM 3 Resistance to Flow in Alluvial Channels........................3.62 

 3.8.4 PROBLEM 4 Beginning of Motion......................................................3.64 

4. SEDIMENT TRANSPORT ......................................................................................... 4.1 

 4.1 INTRODUCTION ............................................................................................. 4.1 

 4.2 DEFINITIONS.................................................................................................. 4.1

 4.3 GENERAL CONSIDERATIONS....................................................................... 4.3 

 4.3.1 Source of Sediment Transport............................................................ 4.3 

 4.3.2 Mode of Sediment Transport .............................................................. 4.4 

 4.3.3 Total Sediment Discharge................................................................... 4.5 

 4.4 SUSPENDED BED SEDIMENT DISCHARGE.................................................. 4.6 

 4.5 BED SEDIMENT DISCHARGE........................................................................4.10 

 4.5.1 Meyer-Peter and Müller Bed Load Equation ......................................4.10 

 4.5.2 Einstein's Method of Computing Bed Sediment Discharge ................4.13 

 4.5.3 Comparison of Meyer-Peter and Müller and Einstein Contact 

 Load Equations..................................................................................4.21 

 4.5.4 Colby's Method of Estimating Bed Sediment Discharge ....................4.22 

 4.5.5 Relative Influence of Variables ..........................................................4.24 
 vi 

 4.6 POWER FUNCTION RELATIONSHIPS ..........................................................4.27 

 4.6.1 Introduction........................................................................................4.27 

 4.6.2 Basic Power Function Relationship....................................................4.27 

 4.6.3 Expanded Power Function Relationship.............................................4.29 

 4.7 YANG'S EQUATIONS .....................................................................................4.30 

 4.8 CONVERSION FACTORS...............................................................................4.31 

 4.9 APPLICATION OF SELECTED SEDIMENT TRANSPORT EQUATIONS .......4.32 

 4.10 SUMMARY ......................................................................................................4.34 

 4.11 SEDIMENT TRANSPORTATION ANALYSIS PROCEDURE...........................4.34 

 4.11.1 Step 1: Determine if Sediment Transport Computations are 

 Necessary ...........................................................................4.34 

 4.11.2 Step 2: Determine Scope of Sediment Transport Analysis...............4.35 

 4.11.3 Step 3: Determine the Type of Analysis...........................................4.35 

 4.12 SOLVED PROBLEMS FOR SEDIMENT TRANSPORT (SI) ............................4.37 

 4.12.1 Introduction.......................................................................................4.37 

 4.12.2 Problem 1 Suspended Sediment Concentration Profile ....................4.37 

 4.12.3 Problem 2 Using the Meyer-Peter and Müller Equation 

 Calculate the Bed Sediment Discharge (Bed Load)..........................4.38 

 4.12.4 Problem 3 Application of the Einstein Method to Calculate 

 Total Bed-Material Discharge ...........................................................4.39 

 4.12.5 Problem 4 Calculation of Total Bed-Material Discharge 

 Using Colby's Method.......................................................................4.39 

 4.12.6 Problem 5 Calculation of Total Bed-Material Discharge Using 

 the Basic Power Function Relationship.............................................4.40 

 4.12.7 Problem 6 Calculation of Total Bed-Material Discharge Using 

 the Expanded Power Function Relationship .....................................4.40 

 4.12.8 Problem 7 Calculate Total Bed-Material Discharge 

 Using Yang's Sand Equation ............................................................4.41 

 4.12.9 Problem 8 Sediment Transport Conversions ....................................4.42 

 4.13 SOLVED PROBLEMS FOR SEDIMENT TRANSPORT (ENGLISH) .................4.43 

 4.13.1 Introduction.......................................................................................4.43 

 4.13.2 Problem 1 Suspended Sediment Concentration Profile ....................4.43 

 4.13.3 Problem 2 Using the Meyer-Peter and Müller Equation 

 Calculate the Bed Sediment Discharge (Bed Load)..........................4.44 

 4.13.4 Problem 3 Application of the Einstein Method to Calculate 

 Total Bed-Material Discharge ...........................................................4.45 

 4.13.5 Problem 4 Calculation of Total Bed-Material Discharge 

 Using Colby's Method.......................................................................4.51 

 4.13.6 Problem 5 Calculation of Total Bed-Material Discharge Using 

 the Basic Power Function Relationship.............................................4.53 

 4.13.7 Problem 6 Calculation of Total Bed-Material Discharge Using 

 the Expanded Power Function Relationship .....................................4.54 

 4.13.8 Problem 7 Calculate Total Bed-Material Discharge Using 

 Yang's Equation ...............................................................................4.54 

 4.13.9 Problem 8 Sediment Transport Conversions ....................................4.55 
 vii 

5. RIVER MORPHOLOGY AND RIVER RESPONSE.................................................... 5.1 

 5.1 INTRODUCTION ............................................................................................. 5.1 

 5.2 FLUVIAL CYCLES AND PROCESSES ........................................................... 5.1 

 5.2.1 Youthful, Mature, and Old Streams..................................................... 5.1 

 5.2.2 Floodplain and Delta Formations ........................................................ 5.2 

 5.2.3 Alluvial Fans ....................................................................................... 5.2 

 5.2.4 Nickpoint Migration and Headcutting .................................................. 5.3 

 5.2.5 Geomorphic Threshold ....................................................................... 5.5 

 5.3 VARIABILITY AND CHANGE IN ALLUVIAL RIVERS ...................................... 5.5 

 5.3.1 Differences Through Time .................................................................. 5.7 

 5.3.2 Differences Between Reaches............................................................ 5.8 

 5.3.3 Summary ...........................................................................................5.12 

 5.4 STREAM FORM AND GEOMETRY OF ALLUVIAL CHANNELS ....................5.12 

 5.4.1 Classification of River Channels ........................................................5.13 

 5.4.2 Straight River Channels.....................................................................5.17 

 5.4.3 Meandering River Channels...............................................................5.17 

 5.4.4 Braided River Channels .....................................................................5.20 

 5.4.5 River Conditions for Meandering and Braiding...................................5.21 

 5.4.6 Hydraulic Geometry of Alluvial Channels ..........................................5.23 

 5.4.7 Dominant Discharge in Alluvial Rivers ...............................................5.27 

 5.4.8 River Profiles and Bed Material..........................................................5.28 

 5.5 QUALITATIVE RESPONSE OF RIVER SYSTEMS ........................................5.28 

 5.5.1 General River Response to Change ..................................................5.29 

 5.5.2 Prediction of Channel Response to Change ......................................5.32 

 5.5.3 River Pattern Thresholds and Response ...........................................5.34 

 5.6 MODELING OF RIVER SYSTEMS.................................................................5.39 

 5.6.1 Physical Modeling..............................................................................5.40 

 5.6.2 Computer Modeling ...........................................................................5.43 

 5.6.3 Data Needs........................................................................................5.47 

 5.7 HIGHWAY PROBLEMS RELATED TO GRADATION CHANGES.................5.48 

 5.7.1 Changes Due to Human Activities .....................................................5.48 

 5.7.2 Natural Causes..................................................................................5.50

 5.7.3 Resulting Problems at Highway Crossings ........................................5.51 

 5.8 STREAM STABILITY PROBLEMS AT HIGHWAY CROSSINGS....................5.52 

 5.8.1 Bank Stability.....................................................................................5.53 

 5.8.2 Stability Problems Associated With Channel Relocation....................5.54 

 5.8.3 Assessment of Stability for Relocated Streams .................................5.59 
 viii 

 5.8.4 Estimation of Future Channel Stability and Behavior .........................5.60 

 5.8.5 Advances in Predicting Meander Migration........................................5.64 

 5.9 SOLVED PROBLEMS RIVER MORPHOLOGY AND RESPONSE.................5.67 

 5.9.1 PROBLEM 1 Meandering and Braiding..............................................5.67 

 5.9.2 PROBLEM 2 Classification of Alluvial Reaches .................................5.68 

 5.9.3 PROBLEM 3 Channel Response to Changes in Watershed 

 Conditions .........................................................................................5.69 

 5.9.4 PROBLEM 4 Channel Migration Rate ................................................5.69 

 5.9.5 PROBLEM 5 At-A-Station and Downstream Hydraulic Geometry 

 Relationships (SI) ..............................................................................5.71 

 5.9.6 PROBLEM 6 At-A-Station and Downstream Hydraulic Geometry 

 Relationships (English) ......................................................................5.73

 5.9.7 PROBLEM 7 Downstream Sediment Size Distribution.......................5.74 

 5.9.8 PROBLEM 8 Scale Ratios for Physical Models (SI)...........................5.75 

 5.9.9 PROBLEM 9 Scale Ratios for Physical Models (English)...................5.76 

6. RIVER STABILIZATION AND BANK PROTECTION................................................. 6.1 

 6.1 OVERVIEW ..................................................................................................... 6.1

 6.2 STREAM BANK EROSION.............................................................................. 6.2 

 6.2.1 Causes of Stream Bank Failure .......................................................... 6.2 

 6.2.2 Bed and Bank Material........................................................................ 6.3 

 6.2.3 Subsurface Flow................................................................................. 6.3 

 6.2.4 Piping of River Banks ......................................................................... 6.5 

 6.2.5 Mass Wasting..................................................................................... 6.5

 6.3 RIVER TRAINING AND STABILIZATION........................................................ 6.6 

 6.3.1 Fixed Points........................................................................................ 6.6 

 6.3.2 Radius of Curvature............................................................................ 6.6 

 6.3.3 Countermeasures for Channel Instability ............................................ 6.7 

 6.3.4 Countermeasure Design Guidelines ................................................... 6.8 

 6.3.5 Protection of Training Works .............................................................. 6.9 

 6.4 FLOW CONTROL STRUCTURES .................................................................6.10 

 6.4.1 Spurs .................................................................................................6.10 

 6.4.2 Bendway Weirs..................................................................................6.12 

 6.4.3 Hardpoints .........................................................................................6.12 

 6.4.4 Retards..............................................................................................6.12 

 6.4.5 Dikes (Floodplain)..............................................................................6.14

 6.4.6 Dikes (Channel).................................................................................6.14

 6.4.7 Jetties ................................................................................................6.16 

 6.4.8 Bulkheads..........................................................................................6.17 

 6.4.9 Fencing..............................................................................................6.18 

 6.4.10 Guide Banks ......................................................................................6.18

 6.4.11 Drop Structures .................................................................................6.19 
 ix 

 6.5 RIPRAP DESIGN AND PLACEMENT.............................................................6.20 

 6.5.1 Factors to Consider ...........................................................................6.20 

 6.5.2 Stability Factor Design Methods ........................................................6.21 

 6.5.3 Velocity Profile and Tractive Force ...................................................6.24 

 6.5.4 U.S. Army Corps of Engineers Design Equation................................6.25 

 6.5.5 Riprap Gradation and Thickness .......................................................6.28 

 6.5.6 Riprap Placement ..............................................................................6.30 

 6.5.7 Riprap Failure Modes.........................................................................6.33 

 6.6 BANK PROTECTION OTHER THAN RIPRAP ...............................................6.36 

 6.6.1 Bioengineering Erosion Control .........................................................6.36 

 6.6.2 Bioengineering Countermeasures......................................................6.37 

 6.6.3 Rock-and-Wire Mattresses ................................................................6.38 

 6.6.4 Gabions .............................................................................................6.39 

 6.6.5 Sacks.................................................................................................6.39 

 6.6.6 Articulated Concrete Block Systems (ACBs)......................................6.40 

 6.6.7 Used Tires .........................................................................................6.41 

 6.6.8 Soil Cement .......................................................................................6.42 

 6.7 FILTERS.........................................................................................................6.43 

 6.8 OVERTOPPING FLOW ON EMBANKMENTS ...............................................6.44 

 6.8.1 Introduction........................................................................................6.44 

 6.8.2 Hydraulics of Overtopping Flow .........................................................6.45 

 6.8.3 Mechanics of Overflow Erosion..........................................................6.48 

 6.8.4 Erosion Protection in Overtoppng Flow..............................................6.50 

 6.8.5 Overtopping Protection Systems........................................................6.51 

 6.9 ENVIRONMENTAL CONSIDERATIONS ........................................................6.53 

 6.9.1 Environmental Impacts ......................................................................6.53 

 6.9.2 Effects of Channelization ...................................................................6.54 

 6.9.3 Channel Restoration and Rehabilitation.............................................6.55 

 6.10 SOLVED PROBLEMS FOR STABILITY OF RIPRAP (SI) ..............................6.55 

 6.10.1 PROBLEM 1 Stability of Particles Under Downslope Flow.................6.55 

 6.10.2 PROBLEM 2 Riprap Design on Embankment Slopes ........................6.56 

 6.10.3 PROBLEM 3 Stability Factors for Riprap Design ...............................6.57 

 6.10.4 PROBLEM 4 Riprap Design on an Abutment.....................................6.60 

 6.11 SOLVED PROBLEMS FOR FILTER DESIGN (SI) .........................................6.61 

 6.11.1 PROBLEM 1 Filter Design .................................................................6.61 

 6.11.2 PROBLEM 2 Filter Design .................................................................6.62 
 x 

 6.12 SOLVED PROBLEMS FOR STABILITY OF RIPRAP (ENGLISH) ..................6.64 

 6.12.1 PROBLEM 1 Stability of Particles Under Downslope Flow.................6.64 

 6.12.2 PROBLEM 2 Riprap Design on Embankment Slopes ........................6.65 

 6.12.3 PROBLEM 3 Stability Factors for Riprap Design ...............................6.67 

 6.12.4 PROBLEM 4 Riprap Design on an Abutment.....................................6.69 

 6.13 SOLVED PROBLEMS FOR FILTER DESIGN (ENGLISH) .............................6.70 

 6.13.1 PROBLEM 1 Filter Design .................................................................6.70 

 6.13.2 PROBLEM 2 Filter Design .................................................................6.71 

7. SCOUR AT BRIDGES............................................................................................... 7.1 

 7.1 INTRODUCTION ............................................................................................. 7.1 

 7.1.1 General............................................................................................... 7.1 

 7.1.2 Costs of Bridge Failure From Scour.................................................... 7.1 

 7.1.3 Bridge Scour Evaluation Program....................................................... 7.2 

 7.1.4 Comprehensive Scour Analysis .......................................................... 7.2 

 7.2 TOTAL SCOUR............................................................................................... 7.4 

 7.3 CLEAR-WATER AND LIVE-BED SCOUR ....................................................... 7.4 

 7.4 ARMORING..................................................................................................... 7.6

 7.5 LONG-TERM BED ELEVATION CHANGES.................................................... 7.6 

 7.6 GENERAL SCOUR.......................................................................................... 7.7 

 7.6.1 Contraction Scour............................................................................... 7.8 

 7.6.2 Computer Models for General Scour .................................................7.12 

 7.7 LOCAL SCOUR AT PIERS.............................................................................7.12 

 7.7.1 Introduction........................................................................................7.12 

 7.7.2 Comparison of Pier Scour Equations.................................................7.13 

 7.7.3 Colorado State University's Equation.................................................7.13 

 7.7.4 FHWA HEC-18 Equation ...................................................................7.16 

 7.7.5 Pier Scour in Cohesive Bed Material..................................................7.19 

 7.7.6 Pier Scour for Other Pier Geometry, Flow Conditions, and Debris.....7.20 

 7.7.7 Topwidth of Scour Holes....................................................................7.20 

 7.7.8 Physical Model Studies......................................................................7.21 

 

 7.8 LOCAL SCOUR AT ABUTMENTS..................................................................7.21 

 7.8.1 Introduction........................................................................................7.21 

 7.8.2 Commentary on Abutment Scour Equations......................................7.23 

 7.8.3 Abutment Site Conditions ..................................................................7.24 

 7.8.4 Abutment Shape................................................................................7.25 

 7.8.5 Froehlich's Live-Bed Scour Equation .................................................7.25 

 7.8.6 1975 and 1990 HIRE Equation ..........................................................7.26 

 7.9 SCOUR PROBLEMS......................................................................................7.27 
 xi 

8. DATA NEEDS AND DATA SOURCES ...................................................................... 8.1 

 8.1 BASIC DATA NEEDS ...................................................................................... 8.1 

 8.1.1 Area Maps .......................................................................................... 8.1 

 8.1.2 Vicinity Maps....................................................................................... 8.1 

 8.1.3 Site Maps............................................................................................ 8.1 

 8.1.4 Aerial and Other Photographs ............................................................ 8.2 

 8.1.5 Field Inspection................................................................................... 8.2 

 8.1.6 Geologic Map ..................................................................................... 8.2

 8.1.7 Climatologic Data................................................................................ 8.3

 8.1.8 Hydraulic Data .................................................................................... 8.3 

 8.1.9 Hydrologic Data .................................................................................. 8.4

 8.1.10 Environmental Data ............................................................................ 8.5 

 8.2 CHECKLIST OF DATA NEEDS....................................................................... 8.5 

 8.3 DATA SOURCES ............................................................................................ 8.5 

 8.4 COMPUTERIZED LITERATURE AND DATA SEARCH..................................8.10 

 8.4.1 COMPENDEX....................................................................................8.10 

 8.4.2 ENVIRONMENTAL BIBLIOGRAPHY.................................................8.10 

 8.4.3 CIVIL ENGINEERING DATABASE....................................................8.11 

 8.4.4 WATER RESOURCES ABSTRACTS................................................8.11 

 8.4.5 TRIS ..................................................................................................8.11 

 8.4.6 GeoRef ..............................................................................................8.12 

9. DESIGN CONSIDERATIONS FOR HIGHWAY ENCROACHMENTS AND 

 RIVER CROSSINGS ................................................................................................. 9.1 

 9.1 INTRODUCTION ............................................................................................. 9.1 

 9.2 PRINCIPAL FACTORS TO BE CONSIDERED IN DESIGN ............................ 9.1 

 9.2.1 Types of Rivers................................................................................... 9.1 

 9.2.2 Location of the Crossing or Encroachment ......................................... 9.2 

 9.2.3 River Characteristics........................................................................... 9.2 

 9.2.4 River Geometry................................................................................... 9.2

 9.2.5 Hydrologic Data .................................................................................. 9.3

 9.2.6 Hydraulic Data .................................................................................... 9.3 

 9.2.7 Characteristics of the Watershed........................................................ 9.3 

 9.2.8 Flow Alignment ................................................................................... 9.4

 9.2.9 Flow on the Floodplain........................................................................ 9.4 

 9.2.10 Site Selection...................................................................................... 9.5 

 9.2.11 Channel Stability Investigations .......................................................... 9.5 

 9.2.12 Short-Term Response ....................................................................... 9.5 

 9.2.13 Long-Term Response ......................................................................... 9.6 

 9.3 PROCEDURE FOR EVALUATION AND DESIGN OF RIVER 

 CROSSINGS AND ENCROACHMENTS ......................................................... 9.6 

 9.3.1 Approach to River Engineering Projects ............................................. 9.6 

 9.3.2 Project Initiation .................................................................................. 9.7 

 9.4 CONCEPTUAL EXAMPLES OF RIVER ENCROACHMENTS......................... 9.8 
 xii 

 9.5 PRACTICAL EXAMPLES OF RIVER ENCROACHMENTS ............................9.20 

 9.5.1 Cimarron River, East of Okeene, Oklahoma (Example 1)..................9.20 

 9.5.2 Arkansas River, North of Bixby, Oklahoma (Example 2)....................9.20 

 9.5.3 Washita River, North of Maysville, Oklahoma (Example 3)................9.20 

 9.5.4 Beaver River, North of Laverne, Oklahoma (Example 4) ...................9.23 

 9.5.5 Powder River, 40 Miles East of Buffalo, Wyoming (Example 5).........9.23 

 9.5.6 North Platte River, Near Guernsey, Wyoming (Example 6) ...............9.25 

 9.5.7 Coal Creek, Tributary of Powder River, Wyoming (Example 7)..........9.26 

 9.5.8 South Fork of Deer River at US-51 Near Halls, Tennessee 

 (Example 8) .......................................................................................9.26 

 9.5.9 Elk Creek at SR-15 Near Jackson, Nebraska (Example 9) ................9.30 

 9.5.10 Big Elk Creek at I-90 Near Piedmont, South Dakota (Example 10)....9.32 

 9.5.11 Outlet Creek at US-101 Near Longvale, California (Example 11).......9.33 

 9.5.12 Nojoqui Creek at US-101 at Buellton, California (Example 12) ..........9.34 

 9.5.13 Turkey Creek at I-10 Near Newton, Mississippi (Example 13) ...........9.35 

 9.5.14 Gravel Mining on the Russian River, California (Example 14)............9.36 

 9.5.15 Nowood River and Ten Sleep Creek Confluence, Wyoming 

 (Example 15) .....................................................................................9.37 

 9.5.16 Middle Fork Powder River, Wyoming (Example 16)...........................9.37 

 9.6 CONCLUDING REMARKS ON DESIGN CONSIDERATIONS .......................9.37 

10. OVERVIEW EXAMPLES OF DESIGN FOR HIGHWAYS IN THE RIVER 

 ENVIRONMENT ......................................................................................................10.1 

 10.1 OVERVIEW EXAMPLE 1 - BIJOU CREEK....................................................10.1 

 10.1.1 Level 1 - Reconnaissance and Geomorphic Analysis .......................10.2 

 10.1.2 Level 2 - Quantitative Engineering Analysis ......................................10.5 

 10.1.3 Riprap Design for Alternatives 1, 2, and 3........................................10.11 

 10.1.4 Design for Alternatives 4 and 5........................................................10.13 

 10.2 OVERVIEW EXAMPLE 2 - RILLITO RIVER ................................................10.17 

 10.2.1 Background .....................................................................................10.17 

 10.2.2 Level 1 - Qualitative Geomorphic Analysis ......................................10.19 

 10.2.3 Level 2 - Engineering Geomorphic Analysis ....................................10.23 

 10.2.4 Level 3 - Sediment Routing for the 100-Year Flood .........................10.31 

 10.2.5 Results of Analysis ..........................................................................10.32 

 10.3 BRI-STARS SEDIMENT TRANSPORT MODELING EXAMPLE..................10.38 

 10.3.1 Background .....................................................................................10.38 

 10.3.2 Level 1 - Reconnaissance and Geomorphic Analysis .....................10.40 

 10.3.3 Level 2 - Hydraulic Engineering Analysis .........................................10.41 

 10.3.4 Level 2 - Scour Analysis Results......................................................10.43 

 10.3.5 Level 3 - BRI-STARS Analysis.........................................................10.45 

11. REFERENCES ........................................................................................................11.1 

APPENDIX A - Metric System, Conversion Factors, and Water Properties ..................... A.1 

APPENDIX B - Analysis of Selected Sediment Transport Relationships........................... B.1 

APPENDIX C - Index........................................................................................................C.1 
 xiii 

LIST OF FIGURES 

Figure 1.1 Geometric properties of bridge crossings .................................................... 1.3 

Figure 1.2 Comparison of the 1884 and 1968 Mississippi River Channel near 

 Commerce, Missouri.................................................................................... 1.4 

Figure 1.3 Sinuosity vs. slope with constant discharge ................................................. 1.8 

Figure 2.1 A river reach as a control volume ................................................................ 2.4 

Figure 2.2 The control volume for conservation of linear momentum............................ 2.6 

Figure 2.3 The streamtube as a control volume............................................................ 2.8 

Figure 2.4 Pressure distribution in steady uniform and in steady nonuniform flow.......2.13 

Figure 2.5 Pressure distribution in steady uniform flow on steep slopes......................2.13 

Figure 2.6 Steady uniform flow in a unit width channel ................................................2.14 

Figure 2.7 Hydraulically smooth boundary...................................................................2.16 

Figure 2.8 Hydraulically rough boundary......................................................................2.17 

Figure 2.9 Velocities in turbulent flow ..........................................................................2.17 

Figure 2.10 Einstein's multiplication factor X in the logarithmic velocity equations.........2.20 

Figure 2.11 Control volume for steady uniform flow.......................................................2.26 

Figure 2.12 Energy and momentum coefficients for a unit width of river........................2.30 

Figure 2.13 The river cross section................................................................................2.30 

Figure 2.14 Definition sketch for small amplitude waves................................................2.32 

Figure 2.15 Sketch of positive and negative surges.......................................................2.34 

Figure 2.16 Hydraulic jump characteristics as a function of the upstream Froude 

 number .......................................................................................................2.36 

Figure 2.17 Various types of hydraulic jump ..................................................................2.36 

Figure 2.18 Roll waves or slug flow ...............................................................................2.37 

Figure 2.19 Transitions in open channel flow.................................................................2.38 

Figure 2.20 Specific energy diagram .............................................................................2.39 
 xiv 

Figure 2.21 Changes in water surface resulting from an increase in bed elevation........2.41 

Figure 2.22 Specific discharge diagram.........................................................................2.41 

Figure 2.23 Change in water surface elevation resulting from a change in width...........2.42 

Figure 2.24 Flow characteristics over a drop structure...................................................2.44 

Figure 2.25 Nappe profiles for supercritical flow ............................................................2.45 

Figure 2.26 Schematic representation of transverse currents in a channel bed.............2.47 

Figure 2.27 Definition sketch of flow around a bend ......................................................2.47 

Figure 2.28 Classification of water surface profiles........................................................2.52 

Figure 2.29 Examples of water surface profiles .............................................................2.54 

Figure 2.30 Definition sketch for the standard step method for computation of 

 backwater curves........................................................................................2.55 

Figure 2.31 Gaging station well and shelters .................................................................2.57 

Figure 2.32 Typical float recording gaging station .........................................................2.58 

Figure 2.33 Typical bubble and manometer recording gaging station............................2.58 

Figure 2.34 Stage vs. time hydrograph..........................................................................2.59 

Figure 2.35 Definition sketch of computing area and discharge at a gaging station.......2.60 

Figure 2.36 Stage-discharge relation for Schoharie Creek, NY......................................2.61 

Figure 2.37 Stage-discharge relation for a sand channel (Los Gatos Creek at 

 El Dorado Avenue, CA) ..............................................................................2.62 

Figure 2.38 Three types of backwater effect associated with bridge crossings..............2.64 

Figure 2.39 Submergence of a superstructure...............................................................2.65 

Figure 2.40 Types of flow encountered..........................................................................2.66 

Figure 2.41 Discharge coefficient for roadway overtopping ...........................................2.69 

Figure 2.42 Weir crest length determinations for roadway overtopping .........................2.69 

Figure 2.43 Sketch of backwater curve over check dam................................................2.77 

Figure 2.44 Sketch of backwater curve over check dam................................................2.87 
 xv 

Figure 3.1 Drag coefficient CD vs. particle Reynolds number Rep for spheres 

 and natural sediments with shape factors Sp equal to 0.3, 0.5, 0.7, 

 and 0.9 ........................................................................................................ 3.5 

Figure 3.2 Nominal diameter vs. fall velocity................................................................. 3.6 

Figure 3.3 Definition sketches for size-frequency characteristics of sediments............. 3.9 

Figure 3.4 Angle of repose of non-cohesive materials .................................................3.13 

Figure 3.5 Forms of bed roughness in sand channels .................................................3.14 

Figure 3.6 Relation between water surface and bed configuration...............................3.15 

Figure 3.7 Change in velocity with stream power for a sand with D50 = 0.19 mm.........3.16 

Figure 3.8 Relation between regime of flow and depth for bed material with a 

 median size equal to or less than 0.35 mm.................................................3.20 

Figure 3.9 Cross-over from lower to upper flow regime based on sand size 

 and Froude Number ...................................................................................3.20 

Figure 3.10 Relation to depth to discharge for Elkhorn River near Waterloo, NE...........3.22 

Figure 3.11 Apparent kinematic viscosity of water-bentonite dispersions ......................3.23 

Figure 3.12 Variation of fall velocity of several sand mixtures with percent 

 bentonite and temperature..........................................................................3.24 

Figure 3.13 Relation between stream power, median fall diameter, and bed 

 configuration and Manning's n values.........................................................3.27 

Figure 3.14 Change in Manning's n with discharge for Padma River in Bangladesh......3.28 

Figure 3.15 Shields Diagram: dimensionless critical shear stress.................................3.36 

Figure 3.16 Shields' relation for beginning of motion .....................................................3.37 

Figure 3.17 Comparison of critical shear stress as a function of grain diameter ............3.39 

Figure 3.18 Critical shear stress as a function of grain diameter ...................................3.44 

Figure 3.19 Critical velocity as a function of stone size..................................................3.45 

Figure 3.20 Definition of sediment discharge (load) components ..................................3.46 

Figure 3.21 Schematic sediment and velocity profiles ...................................................3.48 

Figure 3.22a Suspended sediment sampler-D49.............................................................3.48 
 xvi 

Figure 3.22b Suspended sediment sampler-DH48 ..........................................................3.49 

Figure 3.23 Gage height and suspended sediment concentration hydrograph, 

 Colorado River near San Saba, Texas, May 1-6, 1952...............................3.50 

Figure 3.24 Bed material size distribution curves...........................................................3.53 

Figure 3.25 Size distribution curve for pebble count ......................................................3.56 

Figure 4.1 Classification of sediment transport in streams (rivers)................................ 4.5 

Figure 4.2 Sediment transport capacity and supply curves........................................... 4.6 

Figure 4.3 Schematic sediment and velocity profiles .................................................... 4.7 

Figure 4.4 Graph of suspended sediment distribution..................................................4.10 

Figure 4.5 Einstein's φ* vs. ψ* bed load function ..........................................................4.15 

Figure 4.6 Hiding factor ...............................................................................................4.16

Figure 4.7 Einstein's multiplication factor X in the logarithmic velocity equations.........4.16 

Figure 4.8 Pressure correction ....................................................................................4.17 

Figure 4.9 Integral l1 in terms of E and Z .....................................................................4.18 

Figure 4.10 Integral l2 in terms of E and Z .....................................................................4.19 

Figure 4.11 Friction loss due to channel irregularities, as a function of sediment 

 transport .....................................................................................................4.20 

Figure 4.12 Comparison of the Meyer-Peter and Müller and Einstein methods 

 for computing contact load..........................................................................4.21 

Figure 4.13 Relation of discharge of sands to mean velocity for six median sizes 

 of bed sands, four depths of flow, and a water temperature of 60°F...........4.23 

Figure 4.14 Colby's correction curves for temperature and fine sediment......................4.24 

Figure 4.15 Bed-material size effects on bed material transport ....................................4.25 

Figure 4.16 Effect of slope on bed material transport ....................................................4.26 

Figure 4.17 Effect of kinematic viscosity (temperature) on bed material transport.........4.26 

Figure 4.18 Variation of bed material load with depth of flow.........................................4.26 

Figure 4.19 Description of the average cross section ....................................................4.46 
 xvii 

Figure 4.20 Grain size distribution of bed material.........................................................4.47 

Figure 4.21 Comparison of the Einstein and Colby methods .........................................4.53 

Figure 5.1 Changing slope at fan-head leading to fan-head trenching.......................... 5.4 

Figure 5.2 Headcuts and nickpoints.............................................................................. 5.4 

Figure 5.3 Relationship between flume slope and sinuosity (ratio between 

 channel length and valley length) during flume experiments at 

 constant water discharge............................................................................. 5.7 

Figure 5.4 Sequence of channel changes as water discharge and sediment 

 loads decrease ............................................................................................ 5.7 

Figure 5.5 Variability of sinuosity between 1765 and 1915 for 24 Mississippi 

 River reaches .............................................................................................. 5.8 

Figure 5.6 Mississippi River reaches between Cairo, Illinois and Old River, 

 Louisiana ..................................................................................................... 5.9 

Figure 5.7 Mississippi River valley profile ..................................................................... 5.9 

Figure 5.8 River Nile between Qena and Cairo showing six reaches of steep 

 valley slope.................................................................................................5.11 

Figure 5.9 River Nile valley profile ...............................................................................5.11 

Figure 5.10 Plot of River Nile sinuosity and valley slope................................................5.12 

Figure 5.11 Stream properties for classification.............................................................5.14 

Figure 5.12 Classification of river channels ...................................................................5.15 

Figure 5.13 Channel classification showing relative stability and types of hazards 

 encountered with each pattern....................................................................5.16 

Figure 5.14 Planview and cross section of a meandering stream ..................................5.17 

Figure 5.15 Definition sketch for meanders ...................................................................5.19 

Figure 5.16 Empirical relations for meander characteristics...........................................5.20 

Figure 5.17 Types of multi-channel streams..................................................................5.21 

Figure 5.18 Slope-discharge relationship for braiding or meandering in sandbed 

 streams.......................................................................................................5.22 

Figure 5.19 Variation of discharge at a given river cross section and at points 

 downstream................................................................................................5.25 
 xviii 

Figure 5.20 Schematic variation of width, depth, and velocity with at-a-station 

 and downstream discharge variation ..........................................................5.25 

Figure 5.21 Changes in channel slope in response to an increase in sediment load 

 at Point C....................................................................................................5.30 

Figure 5.22 Changes in channel slope in response to a dam at Point C........................5.31 

Figure 5.23 Map of lower Chippewa River .....................................................................5.36 

Figure 5.24 Leopold and Wolman's (1957) relation between channel patterns, 

 channel gradient, and bankfull discharge....................................................5.37 

Figure 5.25 Lane's (1957) relation between channel patterns, channel gradient, 

 and mean discharge ..................................................................................5.37 

Figure 5.26 Median bank erosion rate in relation to channel width for different 

 types of streams .........................................................................................5.55 

Figure 5.27 Encroachment on a meandering river.........................................................5.56 

Figure 5.28 Erosion index in relation to sinuosity...........................................................5.61 

Figure 5.29 Modes of meander loop development.........................................................5.62 

Figure 5.30 Domains of meander behavior....................................................................5.65 

Figure 5.31 Sinuous point bar stream............................................................................5.67 

Figure 5.32 River channels............................................................................................5.68 

Figure 5.33 Meandering river sketch .............................................................................5.70 

Figure 5.34 Estimated future location of a meandering river..........................................5.70 

Figure 6.1 Typical bank failure surfaces (a) noncohesive, (b) cohesive, 

 (c) composite............................................................................................... 6.4 

Figure 6.2 Placement of flow control structures relative to channel banks, 

 crossing, and floodplain..............................................................................6.11

Figure 6.3 Perspective of hard point with section detail ...............................................6.12 

Figure 6.4 Retard.........................................................................................................6.13 

Figure 6.5 Concrete or timber cribs .............................................................................6.13 

Figure 6.6 Pile dikes ....................................................................................................6.15 
 xix 

Figure 6.7 Vane dike model, ground walnut shell bed, during low stage portion 

 of test run ...................................................................................................6.15 

Figure 6.8 Typical jetty-field layout ..............................................................................6.16 

Figure 6.9 Steel jacks ..................................................................................................6.17 

Figure 6.10 Typical guide bank......................................................................................6.18 

Figure 6.11 Definition sketch for a vertical drop.............................................................6.19 

Figure 6.12 Flow and scour patterns at a sloping sill .....................................................6.20 

Figure 6.13 Diagram for riprap stability conditions.........................................................6.21 

Figure 6.14 Stability numbers for a 1.5 stability factor for horizontal flow along 

 a side slope ................................................................................................6.24 

Figure 6.15 Suggested gradation for riprap ...................................................................6.28 

Figure 6.16 Tie-in trench to prevent riprap blanket from unraveling...............................6.31 

Figure 6.17 Rock-fill trench............................................................................................6.31 

Figure 6.18 Windrow revetment, definition sketch .........................................................6.33 

Figure 6.19 Riprap failure models .................................................................................6.35 

Figure 6.20 Rock and wire mattress ..............................................................................6.38 

Figure 6.21 Typical sand-cement bag revetment...........................................................6.40 

Figure 6.22 Articulated concrete block system ..............................................................6.41 

Figure 6.23 Used tire mattress ......................................................................................6.41 

Figure 6.24 Typical soil-cement bank protection............................................................6.43 

Figure 6.25 Hydraulic flow zones on an embankment during overtopping flow..............6.45 

Figure 6.26 Definition sketch of variables involved in overtopping flow..........................6.46 

Figure 6.27 Definition sketch for application of the momentum equation for 

 embankment overtopping flow....................................................................6.47 

Figure 6.28 Typical embankment erosion pattern with submerged flow.........................6.48 

Figure 6.29 Typical embankment erosion pattern with free flow ....................................6.49 
 xx 

Figure 6.30 Permissible velocities of various protection materials as a 

 function of flow duration..............................................................................6.51 

Figure 6.31 Full-scale test of an embankment overtopping protection system 

 under steep-slope, high-velocity flow conditions .........................................6.52 

Figure 6.32 Definition sketch for riprap on a channel bed..............................................6.56 

Figure 6.33 Stability factors for various rock sizes on a side slope ................................6.59 

Figure 6.34 Safety factors for various side slopes .........................................................6.59 

Figure 6.35 Gradations of filter blanket for Problem 2 ...................................................6.64 

Figure 6.36 Definition sketch for riprap on a channel bed..............................................6.65 

Figure 6.37 Stability factors for various rock sizes on a side slope ................................6.68 

Figure 6.38 Safety factors for various side slopes .........................................................6.68 

Figure 6.39 Gradations of filter blanket for Problem 2 ...................................................6.73 

Figure 7.1 Flow chart for scour and stream stability analysis and 

 countermeasures......................................................................................... 7.3 

Figure 7.2 Local scour depth at a pier as a function of time.......................................... 7.5 

Figure 7.3 Comparison of scour formulas for variable depth ratios (y/a)......................7.14 

Figure 7.4 Comparison of scour formulas with field scour measurements ...................7.14 

Figure 7.5 Comparison of pier scour equations with field measurements ....................7.15 

Figure 7.6 Common pier shapes..................................................................................7.17 

Figure 7.7 Topwidth of scour hole................................................................................7.21 

Figure 7.8 Schematic representation of abutment scour..............................................7.22 

Figure 7.9 Comparison of laboratory flow characteristics to field flow conditions.........7.23 

Figure 7.10 Orientation of embankment angle θ to the flow...........................................7.24 

Figure 7.11 Abutment shape .........................................................................................7.25 

Figure 9.1 Three-level analysis procedure for river engineering studies ....................... 9.7 

Figure 9.2 Cimarron River, east of Okeene, Oklahoma (Example 1) ...........................9.21 

Figure 9.3 Arkansas River, north of Bixby, Oklahoma (Example 2) .............................9.22 
 xxi 

Figure 9.4 Washita River, north of Maysville, Oklahoma (Example 3) .........................9.23 

Figure 9.5 Beaver River, north of Laverne, Oklahoma (Example 4).............................9.24 

Figure 9.6 Powder River, 40 miles east of Buffalo, Wyoming (Example 5) ..................9.25 

Figure 9.7 North Platte River near Guernsey, Wyoming (Example 6)..........................9.26 

Figure 9.8 Coal Creek, tributary of Powder River, Wyoming (Example 7)....................9.27 

Figure 9.9a Map showing South Fork of Deer River at U.S. Highway 51 

 Crossing (Example 8) .................................................................................9.27 

Figure 9.9b Channel modifications to South Fork of Deer River at U.S. Highway 

 51 Crossing near Halls, Tennessee (Example 8)........................................9.28 

Figure 9.9c Elevation sketch of U.S. Highway 51 Bridge (Example 8) ...........................9.29 

Figure 9.10 Stage trends at Sioux City, Iowa, on the Missouri River (Example 9)..........9.31 

Figure 9.11 Deflector arrangement and alignment problem on I-90 Bridge across 

 Big Elk Creek near Piedmont, North Dakota (Example 10).........................9.32 

Figure 9.12 Plan sketch of channel relocation, Outlet Creek (Example 11) ...................9.33 

Figure 9.13 Plan sketch of Nojoqui Creek channel relocation (Example 12)..................9.34 

Figure 9.14 Plan sketch of Turkey Creek channel relocation (Example 13)...................9.35 

Figure 9.15 Case study of sand and gravel mining (Example 14)..................................9.36 

Figure 9.16 Nowood River near Ten Sleep, Wyoming (Example 15) .............................9.38 

Figure 9.17 Middle Fork Powder River at Kaycee, Wyoming (Example 16) ...................9.39 

Figure 10.1 Topographic map of Bijou Creek study area ...............................................10.1 

Figure 10.2 Cross sectional area versus flow depth relation..........................................10.3 

Figure 10.3 Wetted perimeter versus flow depth relation...............................................10.4 

Figure 10.4 Analysis of bed material size of Bijou Creek ...............................................10.4 

Figure 10.5 Gumbel's method of frequency analysis .....................................................10.6 

Figure 10.6 Proposed first alternative ............................................................................10.6 

Figure 10.7 Proposed second alternative ......................................................................10.7 
 xxii 

Figure 10.8 Proposed third alternative...........................................................................10.7 

Figure 10.9 Proposed fifth alternative ............................................................................10.8 

Figure 10.10 The sketch of proposed riprap design.......................................................10.12 

Figure 10.11 Sketch of spur design ...............................................................................10.16 

Figure 10.12 Rillito River system vicinity map................................................................10.18 

Figure 10.13 Slope-discharge relationship for Rillito River.............................................10.21 

Figure 10.14 Sketch of Craycroft Road crossing ...........................................................10.22 

Figure 10.15 Sabino Canyon Road crossing site ...........................................................10.24 

Figure 10.16 Flood events at Rillito River near Tucson, Arizona....................................10.25 

Figure 10.17 Log-normal frequency analysis for Rillito River near Tucson ....................10.26 

Figure 10.18 December 1965 flood in Rillito River and Tanque Verde Creek ................10.26 

Figure 10.19 100-year flood design hydrographs...........................................................10.27 

Figure 10.20 Stage-discharge plot for Rillito River near Tucson ....................................10.27 

Figure 10.21 Rillito-Pantano-Tanque Verde bed sediment distribution ..........................10.28 

Figure 10.22 Bed elevation change of Rillito-Tanque Verde System (Tanque 

 Verde flooding) .........................................................................................10.33 

Figure 10.23 Bed elevation change of Rillito-Tanque Verde System (Pantano 

 flooding)....................................................................................................10.33 

Figure 10.24 Alternative I for Craycroft Road Bridge crossing .......................................10.34 

Figure 10.25 Alternative II for Craycroft Road Bridge crossing ......................................10.35 

Figure 10.26 Alternative III for Craycroft Road Bridge crossing .....................................10.37 

Figure 10.27 General plan view of Sabino Canyon Road crossing for alternative 

 II, III, and IV..............................................................................................10.39 

Figure 10.28 Cross section of proposed bridge .............................................................10.40 

Figure 10.29 Equal conveyance tubes of approach section...........................................10.42 

Figure 10.30 Plan view of equal conveyance tubes showing velocity distribution 

 at approach and bridge sections...............................................................10.42 
 xxiii 

Figure 10.31 Velocity distribution at bridge crossing......................................................10.43 

Figure 10.32 Plot of total scour for example problem.....................................................10.44 

Figure 10.33 Revised plot of total scour for example problem .......................................10.45 

Figure 10.34 Three-day hydrograph for BRI-STARS analysis........................................10.46 

Figure 10.35 Comparison longitudinal profiles from BRI-STARS...................................10.46 

Figure 10.36 Comparison cross-sections from BRI-STARS...........................................10.47 

Figure 10.37 BRI-STARS model profiles for steady-state run........................................10.48 
 xxiv 

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 xxv 

LIST OF TABLES 

Table 1.1 Commonly Used Engineering Terms in SI and English Units.......................1.21 

Table 2.1 Manning's Roughness Coefficients for Various Boundaries.........................2.22 

Table 2.2 Adjustment Factors for the Determination of n Values.................................2.25 

Table 2.3 Characteristics of Water Surface Profiles ....................................................2.53 

Table 2.4 Discharge Measurement Notes....................................................................2.70 

Table 2.5 Observed Velocity Data ...............................................................................2.71 

Table 2.6 Velocity Profile Calculations.........................................................................2.72 

Table 2.7 Detailed Computation of Superelevation in Bends .......................................2.74 

Table 2.8 Discharge Measurement Notes....................................................................2.80 

Table 2.9 Observed Velocity Data ...............................................................................2.81 

Table 2.10 Velocity Profile Calculations.........................................................................2.82 

Table 2.11 Detailed Computation of Superelevation in Bends .......................................2.84 

Table 3.1 Sediment Grade Scale.................................................................................. 3.3 

Table 3.2 Guide to Size Range for Different Types of Size Analysis............................. 3.6 

Table 3.3 Typical Sediment Size Distribution for Gravel-Bed Stream ..........................3.31 

Table 3.4 Maximum Permissible Velocities Proposed by Fortier and Scobey ..............3.43 

Table 3.5 Nonscour Velocities for Soils .......................................................................3.44 

Table 3.6 Sand Bed Material Size Distribution.............................................................3.52 

Table 3.7 Sand Size Bed Material Properties ..............................................................3.53 

Table 3.8 Gravel Bed Material Size Distribution...........................................................3.54 

Table 3.9 Gravel Bed Material Properties ....................................................................3.55 

Table 4.1 Coefficient and Exponents of Equation 4.48 ................................................4.28 

Table 4.2 Range of Parameters Equation 4.48 Developed by Simons et al.................4.29 

Table 4.3 Coefficient and Exponents for Equation 4.50 ...............................................4.30 
 xxvi 

Table 4.4 Summary of Applicability of Selected Sediment Transport Formulas ...........4.33 

Table 4.5 Concentration vs. Elevation Above the Bed ..................................................4.38 

Table 4.6 Concentration vs. Elevation Above the Bed ..................................................4.44 

Table 4.7 Bed Material Information for Sample Problem...............................................4.46 

Table 4.8 Hydraulic Calculations for Sample Problem 3, Application of the 

 Einstein Procedure .......................................................................................4.48 

Table 4.9 Bed-Material Load Calculations for Sample Problem by Applying 

 the Einstein (1950) Procedure .....................................................................4.49 

Table 4.10 Bed Material Load Calculations for Sample Problem by Applying 

 the Colby Method (Median Diameter) ..........................................................4.51 

Table 4.11 Bed Material Discharge Calculations for Sample Problem by 

 Applying the Colby (Individual Size Fraction)...............................................4.52 

Table 5.1 Empirical Relations for Meanders in Alluvial Valleys ....................................5.20 

Table 5.2 At-A-Station and Downstream Hydraulic Geometry Relationships ...............5.26 

Table 5.3 Derived At-A-Station and Downstream Geometry Relationships..................5.26 

Table 5.4 Change of Variables Induced by Changes in Sediment Discharge, 

 Size of Bed Sediment and Wash Load ........................................................5.32 

Table 5.5 Qualitative Response of Alluvial Channels...................................................5.33 

Table 5.6 Chippewa River Morphology ........................................................................5.35 

Table 5.7 Scale Ratios for Similitude ...........................................................................5.42 

Table 5.8 Channel Response to Changes in Watershed and River Condition .............5.52 

Table 5.9 Sediment Size Distribution in the St. Lawrence Seaway ..............................5.75 

Table 6.1 Factors Affecting Erosion of River Banks...................................................... 6.2 

Table 6.2 Data for Suggested Gradation .....................................................................6.28 

Table 6.3 Causes of Riprap Failure and Solutions ......................................................6.36 

Table 6.4 Sizes of Materials.........................................................................................6.61 

Table 6.5 Sizes of Materials.........................................................................................6.62 

Table 6.6 Sizes of Materials.........................................................................................6.70 
 xxvii 

Table 6.7 Sizes of Materials.........................................................................................6.71 

Table 7.1 Correction Factor, K1, for Pier Nose Shape..................................................7.17 

Table 7.2 Correction Factor, K2, for Angle of Attack, θ, of the Flow .............................7.17 

Table 7.3 Increase in Equilibrium Pier Scour Depths, K3, for Bed Condition................7.17 

Table 7.4 Abutment Shape Coefficients for Short Abutment Sections .........................7.25 

Table 8.1 Checklist of Data Needs ............................................................................... 8.6 

Table 8.2 List of Data Sources ..................................................................................... 8.8 

Table 9.1 River Response to Highway Encroachments and to River Development .....9.10 

Table 10.1 Summary of Computer Hydraulic Conditions ...............................................10.8 

Table 10.2 Summary of Hydraulic Designs..................................................................10.10 

Table 10.3 Design of Riprap Sizes ..............................................................................10.11 

Table 10.4 Bank Migration for Alternative 4 .................................................................10.14 

Table 10.5 Summary of Spur Design...........................................................................10.14 

Table 10.6 Dimension of Spurs....................................................................................10.14 

Table 10.7 Size and Filter Design for Riprap at Spur Noses and Spur Shanks............10.15 

Table 10.8 Rillito River Near Tucson, Arizona Log-Pearson Type III Frequency 

 Analysis by USGS ......................................................................................10.25 

Table 10.9 Equilibrium Slope Calculations, Dominant Discharge.................................10.31 

Table 10.10 Low Chord and Total Scour Requirements at Craycroft Road....................10.36 
 xxviii 

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 xxix 

ACKNOWLEDGMENTS

This manual is a major revision of the 1990 manual. The writers wish to acknowledge the 

contributions made by Drs. E.V. Richardson, D.B. Simons, S. Karaki, K. Mahmood, and M.A. 

Stevens as co-authors of the 1975 manual, as well as the review, help, and guidance given by 

the Steering Committee for the 1975 edition consisting of Frank L. Johnson, Roger L. Dean, 

Murray L. Corry, Dah-Chen Woo, Milo Cress, Lawrence J. Harrison, Herbert Gregory, Gene R. 

Fiala of the Federal Highway Administration and Mainard Wacker of the Wyoming Highway 

Department. The writers also wish to acknowledge the help of Colorado State University 

Professors J.F. Ruff and A.G. Mercer and Graduate Research Assistants V.M. Ponce, Larry 

Rundquist, and Tony Melone in the preparation of the 1975 edition. 

For the 1990 manual, the writers acknowledge the contributions of Drs. E.V. Richardson, D.B. 

Simons, and P.Y. Julien as co-authors, the technical assistance of Philip Thompson, J. Sterling 

Jones, and Lawrence J. Harrison, FHWA; Mainard A. Wacker, Wyoming Highway Department, 

and Michael E. Zeller, Simons, Li & Associates, and CSU Graduate Research Assistant Noel 

E. Bormann. 

Several topics have been added or expanded in this 2001 edition of the manual. Specifically, 

the writers acknowledge the contributions of Dr. L.W. Zevenbergen for the expanded coverage 

of sediment transport in Chapter 4 and Dr. L.A. Arneson for his critical review and valuable 

comments on that chapter. In addition, Mr. P.E. Clopper provided a new section for Chapter 6 

on overtopping flow on embankments based on FHWA studies conducted in the late 1980s. 

Copyright © Ayres Associates 2001 
 xxx 

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 xxxi 

LIST OF SYMBOLS 

A = surface area 

A1 = cross-sectional area for the fluid entering the control volume 

A2 = cross-sectional area for the fluid leaving the control volume 

An2 = normal cross-sectional area at the bridge crossing 

Ap = projected area of a pier normal to the flow 

As = area of a sediment particle 

a = acceleration or pier width or embankment length 

a = thickness of the bed layer, the unmeasured zone of flow 

an = embankment length for relief bridge 

an = acceleration component normal to a streamline 

am = embankment length for main bridge 

ao = amplitude of wave 

a = coefficient of width relationship 

B, B' = dimensionless constants in Meyer-Peter, Müller equations 

bs = bridge opening length 

b = exponent of width relationship 

C = sediment concentration 

C = Chezy coefficient 

Co = Chezy coefficient for steady uniform flow 

C1 = constant for free vortex flow 

C2 = constant for forced vortex flow 

CD = drag coefficient on a particle 

Cf = concentration of fine material 

Cr = overtopping discharge coefficient 

CT = bed material sediment concentration 

Cs = coefficient of shear 

C = average suspended sediment concentration 

c = wave celerity 

c = coefficient of the flow depth relationship 

D = culvert diameter or sediment size 

D50 = sediment riprap size for which 50% by weight of the particles are smaller; 

 similarly D65, D84, D90 represent sizes for which 65, 84, and 90% of the particles 

 are smaller 

D50x = mean sediment size a distance x downstream from reference location 

D50o = mean sediment size at the reference section 

Dm = effective mean diameter of sediment or riprap 

Ds = sediment size or riprap size 

dA1 = differential area for the fluid entering the control volume 

dA2 = differential area for the fluid leaving the control volume 

drr = thickness of riprap layer 

ds = displacement of a fluid particle during time dt 

ds = scour depth 

dt = differential time interval 

E = energy of a system 

E = ratio of bed layer thickness to flow depth a/yo

e = energy per unit mass 

e1, e2, 

e3, e4 = moment arms of forces acting on a rock particle 
 xxxii 

ec = eccentricity coefficient 

F1, F2 = pressure force acting at section one and two 

F1(h), 

F2(h) = functions for the transverse velocity distribution in bends 

FB = buoyant force on a particle 

Fb = force exerted at the boundary 

Fc = critical Froude number for the beginning of motion of sediment 

FD = drag force on a particle 

Fd = drag force on a rock particle 

Ff = fluid force on a particle 

Fg = gravitational force on a particle 

FP = lift force on a rock particle 

Fn = external forces between particles 

Fr = Froude number 

Fs = shear force 

Fv = shear force on a particle 

Fx = force acting in the x direction 

f = Darcy-Weisbach friction factor 

f'b = Darcy-Weisbach friction factor for the grain roughness 

fs = seepage force 

f = exponent of the flow depth relationship 

G = gradation coefficient 

g = acceleration due to gravity 

H = total energy 

He = entrance loss of head 

He = average head loss over a cross-section 

Hf = friction loss of head 

Hmin = minimum total energy at a critical section 

HT = total head 

Hv = velocity head 

HW2 = headwater at a culvert 

Hs = horizontal side slope related to one unit vertically 

hL = head loss in a hydraulic jump 

he = head loss 

h1* = total backwater elevation 

Wh = drop in water surface elevation through bridge opening 

I1, I2 = integrals in the Einstein method 

ib = fraction of the bed load for given equation 

is = fraction of the suspended load for a given equation 

iT = fraction of total load for a given equation 

J = ratio of projected area of piers to the gross constricted area An2

j = exponent of the bed sediment discharge relationship 

K = coefficient of the Meyer-Peter and Müller equation 

K1, K2 = coefficient for the area and volume of sediment particles 

KB, Kr = coefficient in Meyer-Peter, Müller equation 

Kb = base backwater coefficient 

Ke = entrance loss coefficient for culverts 

Ku = conversion constant (SI/English units) 

ks = height of roughness elements 

k = coefficient of the velocity relationship 
 xxxiii 

L = length of control volume 

L = length of culvert 

L1, L2, 

L3 = lengths for sloping sill drop structure 

La = length of abutment 

Lc = length of a channel along the thalweg 

Le = length of embankment 

Lj

 = length of hydraulic jump 

Ls = upstream length of a guide bank 

Ls = length of the roadway crest for overspill 

P = pier length 

P = mixing length 

Pa = longest axis of particle 

Pb = intermediate axis of particle 

Pc = shortest axis of particle 

M = bridge opening ratio 

m = exponent of the velocity relationship 

M/N = ratio of lift to drag moments on a particle 

n = Manning's resistance coefficient 

nb = Manning's resistance coefficient of the bed 

no = Manning's resistance coefficient for steady uniform flow 

nw = Manning's resistance coefficient of the walls 

P = wetted perimeter of the flow 

p1, p2 = pressure of fluid at sections one and two 

Pb = wetted perimeter of the bed 

PE = coefficient in the Einstein's total sediment discharge method 

Pi

, Po = pressure of fluid inside and outside the bend 

Pw = wetted perimeter of the banks 

pi

 = percentage by weight of a size fraction of sediment 

p = coefficient of the total bed sediment discharge relationship 

p = average pressure over a cross section 

Q = discharge (flow rate) 

Q = rate at which heat is added to the system 

QB = bed load discharge in weight units 

Qb = water discharge quantity determining the bed load transport 

Qc = approach channel flow discharge 

Qe = approach flow discharge on the embankment 

Qf = formative discharge for a particular phenomena 

Qn = uncorrected bed sediment discharge 

Qo = overtopping discharge 

Qs = total suspended sediment discharge 

Qsmax = maximum suspended sediment discharge 

QT = total bed sediment discharge 

QT = total discharge 

q = unit discharge 

qb = unit bed load discharge 

qn = uncorrected unit bed sediment discharge 

qs = unit suspended sediment discharge 

qT = unit total bed sediment discharge 

R = hydraulic radius 
 xxxiv 

R'b = hydraulic radius due to grain roughness 

Re = Reynolds number 

Rep = Reynolds number of falling particle 

R = vector R of the direction of particle movement 

r = radius of curvature 

rc = radius of curvature at the center of the stream 

ri

, ro = radius of curvature for inner and outer banks in a bend 

r = coefficient of Manning's n relationship 

S = channel slope 

Sc = shape factor of a cross-section 

Sf = friction slope (also called the energy slope) 

S'f = friction slope to overcome grain resistance 

Sm = ratio of tangents of friction angle to side slope angle 

Sn = channel sinuosity 

So = bed slope 

Sp = shape factor of sediment particle 

SR = shape factor of a river reach 

Ss = specific gravity of sediment particle 

Sx = bed slope at a distance x downstream of a reference location 

Sw = slope of water surface 

s = direction tangential to streamline 

T = wave period 

T = time 

Tâ = temperature 

t = coefficient of the friction slope relationship 

u = internal energy associated with fluid temperature 

V = velocity vector 

V = volume 

Vc, Vcr = critical velocity 

Vd = depth-average velocity at the vena contracta 

Vm = mean stream velocity 

Vmax = maximum velocity 

Vn2 = average velocity at the bridge opening 

Vo = tangential velocity in a bend 

Vp = volume of a particle, or average velocity on a spillslope 

Vr = reference velocity 

Vs = velocity against the stone 

VT = depth-average velocity at the toe of an embankment 

Vw = velocity of the wave 

V* = shear velocity 

V*c = shear velocity in the channel 

V'* = shear velocity due to grain roughness 

V''* = shear velocity due to form roughness 

v1, V1 = velocity of the fluid entering a control volume 

v2, V2 = velocity of the fluid leaving a control volume 

vs, vn = velocity components in the s and n directions 

vx, vy = velocity components along x and y 

vx, vy = average velocity components along x and y 

v'x, v'y = velocity fluctuations 

W = width of control volume or channel free surface width 
 xxxv 

W = rate at which a fluid system does work on its surroundings 

Wp = rate of pressure work done by system 

Ws = Buoyant weight of sediment particle 

w = lateral location in a cross-section 

X = Einstein's multiplication factor 

X = function of W/w' in Einstein's method 

Y = pressure correction factor, a function of D65/w' in Einstein's method 

y = vertical distance 

yc = critical depth of flow 

ymax = maximum depth of flow 

yn = normal depth of flow 

yo = total flow depth 

ys = scour depth due to contraction 

y' = vertical distance above bed at which the velocity is zero 

y = exponent of the Manning's n relationship 

Z = exponent of the Rouse equation 

∆Z = superelevation of flow in bends 

∆z = drop in water surface elevation along a channel 

z = elevation above arbitrary reference level 

zi

, zo = water surface elevation at inside and outside of bend 

z = exponent of the friction slope relationship 

GREEK SYMBOLS 

α = energy correction factor 

α = coefficient of the downstream decrease in slope 

α' = energy correction factor for unit width 

β = momentum correction factor 

β = angle between the particle movement direction and the vertical 

β1 = angle of the hydraulic jump in bends 

β' = momentum correction factor for a unit width 

β = coefficient of the downstream decrease in sediment size 

β x = correction term in Einstein's Method 

γ = specific weight of the fluid 

γs = specific weight of the sediment 

∆ = apparent roughness of the bed 

∆o = dimensionless bend angle 

δ = angle between the drag force and the particle movement direction 

δ' = thickness of the laminar sublayer in turbulent flow 

ε = exponent in Laursen's equation 

η = stability number for particles on a plane bed 

η1 = stability number for particles on side slopes 

θ = side slope angle 

θ = inclination angle of a channel 

θ1 = bend angle in supercritical flows 

θe = angle of orientation of the embankment 

κ = von Karman velocity constant 

λ = wave length 

λ = angle between the horizontal and the drag force vector 
 xxxvi 

μ = dynamic viscosity of a fluid 

ν = kinematic viscosity of a fluid 

ξ = correction factor, a function of Ds/X in Einstein's method 

ξ = function of other parameters (Eq. 5.3.31) 

ρ = fluid density 

ρ1 = density of a fluid entering a control volume 

ρ2 = density of a fluid leaving a control volume 

ρs = density of sediment particles 

σ = correction coefficient for piers 

τ = shear stress 

τo = bed shear stress 

τc = critical shear stress 

φ* = dimensionless sediment transport function 

ψ = uncorrected entrainment function 

ψ* = entrainment function 

ω = fall velocity of a sediment particle 

ωa = angle of the abutment 
 xxxvii 

GLOSSARY

abrasion: Removal of stream bank material due to entrained sediment, 

ice, or debris rubbing against the bank. 

aggradation: General and progressive buildup of the longitudinal profile of a 

channel bed due to sediment deposition. 

alluvial channel: Channel wholly in alluvium; no bedrock is exposed in channel 

at low flow or likely to be exposed by erosion. 

alluvial fan: A fan-shaped deposit of material at the place where a stream 

issues from a narrow valley of high slope onto a plain or broad 

valley of low slope. An alluvial cone is made up of the finer 

materials suspended in flow while a debris cone is a mixture of 

all sizes and kinds of materials. 

alluvial stream: A stream which has formed its channel in cohesive or 

noncohesive materials that have been and can be transported 

by the stream. 

alluvium: Unconsolidated material deposited by a stream in a channel, 

floodplain, alluvial fan, or delta. 

alternating bars: Elongated deposits found alternately near the right and left 

banks of a channel. 

anabranch: Individual channel of an anabranched stream. 

anabranched stream: A stream whose flow is divided at normal and lower stages by 

large islands or, more rarely, by large bars; individual islands or 

bars are wider than about three times water width; channels 

are more widely and distinctly separated than in a braided 

stream. 

anastomosing stream: An anabranched stream. 

angle of repose: The maximum angle (as measured from the horizontal) at 

which gravel or sand particles can stand. 

annual flood: The maximum flow in one year (may be daily or 

instantaneous). 

apron: Protective material placed on a streambed to resist scour. 

apron, launching: An apron designed to settle and protect the side slopes of a 

scour hole after settlement. 
 xxxviii 

armor (armoring): Surfacing of channel bed, banks, or embankment slope to 

resist erosion and scour. (a) Natural process whereby an 

erosion- resistant layer of relatively large particles is formed on 

a streambed due to the removal of finer particles by 

streamflow; (b) placement of a covering to resist erosion. 

articulated concrete Rigid concrete slabs which can move without separating 

mattress: as scour occurs; usually hinged together with corrosion- 

 resistant cable fasteners; primarily placed for lower bank 

 protection. 

average velocity: Velocity at a given cross section determined by dividing 

discharge by cross sectional area. 

avulsion: A sudden change in the channel course that usually occurs 

when a stream breaks through its banks; usually associated 

with a flood or a catastrophic event. 

backfill: The material used to refill a ditch or other excavation, or the 

process of doing so. 

backwater: The increase in water surface elevation relative to the elevation 

occurring under natural channel and floodplain conditions. It is 

induced by a bridge or other structure that obstructs or 

constricts the free flow of water in a channel. 

backwater area: The low-lying lands adjacent to a stream that may become 

flooded due to backwater. 

bank: The sides of a channel between which the flow is normally 

confined. 

bank, left (right): The side of a channel as viewed in a downstream direction. 

bankfull discharge: Discharge that, on the average, fills a channel to the point of 

overflowing. 

bank protection: Engineering works for the purpose of protecting stream banks 

from erosion. 

bank revetment: Erosion-resistant materials placed directly on a stream bank to 

protect the bank from erosion. 

bar: An elongated deposit of alluvium within a channel, not 

permanently vegetated. 

base floodplain: The floodplain associated with the flood with a 100-year 

recurrence interval. 

bed: The bottom of a channel bounded by banks. 
 xxxix 

bed form: A recognizable relief feature on the bed of a channel, such as 

a ripple, dune, plane bed, antidune, or bar. Bed forms are a 

consequence of the interaction between hydraulic forces 

(boundary shear stress) and the bed sediment. 

bed layer: A flow layer, several grain diameters thick (usually two) 

immediately above the bed. 

bed load: Sediment that is transported in a stream by rolling, sliding, or 

skipping along the bed or very close to it; considered to be 

within the bed layer (contact load). 

bed load discharge The quantity of bed load passing a cross section of a stream in 

(or bed load): a unit of time. 

bed material: Material found in and on the bed of a stream (May be 

transported as bed load or in suspension). 

bedrock: The solid rock exposed at the surface of the earth or overlain 

by soils and unconsolidated material. 

bed sediment discharge: The part of the total sediment discharge that is composed of 

grain sizes found in the bed and is equal to the transport 

capability of the flow. 

bed shear The force per unit area exerted by a fluid flowing past a 

(tractive force): stationary boundary. 

bed slope: The inclination of the channel bottom. 

blanket: Material covering all or a portion of a stream bank to prevent 

erosion. 

boulder: A rock fragment whose diameter is greater than 250 mm. 

braid: A subordinate channel of a braided stream. 

braided stream: A stream whose flow is divided at normal stage by small 

mid-channel bars or small islands; the individual width of bars 

and islands is less than about three times water width; a 

braided stream has the aspect of a single large channel within 

which are subordinate channels. 

bridge opening: The cross-sectional area beneath a bridge that is available for 

conveyance of water. 

bridge waterway: The area of a bridge opening available for flow, as measured 

below a specified stage and normal to the principal direction of 

flow. 
 xl 

bulk density: Density of the water sediment mixture (mass per unit volume), 

including both water and sediment. 

bulkhead: A vertical, or near vertical, wall that supports a bank or an 

embankment; also may serve to protect against erosion. 

bulking: Increasing the water discharge to account for high 

concentrations of sediment in the flow. 

catchment: See drainage basin. 

causeway: Rock or earth embankment carrying a roadway across water. 

caving: The collapse of a bank caused by undermining due to the 

action of flowing water. 

cellular-block Interconnected concrete blocks with regular cavities placed 

mattress: directly on a stream bank or filter to resist erosion. The 

 cavities can permit bank drainage and the growth of vegetation 

 where synthetic filter fabric is not used between the bank and 

 mattress. 

channel: The bed and banks that confine the surface flow of a stream. 

channelization: Straightening or deepening of a natural channel by artificial 

cutoffs, grading, flow-control measures, or diversion of flow into 

an engineered channel. 

channel diversion: The removal of flows by natural or artificial means from a 

natural length of channel. 

channel pattern: The aspect of a stream channel in plan view, with particular 

reference to the degree of sinuosity, braiding, and 

anabranching. 

channel process: Behavior of a channel with respect to shifting, erosion and 

sedimentation. 

check dam: A low dam or weir across a channel used to control stage or 

degradation. 

choking (of flow): Excessive constriction of flow which may cause severe 

backwater effect. 

clay (mineral): A particle whose diameter is in the range of 0.00024 to 0.004 

mm. 

clay plug: A cutoff meander bend filled with fine grained cohesive 

sediments. 
 xli 

clear-water scour: Scour at a pier or abutment (or contraction scour) when there 

is no movement of the bed material upstream of the bridge 

crossing at the flow causing bridge scour. 

cobble: A fragment of rock whose diameter is in the range of 64 to 250 

mm. 

concrete revetment: Unreinforced or reinforced concrete slabs placed on the 

channel bed or banks to protect it from erosion. 

confluence: The junction of two or more streams. 

constriction: A natural or artificial control section, such as a bridge crossing, 

channel reach or dam, with limited flow capacity in which the 

upstream water surface elevation is related to discharge. 

contact load: Sediment particles that roll or slide along in almost continuous 

contact with the streambed (bed load). 

contraction: The effect of channel or bridge constriction on flow 

streamlines. 

contraction scour: Contraction scour, in a natural channel or at a bridge crossing, 

involves the removal of material from the bed and banks 

across all or most of the channel width. This component of 

scour results from a contraction of the flow area at the bridge 

which causes an increase in velocity and shear stress on the 

bed at the bridge. The contraction can be caused by the 

bridge or from a natural narrowing of the stream channel. 

Coriolis force: The inertial force caused by the Earth's rotation that deflects a 

moving body to the right in the Northern Hemisphere. 

countermeasure: A measure intended to prevent, delay or reduce the severity of 

hydraulic problems. 

crib: A frame structure filled with earth or stone ballast, designed to 

reduce energy and to deflect streamflow away from a bank or 

embankment. 

critical shear stress: The minimum amount of shear stress required to initiate soil 

particle motion. 

crossing: The relatively short and shallow reach of a stream between 

bends; also crossover or riffle. 

cross section: A section normal to the trend of a channel or flow. 

current: Water flowing through a channel. 

current meter: An instrument used to measure flow velocity. 

cut bank: The concave wall of a meandering stream. 
 xlii 

cutoff: (a) A direct channel, either natural or artificial, connecting two 

points on a stream, thereby shortening the original length of 

the channel and increasing its slope; (b) A natural or artificial 

channel which develops across the neck of a meander loop 

(neck cutoff) or across a point bar (chute cutoff). 

cutoff wall: A wall, usually of sheet piling or concrete, that extends down to 

scour-resistant material or below the expected scour depth. 

daily discharge: Discharge averaged over one day (24 hours). 

debris: Floating or submerged material, such as logs, vegetation, or 

trash, transported by a stream. 

degradation (bed): A general and progressive (long-term) lowering of the channel 

bed due to erosion, over a relatively long channel length. 

depth of scour: The vertical distance a streambed is lowered by scour below a 

reference elevation. 

design flow (design flood): The discharge that is selected as the basis for the design or 

evaluation of a hydraulic structure. 

dike: An impermeable linear structure for the control or containment 

of overbank flow. A dike-trending parallel with a stream bank 

differs from a levee in that it extends for a much shorter 

distance along the bank, and it may be surrounded by water 

during floods. 

dike (groin, spur, jetty): A structure extending from a bank into a channel that is 

designed to: (a) reduce the stream velocity as the current 

passes through the dike, thus encouraging sediment 

deposition along the bank (permeable dike); or (b) deflect 

erosive current away from the stream bank (impermeable 

dike). 

discharge: Volume of water passing through a channel during a given 

time. 

dominant discharge: (a) The discharge of water which is of sufficient magnitude and 

frequency to have a dominating effect in determining the 

characteristics and size of the stream course, channel, and 

bed; (b) That discharge which determines the principal 

dimensions and characteristics of a natural channel. The 

dominant formative discharge depends on the maximum and 

mean discharge, duration of flow, and flood frequency. For 

hydraulic geometry relationships, it is taken to be the bankfull 

discharge which has a return period of approximately 1.5 years 

in many natural channels. 
 xliii 

drainage basin: An area confined by drainage divides, often having only one 

outlet for discharge (catchment, watershed). 

drift: Alternative term for vegetative "debris." 

eddy current: A vortex-type motion of a fluid flowing contrary to the main 

current, such as the circular water movement that occurs when 

the main flow becomes separated from the bank. 

entrenched stream: Stream cut into bedrock or consolidated deposits. 

ephemeral stream: A stream or reach of stream that does not flow for parts of the 

year. As used here, the term includes intermittent streams with 

flow less than perennial. 

equilibrium scour: Scour depth in sand-bed stream with dune bed about which 

live bed pier scour level fluctuates due to variability in bed 

material transport in the approach flow. 

erosion: Displacement of soil particles due to water or wind action. 

erosion control matting: Fibrous matting (e.g., jute, paper, etc.) placed or sprayed on a 

stream- bank for the purpose of resisting erosion or providing 

temporary stabilization until vegetation is established. 

fabric mattress: Grout-filled mattress used for stream bank protection. 

fall velocity: The velocity at which a sediment particle falls through a column 

of still water. 

fascine: A matrix of willow or other natural material woven in bundles 

and used as a filter. Also, a stream bank protection technique 

consisting of wire mesh or timber attached to a series of posts, 

sometimes in double rows; the space between the rows may 

be filled with rock, brush, or other materials. 

fetch: The area in which waves are generated by wind having a 

rather constant direction and speed; sometimes used 

synonymously with fetch length. 

fetch length: The horizontal distance (in the direction of the wind) over which 

wind generates waves and wind setup. 

fill slope: Side or end slope of an earth-fill embankment. Where a 

fill-slope forms the streamward face of a spill-through 

abutment, it is regarded as part of the abutment. 
 xliv 

filter: Layer of fabric (geotextile) or granular material (sand, gravel, 

or graded rock) placed between bank revetment (or bed 

protection) and soil for the following purposes: (1) to prevent 

the soil from moving through the revetment by piping, 

extrusion, or erosion; (2) to prevent the revetment from sinking 

into the soil; and (3) to permit natural seepage from the stream 

bank, thus preventing the buildup of excessive hydrostatic 

pressure. 

filter blanket: A layer of graded sand and gravel laid between fine-grained 

material and riprap to serve as a filter. 

filter fabric (cloth): Geosynthetic fabric that serves the same purpose as a 

granular filter blanket. 

fine sediment load: That part of the total sediment load that is composed of particle 

sizes finer than those represented in the bed (wash load). 

Normally, the fine-sediment load is finer than 0.062 mm for 

sand-bed channels. Silts, clays and sand could be considered 

wash load in coarse gravel and cobble-bed channels. 

flanking: Erosion around the landward end of a stream stabilization 

countermeasure. 

flashy stream: Stream characterized by rapidly rising and falling stages, as 

indicated by a sharply peaked hydrograph. Typically 

associated with mountain streams or highly disturbed 

urbanized catchments. Most flashy streams are ephemeral, 

but some are perennial. 

flood-frequency curve: A graph indicating the probability that the annual flood 

discharge will exceed a given magnitude, or the recurrence 

interval corresponding to a given magnitude. 

floodplain: A nearly flat, alluvial lowland bordering a stream, that is subject 

to frequent inundation by floods. 

flow-control structure: A structure either within or outside a channel that acts as a 

countermeasure by controlling the direction, depth, or velocity 

of flowing water. 

flow hazard: Flow characteristics (discharge, stage, velocity, or duration) 

that are associated with a hydraulic problem or that can 

reasonably be considered of sufficient magnitude to cause a 

hydraulic problem or to test the effectiveness of a 

countermeasure. 

flow slide: Saturated soil materials which behave more like a liquid than a 

solid. A flow slide on a channel bank can result in a bank 

failure. 
 xlv 

fluvial geomorphology: The science dealing with the morphology (form) and dynamics 

of streams and rivers. 

fluvial system: The natural river system consisting of (1) the drainage basin, 

watershed, or sediment source area, (2) tributary and 

mainstem river channels or sediment transfer zone, and (3) 

alluvial fans, valley fills and deltas, or the sediment deposition 

zone. 

freeboard: The vertical distance above a design stage that is allowed for 

waves, surges, drift, and other contingencies. 

Froude Number: A dimensionless number that represents the ratio of inertial to 

gravitational forces in open channel flow. 

gabion: A basket or compartmented rectangular container made of wire 

mesh. When filled with cobbles or other rock of suitable size, 

the gabion becomes a flexible and permeable unit with which 

flow- and erosion-control structures can be built. 

general scour: General scour is a lowering of the streambed across the 

stream or waterway at the bridge. This lowering may be 

uniform across the bed or non-uniform. That is, the depth of 

scour may be deeper in some parts of the cross section. 

General scour may result from contraction of the flow or other 

general scour conditions such as flow around a bend. 

geomorphology/ That science that deals with the form of the Earth, the 

morphology: general configuration of its surface, and the changes that take 

 place due to erosion and deposition. 

grade-control structure Structure placed bank to bank across a stream channel 

(sill, check dam): usually with its central axis perpendicular to flow) for the 

 purpose of controlling bed slope and preventing scour or 

 headcutting. 

graded stream: A geomorphic term used for streams that have apparently 

achieved a state of equilibrium between the rate of sediment 

transport and the rate of sediment supply throughout long 

reaches. 

gravel: A rock fragment whose diameter ranges from 2 to 64 mm. 

groin: A structure built from the bank of a stream in a direction 

transverse to the current to redirect the flow or reduce flow 

velocity. Many names are given to this structure, the most 

common being "spur," "spur dike," "transverse dike," "jetty," 

etc. Groins may be permeable, semi-permeable, or 

impermeable. 
 xlvi 

grout: A fluid mixture of cement and water or of cement, sand, and 

water used to fill joints and voids. 

guide bank: A dike extending upstream from the approach embankment at 

either or both sides of the bridge opening to direct the flow 

through the opening. Some guide banks extend downstream 

from the bridge (also spur dike). 

hardpoint: A stream bank protection structure whereby "soft" or erodible 

materials are removed from a bank and replaced by stone or 

compacted clay. Some hard points protrude a short distance 

into the channel to direct erosive currents away from the bank. 

Hard points also occur naturally along stream banks as 

passing currents remove erodible materials leaving nonerodible 

materials exposed. 

headcutting: Channel degradation associated with abrupt changes in the 

bed elevation (headcut) that generally migrates in an upstream 

direction. 

helical flow: Three-dimensional movement of water particles along a spiral 

path in the general direction of flow. These secondary-type 

currents are of most significance as flow passes through a 

bend; their net effect is to remove soil particles from the cut 

bank and deposit this material on a point bar. 

hydraulics: The applied science concerned with the behavior and flow of 

liquids, especially in pipes, channels, structures, and the 

ground. 

hydraulic model: A small-scale physical or mathematical representation of a flow 

 situation. 

hydraulic problem: An effect of streamflow, tidal flow, or wave action such that the 

integrity of the highway facility is destroyed, damaged, or 

endangered. 

hydraulic radius: The cross-sectional area of a stream divided by its wetted 

perimeter. 

hydraulic structures: The facilities used to impound, accommodate, convey or 

control the flow of water, such as dams, weirs, intakes, 

culverts, channels, and bridges. 

hydrograph: The graph of stage or discharge against time. 

hydrology: The science concerned with the occurrence, distribution, and 

circulation of water on the earth. 

imbricated: In reference to stream bed sediment particles, having an 

overlapping or shingled pattern. 
 xlvii 

icing: Masses or sheets of ice formed on the frozen surface of a river 

or floodplain. When shoals in the river are frozen to the bottom 

or otherwise dammed, water under hydrostatic pressure is 

forced to the surface where it freezes. 

incised reach: A stretch of stream with an incised channel that only rarely 

overflows its banks. 

incised stream: A stream which has deepened its channel through the bed of 

the valley floor, so that the floodplain is a terrace. 

invert: The lowest point in the channel cross section or at flow control 

devices such as weirs, culverts, or dams. 

island: A permanently vegetated area, emergent at normal stage, that 

divides the flow of a stream. Islands originate by establishment 

of vegetation on a bar, by channel avulsion, or at the junction 

of minor tributary with a larger stream. 

jack: A device for flow control and protection of banks against lateral 

erosion consisting of three mutually perpendicular arms rigidly 

fixed at the center. Kellner jacks are made of steel struts 

strung with wire, and concrete jacks are made of reinforced 

concrete beams. 

jack field: Rows of jacks tied together with cables, some rows generally 

parallel with the banks and some perpendicular thereto or at an 

angle. Jack fields may be placed outside or within a channel. 

jetty: (a) An obstruction built of piles, rock, or other material 

extending from a bank into a stream, so placed as to induce 

bank building, or to protect against erosion; (b) A similar 

obstruction to influence stream, lake, or tidal currents, or to 

protect a harbor (also spur). 

lateral erosion: Erosion in which the removal of material is extended 

horizontally as contrasted with degradation and scour in a 

vertical direction. 

launching: Release of undercut material (stone riprap, rubble, slag, etc.) 

downslope or into a scoured area. 

levee: An embankment, generally landward of top bank, that confines 

flow during high-water periods, thus preventing overflow into 

lowlands. 

live-bed scour: Scour at a pier or abutment (or contraction scour) when the 

bed material in the channel upstream of the bridge is moving at 

the flow causing bridge scour. 
 xlviii 

load (or sediment load): Amount of sediment being moved by a stream. 

local scour: Removal of material from around piers, abutments, spurs, and 

embankments caused by an acceleration of flow and resulting 

vortices induced by obstructions to the flow. 

longitudinal profile: The profile of a stream or channel drawn along the length of its 

centerline. In drawing the profile, elevations of the water 

surface or the thalweg are plotted against distance as 

measured from the mouth or from an arbitrary initial point. 

lower bank: That portion of a stream bank having an elevation less than the 

mean water level of the stream. 

mathematical model: A numerical representation of a flow situation using 

mathematical equations (also computer model). 

mattress: A blanket or revetment of materials interwoven or otherwise 

lashed together and placed to cover an area subject to scour. 

meander or full A meander in a river consists of two consecutive loops, one 

meander: flowing clockwise and the other counter-clockwise. 

meander amplitude: The distance between points of maximum curvature of 

successive meanders of opposite phase in a direction normal 

to the general course of the meander belt, measured between 

center lines of channels. 

meander belt: The distance between lines drawn tangent to the extreme limits 

of successive fully developed meanders. 

meander length: The distance along a stream between corresponding points of 

successive meanders. 

meander loop: An individual loop of a meandering or sinuous stream lying 

between inflection points with adjoining loops. 

meander ratio: The ratio of meander width to meander length. 

meander radius The radius of a circle inscribed on the centerline of a meander 

of curvature: loop. 

meander scrolls: Low, concentric ridges and swales on a floodplain, marking the 

successive positions of former meander loops. 

meander width: The amplitude of a fully developed meander measured from 

midstream to midstream. 
 xlix 

meandering stream: A stream having a sinuosity greater than some arbitrary value. 

The term also implies a moderate degree of pattern symmetry, 

imparted by regularity of size and repetition of meander loops. 

The channel generally exhibits a characteristic process of bank 

erosion and point bar deposition associated with systematically 

shifting meanders. 

median diameter: The particle diameter of the 50th percentile point on a size 

distribution curve such that half of the particles (by weight, 

number, or volume) are larger and half are smaller (D50.) 

mid-channel bar: A bar lacking permanent vegetal cover that divides the flow in a 

channel at normal stage. 

middle bank: The portion of a stream bank having an elevation 

approximately the same as that of the mean water level of the 

stream. 

migration: Change in position of a channel by lateral erosion of one bank 

and simultaneous accretion of the opposite bank. 

mud: A soft, saturated mixture mainly of silt and clay. 

natural levee: A low ridge that slopes gently away from the channel banks 

that is formed along stream banks during floods by deposition. 

nominal diameter: Equivalent spherical diameter of a hypothetical sphere of the 

same volume as a given sediment particle. 

nonalluvial channel: A channel whose boundary is in bedrock or non-erodible 

material. 

normal stage: The water stage prevailing during the greater part of the year. 

overbank flow: Water movement that overtops the bank either due to stream 

stage or to overland surface water runoff. 

oxbow: The abandoned former meander loop that remains after a 

stream cuts a new, shorter channel across the narrow neck of 

a meander. Often bow-shaped or horseshoe-shaped. 

pavement: Stream bank surface covering, usually impermeable, designed 

to serve as protection against erosion. Common pavements 

used on stream banks are concrete, compacted asphalt, and 

soil-cement. 

paving: Covering of stones on a channel bed or bank (used with 

reference to natural covering). 
 l 

peaked stone dike: Riprap placed parallel to the toe of a stream bank (at the 

natural angle of repose of the stone) to prevent erosion of the 

toe and induce sediment deposition behind the dike. 

perennial stream: A stream or reach of a stream that flows continuously for all or 

most of the year. 

phreatic line: The upper boundary of the seepage water surface landward of 

a stream bank. 

pile: An elongated member, usually made of timber, concrete, or 

steel, that serves as a structural component of a river-training 

structure. 

pile dike: A type of permeable structure for the protection of banks 

against caving; consists of a cluster of piles driven into the 

stream, braced and lashed together. 

piping: Removal of soil material through subsurface flow of seepage 

water that develops channels or "pipes" within the soil bank. 

point bar: An alluvial deposit of sand or gravel lacking permanent vegetal 

cover occurring in a channel at the inside of a meander loop, 

usually somewhat downstream from the apex of the loop. 

poised stream: A stream which, as a whole, maintains its slope, depths, and 

channel dimensions without any noticeable raising or lowering 

of its bed (stable stream). Such condition may be temporary 

from a geological point of view, but for practical engineering 

purposes, the stream may be considered stable. 

probable maximum flood: A very rare flood discharge value computed by 

hydrometeorological methods, usually in connection with major 

hydraulic structures. 

quarry-run stone: Stone as received from a quarry without regard to gradation 

requirements. 

railbank protection: A type of countermeasure composed of rock-filled wire fabric 

supported by steel rails or posts driven into streambed. 

rapid drawdown: Lowering the water against a bank more quickly than the bank 

can drain without becoming unstable. 

reach: A segment of stream length that is arbitrarily bounded for 

purposes of study. 

recurrence interval: The reciprocal of the annual probability of exceedance of a 

hydrologic event (also return period, exceedance interval). 
 li 

regime: The condition of a stream or its channel with regard to stability. 

A stream is in regime if its channel has reached an equilibrium 

form as a result of its flow characteristics. Also, the general 

pattern of variation around a mean condition, as in flow regime, 

tidal regime, channel regime, sediment regime, etc. (used also 

to mean a set of physical characteristics of a river). 

regime change: A change in channel characteristics resulting from such things 

as changes in imposed flows, sediment loads, or slope. 

regime channel: Alluvial channel that has attained, more or less, a state of 

equilibrium with respect to erosion and deposition. 

regime formula: A formula relating stable alluvial channel dimensions or slope 

to discharge and sediment characteristics. 

reinforced-earth A retaining structure consisting of vertical panels and 

bulkhead: attached to reinforcing elements embedded in compacted 

 backfill for supporting a stream bank. 

reinforced revetment: A stream bank protection method consisting of a continuous 

stone toe-fill along the base of a bank slope with intermittent 

fillets of stone placed perpendicular to the toe and extending 

back into the natural bank. 

relief bridge: An opening in an embankment on a floodplain to permit 

passage of overbank flow. 

retard (retarder A permeable or impermeable linear structure in a channel 

structure): parallel with the bank and usually at the toe of the bank, 

 intended to reduce flow velocity, induce deposition, or deflect 

 flow from the bank. 

revetment: Rigid or flexible armor placed to inhibit scour and lateral 

erosion. (See bank revetment). 

riffle: A natural, shallow flow area extending across a streambed in 

which the surface of flowing water is broken by waves or 

ripples. Typically, riffles alternate with pools along the length of 

a stream channel. 

riparian: Pertaining to anything connected with or adjacent to the banks 

of a stream (corridor, vegetation, zone, etc.). 

riprap: Layer or facing of rock or broken concrete which is dumped 

 or placed to protect a structure or embankment from erosion; 

 also the rock or broken concrete suitable for such use. Riprap 

 has also been applied to almost all kinds of armor, including 

 wire-enclosed riprap, grouted riprap, sacked concrete, and 

 concrete slabs. 
 lii 

river training: Engineering works with or without the construction of 

embankment, built along a stream or reach of stream to direct 

or to lead the flow into a prescribed channel. Also, any 

structure configuration constructed in a stream or placed on, 

adjacent to, or in the vicinity of a stream bank that is intended 

to deflect currents, induce sediment deposition, induce scour, 

or in some other way alter the flow and sediment regimes of 

the stream. 

rock-and-wire mattress: A flat wire cage or basket filled with stone or other suitable 

 material and placed as protection against erosion. 

roughness coefficient: Numerical measure of the frictional resistance to flow in 

 a channel, as in the Manning's or Chezy's formulas. 

rubble: Rough, irregular fragments of materials of random size used to 

retard erosion. The fragments may consist of broken concrete 

slabs, masonry, or other suitable refuse. 

runoff: That part of precipitation which appears in surface streams of 

either perennial or intermittent form. 

sack revetment: Sacks (e.g., burlap, paper, or nylon) filled with mortar, 

concrete, sand, stone or other available material used as 

protection against erosion. 

saltation load: Sediment bounced along the streambed by energy and 

turbulence of flow, and by other moving particles. 

sand: A rock fragment whose diameter is in the range of 0.062 to 2.0 

mm. 

scour: Erosion of streambed or bank material due to flowing water; 

often considered as being localized (see local scour, 

contraction scour, total scour). 

sediment or fluvial Fragmental material transported, suspended, or deposited by 

sediment: water. 

sediment Weight or volume of sediment relative to the quantity of 

concentration: transporting (or suspending) fluid. 

sediment discharge: The quantity of sediment that is carried past any cross section 

of a stream in a unit of time. Discharge may be limited to 

certain sizes of sediment or to a specific part of the cross 

section. 

sediment load: Amount of sediment being moved by a stream. 

sediment yield: The total sediment outflow from a watershed or a drainage 

area at a point of reference and in a specified time period. This 

outflow is equal to the sediment discharge from the drainage 

area. 
 liii 

seepage: The slow movement of water through small cracks and pores 

of the bank material. 

shear stress: See unit shear force. 

shoal: A relatively shallow submerged bank or bar in a body of water. 

sill: (a) A structure built under water, across the deep pools of a 

stream with the aim of changing the depth of the stream; (b) A 

low structure built across an effluent stream, diversion channel 

or outlet to reduce flow or prevent flow until the main stream 

stage reaches the crest of the structure. 

silt: A particle whose diameter is in the range of 0.004 to 0.062 

 mm. 

sinuosity: The ratio between the thalweg length and the valley length of a 

 stream. 

slope (of channel Fall per unit length along the channel centerline or thalweg. 

or stream): 

slope protection: Any measure such as riprap, paving, vegetation, revetment, 

brush or other material intended to protect a slope from 

erosion, slipping or caving, or to withstand external hydraulic 

pressure. 

sloughing: Sliding or collapse of overlying material; same ultimate effect 

as caving, but usually occurs when a bank or an underlying 

stratum is saturated. 

slope-area method: A method of estimating unmeasured flood discharges in a 

uniform channel reach using observed high-water levels. 

slump: A sudden slip or collapse of a bank, generally in the vertical 

direction and confined to a short distance, probably due to the 

substratum being washed out or having become unable to bear 

the weight above it. 

soil-cement: A designed mixture of soil and Portland cement compacted at 

a proper water content to form a blanket or structure that can 

resist erosion. 

sorting: Progressive reduction of size (or weight) of particles of the 

sediment load carried down a stream. 

spill-through A bridge abutment having a fill slope on the streamward side. 

abutment: The term originally referred to the "spill-through" of fill at an 

 open abutment but is now applied to any abutment having such 

 a slope. 

spread footing: A pier or abutment footing that transfers load directly to the 

earth. 
 liv 

spur: A permeable or impermeable linear structure that projects into 

a channel from the bank to alter flow direction, induce 

deposition, or reduce flow velocity along the bank. 

spur dike: See guide bank. 

stability: A condition of a channel when, though it may change slightly at 

different times of the year as the result of varying conditions of 

flow and sediment charge, there is no appreciable change from 

year to year; that is, accretion balances erosion over the years. 

stable channel: A condition that exists when a stream has a bed slope and 

cross section which allows its channel to transport the water 

and sediment delivered from the upstream watershed without 

aggradation, degradation, or bank erosion (a graded stream). 

stage: Water-surface elevation of a stream with respect to a reference 

elevation. 

stone riprap: Natural cobbles, boulders, or rock dumped or placed as 

protection against erosion. 

stream: A body of water that may range in size from a large river to a 

small rill flowing in a channel. By extension, the term is 

sometimes applied to a natural channel or drainage course 

formed by flowing water whether it is occupied by water or not. 

stream bank erosion: Removal of soil particles or a mass of particles from a bank 

surface due primarily to water action. Other factors such as 

weathering, ice and debris abrasion, chemical reactions, and 

land use changes may also directly or indirectly lead to bank 

erosion. 

stream bank failure: Sudden collapse of a bank due to an unstable condition such 

as removal of material at the toe of the bank by scour. 

stream bank Any technique used to prevent erosion or failure of a 

protection: stream bank. 

suspended sediment The quantity of sediment passing through a stream cross 

discharge: section above the bed layer in a unit of time suspended by the 

 turbulence of flow (suspended load). 

sub-bed material: Material underlying that portion of the streambed which is 

subject to direct action of the flow. Also, substrate. 

subcritical, Open channel flow conditions with Froude Number less 

supercritical flow: than and greater than unity, respectively. 

tetrahedron: Component of river-training works made of six steel or 

concrete struts fabricated in the shape of a pyramid. 
 lv 

tetrapod: Bank protection component of precast concrete consisting of 

four legs joined at a central joint, with each leg making an 

angle of 109.5 with the other three. 

thalweg: The line extending down a channel that follows the lowest 

elevation of the bed. 

tieback: Structure placed between revetment and bank to prevent 

flanking. 

timber or brush mattress: A revetment made of brush, poles, logs, or lumber interwoven 

or otherwise lashed together. The completed mattress is then 

placed on the bank of a stream and weighted with ballast. 

toe of bank: That portion of a stream cross section where the lower bank 

terminates and the channel bottom or the opposite lower bank 

begins. 

toe protection: Loose stones laid or dumped at the toe of an embankment, 

groin, etc., or masonry or concrete wall built at the junction of 

the bank and the bed in channels or at extremities of hydraulic 

structures to counteract erosion. 

total scour: The sum of long-term degradation, general (contraction) scour, 

and local scour. 

total sediment load: The sum of suspended load and bed load or the sum of bed 

material load and wash load of a stream (total load). 

tractive force: The drag or shear on a streambed or bank caused by passing 

water which tends to move soil particles along with the 

streamflow. 

trench-fill revetment: Stone, concrete, or masonry material placed in a trench dug 

behind and parallel to an eroding stream bank. When the 

erosive action of the stream reaches the trench, the material 

placed in the trench armors the bank and thus retards further 

erosion. 

turbulence: Motion of fluids in which local velocities and pressures fluctuate 

irregularly in a random manner as opposed to laminar flow 

where all particles of the fluid move in distinct and separate 

lines. 

ultimate scour: The maximum depth of scour attained for a given flow 

condition. May require multiple flow events and in cemented or 

cohesive soils may be achieved over a long time period. 
 lvi 

uniform flow: Flow of constant cross section and velocity through a reach of 

channel at a given time. Both the energy slope and the water 

slope are equal to the bed slope under conditions of uniform 

flow. 

unit discharge: Discharge per unit width (may be average over a cross section, 

or local at a point). 

unit shear force The force or drag developed at the channel bed by flowing 

(shear stress): water. For uniform flow, this force is equal to a component of 

 the gravity force acting in a direction parallel to the channel bed 

 on a unit wetted area. Usually in units of stress, Pa (N/m2

) or 

 (lb/ft2

). 

unsteady flow: Flow of variable discharge and velocity through a cross section 

with respect to time. 

upper bank: The portion of a stream bank having an elevation greater than 

the average water level of the stream. 

velocity: The time rate of flow usually expressed in m/s (ft/sec). The 

average velocity is the velocity at a given cross section 

determined by dividing discharge by cross-sectional area. 

vertical abutment: An abutment, usually with wingwalls, that has no fill slope on its 

streamward side. 

vortex: Turbulent eddy in the flow generally caused by an obstruction 

such as a bridge pier or abutment (e.g., horseshoe vortex). 

wandering channel: A channel exhibiting a more or less non-systematic process of 

channel shifting, erosion and deposition, with no definite 

meanders or braided pattern. 

wandering thalweg: A thalweg whose position in the channel shifts during floods 

and typically serves as an inset channel that conveys all or 

most of the stream flow at normal or lower stages. 

wash load: Suspended material of very small size (generally clays and 

colloids) originating primarily from erosion on the land slopes of 

the drainage area and present to a negligible degree in the bed 

itself. 

watershed: See drainage basin. 

waterway opening Width (area) of bridge opening at (below) a specified stage, 

width (area): measured normal to the principal direction of flow. 

weephole: A hole in an impermeable wall or revetment to relieve the 

neutral stress or pore pressure in the soil. 
 lvii 

windrow revetment: A row of stone placed landward of the top of an eroding stream 

bank. As the windrow is undercut, the stone is launched 

downslope, thus armoring the bank. 

wire mesh: Wire woven to form a mesh; where used as an integral part of 

a countermeasure, openings are of suitable size and shape to 

enclose rock or broken concrete or to function on fence-like 

spurs and retards. 
 lviii 

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1.1 

CHAPTER 1 

INTRODUCTION

The purpose of this chapter is to lay the groundwork for application of the concepts of 

open-channel flow, fluvial geomorphology, sediment transport, and river mechanics to the 

design, maintenance, and environmental problems associated with highway crossings and 

encroachments. 

This manual is a basic reference for related Federal Highway Administration (FHWA) hydraulic 

publications and National Highway Institute (NHI) Hydraulics Courses. Some of these 

publications are: "Hydraulics of Bridge Waterways" (Bradley 1978), "Design of Riprap 

Revetment" (Brown and Clyde 1989), "Evaluating Scour at Bridges" (Richardson and Davis 

2001), "Stream Stability at Highway Structures" (Lagasse et al. 2001), "Bridge Scour and 

Stream Instability Countermeasures - Experience, Selection and Design Guidance" (Lagasse 

et al. 2001). Related NHI courses include: (1) River Engineering for Highway Encroachments, 

(2) Stream Stability and Scour at Highway Bridges, and (3) Finite Element Surface Water 

Modeling System (FESWMS). 

Basic definitions of terms and notations adopted for use in this document have been presented 

in the preceding section (Glossary) for rapid reference. Additionally, these important terms and 

variables are defined and explained as they are encountered. 

1.1 CLASSIFICATION OF RIVER CROSSINGS AND ENCROACHMENTS

The objective in this document is to consider the fluvial, hydraulic, geomorphic, sediment 

transport, and environmental aspects of highway encroachments, including bridge locations, 

alignments, longitudinal encroachments, stabilization works and road approaches. 

Encroachment is any occupancy of the river and floodplain for highway use. Encroachments 

usually present no problems during normal stages, but require special protection against 

floods. Flood protection requirements vary from site to site. 

Some bridges and culverts must accommodate the passage of livestock and farm equipment 

during periods of low flow. Other bridges require low embankments for aesthetic appeal, 

especially in populated areas. Still other bridges require short spans with long approaches and 

numerous piers for economic reasons. All of these factors, and many more, contribute to the 

difficulty in generalizing the design for all highway encroachments. 

 

A classification of encroachments based on prominent features is helpful. Classifying the 

regions requiring protection, the possible types of protection, the possible flow conditions, the 

possible channel shapes, and the various geometric conditions aids the engineer in selecting 

the design criteria for the conditions encountered. 
1.2 

1.1.1 Types of Encroachment

 

In the vicinity of rivers, highways generally must impose a degree of encroachment. In some 

instances, particularly in mountainous regions or in river gorges and canyons, river crossings 

can be accomplished with absolutely no encroachment on the river. The bridge and its 

approaches are located far above and beyond any possible flood stage. More commonly, the 

economics of crossings require substantial encroachment on the river and its floodplain, the 

cost of a single span over the entire floodplain being prohibitive. The encroachment can be in 

the form of earth fill embankments on the floodplain or into the main channel itself, reducing 

the required bridge length; or in the form of piers and abutments or culverts in the main 

channel of the river. 

Longitudinal encroachments may exist that are not connected with river crossings. Floodplains 

often appear to provide an attractive low cost alternative for highway location, even when the 

extra cost of flood protection is included. As a consequence, highways, including 

interchanges, often encroach on a floodplain over long distances. In some regions, river 

valleys provide the only feasible route for highways. This is true even in areas where a 

floodplain does not exist. In many locations the highway must encroach on the main channel 

itself and the channel is partly filled to allow room for the roadway. In some instances, this 

encroachment becomes severe, particularly as older highways are upgraded and widened. 

Often, a stretch of the river must be straightened to eliminate meanders to accommodate the 

highway. 

1.1.2 Geometry of Bridge Crossings

The bridge crossing is the most common type of river encroachment. The geometric 

properties of bridge crossings illustrated in Figure 1.1 are commonly used depending on the 

conditions at the site. The approaches may be skewed or normal (perpendicular) to the 

direction of flow or one approach may be longer than the other, producing an eccentric 

crossing. Abutments used for the overbank-flow case may be set back from the low-flow 

channel banks to provide room to pass the flood flow or simply to allow passage of livestock 

and machinery, or the abutments may extend up to the banks or even protrude over the banks, 

constricting the low-flow channel. Piers, dual bridges for multi-lane freeways, channel bed 

conditions, spurs and guide banks add to the list of geometric classifications. 

The design procedures in this document have been derived from laboratory and field 

observations of bridge crossings. The design procedures include allowances made for the 

effects of skewness, eccentricity, scour, abutment setback, channel shape, submergence of 

the superstructure, debris, spurs, wind waves, ice, piers, abutment types, and flow conditions. 

These design procedures take advantage of the large volume of work that has been done by 

many people in describing the hydraulics and scour characteristics of bridge crossings. 

1.2 DYNAMICS OF NATURAL RIVERS AND THEIR TRIBUTARIES

Frequently, hydraulic engineers, and those involved in transportation, navigation, and flood 

control mistakenly consider a river to be static; that is, unchanging in shape, dimensions, and 

pattern. However, an alluvial river generally is continually changing its position and shape as a 
1.3 

consequence of hydraulic forces acting on its bed and banks. These changes may be slow or 

rapid and may result from natural environmental changes or from changes by human activities. 

When an engineer modifies a river channel locally, this local change frequently causes 

modification of channel characteristics both up and down the stream. The response of a river 

to human-induced changes often occurs in spite of attempts by engineers to keep the 

anticipated response under control. The points that must be stressed are that a river through 

time is dynamic and that human-induced change frequently sets in motion a response that can 

be propagated upstream or downstream for long distances. 

Figure 1.1. Geometric properties of bridge crossings. 

In spite of their complexity, all rivers are governed by the same basic forces. The design 

engineer must understand, and work with these natural forces: 

• Geological factors, including soil conditions 

• Hydrologic factors, including possible changes in flows, runoff, and the hydrologic effects of 

changes in land use 

• Geometric characteristics of the stream, including the probable geometric alterations that 

will be activated by the changes a project and future projects will impose on the channel 

• Hydraulic characteristics such as depths, slopes, and velocity of streams and what 

changes may be expected in these characteristics in space and time. 

1.2.1 Historical Evidence of the Natural Instability of Fluvial Systems

In order to emphasize the inherent dynamic qualities of river channels, evidence is cited below 

to demonstrate that most alluvial rivers are not static in their natural state. Indeed, scientists 

concerned with the history of landforms (geomorphologists), vegetation (botanists), and the 

past activities of man (archaeologists), rarely consider the landscape as unchanging. Rivers, 

glaciers, sand dunes, and seacoasts are highly susceptible to change with time. Over a 

relatively short period of time, perhaps in some cases up to 100 years, components of the 
1.4 

landscape may be relatively stable. Nevertheless, stability cannot be automatically assumed. 

Rivers are, in fact, the most actively changing of all geomorphic forms. 

Evidence from several sources demonstrates that river channels are continually undergoing 

changes of position, shape, dimensions, and pattern. In Figure 1.2 a section of the Mississippi 

River as it was in 1884 is compared with the same section as observed in 1968. In the lower 

9.6 km (6 mi) of river, the surface area has been reduced approximately 50 percent during this 

84-year period. Some of this change has been natural and some has been the consequence 

of river development work. 

 Figure 1.2. Comparison of the 1884 and 1968 Mississippi River 

 Channel near Commerce, Missouri. 

In alluvial river systems, it is the rule rather than the exception that banks will erode, sediments 

will be deposited and floodplains, islands, and side channels will undergo modification with 

time. Changes may be very slow or dramatically rapid. Fisk's (1944) report on the Mississippi 

River and his maps showing river position through time are sufficient to convince everyone of 

the innate instability of the Mississippi River. The Mississippi is our largest and most 

impressive river, and because of its dimensions it has sometimes been considered unique. 

This, of course, is not so. Hydraulic and geomorphic laws apply at all scales of comparable 

landform evolution. The Mississippi may be thought of as a prototype of many rivers or as a 

much larger than prototype model of many sandbed rivers. 

Rivers change position and morphology (dimensions, shape, pattern) as a result of changes of 

hydrology. Hydrology can change as a result of climatic changes over long periods of time, or 

as a result of natural stochastic climatic fluctuations (droughts, floods), or by human 

modification of the hydrologic regime. For example, the major climatic changes of recent 

geological time (the last few million years of earth history) have triggered dramatic changes in 

runoff and sediment loads with corresponding channel alteration. Equally significant during this 

time were fluctuations of sea level. During the last continental glaciation, sea level was on the 

order of 120 m (400 ft) lower than at present, and this reduction of baselevel caused major 

incisions of river valleys near the coasts. 
1.5 

In recent geologic time, major river changes of different types occurred. These types are deep 

incision and deposition as sea level fluctuated, changes of channel geometry as a result of 

climatic and hydrologic changes, and obliteration or displacement of existing channels by 

continental glaciation. Climatic change, sea level change, and glaciation, which are interesting 

from an academic point of view, may also be causes of modern river instability, particularly 

when the 50- 100-year design life of a bridge is considered. The movement of the earth's crust 

is one geologic agent causing modern river instability. 

The earth's surface in many parts of the world is undergoing continuous measurable change 

by upwarping, subsidence or lateral displacement. As a result, the study of these ongoing 

changes (called neotectonics) has become a field of major interest for many geologists and 

geophysicists. Such gradual surface changes can affect stream channels dramatically. For 

example, Wallace (1967) has shown that many small streams are clearly offset laterally along 

the San Andreas Fault in California. Progressive lateral movement of this fault on the order of 

25 mm (an inch) per year has been measured. The rates of movement of faults are highly 

variable, but an average rate of mountain building has been estimated by Schumm (1963) to 

be on the order of 7.6 m (25 ft) per 1,000 years. Seemingly insignificant in human terms, this 

rate is actually 7.6 mm (0.3 in) per year or 76 mm (3 in) per decade. For many river systems, a 

change of slope of 76 mm (3 in) would be significant. For example, the slope of the energy 

gradient on the Lower Mississippi River is about 47 to 95 mm/km (3 to 6 in/mi). 

Of course, the geologist is not surprised to see drainage patterns that have been disrupted by 

uplift or some complex warping of the earth's surface. In fact, complete reversals of drainage 

lines have been documented. In addition, convexities in the longitudinal profile of both rivers 

and river terraces (these profiles are concave under normal development) have been detected 

and attributed to upwarping. Further, the progressive shifting of a river toward one side of its 

valley has resulted from lateral tilting. Major shifts in position of the Brahmaputra River toward 

the west are attributed by Coleman (1969) to tectonic movements. Hence, neotectonics should 

not be ignored as a possible cause of local river instability. 

Long-term climatic fluctuations have caused major changes of river morphology. Floodplains 

have been destroyed and reconstructed many times over. The history of semi-arid and arid 

valleys of the western United States is one of alternating periods of channel incision and arroyo 

formation followed by deposition and valley stability which have been attributed to climatic 

fluctuations. 

It is clear that rivers can display a remarkable propensity for change of position and 

morphology in time periods of a century. Hence, rivers from the geomorphic point of view are 

unquestionably dynamic, but does this apply to modern rivers? It is probable that during a 

period of several years, neither neotectonics nor a progressive climate change will have a 

detectable influence on river character and behavior. What then causes a river to appear 

relatively unstable from the point of view of the highway engineer or the environmentalist? It is 

the slow but implacable shift of a river channel through erosion and deposition at bends, the 

shift of a channel to form chutes and islands, and the cutoff of a bend to form oxbow lakes. 

Lateral migration rates are highly variable; that is, a river may maintain a stable position for 

long periods and then experience rapid movement. Much, therefore, depends on flood events, 

bank stability, permanence of vegetation on banks and the floodplain and watershed land use. 

A compilation of data by Wolman and Leopold shows that rates of lateral migration for the Kosi 

River of India range up to approximately 760 m/yr (2,500 ft/yr). Rates of lateral migration for 

two major rivers in the United States are as follows: Colorado River near Needles, California, 3 
1.6 

to 46 m/yr (10 to 150 ft/yr); Mississippi River near Rosedale, Mississippi, 48 to 192 m/yr (158 to 

630 ft/yr). 

Archaeologists have also provided clear evidence of channel changes that are completely 

natural and to be expected. For example, the number of archaeologic sites of the floodplains 

decreases significantly with age because the earliest sites are destroyed as floodplains are 

modified by river migration. Lathrop (1968), working on the Rio Ycayali in the Amazon 

headwaters of Peru, estimates that on the average a meander loop on this river begins to form 

and cuts off in 5,000 years. These loops have an amplitude of 3.2 to 9.7 km (2 to 6 mi) and an 

average rate of meander growth of approximately 12 m/yr (40 ft/yr). 

A study by Schmuddle (1963) shows that about one-third of the floodplain of the Missouri River 

over the 274 km (170 mi) reach between Glasgow and St. Charles, Missouri, was reworked by 

the river between 1879 and 1930. On the Lower Mississippi River, bend migration was on the 

order of 0.6 m/yr (2 ft/yr), whereas in the central and upper parts of the river, below Cairo, it 

was at times 305 m/yr (1,000 ft/yr) (Kolb 1963). On the other hand, a meander loop pattern of 

the lower Ohio River has altered very little during the past thousand years (Alexander and 

Nunnally 1972). 

Although the dynamic behavior of perennial streams is impressive, the modification of rivers in 

arid and semi-arid regions and especially of ephemeral (flowing occasionally) stream channels 

is startling. A study of floodplain vegetation and the distribution of trees in different age groups 

led Everitt (1968) to the conclusion that about half of the Little Missouri River floodplain in 

western North Dakota was reworked in 69 years. 

Historical and field studies by Smith (1940) show that floodplain destruction occurred during 

major floods on rivers of the Great Plains. As exceptional example of this is the Cimarron 

River of Southwestern Kansas, which was 15 m (50 ft) wide during the latter part of the 19th 

and first part of the 20th centuries (Schumm and Lichty 1957). Following a series of major 

floods during the 1930s it widened to 366 m (1,200 ft) and the channel occupied essentially the 

entire valley floor. During the decade of the 1940s a new floodplain was constructed, and the 

river width was reduced to about 152 m (500 ft) in 1960. Equally dramatic changes of channel 

dimensions have occurred along the North and South Platte Rivers in Nebraska and Colorado 

as a result of control of flood peaks by reservoir construction, decrease in annual flow by 

irrigation and restriction of channel width by bridges. In their natural state the rivers were 600 

to 1,500 m (2000 to 5,000 ft) wide but now are less than 300 m (1,000 ft). Changes of this 

magnitude due to changes in flow are perhaps exceptional, but emphasize the mobility of 

rivers and their ability to adapt to changing conditions. 

Another somewhat different type of channel modification, which testifies to the rapidity of fluvial 

processes, is described by Shull (1922, 1944). During a major flood in 1913, a barge became 

stranded in a chute of the Mississippi River near Columbus, Kentucky. The barge induced 

deposition in the chute and an island formed. In 1919, the island was sufficiently large to be 

homesteaded, and a few acres were cleared for agricultural purposes. By 1933, the side 

channel separating the island from the mainland had filled to the extent that the island became 

part of Missouri. The island formed in a location protected from the erosive effects of floods 

but susceptible to deposition of sediment during floods. For these reasons the channel filling 

was rapid and progressive. It cannot be concluded that islands will always form and side 

channels fill at such rapid rates, but island formation and side-channel filling appear to be the 

normal course of events in any river transporting moderate or high sediment loads regardless 

of the river size. These topics are discussed in detail in Chapter 5. 
1.7 

In addition to changes in planform location and size with time, the bed configuration of a river 

can change with temperature, discharge, and concentration of silts and clays. At low flow or 

with warm water the bed of a sand bed stream can be dunes, but at large flows or cold 

temperature the bed may become plane or have antidune flow. With dunes, resistance to flow 

is large and bed material transport is low. Whereas, with plane bed or antidune flow the 

resistance to flow is small and the bed material transport is large. This topic is discussed in 

detail in Chapter 3. 

In summary, archaeological, botanical, geological, and geomorphic evidence supports the 

conclusion that most rivers are subject to constant change as a normal part of their 

morphologic evolution. Therefore, stable or static channels are the exception in nature. 

1.2.2 Introduction to River Hydraulics and River Response

In the previous section, it was established that rivers are dynamic and respond to changing 

environmental conditions. The direction and extent of the change depends on the forces 

acting on the system. The mechanics of flow in rivers is a complex subject that requires 

special study, which is unfortunately not included in basic courses of fluid mechanics. The 

major complicating factors in river mechanics are: (a) the large number of interrelated variables 

that can simultaneously respond to natural or imposed changes in a river system and (b) the 

continual evolution of river channel patterns, channel geometry, bars and forms of bed 

roughness with changing water and sediment discharge. In order to understand the responses 

of a river to the actions of humans and nature, a few simple hydraulic and geomorphic 

concepts are presented here. 

River forms are broadly classified as straight, meandering, braided or some combination of 

these classifications; but any changes that are imposed on a river may change its form. The 

dependence of river sinuosity on the slope, which may be imposed independent of the other 

river characteristics, is illustrated schematically in Figure 1.3. By changing the slope, it is 

possible to change the river from a meandering one that is relatively tranquil and easy to 

control to a braided one that varies rapidly with time, has high velocities, is subdivided by 

sandbars and carries relatively large quantities of sediment. Such a change could be caused 

by a natural or artificial cutoff. Conversely, it is possible that a slight decrease in slope could 

change an unstable braided river into a meandering one. 

The significantly different channel dimensions, shapes, and patterns associated with different 

quantities of discharge and amounts of sediment load indicate that as these independent 

variables change, major adjustments of channel morphology can be anticipated. Further, if 

changes in sinuosity and meander wavelength as well as in width and depth are required to 

compensate for a hydrologic change, then a long period of channel instability can be 

envisioned with considerable bank erosion and lateral shifting of the channel before stability is 

restored. The reaction of a channel to changes in discharge and sediment load may result in 

channel dimension changes contrary to those indicated by many regime equations. For 

example, it is conceivable that a decrease in discharge together with an increase in sediment 

load could actuate a decrease in depth and an increase in width. 
1.8 

Figure 1.3. Sinuosity vs. slope with constant discharge. 

Changes in sediment and water discharge at a particular point or reach in a stream may have 

an effect ranging from some distance upstream to a point downstream where the hydraulic and 

geometric conditions will have absorbed the change. Thus, it is necessary to consider a 

channel reach as part of a complete drainage system. Artificial controls that could benefit the 

reach may, in fact, cause problems in the system as a whole. For example, flood control 

structures can cause downstream flood damage to be greater at reduced flows if the average 

hydrologic regime is changed so that the channel dimensions are actually reduced. Also, 

where major tributaries exert a significant influence on the main channel by introducing large 

quantities of sediment, upstream control on the main channel may allow the tributary to 

intermittently dominate the system with deleterious results. If discharges in the main channel 

are reduced, sediments from the tributary that previously were eroded will no longer be carried 

away and serious aggradation with accompanying flood problems may arise. 

An insight into the direction of change, the magnitude of change, and the time involved to 

reach a new equilibrium can be gained by: 

• Studying the river in a natural condition 

• Having knowledge of the sediment and water discharge 

• Predicting the effects and magnitude of future human activities 

• Applying knowledge of geology, soils, hydrology, and hydraulics of alluvial rivers 

The current interest in ecology and the environment have made people aware of the many 

problems that humans can cause. Prior to this interest in environmental impacts, very few 

people interested in rivers ever considered the long-term changes that were possible. It is 

imperative that anyone working with rivers, either with localized areas or entire systems, have 

an understanding of the many factors involved, and of the potential for change existing in the 

river system. 
1.9 

Two methods of predicting response are physical and mathematical models. Engineers have 

long used small-scale hydraulic models to assist them in anticipating the effect of altering 

conditions in a reach of a river. With proper awareness of the large-scale effects that can 

exist, the results of hydraulic model testing can be extremely useful for this purpose. An 

alternative method of predicting short-term and long-term changes in rivers involves the use of 

mathematical computer models. To study a transient phenomenon in natural alluvial channels, 

the equations of motion and continuity for sediment laden water and the continuity equation for 

sediment can be used as discussed in Chapters 2, 3, 4, and 5. 

 

1.3 EFFECTS OF HIGHWAY CONSTRUCTION ON RIVER SYSTEMS

Highway construction can have significant general and local effects on the geomorphology and 

hydraulics of river systems. Hence, it is necessary to consider induced short-term and 

long-term responses of the river and its tributaries, the impact on environmental factors, the 

aesthetics of the river environment and short-term and long-term effects of erosion and 

sedimentation on the surrounding landscape and the river. The biological response of the river 

system should also be considered and evaluated. 

1.3.1 Immediate Response of Rivers to Encroachment

Let us consider a few of the numerous and immediate responses of rivers to the construction 

of bridges, channel stabilization, and countermeasures. 

Local changes made in the geometry or the hydraulic properties of the river might be of such a 

magnitude as to have an immediate impact upon the entire river system. At bridges, 

contraction due to the construction of encroachments usually cause contraction and local 

scour, and the sediments removed from this location are usually dropped in the immediate 

reach downstream. In the event that the contraction is extended further downstream, the river 

may be capable of carrying the increased sediment load an additional distance, but only until a 

reduction in gradient and a reduction in transport capability is encountered. The increased 

velocities caused by encroachments may also affect the general lateral stability of the river 

downstream. 

In addition, the development of crossings and the contraction of river sections may have a 

significant effect on the water level in the vicinity and upstream of the bridge. Such changes in 

water level upstream of the bridge are called backwater effects. The highway engineer must 

be in a position to accurately assess the effects of the construction of crossings upon the water 

surface profile. 

In many instances, to offset increased velocities and to reduce bank instabilities and related 

problems, the river is stabilized or channelized to some degree. When it is necessary to do 

this, every effort should be made to accomplish the channelization in a manner which does not 

degrade the river environment, including the river's aesthetic value. 

As a consequence of construction, many areas become highly susceptible to erosion. The 

transported sediment is carried from the construction site by surface flow into the minor rills, 

which combine within a short distance to form larger channels leading to the river. The water 
1.10 

flowing from the construction site is usually a consequence of rain. The surface runoff and the 

accompanying erosion can significantly increase the sediment yield to the river channel unless 

careful control is exercised. The large sediment particles transported to the main channel may 

remain in the vicinity of the construction site for a long period of time or may be slowly moved 

away. On the other hand, the fine sediments are easily transported and generally disperse into 

the whole cross section of the river. The fine sediments are transported downstream to the 

nearest reservoir or to the sea. As will be discussed later, the sudden injection of the larger 

sediments into the channel may cause local aggradation, thereby steepening the channel, 

increasing the flow velocities and possibly causing instability in the river at that site. 

The suspended fine sediments can have very significant effects on the biomass of the stream. 

 Certain species of fish can only tolerate large quantities of suspended sediment for relatively 

short periods of time. This is particularly true of the eggs and fry. This type of biological 

response to development normally falls outside of the competence of the engineer. Yet, 

engineering works may be responsible for the discharge of these sediments into the system. 

In this regard, the engineer should utilize adequate technical assistance from experts in 

fisheries, biology, and other related areas to assess the consequences of sediment in a river. 

Only with such knowledge can one develop the necessary arguments to "sell" the case that 

erosion control measures must be exercised to avoid significant deterioration of the stream 

environment not only in the immediate vicinity of the bridge but in many instances for great 

distances downstream. 

Another possible immediate response of the river system to construction is the loss of the 

recreational use of the river. In many streams, there may be an immediate drop in the quality 

of the fishing due to the increase of sediment load or other changed hydraulic characteristics 

within the channel. Some natural rivers consist of a series of pools and riffles. Both form an 

important part of the environment for fish. The introduction of larger quantities of sediment into 

the channel and changes made in the geometry of the channel may result in the loss of these 

pools and riffles. Along the same lines, construction work within the river may cause a loss of 

food essential to fish life and often it is difficult to get the food chain reestablished in the 

system. In contrast, construction of barbs, spurs, and other river control works to stabilize the 

river channel and protect a bridge crossing can decrease erosion, provide a better habitat for 

fish and in general improve the environment. 

Construction and operation of highways in water supply watersheds present very real problems 

and may require special precautionary designs to protect the water supplies from highway 

runoff or accidental spills. Runoff from highway construction may increase sediment and 

turbidity of the water. 

The preceding discussion is related to only a few immediate responses to construction along a 

river. However, they are responses that illustrate the importance of considering the 

environment in the design of highway encroachments. 

1.3.2 Delayed Response of Rivers to Encroachment 

In addition to the examples of possible immediate response discussed above, there are 

important delayed responses of rivers to highway development. As part of this introductory 
1.11 

chapter, consideration is given to some of the more obvious effects that can be induced on a 

river system over a long time period by highway construction. 

Sometimes it is necessary to employ training works in connection with highway encroachments 

to align the flow with bridge or culvert openings. When such training works are used, they 

generally straighten the channel, shorten the flow line, and increase the local velocity within the 

channel. Any such changes made in the system that cause an increase in the gradient may 

cause an increase in local velocities. The increase in velocity increases local and contraction 

scour with subsequent deposition downstream where the channel takes on its normal 

characteristics. If significant lengths of the river are trained and straightened, there can be a 

noticeable decrease in the elevation of the water surface profile for a given discharge in the 

main channel. Tributaries emptying into the main channel in such reaches are significantly 

affected. Having a lower water level in the main channel for a given discharge means that the 

tributary streams entering in that vicinity are subjected to a steeper gradient and higher 

velocities which cause degradation in the tributary streams. In extreme cases, degradation can 

be induced of such magnitude as to cause failure of structures such as bridges, culverts or 

other encroachments on the tributary systems. In general, any increase in transported 

materials from the tributaries to the main channel causes a reduction in the quality of the 

environment within the river. More specifically, as degradation occurs in the tributaries, bank 

instabilities are induced and the sediment loads are greatly increased. Increased sediment 

loads usually result in a deterioration of the environment. 

1.4 EFFECTS OF RIVER DEVELOPMENT ON HIGHWAY ENCROACHMENTS

Some of the possible immediate and delayed responses of rivers and river systems to the 

construction of bridges, approaches, culverts, channel stabilization, longitudinal 

encroachments, and the utilization of training works have been mentioned. It is necessary also 

to consider the effects on highway encroachments of river development works. These works 

may include, for example, water diversions to and from the river system, construction of 

reservoirs, flood control works, cutoffs, levees, navigation works, and the mining of sand and 

gravel. It is essential to consider the probable long-term plans of all agencies and groups as 

they pertain to a river when designing crossings or when dealing with the river in any way. Let 

us consider a few typical responses of a crossing to different types of water resources 

development. 

Cutoffs may develop naturally in the river system or they may be the result of human activity. 

The general consequence of cutoffs is to shorten the flow path and steepen the gradient of the 

channel. The local steepening can significantly increase the velocities and sediment transport. 

Also, this action can induce significant instability such as bank erosion and degradation in the 

reach. The material scoured in the reach affected by the cutoff is probably carried only to an 

adjacent downstream reach where the gradient is flatter. In this region of slower velocities, the 

sediment drops out rapidly. Deposition can have significant detrimental effect on the 

downstream reach of river, increasing the flood stage in the river itself and increasing the base 

level for the tributary stream, thereby causing aggradation in the tributaries. 

Consider a classic example of a cutoff that was constructed on a large bend in one of the 

tributaries to the Mississippi. Along this bend, small towns had developed and small tributary 

streams entered the main channel within the bendway. It was decided to develop a cutoff 
1.12 

across the meander to shorten the flow line of the river, reduce the flood stage and generally 

improve poor hydraulic conditions in that location. Several interesting results developed. 

In the vicinity of the cutoff, the bankline eroded and degradation was initiated. Within the 

bendway, the small tributaries continued to discharge their water and sediment. Because of 

the flat gradient in the bend, this channel section could not convey the sediment from these 

small stream systems and aggradation was initiated. Within a short period of time, sufficient 

aggradation had occurred so as to jeopardize water intakes and sewage outfalls. As a 

consequence of the adverse action in the vicinity of the cutoff and within the bendway itself, it 

was finally decided that it would be more beneficial to restore the river to its natural form 

through the bend. This action was taken and the serious problems were alleviated. 

In such a haphazard program of river development, the highway engineer would be hard 

pressed to maintain and plan for the highway system along and over this reach of river. 

Another common case occurs with the development of reservoirs for storage and flood control. 

These reservoirs serve as traps for the sediment normally flowing through the river system. 

With sediment trapped in the reservoir, essentially clear water is released downstream of the 

dam site. This clear water has the capacity to transport more sediment than is immediately 

available. Consequently the channel begins to supply this deficit with resulting degradation of 

the bed or banks. This degradation may significantly affect the safety of bridges in the 

immediate vicinity. Again, the degraded or widened main channel causes steeper gradients on 

tributary streams in the vicinity of the main channel. The result is degradation in the tributary 

streams. It is entirely possible, however that the additional sediments supplied by the tributary 

streams would ultimately offset the degradation in the main channel. Thus, it must be 

recognized that downstream of storage structures the channel may either aggrade or degrade 

and the tributaries will be affected in either case. 

Significant responses can be induced upstream of reservoirs as well as downstream. When 

the stream flowing into a reservoir encounters the ponded water, its sediment load is deposited 

forming a delta. This deposition in the reservoir flattens the gradient of the channel upstream. 

The flattening of the upstream channel induces aggradation causing the bed of the river to rise, 

threatening highway installations and other facilities. For example, Elephant Butte Reservoir, 

built on the Rio Grande in New Mexico, has caused the Rio Grande to aggrade many miles 

upstream of the reservoir site. This change in bed level can have very significant effects upon 

bridges, other hydraulic structures and all types of training and stabilization works. Ultimately 

the river may be subjected to a flow of magnitude sufficient to overflow existing banks, causing 

the water to seek an entirely new channel. With the abandonment of the existing channel 

there would be a variety of bridges and hydraulic structures that would also be abandoned at 

great expense to the public. 

The clear-water diversion into South Boulder Creek in Colorado is another example of river 

development that affects bridge crossings and encroachments as well as the environment in 

general. Originally the North Fork of South Boulder Creek was a small but beautiful scenic 

mountain stream. The banks were nicely vegetated; there was a beautiful sequence of riffles 

and pools, which had all the attributes of a good fishing habitat. Years ago, water was diverted 

from the Western Slope of the Rockies through a tunnel to the North Fork of South Boulder 

Creek. The normal stage in that channel was increased by a factor of 4 to 5. The extra water 

caused significant bank erosion and channel degradation. In fact, the additional flow gutted 

the river valley, changing the channel to a straight raging torrent capable of carrying large 
1.13 

quantities of sediment. Degradation in the system had reached as much as 5 to 6 m (15 to 20 

ft) before measures were taken to stabilize the creek. 

Stabilization was achieved by flattening the gradient by constructing numerous drop structures 

and by reforming the banks with riprap. The system has stabilized but it is a different system. 

The channel is straight, much of the vegetation has been washed away, and the natural 

sequence of riffles and pools has been destroyed. The valley may never again have the 

natural form and beauty it once possessed. We should bear in mind that diversions to or from 

the natural river system can greatly alter its geometry, beauty and utility. The river may 

undergo a complete change, giving rise to a multitude of problems in connection with the 

design and maintenance of hydraulic structures, encroachments and bridge crossings along 

the affected reach. 

In the preceding paragraphs, possible immediate and long-term responses of river systems to 

various types of river development have been described, but no guidance has been given on 

how to determine the magnitude of these changes. This important aspect of the response of 

rivers to development is treated in detail in later chapters. 

1.5 TECHNICAL ASPECTS

Effects of river development, flood control measures and channel structures built during the 

last century have proven the need for considering delayed and far-reaching effects of any 

alteration humans make in a natural alluvial river system. 

Because of the complexity of the processes occurring with natural channel flows and the 

accompanying erosion and deposition of material, an analytical approach to the problem can 

be very difficult and time consuming. Most of our river process relations have been derived 

empirically. Nevertheless, if a greater understanding of the principles governing the processes 

of river formation is to be gained, the empirically derived relations must be put in the proper 

context by employing an analytical approach. 

Attempts at controlling large rivers have often led to the situation described by J. Hoover 

Mackin (1937) when he wrote: 

"The engineer who alters natural equilibrium relations by diversion or damming 

or channel improvement measures will often find that he has the bull by the tail 

and is unable to let go. . . . . . as he continues to correct or suppress 

undesirable phases of the chain reaction of the stream to the initial 'stress' he 

will necessarily place increasing emphasis on study of the genetic aspects of 

the equilibrium in order that he may work with rivers, rather than merely on 

them." 

Through such experiences, one realizes that, to prevent or reduce the detrimental effects of 

any modification of the natural processes and state of equilibrium on a river, one must gain an 

understanding of the physical laws governing them and become knowledgeable of the 

far-reaching effects of any attempt to control or modify a river's course. 
1.14 

1.5.1 Variables Affecting River Behavior

Variables affecting alluvial river channels are numerous and interrelated. Their nature is such 

that, unlike rigid boundary hydraulic problems, it is extremely difficult to study the role of any 

individual variable. 

Major factors affecting alluvial stream channel forms are: (1) stream discharge, viscosity, 

temperature; (2) sediment discharge; (3) longitudinal slope; (4) bank and bed resistance to 

flow; (5) vegetation; (6) geology, including types of sediments; and (7) human activity. 

The fluvial processes involved are very complicated and the variables of importance are 

difficult to isolate. Many laboratory and field studies have been carried out in an attempt to 

relate these and other variables. The problem has been more amenable to an empirical 

solution than an analytical one. 

In an analysis of flow in alluvial rivers, the flow field is complicated by the constantly changing 

discharge. Significant variables are, therefore, quite difficult to relate mathematically. One 

approach is to list measurable or computable variables which effectively describe the 

processes occurring and then to reduce the list by making simplifying assumptions and 

examining relative magnitudes of variables. When this is done, the basic equations of fluid 

motion may be simplified (on the basis of valid assumptions) to describe the physical 

processes within an acceptable balance between accuracy and limitations of obtaining data. 

It is the role of the succeeding chapters to present these variables, define them, show how they 

interrelate, quantify their interrelations where feasible, and show how they can be applied to 

achieve the successful design of river crossings and encroachments. 

1.5.2 Basic Knowledge Required

In order for engineers to cope successfully with river engineering problems, it is necessary to 

have an adequate background in engineering with an emphasis on hydrology, hydraulics, 

erosion and sedimentation, river mechanics, soil mechanics, structures, economics, the 

environment and related subjects. In all aspects of bridge inspection, maintenance, design and 

construction an interdisciplinary approach is needed. In fact, as the public demands more 

comprehensive treatment of river development problems, the highway engineer must further 

improve his or her knowledge, and the application of it, by soliciting the cooperative efforts of 

the hydraulic engineer, hydrologist, geologist, geomorphologist, meteorologist, mathematician, 

statistician, computer programmer, systems engineer, soil physicist, soil chemist, biologist, 

water management staff and economist. In addition, professional organizations representing 

these disciplines should be encouraged to work cooperatively to achieve long-range research 

needs and goals relative to river development. 

1.5.3 Data Requirements

Large amounts of data pertaining to understanding the behavior of rivers have been acquired 

over a long period of time. Nevertheless, some data collection efforts have been sporadic and 

unfocused. Agencies should take a careful look at present data requirements needed to solve 
1.15 

practical problems along with existing data. A careful analysis of data requirements would 

make it possible to more efficiently utilize funds to collect data in the future. The basic type of 

information that is required includes: water discharge hydrograph, sediment discharge 

hydrograph, the characteristics of the sediments being transported by streams, the 

characteristics of the channels in which the water and sediment are transported, and the 

characteristics of watersheds and how they deliver water and sediment to the stream systems. 

 Environmental data is also needed so that proper assessment can be made of the impact of 

river development upon the environment and vice versa. The problem of data requirements at 

river crossings is of sufficient importance that it is treated in greater detail in Chapter 8. 

1.6 FUTURE TECHNICAL TRENDS

When considering the future, it is essential to recognize the present state of knowledge 

pertaining to river hydraulics and then identify inadequacies in existing theories and our 

understanding of the physical processes involved, and encourage further research to help 

correct these deficits of knowledge. In order to correct such deficits there is a need to take a 

careful look at existing data pertaining to rivers, future data requirements, research needs, 

training programs and methods of developing staff that can apply this knowledge to the 

solution of practical problems. 

Advances have been made in developing computational software to establish hydraulic 

variables for scour computations, including 1- and 2-dimensional, steady and unsteady 

models. Most, if not all, of the commonly used scour prediction equations have been 

incorporated into these models. However, applications methodologies are required to 

facilitate the use of more appropriate hydraulic variables that can be obtained from more 

sophisticated computer models. World wide web sites providing hydraulic models applicable 

to scour computations include: 

www.fhwa.dot.gov/bridge/hydsoft.htm

www.hec.usace.army.mil/software/index.html

1.6.1 Adequacy of Current Knowledge

The basic principles of fluid mechanics involving application of continuity, momentum and 

energy concepts are well known and can be effectively applied to a wide variety of river 

problems. Considerable work has been done on the hydraulics of rigid boundary open 

channels and excellent results can be expected if the principles are applied properly. The 

steady-state sediment transport of nearly uniform sizes of sediment in alluvial channels is well 

understood. There is good understanding of stable channel theory in non-cohesive materials 

of all sizes. The theory is adequate to enable us to design stable systems in the existing bed 

and bank material or, if necessary, designs can be made for appropriate types of stabilization 

treatments so that canals and rivers behave in a stable manner. There have been extensive 

studies of the fall velocity of non-cohesive sediments in static fluids to provide knowledge about 

the interaction between the particle and fluid so essential to the development of sediment 

transport theories. 
1.16 

The use of computers and the development of computer programs have greatly helped the 

hydraulic engineer to solve problems on highway crossings and encroachments. The level of 

effort required to apply these models to bridge and hydraulic structure design has been greatly 

reduced with the availability of graphical user's interfaces. These interfaces are used for model 

development, troubleshooting, output presentation, and review. The one-dimensional 

WSPRO (FHWA) or HEC-RAS (Corps of Engineers) computer programs are extremely useful 

to determine the water surface elevation and the velocity of the flow in a river at a highway 

crossing. Both the Corps' and HEC-RAS (Version 3x) and UNET models are particularly useful 

in unsteady flow problems. FHWA’s BRI-STARS program can be used to determine flow of 

water and sediment through a bridge crossing. FHWA’s FESWMS and the Corps of Engineers 

RMA-2V 2-dimensional models provide water surface elevation and local velocities in two 

dimensions for unsteady flows for complex problems such as tidal flows. The Corps of 

Engineers HEC 1 and related HEC HMS programs are of particular importance in routing water 

from an upstream gaging station downstream to a bridge, taking into account any increase or 

decrease in flow that might occur in between the station and the bridge. 

Thus, available concepts and theories, which can be applied to the behavior of rivers, are 

extensive. However, in many instances only empirical relationships have been developed and 

these are pertinent to specific problems only. Consequently, a more basic theoretical 

understanding of flow in the river systems needs to be developed. 

With respect to many aspects of river mechanics, it can be concluded that knowledge is 

available to cope with the majority of river problems. On the other hand, the number of 

individuals who are cognizant of existing theory and can apply it successfully to the solution of 

river problems is limited. Particularly, the number of individuals involved in the actual solution 

of applied river mechanics problems is very small. There is a specific reason for this deficit of 

trained personnel. Undergraduate engineering educators in the universities in the United 

States, and in the world for that matter, devote only a small amount of time to teaching 

hydrology, river mechanics, channel stabilization, fluvial geomorphology, and related problems. 

 It is not possible to obtain adequate training in these important topics except at the graduate 

level, and only a limited number of universities and institutions offer the required training in 

these subject areas. There is great need for adequately trained and experienced practitioners 

to cope with river problems. 

1.6.2 Research Needs

As knowledge of river hydraulics is reviewed, it becomes quite obvious that many things are 

not adequately known. Some research needs are particularly urgent and promise a rather 

quick return. Stabilization of rivers and bank stability of river systems needs further 

consideration. Also, the study of bed forms generated by the interaction between the water 

and sediment in the river systems deserves further study. The types of bed forms have been 

identified, but theories pertaining to their development are inadequate. Simple terms have 

been used to describe the characteristics of alluvial material of both cohesive and 

non-cohesive types; a comprehensive look at the characteristics of materials is warranted. 

Other important research problems include the fluid mechanics of the motion of particles, 

secondary currents, two-dimensional velocity distributions, fall velocity of particles in turbulent 

flow and the application of remote sensing and geographic information system techniques to 
1.17 

hydrology and river mechanics. The physical modeling of rivers followed by prototype 

verification, mathematical modeling of river response followed by field verification, 

mathematical modeling of water and sediment yield from small watersheds, and studies of 

unsteady sediment transport are areas in which significant advances can be made. 

A primary research need is the collection of field data on the flow variables and depth of scour 

at bridges, embankments and at river control structures. Studies are needed to better define 

the planform of rivers and their response to changing flow conditions. In addition, laboratory 

and field studies are needed to improve the equations for estimating total scour at piers and 

abutments. 

Operational research on decision making, considering cost and risk criteria to determine the 

hydrologic and hydraulic design of highway structures and project alternatives, is another 

pressing research area. Insufficient data is frequently a problem of river mechanics analysis. 

A comprehensive study on information theory is needed to cope with such difficulties. 

Finally the results of these efforts must be presented in such a form that it can be easily taught 

and easily put to practical use. 

1.6.3 Training

As pointed out in Section 1.6.1, engineering training is often inadequate in relation to 

understanding the dynamics of rivers. Better ways to train engineers and to disseminate 

existing knowledge in this important area need to be considered. The curriculum of university 

education made available to engineers should be improved, particularly at the undergraduate 

level. At the very minimum, such a curriculum should strive to introduce concepts of fluvial 

geomorphology, river hydraulics, erosion and sedimentation, environmental considerations and 

related topics. 

Formal training should be supported with field trips and laboratory demonstrations. Laboratory 

demonstrations are an inexpensive method of quickly and effectively teaching the 

fundamentals of river mechanics and illustrating the behavior of structures. These 

demonstrations should be followed by field trips to illustrate similarities and differences 

between phenomena in the laboratory and in the field. 

1.7 OVERVIEW OF MANUAL CONTENTS 

In the following sections a brief overview of each chapter will be given. 

1.7.1 Chapter 2 - Open Channel Flow 

This chapter describes the fundamentals of rigid boundary open channel flow. In open 

channel flow, the water surface is not confined; surface configuration, flow pattern and 

pressure distribution within the flow depend on gravity. In rigid boundary open channel flow, 

no deformation of the bed and banks is considered. Mobile boundary hydraulics refers to 

flow, which can generate deformation of the boundary through scour and fill. Mobile 
1.18 

boundary hydraulics will be discussed in Chapters 3, 4, and 5. Chapter 2 is restricted to a 

1-dimensional analysis of rigid boundary open channel flow, where velocity and acceleration 

are large only in one direction and are so small as to be negligible in all other directions. 

The three basic equations of flow: continuity, energy, and momentum are derived and used. 

The characteristics of uniform or nonuniform, steady or unsteady, laminar or turbulent, and 

tranquil or rapid (subcritical or supercritical) open channel flow are described, equations 

derived and problems for highways in the river environment are solved. Example problems 

are solved at the end of the chapter in both SI and English units of measurement.

1.7.2 Chapter 3 - Fundamentals of Alluvial Channel Flow 

Most streams that a highway will cross or encroach upon are alluvial. That is, the rivers are 

formed in cohesive or non-cohesive materials that have been, and can still be, transported 

by the stream. The non-cohesive material generally consists of silt, sand, gravel, or cobbles, 

or any combination of these sizes. Silt generally is not present in appreciable quantities in 

streams having non-cohesive boundaries. Cohesive material consists of clays (sizes less 

than 0.004 mm) forming a binder with silts and sand 

Chapter 3 presents the fundamentals of alluvial channel flow. It covers properties of alluvial 

material, methods of measuring these properties; describes flow in sandbed channels and 

associated bed forms; presents methods for prediction of bed forms, and Manning's n for 

sandbed and other natural streams; describes how bed-form changes affect highways in the 

river environment; presents equations, tables and figures for determining the beginning of 

motion of non-cohesive sediments; and describes the methods used for the physical 

measurement and calculation of sediment discharge. These fundamentals of alluvial 

channel flow are used in later chapters to develop design considerations for highway 

crossings and encroachments in river environments. Example problems are solved at the 

end of the chapter in both SI and English units of measurement. 

1.7.3 Chapter 4 - Sediment Transport 

The quantity and quality of the sediments that a stream can transport is an important 

consideration for highways in the river environment. Scour at a bridge or culvert is a sediment 

transport process. Streambed erosion and movement is the result of sediment transport. The 

amount of sediment transported or deposited in a stream under a given set of conditions is the 

result of the interaction of two groups of variables. In the first group are those variables that 

determine the quantity and quality of the sediment brought down to that section of the stream 

(bed and bank material). In the second group are the variables that determine the capacity of 

the stream to transport that sediment. 

Chapter 4 presents the terms that describe sediment transport, discusses the methods and 

fluid forces that move sediment, derives and explains the basic equations, and presents 

equations and computer models that other practitioners have developed. Example problems to 

determine the quantity and quality of sediment transport are solved at the end of the chapter in 

SI and English units of measurement. 
1.19 

1.7.4 Chapter 5 - River Morphology and River Response 

Rivers have different alignments and geometry. There are meandering rivers, braided rivers, 

and rivers that are essentially straight. In general, braided rivers are relatively steep and 

meandering rivers have more gentle slopes. Meandering rivers that are not subject to rapid 

movement, are reasonably predictable in behavior; however, meandering rivers are generally 

unstable with eroding banks which may result in destruction of productive land, bridges, bridge 

approaches, control works, buildings, and urban properties during floods. Bank protection 

works are often necessary to stabilize reaches of many rivers and to improve them for other 

aspects of flood control and navigation. 

Chapter 5 presents the fluvial geomorphology of rivers and methods to predict river response 

to external forces such as a bridge crossing or other natural or human induced changes. 

Terms such as fluvial cycles, meandering, alluvial fans, geomorphic thresholds, nick points, 

and head cuts are described. A simple river classification scheme is presented. A variety of 

methods to predict a river's response to change are given. Additional information on stream 

stability at highway structures can be found in HEC-20 (Lagasse et al. 2001). Example 

problems to determine river classification and response are given at the end of Chapter 5. 

1.7.5 Chapter 6 - River Stabilization and Bank Protection 

Numerous types of river control and bank stabilization devices have evolved through past 

experience. Concrete, brick, willow and asphalt mattresses, sacked concrete and sand, riprap 

grouted slope protection, sheet piles, timber piles, steel jack and brush jetties, angled and 

sloped rock-filled, earth-filled, and timber dikes, automobile bodies, and concrete tetrahedrons 

have all been used in the practice of training rivers and stabilizing river banks.

The study of river morphology and river response in Chapter 5 makes it clear that both shortand long-term changes can be expected on river systems as a result of natural and human 

influences. Recommended structures and design methods for river control are presented in 

Chapter 6. The integrated and interactive effects of these structures with the river are 

discussed in Chapter 9. Detailed guidelines for selection and design of stream instability and 

bridge scour countermeasures are presented in HEC-23 (Lagasse et al. 2001). Example 

problems related to riprap are solved at the end of Chapter 6 in SI and English units of 

measurement. However, HEC-23 has detailed design guidelines for river stabilization and 

protection countermeasures. 

1.7.6 Chapter 7 - Scour at Bridges

 

Scour at highway structures is the result of the erosive action of flowing water removing bed 

material from around the abutments and piers which support the bridge and bed and bank 

material of the stream the structure crosses. Both scour at highway structures and stream 

migration (instability) can cause a bridge failure.

All material in a streambed will erode with time. However, some material such as granite may 

take hundred's of years to erode, while sandbed streams may erode to the maximum depth of 

scour in hours. Sandstone, shales, and other sedimentary bedrock materials can erode to the 
1.20 

extent that a bridge will be in danger unless the substructures are founded deep enough. 

Cohesive bed and bank material such as clays, silty clays, silts and silty sands or even coarser 

bed material such as glacial tills, which are cemented by chemical action or compression, will 

erode. The erosion of cohesive and other cemented material is slower than sand bed material, 

but their ultimate scour will be as deep if not deeper than the scour depth in a non-cohesive 

sandbed stream. It might take the erosive action of several major floods, but the effects of 

scour can be cumulative in both cohesive and non-cohesive materials. 

 

Scour at bridge crossings is a sediment transport process. Long-term degradation, general 

scour, and local scour at piers and abutments result when more sediment is removed from 

these areas than is transported into them. If there is no transport of bed material into the 

bridge crossing, clear-water scour exists. Transport of appreciable bed material into the 

crossing results in live-bed scour. In this latter case the transport of the bed material limits 

the scour depth. Whereas, with clear-water scour the scour depths are limited by the critical 

velocity or critical shear stress of a dominant size in the bed material at the crossing. 

Chapter 7 presents the basic definitions of scour at bridges, and develops the basic equations 

for determining the scour components. However, a more detailed analysis of scour at highway 

bridges and detailed example problems in SI and English systems of measurement are 

presented in HEC-18 (Richardson and Davis 2001). 

1.7.7 Chapter 8 – Data Need and Data Sources 

The purpose of Chapter 8 is to identify data needed for calculations and analyses for 

highway crossings and encroachments of rivers. The types and amounts of data needed for 

planning and designing river crossings and lateral encroachments depend upon the class of 

the proposed highway. In addition to identifying data needed, Chapter 8 identifies sources of 

data. 

1.7.8 Chapter 9 - Design Considerations for Highway Encroachment 

 and River Crossings 

Chapter 9 presents applications of the fundamentals of hydraulics, hydrology, fluvial 

geomorphology, sediment transport, and river mechanics to the hydraulic and environmental 

design of river crossings and highway encroachments. The principal factors to be considered 

in design are presented, followed by a discussion of the procedures recommended for the 

evaluation, analysis and design of river crossings and encroachments. The design of most 

complex problems in river engineering can be facilitated by a qualitative evaluation combined 

with a quantitative analyses. In most cases, the systematic approach of a qualitative 

assessment of channel response, followed by a quantitative estimate, is necessary for a 

meaningful analysis of complex river response problems. 

Chapter 9 contains a series of conceptual examples (cases) of river environments and their 

response to crossings and encroachments based upon geomorphic principles given in Chapter 

5. These cases indicate the trend of change in river morphology for given initial conditions. 

The hypothetical cases are followed by practical examples (actual case histories) for river 

crossings in the United States. These histories document river response to highway crossings 

and encroachments and illustrate river response qualitatively. 
1.21 

1.7.9 Chapter 10 - Overview Examples of Design for Highways in the River Environment 

The objective of Chapter 10 is to present overview examples of the use of the principles 

presented in this manual in the design of highway encroachments and crossings in the river 

environment. A three-level design procedure is emphasized in these designs. The designs 

use the geomorphic, hydrologic, hydraulic, and river mechanics principles described in this 

manual to design safe and economical crossings that protect, maintain and restore the river 

environment. 

Three examples are given to illustrate the application of the principles, methods, and 

concepts given in this manual. In the examples, the designs are determined by wellestablished numerical procedures; however, they also depend heavily on the judgment of the 

engineer. 

1.7.10 Chapter 11 - References

Chapter 11 contains an alphabetical listing of the references cited throughout the document. 

1.7.11 Appendices

The appendices present a discussion of the metric (SI) system (Appendix A), analysis of 

additional sediment transport relationships (Appendix B), and an Index (Appendix C).

1.8 DUAL SYSTEM OF UNITS

This manual uses dual units (SI metric and English). In Appendix A, the metric (SI) unit of 

measurement is explained. The conversion factors, physical properties of water in SI system 

of units, sediment particle size grade scale, and some common equivalent hydraulic units 

are also given. This edition uses for the unit of length the meter (m) or foot (ft); of mass the 

kilogram (kg) or slug; of weight/force the newton (N) or pound (lb); of pressure the Pascal 

(Pa, N/m2

) or (lb/ft2); and of temperature the degree centigrade (°C) or Fahrenheit (°F). The 

unit of time is the same in SI as in English system (seconds, s). Sediment particle size is 

given in millimeters (mm), but in calculations the decimal equivalent of millimeters in meters 

is used (1 mm = 0.001 m) or for the English system feet (ft). The value of some hydraulic 

engineering terms used in the text in SI units and their equivalent English units are given in 

Table 1.1. 

Table 1.1. Commonly Used Engineering Terms in SI and English Units. 

Term SI Units English Units 

Length 1 m 3.28 ft 

Volume 1 m3 35.31 ft3

Discharge 1 m3/s 35.31 ft3/s 

Acceleration of Gravity 9.81 m/s2 32.2 ft/s2

Unit Weight of Water 9800 N/m3 62.4 lb/ft3

Density of Water 1000 kg/m3 1.94 slugs/ft3

Density of Quartz 2647 kg/m3 5.14 slugs/ft3

Specific Gravity of Quartz 2.65 2.65 

Specific Gravity of Water 1 1 

Temperature °C = 5/9 (°F - 32) °F 
1.22 

(page intentionally left blank) 
2.1 

CHAPTER 2 

OPEN CHANNEL FLOW

2.1 INTRODUCTION

In this chapter, the fundamentals of rigid boundary open channel flow are described. In open 

channel flow, the water surface is not confined; surface configuration, flow pattern and pressure 

distribution within the flow depend on gravity. In rigid boundary open channel flow, no 

deformation of the bed and banks is considered. Mobile boundary hydraulics refers to flow which 

can generate deformation of the boundary through scour and fill. Mobile boundary hydraulics will 

be discussed in later chapters. In this chapter, the discussion is restricted to a one-dimensional 

analysis of rigid boundary open channel flow where velocity and acceleration are large only in 

one direction and are so small as to be negligible in all other directions. 

Open channel flow can be classified as: (1) uniform or nonuniform flow; (2) steady or unsteady 

flow; (3) laminar or turbulent flow; and (4) tranquil or rapid flow. In uniform flow, the depth and 

discharge remain constant with respect to space. Also, the velocity at a given depth does not 

change. In steady flow, no change occurs with respect to time at a given point. In laminar flow, 

the flow field can be characterized by layers of fluid, one layer not mixing with adjacent ones. 

Turbulent flow on the other hand is characterized by random fluid motion. Tranquil flow is 

distinguished from rapid flow by a dimensionless number called the Froude number, Fr. If Fr < 1, 

the flow is subcritical; if Fr > 1, the flow is supercritical, and if Fr = 1, the flow is called critical. 

Open channel flow can be nonuniform, unsteady, turbulent and rapid at the same time. Because 

the classifying characteristics are independent, sixteen different types of flow can occur. These 

terms, uniform or nonuniform, steady or unsteady, laminar or turbulent, rapid or tranquil, and the 

two dimensionless numbers (the Froude number and Reynolds number) are more fully explained 

in the following sections. 

2.1.1 Definitions 

Velocity: The velocity of a fluid particle is the time rate of displacement of the particle from one 

point to another. Velocity is a vector quantity. That is, it has magnitude and direction. The 

mathematical representation of the fluid velocity is a function of the increment of length ds during 

the infinitesimal time dt; thus, 

dt

ds

v = (2.1) 

Streamline: An imaginary line within the flow which is everywhere tangent to the velocity vector is 

called a streamline. 

Acceleration: Acceleration is the time rate of change in magnitude or direction of the velocity 

vector. Mathematically, acceleration a is expressed by the total derivative of the velocity vector or 

dt

dv

a = (2.2) 
2.2 

The vector acceleration, a, has components both tangential and normal to the streamline, the 

tangential component embodying the change in magnitude of the velocity, and the normal 

component reflecting the change in direction 

( )

s

v

2

1

t

v

dt

dv

a

2

s s s

s ∂

∂

+

∂

∂ == (2.3) 

r

v

t

v

dt

dv

a

2

n n

n +

∂

∂ == (2.4) 

The first terms in Equations 2.3 and 2.4 represent the change in velocity, both magnitude and 

direction, with time at a given point. This is called the local acceleration. The second term in 

each equation is the change in velocity, both magnitude and direction, with distance. This is 

called convective acceleration. 

Uniform flow: In uniform flow the convective acceleration terms are zero. 

0

r

v and0

s

v 2 2

s = = ∂

∂ (2.5) 

Nonuniform flow: In nonuniform flow, the convective acceleration terms are not equal to zero. 

0

r

v and0

s

v 2

s ≠ ≠

∂

∂ (2.6) 

Flow around a bend ) 0r/v( 2 ≠ and flow in expansions or contractions 



 



 ≠

∂

∂ 0

s

v are examples of 

nonuniform flow. 

Steady flow: In steady flow, the velocity at a point does not change with time 

0

t

v and0

t

vs n = ∂

∂ = ∂

∂ (2.7) 

Unsteady flow: In unsteady flow, the velocity at a point varies with time 

0

t

v and0

t

vs n ≠

∂

∂

≠

∂

∂ (2.8) 

Examples of unsteady flow are channel flows with waves, flood hydrographs, and surges. 

Laminar flow: In laminar flow, the mixing of the fluid and momentum transfer is by molecular 

activity. 

Turbulent flow: In turbulent flow the mixing of the fluid and momentum transfer is related to 

random velocity fluctuations. The flow is laminar or turbulent depending on the value of the 

Reynolds number (Re = ρVL/μ), which is a dimensionless ratio of the inertial forces to the viscous 

forces. Here ρ and μ are the density and dynamic viscosity of the fluid, V is the fluid velocity, and 
2.3 

L is a characteristic dimension, usually the depth (or the hydraulic radius) in open channel flow. In 

laminar flow, viscous forces are dominant and Re is relatively small. In turbulent flow, Re is 

large; that is, inertial forces are very much greater than viscous forces. Turbulent flows are 

predominant in nature. Laminar flow occurs very infrequently in open channel flow. 

Tranquil flow: In open channel flow, the free surface configuration, in response to changes in 

channel geometry depends on the Froude number = )gL/VFr( , which is the ratio of inertial 

forces to gravitational forces. The Froude number is also the ratio of the flow velocity V to the 

celerity = )gLc( of a small gravity wave in the flow (this concept is detailed in Section 2.5). 

When Fr < 1, the flow is subcritical (or tranquil), and surface waves propagate upstream as well 

as downstream. The boundary condition that controls the tranquil flow depth is always located at 

the downstream end of the subcritical reach. 

Rapid flow: When Fr > 1, the flow is supercritical (or rapid) and surface disturbances can 

propagate only in the downstream direction. The control section of rapid flow depth is always at 

the upstream end of the rapid flow region. When Fr = 1.0, the flow is critical and surface 

disturbances remain stationary in the flow. 

2.2 THREE BASIC EQUATIONS 

The basic equations of flow in open channels are derived from the three conservation laws. 

These are: (1) the conservation of mass; (2) the conservation of linear momentum; and (3) the 

conservation of energy. The conservation of mass is another way of stating that (except for 

mass-energy interchange) matter can neither be created nor destroyed. The principle of 

conservation of linear momentum is based on Newton's second law of motion which states that a 

mass (of fluid) accelerates in the direction of and in proportion to the applied forces on the mass. 

In the analysis of flow problems, much simplification can result if there is no acceleration of the 

flow or if the acceleration is primarily in one direction, the accelerations in other directions being 

negligible. However, a very inaccurate analysis may occur if one assumes accelerations are 

small or zero when in fact they are not. The concepts explained in this chapter assume 

one-dimensional flow and the derivations of the equations utilize a control volume. A control 

volume is an isolated volume in the body of the fluid, through which mass, momentum, and 

energy can be convected. The control volume may be assumed fixed in space or moving with 

the fluid. 

2.2.1 Conservation of Mass

Consider a short reach of river shown in Figure 2.1 as a control volume. The boundaries of the 

control volume are the upstream cross-section, designated section 1, the downstream 

cross-section, designated section 2, the free surface of the water between sections 1 and 2, and 

the interface between the water and the wetted perimeter (banks and bed). 
2.4 

Figure 2.1. A river reach as a control volume. 

The statement of the conservation of mass for this control volume is 

Mass flux 

out of the 

control volume 

- 

Mass flux 

into the 

control volume 

+ 

Time rate of change 

in mass in the 

control volume 

= 0 

Mass can enter or leave the control volume through any or all of the control volume surfaces. 

Rainfall would contribute mass through the surface of the control volume and seepage passes 

through the interface between the water and the banks and bed. In the absence of rainfall, 

evaporation, seepage and other lateral mass fluxes, mass enters the control volume at section 1 

and leaves at section 2. 

At section 2, the mass flux out of the control volume through the differential area dA2 is ρ2 v2 dA2. 

The values of ρ2 and v2 can vary from position to position across the width and throughout the 

depth of flow at section 2. The total mass flux out of the control volume at section 2 is the 

integral of all ρ2 v2 dA2 through the differential areas that make up the cross-section area A2, and 

may be written as: 

Mass flux 

out of the =  2 ρ dAv 222A (2.9) 

control volume 

Similarly 

Mass flux 

into the =  1 ρ dAv 111A (2.10) 

control volume 

The amount of mass inside a differential volume dV inside the control volume is ρ dV and 
2.5 

Mass inside 

the =  ∀ d (2.11) ∀ρ

control volume 

The statement of conservation of mass for the control volume calls for the time rate of change in 

mass. In mathematical notation, 

Time rate of change 

in mass in the = ∀ρ ∂

∂  ∀ d

t (2.12) 

control volume 

For the reach of river, the statement of the conservation of mass becomes 

  =∀ρ ∂

∂

+ρ−ρ ∀ 0d

t

dAv dAv 2 1 111A222A (2.13) 

It is often convenient to work with average conditions at a cross-section, so we define an average 

velocity V such that 

dAv

A

1 V =  A (2.14) 

The symbol v represents the local velocity whereas the velocity V is the average velocity at the 

cross-section. 

Because water is nearly incompressible the density ρ of the fluid is considered constant, ρ1 = ρ2 = 

ρ. When the flow is steady 

0d

t =∀ρ ∂

∂ 

∀

 (2.15) 

and Equation 2.13 reduces to the statement that inflow equals outflow or 

1122 =ρ−ρ 0AVAV

That is, for steady flow of incompressible fluids 

2211 === VAQAVAV (2.16) 

where Q is the volume flow rate or the discharge. 

Equation 2.16 is the familiar form of the conservation of mass equation for steady flow in rivers. It 

is applicable when the fluid density is constant, the flow is steady and there is no significant 

lateral inflow or seepage. 
2.6 

2.2.2 Conservation of Linear Momentum

The curved reach of the river shown in Figure 2.1 is rather complex to analyze in terms of 

Newton's Second Law because of the curvature in the flow. Therefore, as a starting point, the 

differential length of reach dx is isolated as a control volume. 

For this control volume, shown in Figure 2.2, the pressure terms p1 and p2 are directed toward 

the control volume in a direction normal to the Sections 1 and 2. The shear stress τo is exerted 

along the interface between the water and the wetted perimeter and is acting in a direction 

opposite to the axis x. The statement of conservation of linear momentum is: 

Flux of Momentum 

out of the control 

volume 

- 

Flux of momentum 

into the control 

volume 

+

Time rate of change

of momentum in 

the control volume 

=

Sum of the forces 

acting on the fluid 

in the control volume

Figure 2.2. The control volume for conservation of linear momentum. 

The terms in the statement are vectors so we must be concerned with direction as well as 

magnitude. 

Consider the conservation of momentum in the direction of flow (the x-direction in Figure 2.2). At 

the outflow section (section 2), the flux of momentum out of the control volume through the 

differential area dA2 is: 

ρ2v2 dA2 v2 (2.17) 

Here ρ2 v2 dA2 is the mass flux (mass per unit of time) and ρ2 v2 dA2 v2 is the momentum flux 

through the area dA2. 

Flux of momentum 

out of the control = 2222A vdAv 2  ρ (2.18) 

volume 

Similarly, at the inflow section (section 1), 
2.7 

Flux of momentum 

into the = 1111A vdAv 1  ρ (2.19) 

control volume 

The amount of momentum in the control volume is so  ρ dVv ∀

Time rate of 

change of momentum =  ρ ∂

∂ ∀ }dVv{ t (2.20) 

in the control volume 

At the upstream section, the force acting on the differential area dA1 of the control volume is p1

dA1 where p1 is the pressure from the upstream fluid on the differential area. The total force in 

the x-direction at section 1 is  1 dAp 11A . Similarly, at section 2, the total force is  2 dAp 22A . 

There is a fluid shear stress τo acting along the interface between the water and the bed and 

banks. The shear on the control volume is in a direction opposite to the direction of flow and 

results in a force -τoP dx where τo is the average shear stress on the interface area, P is the 

average wetted perimeter and dx is the length of the control volume. The term P dx is the 

interface area. 

The body force component acting in the x-direction is denoted Fb and will be discussed in a 

subsequent section. The statement of conservation of momentum in the x-direction for the 

control volume is: 

  =ρ −   +τ−

∂

∂ 1 +ρ−ρ ∀ oL22A11A b

2

11A2

2

22A dApdApVdv FdxP

t

dAv dAv 2 1 1 2 (2.21) 

Again, as with the conservation of mass equation, it is convenient to use average velocities 

instead of point velocities. We define a momentum coefficient β so that when average velocities 

are used instead of point velocities, the correct momentum flux is considered. 

The momentum coefficient for incompressible fluids is: 

dAv

AV

1 2 =β 2  A (2.22) 

For steady incompressible flow, Equation 2.21 is combined with Equation 2.22 to give 

ρβ − ρβ =   − oL22A11A1 +τ− b

2

2 11

2

22 dApdApAVAV FdxP 1 2 (2.23) 

The pressure force and shear force terms on the right-hand side of Equation 2.23 are usually 

abbreviated as Σ Fx so: 

 =   1 −  2 dApdApF oL22A11Ax +τ− FdxP b (2.24) 
2.8 

The conservation of momentum equation becomes: 

1 x

2

2 11

2 ρβ 22 − ρβ = FAVAV (2.25) 

for steady flow with constant density. With Equation 2.16 the steady flow conservation of linear 

momentum equation takes on the familiar form 

1122 =β−βρ F)VV(Q x (2.26) 

2.2.3 Conservation of Energy

The First Law of Thermodynamics can be written: 

dt

dE WQ =− ••

 (2.27) 

where: 

•

Q = Rate at which heat is added to a fluid system 

•

W = Rate at which a fluid system does work on its surroundings 

 E = Energy of the system 

Then dE/dt is the rate of change of energy in the system. 

The statement of conservation of energy for a control volume is then: 

Flux of energy 

out of the 

control volume 

- 

Flux of energy 

into the control 

volume 

+

Time rate of change 

of energy in the 

control volume 

= ••

Q − W

 

The choice of a control volume is arbitrary. To illustrate the procedure, the control volume is 

reduced to the size of a streamtube connecting dA1 and dA2 as shown in Figure 2.3. The 

streamtube is bounded by streamlines through which no mass or momentum enters. 

Figure 2.3. The streamtube as a control volume. 
2.9 

For steady flow of an incompressible fluid in the streamtube 

Flux of energy 

out of the = 2222 ρ vdAe (2.28) 

control volume 

Similarly, at the inflow section (section 1), 

Flux of energy 

into the = 1111 ρ vdAe (2.29) 

control volume 

and 

Time rate of 

change of energy = 0 (2.30) 

in the control volume 

Here e is the energy per unit mass. Accordingly, the total energy E in a control volume is: 

E (2.31)  ∀ de ∀ρ=

Unless one is concerned with thermal pollution, evaporation losses, or problems concerning the 

formation of ice in rivers, the rate at which heat is added to the control volume can be neglected; 

that is: 

Q~−0 •

 (2.32) 

The work done by the fluid in the control volume on its surroundings can be in the form of 

pressure work Wp, shear work Wτ, or shaft work (mechanical work) Ws. For the streamtube 

shown in Figure 2.3, no shaft work is involved (Ws = 0). 

The rate at which the fluid pressure does work on the control volume surrounding through the 

boundary dA1 in Figure 2.3 is: 

111 − vdAp

and on boundary dA2, the rate of doing pressure work is 

222 p vdA

At the other boundaries of the streamtube, there is no pressure work because there is no fluid 

motion normal to the boundary. Hence, for the streamtube 

111222 p = − vdApvdApW•

 (2.33) 
2.10 

Along the interior boundaries of the streamtube there is a shear stress resulting from the 

condition that the fluid velocity inside the streamtube may not be the same as the velocity of the 

fluid surrounding the streamtube. The rate at which the fluid in the streamtube does shear work 

on the control volume is: 

τ τ= vdxPW•

 (2.34) 

where τ is the average shear stress on the streamtube boundary, P is the average perimeter of 

the streamtube, dx is the length of the streamtube and v is the fluid velocity at the streamtube 

boundary. The product P dx is the surface of the streamtube subjected to shear stresses. 

Then for steady flow in the streamtube, the statement of the conservation of energy in the 

streamtube shown in Figure 2.3 is: 

ρ ρ− = − 22211111112222 τ− dxPvdAvpdAvpdAvedAve (2.35) 

The conservation of mass for steady flow in the streamtube is (according to Equation 2.16) 

dQdAvdAv 1122 == (2.36) 

Now Equation 2.35 reduces to 

dxPvdQ)pe(dQ)pe( 111 222 τ=+ρ−+ρ (2.37) 

The energy per unit mass e is the sum of the internal, kinetic and potential energies or 

 

gz 2

v

ue

2

++= (2.38) 

where: 

 u = Internal energy associated with the fluid temperature 

 v = Velocity of the mass fluid 

 g = Acceleration due to gravity 

 z = Elevation above some arbitrary reference level 

This expression for e is substituted in Equation 2.37 to yield 

dQ

p Pvdx

gz 2

v

u p gz 2

v

u 2

2

2

2

2

1

1

2

1

1 ρ

τ

+

ρ

+++=

ρ

++ (2.39) 

By dividing through by g and defining the head loss h as follows: 

gdQ

Pvdx

g

uu h 12

ρ

τ

+ −  = (2.40) 
2.11 

The energy equation for the streamtube becomes 

hz 

p

g2

v

z p

g2

v

2

2

2

2

1

1

2

1 ++

γ

+=+

γ

+ (2.41) 

If there is no shear stress on the streamtube boundary and if there is no change in internal 

energy (u1 = u2), the energy equation reduces to: 

ttanconsz p

g2

v

z p

g2

v 2

2

2

2

1

1

2

1 =+

γ

+=+

γ

+ (2.42) 

which is the Bernoulli Equation. 

Generally, there is not sufficient information available to do a differential streamtube analysis of a 

reach of river, so appropriate changes must be made in the energy equation. A reach of river 

such as that shown in Figure 2.1 can be pictured as a bundle of streamtubes. We know the 

statement of the conservation of energy for a streamtube. It is Equation 2.41 which can be 

written: 

vdAhdAvz p

g2

v dAvz p

g2

v 2

2

2

2

1

1

2

1

 + l 













+

γ

+= 















+

γ

+ (2.43) 

because v1 dA1 = v2 dA2 = vdA for the streamtube. 

The common form of the energy equation used in open channel flow is derived by integrating 

Equation 2.43 over the cross-section area: 

Qz p

g2

V vdAz p

g2

v 2 2

A

















+

γ

+

α = 



	













+

γ  + (2.44) 

where: 

α = Kinetic energy correction factor defined by the expression 

=α  dAv

V A

1 3

3 A (2.45) 

to allow the use of average velocity V rather than point velocity v. The average pressure over the 

cross-section is p , defined as: 

=  vdAp VA

1

p A (2.46) 
2.12 

The term z is the average elevation of the cross-section defined by the expression: 

=  zvdA

VA

1

z A (2.47) 

and Q is the volume flow rate or the discharge. By definition 

=  A vdAQ (2.48) 

Also 

=  vdah

VA

1 HL A  (2.49) 

In summary, the expression for conservation of energy for steady flow in a reach of river is written 

L 2 2

2

22 1 1

2

11 Hz p

g2

V

z p

g2

V

++

γ

+

α

=+

γ

+

α (2.50) 

The tendency in river work is to neglect the energy correction factor even though its value may 

be as large as 1.5. Usually it is assumed that the pressure is hydrostatic and the average 

elevation head z is at the centroid of the cross-sectional area. However, it should be kept in 

mind that Equations 2.45, 2.46, and 2.47 are the correct definitions of the terms in the energy 

equation. An example problem illustrating the calculation of α and β for a stream is provided in 

Section 2.14 (SI) and 2.15 (English). 

2.3 HYDROSTATICS 

When the only forces acting on the fluid are pressure and fluid weight, the differential equation of 

motion in an arbitrary direction x is 

g

a

z p

x

x =















+

γ∂

∂ (2.51) 

In steady uniform flow (and for zero flow), the acceleration is zero and we obtain the equation of 

hydrostatics 

ttanConsz p =+

γ

 (2.52) 

However, when there is acceleration, the piezometric head varies in the flow field. That is, the 

piezometric head is not constant in the flow. This is illustrated in Figure 2.4. In Figure 2.4a the 

pressure at the bed is hydrostatic and equal to γyo whereas in the curvilinear flow (Figure 2.4b) 

the pressure is larger than γyo because of the acceleration resulting from a change in direction. 
2.13 

Figure 2.4. Pressure distribution in steady uniform and in steady nonuniform flow. 

In general, when fluid acceleration is small (as in gradually varied flow) the pressure distribution 

is considered hydrostatic. However, for rapidly varying flow where the streamlines are 

converging, expanding or have substantial curvature (curvilinear flow), fluid accelerations are not 

small and the pressure distribution is not hydrostatic. 

In Equation 2.52, the constant is equal to zero for gage pressure at the free surface of a liquid. 

For flow with hydrostatic pressure throughout (steady, uniform flow or gradually varied flow), it 

follows that the pressure head p/γ is equal to the vertical distance below the free surface. In 

sloping channels with steady uniform flow, the pressure head p/γ at a depth y below the surface 

is equal to 

θ=

γ

cosy p (2.53) 

Note that y is the depth (perpendicular to the water surface) to the point, as shown in Figure 2.5. 

For most channels, θ is small and cosθ ~ 1. 

Figure 2.5. Pressure distribution in steady uniform flow on steep slopes (after Chow 1959). 
2.14 

2.4 STEADY UNIFORM FLOW

2.4.1 Introduction 

In steady, uniform open channel flow there are no accelerations, streamlines are straight and 

parallel, and the pressure distribution is hydrostatic. The slope of the water surface Sw and the 

bed surface So and the energy gradient Sf are equal. Consider the unit width of channel shown in 

Figure 2.6 as a control volume. According to Equation 2.50, the conservation of energy for this 

control volume is: 

Figure 2.6. Steady uniform flow in a unit width channel. 

L 2 2

2

22 1 1

2

11 Hz p

g2

V

z p

g2

V

++

γ

+

α

=+

γ

+

α (2.54) 

The pressure at any point y below the surface is y cosθ. Then according to Equation 2.46 

=  θγ dyvcosy yV

1

p 1

y

o

11

1

1 (2.55) 

Assuming only small variations in the point velocity v with y we have: 

2

cosy p ~ 1

1

γ θ − (2.56) 

Similarly, 

2

cosy p ~ 2

1

γ θ − (2.57) 
2.15 

Also according to Equation 2.47 

 

2

cosy z ~ z 1

1 1

θ

+− (2.58) 

and 

 

2

cosy z ~ z 2

2 2

θ

+− (2.59) 

With the above expressions for 1 2 p the energy equation for this control volume 21 zand,z,p,

reduces to 

L

2

2

2

2

1 22

1

1

2

11 H

2

cosy

z

2

cosy

g2

V

2

cosy z

2

cosy

g2

V

+

θ

++

θ

+

α = θ

++

θ

+

α (2.60) 

or 

 

2 L2

2

22

1 1

2

11 Hzcosy g2

V

zcosy g2

V ++θ+ α =+θ+ α (2.61) 

For most natural channels θ is small and y cosθ  y. The velocity distribution in the vertical is 

normally a log function for which α1  α2  1. Then the energy equation becomes: 

L22

2

2

11

2

1 Hzy g2

V

zy g2

V

+++=++ (2.62) 

and the slopes of the bed, water surface and energy grade line are respectively 

L

)zz( sinS 21

o ∆

− =θ= (2.63) 

L

)yz()yz( S 11 22

w ∆

+−+ = (2.64) 

and 

L

zy g2

V

zy g2

V

L

H S

22

2

2

11

2

1

L

f ∆

















++− 















++

= ∆ = (2.65) 

Steady uniform flow is an idealized concept for open channel flow and is difficult to obtain even in 

laboratory flumes. For many applications, the flow is steady and the changes in width, depth or 

direction (resulting in nonuniform flow) are so small or occur over such a long distance that the 

flow can be considered uniform. 
2.16 

Variables of interest for steady uniform flow are: (1) the mean velocity V, (2) the discharge Q, 

(3) the velocity distribution v(y) in the vertical, (4) the head loss HL through the reach, and (5) the 

shear stress, both local τ and at the bed τo. These variables are interrelated. 

2.4.2 Shear Stress, Velocity Distribution, and Average Velocity

Shear stress τ is the internal fluid stress which resists deformation. The shear stress exists only 

when fluids are in motion. It is a tangential stress in contrast to pressure, which is a normal 

stress. 

The local shear stress at the interface between the boundary and the fluid can be determined 

quite easily if the boundary is hydraulically smooth; that is, if the roughness at the boundary is 

submerged in a viscous sublayer as shown in Figure 2.7. 

Figure 2.7. Hydraulically smooth boundary. 

Here, the thickness of the laminar sublayer is denoted δ′. In laminar flow, the shear stress at the 

boundary is: 

0yat dy

dv o = 















μ=τ (2.66) 

The velocity gradient is evaluated at the boundary. The dynamic viscosity μ is the proportionality 

constant relating boundary shear and velocity gradient in the viscous sublayer. 

When the boundary is hydraulically rough, the thickness of the laminar sublayer is very small 

compared to the roughness height. The paths of fluid particles in the vicinity of the boundary are 

shown in Figure 2.8. 

The velocity at a point near the boundary fluctuates randomly about a mean value. The random 

fluctuation in velocity characterizes turbulent flows. As shown in Figure 2.9a, the particle has a 

vertical component of velocity vy as well as a horizontal component vx. 
2.17 

Figure 2.8. Hydraulically rough boundary. 

Figure 2.9. Velocities in turbulent flow. 

The two components of velocity in Figure 2.9a can be written as: 

 

x x x += vvv ′ (2.67) 

and 

 

y y y += vvv ′ (2.68) 

where v and x v are the time-averaged mean velocities in the x and y direction and v y x′ and vy′

are the fluctuating components. 

Through theoretical investigation it has been found that turbulence generates shear stress given 

by 

yx ρ−=τ vv ′′ (2.69) 
2.18 

The term yx vv ′′ is the time-average of the product of x v′ and y v′ at a point in the flow. It is 

called the Reynolds shear stress. 

Prandtl (1925) suggested that x v′ and y v′ are related to the velocity gradient dv/dy shown in 

Figure 2.9b. He proposed to characterize the turbulence with a dimension called the "mixing 

length" ,  which is assumed to be the same in both the x and y directions. Accordingly, 

dy

dv

~v x

′  (2.70) 

dy

dv

~v y

′  (2.71) 

and 

 2

2

dy

dv ~ 















ρτ  (2.72) 

If it is assumed that the mixing length can be represented by the product of a constant κ and y 

(i.e.,  = κy), then for steady uniform turbulent flow, 

2

22

dy

dv

y 















=τ ρκ (2.73) 

Using different reasoning von Karman (1930) derived the same equation. Equation 2.73 can be 

rearranged to the form: 

y

/

dy

dv o

κ

ρτ = (2.74) 

where κ is the von Karman universal velocity coefficient. For rigid boundaries κ has the average 

value of 0.4. The term τo is the bed shear stress. The term (τo/ρ)

1/2 has the dimensions of 

velocity and is called the shear velocity, V*. Integration of Equation 2.74 yields 

y

y log 31.2

y

y In 1v

o κ ′ = κ ′ =

ρ

τ

 (2.75) 

Here In is the logarithm to the base e and log is the logarithm to the base 10. The term y′ results 

from evaluation of the constant of integration assuming v = 0 at some distance y′ above the bed. 

The term y′ depends on the flow and has been experimentally determined. The many 

experiments have resulted in characterizing turbulent flow into three general types: 
2.19 

(1) Hydraulically smooth boundary turbulent flow where the velocity distribution, mean velocity 

and resistance to flow are independent of the boundary roughness of the bed but depend 

on fluid kinematic viscosity. Then with 

107 yfindwe 6.11

o

δ′ ′ =

ρ

τ

υ ′=δ (2.76) 

(2) Hydraulically rough boundary turbulent flow where velocity distribution, mean velocity and 

resistance to flow are independent of viscosity and depend entirely on the boundary 

roughness. For this case, y′ = ks/30.2 where ks is the height of the roughness element. 

(3) Transition where the velocity distribution, mean velocity and resistance to flow depend on 

both fluid viscosity and boundary roughness. Then 

2.30

k

y 107

s < ′ <

δ′ (2.77) 

The boundary roughness effects can be merged into one equation by using y′ = ks/(30.2X) where 

X is determined from Figure 2.10. As a result, the velocity distribution v, mean velocity V, and 

resistance to flow equations can be written in the following dimensionless form which is related to 

the above flow types by Figure 2.10. 















 = 













 =

s s k

Xy 2.30In5.2

k

Xy 2.30log75.5 V

v

*

 (2.78) 

and 

s

o

s

o

k

Xy 27.12In5.2

k

Xy 27.12log75.5

g

C

V

V

*

== = (2.79) 

Note that any system of units can be used as long as yo and ks (and V, v and V*) have the same 

dimensions. The symbols of Equations 2.78 and 2.79 denote: 

 X = Coefficient given in Figure 2.10 

 ks = Height of the roughness elements, for sand channels 

 v = Local mean velocity at depth y 

 yo = Depth of flow 

 V = Depth-averaged velocity 

V* = Shear velocity / ρτo which for steady uniform flow is gRS f

τo = Shear stress at the boundary and for steady uniform flow the average is 

 o RSf γ=τ (2.80)

 R = Hydraulic radius, equal to the cross-sectional area A divided by the 

wetted perimeter P 
2.20 

 Figure 2.10. Einstein's multiplication factor X in the logarithmic velocity equations 

 (Einstein 1950). 

 Sf = Slope of the energy gradeline 

δ′ = Thickness of the viscous sublayer 

ν = Kinematic viscosity of fluid 

ν=δ V/6.11 * ′

C = Chezy discharge coefficient in the equation g/

2/1

f

g8 CorRSCV 



 



 = = (2.81) 

 f = Darcy-Weisbach resistance coefficient which is given by the expression 

2

o

V

8f

ρ

τ = (2.82) 

Any consistent set of dimensions can be used (m or ft). 
2.21 

2.4.3 Other Velocity Equations

Because of the difficulties involved in determining the shear stress and hence the velocity 

distribution in turbulent flows, other approaches to determine mean velocities in rivers has been 

prevalent. Two such equations are in common use. They are Manning's equation: 

2/1

f

u 3/2 SR

n

K V = (2.83) 

and Chezy's equation 

2/1

f

2/1 = SCRV (2.84) 

where: 

 V = Average velocity in the waterway cross-section in m/s, ft/s 

 n = Manning's roughness coefficient 

 R = Hydraulic radius in m or ft equal to the cross-sectional area A divided by the 

wetted perimeter P of the waterway m, ft 

 Sf = Friction slope m/m, ft/ft 

 C = Chezy's discharge coefficient known as Chezy's C 

 Ku = 1.0 (SI) 

 Ku = 1.486 (English) 

In these equations, the boundary shear stress is expressed implicitly in the roughness coefficient 

"n" or in the discharge coefficient C. By equating the velocity determined from Manning's 

equation with the velocity determined from Chezy's equation, the relation between the 

coefficients is 

u 6/1 R

n

K C = (2.85) 

If the flow is gradually varied, Manning's and Chezy's equations are used with the average 

friction slope . S ave f The term ave S is determined by averaging over a short time increment at a f

station or over a short length increment 300 m (1,000 ft ) for example at an instant of time, or 

both. 

Over many decades, a catalog of values of Manning's n and Chezy's C has been assembled so 

that an engineer can estimate the appropriate value by knowing the general nature of the 

channel boundaries. An abbreviated list of Manning's roughness coefficients is given in Table 

2.1. Additional values are given by Barnes (1967) and Chow (1959). Manning's n for sandbed 

and gravel-bed channels is discussed in detail in Chapter 3. 
2.22 

Table 2.1. Manning's Roughness Coefficients for Various Boundaries. 

Manning's n 

Rigid Boundary Channels 

Very smooth concrete and planed timber 

Smooth concrete 

Ordinary concrete lining 

Wood 

Vitrified clay 

Shot concrete, untrowelled, and earth channels in best condition 

Straight unlined earth canals in good condition 

Mountain streams with rocky beds 

0.011 

0.012 

0.013 

0.014 

0.015 

0.017 

0.020 

0.040 - 0.050 

Minor Streams (top width at flood stage < 30 m (100 ft) 

Streams on Plain 

1. Clean, straight, full stage, no rifts or deep pools 

2. Same as above, but more stones and weeds 

3. Clean, winding, some pools and shoals 

4. Same as above, but some weeds and stones 

5. Same as above, lower stages, more ineffective slopes and sections 

6. Same as 4, but more stones 

7. Sluggish reaches, weedy, deep pools 

8. Very weedy reaches, deep pools, or floodways with heavy stand of timber 

and underbush 

Mountain Streams, no vegetation in channel, banks usually steep, trees and 

brush along banks submerged at high stages 

1. Bottom: gravels, cobbles, and few boulders 

2. Bottom: cobbles with large boulders 

0.025-0.033 

0.030-0.040 

0.033-0.045 

0.035-0.050 

0.040-0.055 

0.045-0.060 

0.050-0.080 

0.075-0.150 

0.030-0.050 

0.040.0.070 

Floodplains 

Pasture, No Brush 

1. Short Grass 

2. High Grass 

Cultivated Areas 

1. No Crop 

2. Mature Row Crops 

3. Mature Field Crops 

0.025-0.035 

0.030-0.050 

0.020-0.040 

0.025-0.045 

0.030-0.050 
2.23 

Table 2.1. Manning's Roughness Coefficients for Various Boundaries. 

Manning's n 

Floodplains (continued) 

Brush 

1. Scattered brush, heavy weeds 

2. Light brush and trees in winter 

3. Light brush and trees in summer 

4. Medium to dense brush in winter 

5. Medium to dense brush in summer 

Trees 

1. Dense willows, summer, straight 

2. Cleared land with tree stumps, no sprouts 

3. Same as above, but with heavy growth of sprouts 

4. Heavy stand of timber, a few down trees, little undergrowth, flood stage 

below branches 

5. Same as above, but with flood stage reaching branches 

0.035-0.070 

0.035-0.060 

0.040-0.080 

0.045-0.110 

0.070-0.160 

0.110-0.200 

0.030-0.050 

0.050-0.080 

0.080-0.120 

0.100-0.160 

Major Streams (Top width at flood stage > 30 m (100 ft)

The n value is less than that for minor streams of similar description, 

because banks offer less effective resistance. 

Regular section with no boulders or brush 

Irregular and rough section 

0.025-0.060 

0.035-0.100 

Alluvial Sandbed Channels (no vegetation)1 

Tranquil flow, Fr < 1 

1. Plane bed 

2. Ripples 

3. Dunes 

4. Washed out dunes or transition 

5. Plane bed 

Rapid Flow, Fr > 1 

1. Standing waves 

2. Antidunes 

0.014 - 0.020 

0.018 - 0.030 

0.020 - 0.040 

0.014 - 0.025 

0.010 - 0.013 

0.010 - 0.015 

0.012 - 0.020 

1

Data is limited to sand channels with D50 < 1.0 mm, details to be discussed in Chapter 3. 
2.24 

The general approach for estimating n values consists of the selection of a base roughness value 

for a straight, uniform, smooth channel in the materials involved, then additive values are 

considered for the channel under consideration: 

++++= m)nnnnn(n 54321o (2.86) 

where: 

 no = Base value for straight uniform channels 

 n1 = Additive value due to cross-section irregularity 

 n2 = Additive value due to variations of the channel 

 n3 = Additive value due to obstructions 

 n4 = Additive value due to vegetation 

 m5 = Mulitiplication factor due to sinuosity 

Detailed values of the coefficients are found in Cowan (1956), Chow (1959), Benson and 

Dalrymple (1967) and Aldridge and Garrett (1973). Typical values are given in Table 2.2. 

Arcement and Schneider (1984) proposed a guide for selecting Manning's roughness coefficients 

for floodplains. For steeper streams, the reader is also referred to the work of Jarrett (1984, 

1985). 

The roughness characteristics on the floodplain are complicated by the presence of vegetation, 

natural and artificial irregularities, buildings, undefined direction of flow, varying slopes and other 

complexities. Resistance factors reflecting these effects must be selected largely on the basis of 

past experience with similar conditions. In general, resistance to flow is large on the floodplains. 

In some instances, conditions are further complicated by deposition of sediment and 

development of dunes and bars which affect resistance to flow and direction of flow. 

The presence of ice affects channel roughness and resistance to flow in various ways. When an 

ice cover occurs, the open channel is more nearly comparable to a closed conduit. There is an 

added shear stress developed between the flowing water and the ice cover. This surface shear 

is much larger than the normal shear stresses developed at the air-water interface. The ice-water 

interface is not always smooth. In many instances, the underside of the ice is deformed so that it 

resembles ripples or dunes observed on the bed of sandbed channels. This may cause overall 

resistance to flow in the channel to be further increased. 

With total or partial ice cover, the drag of ice retards flow, decreasing the average velocity and 

increasing the depth. Another serious effect is its influence on bank stability, in and near water 

structures such as docks, loading ramps, and ships. For example, the ice layer may freeze into 

bank stabilization materials, and when the ice breaks up, large quantities of rock and other 

material embedded in the ice may be floated downstream and subsequently thawed loose and 

dumped randomly leaving banks raw and unprotected. 

2.4.4 Average Boundary Shear Stress

The average shear stress at the boundary τo for steady uniform flow is determined by applying 

the conservation of mass and momentum principles to the control volume shown in Figure 2.11. 

The conservation of mass Equation 2.16 is then: 
2.25 

Table 2.2. Adjustment Factors for the Determination of n Values. 

Conditions n Value Remarks 

Cross-Section 

Irregularity 

n1 Smooth 

 Minor 

 Moderate 

 Severe 

0 

0.001-0.005 

0.006-0.010 

0.011-0.020 

Smoothest Channel 

Slightly Eroded Side Slopes 

Moderately Rough Bed and Banks 

Badly Sloughed & Scalloped Banks 

Variations 

In Channel 

Section 

n2 Gradual 

 Alternating 

 Occasionally 

 Alternating 

 Frequently 

0 

0.001-0.005 

0.010-0.015 

Gradual Changes 

Occasional Shifts From 

Large to Small Sections 

Frequent Changes in 

Cross-Sectional Shape 

Obstructions 

n3 Negligible 

 Minor 

 Moderate 

 Severe 

0-0004 

0.005-0.015 

0.020-0.030 

0.040-0.060 

Obstructions < 5% of 

Cross-Section Area 

Obstructions < 15% of 

Cross-Section Area 

Obstructions 15-50% of 

Cross-Section Area 

Obstructions > 50% of 

Cross-Section Area 

Vegetation 

n4 Small 

 Medium 

 Large 

 Very Large 

0.002-0.010 

0.010-0.025 

0.025-0.050 

0.050-0.100 

Flow Depth > 2 x Vegetation Height 

Flow Depth > Vegetation Height 

Flow Depth < Vegetation Height 

Flow Depth < 0.5 Vegetation Height 

Sinuosity 

m5 Minor 

 Moderate 

 Severe 

1.00 

1.15 

1.30 

Sinuosity < 1.2 

1.2 < Sinuosity < 1.5 

Sinuosity > 1.5 
2.26 

Figure 2.11. Control volume for steady uniform flow. 

0WVyWVy 1o2o =ρ−ρ (2.87) 

or 

= VV 21 (2.88) 

The conservation of momentum in the downstream direction is described from Equation 2.25 with 

A1 = A2 = Wyo and V1 = V2. The pressure forces acting on the control boundary are approximated 

by: 

2

Wy FF

2

o

21

γ == (2.89) 

The downstream component of the body force γAL (equal to the weight of fluid in the control 

volume) in the X direction is: 

b sinALF θγ= (2.90) 

where θ is the slope angle of the channel bed. The average boundary shear stress is τo acting 

on the wetted perimeter P. The shear force Fs in the x-direction is: 

τ= os PLF (2.91) 

With the above expressions for the components, the statement of conservation of linear 

momentum becomes: 

PL

2

Wy

2

Wy sinALVWyVWy o

2

o

2

2 o

o

2

o τ− γ − γ ρβ − ρβ +θγ= (2.92) 

which reduces to 

sinθγ=τ P

A o (2.93) 
2.27 

The term A/P is the hydraulic radius R. If the channel slope angle is small, 

So sin ~−θ (2.94) 

and for steady uniform flow the average shear stress on the boundary is 

o RSo γ=τ (2.95) 

If the flow is gradually varied nonuniform flow, the average boundary shear stress is 

o RSf γ=τ (2.96) 

where Sf is the slope of the energy grade line. 

2.4.5 Relation Between Shear Stress and Velocity

Measuring the average bottom shear stress directly in the field is tenuous. However, the average 

bottom shear stress can be computed from the expression 

o RSf γ=τ (2.97) 

For steady uniform flow, the local and average shear stress on the bed can be estimated by 

employing the velocity profile equations in Section 2.4.2. If the local velocity v1 at depth y1 is known 

and X = 1 for hydraulically rough boundary then, from Equation 2.78 the local shear stress can be 

determined. The equation is: 

2

s

1

2

1

o

k

y 2.30log75.5

v























	











ρ =τ (2.98) 

 

This equation and the ones given below are valid for fully turbulent uniform flow over a hydraulically 

rough boundary in wide channels with a plane bed. Alternatively, if two point velocities in a vertical 

profile are known (preferably in the lower 15 percent of the depth) the local shear stress on the bed 

can be determined from the following equation: 

2

2

1

2

21

o

y

y log75.5

)vv(























	











−ρ =τ (2.99) 

If the depth of flow yo, the grain roughness ks and the average velocity in the vertical V are known, 

then the average shear stress can be determined from Equation 2.79. The equation is: 
2.28 

2

s

o

2

o

k

y 27.12log75.5

V























	











ρ =τ (2.100) 

2.4.6 Energy and Momentum Coefficients for Rivers

In prismatic or constructed channels it is common to assume that the energy coefficient α and the 

momentum coefficient β are unity. In river channels, this is usually not the case. From Equations 

2.45 and 2.22: 

=α  dAv

AV

1 3

3 A (2.45) 

and 

dAv

AV

1 2 =β 2  A (2.22) 

The velocity distribution in wide channels for turbulent flow over a rough boundary is given by 

Equation 2.78 with X = 1.0 

)k/y2.30(In5.2 V

v s

*

= (2.101) 

The average velocity in the vertical is 

dy)y/y(In

yy

V5.2 dyv ~ y

1 V o o y

y

o

y

0

o

* ′ − ′ =  −  ′ (2.102) 

Here, the upper limit of integration is yo, the depth of flow and the lower limit is 

2.30

k

y s ′ = (2.103) 

the value of y for which Equation 2.78 gives a zero velocity. The integration of Equation 2.102 

yields 





















	











 −



	













− ′′ = 1y

y In

yy

y 5.2

V

V o

o

o

*

 (2.104) 

For a vertical section of unit width, the momentum coefficient β′ is 
2.29 

 − ′ ′ =β dyv )yy(V

1 y 2

'y

o

2

o (2.105) 

If we substitute Equations 2.78 and 2.104 into Equation 2.107 and integrate, the result is the 

expression 















 ′ −+



	













′ − 



	













′ 



	













− ′





	













′ =β

o

o

2

o

o

o

2

s

o y

y2 2

y

y In2

y

y In

yy

y

k

y 11.11In

1 (2.106) 

Similarly, the energy coefficient for a vertical section unit width is 

dyv )yy(V

1 y 3

y

o

3

o

 ′

− ′ ′ =α (2.107) 

or 















 ′ +−



	













′ + 



	













′ − 



	













′ 



	













− ′





	













′ =α

o

o

2

o

3

o

o

o

2

s

o y

y6 6

y

y In6

y

y In3

y

y In

yy

y

k

y 11.11In

1 (2.108) 

These equations (Equations 2.106 and 2.108 are rather complex, so a graph of α′ and β′ vs yo/ks

has been prepared. The relations are shown in Figure 2.12. 

For the entire river cross-section (shown in Figure 2.13) Equation 2.45 can be written 

[ ]  





 





=α dwdyv

A

A

Q

1 y 3

0

W

3 0 o (2.109) 

where W is the top width of the section, w is the lateral location of any vertical section, yo is the 

depth of flow at location w, and v is the local velocity at the position y, w. The total discharge is Q 

and the total cross-sectional area is A. 
2.30 

Figure 2.12. Energy and momentum coefficients for a unit width of river. 

Figure 2.13. The river cross section. 
2.31 

The second integral term in brackets in Equation 2.109 can be written 

 = ′ −α ′ ′ )yy(Vdyv o

y 3 3

y

o (2.110) 

Here α′ is the energy coefficient for the vertical section dw wide and yo deep, V is the 

depth-averaged velocity in this vertical section and y′ = ks/30.2. 

Now, Equation 2.109 can be written 

dw)yy(V Q

A o

W 3

3 0

2

=α ′ −α ′  (2.111) 

Except for cases of low flow in gravel bed rivers, the term y′ is very small compared to yo so 

 

dwyV Q

A o

W 3

3 0

2

=α α′  (2.112) 

The discharge at a river cross-section is determined in the field by measuring the local depth and 

two local velocities at each of approximately 20 vertical sections. In accordance with this general 

stream gaging procedure, Equation 2.112 could be written as 

ioi

3

3 iii

2

wyv Q

A

=α  ′ ∆α

or 

i

2

3 iii

2

Qv Q

A Σ=α ′ ∆α (2.113) 

Here, the subscript i refers to the i-th vertical section, and ∆Qi

 is the river discharge associated 

with the i-th vertical or 

∆=∆ wyVQ ioiii

In a similar manner, the expression for β is 

2 QV iiii Q

A =β  ′ ∆β (2.114) 

Now, with Equations 2.113 and 2.114, and Figure 2.12 we are in a position to compute α and β

for any river cross-section given the discharge measurement notes. A calculation example is 

presented in Section 2.14 (SI) and 2.15 (English). It is important to recognize that for river flows 

over floodplains, the correction factors α and β can be significantly larger than α′ and β′. 
2.32 

2.5 UNSTEADY FLOW

Unsteady flows of interest to the designer of waterway crossings and encroachments are: (1) 

waves resulting from disturbances of the water surface by wind and boats; (2) waves resulting 

from the surface instability that exists for flows with Froude numbers close to 1.0; (3) waves 

resulting from flow disturbance due to change in direction of flow with Froude numbers greater 

than about 2.0; (4) surges or bores resulting from sudden increase or decrease in the flow by 

opening or closing of gates or the movement of tides on coastal streams; (5) standing waves and 

antidunes that occur in alluvial channel flow; and (6) flood waves resulting from the progressive 

movement downstream of stream runoff or gradual release from reservoirs. 

Waves are an important consideration in bridge hydraulics when designing slope protection of 

embankments and dikes, and channel improvements. In the following paragraphs, only the basic 

one-dimensional analysis of waves and surges is presented. Other aspects of waves are 

presented in other sections. 

2.5.1 Gravity Waves

The general equation for the celerity c (velocity of the wave relative to the velocity of flow) of a 

small amplitude gravity wave (ao << λ) is 

2/1

o y2 htan

2

g c













λ

π

π

λ = (2.115) 

where the terms are defined in Figure 2.14. 

Figure 2.14. Definition sketch for small amplitude waves. 

For deep water waves (short waves) defined as 

2

y o 1

>

λ (2.116) 
2.33 

the celerity relationship (Equation 2.115) reduces to 

2/1

2

g c 











π

λ = (2.117) 

For shallow water waves (long waves) 

20

y o 1

<

λ (2.118) 

Then Equation 2.115 reduces to 

= gyc o (2.119) 

The time of travel of one water crest to another at a given point is called the period T and can be 

defined from the celerity and wave length 

λ= T/c (2.120) 

In Equation 2.117, the celerity is independent of depth and depends on gravity g and wave 

length λ. This is the celerity of ocean waves. In Equation 2.119, the celerity is a function of 

gravity and depth which describes small amplitude waves in open channels. These two 

equations apply only to small amplitude waves; that is ao/λ << 1. 

The celerity of finite amplitude shallow water waves has been determined both analytically using 

Bernoulli's equation and experimentally, and is given by the expression 

2/1

o

ooo

2

o o gy y)ay(

)a2y( c













+

+ = (2.121) 

When 2ao is small in comparison to yo

2/1

o

o

o gy

y

a2 1c













+= (2.122) 

Generally as 2ao/yo approaches unity the crest develops a sharp peak and breaks. 

In the above equations, c is measured relative to the fluid. If the wave is moving opposite to the 

flow then, when c > V, the waves move upstream; when c = V, the wave is stationary; and when c 

< V, the wave moves downstream. When V = c for small amplitude flow, 

== gycV o (2.123) 

The ratio of the flow velocity to the celerity of a shallow water wave of small amplitude is defined 

by the Froude number: 
2.34 

gy o

V Fr = (2.124) 

When Fr < 1 (subcritical or tranquil), a small amplitude wave moves upstream. When Fr > 1 

(supercritical or rapid flow), a small amplitude wave moves downstream and when Fr = 1 (critical 

flow), a small amplitude wave is stationary. The fact that waves or surges cannot move upstream 

when the Froude number is equal to or greater than 1.0 is important to remember when 

determining when the stage-discharge relation at a cross-section can be affected by downstream 

conditions. The Froude number is not only the ratio of the flow velocity to the celerity of a shallow 

water wave, but is also the ratio of the inertia forces to the gravity forces. 

2.5.2 Surges

A surge is a rapid increase in the depth of flow. A surge may result from sudden release of water 

from a dam, or from an incoming tide. If the ratio of wave height 2ao to the depth yo is less than 

unity, the surge has an undulating wave form. If 2ao/yo is greater than one, the first wave breaks 

and produces a discontinuous surface. The breaking wave dissipates energy and the previous 

equations for wave celerity are invalid. However, by applying the momentum and continuity 

equations for a control volume encompassing the surge shown in Figure 2.15, the equation for 

the celerity of a surge can be derived for flat slopes as: 

2/1

1

2

1

2

1 1

y

y

y

y

2

1

gyc 



















	



























 = + (2.125) 

Figure 2.15. Sketch of positive and negative surges. 

Equation 2.125 gives the velocity of a surge as it moves upstream as the result of a sudden total 

or partial closure of gates, or of an incoming tide, or of a surge that moves downstream as the 

result of a sudden opening of a gate. The lifting of a gate in a channel sketched in Figure 2.15b 

not only causes a positive surge to move downstream, it also causes a negative surge to move 

upstream. Equation 2.125 is approximately correct for the celerity of the negative surge if the 

height of the surge is small compared to the depth. As it moves upstream a negative surge 

quickly flattens out. 
2.35 

2.5.3 Hydraulic Jump

When the flow velocity V1 is rapid or supercritical the surge dissipates energy through a moving 

hydraulic jump. When V1 equals the celerity c of the surge the jump is stationary and Equation 

2.125 is the equation for a hydraulic jump on a flat slope. Solutions for a hydraulic jump on a 

sloping channel are given in HEC-14. Equation 2.125 can be rearranged to the form: 

2/1

1

2

1

2

1

1

1 1

y

y

y

y

2

1 Fr

gy

V





















	











 == + (2.126) 

or 

{ } ( ) 1Fr81

2

1

y

y 2/1 2

1

1

2 += − (2.127) 

The corresponding energy loss in a hydraulic jump is the difference between the two specific 

energies. It can be shown that this head loss is: 

21

3

12

L yy4

)yy( h − = (2.128) 

Equation 2.128 has been experimentally verified along with the dependence of the jump length Lj

and energy dissipation (head loss hL) on the Froude number of the approaching flow. The results 

of these experiments are given in Figure 2.16. 

When the Froude number for rapid flow is less than 1.7, an undulating jump with large surface 

waves is produced. The waves are propagated for a considerable distance downstream. In 

addition, when the Froude number of the approaching flow is less than three, the energy 

dissipation of the jump is not large and jets of high velocity flow can exist for some distance 

downstream of the jump. These waves and jets can cause erosion a considerable distance 

downstream of the jump. For larger values of the Froude number, the rate of energy dissipation 

in the jump is very large and Figure 2.16 is recommended. The U.S. Bureau of Reclamation 

(Chow 1959) classifies the hydraulic jump on a flat slope into various types as illustrated in Figure 

2.17. For additional information on hydraulic jumps, see HEC-14, Chow (1959) and Rouse 

(1950). 

2.5.4 Roll Waves

Under certain conditions on steep slopes, surges of an intermittent nature may occur which are 

called roll waves or slug flow (Figure 2.18). Such flow is not at all uncommon with harmless thin 

sheets of flow on sloping sidewalks, for example. When these roll waves occur in large open 

channels, however, they may cause considerable damage, or force the operation of the channel 

at inefficient discharges in order to prevent damage. 

Roll waves can be superposed over the normal flow in an open channel. They travel at velocities 

greater than the normal flow and grow in size as they progress downstream. 
2.36 

Figure 2.16. Hydraulic jump characteristics as a function of the upstream Froude number. 

Figure 2.17. Various types of hydraulic jump (Chow 1959). 
2.37 

Figure 2.18. Roll waves or slug flow. 

There is no simple criterion for determining the size of roll waves, since their size depends upon 

the magnitude of the discharge, the type of flow (laminar or turbulent), the roughness and slope 

of the channel, the length of the channel, and the nature and frequency of the initial disturbances 

which cause the waves to form. However, a necessary condition required to generate instability 

of the free surface and induce the formation of roll waves in turbulent flows when Chezy's 

equation is applicable is: 

2

gy

V Fr

o

>= (2.129) 

which can be expressed in alternate form for a wide channel as 

2 C

g ≥ 4S (2.130) 

for turbulent flow with a rough boundary in which yo is the normal depth, S is the slope of the 

channel, and C is the Chezy discharge coefficient. 

When the flow in a wide channel is turbulent with a smooth boundary, roll waves can form if 

≥ 5.1Fr (2.131) 

2 C

g ≥ 25.2S (2.132) 

and when the flow in a wide channel is laminar, roll waves can form if 

≥ 5.0Fr (2.133) 

These conditions indicate that, for turbulent flow in a wide channel with a rough boundary, roll 

waves can occur when the flow velocity is greater than twice the celerity of a wave that is, the 

Froude number is greater than 2, or when the slope is four times as great as the slope required 

for critical depth. They can also form for turbulent flow in a wide channel with a smooth boundary 

if the velocity of flow is greater than 1.5 times the celerity of a wave, or the slope is 2.25 times the 

slope required for critical depth. By way of contrast, roll waves can form in laminar flow in a wide 

channel if the velocity is half the celerity of a gravity wave; in other words, the flow may never 

pass through critical flow (Fr = 1.0). 
2.38 

2.6 STEADY RAPIDLY VARYING FLOW

2.6.1 Flow Through Transitions 

Steady flow through relatively short transitions where the flow is uniform before and after the 

transition can be analyzed using the Bernoulli equation. Energy loss due to friction may be 

neglected, at least as a first approximation. Refinement of the analysis can be made in a second 

step by including friction loss (see HEC-14, Chapter 4). For example, the water surface elevation 

through a transition is determined using the Bernoulli equation and then modified by determining 

the friction loss effects on velocity and depth in short reaches through the transition. Energy 

losses resulting from flow separation cannot be neglected, and transitions where separation may 

occur need special treatment which may include model studies. Contracting flows (converging 

streamlines) are less susceptible to separation than for expanding flows. Also, any time a 

transition changes velocity and depth such that the Froude number approaches unity, problems 

such as waves, blockage, or choking of the flow may occur. If the approaching flow is 

supercritical, a hydraulic jump may result. Unsubmerged flow through bridges or culverts can be 

considered as flow through transitions. 

Transitions are used to contract or expand a channel width (Figure 2.19a); to increase or 

decrease bottom elevation (Figure 2.19b); or to change both the width and bottom elevation. The 

first step in the analysis is to use the Bernoulli equation (neglecting any head loss resulting from 

friction or separation) to determine the depth and velocity changes of the flow through the 

transition. Further refinement depends on importance of freeboard, whether flow is supercritical 

or approaching critical conditions. 

Figure 2.19. Transitions in open channel flow. 

The Bernoulli equation for flow in Figure 2.19b is: 

zy g2

V

y g2

V 2

2

2

1

2

1 ∆++=+ (2.134) 

or 
2.39 

21 ∆+= zHH (2.135) 

where 

y g2

V H

2

+= (2.136) 

The term H is called the specific head, and is the height of the total head above the channel 

bed. 

2.6.2 Specific Energy Diagram

For simplicity, the following specific energy (often referred to as specific head) analysis is done 

on a unit width of channel so that Equation 2.136 becomes: 

y gy2

q H 2

2

+= (2.137) 

For a given q, Equation 2.137 can be solved for various values of H and y. When y is plotted as 

a function of H, Figure 2.20 is obtained (Rouse 1946). There are two possible depths called 

alternate depths for any H larger than a specific minimum. Thus, for specific energy larger than 

the minimum, the flow may have a large depth with small velocity or small depth with large 

velocity. Flow cannot occur with specific energy less than the minimum. The single depth of flow 

at the minimum specific energy is called the critical depth yc and the corresponding velocity, the 

critical velocity Vc = q/yc. To determine yc the derivative of H with respect to y is set equal to 0. 

Figure 2.20. Specific energy diagram. 

0

gy

q 1

dy

dH 3

2

=−= (2.138) 
2.40 

and 

( ) 2/1 3 = gyq c (2.139) 

or 

g2

V2

g

q y

2

c

3/1 2

c = 













 = (see Equation 2.147) (2.140) 

Note that 

cgyV2c = (2.141) 

or 

1

gy

V

c

c = (2.142) 

but 

Fr

gy

V = (2.143) 

also 

c c

2

c

min y 2

3

y g2

V H =+= (see Equation 2.147) (2.144) 

Thus, flow at minimum specific energy has a Froude number equal to one. Flows with velocities 

larger than critical (Fr > 1) are called rapid or supercritical and flow with velocities smaller than 

critical (Fr < 1) are called tranquil or subcritical. These flow conditions are illustrated in Figure 

2.21, where a rise in the bed causes a decrease in depth when the flow is tranquil and an 

increase in depth when the flow is rapid. Furthermore, there is a maximum rise in the bed for a 

given H1 where the given rate of flow is physically possible. If the rise in the bed is increased 

beyond ∆zmax for Hmin then the approaching flow depth y1 would have to increase (increasing H) 

or the flow would have to be decreased. Thus, for a given flow in a channel, a rise in the bed 

level can occur up to a ∆zmax without causing backwater. 

2.6.3 Specific Discharge Diagram

For a constant H, Equation 2.137 can be solved for y as a function of q. By plotting y as a 

function of q, Figure 2.22 is obtained and for any discharge smaller than a specific maximum, two 

depths of flow are possible (Rouse 1946). 
2.41 

Figure 2.21. Changes in water surface resulting from an increase in bed elevation. 

Figure 2.22. Specific discharge diagram. 

To determine the value of y for qmax Equation 2.137 is rearranged to obtain: 

−= )yh(g2yq (2.145) 

The differential of q with respect to y is set equal to zero. 

2/1)yH(

)y3H2(

2

q 0

dy

dq

−

− == (2.146) 
2.42 

from which 

g2

V2 H

3

2

y

2

c

c == (2.147) 

or 

c = gyV c (2.148) 

Thus for maximum discharge at constant H, the Froude number is 1.0, and the flow is critical. 

From this: 

3/1 2

max

2

c

c g

q

g2

V2 H

3

2

y 













 === (see Equation 2.140) (2.149) 

For critical conditions, the Froude number is 1.0, the discharge is a maximum for a given specific 

head and the specific head is a minimum for a given discharge. 

Flow conditions for constant specific head for a width contraction are illustrated in Figure 2.23 

assuming a fixed bed and no geometrical effects such as eccentricity, skew, piers, and 

expansion. The contraction causes a decrease in flow depth when the flow is tranquil and an 

increase when the flow is rapid. The maximum possible contraction without causing backwater 

effects occurs when the Froude number is one, the discharge per foot of width q is a maximum, 

and yc is 2H/3. A further decrease in width will cause backwater. That is, an increase in depth 

upstream will occur to produce a larger specific energy and increase yc in order to get the flow 

through the decreased width. 

Figure 2.23. Change in water surface elevation resulting from a change in width. 
2.43 

The flow in Figure 2.23 can go from point A to C and then either back to D or down to E 

depending on the downstream boundary conditions. An increase in slope of the bed downstream 

from C and no separation would allow the flow to follow the line A to C to E. Similarly the flow 

can go from B to C and back to E or up to D depending on boundary conditions. Figure 2.23 is 

drawn with the side boundary forming a smooth streamline. If the contraction were due to bridge 

abutments, the upstream flow would follow a natural streamline to a vena contracta, but then 

downstream, the flow would probably separate. Tranquil approach flow could follow line A-C but 

the downstream flow probably would not follow either line C-D or C-E but would have an 

undulating hydraulic jump. There would be interaction of the flow in the separation zone and 

considerable energy would be lost. If the slope downstream of the abutments was the same as 

upstream, then the flow could not be sustained with this amount of energy loss. Backwater would 

occur, increasing the depth in the constriction and upstream, until the flow could go through the 

constriction and establish uniform flow downstream. 

2.6.4 Transitions With Super Critical Flows 

Contractions and expansions in rapid flows produce cross wave patterns similar to those 

observed in curved channels (Ippen 1950 and Chow 1959). The cross waves are symmetrical 

with respect to the centerline of the channel. Ippen and Dawson (1951) have shown that in order 

to minimize the disturbance downstream of a contraction, the length of the contraction should be: 

θ

− = tan2WW L 21 (2.150) 

where W is the channel width and the subscripts 1 and 2 refer to sections upstream and 

downstream from the contraction. The contraction angle is θ and should not exceed 12°. This 

requires a long transition and should not be attempted unless the structure is of primary 

importance. A model study should be used to determine transition geometry where a hydraulic 

jump is not desired. If a hydraulic jump is acceptable, the inlet structure can be designed using 

the procedure in HEC-14, Chapter 4B. 

For an expansion, Rouse et al. (1951) found experimentally that the most satisfactory boundary 

form is given by: 

2

1

FrW

x

2

1

W

w

2/3

1 11

+ 













 = (2.151) 

where x is the longitudinal distance measured from the start of the expansion or outlet section 

and w is the lateral coordinate measured from the channel centerline. A boundary developed 

from this equation diverges indefinitely. Therefore, for practical purposes, the divergent walls are 

followed by a transition to parallel lines. A satisfactory straight transition can be created by flaring 

the walls so that tan θ = 1/3 Fr. This criteria recommended by Blaisdel (1949) avoids creating an 

abrupt expansion. 
2.44 

2.6.5 Flow Over Drop Structures

Subcritical flow over vertical wall drop structures at flat slopes is illustrated in Figure 2.24. As 

illustrated, the flow impinging on an apron will continue downstream but at supercritical depth and 

velocity. Depending on downstream conditions (bed elevation and slope and down stream flow 

depth) the flow will continue to be supercritical or go through a hydraulic jump. If the flow 

impinges on an erodible bed it will scour a deep hole. Methods to calculate the depth of this 

scour hole are given in Chapter 3. Flow geometry and conditions of interest are: 

• discharge for unit width, q 

• drop height, h 

• distance from drop wall to point of impingement, Ld

• depth and velocity at impingement, y1, V1 

• pool depth under nape, yp

• sequent depth if a jump forms, y2

Figure 2.24. Flow characteristics over a drop structure (Chow 1959). 

Chow (1959) gives equations to determine the geometric terms as functions of a drop number D. 

 In English units of measurement these are: 

h

q

3

2

g

D = (2.152) 

D 27.0 D30.4

h

L = (2.153) 

p 22.0 D00.1

h

y = (2.154) 

 

1 425.0 D54.0

h

y = (2.155) 

2 27.0 D66.1

h

y = (2.156) 
2.45 

Note: 

1. The equations are for a well-aerated nappe. If the nape is poorly or not aerated, the nappe 

will move upstream until in contact with the vertical wall. 

2. If the tailwater depth, y2, is less than yc, the hydraulic jump will move downstream. 

3. If the tailwater depth, y2, is larger than yc, the jump will move upstream and the jump will be 

submerged. 

4. For upstream subcritical flow the brink serves as the control and the subcritical approach flow 

goes through critical 3 to 4 yc upstream of the brink (Chow, 1959, p. 44). 

5. For upstream supercritical flow the control for the flow is upstream and the profile for the 

nappe is a function of the approach Froude number. Ippen (1950, p. 533) gives a composite 

plot of nappe profiles for different values of the Froude number (Figure 2.25). 

Figure 2.25. Nappe profiles for supercritical flow (Ippen 1950). 

2.7 FLOW IN BENDS 

2.7.1 Types of Bends

Two principal types of bends are deepened or entrenched bends and meandering surface bends. 

The first type includes those in which the river bends follow the curves of the valley so that each 

river bend includes a promontory of the parent plateau. The second type includes bends which 

are formed only by the river on a flat, alluvium covered valley floor, and where the slopes of the 

valley are not involved in the formation of such bends. This division of bends is correct and 

sufficiently definitive with respect to external forms of the relief and the processes of formation 

and development of bends. It is, however, incomplete from the standpoint of the work of the river 

and of the physical nature of this phenomenon. Both of these types of bends can be put into one 

category--the category of freely meandering channels, i.e., meandering determined only by the 

interaction of the stream and the bed material. Such meandering when not disturbed by the 

influence of external factors, proceeds at an approximately equal rate along the length of the 

river. 
2.46 

Under natural conditions, a third type of bend is often encountered. This bend occurs when the 

stream impinging on a erosion-resistant bank forms a forced curve which is gradually 

transformed into a river bend of a more constricted shape. In all cases, the effect of the 

character (density) of the bank material is important and, to a certain degree, determines the 

radius of curvature of the channel in a free bend. The radius of curvature increases with the 

density of the material. Considering both the action of the stream and the interaction between 

the stream and the channel, as well as the general laws of their formation, one can distinguish 

the following three types of bends of a natural river channel: 

1. Free bends - Both banks are composed of alluvial floodplain material which is usually quite 

mobile; the free bend corresponds to the common concept of a surface bend; 

2. Limited bends - The banks of the stream are composed of consolidated parent material 

which limits the lateral erosion by the stream. Limited bends are entrenched bends; and 

3. Forced bends - The stream impinges onto an almost straight parent bank at a large angle 

(60° to 90°). 

A typical feature of bends is a close relationship between the type of stream bend and the radius 

of curvature. The forced bend has the smallest radius of curvature. Next in size are the radii of 

free bends. The limited bends have the greatest radii. The average values of the ratios of the 

radii of curvature to the width of the stream at bankfull stage for the three types of bends are: (1) 

free bends 4.5 to 5.0; (2) limited bends 7.0 to 8.0; and (3) forced bends 2.5 to 3.0. 

A second characteristic feature of bends is the distribution of depths along the length of the bend. 

In free bends and limited bends, the depth gradually increases and the maximum depth is found 

some distance below the apex of the bend. In the forced bend, the depth sharply increases at 

the beginning of the bend and then gradually diminishes. In forced bends the greatest depth is 

located in the middle third of the bend, where there appears to be a concentrated deep scour 

hole. 

2.7.2 Transverse Velocity Distribution in Bends

The transverse velocities in bends result from an imbalance of radial pressures on a particle of 

fluid traveling around the bend. In Figure 2.26, a cross section through a typical bend is shown. 

The radial forces acting on the shaded control volume are the centrifugal force mv2

/r in which r is 

the radius of curvature, and the differential hydrostatic force γdz caused by the superelevation of 

the water surface dz. As shown in Figure 2.26a, the centrifugal force is greater near the surface 

where the fluid velocity v is greater and less at the bed where v is small. The differential 

hydrostatic force is uniform throughout the depth of the control volume. As shown in Figure 

2.26b, the sum of the centrifugal and excess hydrostatic forces varies with depth and can cause 

a lateral velocity component. The magnitude of the transverse velocity is dependent on the 

radius of curvature and on the proximity of the banks. In the immediate vicinity of the banks, 

there can be no lateral velocity if the river is narrow and deep. 
2.47 

Figure 2.26. Schematic representation of transverse currents in a channel bed. 

2.7.3 Subcritical Flow in Bends

Because of the change in flow direction which results in centrifugal forces, there is a 

superelevation of the water surface in river bends. The water surface is higher at the outside 

bank than at the inside bank. The resulting transverse slope can be evaluated quantitatively. 

Using cylindrical coordinates (Figure 2.27), the differential pressure in the radial direction arises 

from the radial acceleration or: 

r

V

r

p1 2

θ = ∂

∂

ρ

 (2.157) 

To calculate the total superelevation between the outer and inner bend two assumptions are 

made: (1) radial and vertical velocities are small compared to the tangential velocities such that 

Vθ ≈ V; and (2) pressure distribution in the bend is hydrostatic, i.e., p = γy. 

Figure 2.27. Definition sketch of flow around a bend. 
2.48 

Then 

dr

r

V

g

1 Z

2

r

r

o

i =∆  (2.158) 

To solve Equation 2.158, the transverse velocity distribution along the radius of the bend must be 

known or assumed. The results obtained assuming various velocity distributions follow. 

Woodward (1920) assumed V equal to the average velocity Q/A and r equal to the radius to the 

center of the stream rc, and obtained: 

dr

r

V

g

1 Z

c

2

r

r

o

i =∆  (2.159) 

or 

)rr(

rg

V zzZ io

c

2

io −=−=∆ (2.160) 

in which zi and ri are the water surface elevation and the radius at the inside of the bend, and zo

and ro are the water surface elevation and the radius at the outside of the bend. 

By assuming the velocity distribution to approximate that of a free vortex (Vθ = C1/r), Shukry 

(1950) obtained: 

















=∆  −= 2

o

2

i

2

1

3

2

r 1

r r

1

r

1

g2

C dr

r

C

g

1 Z o

i

 (2.161) 

in which C1 = rV is the free vortex constant. By assuming flow depth of flow upstream of the 

bend equal to the average depth in the bend, Ippen and Drinker (1962) reduced Equation 2.161 

to: 

































	











 −

=∆ 2

c

c

2

r2

W 1

1

r

W2

g2

V Z (2.162) 

For situations where high velocities occur near the outer bank of the channel, a forced vortex 

may approximate the flow pattern. With this assumption and assuming a constant average 

specific head, Ippen and Drinker (1962) obtained: 
2.49 



































	









 −

=∆

2

c

2

c

2

r12

W 1

1

r

W2

g2

V Z (2.163) 

By assuming that the maximum velocities are close to the centerline of the channel in the bend 

and that the flow pattern inward and outward from the centerline can be represented as forced 

and free vortices, respectively, then: 

dr

r

C

g

1 dr

r

rC

g

1 Z 3

2

r 1

r

22

r 2

r

o

c

c =∆  i +  (2.164) 

and when r = rc, V = Vmax

Therefore, 1 cmax

c

max

2 rVCand

r

V C = =

and Equation 2.164 becomes: 





















	











 − 



	











 =∆ −

2

o

c

2

c

i

2

max

r

r

r

r 2

g2

V Z (2.165) 

The differences in superelevation that are obtained by using the different equations are small, 

and in alluvial channels the resulting erosion of the outside bank and deposition on the inside 

bank leads to further error in computing superelevation. Therefore, it is recommended that 

Equation 2.158 be used to compute superelevation in alluvial channels. For lined canals with 

strong curvature, superelevation should be computed using Equations 2.162 or 2.165. 

An example showing how to calculate superelevation in bends from velocity measurements is 

presented in Section 2.14 (SI) and 2.15 (English) at the end of this chapter. The example also 

compares the various approximate equations included in this section. 

2.7.4 Supercritical Flow in Bends

Rapid flow or supercritical flow in a curved prismatic channel produces cross wave disturbance 

patterns which persist for long distances in a downstream direction. These disturbance patterns 

are the result of non-equilibrium conditions which persist because the disturbances cannot 

propagate upstream or even propagate directly across the stream. Therefore, the turning effect 

of the walls is not felt on all filaments of the flow at the same time and the equilibrium of the flow 

is destroyed. The waves produced form a series of troughs and crests in the water surface along 

the channel walls. 

Two methods have been used in the design of curves for rapid flow in channels. One method is 

to bank the floor of the channel and the other is to provide curved vanes in the flow. Banking on 

the floor produces lateral forces which act simultaneously on all filaments and causes the flow to 
2.50 

turn without destroying the flow equilibrium. Curved vanes break up the flow into a series of 

small channels and since the superelevation is directly proportional to the channel width, each 

small channel has a smaller superelevation. If the bend is not properly shaped or designed a 

hydraulic jump may occur or the cross-waves can be amplified. There are design methods in the 

literature (Rouse 1950, Ippen 1950, Chow 1959). In most cases a physical model study is 

recommended. 

2.8 GRADUALLY VARIED FLOW

2.8.1 Introduction

Thus far, two types of steady flow have been considered. They are uniform flow and rapidly 

varied nonuniform flow. In uniform flow, acceleration forces are zero and energy is converted to 

heat as a result of viscous forces within the flow. There are no changes in cross-section or flow 

direction, and the depth (called normal depth) is constant. In rapidly varied flow, changes in 

cross-section, direction, or depth take place in relatively short distances; acceleration forces are 

not zero; and viscous forces can be neglected (at least as a first approximation). 

Different conditions prevail for each of these two types of steady flow. In steady uniform flow, the 

slope of the bed, the slope of the water surface and the slope of the energy gradeline are all 

parallel and are equal to the head loss divided by the length of the channel in which the loss 

occurred. In rapidly varied flow through short streamlined transitions, resistance is neglected and 

changes in depth due to acceleration are dominant. In this section, a third type of steady flow is 

considered. In this type of flow, changes in depth and velocity take place slowly over large 

distances, resistance to flow dominates and acceleration forces are neglected. This type of flow 

is called gradually varied flow. 

In gradually varied flow, the actual flow depth y is either larger or smaller than the normal depth yo

and either larger or smaller than the critical depth yc. The water surface profiles, which are often 

called backwater curves, depend on the magnitude of the actual depth of flow y in relation to the 

normal depth yo and the critical depth yc. Normal depth yo is the depth of flow that would exist for 

steady-uniform flow as determined using the Manning or Chezy velocity equations, and the 

critical depth is the depth of flow when the Froude number equals 1.0. Reasons for the depth 

being different than the normal depth are changes in slope of the bed, changes in cross-section, 

obstruction to flow and imbalances between gravitational forces accelerating the flow and shear 

forces retarding the flow. 

In working with gradually varied flow, the first step is to determine of the general characteristics of 

the water surface and what type of backwater curve would exist. The second step is to perform 

the numerical computations to determine the elevation of the water surface or depth of flow. 

2.8.2 Classification of Flow Profiles

The classification of flow profiles is obtained by analyzing the change of the various terms in the 

total head equation in the x-direction. The total head is: 
2.51 

zy g2

V H

2

T ++= (2.166) 

or 

zy gA2

Q H 2

2

T ++= (2.167) 

Then assuming a wide channel for simplicity 

dx

dz

dx

dy

dx

dy

gy

q

dx

dH

3

2

T + += (2.168) 

The term dHT/dx is the slope of the energy gradeline Sf, by assumption. For short distances and 

small changes in y the energy gradient can be evaluated using the Manning or Chezy velocity 

equations. 

When Chezy's equation (Equation 2.84) is used the expression for dHT/dx is 

32

2

f

T

yC

q S

dx

dH ==− (2.169) 

The term dy/dx is the slope of the water surface Sw, and dz/dx is the bed slope -So. For steady 

flow, the bed slope is (from Equation 2.84) 

3

o

2

o

2

o yC

q S − = (2.170) 

where the subscript "o" indicates the steady uniform flow values. 

When Equations 2.169 and 2.170 are substituted into Equation 2.168, the familiar form of the 

gradually varied flow equation is obtained: 

































	











 −





	











 

	


 




 −

= 3

c

3

o

2

o

o

y

y 1

y

y

C

C 1

S

dx

dy (2.171) 

If Manning's equation is used to evaluate Sf and So, Equation 2.171 becomes 









































	











 −





	

















	











 −

=

3/10

3

c

o

2

o

o

y

y 1

y

y

n

n 1

S

dx

dy (2.172) 
2.52 

The slope of the water surface dy/dx depends on the slope of the bed So, the ratio of the normal 

depth yo to the actual depth y and the ratio of the critical depth yc to the actual depth y. The 

difference between flow resistance for steady uniform flow no to flow resistance for steady 

nonuniform flow n is small and the ratio is taken as 1.0. With n = no, there are twelve types of 

water surface profiles. These are illustrated in Figure 2.28 and summarized in Table 2.3. 

When y → yc, the assumption that acceleration forces can be neglected no longer holds. 

Equations 2.171 and 2.172 indicate that dy/dx is perpendicular to the bed slope when y → yc. For 

cross-sections close to the cross-section where the flow is critical (a distance from [3 to 15 m (10 

to 50 ft)]), curvilinear flow analysis and experimentation must be used to determine the actual 

values of y. 

Figure 2.28. Classification of water surface profiles. 

When analyzing long distances 30 to 300 m (100 to 1,000 ft) or longer one can assume 

qualitatively that y reaches yc. In general, when the flow is rapid (Fr ≥ 1), the flow cannot become 

tranquil without a hydraulic jump occurring. In contrast, tranquil flow can become rapid (cross the 

critical depth line). This is illustrated in Figure 2.29. 

When there is a change in cross-section or slope or an obstruction to the flow, the qualitative 

analysis of the flow profile depends on locating the control points, determining the type of curve 

upstream and downstream of the control points, and then sketching the backwater curves. It 

must be remembered that when flow is rapid (Fr > 1), the control of the depth is upstream and 

the backwater proceeds in the downstream direction. When flow is tranquil (Fr < 1), the depth 

control is downstream and the computations must proceed upstream. The backwater curves that 

result from a change in slope of the bed are illustrated in Figure 2.29. 
2.53 

Table 2.3. Characteristics of Water Surface Profiles. 

Class Bed Slope Depth Type Classification 

Mild So>0 y>yo>yc 1 M1 

Mild So>0 yo>y>yc 2 M2 

Mild So>0 yo>Yc>y 3 M3 

Critical So>0 y>yo=yc 1 C1 

Critical So>0 y<yo=yc 3 C2 

Steep So>0 y>yc>yo 1 S1 

Steep So>0 yc>y>yo 2 S2 

Steep So>0 yc>yo>y 3 S3 

Horizontal So = 0 y>yc 2 H2 

Horizontal So = 0 yc>y 3 H3 

Adverse So<0 y>yc 2 A2 

Adverse So<0 yc>y 3 A3 

Note: 

(1) With a type 1 curve (M1, S1, C1), the actual depth of flow y is greater than both the normal 

depth yo and the critical depth yc. Because flow is tranquil, control of the flow is 

downstream. 

(2) With a type 2 curve (M2, S2, A2, H2), the actual depth y is between the normal depth yo and 

the critical depth yc. The flow is tranquil for M2, A2 and H2 and thus the control is 

downstream. Flow is rapid for S2 and the control is upstream. 

(3) With a type 3 curve (M3, S3, C3, A3, H3), the actual depth y is smaller than both the normal 

depth yo and the critical depth yc. The flow is rapid and control is upstream. 

(4) For a mild slope, So is smaller than Sc and yo > yc. 

(5) For a steep slope, So is larger than Sc and yo < yc. 

(6) For a critical slope, So equals Sc and yo = yc. 

(7) For an adverse slope, So is negative. 

(8) For a horizontal slope, So equals zero. 

(9) The case where y → yc is of special interest. See paragraph preceding Figure 2.28. 
2.54 

Figure 2.29. Examples of water surface profiles. 
2.55 

2.8.3 Standard Step Method for the Computation of Water Surface Profiles

The standard step method is a simple computational procedure to determine the water surface 

profile in gradually varied flows. Prior knowledge of the type of backwater curve as classified in 

Section 2.8.2 is useful to determine whether the analysis should proceed upstream or 

downstream. 

The standard step method is derived from the energy equation 

L2

2

2

1

2

1 Hy g2

V

zy g2

V ++=∆++ (2.173) 

From Figure 2.30 

LSy g2

V LSy g2V f2

2

2

o1

2

1 ∆++=∆++ (2.174) 

o1 f2 ∆+=∆+ LSHLSH (2.175) 

and 

fo

12

SS

HH L −− =∆ (2.176) 

Figure 2.30. Definition sketch for the standard step method for computation of backwater 

 curves. 

The procedure is to start from some known y, which can be obtained from a stage-discharge 

relationship, assume another y either upstream or downstream depending on whether the flow is 

tranquil or rapid, and compute the distance ∆L to the assumed depth using Equation 2.176. It is 

recommended that the assumed depth be kept relatively close to the known value of y1 in order 

to keep the interval ∆L as short as possible to obtain better accuracy in the calculation. 
2.56 

2.9 STREAM GAGING 

2.9.1 Introduction 

The hydrology of the stream at a bridge crossing is determined from records obtained by State 

and Federal agencies. The most extensive records of streamflow are by the U.S. Geological 

Survey. Their records of stream flow and analysis can be obtained for any State at their District 

offices or from their Web Site. The USGS has been collecting stream gaging records since 1888 

(Corbet 1962) and maintain over 4,000 sites in the United States in cooperation with other 

Federal, State, Counties, and Cities. Often a gaging station will be located near the site of a 

bridge crossing. If not, regional or other hydrologic tools can be used to analyze the stations in 

the region to obtain the hydrology for the bridge site. 

The quality of the hydrologic records at a stream gaging site depends on obtaining an accurate 

gage height record and stage-discharge relation. These depend on stream characteristics (bed 

material, cross-section, and control) and, in many cases, the magnitude of the sediment 

transported by the stream. These terms are defined in the glossary and will be defined later in 

this section. For example, the hydrology used for the analysis of the 1987 I-90 bridge failure in 

upstate New York was excellent. There was a gaging station located 22.5 km (14 mi) upstream 

of the bridge. The stream at the gaging site was in bedrock with an excellent control. Thus, the 

stage-discharge relation was well defined with a single curve established with actual discharge 

measurements made over many years and large flows. The gage height record was excellent 

with few time gaps. The stage record for the 1987 storm was excellent. The discharge was 

routed to the bridge from the gage site using the U.S. Corps of Engineers HEC-1 model to obtain 

the hydrology of the flow at the bridge. 

In contrast, the hydrology for the 1997 I-5 bridge failure in California was not well defined. The 

streambed was sand and there was no bed rock control, only channel control. The gage height 

record was good, but the stage discharge relation was poorly defined with no consistent curve. 

The peak discharge for the flood had to be determined using indirect methods with results that 

ranged from to 420 to 1,140 m3

/s (14,800 to 40,300 cfs). The range in discharge for the slope 

area measurement resulted from assumptions on Manning's n and the amount of degradation. 

The discharge used in the final analysis 773 m3

/s (27,300 cfs) was determined from the slopearea measurement, study of the rainfall records and discharge records of other gages in the 

drainage basin. 

A program to obtain a systematic record of the stream flow consist of (1) establishing and 

constructing a streamflow measurement station, (2) operating and maintaining the station, and 

(3) computing, compiling and publishing the stream flow data. In addition, analyses of the longterm stream flow data are made (flood frequency, low flow analysis, trends). The methods of 

obtaining water discharge records are described in publications of the USGS (Corbet 1962, 

Carter and Davidian 1968, Buchanan and Somers 1968a,b, and Kennedy 1983). In the following 

sections a typical gaging station, discharge measurement, stage discharge relation and the 

determination of the daily discharge will briefly be explained. 

2.9.2 Gaging Station 

A gaging station consists of structures and equipment to measure and record the stage and 

discharge as a function of time at a given site on a stream. Stage or gage height is the height of 

the water surface above a chosen arbitrary datum corresponding to the zero of the gage. The 

zero of the gage is related to sea-level elevation. An accurate record of stage is essential for 

computing the discharge for any period of time. Discharge is the rate of flow in m3

/s or cfs. 
2.57 

Typical gaging stations are illustrated in Figure 2.31 and 2.32. In many gaging stations, a 

manometer and gas under pressure bubbling into the stream are used to measure stage instead 

of a float, well and pipe intakes (Figure 2.33). The stage versus time is recorded by pen on 

paper, or electronically on tape, or both. Figure 2.34 illustrates a stage versus time chart. Also, 

at many stations the stage is transmitted over phone lines or satellite to a central location. 

Discharge is measured using a current (velocity) meter by wading at low flow and from a 

cableway or highway bridge at high flows. The procedure for an actual current meter 

measurement is described in the next section. In many cases, large flows are measured by 

indirect methods such as a slope-area measurement using Manning's equation; however, the 

preferred method is to use a current meter measurement. 

Figure 2.31. Gaging station well and shelters (from Buchanan and Somers 1968a). 
2.58 

Figure 2.32. Typical float recording gaging station (from Buchanan and Somers 1968a). 

Figure 2.33. Typical bubble and manometer recording gaging station (from Buchanan and 

 Somers 1968a). 
2.59 

Figure 2.34. Stage vs. time hydrograph (Kennedy 1983). 
2.60 

2.9.3 Measuring Water Discharge 

Generally the stream cross-section is divided into 20 sections (verticals). At each vertical the 

depth and velocity are measured. The velocity is measured using a vertical-axis rotor with cups 

(Price is typical), horizontal-axis rotor with vanes (Ott is typical) or a magnetic (Brien-Macnertry is 

typical) velocity meter. If the depth is less than 0.8 m (2.5 ft) the velocity is measured at 0.6 the 

depth below the water surface. If over 0.8 m (2.5 ft) the velocity is the average of the velocity 

measured at 0.2 and 0.8 the depth below the water surface. In complex velocity distributions, 

additional point velocities are taken, plotted, and the average velocity determined. The average 

velocity in the vertical (one point, two points, and etc measurement) is multiplied by the sub-area 

to obtain a unit discharge. The sub-area is determined by multiplying the measured depth by 0.5 

times the width to the adjacent sections (Figure 2.35). Total Discharge is the sum of all the unit 

discharges. 

 Figure 2.35. Definition sketch of computing area and discharge at a gaging station (from 

 Buchanan and Somers 1968b). 
2.61 

2.9.4 Stage-Discharge Relation 

From the recorded stage and corresponding discharge measurement of many measurements 

over time at a gaging station, a stage vs. discharge relation is developed. Figure 2.36 is an 

example of an excellent well-defined stage-discharge relation for a station with a good control. 

Figure 2.37 is an example of a poorly defined stage-discharge relation for a gage with a sand 

channel and channel control. The discharge for the first would be rated from good to excellent, 

and the latter from fair to poor. An additional example of a sand channel stage-discharge 

relation is given in Chapter 3. 

Figure 2.36. Stage-discharge relation for Schoharie Creek, NY (Butch 2000). 

From the recorded stage-time record and the stage-discharge relation the daily, yearly peak, and 

minimum flow at the station are determined. If for a given day the stage doesn’t change 

appreciably, the daily discharge is determined using the average gage elevation for that day. If 

there is a rapid change in stage with time, daily discharge is determined by converting the timestage curve to time-discharge and by integrating over the 24 hours to determine the average 

daily discharge. 
2.62 

Figure 2.37. Stage-discharge relation for a sand channel (Los Gatos Creek at El Dorado 

 Avenue, CA). 

2.10 HYDRAULICS OF BRIDGE WATERWAYS

2.10.1 Introduction 

As flow passes through a channel constriction most of the energy losses occur as expansion 

losses downstream of the contraction. This loss of energy is reflected by a rise in the water 

surface and the energy line upstream from the bridge. The rise in water level is referred to as the 

bridge backwater. Hydraulic engineers are concerned about the computation of backwater with 

respect to flooding upstream of the bridge. Other concerns discussed in the following chapters 

include the stability and scour around embankments, general scour depths due to constriction, 

and local scour around piers. 
2.63 

2.10.2 Backwater Effects on Waterway Openings

It is necessary to distinguish between the following types of backwater effects. 

• Backwater on a floodplain resulting from construction of a long, skewed or curved road 

embankment as sketched in Figure 2.38a, where the bridge opening is, in effect, located 

up-valley from one end of the embankment. The backwater effect along the embankment 

arises from ponding of water along a line running obliquely down-valley. In the case of steep 

rivers with wide floodplains this effect can be very large, since a large pond is created. This 

type of effect can be prevented by choosing a suitable location and alignment, or by providing 

dikes (shown on the figure) to close off the affected part of the floodplain from flood waters, 

or possibly by providing a relief span. 

• Backwater in an incised river channel without substantial overbank flow, resulting in part from 

constriction of flow through an opening somewhat smaller than the natural cross section, and 

in part from obstructive effects of piers (Figure 2.38b). The backwater effect arising from this 

type is seldom large, but may be significant in developed areas. 

• Backwater in a river with floodplain where the road crossing is more or less normal to the 

valley but the road approaches block off overbank flow (Figure 2.38c). In these cases the 

backwater may be significantly greater than in type b. The effect of guide banks shown in 

Figure 2.38c is to reduce the backwater effects by improving the hydraulic efficiency of the 

opening. 

It is advisable to be aware of other unusual backwater effects that might occur in special 

circumstances, although they might never arise in ordinary bridge design practice. 

2.10.3 Effects of a Submerged Superstructure 

If the high-water level reaches the bottom of the superstructure, the bridge will act as a short 

culvert. For bridges which are designed to be submersible under certain conditions, it is 

advisable to provide a rounded nosing on the leading edge of the girder, in order to improve the 

hydraulic efficiency and to reduce the tendency to catch driftwood and ice as illustrated in Figure 

2.39. Also, the superstructure must be anchored to counter buoyancy. 

2.10.4 Effects of Supercritical Flow 

In contrast to the usual drop at a constriction in subcritical flow, in supercritical flow water levels 

may rise suddenly at the contracted section. The phenomenon of "choking" is particularly likely if 

the Froude number only slightly exceeds 1.0. "Choking" may occur even in subcritical flow if the 

constriction is severe enough. Wider or additional openings should be designed if choking 

effects are expected to occur. 

2.10.5 Types of Flow in Bridge Openings

Three types of flow, I through III, illustrated in Figure 2.40 are often encountered in bridge 

waterway design. As the scale of the normal depth is the same for all flow profiles, the 

discharge, boundary roughness and slope of the channel must increase from Type I to Type II 

and to Type III. 
2.64 

Figure 2.38. Three types of backwater effect associated with bridge crossings; (a) effect of a 

 skewed embankment across a floodplain; (b) effect due to constriction of the 

 channel flow; (c) effect due to constriction of the overbank flow, both without and 

 with guide banks (after Neill 1975). 
2.65 

Figure 2.39. Submergence of a superstructure. 

In Type I flow, the normal water surface is everywhere above critical depth and the flow is 

subcritical. Backwater calculations are obtained by applying the conservation of energy principle 

between Sections 1 and 4. 

In Type II flow, subcritical flow upstream of the bridge passes through critical depth in the 

constriction. The backwater curve for the water surface elevation upstream from the constriction 

is independent of the water surface elevation downstream. An undulating hydraulic jump (with Fr 

< 2) is formed when the water surface elevation dips below critical depth downstream from the 

contracted section (Type IIB). 

Referring to Type III flow, the flow is supercritical throughout the reach as the normal water 

surface is everywhere below critical depth. Such conditions require steep channels as 

experienced in, but not limited to, mountainous regions. Backwater should not occur as long as 

the flow remains supercritical since the flow is controlled from upstream conditions. However, 

significant rise in the water surface might occur in the vicinity of the constriction due to: (1) 

changes in the specific energy or specific discharge diagram as indicated in Figures 2.19 and 

2.21; (2) cross waves and transitions; and (3) possible hydraulic jumps near the embankments. 

Solved problems are presented in Section 2.14 (SI) and 2.15 (English) to illustrate how to 

calculate maximum constrictions without causing backwater and to calculate water surface 

elevation upstream of a grade control structure. 

2.11 COMPUTER MODELS FOR HIGHWAY BRIDGES 

2.11.1 One-Dimensional Computer Models 

Many one-dimensional computer models are available for computing water surface profiles, 

average depth and velocity for open channel flow. However, three have the most utility for 

highway bridge analysis. These are FHWA's WSPRO (Arneson and Sherman 1998). The U.S. 

Army Corps of Engineers (2001) HEC-RAS (River Analysis System) and UNET (Barkau 1993). 

HEC-RAS is the successor the Corps of Engineers HEC-2 water surface profile program. 

WSPRO is for steady, nonuniform flow. 
2.66 

Figure 2.40. Types of flow encountered (HDS 1, Bradley 1978). 
2.67 

HEC-RAS performs for steady or unsteady, uniform or nonuniform flow. UNET is for unsteady, 

uniform, and nonuniform flow. Although one-dimensional, they can give an approximate 

distribution of the velocity in a cross-section. Embankment overtopping flows, in conjunction with 

either free surface or pressure flow through the bridge, can be computed. The programs are 

capable of computing profiles at stream crossings with multiple openings (including culverts), at 

river confluences, and mixed flow regimes. They also, incorporate the effect of wide, wooded, 

floodplains into the bridge backwater calculations. In addition, there are two one-dimensional 

computer models which include sediment transport. These are FHWA's BRI-STARS (Molinas 

2000) and U.S. Army Corps of Engineers HEC-6 (1993) computer programs. 

2.11.2 Two-Dimensional Computer Models 

Two-dimensional computer models give the water surface profile, and the depth and velocity 

along and across the stream. Of the many models, two have the most utility for highway bridge 

analysis. They are the FHWA’s FESWMS (Froehlich 1996) and the U.S. Army Corps of 

Engineer’s (1997) RMA-2V (Thomas and McAnally 1985) models. The models are for steady or 

unsteady and nonuniform open channel flow. They have all the capabilities of the onedimensional programs and have the utility of giving the velocity and depth distribution along and 

across the channel as a function of time and distance. They can and have been used to analyze 

tidal flows (Zevenbergen et al. 1997, Ayres Associates 1994 and 1997, and Richardson and 

Lagasse 1999 pages 701 to 824) Both models require that a grid system be created for the river 

system. However, constructing the grid is greatly aided by the use BYU’s (2000) SMS modeling 

system. 

2.12 HYDRAULICS OF CULVERT FLOW 

2.12.1 Introduction

A culvert is a conduit which conveys stream flow through a roadway embankment. Most culverts 

are constructed of concrete, corrugated aluminum, corrugated steel, and sometimes corrugated 

plastics. Culvert shapes vary from circular to rectangular, and elliptical, pipe arch, arch and metal 

box sections are commonly used. 

Two basic types of flow control are recognized depending on the location of the control section: 

inlet control or outlet control. The characterization of pressure, as well as subcritical and 

supercritical flow regimes play an important role in determining the location of the control section. 

Inlet control occurs when the culvert barrel is capable of carrying more flow than the inlet will 

accept. Critical flow depth is located at the inlet and the flow is supercritical in the barrel. 

Outlet control flow occurs when the culvert barrel is not capable of conveying as much flow as the 

inlet opening will accept. Under outlet control conditions, either subcritical or pressure flow exists 

in the culvert barrel. 

The hydraulic design of culverts is given in HDS 5 (FHWA 1985) and HY8 (FHWA 1998) is a 

computer program for the design and analysis of culvert flows. 
2.68 

2.13 ROADWAY OVERTOPPING 

Roadway overtopping will begin as the headwater rises to the elevation of the lowest point of the 

roadway. This type of flow is similar to flow over a broad crested weir. The length of the weir can 

be taken as the horizontal length across the roadway. The flow across the roadway is calculated 

from the broad crested weir equation 

5.1

rsrtuo = )HW(LCkKQ (2.177) 

where: 

 Qo = Overtopping discharge in m3

/s (ft3/s) 

 Cr = Overtopping discharge coefficient 

 HWr = Flow depth above the roadway in m (ft) 

 kt = Submergence factor 

 Ls = Length of the roadway crest along the roadway in m (ft) 

 Ku = 1.0 (English) 

 Ku = 9 ( 552.0or2.32/81. SI)

The charts in Figure 2.41 indicate how to evaluate the correction factors kt and Cr. 

If the elevation of the roadway crest varies, for instance where the crest is defined by a roadway 

sag vertical curve, the vertical curve can be approximated as a series of horizontal segments. 

The flow over each is calculated separately and the total flow across the roadway is the sum of 

the incremental flows for each segment (Figure 2.42). 

The total flow across the roadway then equals the sum of the roadway overflow plus the culvert 

flow. A trial and error procedure is necessary to separate the amount of water passing through 

the culvert, if any, from the amount overtopping the roadway. Performance curves must then 

include both culvert flow and road overflow. 

2.14 SOLVED PROBLEMS OPEN CHANNEL FLOW (SI)

2.14.1 PROBLEM 1 Evaluation of Correction Factors α and β

Calculate the correction factors α and β for a cross-section given the discharge measurement 

during the peak flood event for the year. From Table 2.4, the following values are obtained: 

 Q = 152.41 cms 

 A = 138.20 m2 

 W = 49.71 m 

Vi∆Qi = 180.46 m4/sec2

Vi

2

∆Qi = 220.88 m5/sec3
2.69 

Figure 2.41. Discharge coefficient for roadway overtopping (after Petersen 1986). 

Figure 2.42. Weir crest length determinations for roadway overtopping. 
2.70 

Table 2.4. Discharge Measurement Notes1

. 

yoi

(m) 

∆zI

(m) 

VI

(mps) 

∆AI

(sq m) 

∆QI

(cms) 

Vi∆QI 

(m4/sec2) 

Vi

2

∆QI 

(m5/sec3) 

0.80 1.22 0.00 0.0 0.000 0.000 0.000 

0.335 2.44 0.300 0.818 0.245 0.074 0.022 

.793 2.44 0.165 1.93 .319 .053 .009 

1.37 2.44 .195 3.34 .651 .127 .025 

2.59 2.44 .732 6.32 4.626 3.386 2.479 

3.35 2.44 .966 8.17 7.892 7.624 7.365 

3.54 2.44 1.23 8.64 10.627 13.071 16.078 

3.66 2.44 1.24 8.93 11.073 13.731 16.026 

3.90 2.44 1.15 9.52 10.948 12.590 14.479 

3.84 2.44 1.14 9.37 10.682 12.178 13.882 

3.78 2.44 1.15 9.22 10.603 12.194 14.023 

3.54 2.44 1.44 8.64 12.442 17.917 25.800 

3.47 2.44 1.31 8.47 11.096 12.539 14.169 

3.29 2.44 1.49 8.03 11.965 17.828 26.564 

3.23 2.44 1.41 7.88 11.111 15.667 22.090 

3.32 2.44 1.32 8.10 10.692 14.113 18.630 

3.54 2.44 1.19 8.64 10.282 12.236 14.560 

3.60 2.44 .945 8.78 8.297 7.841 7.409 

2.99 2.44 .921 7.30 6.723 6.192 5.703 

1.95 2.13 .515 4.15 2.137 1.101 .567 

1.16 1.68 0.0 1.95 0.0 0.0 0.0 

0.0 .76 0.0 0.0 0.0 0.0 0.0 

TOTAL 49.71 138.20 152.411 180.462 220.88 

1

Simons, D.B., E.V. Richardson, M.A. Stevens, J.H. Duke, and V.C. Duke, "Stream flow, groundwater 

 and ground response data, Hydrology Report, Vol. II, Venezuelan International Meteorological and 

 Hydrological Experiment, Civil Engineering Dept., Colorado State University, August 1971. 

The bed material at this gaging station has a D50 of 0.33 mm and a D65 of 0.45 mm and a gradation 

coefficient G of 3.27. If the value of D65 is used for ks, then for yo = 3.90 m (the maximum depth) 

700,8 00045.0

9.3

k

y

s

0 = ≅

and for yo = .335 m (the smallest non-zero depth) 

 

750

00045.0

335.0

k

y

s

0 = ≅

Using a mean yo/ks of approximately 5,000, then from Figure 2.12 the average values for the energy 

and momentum coefficients are: 

α' = 1.024 

and 

β' = 1.008 
2.71 

As it has been assumed that α' and β' are constant across the river (for convenience), Equations 

2.113 and 2.114 become: 

=α α′  ∆

i

i

2

3 i

2

V Q Q

A

and 

=β β′  ∆

i

i

2

2 Vi Q Q

A

With the values in Table 2.4 

221.1)9.220( )4.152(

)2.38.1( 024.1 3

2

=α =

083.1)5.180( )4.152(

)2.138 008.1 2 =β =

These values for α (1.22) and β (1.094) differ from unity by appreciable amounts. The difference 

may be important in many river channel calculations. If no data are available, the assumptions that 

α = 1.22 and β = 1.1 should be used for river channels. 

2.14.2 PROBLEM 2 Velocity Profiles and Shear Stress 

A velocity and discharge measurement is made on the Missouri River at Sioux City. The total 

discharge is 923.1 cms. The average depth is 2.38 meters and the average velocity is 0.738 m/sec 

in a vertical section at a point 244 meters from the right bank. The velocity measurements at 

various distances from the bottom (y) are shown in Table 2.5. 

Also during this measurement the slope of the energy grade line was observed as 0.000206 and 

the bed material gradation was determined with D50 = 0.270 mm, D65 = 0.315 mm and D84 = 0.42 

mm. 

Table 2.5. Observed Velocity Data. 

y 

(m) 

Observed Velocity 

(m/sec) 

0.152 0.518

0.305 0.633

0.457 0.701

0.762 0.762

1.07 0.838

1.37 0.884

1.68 0.945

1.90 0.945

2.29 0.991
2.72 

Velocity Profile Analysis 

The log and power function velocity distribution functions will be used to obtain a mathematical 

description of the velocity profile. 

(a) Logarithmic Velocity Distribution Equation 

The log velocity relation is Equation 2.75: 

v/V* = 1/k ln (y/y’) = 2.31/k log (y/y’) 

Determine the value of k and y': 

 V* = (g yoSf )1/2 = (9.81 x 2.38 x 0.000206)1/2 = 0.069 m/sec 

 

Table 2.6 is prepared for both log and power forms with the values shown. 

Using regression techniques for the logarithmic form of the equation on the values of v/V* and ln y' 

with the data in Table 2.6, obtain: 

v/V* = 2.564 (ln y - ln 0.009) with a regression coefficient R = 0.996 giving the values of k = 0.39 and 

y' = 0.009. The final equation for the data is 

v/V* = 2.564 ln (y/0.009) 

Table 2.6. Velocity Profile Calculations. 

y (m) v (m/s) v/V* v/V y/y0

0.152 0.518 7.51 0.702 0.064

0.305 0.633 9.17 0.858 0.128

0.457 0.701 10.16 0.950 0.192

0.762 0.762 11.04 1.033 0.321

1.07 0.838 12.14 1.136 0.449

1.37 0.884 12.81 1.198 0.577

1.68 0.945 13.70 1.281 0.705

1.90 0.945 13.70 1.281 0.801

2.29 0.991 14.36 1.340 0.962

(b) Power Velocity Distribution Equation 

The power form of the velocity distribution equation is: 

v/V = a (y/yo)

b

. 

To simplify the regression calculations take the ln transform giving: 

ln v/V = ln a+ b ln (y/yo) using linear regression of ln v/V vs ln (y/yo) calculate the values of a and b. 

The results are: 

 a = 1.368 

 b = 0.240 

 R = 0.998 
2.73 

The resulting power equation is: 

 

v/0.738 = 1.37 (y/yo)0.24 

Shear Stress Analysis 

Using the above information calculate the shear stress on the bed using the several methods given 

in Section 2.4.5. 

(a) Average Shear Stress on the Bed 

Calculate the average shear stress on the bed using Equation 2.97 

τ0 = γ R Sf = 9800 x 2.38 x 0.000206 = 4.805 N/m2

 

Using R = yo (for a wide channel) 

(b) Local Shear Stress on the Bed Using Single Point Velocity near the Bed 

Using the above information and the logarithmic velocity distribution equation derived from Equation 

2.75 (v/V* = 2.564 ln (y/0.009)) the shear stress on the bed using a single point velocity near the 

bed is: 

2

2

2

2

2

0 m/N11.5

009.0

152.0

ln564.2

)518.0(1000

009.0

y ln564.2

v = = ρ =





 



 



 



 τ

(c) Local Shear Stress on the Bed Using Two Point Velocities near the Bed 

For this determination use Equation 2.99 

( ) ( ) 2

2

2

2

1

2

21

2

0 m/N37.4

152.0

305.0

log75.5

1000 518.0633.0

y

y log75.5

vv = − = ρ =





 





















− τ

(d) Shear Stress on the Bed Using Average Vertical Velocity 

Using Equation 2.100 and the average depth 2.38 m, average velocity 0.738 m/sec, and a value 

for ks determine the average shear stress on the bed. 

Taking ks as (30.2) y' = 0.27 m (see Section 2.4.2) 

( ) ( ) 2

2

2

s

0

2

2

0 m/N98.3

27.0

38.2x27.12

log75.5

1000 738.0

k

27.12 y log75.5

V = = ρ =





 



 



 



 τ 
2.74 

(e) Summary 

The average shear stress on the boundary from (a) is 4.805 newtons per square meter. This is the 

average stress on the entire boundary (bed and banks). The shear stress computed using velocity 

is the local shear stress on the bed where the velocity profile was taken. This local shear stress is 

calculated as 5.11 in (b), 4.37 in (c), and 3.98 in (d). Normally the use of the two point velocity 

equation with the velocities close to the bed is the more accurate, however, very accurate velocity 

measurements are required. For this example, the value from the two velocity method appears to 

be low. For wide channels the centerline local shear stress on the bed should generally not be 

lower than the average shear stress on the boundary (i.e., the value of 4.805 newtons per square 

meter). 

2.14.3 PROBLEM 3 Superelevation in Bends 

Calculate the superelevation of the water surface in a river bend given the velocity profile from 

Table 2.4. The river radius of curvature ri

 is measured equal to 106.7 meters and the outer radius 

of curvature ro is 156.4 meters. The detailed calculations based on Equation 2.158 are presented in 

Table 2.7. 

Table 2.7. Detailed Computation of Superelevation in Bends. 

∆rI (m) rI (m) VI (m/s) i

i

2

i

i r

gr

V Z ∆=∆ (m) 

0.61 107.3 0.00 .0000 

1.82 109.1 0.300 .0002 

2.44 111.6 .165 .0001 

2.44 114.0 .195 .0001 

2.44 116.4 .732 .0011 

2.44 118.9 .966 .0020 

2.44 121.3 1.23 .0031 

2.44 123.7 1.24 .0031 

2.44 126.2 1.15 .0026 

2.44 128.6 1.14 .0025 

2.44 131.1 1.15 .0025 

2.44 133.5 1.44 .0038 

2.44 136.0 1.31 .0031 

2.44 138.4 1.49 .0040 

2.44 140.9 1.41 .0035 

2.44 143.3 1.32 .0030 

2.44 145.8 1.19 .0024 

2.44 148.2 .945 .0015 

2.44 150.6 .921 .0014 

2.44 153.1 .515 .0004 

1.68 154.8 0.00 .0000 

0.76 155.6 0.00 .0000 

Total ∆Z 0.0404 m 
2.75 

r

rg

1 Z

r

r

0

i

v

2

=∆  ∆

Total superelevation is 0.040 meters. 

Based on relationships, V = Q/A = 1.10 m/s, the following are obtained: 

Woodward's Equation 2.160 

( ) ( )

( ) ( ) m047.07.1064.156

4.13181.9

10.1

g

V Z

2

i0

c

2

rr

r

=∆ =− =−

Ippen and Drinker's Equation 2.162 

m042.0

4.131x2

7.49 1

1

4.131

7.49x2

x

81.9x2

10.1

r2

W 1

1W2

g2

V Z 2

2

2

c

c

2

r =

































−

=



































	









 −

=∆







 







 

Ippen and Drinker’s Equation 2.163 

m041.0

4.131x12

97.4 1

1

4.131

7.49x2

x

81.9x2

10.1

12

1 W

1W2

g2

V Z

2

2

2

2

c

2

c

2

r

r =





























−

=





























−

=∆ 

Combined Free and Forced Vortex Equation 2.165 

Vmax = 1.49 m/s 

( ) m0718.0

4.156

4.131

4.131

7.106 2

81.9x2

49.1 2

g2

V Z

2 2 2

o

c

2

c

i

2 2

r

r

r

max r =

















= − −





















=∆ −− 



 



 



 



 





























 

The equations give comparable results (0.040 to 0.0718 m). Equation 2.158 which integrates 

across the section using the velocity distribution is the most exact. But using the value 0.072 m 

provides a safety factor. 

2.14.4 PROBLEM 4 Maximum Stream Constriction Without Causing Backwater 

(Neglecting Energy Losses) 

A stream is rectangular in shape and 30.48 m wide. The design discharge is 141.6 cms and the 

uniform depth for this discharge is 3.05 m. Neglecting energy losses what is the maximum 

amount of constriction that a bridge can impose without causing backwater. 
2.76 

The upstream flow rate per unit width (q) is: 

ft/cfs65.4

48.30

6.141

W

Q

q === 

The average velocity (V) is: 

s/m52.1

96.92

6.141

A

Q V ===

The specific head (H) is: 

m17.305.3

62.19

52.1

y g2H

2 2

V =+=+=

According to Section 2.6.2 the maximum unit discharge for a given specific head (specific head 

equals to a constant) occurs at the critical depth yc where the Froude number is 1. From 

Equation 2.149 the critical depth yc for this specific head is: 

g2

2 H

3

2

g

q V y

2

c

2

max

3/1

c = == 













 

17.3x m/s/m63.9

3

2

H 81.9

3

2

q g 3

3

2/1

3

2/1

max = = =





















	 


























	 






Therefore, the width of the channel can be contracted until the unit discharge q is a maximum. 

The minimum width (maximum constriction) is: 

m70.14

63.9

6.141Q

q

W max

min ===

and the maximum constriction is 30.48 -14.70 = 15.8 m 

This contraction causes the flow to go to critical. This results in an undulating hydraulic jump 

downstream. Also, when energy losses are considered there will be some backwater at the 

constriction. 

2.14.5 PROBLEM 5 Maximum Water Surface Elevation Upstream of a Grade Control 

Structure Without Backwater (Neglecting Energy Losses) 

A low grade control structure (check dam) is to be placed across a stream downstream of a 

highway bridge. The stream is degrading. The purpose of the check dam is to maintain the 

elevation of the water surface at the bridge, cause deposition of bed material, protect the abutments 

and piers from long term degradation and contraction scour and increase the elevation of the be for 
2.77 

local scour. There is to be no increase in water surface elevation upstream of the bridge at the 

design discharge. The energy losses are expected to be small so they can be neglected. 

At the bridge, the design flood discharge is 141.6 cms. The river is 30.48 m wide and has a uniform 

flow depth of 3.05 m for the design discharge. 

What is the maximum height of the structure that will not cause backwater at the 

bridge at the design discharge? 

The unit discharge (q) in the river at design flood discharge is: 

m/s/m65.4

48.30

6.141

W

Q

q 3 == =

 

The average velocity (V) is 

s/m52.1

96.92

6.141

A

Q V == =

The specific head (H) is 

 

m17.305.3

62.19

52.1

y g2H

2 2

V =+=+=

As a first approximation assume no energy loss in the bridge reach. Then at the check dam, the 

elevation of the total energy line is 3.17 m above the bed (Figure 2.43). At the dam: 

Hmin + ∆Zmax = 3.17 m 

Figure 2.43. Sketch of backwater curve over check dam. 

That is, the dam can be built to a height of ∆Zmax which decreases the specific head at the dam to 

Hmin. From Section 2.6.3 and Equation 2.147 

Hmin yc 2

3 =
2.78 

From Equation 2.140 

m30.1

81.9

65.4

g

q

2

3/1

2

3/1

c y = = =































 

Then 

m95.1m30.1

2

3

2

3 H y min c == =

and 

∆Zmax = 3.17-1.95 = 1.22 m 

If the structure is built to a crest elevation 1.22 m above the bed, critical flow will occur at the dam 

for a flow of 141.6 cms and the dam will cause no backwater. At lower discharges the check dam 

will cause backwater. However, discharges larger than 141.6 cms at the check dam will not cause 

backwater. Local scour downstream of the check dam should be evaluated (see HEC-23). 

How much backwater will the dam cause for a flow of 28.37 m3/s if the normal depth for this 

discharge is 1.52 m and the dam height is 1.22 m? 

 Upstream of the dam the discharge for foot of width (q) is, 

m/s/m93.0

48.30

32.28

W

Q

q 3 == = 

The average velocity (V) is 

sec/m61.0

52.1

93.0

y

q V ===

 

At the check dam the flow is critical so from Equation 2.140 

m45.0

81.9

93.0

g

q y

3/1 2 3/1 2

c  =



 



  =



 



 =

and from Equation 2.144 

m67.0m45.0

2

3

2

3 H y min c == =

 

Assuming no energy loss, the specific head upstream of the dam is 

H = Hmin + ∆Z = 0.67 + 1.22 = 1.89 m 
2.79 

To determine the depth of flow upstream of the dam (y) solve the specific head equation (Equation 

2.137) 

m89.1

y62.19

93.0

y gy2

q H 2

2

2

2

=+= =

or 

0

62.19

93.0 y89.1y

2

3 2 =+−

The solution is 

y = 1.87 m 

As the normal depth is only 1.52 m, the backwater is 

∆y = 1.87 - 1.52 = 0.35 m 

That is, the depth upstream of the dam is increased 0.35 m by the 1.22 m high dam when the flow 

is 28.32 cms. 

2.15 SOLVED PROBLEMS OPEN CHANNEL FLOW (ENGLISH)

2.15.1 PROBLEM 1 Evaluation of Correction Factors α and β

Calculate the correction factors α and β for a cross-section given the discharge measurement 

during the peak flood event for the year. From Table 2.8, the following values are obtained: 

 Q = 5,370 cfs 

 A = 1,485 ft2

 W = 163 ft 

Vi∆Qi = 21,070 ft4

/sec2 

Vi

2

∆Qi = 85,500 ft5

/sec3 

The bed material at this gaging station has a D50 of 0.33 mm and a D65 of 0.45 mm and a gradation 

coefficient G of 3.27. If the value of D65 is used for ks, then for yo = 12.8 ft (the maximum depth) 

( ) 700,88.304 45.0

8.12

k

y

s

0 = ≅

and for yo = 1.1 ft (the smallest non-zero depth) 

 

( ) 7508.304

45.0

1.1

k

y

s

0 = ≅
2.80 

Using a mean yo/ks of approximately 5,000, then from Figure 2.12 the average values for the energy 

and momentum coefficients are: 

α' = 1.024 

and 

β' = 1.008 

As it has been assumed that α' and β' are constant across the river (for convenience), Equations 

2.113 and 2.114 become: 

=α α′  ∆

i

i

2

3 i

2

V Q Q

A

Table 2.8. Discharge Measurement Notes1

. 

yoi

(ft) 

∆zI

(ft) 

VI

(fps) 

∆AI

(sq ft) 

∆QI

(cfs) 

VI∆QI 

(ft4/sec2) 

Vi

2

∆QI 

(ft5/sec3) 

0.0 4.0 0.00 0.0 0.00 0.00 0.00 

1.1 8.0 0.98 8.8 8.62 8.45 8.28 

2.6 8.0 0.54 20.8 11.23 6.06 3.27 

4.5 8.0 0.64 36.0 23.04 14.75 9.44 

8.5 8.0 2.40 68.0 163.20 391.68 940.03 

11.0 8.0 3.17 88.0 278.96 884.30 2803.24 

11.6 8.0 4.02 92.8 373.06 1499.70 6028.80 

12.0 8.0 4.06 96.0 389.76 1582.43 6424.65 

12.8 8.0 3.78 102.4 387.07 1463.12 5530.61 

12.6 8.0 3.74 100.8 376.99 1409.94 5273.19 

12.4 8.0 3.78 99.2 374.98 1417.42 5357.86 

11.6 8.0 4.71 92.8 437.09 2058.69 9696.45 

11.4 8.0 4.30 91.2 392.16 1686.29 7251.04 

10.8 8.0 4.90 86.4 423.36 2074.46 10164.87 

10.6 8.0 4.63 84.8 392.62 1817.83 8416.56 

10.9 8.0 4.32 87.2 376.70 1627.34 7030.13 

11.4 8.0 3.89 91.2 354.77 1380.06 5368.42 

11.8 8.0 3.10 94.4 292.64 907.18 2812.27 

9.8 8.0 3.02 78.4 236.77 715.05 2159.44 

6.4 7.0 1.69 44.8 75.71 127.95 216.24 

3.8 5.5 0.0 20.9 0.0 0.0 0.0 

0.0 2.5 0.0 0.0 0.0 0.0 0.0 

TOTAL 163.0 1484.9 5368.74 21072.71 85494.77 

1

Simons, D.B., E.V. Richardson, M.A. Stevens, J.H. Duke, and V.C. Duke, "Stream flow, groundwater 

 and ground response data, Hydrology Report, Vol. II, Venezuelan International Meteorological and 

 Hydrological Experiment, Civil Engineering Dept., Colorado State University, August 1971. 
2.81 

and 

=β β  ∆ i i

2

2 i

'

V Q Q

A

With the values in Table 2.8 

( )

( ) ( ) 247.1490,85

369,5

485,1 024.1 3

2

=α =

( )

( ) ( ) 094.1070,21

369,5

485,1 008.1 2 =β =

These values for α (1.247) and β (1.094) differ from unity by appreciable amounts. The difference 

may be important in many river channel calculations. If no data are available, the assumptions that 

α = 1.25 and β = 1.1 should be used for river channels. 

2.15.2 PROBLEM 2 Velocity Profiles and Shear Stress 

A velocity and discharge measurement is made on the Missouri River at Sioux City, Iowa. The total 

discharge is 32,600 cfs. The average depth is 7.80 feet and the average velocity is 2.42 ft/sec in a 

vertical section at a point 800 feet from the right bank. The velocity measurements at various 

distances from the bottom (y) are shown in Table 2.9. 

Table 2.9. Observed Velocity Data. 

y 

(ft) 

Observed Velocity 

(ft/sec) 

0.5 1.70 

1.0 2.08 

1.5 2.30 

2.5 2.50 

3.5 2.75 

4.5 2.90 

5.5 3.10 

6.25 3.10 

7.5 3.25 

Also during this measurement the slope of the energy grade line was observed as 0.000206 and 

the bed material gradation was determined with D50 = 0.270 mm, and D65 = 0.315 mm. 

Velocity Profile Analysis 

The log and power function velocity distribution functions will be used to obtain a mathematical 

description of the velocity profile. 

(a) Logarithmic Velocity Distribution Equation 
2.82 

The log velocity relation is Equation 2.75: 

 v/V* = 1/k ln (y/y’) = 2.31/k log (y/y’) 

Determine the value of k and y': 

 V* = (g yoSf )1/2 = (32.2 x 7.80 x 0.000206)1/2 = 0.228 ft/sec 

 

Table 2.10 is prepared for both log and power forms with the values shown. 

Using regression techniques, for the logarithmic form of the equation on the values of V/V* and ln y' 

with the data in Table 2.10 obtain: 

v/V* = 2.564 (ln y - ln 0.03) with a regression coefficient R = 0.996 giving the values of k = 0.39 and 

y' = 0.03. The final equation for the data is 

v/V* = 2.564 ln (y/0.03) 

Table 2.10. Velocity Profile Calculations. 

y (ft) v (ft/s) v/V* v/V y/y0

0.50 1.70 7.46 0.702 0.064 

1.00 2.08 9.12 0.860 0.128 

1.50 2.30 10.09 0.950 0.192 

2.50 2.50 10.96 1.033 0.321 

3.50 2.75 12.06 1.136 0.449 

4.50 2.90 12.71 1.198 0.577 

5.50 3.10 13.60 1.281 0.705 

6.25 3.10 13.60 1.281 0.801 

7.50 3.25 14.25 1.340 0.962 

(b) Power Velocity Distribution Equation 

The power form of the velocity distribution equation is: 

v/V = a (y/yo)

b

. 

To simplify the regression calculations take the ln transform giving: 

ln v/V = ln a+ b ln (y/yo) using linear regression of ln v/V vs ln (y/yo) calculate the values of a and b. 

The results are: 

 a = 1.368 

 b = 0.240 

 R = 0.998 

The resulting power equation is: 

 

v/2.42 = 1.38 (y/yo)0.24 
2.83 

Shear Stress Analysis 

Using the above information calculate the shear stress on the bed using the several methods given 

in Section 2.4.5. 

(a) Average Shear Stress on the Bed 

Calculate the average shear stress on the bed using Equation 2.97 

τ0 = γ R Sf = 62.4 x 7.8 x 0.000206 = 0.100 lb/ft2

Using R = yo (for a wide channel) 

(b) Local Shear Stress on the Bed Using Single Point Velocity near the Bed 

Using the above information and the logarithmic velocity distribution equation derived from Equation 

2.75 (v/V* = 2.564 ln (y/0.03)) the shear stress on the bed using a single point velocity near the bed 

is: 

( ) 2

2

2

2

2

0 ft/lb108.0

03.0

5.0 In564.2

)70.194.1

03.0

y In564.2

V =





 





=





 





ρ =τ

(c) Local Shear Stress on the Bed Using Two Point Velocities near the Bed 

For this determination use Equation 2.99 

( ) 2

2

2

2

2

1

21

0 ft/lb094.0

5.0

0.1 Iog75.5

)70.108.294.1

y

y Iog75.5

)vv( =





 





− =





 





−ρ =τ

(d) Shear Stress on the Bed Using Average Vertical Velocity 

Using Equation 2.100 and the average depth 7.80 ft, average velocity 2.42 ft/sec, and a value for 

ks determine the average shear stress on the bed. 

Taking ks as (30.2) y' = 0.91 (see Section 2.4.2) 

2

2

2

2

s

o

2

0 ft/lb084.0

91.0

8.7x27.12 Iog75.5

)42.2(94.1

k

y27.12 Iog75.5

)V( =





 





=





 





ρ =τ
2.84 

(e) Summary 

The average shear stress on the boundary from (a) is 0.10 pound per square foot. This is the 

average stress on the entire boundary (bed and banks). The shear stress computed using 

velocity is the local shear stress on the bed where the velocity profile was taken. This local shear 

stress is calculated as 0.108 in (b), 0.094 in (c), 0.084, in (d). Normally the use of the two point 

velocity equation with the velocities close to the bed is the more accurate, however, very accurate 

velocity measurements are required. For this example, the value from the two velocity method 

appears to be low. For wide channels for the centerline local shear stress on the bed should 

generally not be lower than the average shear stress on the boundary, i.e., the value of 0.10 

pounds per square foot. 

2.15.3 PROBLEM 3 Superelevation in Bends 

Calculate the superelevation of the water surface in a river bend given the velocity profile from 

Table 2.8. The river inner radius of curvature ri

 is measured equal to 350 feet and the outer radius 

of curvature ro is 513 feet. The detailed calculations based on Equation 2.158 are presented in 

Table 2.11. 

Table 2.11. Detailed Computation of Superelevation in Bends. 

∆rI (ft) rI (ft) VI (ft/s) i

i

2

i

i r

gr

V Z ∆=∆ (ft) 

2.0 352 0.00 .0000 

6.0 358 0.98 .0005 

8.0 366 0.54 .0002 

8.0 374 0.64 .0003 

8.0 382 2.40 .0037 

8.0 390 3.17 .0064 

8.0 398 4.02 .0101 

8.0 406 4.06 .0101 

8.0 414 3.78 .0086 

8.0 422 3.74 .0082 

8.0 430 3.78 .0083 

8.0 438 4.71 .0126 

8.0 446 4.30 .0103 

8.0 454 4.90 .0131 

8.0 462 4.63 .0115 

8.0 470 4.32 .0099 

8.0 473 3.89 .0079 

8.0 486 3.10 .0049 

8.0 494 3.02 .0046 

7.0 502 1.69 .0014 

5.5 507 0.00 .0000 

2.5 512 0.00 .0000 

Total ∆Z 0.1330 ft 
2.85 

r

rg

1 Z

r

r

0

i

v

2

=∆  ∆

Total superelevation is 0.133 feet. 

Based on relationships, V = Q/A = 3.61 ft/s, the following are obtained: 

Woodward's Equation 2.160 

( ) ( )

( ) ( ) ft153.0350513

4312.32

61.3

g

Z

2

i0

c

2

rr

r

V =∆ =− =−

Ippen and Drinker's Equation 2.162 

ft158.0

431x2

163 1

1

431

163x2

x

2.32x2

61.3

r2

W 1

1W2

g2

Z 2

2

2

c

c

2

r

V =

































−

=



































	









 −

=∆







 







Ippen and Drinker’s Equation 2.163 

ft151.0

x12

163 1

1

431

163x2

x

2.32x2

61.3

12

1 W

1W2

g2

Z

r 431

r

V

2

2

2

2

c

2

c

2

=





























−

=





























−

=∆ 

Combined Free and Forced Vortex Equation 2.165 

Vmax = 4.90 ft/s 

( ) ft236.0

513

431

431

350 2

2.32x2

90.4 2

g2

V Z

2 2 2

o

c

2

c

i

2

max

2

r

r

r

r =

















= −−





















=∆ −− 



 



 



 



 





























 

The equations give comparable results (0.133 to 0.236 ft). Equation 2.158, which integrates 

across the section using the velocity distribution is the most exact. But using the value 0.24 ft 

provides a safety factor. 

2.15.4 PROBLEM 4 Maximum Stream Constriction Without Causing Backwater 

(Neglecting Energy Losses) 

A stream is rectangular in shape and 100 ft wide. The design discharge is 5,000 cfs and the 

uniform depth for this discharge is 10 ft. Neglecting energy losses what is the maximum amount 

of constriction that a bridge can impose without causing backwater. 
2.86 

The upstream flow rate per unit width (q) is: 

ft/cfs50

100

000,5

W

Q

q === 

The average velocity (V) is: 

s/ft00.5

000,1

000,5

A

Q V ===

The specific head (H) is: 

ft39.1010

4.64 y g2

H V 5 2 2=+=+=

According to Section 2.6.2 the maximum unit discharge for a given specific head (specific head 

equals to a constant) occurs at the critical depth yc where the Froude number is 1. Equation 

2.149 the critical depth yc for this specific head is: 

g2

2 H

3

2

g

q V y

2

c

2

max

3/1

c = == 













 

Solving for qm

39.10x ft/cfs4.103

3

2 H 2.32

3

2

gq

2/1 3 2/1 3

max = 













 



	 




 = 













 



	 




 =

Therefore, the width of the channel can be contracted until the unit discharge q is a maximum. 

The minimum width (maximum constriction) is: 

ft3.48

4.103

000,5Q

q

W max

min ===

and the maximum constriction is 100 – 48.3 = 51.7 ft 

This contraction causes the flow to go to critical. This results in an undulating hydraulic jump 

downstream. Also, when energy losses are considered there will be some backwater at the 

constriction. 

2.15.5 PROBLEM 5 Maximum Elevation of a Grade Control Structure 

Without Backwater (Neglecting Energy Losses) 

A low grade control structure (check dam) is to be placed across a stream downstream of a 

highway bridge. The stream is degrading. The purpose of the check dam is to maintain the 
2.87 

elevation of the water surface at the bridge, cause deposition of bed material, protect the abutments 

and piers from long term degradation and contraction scour and increase the elevation of the be for 

local scour. There is to be no increase in water surface elevation upstream of the bridge at the 

design discharge. The energy losses are expected to be small so they can be neglected. 

At the bridge, the design flood discharge is 5000 cfs. The river is 100 ft wide and has a uniform 

flow depth of 10 ft for the design discharge. 

What is the maximum height of the structure that will not cause backwater at the bridge at 

the design discharge? 

The unit discharge (q) in the river at design flood discharge is: 

ft/cfs50

100

000,5

W

Q

q ===

 

The average velocity (V) is 

sec/ft00.5

000,1

000,5

A

Q V ===

The specific head (H) is 

 

ft39.1010

4.64 y g2

H V 5 2 2=+=+=

As a first approximation assume no energy loss in the bridge reach. Then at the check dam, the 

elevation of the total energy line is 10.39 ft above the bed (Figure 2.44). At the dam: 

Hmin + ∆Zmax = 10.39 ft 

Figure 2.44. Sketch of backwater curve over check dam. 
2.88 

That is, the dam can be built to a height of ∆Zmax which decreases the specific head at the dam to 

Hmin. From Section 2.6.3 and Equation 2.147 

H y min c 2

3 =

From Equation 2.140 

ft27.4

2.32

50

g

q 2

3/1 2

3/1

c y = = = 



 



















 

Then 

ft40.6ft27.4

2

3

2

3 Hmin yc == =

and 

∆Zmax = 10.39 - 6.40 = 4.0 ft 

If the structure is built to a crest elevation 4.0 ft above the bed, critical flow will occur at the dam for 

a flow of 5,000 cfs and larger and the dam will cause no backwater. At lower discharges the at the 

check dam will cause backwater. However, discharges larger than 5,000 cfs check dam will not 

cause backwater. Local scour downstream of the check dam should be evaluated. HEC-23 has 

am problem for determining this scour. 

How much backwater will the dam cause for a flow of 1000 cfs if the normal depth for this 

discharge is 5 ft and the dam height is 4.0 ft? 

Upstream of the dam the discharge for foot of width (q) is, 

ft/cfs10

100

000,1

W

Q

q === 

The average velocity (V) is 

sec/ft00.2

5

10

y

q V ===

 

At the check dam the flow is critical so from Equation 2.142 

ft46.1

2.32

10

g

q 2

3/1 2

3/1

c y = = = 



 




















2.89 

and from Equation 2.144 

ft19.2ft46.1

2

3

2

3 Hmin yc == =

 

Assuming no energy loss, the specific head upstream of the dam is 

H = Hmin + ∆Z = 2.19 + 4.0 = 6.19 ft 

 

To determine the depth of flow upstream of the dam (y) solve the specific head equation (Equation 

2.137) 

ft19.6y

4.64

y

g2

H

y

10

y

q

2

2

2

2

=+= =+

or 

y

3

 - 6.19 y2 + 102/64.4 = 0 

The solution is 

y = 6.14 ft 

As the normal depth is only 5 ft, the backwater is 

∆y = 6.14 - 5.00 = 1.14 ft 

That is, the depth upstream of the dam is increased 1.14 ft by the 4.0 ft high dam when the flow is 

1,000 cfs. 
2.90 

(page intentionally left blank) 
 3.1 

CHAPTER 3

FUNDAMENTALS OF ALLUVIAL CHANNEL FLOW

3.1 INTRODUCTION

Most streams that a highway will cross or encroach upon are alluvial. That is, the rivers are 

formed in cohesive or non-cohesive materials that have been, and can still be, transported by 

the stream. The non-cohesive material generally consists of silt, sand, gravel, or cobbles, or 

any combination of these sizes. Silt generally is not present in appreciable quantities in 

streams having non-cohesive boundaries. Cohesive material consists of clays (sizes less 

than 0.004 mm) forming a binder with silts and sand. Because of the electro-chemical 

bonding between clay particles, clays are more resistant to erosion than silts. 

In alluvial rivers, bed configuration and resistance to flow are a function of the flow and can 

change to increase or decrease the velocity and water surface level. The river channel can 

shift its location so that the crossing or encroachment is unfavorably located with respect to 

the direction of flow. The moveable boundary of the alluvial river adds another dimension to 

the design problem and can compound environmental concerns. Therefore, the design of 

highway crossings and encroachments in the river environment requires knowledge of the 

mechanics of alluvial channel flow. 

This chapter presents the fundamentals of alluvial channel flow. It covers properties of 

alluvial material, methods of measuring properties of alluvial materials, flow in sandbed 

channels, prediction of bed forms, Manning's n for sandbed and other natural streams, how 

bed-form changes affect highways in the river environment, beginning of motion, flow in 

coarse-material streams, and physical measurement and determination of sediment 

discharge in the field. These fundamentals of alluvial channel flow are used in later chapters 

to develop design considerations for highway crossings and encroachments in river 

environments. 

In Chapter 4, sediment transport and physical and computer modeling of sediment transport 

will be covered. 

3.2 SEDIMENT PROPERTIES AND MEASUREMENT TECHNIQUES

A knowledge of the properties of the bed material particles is essential, as they indicate the 

behavior of the particles in their interaction with the flow. Several of the important bed 

material properties are discussed in the following sections (U.S. Interagency Subcommittee 

1941, 1943, 1957; Richardson 1971). 

3.2.1 Particle Size

Of the various sediment properties, physical size has by far the greatest significance to the 

hydraulic engineer. The particle size is the most readily measured property, and other 

properties such as shape, fall velocity and specific gravity tend to vary with size in a roughly 

predictable manner. In general, size represents a sufficiently complete description of the 

sediment particle for many practical purposes. 
 3.2 

Particle size Ds may be defined by its volume, diameter, weight, fall velocity, or sieve mesh 

size. Except for volume, these definitions also depend on the shape and density of the 

particle. The following definitions are commonly used to describe the particle size (U.S. 

Interagency Subcommittee 1943): 

1. Nominal diameter - The diameter of a sphere having the same volume as the particle. 

2. Sieve diameter - The diameter of a sphere equal to the length of the side of a square 

sieve opening through which measured quantities (by weight) of the sample will pass. As 

an approximation, the sieve diameter is equal to the nominal diameter. 

3. Sedimentation diameter - The diameter of a sphere with the same fall velocity and 

specific gravity as the particle in the same fluid under the same conditions. 

4. Standard fall diameter - The diameter of a sphere that has a specific gravity of 2.65 and 

also has the same terminal settling velocity as the particle when each is allowed to settle 

alone in quiescent, distilled water of infinite extent and at a temperature of 24°C. 

In general, sediments have been classified into boulders, cobbles, gravels, sands, silts, and 

clays on the basis of their nominal or sieve diameters. The size range in each general class 

is given in Table 3.1. The non-cohesive material generally consists of silt (0.004-0.062 mm), 

sand (0.062 - 2.0 mm), gravel (2.0 - 64 mm), or cobbles (64-250 mm). 

The boulder class (250 - 4000 mm) is generally of little interest in sediment problems. The 

cobble and gravel class plays a considerable role in the problems of local scour and 

resistance to flow and to a lesser extent in bed load transport. The sand class is one of the 

most important in alluvial channel flow. The silt and clay class is of considerable importance 

in the evaluation of stream sediment loads, bank stability and problems of seepage and 

consolidation. 

3.2.2 Particle Shape

Generally speaking, shape refers to the overall geometrical form of a particle. Sphericity is 

defined as the ratio of the surface area of a sphere of the same volume as the particle to the 

actual surface area of the particle. Roundness is defined as the ratio of the average radius 

of curvature of the corners and edges of a particle to the radius of a circle inscribed in the 

maximum projected area of the particle. However, because of simplicity and effectiveness of 

correlation with the behavior of particles in flow, the most commonly used parameter to 

describe particle shape is the Corey shape factor, Sp, (Albertson 1953) defined as: 

ba

c Sp



 = (3.1) 

where: a, b, and c are the dimensions of the three mutually perpendicular axes of a particle 

a the longest; b the intermediate; and c the shortest axis. 
 3.3 

Table 3.1. Sediment Grade Scale (Brown 1950). 

Size 

Approximate 

Sieve Mesh 

Openings 

Per Inch 

Millimeters Microns Inches Tyler 

U.S. 

Standard 

Class 

4000-2000 ----- ----- 160-80 ----- ----- Very large boulders 

2000-1000 ----- ----- 80-40 ----- ----- Large boulders 

1000-500 ----- ----- 40-20 ----- ----- Medium boulders 

500-250 ----- ----- 20-10 ----- ----- Small boulders 

250-130 ----- ----- 10-5 ----- ----- Large cobbles 

130-64 ----- ----- 5-2.5 ----- ----- Small cobbles 

64-32 ----- ----- 2.5-1.3 ----- ----- Very coarse gravel 

32-16 ----- ----- 1.3-0.6 ----- ----- Coarse gravel 

16-8 ----- ----- 0.6-0.3 2 1/2 ----- Medium gravel 

8-4 ----- ----- 0.3-0.16 5 5 Fine gravel 

4-2 ----- ----- 0.16-0.08 9 10 Very fine gravel 

2-1 2.00-1.00 2000-1000 ----- 16 18 Very coarse sand 

1-1/2 1.00-0.50 1000-500 ----- 32 35 Coarse sand 

1/2-1/4 0.50-0.25 500-250 ----- 60 60 Medium sand 

1/4-1/8 0.25-0.125 250-125 ----- 115 120 Fine sand 

1/8-1/16 01.25-0.062 125-62 ----- 250 230 Very fine sand 

1/16-1/32 0.062-0.031 62-31 ----- Coarse silt 

1/32-1/64 0.031-0.016 31-16 ----- Medium silt 

1/64-1/128 0.016-0.008 16-8 ----- Fine silt 

1/128-1/256 0.008-0.004 8-4 ----- Very fine silt 

1/256-1/512 0.004-0.0020 4-2 ----- Coarse clay 

1/512-1/1024 0.0020-0.0010 2-1 ----- Medium clay 

1/1024-1/2048 0.0010-0.0005 1-0.5 ----- Fine clay 

1/2048-1/4096 0.0005-0.0002 0.5-0.24 ----- Very fine clay 
 3.4 

3.2.3 Fall Velocity

The prime indicator of the interaction of sediments in suspension within the flow is the fall 

velocity of sediment particles. The fall velocity of a particle is defined as the velocity of that 

particle falling alone in quiescent, distilled water of infinite extent. In most cases, the particle 

is not falling alone, and the water is not distilled or quiescent. Measurement techniques are 

available for determining the fall velocity of groups of particles in a finite field in fluid other 

than distilled water. However, little is known about the effect of turbulence on fall velocity. 

A particle falling at terminal velocity in a fluid is under the action of a driving force due to its 

buoyant weight and a resisting force due to the fluid drag. Fluid drag is the result of either 

the tangential shear stress on the surface of the particle, or a pressure difference on the 

particle or a combination of the two forces. The fluid drag on the falling particle FD is given 

by the drag equation 

2/ACF 2

sDD ωρ= (3.2) 

The buoyant weight of the particle Ws is: 

ss ρ−ρ= gV)(W p (3.3) 

where: 

CD = Coefficient of drag

ω = Terminal fall velocity of the particle 

As = Projected area of the particle normal to the direction of flow 

ρ = Fluid density 

ρs = Particle density 

g = Acceleration due to gravity 

Vp = Volume of the particle 

The area and volume can be written in terms of the characteristic diameter of the particle Ds

or: 

2 = DKA s1s (3.4) 

and 

3 = DKV s2p (3.5) 

Where the coefficients K1 and K2 depend on the shape of sediment particles. For example, 

K1 = π/4 and K2 = π/6 for spherical particles. If the particle is falling at its terminal velocity, FD

= Ws

2/ACgV)( 2

s sDp ωρ=ρ−ρ (3.6) 
 3.5 

By substituting Equations 3.4 and 3.5 into Equation 3.6, the expression: 

s

s

1

2

D

2 gD1 K

K

C

2















 − ρ

ρ =ω (3.7) 

is obtained. 

Four dimensionless variables describing fall velocity result from dimensional analysis of 

Equation 3.7: 

0

K

K

,,C, gDf

1

2s

D

s

2

=















ρ

ω ρ (3.8) 

The drag coefficient CD is dependent on the particle Reynolds number (Re = ρ ωDs/μ), the 

shape, and the surface texture of the particle, where μ is dynamic viscosity of the fluid. The 

ratio K2/K1 is usually replaced by the Corey shape factor Sp (Equation 3.1). 

The relations between the fall velocity of particles and the other variables are given in 

Figures 3.1 and 3.2. 

Figure 3.1. Drag coefficient CD vs particle Reynolds number Rep for spheres and natural 

sediments with shape factors Sp equal to 0.3, 0.5, 0.7, and 0.9. Also, sediment 

diameter Ds vs. fall velocity ω and temperature To (Brown 1950, p. 781). 
 3.6 

 Figure 3.2. Nominal diameter vs. fall velocity (Temperature = 24°C) (Guy 1977, 

 U.S. Interagency 1957b). 

3.2.4 Sediment Size Distribution

Four methods of obtaining sediment size distribution are described below: Sieve analysis, 

visual accumulation (VA) tube analysis, pebble count method, and pipette analysis. Each 

method for size distribution analysis is appropriate for only a particular range of particle sizes 

(Table 3.2). All together the four methods provide a means of obtaining particle size 

distributions for most bed material samples. 

Table 3.2. Guide to Size Range for Different Types of Size Analysis. 

 

Size 

Range 

Analysis 

Concentration 

(mg/l) 

Quantity of 

Sediment (g) 

or Pebbles 

Sieves 0.062-32 mm ----- 100-500 

VA tube 0.062-2 mm ----- 0.05-15.0 

Pipette 0.002-0.062 mm 2,000-5,000 1.0-5.0 

Pebble Count 12-1000 mm ----- 100 pebbles 

If the sediment sample to be analyzed (bed material or suspended sediment) has 

considerable fine material (Ds < .062 mm) it must be separated prior to analysis. To separate 

the coarser from the finer sediment, the sediment should be wet-sieved using distilled water 

and a 250-mesh (0.062 mm) sieve. The material passing through the sieve can be analyzed 

by pipette analysis if further breakdown of the fine sediment is desired, or dried and included 
 3.7 

as percent finer than 0.062 mm with the analysis of the coarser material. If it is going to be 

dry-sieved, the material retained on the sieve is oven dried for one hour after all visible water 

has been evaporated. If the material is to be analyzed by wet-sieving or with the 

accumulation tube, it is not dried. 

Sieve Size Distribution in the sand and gravel range is generally determined by passing the 

sample through a series of sieves of mesh size ranging from 32 mm to 0.062 mm. A 

minimum of about 100 grams of sand is required for an accurate sieve analysis. More is 

required if the sample contains particles of 1.0 mm or larger. Standard methods employed in 

soil mechanics are suitable for determining the sieve sizes of sand and gravel sediment 

samples. 

Visual Accumulation (VA) Tube is used for determining the size distribution of the sand 

fraction of sediment samples (0.062 mm < Ds < 2.0 mm). It is a fast, economical, and 

accurate means of determining the fall velocity or fall diameter of the sediment. The 

equipment for the visual accumulation tube analysis consists of: (1) a glass funnel about 250 

mm (9.8 in.) long; (2) a rubber tube connecting the funnel and the main sedimentation tube, 

with a special clamping mechanism serving as a "quick acting" valve; (3) glass sedimentation 

tubes having different sized collectors; (4) a tapping mechanism that strikes against the glass 

tube and helps keep the accumulation of sediment uniformly packed; (5) a special recorder 

consisting of a cylinder carrying a chart that rotates at a constant rate and a carriage that can 

be moved vertically by hand on which is mounted a recording pen and an optical instrument 

for tracking the accumulation; and (6) the recorder chart which has a printed form 

incorporating the fall diameter calibration. 

In the visual accumulation tube method, the particles start falling from a common source and 

become stratified according to settling velocities. At a given instant, the particles coming to 

rest at the bottom of the tube are of one "sedimentation size" and are finer than particles that 

have previously settled out and are coarser than those remaining in suspension. 

It has been shown that particles of a sample in the visual tube settle with greater velocities 

than the same particles falling individually because of the effect of mutual interaction of the 

particles. The visual accumulation tube apparatus is calibrated to account for the effects of 

this mutual interaction and the final results are given in terms of the standard fall diameter of 

the particles. 

Visual accumulation tube method may not be suitable for some streams that transport large 

quantities of organic materials such as root fibers, leaf fragments, and algae. Also, extra 

care is needed when a stream transports large quantities of heavy or light minerals such as 

taconite or coal. The method is explained in detail by Guy (1977). 

Pebble Count Method is used to obtain the size distribution of coarse bed materials (gravel 

and cobbles) which are too large to be sieved. Very often the coarser material is underlain 

by sands. Then, the underlying sands are analyzed by sieving. The two classes of bed 

material are either combined into a single distribution or used separately. The large material 

sizes are measured in situ by laying out a square grid. Within the grid, all the particle sizes 

are measured and counted by size intervals. For large samples a random selection of 

particles in the various classes is appropriate to develop frequency histograms of sediment 

sizes. 
 3.8 

Square-Surface Sample is obtained by picking up and counting all the surface pebbles in 

each predetermined size class within a small enclosed area of the bed. The area is taken to 

be representative of the whole channel bed. 

The pebble count method entails measurement of randomly selected particles in the field, 

often under difficult conditions. Therefore, use of the Zeiss Particle-Size Analyzer should be 

considered (Ritter and Helley 1968). For this method, a photograph of the stream bed is 

made, preferably at low flow, with a 35 mm camera supported by a tripod about 2 m (6.6 ft) 

above the stream bed, the height depending on the size of the bed material. A reference 

scale, such as a steel tape or a surveyor's rod must appear in the photograph. The 

photographs are printed on the thinnest paper available. An iris diaphragm, illuminated from 

one side, is imaged by a lens onto the plane of a Plexiglass plate. By adjusting the iris 

diaphragm the diameter of the sharply defined circular light spot appearing on the 

photograph can be changed and its area made equal to that of the individual particles. As 

the different diameters are registered, a puncher marks the counted particle on the 

photograph. An efficient operation can count up to 1,000 particles in 30 minutes. 

In the line sampling method of pebble count sampling, a line is laid out or placed either 

across or along the stream. Particles are picked at random intervals along the line and 

measured. The measured particles are classified as to size or weight and a percent finer 

curve or table is prepared. Usually 100 particles are sufficient to give an accurate 

classification of the size distribution of coarse materials. 

Pipette Method of determining gradation of sizes finer than 0.062 mm is one of the most 

widely accepted techniques utilizing the Oden theory and the dispersed system of 

sedimentation. The upper size limit of sediment particles which settle in water according to 

Stokes is about 1/16 mm or 0.062 mm. This corresponds to the lower size limit which can be 

determined readily by sieves. This size is the division between sand and silt (Table 3.1) and 

is an important division in many phases of sediment phenomena. 

The fundamental principle of the pipette method is to determine the concentration of a 

suspension in samples withdrawn from a predetermined depth as a function of settling time. 

Particles having a settling velocity greater than that of the size at which separation is desired 

will settle below the point of withdrawal after elapse of a certain time. The time and depth of 

withdrawal are predetermined on the basis of Stokes law. 

Satisfactory use of the pipette method requires careful and precise operation to obtain 

maximum accuracy in each step of the procedure. Also, for routine analysis, special 

apparatus can be set up for the analysis of a large number of samples. A complete 

description of a laboratory setup and procedure for this method is given by Guy (1969). 

Frequency Curves. The presentation of sediment size analysis is made in various formats. 

A histogram is a graphical representation of the number, weight, or volume percentage of 

items in given class intervals. An example of a histogram is shown in Figure 3.3a. The 

abscissa scale represents the class intervals, usually in geometric progression, and the 

ordinate scale represents either actual concentration or percent (by number, volume, or 

weight) of the total sample contained in each class interval. If the class intervals are small, 

the shape of the histogram will approach a continuous curve. The successive sizes 

employed in the size analysis of sediment are usually in ratios of 2 or 2 . 
 3.9 

Figure 3.3. Definition sketches for size-frequency characteristics of sediments (Rouse 1940). 
 3.10 

When the ordinates of successive classes are added and plotted against the upper limit of 

the size class, the cumulative percent finer distribution diagram is obtained (Figure 3.3). In 

this diagram, the abscissa scale (usually logarithmic) represents the intervals of the size 

scale and the ordinate scale is the cumulative percent by weight of the sample up to (or 

percent finer than) the size in question. From the cumulative percent finer the D35, D50, D85, 

Di

, etc. sizes can be determined. 

In a size frequency distribution curve, it is possible to choose certain particle sizes as 

representing significant values, such as particles just larger than one-fourth of the distribution 

D25 (the first quartile), and particles just larger than three-fourths of the distribution D75 (the 

third quartile). Measures of spread are based on differences or ratios between the two 

quartiles. Quartile measures are confined to the central half of the frequency distribution and 

the values obtained are not influenced by larger or smaller sizes. Quartile measures are very 

readily computed, and most of the data may be obtained directly from the cumulative curve 

by graphic means. 

In contrast to quartile measures, moment measures are influenced by each individual size 

class in the distribution. The first moment of a frequency curve is its center of gravity and is 

called the arithmetic mean and is the average size of the sediment. The second moment is a 

measure of the average spread of the curve and is expressed as the standard deviation of 

the distribution. 

Commonly, the size distribution of natural sediments plots as a straight line on log probability 

paper. If this is true, then a natural sediment is completely described by the median diameter 

(D50 the size of sediment of which 50 percent is finer) and the slope of the cumulative 

frequency line on log probability paper. The slope of this line is proportional to the spread of 

the size distribution in a sediment sample. It is computed with the expression 





 





+=

50

84

16

50

D

D

D

D

2

1 G (3.9) 

where: 

G = Gradation coefficient 

Dx = Sediment diameter particle of which x percent of sample is finer 

The grain roughness used in velocity equations is taken as the D80, D85, or D90. 

In studies of scour below culvert outlets, Stevens (1968) was able to consolidate a wide 

range of scour data by employing the expression 

3/1

3

i

10

1i

10

D

k



























= = (3.10) 

for the effective or representative grain size of well-graded materials. 
 3.11 

Here 

2

DD )1i(D 100

i

+ == (3.11) 

2

DD )2i(D 10 20

i

+ ==

2

DD )10i(D 90 100

i

+ ==

The terms D0, D10, ..., D100, are the sieve diameters of the riprap for which 0 percent, 10 

percent, ..., 100 percent of the material (by weight) is finer. Stevens’ equation is the 

equivalent to utilizing the arithmetic average of the sum of the weights of the individual 

particles. 

The effective diameter Dm is also used in sediment studies. It is defined by the following 

equation: 

100

Dp D

n

1i sii

m

= = (3.12) 

Where pI is the percentage by weight of that fraction of the sediment with geometric mean 

size Dsi. The geometric mean size is the square root of the product of the end points of a 

given size range. 

3.2.5 Specific Weight 

Specific weight is weight per unit volume. In the English system of units, specific weight is 

usually expressed in units of pounds per cubic foot and in the metric system, in grams per 

cubic centimeter. In connection with granular materials such as soils, sediment deposits, or 

water sediment mixtures, the specific weight is the weight of solids per unit volume of the 

material including its voids. The measurement of the specific weight of sediment deposits is 

determined simply by measuring the dry weight of a known volume of the undisturbed 

material. 

3.2.6 Porosity 

The porosity of granular materials is the ratio of the volume of void space to the total volume 

of an undisturbed sample. To determine porosity, the volume of the sample must be 

obtained in an undisturbed condition. Next, the volume of solids is determined either by 

liquid displacement or indirectly from the weight of the sample and the specific gravity of 

material. The void volume is then obtained by subtracting the volume of solids from the total 

volume. 
 3.12 

3.2.7 Cohesion 

Cohesion is the force by which particles of clay are bound together. This force is the result of 

ionic attraction among individual particles, and is a function of the type of mineral, particle 

spacing, salt concentration in the fluid, ionic valence, and hydration and swelling properties 

of the constituent minerals. Cohesive soils are composed of silts and clay. Their size 

classification is given in Table 3.1. In the unified soil classification system, silt and clay soils 

have more than 50 percent by weight of particles passing the 0.075 mm sieve opening. 

However, cohesive soils are not classified by grain size but by their degree of plasticity, 

which is measured by Atterberg limits. 

Clays are alumino-silicate crystals composed of two basic building sheets, the tetrahedral 

silicate sheet and the octahedral hydrous aluminum oxide sheet. Various types of clays 

result from different configurations of these sheets. The two main types of clays are kaolinite 

and montmorillonite. Kaolinite crystals are large (70 to 100 layers thick), held together by 

strong hydrogen bonds, and are not readily dispersible in water. Montmorillonite crystals are 

small (3 layers thick) held together by weak bonds between adjacent oxygen layers and are 

readily dispersible in water into extremely small particles. 

Several laboratory and field measurement techniques are available for determining the 

magnitude of cohesion, or shear strength, of clays. Among these, the vane shear test, which 

is performed in the field is one of the simplest. The vane is forced into the ground and then 

the torque required to rotate the vane is measured. The shear strength is determined from 

the torque required to shear the soil along the vertical and horizontal edges of the vane. 

Briaud et al. (1999) describe equipment and methods developed to determine the erosion 

rate of cohesive soils. Erosion rate is defined as the vertical distance scoured per unit of 

time. The erosion rate, expressed as mm/hr, is related to shear stress, given in N/m2

, 

imposed at the soil water interface using an erosion function apparatus (EFA). The EFA 

develops an erosion rate-shear stress curve for a soil sample taken with an ASTM standard 

Shelby tube with a 3.0 inches (76.2) mm outside diameter. The erosion rate-shear stress 

curve for samples taken at a site can then be used to determine the scour depth as a 

function of time. 

3.2.8 Angle of Repose 

The angle of repose is the maximum slope angle upon which non-cohesive material will 

reside without moving. It is a measure of the intergranular friction of the material. Simons 

developed Figure 3.4 for the angle of repose for dumped granular material. 

3.3 FLOW IN SANDBED CHANNELS

Most larger streams flow on sandbeds for the greater part of their length. Thus, there are 

potentially many more opportunities for highway crossings or encroachments on sandbed 

streams than in cohesive or gravel streams. In sandbed rivers, the sand material is easily 

eroded and is continually being moved and shaped by the flow. The mobility of the sandbed 

creates problems for the safety of any structure placed in or over the stream, for the 

protection of private property along these streams, and in the preservation and enhancement 

of the stream environment. 
 3.13 

Figure 3.4. Angle of repose of non-cohesive materials (Simons 1955). 

The interaction between the flow of the water-sediment mixture and the sandbed creates 

different bed configurations which change the resistance to flow and rate of sediment 

transport. The gross measures of channel flow, such as the flow depth, river stage, bed 

elevation and flow velocity, change with different bed configurations. In the extreme case, 

the change in bed configuration can cause a three-fold change in resistance to flow and a 

10-to-15 fold change in concentration of bed sediment transport. For a given discharge and 

channel width, a three-fold increase in Manning's n results in a doubling of the flow depth. 

The interaction between the flow and bed material and the interdependency among the 

variables makes the analysis of flow in alluvial sandbed streams extremely complex. 

However, with an understanding of the different types of bed forms that may occur and a 

knowledge of the resistance to flow and sediment transport associated with each bed form, 

the river engineer can better analyze alluvial channel flow. 

3.3.1 Regimes of Flow in Alluvial Channels

The flow in alluvial channels is divided into lower and upper flow regimes separated by a 

transition zone (Simons and Richardson 1963, 1966). These two flow regimes are 

characterized by similarities in the shape of the bed configuration, mode of sediment 

transport, process of energy dissipation, and phase relation between the bed and water 

surfaces. The two regimes and their associated bed configurations shown in Figure 3.5 are: 

Lower flow regime: (1) ripples; (2) dunes with ripples superposed; and (3) dunes 

Transitional Flow Regime: The bed roughness ranges from dunes to plane bed or 

antidunes. 

Upper Flow Regime: (1) plane bed; (2) antidunes, a) with standing waves, b) with 

breaking antidunes; and (3) chutes and pools. 
 3.14 

 Lower Flow Regime Upper Flow Regime 

Figure 3.5. Forms of bed roughness in sand channels (Simons and Richardson 1963, 1966). 

Lower Flow Regime. In the lower flow regime, resistance to flow is large and sediment 

transport is small. The bed form is either ripples or dunes or some combination of the two. 

The water-surface undulations are out of phase with the bed surface, and there is a relatively 

large separation zone downstream from the crest of each ripple or dune. The most common 

mode of bed material transport is for the individual grains to move up the back of the ripple or 

dune and avalanche down its face. After coming to rest on the downstream face of the ripple 

or dune, the particles remain there until exposed by the downstream movement of the dunes; 

they repeat this cycle of moving up the back of the dune, avalanching, and storage. Thus, 

most movement of the bed material particles is in steps. The velocity of the downstream 

movement of the ripples or dunes depends on their height and the velocity of the grains 

moving up their backs. 

Transition. The bed configuration in the transition zone is erratic. It may range from that 

typical of the lower flow regime to that typical of the upper flow regime, depending mainly on 

antecedent conditions. If the antecedent bed configuration is dunes, the depth or slope can 

be increased to values more consistent with those of the upper flow regime without changing 

the bed form; or, conversely, if the antecedent bed is plane, depth and slope can be 

decreased to values more consistent with those of the lower flow regime without changing 

the bed form. Often in the transition from the lower to the upper flow regime, the dunes 

decrease in amplitude and increase in length before the bed becomes plane (washed-out 

dunes). Resistance to flow and sediment transport also have the same variability as the bed 

configuration in the transition. This phenomenon can be explained by the changes in 

resistance to flow and, consequently, the changes in depth and slope as the bed form 

changes. Resistance to flow is small for flow over a plane bed; so the shear stress 

decreases and the bed form changes to dunes. The dunes cause an increase in resistance 

to flow which increases the shear stress on the bed and the dunes wash out forming a plane 
 3.15 

bed, and the cycle continues. It was the transition zone, which covers a wide range of shear 

values, that Brooks (1958) was investigating when he concluded that a single-valued function 

does not exist between velocity or sediment transport and the shear stress on the bed. 

Upper Flow Regime. In the upper flow regime, resistance to flow is small and sediment 

transport is large. The usual bed forms are plane bed or antidunes. The water surface is in 

phase with bed surface except when an antidune breaks, and normally the fluid does not 

separate from the boundary. A small separation zone may exist downstream from the crest 

of an antidune prior to breaking. Resistance to flow is the result of grain roughness with the 

grains moving, of wave formation and subsidence, and of energy dissipation when the 

antidunes break. The mode of sediment transport is for the individual grains to roll almost 

continuously downstream in sheets one or two grain diameters thick; however, when 

antidunes break, much bed material is briefly suspended, then movement stops temporarily 

and there is some storage of the particles in the bed. 

3.3.2 Bed Configuration

The bed configurations (roughness elements) that commonly form in sand bed channels are 

plane bed without sediment movement, ripples, ripples on dunes, dunes, plane bed with 

sediment movement, antidunes, and chutes and pools. These bed configurations are listed 

in their order of occurrence with increasing values of stream power (VγyoS) for bed materials 

having D50 less than 0.6 mm. For bed materials coarser than 0.6 mm, dunes form instead of 

ripples after beginning of motion at small values of stream power. The relation of bed form to 

water surface is shown in Figure 3.6. 

The different forms of bed-roughness are not mutually exclusive in time and space in a 

stream. Different bed-roughness elements may form side-by-side in a cross-section or reach 

of a natural stream, giving a multiple roughness; or they may form in time sequence, 

producing variable roughness. 

Figure 3.6. Relation between water surface and bed configuration (Richardson et al. 1975). 
 3.16 

Multiple roughness is related to variations in shear stress (γyoS) in a channel cross-section. 

The greater the width-depth ratio of a stream, the greater is the probability of a spatial 

variation in shear stress, stream power or bed material. Thus, the occurrence of multiple 

roughness is closely related to the width-depth ratio of the stream. Variable roughness is 

related to changes in shear stress, stream power, or reaction of bed material to a given 

stream power over time. A commonly observed example of the effect of changing shear 

stress or stream power is the change in bed form that occurs with changes in depth during a 

runoff event. Another example is the change in bed form that occurs with change in the 

viscosity of the fluid as the temperature or concentration of fine sediment varies over time. It 

should be noted that a transition occurs between the dune bed and the plane bed; either bed 

configuration may occur for the same value of stream power (Figure 3.7). 

A relation between stream power, velocity, and bed configuration is shown in Figure 3.7. 

The relation pertains to one sand size and was determined in the 2.4 m (8-foot) flume at 

Colorado State University. 

 Figure 3.7. Change in velocity with stream power for a sand with D50 = 0.19 mm 

 (Simons and Richardson 1966). 

In the following paragraphs bed configurations and their associated flow phenomena are 

described in the order of their occurrence with increasing stream power. 
 3.17 

3.3.3 Plane Bed Without Sediment Movement

Plane bed without movement has been studied to determine the bed configuration that would 

form after beginning of motion. After the beginning of motion, for flat slopes and low velocity, 

the plane bed will change to ripples for sand material smaller than 0.6 mm, and to dunes for 

coarser material. Resistance to flow is small for a plane bed without sediment movement 

and is due solely to the sand grain roughness. Values of Manning's n range from 0.012 to 

0.014 depending on the size of the bed material. 

If the bed material of a stream is not moving, the bed configuration will be a remnant of the 

bed configuration formed when sediment was moving. The bed configurations after the 

beginning of motion may be those illustrated in Figure 3.5, depending on the flow and bed 

material. Prior to the beginning of motion, the problem of resistance to flow is one of 

rigid-boundary hydraulics. After the beginning of motion, the problem relates to defining bed 

configurations and resistance to flow. 

3.3.4 Ripples

Ripples are small triangle-shaped elements having gentle upstream slopes and steep 

downstream slopes. Length ranges from 0.12 m to 0.6 m (0.4 ft to 2 ft) and height from 0.01 

m to 0.06 m (0.03 ft to 0.2 ft) (Figure 3.5). Resistance to flow is relatively large (with 

Manning's n ranging from 0.018 to 0.030). There is a relative roughness effect associated 

with a ripple bed and the resistance to flow decreases as flow depth increases. The ripple 

shape is independent of sand size and at large values of Manning's n the magnitude of grain 

roughness is small relative to the form roughness. The length of the separation zone 

downstream of the ripple crest is about ten times the height of the ripple. Ripples cause very 

little, if any, disturbance on the water surface, and the flow contains very little suspended bed 

material. The bed material discharge concentration is small, ranging from 10 to 200 ppm. 

3.3.5 Dunes

When the shear stress or the stream power is increased for a bed having ripples (or a plane 

bed without movement, if the bed material is coarser than 0.6 mm), sand waves called dunes 

form on the bed. At smaller shear-stress values, the dunes have ripples superposed on their 

backs. These ripples disappear at larger shear values, particularly if the bed material is 

coarse sand with D50 > 0.4 mm. 

Dunes are large triangle-shaped elements similar to ripples (Figure 3.5). Their lengths range 

from 0.6 m (2 ft) to many tens of meters (hundreds of feet), depending on the scale of the 

flow system. Dunes that formed in the 2.4 m (8-foot) wide flume used by Simons and 

Richardson (1963, 1966) ranged from 0.6 to 3 m (2 to 10 ft) in length and from 0.06 to 0.3 m 

(0.2 to 1 ft) in height; whereas, those described by Carey and Keller (1957) in the Mississippi 

River were 100 to 200 m (300 to 700 ft) long and as much as 12 m (40 ft) high. The 

maximum amplitude to which dunes can develop is approximately the average depth. 

Hence, in contrast with ripples, the amplitude of dunes can increase with increasing depth of 

flow. With dunes, the relative roughness can remain essentially constant or even increase 

with increasing depth of flow. 
 3.18 

Field observations indicate that dunes can form in any sand channel, irrespective of the size 

of bed material, if the stream power is sufficiently large to cause general transport of the bed 

material without exceeding a Froude number of unity. 

Resistance to flow caused by dunes is large. Manning's n ranges from 0.020 to 0.040. The 

form roughness for flow with dunes is equal to or larger than the sand grain roughness. 

Dunes cause large separation zones in the flow. These zones, in turn, cause boils to form 

on the surface of the stream. Measurements of flow velocities within the separation zone 

show that velocities in the upstream direction exist that are l/2 to l/3 the average stream 

velocity. Boundary shear stress in the dune trough is sometimes sufficient to form ripples 

oriented in a direction opposite to that of the primary flow in the channel. With dunes, as with 

any tranquil flow over an obstruction, the water surface is out of phase with the bed surface 

(Figure 3.6). 

3.3.6 Plane Bed With Movement

As the stream power of the flow increases further, the dunes elongate and reduce in 

amplitude. This bed configuration is called the transition or washed out dunes. The next bed 

configuration with increased stream power is plane bed with movement. Dunes of fine sand 

(low fall velocity) are washed out at lower values of stream power than are dunes of coarser 

sand. With coarse sands larger slopes are required to affect the change from transition to 

plane bed and the result is larger velocities and larger Froude numbers. In flume studies 

with fine sand, the plane-bed condition commonly exists after the transition and persists over 

a wide range of Froude numbers (0.3 < Fr < 0.8). If the sand is coarse and the depth is 

shallow, however, transition may not terminate until the Froude number is so large that the 

subsequent bed form may be antidunes rather than plane bed. In natural streams, because 

of their greater depths, the change from transition to plane bed may occur at a much lower 

Froude number than in flumes. Manning's n for plane bed sand channels range from 0.010 

to 0.013. 

3.3.7 Antidunes

Antidunes form as a series or train of inphase (coupled) symmetrical sand and water waves 

(Figure 3.5). The height and length of these waves depend on the scale of the flow system 

and the characteristics of the fluid and the bed material. In a flume where the flow depth was 

about 0.15 m (0.5 ft) deep, the height of the sand waves ranged from 0.01 to 0.15 m (0.03 to 

0.5 ft). The height of the water waves was 1.5 to 2 times the height of the sand waves and 

the length of the waves, from crest to crest, ranged from 1.5 to 3 m (5 to 10 ft). In natural 

streams, such as the Rio Grande or the Colorado River, much larger antidunes form. In 

these streams, surface waves 0.6 to 1.5 m (2 to 5 ft) high and 3 to 12 m (10 to 40 ft) long 

have been observed. 

Antidunes form as trains of waves that gradually build up from a plane bed and a plane water 

surface. The waves may grow in height until they become unstable and break like the sea 

surf or they may gradually subside. The former have been called breaking antidunes, or 

antidunes; and the latter, standing waves. As the antidunes form and increase in height, 

they may move upstream, downstream, or remain stationary. Their upstream movement led 

Gilbert (1914) to name them antidunes. 
 3.19 

Resistance to flow due to antidunes depends on how often the antidunes form, the area of 

the stream they occupy, and the violence and frequency of their breaking. If the antidunes 

do not break, resistance to flow is about the same as that for flow over a plane bed. If many 

antidunes break, resistance to flow is larger because the breaking waves dissipate a 

considerable amount of energy. With breaking waves, Manning's n may range from 0.012 to 

0.020. 

3.3.8 Chutes and Pools

At very steep slopes, alluvial-channel flow changes to chutes and pools (Figure 3.5). In the 

2.4 m (8-foot) wide flume at Colorado State University, this type of flow and bed configuration 

was studied using fine sands. The flow consisted of a long chute 3 to 9 m (10 to 30 ft) in 

which the flow was rapid and accelerating followed by a hydraulic jump and a long pool. The 

chutes and pools moved upstream at velocities of about 0.3 to 0.6 m (1 to 2 ft) per minute. 

The elevation of the sandbed varied within wide limits. Resistance to flow was large with 

Manning's n of 0.018 to 0.035. 

3.3.9 Regime of Flow, Bed Configuration, and Froude Number 

The change from lower to upper regime flow or the reverse (that is a change from dune bed 

to a plane bed or plane bed to a dune bed) is not related to the Froude number. However, 

standing wave and antidune bed configuration in the upper flow regime only occurs with a 

Froude number greater than 1.0 (Fr > 1.0), and ripples and dunes only occur in the lower 

flow regime at a Froude number less than 1.0 (Fr < 1.0) (Vanoni 1977). 

The misconception that the lower flow regime shifts to the upper flow regime at a Froude 

number of 1.0 (Fr = 1.0) results from studies made in small flumes where the depth is 

shallow and large velocities are needed in order to shift from a dune bed to a plane bed. In 

larger flumes and in rivers the shift occurs at Froude numbers as low as 0.2 (Simons and 

Richardson 1966, Richardson and Simons 1967, Nordine 1964, Richardson 1965, Dawdy 

1961). Figure 3.8 illustrates the relation between flow depth, Froude number and regimes of 

flow and Figure 3.9 conceptualizes the crossover from lower to upper flow regime in natural 

rivers. 

3.3.10 Bars

In natural channels, some other bed configurations are also found. These bed configurations 

are generally called bars and are related to the plan form geometry and the width of the 

channel (Figure 5.14). 

Bars are bed forms having lengths of the same order as the channel width or greater and 

heights comparable to the mean depth of the generating flow. Several different types of bars 

are observed. They are classified as: 

(1) Point Bars which occur adjacent to the inside banks of channel bends. Their shape 

may vary with changing flow conditions and motion of bed particles but they do not 

move relative to the bends; 
 3.20 

 Figure 3.8. Relation between regime of flow and depth for bed material with a median 

 size equal to or less than 0.35 mm. 

 Figure 3.9 Cross-over from lower to upper flow regime based on sand size and 

 Froude Number. 
 3.21 

(2) Alternate Bars which occur in somewhat straighter reaches of channels and tend to be 

distributed periodically along the reach, with consecutive bars on opposite sides of the 

channel. Their lateral extent is significantly less than the channel width. Alternate bars 

move slowly downstream; 

(3) Transverse Bars which also occur in straight channels. They occupy nearly the full 

channel width. They occur both as isolated and as periodic forms along a channel, and 

move slowly downstream; and 

(4) Tributary Bars which occur immediately downstream from points of lateral inflow into a 

channel. 

In longitudinal section, bars are approximately triangular, with very long gentle upstream 

slopes and short downstream slopes that are approximately the same as the angle of 

repose. Bars appear as small barren islands during low flows. Portions of the upstream 

slopes of bars are often covered with ripples or dunes. 

3.4 RESISTANCE TO FLOW IN ALLUVIAL CHANNELS

Resistance to flow in alluvial channels is complicated by the large number of variables and 

by the interdependency of these variables. It is difficult, especially in field studies, to tell 

which variables are governing the flow and which variables are the result of this flow. 

The slope of the energy grade line for an alluvial stream illustrates the changing role of a 

variable. If a stream is in equilibrium with its environment, slope is an independent variable. 

In such a stream, the average slope over a period of years has adjusted so that the flow is 

capable of transporting only the amount of sediment supplied at the upper end of the stream 

and by the tributaries. If for some reason a larger or smaller quantity of sediment is supplied 

to the stream than the stream is capable of transporting, the slope would change and would 

be dependent on the amount of sediment supplied until a new equilibrium is reached. 

In the following sections the variables affecting resistance to flow are discussed. The effects 

produced by different variables change under different conditions. These changing effects 

are discussed along with approximations to simplify the analysis of alluvial channel flow. 

The variables that describe alluvial channel flow are: 

 V = velocity 

 yo = depth 

 Sf = slope of the energy grade line 

ρ = density of water-sediment mixture 

ρs = density of sediment 

μ = apparent dynamic viscosity of the water-sediment mixture 

 g = gravitational acceleration 

 Ds = representative fall diameter of the bed material 

 G = gradation coefficient of the bed material 

 Sp = shape factor of the particles 

 SR = shape factor of the reach of the stream 

 Sc = shape factor of the cross-section of the stream 

 fs = seepage force in the bed of the stream 
 3.22 

 CT = bed material concentration 

 Cf = fine-material concentration 

ω = terminal fall velocity of the particles 

τc = critical shear stress 

 

In general, analysis of river problems is confined to flow of water over beds consisting of 

quartz particles with constant ρs. The gravitational acceleration g is also constant in the 

present context. The effect of other variables on the flow in alluvial channels is qualitatively 

discussed in the following sections. Most of this presentation is based on laboratory studies 

which have been supplemented by data from field experience when available. 

3.4.1 Depth

With a constant slope, Sf, and bed material, Ds, an increase in depth, yo, can change a plane 

bed (without movement) to ripples, and ripple-bed configuration to dunes, and a dune bed to 

a plane bed or antidunes. Also, a decrease in depth may cause plane bed or antidunes to 

change to a dune-bed configuration. A typical break in a depth-discharge relation caused by 

a change in bed form from dunes to plane bed or from plane bed to dunes is shown in Figure 

3.10. 

Often there is a gradual change in bed form and a gradual reduction in resistance to flow and 

this type of change prevents the break in the stage-discharge relation. Nevertheless, it is 

possible to experience a large increase in discharge with little or no change in stage. For this 

and related reasons, development of dependable stage-discharge relations in alluvial 

channels is very difficult. 

Resistance to flow varies with depth even when the bed configurations do not change. When 

the bed configuration is plane bed, either with or without sediment movement or ripples, 

there is a decrease in resistance to flow with an increase in depth. This is a relative 

roughness effect. 

Figure 3.10. Relation of depth to discharge for Elkhorn River near Waterloo, Nebraska 

 (after Beckman and Furness 1962). 
 3.23 

When the bed configuration is dunes, field and laboratory studies indicate that resistance to 

flow may increase or decrease with an increase in depth, depending on the size of bed 

material and magnitude of the depth. Additional studies are needed to define the variation of 

resistance to flow for flow over dune beds. 

When the bed configuration is antidunes, resistance to flow increases with an increase in 

depth to some maximum value, then decreases as depth is increased further. This increase 

or decrease in flow resistance is directly related to changes in length, amplitude, and activity 

of the antidunes as depth is increased. 

3.4.2 Slope

The slope, Sf, is an important factor in determining the bed configuration which will exist for a 

given discharge. The slope provides the downstream component of the fluid weight, which in 

turn determines the fluid velocity and stream power. The relation between stream power, 

velocity and bed configuration has been illustrated in Figure 3.7. 

 

Even when bed configurations do not change, resistance to flow is affected by a change in 

slope. For example, with shallow depths and the ripple-bed configuration, resistance to flow 

increases with an increase in slope. With the dune-bed configuration, an increase in slope 

increases resistance to flow for bed materials having fall velocities greater than 0.06 m/s 

(0.20 ft/s). For those bed materials having fall velocities less than 0.06 m/s (0.20 ft/s), the 

effect is uncertain. 

3.4.3 Apparent Viscosity and Density

The effect of fine sediment (bentonite) on the apparent kinematic viscosity of the mixture is 

shown in Figure 3.11. The magnitude of the effect of fine sediment on viscosity is large and 

depends on the chemical make up of the fine sediment. 

 Figure 3.11. Apparent kinematic viscosity of water-bentonite dispersions 

 (Simons and Richardson 1966). 
 3.24 

In addition to increasing the viscosity, fine sediment suspended in water increases the mass 

density of the mixture (ρ) and, consequently, the specific weight (γ). The specific weight of a 

sediment-water mixture is computed from the relation, 

wsss )(C

sw

γ−γ−γ

γγ

=γ (3.13) 

where: 

γw = Specific weight of the water [9810 N/m3

 (62.4 lb per cu ft)] 

γs = Specific weight of the sediment [about 26,000 N/m3

 (165.4 lb per cu ft)] 

 Cs = Concentration by weight (in fraction form) of the suspended sediment 

A sediment-water mixture, where Cs = 10 percent, has a specific weight (γ) of about 10,460 

N/m3

 (66.5 lb per cu ft). It is clear any change in γ affects the boundary shear stress and the 

stream power. 

Changes in the fall velocity of a particle caused by changes in the viscosity and the fluid 

density resulting from the presence of suspended bentonite clay in the water are shown in 

Figure 3.12a. For comparative purposes, the effect of temperature on the fall velocity of two 

sands in clear water is shown in Figure 3.12b. 

Figure 3.12. Variation of fall velocity of several sand mixtures with percent bentonite and 

 temperature (Simons and Richardson 1966). 

3.4.4 Size of Bed Material

The effects of the physical size of the bed material, Ds, on resistance to flow are: (1) its 

influence on the fall velocity, ω, which is a measure of the interaction of the fluid and the 
 3.25 

particle in the formation of the bed configurations; (2) its effect on grain roughness, Ks; and 

(3) its effect on the turbulent structure and the velocity field of the flow. 

The physical size of the bed material, as measured by the fall diameter or by sieve diameter, 

is a primary factor in determining fall velocity. Use of the fall diameter instead of the sieve 

diameter is advantageous because the shape factor and density of the particle can be 

eliminated as variables. That is, if only the fall diameter is known, the fall velocity of the 

particle in any fluid at any temperature can be computed; whereas, to do the same 

computation when the sieve diameter is known, knowledge of the shape factor and density of 

the particle are also required. 

The physical size of the bed material determines the friction factor mainly for the plane-bed 

condition and for antidunes when they are not actively breaking. The breaking of the waves, 

which increases with a decrease in the fall velocity of the bed material, causes additional 

dissipation of energy. 

The physical size of the bed material for a dune-bed configuration also has an effect on 

resistance to flow. The flow of fluid over the back of dunes is affected by grain roughness, 

although the dissipation of energy by the form roughness is the major factor. The form of the 

dunes is also related to the fall velocity of the bed material. 

3.4.5 Size Gradation 

The gradation, G, of sizes of the bed material affects bed form and resistance to flow. Flume 

experiments indicate that uniform sands (sands of practically the same size) have larger 

resistance to flow (except plane bed) than graded sands for the various bed forms. Also the 

transition from upper flow regime to lower flow regime occurs over a narrower range of shear 

values for the uniform sand. For a plane bed with motion, resistance to flow is about the 

same for both uniform or graded sand. 

3.4.6 Fall Velocity

Fall velocity, ω, is the primary variable that determines the interaction between the bed 

material and the fluid. For a given depth, and slope, the fall velocity determines the bed form 

that will occur, the actual dimensions of the bed form and, except for the contribution of the 

grain roughness, the resistance to flow. 

Observations of natural streams have shown that the bed configuration and resistance to 

flow change with changes in fall velocity when the discharge and bed material are constant. 

For example, the Loup River near Dunning, Nebraska has bed roughness in the form of 

dunes in the summer when the water is warm and less viscous but has a nearly plane bed 

during the cold winter months. Similarly, two sets of data collected by Harms and 

Fahnestock (1965) on a stable branch of the Rio Grande at similar discharges show that 

when the water was cold, the bed of the stream was plane, the resistance to flow was small, 

the depth was relatively shallow, and the velocity was large; but when the water was warm, 

the bed roughness was dunes, the resistance to flow was large, the depth was large, and the 

velocity was low. 
 3.26 

3.4.7 Shape Factor for the Reach and Cross-Section

The configuration of the reach, SR, and the shape of the cross-section, Sc, affect the energy 

losses resulting from the nonuniformity of the flow in a natural stream caused by the bends 

and the nonuniformity of the banks. Study of these losses in natural channels has long been 

neglected. Also, flow phenomena, bed configuration, and resistance to flow vary with the 

width of the stream. In narrow channels dunes and antidunes vary mainly in the downstream 

direction and resistance to flow is larger than for a wide channel. Also, in wide channels 

more than one bed form can occur in the cross-section. 

3.4.8 Seepage Force

A seepage force, fs, occurs whenever there is inflow or outflow through the bed and banks of 

a channel in permeable alluvium. The seepage flow affects the alluvial channel phenomena 

by altering the velocity field in the vicinity of the bed particles and by changing the effective 

weight of the bed particles. Seepage may have a significant effect on bed configuration and 

resistance to flow. If there is inflow, the seepage force acts to reduce the effective weight of 

the sand and, consequently, the stability of the bed material. If there is outflow, the seepage 

force acts in the direction of gravity and increases the effective weight of the sand and the 

stability of the bed material. As a direct result of changing the effective weight, the seepage 

forces can influence the form of bed roughness and the resistance to flow for a given channel 

flow. For example, under shallow flow a bed material with median diameter of 0.5 mm will be 

molded into the following forms as shear stress is increased: dunes, transition, plane bed, 

and antidunes. If this same material was subjected to a seepage force that reduced its 

effective weight to a value consistent with that of medium sand (median diameter, Ds = 0.3 

mm), the forms of bed roughness would be transition, plane bed, and antidunes for the same 

range of flow conditions. 

A common condition is outflow from the channel during the rising stage; this process 

increases the stability of the bed and bank material but stores water in the banks. During the 

falling stage, the situation is reversed; inflow to the channel reduces the effective weight and 

stability of the bed and bank material and influences the form of bed roughness and the 

resistance to flow. 

3.4.9 Bed Material Concentration 

The bed material concentration, CT, affects the fluid properties by increasing the apparent 

viscosity and the density of the water-sediment mixture. However, the effect of the sediment 

on viscosity μ and the density ρ in any resistance to flow relation is accounted for by using 

their values for the water-sediment mixture instead of their values for pure water. The 

presence of sediment in the flow causes a small change in the turbulence characteristics, 

velocity distribution and resistance to flow. 

3.4.10 Fine-Material Concentration

Fine-material concentration, cf, or washload is that part of the total sediment discharge that is 

not found in appreciable quantities on the bed. If significant amounts of sediment is in 

suspension, its effect on the viscosity of the water-sediment mixture should be taken into 
 3.27 

account. The effect of fine sediment on resistance to flow is a result of its effect on the 

apparent viscosity and the density of the water-sediment mixture. Generally, the fine 

sediment is uniformly distributed in the stream cross-section. The method of defining and 

treating the fine-material load computations is discussed in Chapter 4. 

3.4.11 Bedform Predictor and Manning's n Values for Sand-Bed Streams

In Figure 3.13, the relation between stream power, median fall diameter of bed material, and 

form roughness is shown. This relation gives an indication of the form of bed roughness one 

can anticipate if the stream power and fall diameter of bed material are known. Flume data 

were utilized to establish the boundaries separating plane bed without sediment movement 

and ripples, and ripples and dunes for bed material with D50 finer than 0.64 mm, and plane 

bed without sediment movement and dunes for D50 coarser than 0.64 mm. The lines dividing 

dunes and transition and dividing transition and upper regime are based on flume data and 

the following field data: (1) Elkhorn River, near Waterloo, Nebraska (Beckman and Furness 

1962); (2) Rio Grande, 32 km (20 mi) above El Paso, Texas; (3) Middle Loup River at 

Dunning, Nebraska (Hubbell and Matejka 1959); (4) Rio Grande at Cochiti, near Bernalillo 

and at Angostura heading, N. Mexico (Culbertson and Dawdy 1964); and (5) Punjab canal 

data upper regime flows that have been observed in large irrigation canals that have fine 

sandbeds. 

Figure 3.13. Relation between stream power, median fall diameter, and bed configuration 

 and Manning's n values. 

Observations by the authors on natural sandbed streams with bed material having a median 

diameter ranging from 0.1 mm to 0.4 mm indicate that the bed planes out and resistance to 

flow decreases whenever high flow occurs. Manning's n changes from values as large as 

0.040 at low flow to as small as 0.012 at high flow. An example is given in Figure 3.14. 

These observations are substantiated by Dawdy (1961), Colby (1960), U.S. Army Corps of 

Engineers (1968) and Beckman and Furness (1962). 
 3.28 

Figure 3.14. Change in Manning's n with discharge for Padma River in Bangladesh. 

3.4.12 How Bedform Changes Affect Highways in the River Environment

At high flows, most sandbed channel streams shift from a dune bed to a transition or a plane 

bed configuration. The resistance to flow is then decreased two to threefold. The 

corresponding increase in velocity can increase scour around bridge piers, abutments, spur 

dikes or banks and also increases the required size of riprap. On the other hand, the 

decrease in stage resulting from the planing out of the bed will decrease the required 

elevation of the bridge crossing, the height of embankments across the floodplain, the height 

of any dikes, and the height of any channel control works that may be needed; and the 

converse is also true. 

Another effect of bed form on highway crossings is that with dunes on the bed there is a 

fluctuating pattern of scour on the bed and around the piers, abutments, guide banks and 

spurs. The average height of dunes is approximately 1/2 to 1/3 the average depth of flow 

and the maximum height of a dune may approach the average depth of flow. If the depth of 

flow is 3 m (10 ft), the maximum dune height may be of the order of 3 m (10 ft), and half of 

this would be below the mean elevation of the bed. With the passage of this dune through a 

bridge section, an increase of 1.5 m (5 ft) in the local scour would be anticipated when the 

trough of the dune arrives at the bridge. 

A very important effect of bed forms and bars is the change of flow direction in channels. At 

low flow the bars can be residual and cause high velocity flow along or at a pier or abutment 

or any of the other structures in the stream bed, causing deeper than anticipated scour. As 

stated previously large discharges normally experience smaller resistance to flow in a 

sandbed stream due to the change in bed form. However, if the bridge crossing or 

encroachment causes appreciable backwater, the dune bed may not plane out at large 

discharges and a higher resistance to flow results. This increase in resistance to flow can 

decrease the velocity of flow and also decrease the transport capacity of the channel so that 

aggradation occurs upstream of the crossing. The aggradation and the roughness increases 

the river stage and thus the height of any control structure or the levees. Thus, the bridge 

crossing can adversely affect the floodplain, due to the change in bed form that would occur. 
 3.29 

With highways in the sandbed river environment, care must be taken in analyzing the 

crossing in order to foresee possible changes that may occur in the bed form and what these 

changes may do to the resistance coefficient, to sediment transport, and to the stability of the 

reach and its structures. 

3.4.13 Alluvial Processes and Resistance to Flow in Coarse Material Streams

The preceding discussion of alluvial channel flow is mainly related to sandbed channels; that 

is, channels with noncohesive bed materials of size less than 2 mm. The analysis of 

coarse-material channels is also pertinent to highway engineering. This classification 

includes all channels with noncohesive bed materials coarser than 2 mm size. 

The behavior of coarse material channels is somewhat different from sandbed channels. 

The main distinction between the two channels lies in the spread of their bed material size 

distribution. In sandbed channels, for example, the bed may consist of particles from 0.02 to 

2 mm; i.e., a 100-fold size range. In coarse-material channels, even if the maximum size is 

limited to cobbles (250 mm), the size range of particles may be 0.10 to 250 mm, which is a 

2,500-fold size range. In general, coarse-material channels are less active and have slower 

rates of bank shifting than sandbed channels. However, the tendency for channel armoring is 

more pronounced in coarse material channels as discussed next. 

Armoring. The phenomenon of armoring in mobile bed channels occurs by the 

rearrangement of bed material during movement. The bed is covered by a one particle thick 

layer of the coarser material underlain by the finer sizes. The absence of finer sizes from the 

surface layer is caused by the winnowing away of these sizes by the flow. As the spread of 

particle sizes available in the bed of coarse-material channels is large, these channels can 

armor their beds and behave as rigid boundary channels for all except the highest flows. 

The bed and bank forming activity in these channels is therefore limited to much smaller 

intervals of the annual hydrographs than the sandbed channels. 

The general lack of mobility in coarse material channels also means the bed forms do not 

change as much or as rapidly as in sandbed channels. The roughness coefficients in coarse 

material channels are therefore more consistent during the annual hydrographs than in 

sandbed channels. Most of the resistance to flow in coarse material channels comes from 

the grain roughness and from bars. The river bed forms (dunes) are less important in the 

hydraulic behavior of coarse bed channels. 

Sampling. The purpose of bed material sampling in coarse-material channels is: (1) to 

determine the conditions of incipient movement; (2) to assess the bed roughness related to 

the resistance to flow; (3) to determine the bed material load for a given flow; and (4) to 

determine the long and short time response of the channel to specific activities. For 

objectives (1) to (3), the properties of the surface layer are needed. If it is anticipated that 

the bed layer will be disrupted at any given stage, it is necessary to take an adequate sample 

of both the surface and subsurface material. 

The surface sampling can be easily done on the channel bed by counting particles on a grid, 

as explained earlier in this chapter. However, special effort should be made to obtain an 

objective sample. There is a tendency to select too many large particles. The scoop sample 

with bed material sizes larger than 25.4 mm (an inch) or so is difficult to obtain and such 

samples may have to be collected from bars and other exposed areas on channel perimeter. 
 3.30 

In the size distribution analysis of coarse-bed materials, it is sometimes necessary to obtain 

particle counts by number, rather than by sieving or visual accumulation tube analysis for a 

part of the sample. Care must be taken in the interpretation of frequency distribution of part 

of a sample obtained by sieving. Only if the size distribution in a sample follows log-normal 

probability distribution can number counts be transferred to distributions by size, volume, 

weight or surface areas directly. For other distributions, special numerical techniques have 

to be used to transform the number distributions to weight or size distributions. 

If the objectives of bed material sampling include bed roughness and channel response, then 

the particles coarser than D84 or D90 need to be analyzed with more care. These sizes also 

require large samples for their determination. 

Resistance to Flow. In sandbed channels, the form roughness is the primary component of 

channel roughness. Form roughness can be much greater than the grain roughness when 

the bed forms are ripples and dunes. In coarse-material channels, the ripples never form 

and dunes are rare. The main type of bed form roughness in such channels is the pool and 

riffle configuration. With coarse material channels the grain roughness is the main 

component of the channel roughness. 

A coarse-material channel may have bed material that is only partly submerged during most 

of the flows. It is difficult to determine the channel roughness for such beds. For other 

cases, analysis of data from many rivers, canals and flumes (Anderson et al. 1968) shows 

that the channel roughness can be predicted by various forms of Strickler's equation: 

6/1 = DKn xu (3.14) 

V.T. Chow (1959) (3.15) 

Ku = 0.0417 (D50 in meters) 

Ku = 0.0342 (D50 in feet) 

Anderson et al. (1970) (3.16) 

Ku = 0.0482 (D50 in meters) 

Ku = 0.0395 (D50 in feet) 

Lane and Carlson (1955) (3.17) 

Ku = 0.0473 (D75 in meters) 

Ku = 0.0388 (D75 in feet) 

Ku = 0.0256 (D75 in inches) 

U.S. Army Corps of Engineers (1991) (3.18) 

Ku = 0.046 (D90 in meters) 

Ku = 0.038 (D90 in feet) 

Equation 3.19 has been proposed by Limerinos (1970) and involves flow depth yo as a 

parameter. Comparisons of several equations have been reported by Bray (1982) and 

Simons and Senturk (1992). 
 3.31 

84

o

6/1

ou

D

y log216.1

yK n

+

= (3.19) 

Ku = 0.113 (yo and D84 in meters) 

Ku = 0.0927 (yo and D84 in feet)

An alternative approach, valid when R/D50 > 10, is to evaluate the Darcy-Weisbach friction 

factor, f, as a function of the hydraulic radius, R, with: 

4/1

D84

R 9.1

f

1















 = (3.20) 

1.1

D

R log0.2 f1

84

+













 = (3.21) 

Note: R and D84 in same units (meters, feet) 

Manning's n can then be calculated from the relationship between R and f : 

fRKn 6/1 = u (3.22) 

Ku = 0.113 (R in meters) 

Ku = 0.0927 (R in feet)

This approach should not be applied to river reaches with active beds or significant sediment 

transport. 

Charlton et al. (1978) and Bray (1979) have proposed the guidelines in Table 3.3 to estimate 

D90, D84, and D65, for gravel-bed material when size distribution curves cannot be obtained 

from field data. 

Table 3.3. Typical Sediment Size Distribution for Gravel-Bed Stream. 

Ratio Mean Value Standard Deviation 

D90/D50 2.1 0.46 

D84/D50 1.9 0.36 

D65/D50 1.3 0.08 
 3.32 

3.5 BEGINNING OF MOTION 

3.5.1 Introduction 

The initiation or ceasing of motion of sediment particles is involved in many geomorphic and 

hydraulic problems including stream stability and scour at highway bridges, sediment 

transport, erosion, slope stability, stable channel design, and design of riprap. These 

problems can only be handled when the threshold of sediment motion is fully understood. 

Beginning of motion can be related to either shear stress on the grains or the fluid velocity in 

the vicinity of the grains. When the grains are at incipient motion, these values are called the 

critical shear stress or critical velocity. The choice of shear stress or velocity depends on: (1) 

which is easier to determine in the field; (2) the precision with which the critical value is 

known or can be determined for the particle size; (3) the type of problem. In sediment 

transport analysis most equations use critical shear stress. In stable channel design either 

critical shear stress or critical velocity is used; whereas, in riprap design critical velocity is 

generally used. 

Equations for determining the shear stress on the bed of a stream are given in Chapter 2, 

Section 2.4.5. The average shear stress on the boundary is given by τ0 = γ RS. Where γ is 

the unit weight of water, R is the hydraulic radius and S is the slope of the energy grade line. 

In wide channels (width equal to or greater than 10 times the depth) R ≈ y, the depth. Other 

relations give the shear stress in terms of the velocity of flow. 

It may not be sufficient to determine the average value of the critical shear stress or velocity 

because both quantities are fluctuating. For the same mean values, they may have larger 

values that act for a sufficiently long enough time to cause a particle to move. In addition, 

the forces on the particle resulting from the flowing water, waves, and seepage into or out of 

the bed or banks affect the beginning of motion. 

In this section, the following topics are discussed: theory of beginning of motion, Shields 

experimental relationship and its modifications, equations to determine the relation between 

flow variables (depth, velocity or discharge) and sediment size, tables giving observe values 

between flow variables and sediment size, and figures for determining the flow or sediment 

variables at beginning of motion. 

3.5.2 Theoretical Considerations 

When the force of the flowing water (as measured by the shear stress or velocity) is less than 

some critical value, the bed material of a channel remains motionless. Then, the alluvial bed 

can be considered as immobile. But when the shear stress or velocity over the bed attains or 

exceeds its critical value, particle motion begins. In general, the observation of particle 

movement is difficult in nature. The most dependable data available have resulted from 

laboratory experiments. 

The beginning of motion is difficult to define. This difficulty is a consequence of a 

phenomenon that is random in time and space. When the shear stress is near its critical 

value, it is possible to observe a few particles moving on the channel bottom. The time 

history of the movement of a particle involves long rest periods. In fact, it is difficult to 

conclude that particle motion has begun. Kramer (1935) and Buffington (1999) proposed 

four levels of motion of bed material. 
 3.33 

1. None. 

2. Weak movement: Only a few particles are in motion on the bed. The grains moving on 

one square centimeter of the bed can be counted. 

3. Medium movement: The grains of mean diameter begin to move. The motion is not local 

in character but the bed continues to be plane. 

4. General movement: All the mixture is in motion; the movement is occurring in all parts of 

the bed at all times. It is sufficiently vigorous to change the bed configuration. 

Whether or not a plane bed can exist with weak sediment motion is debated; though positive 

evidence of its existence has been presented by Liu (1957) and Senturk (1969). Liu’s 

observations may have involved such shallow flow that the Froude number was larger than 

1, which would indicate antidunes not ripples. Many researchers such as Kramer (1935), 

Shields (1935), White (1940), Tison (1953), Simons and Richardson (1966), Gessler (1971), 

Vanoni (1977), have studied the problem of initiation of motion. Buffington (1999) provides 

an excellent review of Shields research. The studies involve both theory and 

experimentation. The complexity of the problem explains the diversity of experimental 

results. In reality, there is no truly critical condition for initiation of motion for which motion 

begins suddenly as the condition is reached. Data available on critical shear stress are 

based on arbitrary definitions of critical conditions and most definitions used have relied on 

subjective visual observations. Also, no evidence has shown that the mean diameter 

represents the composition of a mixture correctly. The engineer facing this dilemma of 

dealing with beginning of motion in a mixture of sediment sizes should analyze the problem 

very carefully. However, the equations presented in this section are theoretically sound and 

have proven to give reliable results. This is true, even though there is a diversity in the 

experimentally determined coefficients. 

Water flowing over a bed of sediment exerts forces on the grains. These forces tend to 

move or entrain the particles. The forces that resist the entraining action of the flowing water 

differ depending upon the properties of bed material. For coarse sediments such as sands 

and gravels, the resisting forces relate mainly to the weight of the particles, but also are a 

function of shape of particle, its position relative to other particles, and the form of bed 

roughness. For cohesive bed material (generally silts and clays), chemical bonding between 

particles resists the beginning of motion. 

When the hydrodynamic forces acting on a grain of sediment have reached a value that, if 

increased even slightly the grain will move, critical or threshold conditions are said to have 

been reached. Under critical conditions, the hydrodynamic forces acting upon a grain are 

just balanced by the resisting force of the particle. 

3.5.3 Theory of Beginning of Motion 

The forces acting on an individual particle on the bed of an alluvial channel are: 

1. Body force Fg due to the gravitational field, 

2. External forces Fn acting at the points of contact between the grain and its neighboring 

grains, and 
 3.34 

3. Fluid force Ff (lift and drag) acting on the surface of the grain. The fluid force varies with 

the velocity field and with the properties of the fluid. 

The relative magnitude of these forces determines whether the grain moves or not. 

For the individual grain, the body force is: 

3 ρ= DKgF s2sg (3.23) 

where: 

ρs = Density of the grain 

 K2 = Volumetric coefficient 

 Ds = Grain diameter 

3 DK s2 = Volume of the grain 

For convenience, the fluid forces acting on the grain are divided into three components: 

(1) The form drag component FD 

)2/V(DKCF 22

s1DD ρ= (3.24) 

(2) The viscous drag component FV 

τ=

2 DKCF s1SV (3.25) 

(3) The buoyant force component FB 

3

B ρ= DKgF s2 (3.26) 

where: 

 CD = Drag coefficient 

 K1 = Coefficient associated with the area of the grain subjected to form drag and 

shear (the term 2 DK s1 represents the cross-sectional area of the grain) 

 K2 = Volumetric coefficient of the grain 

 Ds = Diameter of the grain 

 v = Velocity in the vicinity of the grain 

 Cs = Coefficient of shear 

τ = Average viscous shear stress 

The external forces Fn depend on the values of the fluid and body forces FB. Under 

conditions of no flow, the fluid force is Ff = FB. There is no form or viscous drag. Then the 

external force Fn is Fn = Fg – FB or 

3

sn ρ−ρ= DgK)(F s2 (3.27) 
 3.35 

That is, the external force is equal to the submerged weight of the grain. 

The form drag can be written in terms of the shear velocity. For turbulent flow, the local 

velocity, v is directly proportional to the shear velocity V*. Then, Equation 3.24 reduces to: 

2

*

2

D ρ s VD~F (3.28) 

The viscous drag is also related to the shear velocity, but it is the shear velocity for laminar 

flow. For laminar flow: 

dy

dv

μ=τ (3.29) 

Again, by replacing V with V* and y with Ds we can write 

μ≈τ * D/V s (3.30) 

 

With this expression for viscous shear, the shear force Fv becomes 

V μ S VD~F * (3.31) 

Now, consider the ratio of the form drag force FD to the viscous shear force Fv. According to 

Equations 3.28 and 3.31: 

*

*

VD

VD ~ F

F

s

22

s

v

D

μ

ρ

or 

υ

VD * ~ F

F s

v

D (3.32) 

When the flow over the grain is turbulent, the form drag is predominant and the term DsV*/ν 

is large. When the flow over the grain is laminar the viscous shear force is predominant and 

the term DsV*/ν is small. Thus, the Reynolds’ number of particle DsV*/ν is an indicator of the 

characteristics of the flow in the vicinity of the grain. 

As both the form drag and viscous shear are proportional to the shear velocity, the ratio of 

the forces tending to move the grain to the forces resisting movement is: 

s s

o

3

s s

22

s

n

D

gD)( D)(

VD ~ F

F *

γ−γ

τ = ρ−ρ

ρ (3.33) 
 3.36 

Recall that V*

2

 = τo/ρ. The relation between τo/(γs - γ) Ds and DsV*/ν for the condition of 

incipient motion has been determined experimentally by Shields and others. As the relation is 

determined experimentally, the relation considers viscous effects. In functional form the 

relation (called the Shields relation) is as follows: 

















υ = γ−γ

τ s

s s

0 DV

f

D)(

c * (3.34) 

3.5.4 Shields Diagram 

Many experiments have been conducted to develop an explicit solution of the Shields 

relation. The earliest one is the graphical presentation given by Shields (1935). The concept 

of the Shields Diagram (Figure 3.15) is widely accepted and τc/(γs - γ) Ds (often referred to as 

Shields parameter) is widely used to determine the shear stress at beginning of motion. For 

example, Gessler (1971) gave Figure 3.16 for the relation given in Equation 3.34. 

Figure 3.15. Shields Diagram: dimensionless critical shear stress. 
 3.37 

Figure 3.16. Shields’ relation for beginning of motion (after Gessler 1971). 

Most authors report that Shields determined this relationship (Figure 3.15) by measuring 

bed-load transport for various values of τ/(γs - γ) Ds and then extrapolated to the point of 

vanishing bed load. Simons and Senturk (1992) state the values of τ/(γs - γ) Ds were at least 

twice as large as the critical value. This indirect procedure was used to avoid the 

implications of the random orientation of grains and variations in local flow conditions that 

may result in grain movement even when τ/(γs - γ) Ds is considerably below the critical value. 

However, Buffington (1999) discusses in great detail Shields' research and states the 

following: 

"Nevertheless, these two passages from Shield’s dissertation offer two definitions of incipient 

motion (bed-load extrapolation for uniform sizes versus visual observation for mixed grains. 

Because Shields neglected to explain which method he used and did not present sufficient 

data to recreate his calculations, the matter of his experimental procedure remains open to 

debate. However, throughout his dissertation he discussed his approach and results as 

being representative of uniform grains (Shields 1936c pp. 11, 14, and 16), suggesting that he 

employed bed-load extrapolation (the method he described for uniform sediment." 

The Shields Diagram (Figure 3.15) was divided into three regions by Simons and Senturk 

(1992) as illustrated in the following. 

Region 1: * s <ν 0.5~63.3/DV

In the region Ds < 3δ, where δ = 11.6ν/V*, and the boundary is considered hydraulically 

smooth (δ is the thickness of the laminar boundary layer, Chapter 2). Shields estimated the 

portion of the diagram for V*Ds/ν < 2. He did not perform any experiments in that region. 

According to Shields, when the value of 

1.0

D)( s s

c ≅

γ−γ

τ (3.35) 

then (approximately) 
 3.38 

00.1 *DV s = ν 

Region 2: * s <ν< 0.70~0.68/DV0.5~63.3

In this region, the boundary is in a transitional state and δ/3 < Ds < 6 δ. For this region: 

ν = δ

Vk *

6.11

ks 1 s (3.36) 

The Shields Diagram has a form similar to Darcy-Weisbach's resistance coefficient f versus 

Reynolds number Re. Also, it is similar in form to the relation between the drag coefficient CD

and the Reynolds number Re for cylindrical bodies and to the relation between V*ks/ν and B = 

f (V*ks/ν) proposed by Nikuradse (1933). 

The minimum value of F* = τc/(γs - γ)Ds is 0.032 ~ 0.033 and the corresponding value of R* = 

V*Ds/ν is about 10. If Ds is computed from these values of R* and F*, it can be seen that Ds = 

0.0006 m = 0.6 mm = 0.002 ft. For larger diameter particles, ripples do not form; dunes form 

on the bed. 

Region 3: * s >ν 500to70/DV

In this region, the boundary is completely rough and F* is independent of Reynolds number 

R* and is equal to 

06.0

D)(

V

s s

2

* = γ−γ

ρ (3.37) 

The upper limit of R* in Region 3 is subject to discussion. Some researchers have given 

values as high as 1,000 for R*. Considering F*, Meyer-Peter and Müller (1948) suggest a 

value of 0.047 instead of 0.06. 

Experimental values of the Shields parameter for sand size (0.062 mm to 1.0 mm) bed 

material range from 0.030 to 0.047. For coarser bed material (gravels, cobbles and larger) 

the experimentally determined Shield's parameter ranges from 0.020 to 0.10. An average 

value of 0.039 is a good compromise for all sizes (Fiuzat and Richardson 1983, Ruff et al., 

1985, 1987). 

It is suggested by Simons (Simons and Senturk 1992) that collecting data on initiation of 

particle motion under field conditions permits selection of a more precise value for the 

particular channel. However, identifying initiation of particle sizes by utilizing observed 

values or by trapping particles in motion over a range of discharges is extremely difficult. It 

must be done with considerable care and with knowledge of channel geometry and hydraulic 

conditions at the cross section and upstream of the selected cross section. This is 

particularly true for gravel- and cobble-bed streams. 
 3.39 

The variation of the critical shear stress for beginning of motion as a function of sediment 

size given by various investigators is shown in Figure 3.17 (Chien 1954). The spread of the 

representative curves shows the diversity of experimental and theoretical results. 

Nevertheless, equations, tables and figures given in this section provide useful tools for 

determining beginning of motion for sediment particles and the design of riprap. 

 Figure 3.17. Comparison of critical shear stress as a function of grain diameter 

 (after Chien 1954). 

3.5.5 Equations for Flow and Sediment Variables for Beginning of Motion 

Particles start to move in steady, uniform flow when the shear stress applied by the flow 

equals the resistance to movement of the particles. A useful relation can be developed 

between the flow velocity, depth, and resistance and bed material size by equating the 

applied shear stress to a resistance to motion shear stress. In equation form it is as follows: 
 3.40 

c0 τ=τ (3.38) 

where: 

τ0 = Average bed shear stress, N/m2

, lb/ft2

τc = Critical bed shear stress at incipient motion, N/m2

, lb/ft2

The average bed shear stress applied by the steady uniform flow, derived in Chapter 2, is as 

follows: 

0 SR f γ=τ (3.39) 

Using y for the hydraulic radius (R) and the Manning equation to determine the slope (Sf) 

The average shear stress can be expressed as follows: 

3/1

22

0 f

y

Vng Syg ρ =ρ=τ (SI) (3.40) 

3/12

22

o y)49.1(

ρ Vng =τ (English) (3.41) 

For noncohesive bed materials, the Shields relation (Vanoni 1975) can be used to determine 

the relation between the critical shear stress and bed material size for beginning of bed 

material movement. The relation is as follows: 

ssc ρ−ρ=τ Dg)(K (3.42) 

At the beginning of sediment movement the applied shear stress is equal to the critical shear 

stress as given in Equation 3.36 (τ0 = τc) resulting in the following: 

3/1 ss s

22

Dg)(K

y

Vng ρ−ρ= ρ (SI) (3.43) 

3/1 ss s

22

Dg)(K y22.2

Vng ρ−ρ= ρ (English) (3.44) 

where: 

 y = Average depth of flow, m, ft 

 Sf = Slope of the energy grade line, m/m, ft/ft 

 V = Average velocity, m/s, ft/s 

 Ds = Diameter of smallest non transportable bed material particle, m, ft 

γ = Unit weight of water (9,800 N/m3

, 62.4 lb/ft3) 

 n = Manning's roughness coefficient 

 Ks = Shield's coefficient 
 3.41 

 Ss = Specific gravity (2.65 for quartz) 

ρ = Density of water (999 kg/m3

, 1.94 slugs/ ft3) 

ρs = Density of sediment (quartz 2,647 Kg/m3

, 165.4 lb/ft3) 

 g = Acceleration of gravity (9.81 m/s2

, 32.2 ft/s2) 

The relationships in Equations 3.43 and 3.44 are the fundamental relations between velocity, 

depth, resistance to flow (Manning's n), density and a coefficient determined experimentally 

for the beginning of movement of the sediment particles. This coefficient is called the 

Shields coefficient. The equation can be solved for the following: 

1. Critical velocity for beginning of sediment movement for a given depth, roughness, 

Shields coefficient, and bed material size and density. 

2. Critical size for a given velocity, depth, roughness, Shields coefficient, and bed material 

density. 

3. Clear-water scour depth for a given velocity, roughness, Shields coefficient, and bed 

material size and density. This depth is the contraction scour depth at the end of a long 

contraction. 

Critical Velocity for the Beginning of Bed Material Movement. Rearranging Equations 3.43 

and 3.44 to give the critical velocity for beginning of motion of bed material of size D for 

depth y, Shield's parameter, and Manning's n results in: 

( )

n

yD1SK V

6/12/1

s

2/1

s

2/1

s

c

− = (SI) (3.45) 

n

yD)1S(k49.1 V

6/12/1

s

2/1

s

2/1

s

c

− = (English) (3.46) 

6/1 = DKn snu (3.47) 

 Knu = 0.041 (SI) 

 Knu = 0.0336 (English) 

6/13/1

suc = yDKV (3.48) 

( )

nu

2/1

s

2/1

s

u K

1SK K − = (SI) (3.49) 

nu

2/1

s

2/1

s

u K

)1S(K49.1 K − = (English) 

where: 

 Vc = Critical velocity above which bed material of size D and smaller will be 

transported, m/s, ft/s 

 Ks = Shields parameter 
 3.42 

 Ss = specific gravity of the bed material 

 Ds = Size of bed material, m, ft 

 y = Depth of flow, m, ft 

 n = Manning's roughness coefficient 

The following values for Ku in Equation 3.48 can be obtained: 

Metric (SI) Units (m)

Using Ks = 0.039, Ss = 2.65, Knu = 0.041 

Ku = 6.19 

English Units (ft)

Using Ks = 0.039, Ss = 2.65, Knu = 0.0336 

Ku = 11.25 

Critical Size for the Beginning of Bed Material Movement. Rearranging Equation 3.48 to give 

the critical bed material size Ds for beginning of motion for velocity V, depth y, Shield's 

parameter and Manning's n results in: 

2/1

3

u

c y

VK D = (3.50) 

0042.0

19.6

1 Ku 3 == (SI) 

0007.0

25.11

1 Ku 3 == (English) 

Depth of Flow For No Bed Material Movement as a function of V and D. Equation 3.48 can 

be solved, for the depth y for no bed material movement for a given velocity, specific density, 

size of bed material, Shields parameter and bed roughness. This equation is useful to 

determine clear water-scour at a contraction. This is particularly true if the velocity is 

converted to discharge using the continuity equation. 

2

s

6

yu

2

s

6

u

6

D

VK

DK

V

y == (3.51) 

5

yu 10x78.1K − = (SI) 

7

yu 10x94.4K − = (English) 
 3.43 

Depth of Flow for No Bed Material Movement as Function of Q and D. The velocity V in 

Equation 3.51 can be replaced with the discharge Q using the continuity equation (Q = W y 

V). From continuity V = Q/(W y) The relation between y, Q and D for the condition of no bed 

material transport is very useful in determining clear-water contraction scour as explained in 

Chapter 7. 

Replacing V with Q in Equation 3.51 to determine the depth (y) when there is no bed material 

movement of size D, gives the following equation: 

7/67/2

7/6

u WD

Q = Ky (3.52) 

210.0KK 7/1

yuu == (SI) 

126.0KK 7/1

yuu == (English) 

3.5.6 Tables for Determining Critical Velocities 

Table 3.4 gives the permissible velocity recommended by Fortier and Scobey (1926). for 

channels at small slope for bed material ranging from fine sand to cobbles. Table 3.5 gives 

non-scour velocities for noncohesive and compact cohesive soils suggested by Keown et al. 

(1977) for bed material ranging from loess soils to boulders. 

Table 3.4. Maximum Permissible Velocities Proposed by Fortier and Scobey (1926). 

Mean velocity, after aging of canals [y ≤ 0.9 m (3 ft)] 

Original 

Material excavated for 

canals n 

Clear water, no 

detritus 

Water transporting 

colloidal silt 

Water transporting 

noncolloidal silts, 

sands, gravels or rock 

fragments 

 ft/sec M/sec ft/sec m/sec ft/sec m/sec 

Fine sand (colloidal) 0.02 1.50 0.46 2.50 0.76 1.50 0.46 

Sandy loam (noncolloidal) 0.02 1.75 0.53 2.50 0.76 2.00 0.61 

Silt loam (noncolloidal) 0.02 2.00 0.61 3.00 0.91 2.00 0.61 

Alluvial silt 

(noncolloidal) 0.02 2.00 0.61 3.50 1.07 2.00 0.61 

Ordinary firm loam 0.02 2.50 0.76 3.50 1.07 2.25 0.69 

Volcanic ash 0.02 2.50 0.76 3.50 1.07 2.00 0.61 

Fine gravel 0.02 2.50 0.76 5.00 1.52 3.75 1.14 

Stiff clay 0.025 3.75 1.14 5.00 1.52 3.00 0.91 

Graded loam to 

Cobbles (noncolloidal) 0.03 3.75 1.14 5.00 1.52 5.00 1.52 

Alluvial silt (colloidal) 0.025 3.75 1.14 5.00 1.52 3.00 0.91 

Graded, silt to cobbles 

(colloidal) 0.03 4.00 1.22 5.50 1.68 5.00 1.52 

Coarse gravel 

(noncolloidal) 0.025 4.00 1.22 6.00 1.83 6.50 1.98 

Cobbles and shingles 0.035 5.00 1.52 5.50 1.68 6.50 1.98 

Shales and hard pans 0.025 6.00 1.83 6.00 1.83 5.00 1.52 
 3.44 

Table 3.5. Nonscour Velocities for Soils (Modified from a report by Keown et al. 1977).

Grain Dimensions 

Approximate Nonscour Velocities 

Kind of Soil (feet per second) 

(mm) (ft) Mean Depth 

1.3 ft 3.3 ft 6.6 ft 9.8 ft 

For Noncohesive Soils 

Boulders >256 >0.840 15.1 16.7 19.0 20.3 

Large cobbles 256-128 0.840-0.420 11.8 13.4 15.4 16.4 

Small cobbles 128-64 0.420-0.210 7.5 8.9 10.2 11.2 

Very coarse gravel 64-32 0.210-0.105 5.2 6.2 7.2 8.2 

Coarse gravel 32-16 0.105-0.0525 4.1 4.7 5.4 6.1 

Medium gravel 16-8.0 0.0525-0.0262 3.3 3.7 4.1 4.6 

Fine gravel 8.0-4.0 0.0262-0.0131 2.6 3.0 3.3 3.8 

Very fine gravel 4.0-2.0 0.0131-0.00656 2.2 2.5 2.8 3.1 

Very coarse sand 2.0-1.0 0.00656-0.00328 1.8 2.1 2.4 2.7 

Coarse sand 1.0-0.50 0.00328-0.00164 1.5 1.8 2.1 2.3 

Medium sand 0.50-0.25 0.00164-0.000820 1.2 1.5 1.8 2.0 

Fine sand 0.25-0.125 0.000820-0.000410 .98 1.3 1.6 1.8 

For Compact Cohesive Soils 

Sandy loam (heavy) 3.3 3.9 4.6 4.9 

Sandy loam (light) 3.1 3.9 4.6 4.9 

Loess soils in the 

Conditions of finished 

Settlement 

 2.6 3.3 3.9 4.3 

3.5.7 Figures for Determining Critical Shear Stress or Velocity

Figure 3.18 shows the relationship between critical shear stress and mean diameter as 

determined and/or recommended by different investigators for different soil types. The 

difference between investigators is possibly due to the effects of cohesion, when present, 

causing the particles to aggregate and not act as individual particles. 

Figure 3.18. Critical shear stress as a function of grain diameter (after Lane 1953). 
 3.45 

Figure 3.19 gives the relationship between maximum allowable velocity (critical velocity) 

against stone or minimum stone size that can sustain hydraulic forces without motion. 

Figure 3.19. Critical velocity as a function of stone size. 

3.5.8 Summary 

Equations 3.45 and 3.46 determine critical velocity. These equations can be used to 

determine the critical size, and critical depth for the beginning of bed material movement 

based on the Manning equation, specific gravity of the bed material and Shield’s parameter 

in metric and English units, respectively. They are for steady, uniform flow. In addition, 

Equation 3.52 determines the depth of flow as a function of discharge in metric and English 

units for these same conditions. To solve the equations Manning's n, specific gravity, and 

Shield’s parameter must be determined as well as the other variables. 

In these basic equations, reasonable values of Manning's n, specific gravity, and Shield’s 

parameter are substituted to obtain equations for the dependent variables. These values are 

(1) Shield's parameter Ks = 0.039, (2) specific gravity Ss = 2 .65, and (3) Manning's n = 0.041 

6/1 Ds for metric units and n = 0.0336 6/1 Ds for English units. The derivations are given in 

sufficient detail that engineers can substitute other values for Ks, Ss, and Manning's n to fit 

their specific data. 

Note that the critical sediment size, DC, is a function of V3 and the critical depth, y, is a 

function of V6 and Q6/7.
 3.46 

3.6 SEDIMENT DISCHARGE MEASUREMENT 

3.6.1 Introduction 

In this section the basic terms and methods of measuring sediment discharge (sediment 

load) are described. In Chapter 4, the theory, equations, and methods of computing bed 

material transport are described. 

3.6.2 Terminology 

There are many terms that have developed over time to describe the many aspects of the 

transport of sediment by water. Sediment transport is the time rate of a quantity of sediment 

(by weight) moving past a cross-section of the stream. It is a discharge. Because it is 

measured by weight (often as tons), discharge is often referred to as sediment load. Other 

terms associated with sediment transport are suspended sediment discharge, bed load, bed 

material load, bed material discharge, and wash load. 

The many terms describing sediment transport result from using or misusing and mixing 

terms that describe (1) the source of the sediment (bed material load and wash load), (2) 

mode of particle movement (suspended bed material load and bed load) and (3) 

measurement of sediment discharge (measured sediment discharge and unmeasured 

sediment discharge). The terms are defined in Chapter 4, (Section 4.2) and illustrated in 

Figure 3.20 from HEC-20 (Lagasse et al. 2001). 

Velocity

Profile

Composed of particles

finer than those found in

appreciable quantities in

the bed. Washload moves

in suspension and is

provided by bank and

watershed erosion.

WASH LOAD

Composed of particle

sizes found in the bed

that move by surface

creep, sliding, saltation

or rolling within the

boundary layer.

BED LOAD

Composed of

particles typically

found in the bed that

are transported in

suspension.

SUSPENDED BED

MATERIAL LOAD

BED MATERIAL LOAD

TOTAL SEDIMENT LOAD

Sediment

Concentration

Profiles

Figure 3.20. Definition of sediment discharge (load) components (Lagasse et al. 2001). 
 3.47 

3.6.3 Suspended Sediment Discharge Measurement 

The measurement of the suspended sediment discharge of a stream requires the time 

dependent measurement of the water discharge (discharge hydrograph) and velocity 

weighted measurement of the concentration of sediment particles moving past the crosssection. In equation form: 

 

= wus CQKQ (3.53) 

Ku = 0.086 (SI) 

Ku = 0.0027 (English) 

where: 

Qs = Suspended sediment discharge, metric or English tons per day 

Qw = Water discharge, m3/s or cfs 

C = Velocity-weighted mean concentration (by weight) of sediment, mg/l 

ku = Coefficient to convert to metric or English tons per day 

 

The determination of water discharge was described in Chapter 2. The measurement of 

suspended-sediment discharge is described in detail in Techniques of Water-Resources 

Investigations of the United States Geological Survey (Guy 1970, 1977; Guy and Norman 

1970; and Porterfield 1977). The essence of the procedure is as follows. 

1. Time dependent measurement of the water discharge (discharge hydrograph). Standard 

stream gaging procedure described in Chapter 2. 

2. Measure the velocity weighted mean suspended-sediment concentration of the flow. 

3. Develop a time suspended-sediment concentration graph similar to the stage hydrograph 

at a gaging station (suspended sediment hydrograph). 

4. Determine the daily suspended-sediment discharge in English or metric tons per day, 

using Equation 3.53. 

3.6.4 Velocity Weighted Mean Suspended-Sediment Concentration 

As illustrated in Figure 3.21 the velocity decreases with depth and the sediment 

concentration increases with depth. Very fine sediment or coarser sediment in a very 

turbulent stream may not decrease in concentration with depth (uniformly distributed). 

Whereas, very coarse sediment, or finer sediment in placid flow may have a very large 

increase in concentration with depth. In addition to the velocity and concentration varying 

with depth, they vary across the stream. The Federal Government through the Interagency 

Subcommittee on Sedimentation (U.S. Interagency Subcommittee 1940a,b, 1941a,b,c, 1943, 

1948, and 1952; Vanoni 1977; Richardson 1994) developed samplers that take a velocity 

averaged suspended-sediment concentration in the vertical to 0.09 m (0.3 ft) above the bed 

(Figure 3.22). These samplers are used to take samples across the stream or to obtain a 

coefficient to be applied to a sample at a single vertical to obtain the mean discharge 

weighted concentration in the cross-section. 
 3.48 

Figure 3.21. Schematic sediment and velocity profiles. 

 

 Figure 3.22a. Suspended sediment sampler-D49 (Guy and Norman 1970, U.S. 

 Interagency 1952). 
 3.49 

 Figure 3.22b. Suspended sediment sampler-DH48 (Guy and Norman 1973, U.S. 

 Interagency 1952). 

3.6.5 Suspended Sediment Hydrograph 

The measured suspended-sediment concentration is used to plot a concentration hydrograph 

(Figure 3.23). Engineering judgment, the stage hydrograph and knowledge of the stream is 

used in drawing the suspended-sediment hydrograph. The Techniques of Water-Resources 

Investigations of the United States Geological Survey, Book 3, Chapters C3 (Porterfield 

1977) Computation of Fluvial-Sediment Discharge describes methods to construct a 

suspended-sediment hydrograph. 

3.6.6 Determination of Daily Suspended-Sediment Discharge 

Using the concentration hydrograph an average daily suspended-sediment concentration is 

determined. Using this concentration and the average daily water discharge in Equation 3.53 

the average daily suspended-sediment discharge is determined. However, If the 

concentration and water discharge are changing rapidly during the day, as illustrated in 

Figure 3.23, the day will have to be subdivided into short time increments. The suspendedsediment discharge is determined for each time increment using Equation 3.53. Then, the 

suspended-sediment discharge for each time increment for the day are converted to a 

common time base, added up and divided by the number of common time increments to 

obtain the average suspended-sediment discharge for the day. 
 3.50 

Figure 3.23. Gage height and suspended sediment concentration hydrograph, Colorado 

 River near San Saba, Texas, May 1-6, 1952 (Porterfield 1977). 
 3.51 

3.6.7 Total Sediment Discharge 

The suspended-sediment discharge determination described above is the measured 

sediment discharge. There is the unmeasured sediment discharge composed of the 

sediment moving in contact with the bed (contact sediment discharge or load) and the 

suspended sediment discharge that the sampler doesn’t sample. This unmeasured sediment 

discharge can be as low as 10 percent of the total sediment discharge (or total load) to as 

high as 50 percent or more of the total sediment discharge. The percent of the total sediment 

discharge that is measured depends on the turbulence of the stream, bed material size, and 

concentration of wash load. Streams with very large concentrations of wash load or very 

large turbulence and fine bed material will have very low percent of unmeasured sediment 

discharge. 

 

In some stream reaches the turbulence is so large that the total sediment discharge is in 

suspension and each size fraction of the total sediment discharge is uniformly distributed in 

the vertical. Here, the suspended-sediment discharge is the total sediment discharge. Many 

of these streams and in some cases streams where the turbulence was artificially created 

were used by Colby et al. (1955, 1956, 1962) to develop the modified Einstein procedure to 

determine the total sediment discharge of a stream as described in Chapter 4. Burkham and 

Dawdy (1980) and others have further modified Colby's procedure.

As discussed in Chapter 4, the total sediment discharge has three classifications. These are: 

1. By source: The bed material discharge and the wash load discharge (Figure 3.20). 

2. By mode of transport: The suspended-sediment discharge and the bed (contact) 

sediment discharge (Figure 3.20). 

3. By measurement: The measured sediment discharge and the unmeasured sediment 

discharge. 

To determine the total sediment discharge various methods are used. Some of these are: 

1. Where there is a suspended-sediment sampling program, the unmeasured sediment 

discharge is determined using methods described in Chapter 4 and added to the 

measured suspended-sediment discharge. 

2. A suspended-sediment sampling program is established and the unmeasured sediment 

discharge determined as in 1 above and added to the measured suspended-sediment 

discharge. 

3. The total sediment discharge is determined by the modified Einstein method described in 

Chapter 4 for a range of discharges and a sediment rating curve developed. The curve 

and a record of daily water discharge are used to determine daily and yearly total 

sediment discharge. 

4. Computer programs are used to determine the total bed material discharge. The wash 

load discharge may be ignored or estimated from periodic suspended sediment 

measurements. 

 
 3.52 

3.7 SOLVED PROBLEMS FOR ALLUVIAL CHANNEL FLOW (SI) 

3.7.1 PROBLEM 1 Sediment Properties and Fall Velocities 

Sand Bed Channel 

The bed material size distribution of a sand bed channel is shown in Table 3.6. 

Table 3.6. Sand Bed Material Size Distribution. 

Size Range 

(mm) 

Percent of Total Weight 

in Size Range 

.002 - .0625 0.8 

.0625 - .125 4.4 

.125 - .250 14.2 

.250 - .500 74.9 

.500 - 1.00 5.0 

1.00 - 2.00 0.5 

2.00 - 4.00 0.2 

From the grain size analysis calculate the following statistics: the geometric mean of each 

size range; effective diameter Dm; the D16, D50, D84, D90 sizes; the gradation coefficient; and the 

fall velocity of each size range. 

The geometric mean is calculated as the square root of the product of the end points of a given 

size range. 

For the first size range: 

Di

 = [(0.002) (.0625)]1/2 = 0.011 mm. 

Likewise for the remainder of the size ranges, the results are summarized in Table 3.7. 

The effective diameter (Dm) of the sample distribution is calculated with the use of Equation 

3.12 in Section 3.2.4. 

100

p D D si i

n

1i

m

= =

( )( ) ( )( ) ( )( )

100

mm83.22.0....mm088.04.4mm011.08.0

100

p D D si i

n

1i

m

+ + = = =

mm34.0

100

2.34 Dm == ; which lies in the sand size. 
 3.53 

Table 3.7. Sand Size Bed Material Properties. 

Size Range 

(mm) 

Geometric 

Mean Size, Di

(mm) 

Percent of 

Bed Material 

in this Size 

(pi

) 

Percent 

Finer 

pi

Di

Fall Velocity 

(m/sec) 

.002 - .0625 0.011 0.8 0.8 0.01 0.0001 

.0625 - .125 0.088 4.4 5.2 0.39 0.0061 

.125 - .250 0.177 14.2 19.4 2.51 0.0183 

.250 - .500 0.354 74.9 94.3 26.51 0.0457 

.500 - 1.00 0.707 5.0 99.3 3.54 0.0884 

1.00 - 2.00 1.41 0.5 99.8 0.71 0.198 

2.00 - 4.00 2.83 0.2 100.0 0.56 0.335 

TOTAL 100 34.2 

 

The fall velocity is determined with the use of Figure 3.1 at 60o

 F. For the geometric mean size 

of the first size range Di

 = 0.011mm, ω = .0001 m/s, similarly Di = 0.707 mm, ω = 0.0884 m/s 

etc. The results are shown in Table 3.7. 

The sand size distribution is plotted on log-probability paper in Figure 3.24 from which the 

following values can be obtained: 

D50 = 0.33 mm 

D16 = 0.24 mm 

D90 = 0.46 mm 

D84 = 0.44 mm 

0.2

0.3

0.4

0.5

0.6

0.8

2

3

4

5

6

8

0.1

1

10

100 90 80 10203040506070 0

Percent Finer

0.1

1

10

Diameter, mm

Figure 3.24. Bed material size distribution curves. 
 3.54 

Finally, the gradation coefficient G is calculated using Equation 3.9. 

 





 



  = +



 





+=

mm33.0

mm44.0

mm24.0

mm33.0

2

1

2

1 G

D

D

D

D

50

84

16

50

G = 1.35 

Gravel Bed Channel 

A gravel bed stream channel was sampled and the grain size analysis yielded the following 

results (Table 3.8): 

 

Table 3.8. Gravel Bed Material Size Distribution. 

Size Range 

(mm) 

Percent of Total Weight 

in Size Range 

.125 - .250 0.1 

.250 - .500 1.6 

.500 - 1.00 1.7 

1.00 - 2.00 6.3 

2.00 - 4.00 31.8 

4.00 - 8.00 58.5 

From this size distribution calculate the following statistics: the geometric mean of each size 

range; the effective diameter Dm; the fall velocity of each size range; the Dm, D16, D50, D84, D90

sizes; and the gradation coefficient. 

The geometric mean size is calculated as the square root of the product of the end points of a 

given size range. For the largest size range: 

Di

 = [(4) (8)]1/2 = 5.66 mm. 

Likewise the rest of the size ranges are calculated, with the results shown in Table 3.9. 

The effective diameter of the sample distribution is calculated with Equation 3.12: 

( )( ) ( )( )

100

mm66.55.58....mm177.01.0

100

p D D si i

n

1i

m

+ = = =

mm32.4

100

9.431 Dm ==

The fall velocity is calculated with the use of Figure 3 .1 at 60o

 F, for each geometric mean size. 

For the largest geometric mean size: Di = 5.66 mm, ω = 0.518 m/s. Likewise the rest of the 

size ranges, the results are shown in Table 3.9. 

The gravel bed material size distribution was plotted on log-probability paper (Figure 3.24) from 

which the following values were obtained: 
 3.55 

D16 = 2.4 mm D84 = 6.7 mm 

D50 = 4.5 mm D90 = 7.1 mm 

The gradation coefficient is calculated by Equation 3.9: 





 



  = +



 





+=

mm5.4

mm7.6

mm4.2

mm5.4

2

1

2

1 G

D

D

D

D

50

84

16

50

G = 1.67 

Table 3.9. Gravel Bed Material Properties. 

Size Range 

(mm) 

Geometric 

Mean Size, Di

(mm) 

Percent of Bed 

Material in this 

Size 

(pi

) 

Percent 

Finer pi

Di Fall Velocity 

(m/s) 

.125 - 0.250 0.177 0.1 0.1 0.018 0.018 

.250 - 0.500 0.354 1.6 1.7 0.566 0.046 

.500 - 1.00 0.707 1.7 3.4 1.20 0.088 

1.00 - 2.00 1.41 6.3 9.7 8.88 0.198 

2.00 - 4.00 2.83 31.8 41.5 90.0 0.335 

4.00 - 8.00 5.66 58.5 100.0 331.1 0.518 

TOTAL 100 431.9 

Pebble Count 

The b axis (Intermediate length) of 100 particles were randomly picked up and measured on 

a line in and along the flow of the Cache la Poudre River. The sampling was done by 

walking in a straight line and picking up and measuring the particle at the big toe at each 

step. The b axis lengths in meters are; 0.046, .189, .146, .436, 137, .280, .207 .226, .128, 

.256, .262, .067, .088, .195, .259, .216, .177, .055, .226, .113, .201, .098, .134, .082, .016, 

.140, .210, .159, .028, .195, .104, .192, .155, .098, .177, .146, .034, .040, .018, .256, .171, 

.146, .104, .226, .192, .049, .226, .015, .110, .128, .082, .180, .067, .037, .162, .195, .110, 

.128, .232, .021, .046, .159, .226, .177, .235, .192, .122, .015, .082, .223, .195, .070, .040, 

.265, .192, .113, .143, .113, .155, .110, .098, .119, .226, .073, .113, .174, .241, .012, .049, 

.192, .110, .082, .131, .043, .137, .171, .046, .131, .104, .192 

Determine the D50, D16, D84, D90, and G of the bed material. 

First decide on discrete size groups and then determine the number of particles in each 

group. 

Size Range 

(m) 

Class 

No. 

Percent 

Finer 

0.305 - 0.436 1 100 

0.244 - 0.305 6 94 

0.198 - 0.244 14 80 

0.152 - 0.198 24 56 

0.091 - 0.152 27 29 

0.030 - 0.091 20 9 

0.012 - 0.030 8 1 

 TOTAL 100 
 3.56 

Plot on log-probability paper to determine D50, D16, D84, D90, and G of the bed material (Figure 

3.25). 

 

D50 = 0.19 m (190 mm) 

D16 = 0.23 m (120 mm) 

D90 = 0.28 m (280 mm) 

D84 = 0.26 m (260 mm) 

D75 = 0.23 m (230 mm) 

 

48.1

m19.0

m26.0

m12.0

m19.0

2

1

2

1 G

D

D

D

D

50

84

16

50 = 



 



  = +



 





+=

20

30

40

50

60

80

200

300

400

500

600

800

100 30405060708090 20 10 0

Percent Finer

10

100

1000

Diamter (mm)

Figure 3.25. Size distribution curve for pebble count (1,000 mm = 1 m = 3.28 ft). 

3.7.2 PROBLEM 2 Angle of Repose 

Determine the angle of repose for well rounded riprap with a D30 of 1 ft = 305 mm. From 

Figure 3.4 the angle of repose is 40o

. 
 3.57 

3.7.3 PROBLEM 3 Resistance to Flow in Alluvial Channels 

Manning's n in Sand Bed Streams 

(a) Plane bed 

Considering only the bed of the stream determine the following: 

What is the range of Manning's n values for a plane bed sand channel stream? 

Manning's n ranges from 0.010 to 0.013. 

What is the Manning's n for a plane bed stream with a D50 of 0.32 mm. 

Using the Strickler equation as an approximation: 

n = 0.0482 D1/6 with D in m n = 0.0482 (0.00032 m)1/6 = 0.013 

(b) Antidune Flow 

A sandbed channel is observed to have an undulating water surface, a discharge of 24.07 

m3

/s, an average velocity of 1.06 m/s, a channel width of 32.00 m, and a bed slope of 0.003. 

The stream bed has a D50 of 0.35 mm. An estimate of the bed form and n-value of the channel 

is desired. 

 

Assuming the sieve diameter equals the fall diameter, and the bed slope equals the friction 

slope. Determine the stream power. 

Stream power equals Vγyo So = (1.06) (9800) (24.07/32.00 x 1.06) (0.003) = 22.126 N/sec-m. 

Figure 3.13 is in English units. Therefore convert velocity and stream power to English units. 

V = 1.06 x 3.281 = 3.48 ft/sec 

ftsec

lb 152

ft281.3

m1

x

N

lb225.0

x

Nsec

N Stream Power 126.22 − = − =

Figure 3.13 for this stream power indicates that upper flow regime is expected. The bed 

configuration should be antidunes with standing waves. 

Based upon the bedform the n-value is estimated to be 0.013. 

Manning's n in Gravel Bed Streams 

Use the following gravel bed material size analysis to estimate the Manning's n-value of the 

stream. 

D16 = 1.8 mm D75 = 4.0 mm D90 = 4.9 mm 

D50 = 3.1 mm D84 = 4.4 mm 
 3.58 

Using Equations 3.15 through 3.22 

( ) 0031.0417.0D0417.0n 016.0 167. 167. = 50 = = (3.15) 

( ) 0031.0482.0D0482.0n 018.0 167. 167. = 50 = = (3.16) 

( ) 004.00473.0D0473.0n 019.0 167. 167. = 75 = = (3.17) 

( ) 0049.0046.0D046.0n 019.0 167. 167. = 90 = = (3.18) 

(Assume yo = 5.0 ft = 1.52 m and R = yo) 

019.0

0044.0

52.1 log0.216.1

)52.1(113.0

log0.216.1

113.0

n

167.0

84

0

6/1

0

D

y

y =





 



 +

=

















+

= (3.19) 

 

19.8

0044.0

52.1 9.1

D

R 9.1

f

1

25.0

84

25.0

2/1 = =  =



 



 













 (3.20) 

f

1/2 = 0.122

015.0)122.0()52.1(113.0f113.0n 2/16/1 6/1 = R = = (3.22) 

18.61.1

0044.0

52.1 log0.21.1

D

R log0.2

f

1

84

2/1 = =+  =+



 



 













 (3.21) 

f

1/2 = 0.162 

020.0)162.0()52.1(113.0f113.0n 2/16/1 6/1 = R = = (3.22) 

Based on the range of n-values indicated above a n-value of 0.018 is selected. Alternatively, 

the maximum and minimum n value could be used and a decision made on the basis of the 

velocity and discharge. 

Manning's n in Cobble Bed Streams 

What are the Manning's n values for a cobble bed stream with the following size distribution: 

D50 = 0.19 m (190 mm) 

D16 = 0.12 m (120 mm) 

D90 = 0.28 m (280 mm) 

D84 = 0.25 m (250 mm) 

D75 = 0.23 m (230 mm) 
 3.59 

( ) 032.0m19.00417.0nmetersinDwithD0417.0n 6/1

50

6/1 = = = (3.15) 

( ) 037.0m19.00482.0nmetersinDwithD0482.0n 6/1 6/1 = 50 = = (3.16) 

( ) 037.0m23.00473.0nmetersinDwithD0473.0n 6/1

75

6/1 = 75 = = (3.17) 

( ) 037.0m28.0046.0nmetersinDwithD046.0n 6/1

90

6/1 = 90 = = (3.18) 

3.7.4 PROBLEM 4 Beginning of Motion 

Bed Material Movement Using Shields Figure 

To establish if the bed material of a channel is in motion the Shields' relationship can be 

used. Recall that τo = γRS and V* = / ρτo

(a) For the following conditions determine if the bed material in a sand bed channel is in 

motion. 

R = 1.22 m 

S = 0.00038 m/m 

D50 = 0.31mm 

D16 = 0.24mm 

D90 = 0.46mm 

D84 = 0.42mm 

The shear stress on the bed, at a single vertical in the cross-section, is 9800 N/m3

 x 1.22 m x 

0.00038 = 4.543 N/m2

τo/(γs - γ) Ds = 4.543 / (25970-9800) 0.00031 = 0.91 

V* Ds/ν = (4.543/1000)1/2 (0.00031)/1.31 x 10-6 = 15.95 

This point plots above the incipient motion line in Shields (Figure 3.15) and indicates the bed of 

the channel is in motion. 

(b) For the gravel sized material with the same values of τo = 4.543 N/m2

; Ds = 3.1 mm we 

have: 

τo/(γs - γ) Ds = 4.543 /(25970-9800) (0.0031) = 0.091 

V* Ds/ ν= (4.543/1000)1/2 (0.0031)/1.31 x 10-6 = 159 

This point plots above the incipient motion line in Shields (Figure 3.15) and indicates this 

channel bed is in motion. 
 3.60 

Critical Velocity for Beginning of Bed Material Movement 

(a) Sand Size Bed Material 

Given depth y is 3.66 m and bed material size D50 is 0.31 mm, what is the critical velocity Vc? 

00031.0x66.3x19.6Dy19.6V s/m52.0 3/16/1 6/1 3/1

c = = =

(b) Gravel Size Bed Material 

Given depth y is 3.66 m and bed material size D50 Is 3.1 mm, what is the critical velocity Vc? 

s/m12.10031.0x66.3x19.6Dy19.6V 3/16/1 6/1 3/1

c = = =

(c) Cobble size Bed Material 

Given depth y is 3.66 and bed material size D50 is 128 mm, what is the critical velocity Vc? 

s/m88.3128.0x66.3x19.6Dy19.6V 3/16/1 6/1 3/1

c = = =

Critical Size for Beginning of Bed Material Movement 

Given depth y is 3.66 m and velocity V is 0.61 m/s, 1.22 m/s, 2.44 m/s, 3.66 m/s, and 4.88 

m/s, respectively. 

Determine the critical bed material size D50. 

V = 0.61 m/s

)mm50.0(m0005.0 66.3

61.0)0042.0(

y

V0042.0 D 2/1

3

2/1

3

c = = =

V = 1.22 m/s

)mm99.3(m00399.0 66.3

22.1)0042.0(

y

V0042.0 D 2/1

3

2/1

3

c = = =

V = 2.44 m/s 

)mm9.31(m0319.0 66.3

44.2)0042.0(

y

V0042.0 D 2/1

3

2/1

3

c = = =

V = 3.66 m/s 

)mm108(m108.0 66.3

66.3)0042.0(

y

V0042.0 D 2/1

3

2/1

3

c = = =
 3.61 

V = 4.88 m/s 

)mm255(m266.0 66.3

88.4)0042.0(

y

V0042.0 D 2/1

3

2/1

3

c = = =

Critical Depth When the Bed Material Movement Would Stop 

(a) Given velocity V of 0.61 m/s and bed material size D50 of 0.305 mm, determine the 

critical depth y for no bed material movement. 

m86.9

000305.0

61.0)10x78.1(

D

V10x78.1

y 2

5 6

2

65

= = =

− −

(b) Given velocity V of 0.61, 2.44, and 4.88 m/s and bed material size D50 of 152 mm, 

determine the critical depth y when bed material movement would stop. 

V = 0.61 m/s 

m00004.0

152.0

61.0)10x78.1(

D

V10x78.1

y 2

5 6

2

65

= = =

− −

There would not be any bed material movement at this velocity. 

V = 2.44 m/s

m16.0

152.0

44.2)10x78.1(

D

V10x78.1

y 2

5 6

2

65

= = =

− −

V = 4.88 m/s 

m4.10

152.0

88.4)10x78.1(

D

V10x78.1

y 2

5 6

2

65

= = =

− −

Would this depth occur? No. Why not? The coarser material would armor the bed. 

3.8 SOLVED PROBLEMS FOR ALLUVIAL CHANNEL FLOW (ENGLISH) 

3.8.1 PROBLEM 1 Sediment Properties and Fall Velocities 

Generally, sediment sizes are given in mm for the English system. Therefore, the solution to 

this problem is identical to the SI solution in Section 3.7.1. 
 3.62 

3.8.2 PROBLEM 2 Angle of Repose 

Determine the angle of repose for well rounded riprap with a D30 of 1 ft = 305 mm. 

From Figure 3.4 the angle of repose is 40o

. 

3.8.3 PROBLEM 3 Resistance to Flow in Alluvial Channels 

Manning's n in Sand Bed Streams 

(a) Plane bed 

 Considering only the bed of the stream determine the following: 

 What is the range of Manning's n values for a plane bed sand channel stream? 

 Manning's n ranges from 0.010 to 0.013. 

 What is the Manning n for a plane bed stream with a D50 of 0.32 mm. 

 Using the Strickler equation as an approximation: 

 

 n = 0.0395 D1/6 with D in ft. n = 0.0395 (0.0010 ft)1/6 = 0.013 

(b) Antidune Flow 

A sandbed channel is observed to have an undulating water surface, a discharge of 850 cfs, 

an average velocity of 3.48 ft/sec, a channel width of 105 ft, and a bed slope of 0.003. The 

stream bed has a D50 of 0.35 mm. An estimate of the bed form and n-value of the channel is 

desired. 

 

Assuming the sieve diameter equals the fall diameter, and the bed slope equals the friction 

slope. Determine the stream power. 

Stream power equals Vγyo So = (3.48) (62.4) (850/105 x 3.48) (0.003) = 1.53 lb/sec-ft. Figure 

3.13 for this stream power indicates that upper flow regime is expected. The bed configuration 

should be antidunes with standing waves. 

Based upon the bedform the n-value is estimated to be 0.013. 

Manning's n in Gravel Bed Streams 

Use the following gravel bed material size analysis to estimate the Manning's n value of the 

stream.

D16 = 1.8 mm D75 = 4.0 mm D90 = 4.9 mm 

D50 = 3.1 mm D84 = 4.4 mm 

Using Equations 3.15 through 3.22 
 3.63 

016.0)0102.0()0342.0(D342.0n 167. 167. = 50 = = (3.15) 

018.0)0102.0()0395.0(D395.0n 167. 167. = 50 = = (3.16) 

019.0)0131.0()0388.0(D388.0n 167. 167. = 75 = = (3.17) 

019.0)0161.0(038.0D038.0n 167. 167. = 90 = = (3.18) 

(Assume yo = 5.0 ft and R = yo) 

019.0

0144.0

0.5 log0.216.1

)0.5(0927.0

D

y log0.216.1

y0927.0 n

167.0

84

0

6/1

0 =





 



 +

=

















+

= (3.19) 

 

20.8

0144.0

00.5 9.1

D

R 9.1

f

1 25.0

84

2/1  = 

 



 =













 = (3.20) 

f

1/2 = 0.122 

015.0)122.0()0.5(0927.0fR0927.0n 2/16/1 6/1 = = = (3.22) 

18.61.1

0144.0

0.5 log0.21.1 DR log0.2 f

1

84

2/1  =+

 



 =+













 = (3.21) 

f

1/2 = 0.162 

020.0)162.0()0.5(0927.0f0927.0n 2/16/1 6/1 = R = = (3.22) 

Based on the range of n-values indicated above a n-value of 0.018 is selected. Alternatively, 

the maximum and minimum n value could be used and a decision made based of the velocity 

and discharge. 

Manning's n in Cobble Bed Streams 

What are the Manning's n values for a cobble bed stream with the following size distribution: 

D50 = 0.62 ft (190 mm) 

D16 = 0.39 ft (120 mm) 

D90 = 0.92 ft (280 mm) 

D84 = 0.82 ft (250 mm) 

D75 = 0.75 ft (230 mm) 

032.0)62.0(0342.0n.ftinDwithD342.0n 6/1

50

6/1 = 50 = = (3.15) 

037.0)ft62.0(0395.0n.ftinDwithD395.0n 6/1

50

6/1 = 50 = = (3.16) 
 3.64 

037.0)ft75.0(0388.0n.ftinDwithD388.0n 6/1

75

6/1 = 75 = = (3.17) 

037.0)ft92.0(038.0n.ftinDwithD038.0n 6/1

75

6/1 = 90 = = (3.18) 

 

3.8.4 PROBLEM 4 Beginning of Motion 

Bed Material Movement Using Shields Figure 

To establish if the bed material of a channel is in motion the Shields' relationship can be 

used. Recall that τo = γRS and V* = / ρτo .

(a) For the following conditions determine if the bed material in a sand bed channel is in 

motion. 

R = 4.0 ft 

S = 0.00038 ft/ft 

D50 = 0.31mm = 0.00102 ft 

D16 = 0.24mm = 0.0008 ft 

D90 = 0.46mm = 0.0015 ft 

D84 = 0.42mm = 0.0014 ft 

The shear stress on the bed, at a single vertical in the cross-section, is 62.4 lb/ft3

 x 4.0 ft x 

0.00038 = 0.0948 lb/ft2

. 

τo/(γs - γ) Ds = 0.095/ (165 - 62.4) (0.00102) = 0.92 

V* Ds/ν = (0.095/1.94)1/2 (.001)/ 1.41 x 10-5 = 15.69 

This point plots above the incipient motion line in Shields (Figure 3.15) and indicates the bed of 

the channel is in motion. 

(b) For the gravel sized material with the same values of τo = 0.095 lb/ft2

; Ds = 3.1 mm = 

0.0102 ft we have: 

τo/(γs - γ) Ds = 0.095/ (165 - 62.4) (0.0102) = 0.091 

V* Ds/ ν= (0.095/1.94)1/2 (0.0102)/ 1.41 x 10-5 = 160 

This point plots above the incipient motion line in Shields (Figure 3.15) and indicates this 

channel bed is in motion. 

Critical Velocity for Beginning of Bed Material Movement 

(a) Sand Size Bed Material 

Given depth y is 12 ft and bed material size D50 is 0.31 mm = 0.00102 ft, what is the critical 

velocity Vc? 

00102.0x12x25.11Dy25.11V s/ft72.1 3/16/1 6/1 3/1

c = = =
 3.65 

 (b) Gravel Size Bed Material 

Given depth y is 12 ft and bed material size D50 Is 3.1 mm = 0.0102 ft, what is the critical 

velocity Vc? 

s/ft70.30102.0x1225.11Dy25.11V 3/16/1 6/1 3/1

c = = =

(c) Cobble size Bed Material 

Given depth y is 12 ft and bed material size D50 is 128 mm = 0.42 ft, what is the critical 

velocity Vc? 

s/ft75.1242.0x12x25.11Dy25.11V 3/16/1 6/1 3/1

c = = =

Critical Size for Beginning of Bed Material Movement 

Given depth y is 12 ft and velocity V is 2.0 ft/s, 4 ft/s, 8.0 ft/s, 12 ft/s, and 16 ft/s, respectively. 

Determine the critical bed material size D50. 

V = 2.0 ft/s

ft00162.0

12

0007.0 0.2)0007.0(

2/1

3

2/1

3

c

y

V D = = = (0.49 mm)

V = 4.0 ft/s

ft0129.0

12

0007.0 0.4)0007.0(

2/1

3

2/1

3

c

y

V D = = = (3.93 mm)

V = 8 ft/s

ft104.0

12

0007.0 0.8)0007.0(

2/1

3

2/1

3

c

y

V D = = = (31.7 mm) 

V = 12 ft/s

ft349.0

12

0007.0 0.12)0007.0(

2/1

3

2/1

3

c

y

V D = = = (106 mm) 

V = 16 ft/s

ft828.0

12

0007.0 0.16)0007.0(

2/1

3

2/1

3

c

y

V D = = = (252 mm)
 3.66 

Critical Depth When the Bed Material Movement Would Stop 

(a) Given velocity V of 2.0 ft/s and bed material size D50 of 0.305 mm = 0.001 ft, determine 

the critical depth y for no bed material movement. 

ft6.31

001.0

0.2)10x94.4(

D

V10x94.4

y 2

67

2

67

= = =

− −

(b) Given velocity V of 2.0, 8.0 and 16 ft/s and bed material size D50 of 152 mm = 0.5 ft, 

determine the critical depth y when bed material movement would stop. 

V = 2 ft/s

ft0001.0

5.0

0.2)10x94.4(

D

V10x94.4

y 2

67

2

67

= = =

− −

There would not be any bed material movement at this velocity. 

V = 8 ft/s

ft52.0

5.0

0.8)10x94.4(

D

V10x94.4

y 2

67

2

67

= = =

− −

V =16 ft/s

ft28.33

5.0

0.16)10x94.4(

D

V10x94.4

y 2

7 6

2

67

= = =

− −

Would this depth occur? No. Why not? The coarser material would armor the bed. 
4.1 

CHAPTER 4 

SEDIMENT TRANSPORT 

4.1 INTRODUCTION 

This chapter describes key terms and several methods of computing sediment transport in 

alluvial channels. Since a significant portion of understanding sediment transport processes 

centers on correct use of the terminology, the first two sections deal with definitions and 

general concepts. The chapter also includes the derivation of the basic suspended bed 

sediment transport equation to illustrate the significant physical processes of sediment 

entrainment into the flow. Three classic sediment transport formulae are then discussed to 

illustrate the application of sediment transport theory. 

Simple sediment transport equations, in the form of power functions, are also presented in 

this chapter. The power function equations are well suited for quick estimates of sediment 

transport capacity and are easily adapted to field measurements. The Yang sand and gravel 

total load equations are also suggested as a basic approach for hand calculation. In 

addition, an overview of selected sediment transport equations is presented. The overview 

suggests the range of applicability of these equations (based on bed-material size), and 

provides references to those equations not discussed in detail in this chapter. Finally, a stepwise application procedure for sediment transport calculations is outlined. 

Numerous sediment transport formulae have been developed with a wide range of laboratory 

and field conditions. Appendix B includes the results of testing several widely used sediment 

transport equations using a large compilation of river data. 

4.2 DEFINITIONS 

In this section the basic terms for describing sediment load in alluvial channels are 

summarized. 

Bed layer: The flow layer, several grain diameters thick (usually taken as two grain diameters 

thick), immediately above the bed. 

Bed load: Sediment that moves by rolling or sliding along the bed and is essentially in contact 

with the stream bed in the bed layer, i.e. contact load. 

Bed material: The sediment mixture of which the stream bed is composed. 

Bed material discharge (load): That part of the total sediment discharge which is composed of 

grain sizes found in the bed, i.e. bed load (contact load) plus suspended bed material 

discharge (load). The total transported bed material discharge (or bed sediment discharge) is 

assumed equal to the transport capacity of the flow. 
4.2 

Contact load: Sediment particles that roll or slide along in almost continuous contact with the 

stream bed, i.e., bed load. 

Density of water-sediment mixture: The mass per unit volume including both water and 

sediment. 

Discharge-weighed concentration: The dry weight of sediment in a unit volume of stream 

discharge, or the ratio of the discharge of dry weight of sediment to the discharge by weight of 

water-sediment mixture normally reported in parts per million (ppm) or parts per liter (ppl). 

Load (or sediment load): The sediment that is being moved by a stream. 

Sediment (or fluvial sediment): Fragmentary material that originates from weathering of rocks 

and is transported by, suspended in, or deposited from water. 

Sediment concentration (by weight or by volume): The quantity of sediment relative to the 

quantity of transporting fluid, or fluid-sediment mixture. The concentration may be by weight or 

by volume. When expressed in ppm or mg/l, the concentration is always in ratio by weight. 

Sediment discharge: The quantity of sediment that is carried past any cross-section of a 

stream in a unit of time. 

Sediment yield: The dry weight of sediment per unit volume of water-sediment mixture in 

place, or the ratio of the dry weight of sediment to the total weight of water-sediment mixture in 

a sample or a unit volume of the mixture. 

Suspended load (or suspended sediment): Sediment that is supported by the upward 

components of turbulence in a stream and that stays in suspension for an appreciable length of 

time. 

Suspended bed load: The portion of suspended load which is composed of grain sizes found 

in the bed. 

Suspended-sediment discharge: The quantity of suspended sediment passing through a 

stream cross-section outside the bed layer in a unit of time. 

Total sediment discharge: The total sediment discharge of a stream. It is the sum of the 

suspended-sediment discharge and the contact sediment discharge, or the sum of the bed 

sediment discharge and washload, or the sum of the measured sediment discharge and the 

unmeasured sediment discharge. 

Unmeasured sediment discharge: Sediment discharge close to the bed that is not sampled by 

a suspended sediment sampler. 

Washload: That part of the total sediment discharge (load) which is composed of particle sizes 

finer than those found in appreciable quantities in the bed and is determined by available bank 

and upstream supply rate. Also called fine sediment discharge (load). 
4.3 

4.3 GENERAL CONSIDERATIONS 

The amount of material transported or deposited in the stream under a given set of conditions 

is the result of the interaction of two groups of variables. In the first group are those variables 

that influence the quantity and quality of the sediment brought down to that section of the 

stream. In the second group are variables that influence the capacity of the stream to transport 

that sediment. A list of these variables is given as follows. 

Group 1 - Sediment brought down to the stream depends on the geology and topography of 

watershed; magnitude, intensity, duration, distribution, and season of rainfall; soil moisture 

conditions; vegetal cover; cultivation and grazing; surface erosion and bank cutting. 

Group 2 – The capacity of a stream to transport sediment depends on hydraulic properties of 

the stream channel. These are fluid properties, slope, roughness, hydraulic radius, 

discharge, velocity, velocity distribution, turbulence, tractive force, viscosity and density of 

the fluid sediment mixture, and size and gradation of the sediment. 

These variables are not all independent and, in some cases, their effect is not definitely 

known. The variables which control the amount of sediment brought down to the stream are 

subject to so much variation, not only between streams but at a given point of a single 

stream, that the quantitative analysis of any particular case is extremely difficult. It is 

practicable, however, to measure the sediment discharge over a long period of time and 

record the results, and from these records to determine a soil loss from the area. 

The variables that deal with the capacity of the stream to transport solids are subject to 

mathematical analysis. These variables are closely related to the hydraulic variables 

controlling the capacity of the stream to carry water. 

4.3.1 Source of Sediment Transport

Einstein (1964) stated that: 

Every sediment particle which passes a particular cross-section of the stream 

must satisfy the following two conditions: (1) it must have been eroded 

somewhere in the watershed above the cross-section; (2) it must be 

transported by the flow from the place of erosion to the cross-section. 

Each of these two conditions may limit the sediment rate at the cross-section, 

depending on the relative magnitude of two controls: the availability of the 

material in the watershed and the transporting ability of the stream. In most 

streams, the finer part of the load, i.e., the part which the flow can easily carry 

in large quantities, is limited by its availability in the watershed. This part of 

the load is designated as washload. The coarser part of the load, i.e., the part 

that is more difficult to move by flowing water, is limited in its rate by the 

transporting ability of the flow between the source and the section. This part 

of the load is designated as bed sediment load. 

Thus, for engineering purposes, the two sources of sediment transported by a stream are: (1) 

bed material that makes up the stream bed; and (2) fine material that comes from the banks 

and the watershed (washload). Geologically both materials come from the watershed. But for 
4.4 

the engineer, the distinction is important because the bed material is transported at the 

capacity of the stream and is functionally related to measurable hydraulic variables. The 

washload is not transported at the capacity of the stream. Instead, the washload depends on 

availability and is not functionally related to measurable hydraulic variables. 

No sharp demarcation exists between washload discharge and bed material discharge. As a 

rule of thumb, many engineers assume that the bed material discharge is composed of sizes 

(Ds) equal to or greater than 0.062 mm which is also the division point between sand and silt. 

The sediment discharge consisting of grain sizes smaller than 0.062 mm is considered as 

washload. A more reasonable criterion is to choose a sediment size finer than the smallest 10 

percent of the bed material as the dividing size between washload and bed sediment load. It is 

important to note that in a fast flowing mountain stream with a bed of cobbles the washload 

may consist of coarse sand sizes. For these conditions, the transport of sand sizes is supply 

limited. In contrast, if the bed of a channel is silt, the rate of bed load transport of the silt sizes 

is less a question of supply than of capacity. The sediment transport capacity of silts has been 

addressed in some transport equations. This invalidates the criterion based on Ds < 0.062 mm 

for coarse bed sediment streams, however, the criterion based on D10 might still be applicable. 

4.3.2 Mode of Sediment Transport 

Sediment particles are transported by rolling or sliding on the bed (bed load or contact load) or 

by suspension by the turbulence of the stream. Even as there is no sharp demarcation 

between bed material discharge and washload there is no sharp line between contact load and 

suspended sediment load. A particle may move part of the time in contact with the bed and at 

other times be suspended by the flow. The distinction is important because the two modes of 

transport follow different laws. The equations for estimating the total bed material discharge of 

a stream are based on these laws. 

A further subdivision of mode of transport of sediment, including a pictorial representation of 

measured load and unmeasured load follows. 

When a river reaches equilibrium, its transport capacities for water and sediment are in 

balance with the rates supplied. In fact, most rivers are subject to some kind of control or 

disturbance, natural or human-induced, that give rise to non-equilibrium conditions. 

Total sediment load can be expressed by three equations as illustrated in Figure 4.1 (Julien 

1995): 

1. By type of movement 

+= LLL sbT (4.1a) 

2. By method of measurement 

+= LLL umT (4.1b) 

3. By source of sediment 

wT += LLL bm (4.1c) 
4.5 

where: 

 LT = Total load 

 Lb = Bed load which is defined as the transport of sediment particles that are 

close to or maintain contact with the bed 

 Ls = Suspended load defined as the suspended sediment passing through a 

stream cross-section above the bed layer 

 Lm = Measured sediment 

 Lu = Unmeasured sediment that is the sum of bed load and a fraction of 

suspended load below the lowest sampling elevation 

 Lw = Wash load which is the fine particles not found in the bed material 

(Ds < D10), and originates from available bank and upstream supply 

 Lbm = Capacity limited bed material load 

Figure 4.1. Classification of sediment transport in streams (rivers). 

4.3.3 Total Sediment Discharge 

The total sediment discharge of a stream is the sum of the bed sediment discharge (bed 

material load) and the fine sediment discharge (washload), or the sum of the contact 

sediment discharge (bed load) and suspended sediment discharge. In the former sum, the 

total sediment discharge is based on source of the sediments and the latter sum is based on 

the mode of sediment transport. Whereas suspended sediment load consists of both bed 

sediments and fine sediments (washload), only the bed sediment discharge can be 

estimated by the various equations that have been developed. The fine sediment discharge 

(washload) depends on its availability not on the transporting capacity of the flow and must 

be measured (Figure 4.2). The presence of high concentrations of wash load affects the 

apparent viscosity of the water-sediment mixture and reduces the fall velocity of silt and sand 

grains. 

The sediment load that is measured by suspended-sediment samplers consists of both the 

washload (fine sediment load) and suspended-sediment load. The contact load is not 

measured, and because suspended sediment samplers cannot travel the total distance in the 

vertical to the bed, part of the suspended sediment in a vertical is not measured. Generally, 

the amount of bed material moving in contact with the bed of a large sand-bed river is from 5 

to 10 percent of the bed material moving in suspension. In general, the measured 

suspended-sediment discharge is from 90 to 95 percent of the total sediment discharge. 

However, in shallow sand-bed streams with little or no washload the measured 

suspended-sediment load may be as small as 50 percent of the total load. 
4.6 

 Figure 4.2. Sediment transport capacity and supply curves (Shen 1971; Simons and 

 Sentürk 1992; Julien 1995). 

The magnitude of the suspended or bed sediment discharge can be very large. 

Suspended-sediment concentrations as large as 600,000 ppm or 60 percent by weight have 

been observed. Concentrations of this magnitude are largely fine sediments. By increasing 

fluid properties (viscosity and density), the fine material in the flow increases the capacity of 

the flow to transport bed material. 

The sediment load of a stream at a cross section or through a reach of a stream can be 

determined by measuring the suspended-sediment portion of the load using samplers and 

estimating the unmeasured discharge or by using one of the many methods that have been 

developed for computing the bed sediment load and estimating the washload. In many 

problems, only the bed sediment load, both in suspension and in contact with the bed, is 

important. In these cases, the washload can be eliminated from the measured 

suspended-sediment load if the size distribution of the material is known. 

Many equations have been developed for the estimation of bed sediment transport. The 

variation between the magnitude of the bed sediment discharge predicted by different 

equations under the same conditions can be significant. For the same water discharge, the 

predicted sediment discharge can have a 100-fold difference between the smallest and the 

largest value. This can be expected given the number of variables, the interrelationships 

among them, the difficulty of measuring many of the variables and the statistical nature of 

bed material transport. Nevertheless, with proper use, knowledge of the river, and 

knowledge of the limitations of each method, useful bed material discharge information can 

be obtained. 

4.4 SUSPENDED BED SEDIMENT DISCHARGE 

The sediment transport process is best described through a discussion of suspended bed 

sediment discharge, which usually accounts for the majority of the total load. The amount of 

material transported in suspension varies with depth, with the highest concentrations 

occurring near the bed. At equilibrium transport, the vertical exchange of sediment is 

balanced between particle settling (due to gravity) and turbulence (that mixes higher 

concentration flow up into the water column). For very fine particles, settling is small in 

comparison to turbulent mixing and the sediment concentration is vertically uniform. For 
4.7 

coarser particles, settling is rapid and particles that are in suspension tend to concentrate 

nearer the bed. 

The suspended bed sediment discharge in Newtons or lbs per second per unit width of 

channel, qs, for steady, uniform two-dimensional flow is 

q vcdy yo

 asS γ= (4.2) 

where v and c vary with y and are the time-averaged flow velocity and volumetric 

concentrations, respectively, and γs is the weight per unit volume of the suspended sediment. 

The integration is taken over the depth between the distance "a" above the bed and the 

surface of the flow "yo." The level "a" is generally assumed to be 2-grain diameters above the 

bed layer. Sediment movement below this level is considered bed load rather than suspended 

load. 

The discharge of suspended sediment for the entire stream cross-section, QS, is obtained by 

integrating Equation 4.2 over the cross section to give 

γ= SS CQQ (4.3) 

where: 

C = Average suspended-sediment concentration by volume 

The vertical distribution of both the velocity and the concentration vary with the mean velocity 

of the flow, bed roughness and size of bed material. The distributions are illustrated in 

Figure 4.3. Also v and c are interrelated. That is, the velocity and turbulence at a point is 

affected by the sediment at the point, and the sediment concentration at the point is affected 

by the point velocity. Normally this interrelation is neglected for low sediment concentrations 

or a coefficient applied to compensate for it. 

 Figure 4.3. Schematic sediment and velocity profiles. 

To integrate Equation 4.2, v and c must be expressed as functions of y. The one-dimensional 

gradient type diffusion equation is employed to obtain the vertical distribution for c and the 

logarithm velocity distribution is assumed for v in turbulent flows. 
4.8 

The one-dimensional diffusion equation describes the equilibrium condition when the quantity 

of sediment settling across a unit area due to the force of gravity is equal to the quantity of 

sediment transported upwards resulting from the vertical component of turbulence and the 

concentration gradient. The resulting equation for a given particle size is 

dy/dcc S ε−=ω (4.4) 

where: 

ω = Fall velocity of the sediment particle at a point

c = Concentration of particles at elevation y above the bed

εS = Exchange coefficient, also called the mass transfer coefficient, which 

characterizes the magnitude of the exchange of particles across any 

arbitrary boundary by the turbulence

dc/dy = Concentration gradient

ωc = Average rate of settling of the sediment particles

εS dc/dy = Average rate of upward sediment flow by diffusion

Integrating Equation 4.4 yields 

{  εω−= s } y

a a expcc /dy o (4.5) 

where ca is the concentration of sediment with settling velocity equal to ω at a level y = a. 

In order to determine the value of c at a given y, the value of ca and the variation of εs with y 

must be known. To obtain an expression for εs the assumption is made that 

s =ε βε m (4.6) 

where: 

εm = Kinematic eddy viscosity or the momentum exchange coefficient defined by

( ))dy/dv m ρ

τ

=ε (4.7) 

where: 

τ and dv/dy = Shear stress and velocity gradient, respectively, at point y

For two-dimensional steady uniform flow 

o o −τ=−γ=τ o )y/y1()yy(S (4.8) 
4.9 

and from Equation 2.74 

y

V

y

/

dy

dv o *

κ = κ

ρτ = (4.9) 

Thus 

s m o )y/y1(yV* =ε βε = βκ − (4.10) 

where: 

β = Coefficient relating εs to εm

κ = Von Karman's velocity coefficient assumed equal to 0.4

V* = Shear velocity equal to gRS in steady uniform flow

Equation 4.10 indicates that εm and εs are zero at the bed and at the water surface, and have a 

maximum value at mid-depth. The substitution of Equation 4.10 into Equation 4.4 gives 

)

y

y 1(y

dy

Vc

dc

o

* − βκ

ω− = (4.11) 

and after integration 

Z

o

o

a ay

a

y

yy

c

c 



 





−

− = (4.12) 

where: 

 c = Concentration at a distance y from the bed 

 ca = Concentration at a point a above the bed 

 z = ω/βκV*, the Rouse number, named after the engineer who developed the 

equation in 1937 

Figure 4.4 shows a family of curves obtained by plotting Equation 4.12 for different values of 

the Rouse number z. An evaluation of the Rouse number, z, shows that for small values, the 

sediment distribution is nearly uniform. For large z values, little sediment is found near the 

water surface. The value of z is small for large shear velocities V* or small fall velocities ω. 

Thus, for small particles or for extremely turbulent flows, the concentration profiles are uniform. 

The values of β and κ have been investigated. For fine particles, β is approximately 1.0. 

Also, it is well known that in clear water κ = 0.4, but apparently decreases with increasing 

sediment concentration. 

Using the logarithmic velocity distribution for steady uniform flow and Equation 4.12, the 

equation for suspended sediment transport becomes 
4.10 

dy k

Xy 2.30ln5.2

y

yy

ay

a cVq

S

Z

o

o

y

S aa

o

* 





















	









 



 



 −

− γ=  (4.13) 

Many investigators have solved this equation through integration. The assumptions and 

integration made by Einstein are presented in Section 4.5.2. An example calculation for a 

suspended sediment concentration profile is given in Section 4.12 (SI) and 4.13 (English). 

Figure 4.4. Graph of suspended sediment distribution (Rouse 1937). 

4.5 BED SEDIMENT DISCHARGE 

In this section, three classic sediment transport formulae are discussed in detail to illustrate 

sediment transport processes. These are the Meyer-Peter and Müller (1948), Einstein (1950), 

and Colby (1964) methodologies. The Meyer-Peter and Müller bedload equation (Section 

4.5.1) is applicable to streams with bed material consisting of sand, gravel, and cobbles. The 

Einstein (Section 4.5.2) and Colby (Section 4.5.4) methods are bed material load equations 

used for sand-bed streams. In Section 4.6, power function sediment transport relationships are 

discussed to provide a practical method for quick sediment transport calculations. The Yang 

sand and gravel total load equations are also included (Section 4.7) because of their frequent 

application and wide acceptance. 

4.5.1 Meyer-Peter and Müller Bed Load Equation 

Meyer-Peter and Müller (1948) developed the following bed load equation based on 

experiments with sand particles of uniform sizes, sand particles of mixed sizes and density, 

natural gravel, lignite, and barite: 

( ) 3/2

B

3/2

s

s

3/1

fo s m

2/3

r

Bb q

g

BDBSy K

K

Q

Q

















γ

γ−γ















 γ =γ ′ +γ−γ 





















 





 (4.14) 
4.11 

where: 

 qB = Bed-load rate in weight per unit time and per unit width 

 Qb = Water discharge quantity determining bed load transport 

 Q = Total water discharge 

 yo = Depth of flow 

 Sf = Energy slope 

B′, B = Dimensionless constants 

B′ has the value 0.047 for sediment transport and 0.034 for the case of no sediment 

transport. B has a value of 0.25 for sediment transport and is meaningless for no transport 

since qB is zero and the last term drops out. The quantities KB and Kr are defined by the 

expressions 

2/1

f

B 3/2 SR

V K = (4.15) 

and 

2/1

f

3/2 r

SR

V K ′ = (4.16) 

where: 

Sf = Total energy slope

Sf′ = Part of the total slope required to overcome grain resistance

Sf - Sf′ = Part of the total slope required to overcome form resistance

Therefore 

f

b

r

B

gRS

V

8

f

K

K ′ = (4.17) 

Where fb′, the Darcy-Weisbach bed friction factor for the grain roughness, fb′ is determined 

from the Nikuradse pipe friction data with pipe diameter equal to four times the hydraulic 

radius and Ks = D90. If the boundary is hydraulically rough, (V*D90/ν≥ 100), Kr is given by 

6/1

90

r

D

26 K = (4.18) 

where D90 is in meters. 

Equation 4.14 is dimensionally homogeneous so that any consistent set of units may be 

used. Equation 4.14 was converted to units generally used in the United States in the field of 

sedimentation for water and quartz particles by the U.S. Bureau of Reclamation in 1960 and 

is expanded here for application in SI units. This equation is 

2/3

fo m

2/3

b

6/1

90b

2u1uB D627.0Sy

n

D

Q

Q KKq 















− 





	

















	 







 = (4.19) 
4.12 

where: 

 Ku1 = 4.780 in SI units 

 Ku1 = 1.606 in English units 

 Ku2 = 10.846 in SI units 

 Ku2 = 3.306 in English units 

 qB = Metric-tons/day/meter (Tons/day/foot) 

 Qb = Water discharge quantity determining the bed-load transport, m3

/s (cfs) 

 Q = Total water discharge m3

/s (cfs) 

 D90, Dm = In millimeters (both SI and English units) 

The quantity Dm is the effective diameter of the sediment given by 

100

Dp D siii

m

Σ = (4.20) 

where: 

 pi = Percentage by weight of that fraction of the bed material with geometric 

mean size, Dsi

The term nb is the Manning's roughness coefficient for the bed of rectangular channels 

3/2 2/3

o w

b n

n 1

W

y2 1nn 





















	









 



	 






−+= (4.21) 

and for trapezoidal channels 

( ) 3/2 2/3

w

2/1 2

o s

b n

n 1

W

H1y2 1nn





















	











 



 



 − +

+= (4.22) 

where: 

 n, nb, nw = Roughness coefficients of the total stream, of the bed, and of the 

banks, respectively 

 Hs = Horizontal side slope related to one unit vertically 

 W = Bottom width 

The ratio Qb/Q for rectangular channels is given by 

2/3

b

wo

b

n

n

W

y2 1

1

Q

Q





















 





+

= (4.23) 
4.13 

and for trapezoidal channels is 

2/3

b

w

2/12

o s

b

n

n

W

)H1(y2 1

1

Q

Q















+ 

+

= (4.24) 

The Meyer-Peter and Müller formula (Equation 4.14) is often written in the form 

2/3

b c τ−τ= )(Kq (4.25) 

where: 

ργ

≅



































	











γ

γ−γ







	









 γ

=

s

2/3

3/2

s

s

3/1

9.12

g

B

1 K (4.26a) 

fo

2/3

r

b B Sy K

K

Q

Q

γ 





















 







=τ (4.26b) 

( ) c s DB m =τ ′ γ−γ (4.26c) 

An example of sediment transport calculations using the Meyer-Peter and Müller equation is 

given in Section 4.12 (SI) and 4.13 (English). 

4.5.2 Einstein's Method of Computing Bed Sediment Discharge 

Einstein’s (1950) method is included in detail because the formulation provides an excellent 

discussion of sediment transport processes. The equations describe and incorporate the 

physical processes of contact load and suspended load. The method computes a contact 

load concentration and uses this concentration as the starting point for integrating the 

suspended load as presented in Section 4.4. The method also incorporates the vertical 

velocity distribution based on the roughness of the boundary. 

The total bed sediment discharge is the sum of the contact load and the suspended load. As 

mentioned earlier, there is no sharp demarcation between the contact bed sediment load and 

the suspended bed sediment. However, this division is warranted by the fact that there is a 

difference in behavior of the two different loads which justifies two physical equations. 

Einstein's bed sediment discharge function gives the rate at which flow of any magnitude in a 

given channel transports the individual sediment sizes which make up the bed material. This 

makes his equations extremely valuable where it is necessary to determine the change in 

bed material with time. Each size moves at its own rate. For each size Ds of the bed 

material, the contact load is given as 
4.14 

qi BB (4.27) 

and the suspended sediment load is given by 

qi ss (4.28) 

and the total bed material discharge is 

+= qiqiqi BBssTT (4.29) 

and finally 

T Σ= qiQ TT (4.30) 

where iT, is, and iB are the fractions of the total, suspended and contact bed sediment 

discharges qT, qs and qB for a given grain size Ds. The term QT is the total bed sediment 

transport. The suspended sediment load is related to the contact load because there is a 

continuous exchange of particles between the two modes of transport. 

With suspended sediment load related to the contact load, Equation 4.29 becomes 

( ) BBTT 21E ++= IIP1qiqi (4.31) 

where: 

E = o ∆)/y2.30(log3.2P (4.32) 

I1 and I2 are integrals of Einstein's form of the suspended sediment Equation 4.12 

2/1

3 s s

sB

BB

gD

1

i qi *

















ρ−ρ

ρ

γφ = (4.33) 

and 

γs = Unit weight of sediment 

ρ = Density of the water 

ρs = Density of the sediment 

 g = Gravitational acceleration 

φ* = Dimensionless sediment transport function = f(ψ*) given in Figure 4.5 

ξ=ψββξ=ψ ψβ 2

x

2

x * )/6.10(logY)/(Y (4.34) 

fb

s s

SR

D

′ 















ρ

ρ−ρ =ψ (4.35) 

ξ = Correction factor given as a function of s X/D in Figure 4.6 
4.15 

X = 0.77∆, if ∆/δ′ > 1.8 (4.36a) 

X = 01.39δ′, if ∆/δ′ < 1.8 (4.36b) 

∆ = Apparent roughness of the bed, X/ks

 X = Correction factor in the logarithmic velocity distribution equation given 

as a function of /k δ′ s in Figure 4.7 

δ′ = * ν V/6.11 ′ (4.37) 

* V/v ′ = Einstein’s velocity distribution equation = 75.5 log ∆)/y2.30( (4.38a) 

* V′ = Shear velocity due to grain roughness = ′

bSRg (4.38b) 

Rb

′ = Hydraulic radius of the bed due to grain roughness = RR bb − ′′

Rb

′′ = Hydraulic radius of the bed due to channel irregularities 

Sf = Slope of the energy grade line normally taken as the slope of the 

water surface 

 Y = Another correction term given as a function of /D δ′ 65 in Figure 4.8 

βx = log ∆)/X6.10(

 



 



 −

− =

−

dy

y

y1

)E1(

E 216.0I

Z

1

Z E

1Z

1 (4.39) 

ydyIn

y

y1

)E1(

E 216.0I

Z

1

Z E

1Z

2 



 



 −

− = 

−

 (4.40) 

where: 

 Z = * V4.0/V/ * κβω ′ ω= ′ (4.41) 

ω = Fall velocity of the particle of size Ds

 E = Ratio of bed layer thickness to flow depth, a/yo

 yo = Depth of flow 

 a = Thickness of the bed layer, 2D65

The two integrals I1 and I2 are given in Figures 4.9 and 4.10 as a function of z and E. 

Figure 4.5. Einstein's φ* vs ψ* bed load function (Einstein 1950). 
4.16 

Figure 4.6. Hiding factor (Einstein 1950). 

 

 Figure 4.7. Einstein’s multiplication factor X in the logarithmic velocity equations 

 (Einstein 1950). 
4.17 

Figure 4.8. Pressure correction (Einstein 1950). 

In the preceding calculations for the bed sediment load, the shear velocity is based on the 

hydraulic radius of the bed due to grain roughness Rb′. Its computation is explained in the 

following paragraph. 

Total resistance to flow is composed of two parts, surface drag and form drag. The 

transmission of shear to the boundary is accompanied by a transformation of flow energy into 

energy turbulence. The part of energy corresponding to grain roughness is transformed into 

turbulence that stays at least for a short time in the immediate vicinity of the grains and has a 

great effect on the bed load motion. Whereas, the other part of the energy which 

corresponds to the form resistance is transformed into turbulence at the interface between 

wake and free stream flow, or at a considerable distance away from the grains. This energy 

does not contribute to the bed load motion of the particles and may be largely neglected in 

the sediment transportation. 

Einstein's equation for mean flow velocity V in terms of * V′ is 

′ = ( ) ′

b /R26.12log75.5V/V ∆ * (4.42a) 

or 

( ) sb Xk/R26.12log75.5V/V *

′ = ′ (4.42b) 
4.18 

Figure 4.9. Integral I1 in terms of E and Z (Einstein 1950). 
4.19 

Figure 4.10. Integral I2 in terms of E and Z (Einstein 1950). 
4.20 

Furthermore, Einstein suggested that 

′′ θ= [ ] ψ′ * V/V (4.43) 

where 

fb

s 35

SR

D

ρ ′

ρ−ρ ′ =ψ (4.44) 

The relation for Equation 4.44 is given in Figure 4.11. The procedure to follow in computing 

Rb′ depends on the information available. If mean velocity V, slope S, hydraulic radius Rb

and the size of bed material are known, then Rb′ is computed by trial and error using 

Equation 4.42 and Figure 4.11. 

 Figure 4.11. Friction loss due to channel irregularities, as a function of sediment 

 transport (after Einstein and Barbarossa 1952). 

The procedure for computing total bed sediment discharge in terms of different size fractions 

of the bed material is: 

(1) Calculate ψ* using Equation 4.33 for each size fraction 

(2) Find φ* from Figure 4.5 for each size fraction 

(3) Calculate iBqB for each size fraction using Equation 4.32 

(4) Sum up the qB across the flow to obtain iBQb 

(5) Sum up the size fractions to obtain Qb 
4.21 

For the suspended sediment discharge: 

(6) Calculate Z for each size fraction using Equation 4.41 

(7) Calculate E = 2Ds / yo for each fraction 

(8) Determine l1 and l2 for each fraction from Figures 4.9 and 4.10 

(9) Calculate PE using Equation 4.32 

(10) Compute the suspended discharge from IB qB (1 + PE I1 + I2) from Equation 4.31 

(11) Sum up all the qB and all the iB to obtain the total suspended discharge Qss 

Thus, the total bed sediment discharge is computed as: 

(12) Add the results of Step 5 and 11. 

A sample problem showing the calculation of the total bed sediment discharge using 

Einstein's procedure is presented in Section 4.13. 

4.5.3 Comparison of Meyer-Peter and Müller and Einstein Contact Load Equations

Chien (1954) has shown that the Meyer-Peter and Müller, equation can be modified into the 

form 

2/3

188.0 4

*

* 













 − ψ

=φ (4.45) 

Figure 4.12 shows the comparison of Equation 4.45 with Einstein's ψ* vs. φ* relation for 

uniform bed sediment size and for sediment mixtures using D35 in the Einstein relation and 

D50 in the Meyer-Peter and Müller relation. They show good agreement for coarse sands, 

but diverge for fine sands. This supports the premise that the Meyer-Peter and Müller 

equation is most applicable to coarse grain sizes with little or no suspended load. 

 

 Figure 4.12. Comparison of the Meyer-Peter and Müller and Einstein methods for 

 computing contact load (Chien 1954). 
4.22 

4.5.4 Colby's Method of Estimating Bed Sediment Discharge

Colby’s (1964) method is a graphical method for estimating total load, which can also be 

used to check the reasonableness of other sediment transport calculations. One very 

important feature of the Colby method is the inclusion of a correction factor for fine sediment 

(washload) concentration effects. Colby’s method illustrates that washload can have a 

significant impact on bed material load transport capacity. 

After investigating the effect of all the pertinent variables, Colby (1964) developed four 

graphical relations shown in Figures 4.13 and 4.14 for determining the bed sediment 

discharge. In arriving at his curves, Colby was guided by the Einstein bed-load function 

(Einstein 1950) and a large amount of data from streams and flumes. However, it should be 

understood that all curves for 100-foot depth, most curves of 10-foot depth and part of the 

curves of 1.0 and 0.1 ft are extrapolated from limited data and theory. Extrapolated curves 

are shown as dashed lines in Figure 4.13. 

In applying Figures 4.13 and 4.14 to compute the bed sediment discharge, the following 

procedure is used: 

1. Required data are the mean velocity V, the depth yo, the median size of bed material 

D50, the water temperature °F and the measured fine sediment concentration Cf. 

2. Uncorrected sediment discharge qn for the given V, yo, and D50 can be found from 

Figure 4.13 by first reading qn knowing V and D50 for two depths that bracket the 

desired depth and then interpolating on a logarithmic graph of depth versus qn to get 

the bed sediment discharge per unit width. 

3. Two correction factors k1 and k2 shown in Figure 4.14 account for the effect of water 

temperature and fine suspended sediment on the bed sediment discharge. If the bed 

sediment size falls outside the 0.20 mm to 0.30 mm range, the factor k3 from Figure 

4.14 is applied to correct for the effect of sediment size. 

4. Unit bed sediment discharge qT corrected for the effect of water temperature, 

presence of fine suspended sediment and sediment size is given by the equation 

[ ] ( ) T 21 −+= qk1kk1q n3 (4.46) 

As Figure 4.14 shows, k1 = 1 when the temperature is 60°F, k2 = 1 when the concentration of 

fine sediment is negligible and k3 = 1 when D50 lies between 0.2 mm and 0.3 mm. The total 

sand discharge is 

T = WqQ T (4.47) 

where: 

W = Width of the stream 

Colby (1964) found that: "The agreement of computed and observed discharges of sands for 

sediment stations whose records were not used to define the graphs seemed to be about as 

good as that for stations whose records were used." An example showing bed sediment 

discharge calculations by the Colby method is presented in Section 4.13. 
4.23 

Discharge of Sands, qn in tons/day/ft

 Figure 4.13. Relation of discharge of sands to mean velocity for six median sizes of bed 

 sands, four depths of flow, and a water temperature of 60°F (Colby 1964). 
4.24 

k3

0.01

0.1

1.0

k2

k1

k3

0.01

0.1

1.0

k2

k1

Figure 4.14. Colby's correction curves for temperature and fine sediment (Colby 1964). 

4.5.5 Relative Influence of Variables 

The study of the relative influence of viscosity, slope, bed sediment size and depth on bed 

sediment and water discharge is examined in this section using Einstein's bed-load function 

(1950) and Colby's (1964) relationships. Einstein's bed-load function is chosen because it is 

the most detailed and comprehensive treatment from the point of fluid mechanics. Colby's 

relations are chosen because of the large amount and range of data used in their 

development. 

The data required to compute the bed material discharge using Einstein's relations are: S = 

channel slope; D65 = size of bed material for which 65 percent is finer; D35 = size of bed 

material for which 35 percent is finer; Di

 = size of bed sediment in fraction i; ν = kinematic 

viscosity; nw = Manning's wall friction coefficient; A = cross-sectional area; Pb = wetted 

perimeter of the bed; Pw = wetted perimeter of the banks; iB = percentage of bed sediment in 

fraction i; γs = specific weight; and V = average velocity. 

To study the relative influence of variables on bed material and water discharges, the data 

taken by the U.S. Geological Survey from October 1, 1940 to October 1, 1970 on the Rio 

Grande near Bernalillo, New Mexico are used. The width of the channel reach was 82.3 m 

(270 ft). In the analysis, the energy slope was varied from 0.7S to 1.5S , in which S is the 

average bed slope assumed to be equal to the average energy slope. Further, the kinematic 

viscosity was varied to correspond with variations in temperature from 39.2° to l00°F 

inclusive. The variation of D65, D50, Di

, and iB was accomplished by using the average bed 

material distribution given by Nordin (1964) and shifting the curve representing the average 
4.25 

bed sediment distribution along a line parallel to the abscissa drawn through D50. The 

average water temperature was assumed to be equal to 70°F and the average energy 

gradient of the channel was assumed to be equal to 0.00095. The water and sediment 

discharges were computed independently for each variation of the variables and for three 

subreaches of the Rio Grande of differing width near Bernalillo. The applicability of the 

results depends on the reliability of the modified Einstein bed-load function and Colby's 

relationships used in the analysis rather than on the choice of data. 

The computed water and sediment discharges are plotted in Figures 4.15, 4.16, and 4.17 

and show the variation of sediment discharge due to changes in bed material size, slope and 

temperature for any given water discharge. Figure 4.15 shows that when the bed sediment 

becomes finer, the sediment discharge increases considerably. The second most important 

variable affecting sediment discharge is the slope variation (Figure 4.16). Temperature is 

third in importance (Figure 4.17). The effects of variables on sediment discharge were 

studied over approximately the same range of variation for each variable. 

Figure 4.18 shows the variation of the sediment discharge due to changes in the depth of 

flow for any given discharge, computed using Colby's (1964) relations. The values of depth 

of flow varied from 1.0 to 10.0 ft, the median diameter of the bed sediment is maintained 

constant equal to 0.030 mm, the water temperature is assumed constant and the 

concentration of fine sediment is assumed less than 10,000 ppm. The channel width is also 

maintained constant at 82.3 m (270 ft). In Figure 4.18, the curves for constant depth of flow 

show a steep slope. This indicates that the capacity of the stream to transport sands 

increases very fast for a small increase of discharge at constant depth. Similar figures can 

be developed for other sizes of bed material, and the relations can be modified to include the 

effect of washload and viscosity effects. 

Figure 4.15. Bed-material size effects on bed material transport. 
4.26 

 Figure 4.16. Effect of slope on bed 

 material transport. 

Figure 4.17. Effect of kinematic viscosity 

 (temperature) on bed material 

 transport. 

Figure 4.18. Variation of bed material load with depth of flow. 
4.27 

4.6 POWER FUNCTION RELATIONSHIPS 

In Section 4.5, three classic sediment transport equations were presented. Of the three, the 

Meyer-Peter and Müller bedload equation is still widely used, and the Colby method provides 

a graphical approach that can be used as a quick check on the reasonableness of the results 

of other sand-bed sediment transport calculations. Both are simple enough in their 

application to be considered "basic" approaches for hand calculation, although, as indicated 

above, they can be integrated into computer solutions. To provide additional basic 

approaches, power function relationships are discussed in this section. These provide a 

practical method for quick sediment transport calculations and are readily adaptable to 

computer solutions. 

4.6.1 Introduction 

Power relationships relate sediment transport rates to hydraulic conditions and sediment 

characteristics. The value of power relationships is in (1) their ease of use, (2) application to 

specific flow regimes, and (3) application to site-specific conditions. Most power function 

relationships correlate sediment transport to velocity and flow depth. This simple relationship 

can be used as a first estimate of sediment transport capacity or as a comparison with the 

more rigorous equations. Power relationships can be developed by computing sediment 

transport with a more rigorous approach and fitting the results to a power function form. The 

Simons et al. (1981) power function was developed specifically for supercritical flow 

conditions on sand and fine gravel bed channels. Power relationships are also well suited for 

correlating measured sediment transport with observed hydraulic conditions. The resulting 

equation is site specific, but can be used, with judgment, to estimate sediment transport rates 

for hydraulic conditions outside the range of observations. HEC-20 (Chapter 6) illustrates the 

application of power relationships to predicting the equilibrium slope for a channel that is 

either sediment deficient or has an excessive sediment supply. 

The following sections present the Simons et al. (1981) power function that was developed 

for steep sand and gravel bed channels. A more general power function relationship, by 

Kodoatie et al. (1999), is also presented. The Kodoatie equation can be used for silt, fine 

sand, medium to coarse sand, and gravel bed rivers by selecting appropriate coefficients and 

exponents. HEC-20 (Lagasse et al. 2001) includes a power function equation developed 

using the Yang (1996) sand transport equation. 

In summary, power relationships empirically relate sediment transport with hydraulic 

conditions and sediment characteristics. They can be developed by fitting the coefficients to 

computed sediment transport data from more sophisticated equations or to measured data. 

Their utility is in their ease of use and, when developed from measured data, their sitespecific accuracy. 

4.6.2 Basic Power Function Relationship 

In 1981, Simons et al. proposed an efficient method of evaluating sediment discharge. The 

method is based on the variables flow depth, velocity, and particle diameter, and gradation 

coefficient. The basic power function relationship is: 

cc 3s2s = 1ss Vycq (4.48) 
4.28 

where: 

 qs = Unit sediment transport rate, ft2/s (m2/s) 

 cs2, cs3 = Exponents based on mean particle diameter (D50) ranging from sand to 

fine gravel 

 cs1 = Coefficient based on mean particle diameter (note that cs1 must be 

adjusted for SI units) 

 y = Mean flow depth, ft (m) 

 V = Mean flow velocity, ft/s (m/s) 

This relationship can be applied in steep streams with sand and fine gravel beds that 

normally exhibit critical or supercritical flow. This is the only transport relationship specifically 

developed for upper flow regime conditions. These power relationships were developed by 

Simons et al., from a computer solution of the Meyer-Peter and Müller bed load transport 

equation and the integration of the Einstein method for suspended bed material discharge 

(Julien 1995). 

Table 4.1 provides the coefficient and exponents for Equation 4.48 (for English units) for 

different gradation coefficients and sizes of bed material. Note that if sediment transport 

in SI units is desired, CS1 needs to be multiplied by a factor of 0.3048(2 - Cs2 - Cs3). The 

term Gr in Table 4.1 is defined as the gradation coefficient of the bed material and is: 





 





+=

50

84

16

50

r D

D

D

D

2

1 G (4.49) 

where: 

 Dn = Size of the bed material for which n percent of a sediment sample is finer. 

In this case, n = 84, 50, and 16, respectively. 

Table 4.1. Coefficient and Exponents of Equation 4.48 (Simons et al.).* 

 D50 (mm) 

0.1 0.25 0.5 1.0 2.0 3.0 4.0 5.0 

Gr = 1 cs1 3.30x10-5 1.42 x10-5 7.60 x10-6 5.62 x10-6 5.64 x10-6 6.32 x10-6 7.10 x10-6 7.78 x10-6

cs2 0.715 0.495 0.28 0.06 -0.14 -0.24 -0.3 -0.34 

cs3 3.3 3.61 3.82 3.93 3.95 3.92 3.89 3.87 

Gr = 2 cs1 1.59 x10-5 9.80 x10-6 6.94 x10-6 6.32 x10-6 6.62 x10-6 6.94 x10-6

cs2 0.51 0.33 0.12 -0.09 -0.196 -0.27 

cs3 3.55 3.73 3.86 3.91 3.91 3.9 

Gr = 3 cs1 1.21 x10-5 9.14 x10-6 7.44 x10-6 

cs2 0.36 0.18 -0.02 

cs3 3.66 3.76 3.86 

Gr = 4 cs1 1.05 x10-5 

cs2 0.21 

Cs3 3.71 

3048.0)unitsSI(c* )c( 1s

)CC2(

1s 3s2s −− =
4.29 

Equation 4.48 was developed for channels with steep slopes for sand and fine gravel beds 

experiencing critical and super-critical flows [Simons, et al. (1981), Julien (1995)]. The range 

of parameters utilized to develop this equation is shown in Table 4.2. 

Table 4.2. Range of Parameters Equation 4.48 Developed by Simons et al. 

Parameter Value Range SI Units 

Froude Number 1 – 4 -- 

Velocity 1.98 – 7.92 m/s 

Bed Slope 0.005 – 0.040 m/m 

Unit Discharge, q 3.05 – 60.96 m2/s 

Particle Size, D50 > 0.062 mm 

As the depth increases for a given velocity, the intensity of the turbulent transfer properties 

decreases for these sizes. The increase in area available for suspended sediment with the 

increased depth does not totally counterbalance the reduced turbulent transfer 

characteristics. The result is an inverse dependence of transport rate on depth for these 

larger sizes. The sizes with no dependence (Cs2 = 0) on depth in their transport rate fall 

between these two extremes. 

When applying the equations resulting from Table 4.1, care should be taken to assure that 

the range of parameters being used is not out of range with those used to develop the 

equations (Table 4.2). If conditions are within the ranges outlined in Table 4.2 the regression 

equations should provide results within ten percent of the values computed using the MeyerPeter and Müller bedload and Einstein suspended load equations for the steep conditions in 

Table 4.2. 

There are several other checks that should be made in order to ensure the equations are 

applicable to a given problem. The equations are based on the assumption that all the 

sediment sizes present can be moved by the flow. If this is not true, armoring will take place. 

The equations are not applicable when armoring occurs. Since the equations were 

developed for sand-bed channels, they do not apply to conditions when the bed material is 

cohesive. Equation 4.48 would overpredict transport rates in a cohesive channel. 

An example of the application of the basic power function relationship is given in Section 

4.12 (SI) and Section 4.13 (English). 

4.6.3 Expanded Power Function Relationship 

Kodoatie et al. (1999) modified a Posada (1995) equation using nonlinear optimization and 

the field data for different sizes of riverbed sediment. A description of the development and 

validation of this equation is presented in Appendix B. The resulting equation is: 

dcb

t = SyaVq (4.50) 

where: 

 qt = Sediment transport rate, metric tons/m/day (tons/ft/day) 

 V = Mean flow velocity, m/s (ft/s) 

 y = Mean flow depth, m (ft) 

 S = Energy slope 

 a, b, c, and d = Regression coefficients 
4.30 

A summary of coefficient and exponents (for SI units) is presented in Table 4.3, depending 

on size of bed material. Note that the values of "a" must be multiplied by a factor of 1.1 

x 0.3048(1+b+c) for input and results in English units. 

Table 4.3. Coefficient and Exponents for Equation 4.50. 

 a* b c d 

Silt-bed rivers 281.4 2.622 0.182 0

Very fine to fine-bed rivers 2,829.6 3.646 0.406 0.412

Medium to very coarse sand-bed rivers 2,123.4 3.300 0.468 0.613

Gravel-bed rivers 431,884.8 1.000 1.000 2.000

*a (English Units) = 1.1 x 0.3048(1+b+c) (a) 

An example of the application of the expanded power function relationship is given in Section 

4.12 (SI) and 4.13 (English) 

4.7 YANG'S EQUATIONS 

The Yang sand and gravel total load equations are presented because of their frequent 

application and wide acceptance. The Yang equations are also readily adaptable to 

computer solutions. Yang (1996) related total load to excess unit stream power, expressed 

as the product of velocity and slope. Separate equations were developed for sand and 

gravel bed material and solved for sediment concentration in ppm by weight. The regression 

equations are developed based on dimensionless combinations of unit stream power, critical 

unit stream power, shear velocity, fall velocity, kinematic viscosity and sediment size. Yang 

also developed critical velocity formulas for use with his equations. The total load equations 

can be used to compute sediment transport by size fraction by using the geometric mean of 

the size class and weighting the computed concentrations by the class interval. The sand 

equation, which should be used for median sizes less than 2.0 mm, is: 

log . . log . log *

. . log . log * log

Ct

D V

D V VS VcrS

=− −

+− −















 − 



 





5 435 0 286 50 0 457

1799 0 409 50 0 314

ω

ν ω

ω

ν ω ωω (4.51) 

and the gravel equation, which should be limited to median sizes between 2.0 and 10.0 mm, 

is: 

log . . log . log *

. . log . log * log

Ct

D V

D V VS VcrS

=− −

+− −















 − 



 





6 681 0 633 50 4 816

2784 0 305 50 0 282

ω

ν ω

ω

ν ω ωω (4.52) 
4.31 

where: 

 Ct = Sediment concentration in parts per million by weight 

ω = Fall velocity of the sediment, m/s (ft/s) 

ν = Kinematic viscosity, m2

/s (ft2/s) 

V* = Shear velocity )gRS( , m/s (ft/s) 

 V = Velocity, m/s (ft/s) 

 Vcr = Critical Velocity, m/s (ft/s) 

 S = Energy slope 

In the above equations, the dimensionless critical velocity is given by: 

V

V D for cr V D

ω

ν

ν =

−

+ <<

2 5

0 06

0 66 12 70

50

50 .

log .

. . *

* (4.53)

and 

V for cr V D

ω ν

= ≥ 2 05 70 50 . * (4.54)

 

An example of calculating total bed-material discharge using Equation 4.51 is given in 

Section 4.12 (SI) and Section 4.13 (English). 

4.8 CONVERSION FACTORS 

There are numerous ways of expressing sediment transport. These include rate of transport 

and concentration. The rate of sediment transport can be expressed as a volumetric rate 

(m3

/s, ft3/s), or mass or weight rates (metric-tons/day, tons/day). The concentration relates 

volume or weight of sediment to the total volume or weight of the sediment-water mixture and 

is converted to a rate of sediment transport by including the water discharge. 

The following equations can be used to convert from a volumetric rate to a mass transport. 

Qs (metric-tons/day) = Qs (m3

/s) x γs / g x 86400 / 1000 (4.55) 

Qs (tons/day) = Qs (ft3

/s) x γs x 86400 / 2000 = Qs (ft3/s) x γs x 43.2 (4.56) 

A common unit for sediment concentration is milligrams per liter (Cmg/l), which is the ratio of 

the mass of sediment to the total volume of water and sediment. Concentrations are often 

expressed as parts per million by multiplying the concentration by 106

. Four ways of 

expressing concentration are: 

 Concentration by volume: Cv = sediment volume/total volume (4.57) 

 Concentration by weight: Cw = sediment weight/total weight = CvSg / [1+(Sg-1)Cv] (4.58) 

 Concentration as parts per millions by weight: Cppm = 106

Cw (4.59) 

 Concentration as mg/l: Cmg/l = 106

CvSg (4.60) 

(Note that Cmg/l equals the ratio of the mass of sediment in milligrams to the volume of the 

water-sediment mixture in liters.) 
4.32 

In these equations, Sg is the specific gravity of the sediment (γs/γ). For sediment 

concentrations (Cppm) less than 100,000 there is less than a 7 percent difference between 

Cppm and Cmg/l. The difference is less than 1 percent for concentrations less than 10,000 

ppm or mg/l. For low concentrations, they are often considered equivalent. This is because 

the weight of the sediment is small compared to the weight of the water. At these low 

concentrations, an approximate rate of sediment transport can be estimated by 

Qs = Cppm x 10-6 x Qwater / Sg 

where Qs and Qwater are in m3/s or ft3/s. 

The exact conversion is: 

))10xC1(S/(Qx10xCQ 6

water g ppm

6

s ppm

− − = − (4.62) 

When sediment is eroded or deposited, the volumetric transport rate must include the void 

space between the sediment particles. The void space, or porosity, is the ratio of voids to 

the total volume and often ranges from 35 to 45 percent. The volume of eroded or deposited 

material for a time interval, ∆t, is 

V = Qs x ∆t / (1 - η) (4.63) 

where Qs is the rate excess (or deficit) sediment transport and η is the porosity of the bed 

material. 

The use of conversion factors is illustrated in Section 4.12 (SI) and 4.13 (English). 

4.9 APPLICATION OF SELECTED SEDIMENT TRANSPORT EQUATIONS 

No single sediment transport equation can encompass all alluvial channel conditions. 

Therefore, an equation should be selected based on the particular river bed material and flow 

characteristics. When possible, the results of the sediment transport calculations should be 

compared with measured sediment transport. The equations presented in this section have 

broad application; however, other equations can, and should also be considered. 

There are several sources for obtaining recommendations on the applicability of sediment 

transport equations. Appendix B includes a review of ten sediment transport equations. The 

review tested the equations with a large compilation of field data that encompasses a wide 

range of bed material, from silts to gravel, and a wide range of river sizes, from 1 m width to 

over 1000 meters in width. The BRI-STARS manual (Molinas 2000) presents a comparison 

of several sediment transport equations with a variety of measured data. The HEC-6 manual 

(U.S. Army Corps of Engineers) refers to Vanoni (1975) as a guide to the applicability of 

sediment transport equations. Another source for recommendations on suitability of various 

transport functions is the SAM Hydraulic Design Package for Channels (Thomas et al. 2000). 

If more than one equation appears suitable, one should apply the equations and determine 

which one provides the most reasonable results. This is most easily done using sediment 

transport programs such as BRI-STARS, HEC-6 and SAM. Table 4.4 includes a list of 

sediment transport relationships and the application of each equation. 
4.33 

Table 4.4. Summary of Applicability of Selected Sediment Transport Formulas.

Formula Gravel Sand Very Fine Sand 

and Silt 

See 

Footnotes 

Ackers-White X 1, 2, 3, 4 

Bagnold X X 4 

Brownlie X 3, 4 

Colby X 2, 3 

Duboys X (fine) X 2 

Einstein X X X 3, 4 

Engelund-Hansen X 1, 3 

Karim X X 4 

Karim and Kennedy X X 4 

Kodoatie et al. (Power) X X X 4 

Laursen X (fine) X 2, 3, 4 

Meyer-Peter and Müller X 1, 2, 3 

Molinas and Wu X 1 

Parker X 3 

Schoklitsch X X (coarse) 2, 3 

Shen and Hung X 4 

Simons et al. (Power) X (fine) X 4 

Toffaleti X 2, 3, 4 

Yang X X 1, 2, 3, 4 

Yang, Molinas and Wu X 1 

1. The BRI-STARS model includes one fine sand equation (Yang, Molinas and Wu 1996), 

four sand equations (Engelund and Hanson 1972, Ackers and White 1973, Yang 1973, 

and Molinas and Wu 1996), two gravel equations (Meyer-Peter and Müller 1948, and 

Yang 1984) and a user specified power relationship. The power relationship can include 

depth, velocity, slope, sediment size and discharge. 

2. The HEC-6 model includes the Toffaleti (1966), Yang (1973), Duboys (in Vanoni, 1975), 

Ackers and White (1973), Colby (1964), Meyer-Peter and Müller (1948), Schoklitsch 

(1937), several modifications to Laursen (1958), and a user specified power relationship 

that is a power function of the product of depth and slope. In HEC-6, the Colby (1964) 

correction factor for wash load concentration can be applied to most of the transport 

functions. 

3. The U.S. Army Corps of Engineers SAM package (Thomas et al. 2000) includes Ackers 

and White (1973), Brownlie (1981), Colby (1964), Einstein (1950), Engelund-Hansen 

(1972), modified Laursen equations (1958), Meyer-Peter and Müller (1948), Parker 

(1990), Schoklitsch (1937), Toffaleti (1966) and Yang (1996). 

4. Appendix B of this manual includes an evaluation by Kodoatie (1999) of ten equations, 

including Ackers and White (1973), Bagnold (1966), Brownlie (1981), Einstein (1950), 

Karim (1998), Karim and Kennedy (1981), Laursen (1958), Shen and Hung (1972), 

Toffaleti (1969), and Yang (1996). Appendix B also discusses the power function 

relationships by Simons et al. (1981), Posada (1995), and Kodoatie (1999). 
4.34 

4.10 SUMMARY 

Computing the bed material sediment transport capacity for a river should be grounded on a 

firm understanding of the river channel characteristics, sources of sediment and modes of 

sediment movement. Equations that are well suited for the particular river conditions should 

be used and, if more than one are well suited, the results should be compared to assess the 

range of possible outcomes. For a specific river, it is recommended that the results be 

compared with actual measurements. 

Selection of an equation for use depends on the data available and scope, objectives, and 

resources available for a specific project. The Colby procedure (Section 4.5.4) provides a 

reliable quick estimate of the bed material discharge in sand bed streams and the MeyerPeter and Müller bed load equation (Section 4.5.1) has been widely used to estimate bed 

load (contact load) in coarser bed systems. The power function relationships (Section 4.6) 

provide a practical method for quick sediment transport calculations for sites within the range 

of conditions for which they were developed. The Yang sand and gravel equations (Section 

4.7) are widely accepted and adaptable to both hand calculation and computer solutions. 

Finally, many of the sediment transport relations of Table 4.4 are frequently used in sediment 

transport investigations and several are represented as options in the code of sediment 

transport models such as HEC-6 (U.S. Army Corps of Engineers 1993), BRI-STARS (Molinas 

1990, 2000), and SAM (Thomas et al. 2000). 

Typical applications of sediment transport relationships include equilibrium slope and 

sediment continuity analyses to determine long-term trends in bed elevation change 

(aggradation/degradation). These applications are illustrated in HEC-20 (Lagasse et al. 

2001). The next section outlines a general sediment transport analysis procedure. 

4.11 SEDIMENT TRANSPORTATION ANALYSIS PROCEDURE 

4.11.1 Step 1: Determine if Sediment Transport Computations are Necessary 

Determine qualitatively if the sediment problem at the bridge is aggradation or degradation. 

If degradation, previous scour calculations may be sufficient or a sediment transport analysis 

using equilibrium slope or sediment continuity could provide an initial estimate of long-term 

degradation (see HEC-20, Lagasse et al. 2001). If aggradation or if more refined 

degradation estimates are required, then calculating the quantity and gradation of the 

sediment being transported may be necessary. 

For important bridges, determining the quantity and gradation of sediment being transported 

can be used to: 

• Check on the previous determination of long-term aggradation or degradation and 

contraction scour. 

• Determine the type of countermeasures to solve a sedimentation problem. 

• Design debris basins for aggradation problems. 

• Design a check dam for degradation problems. 

• Determine if there will be environmental problems upstream or downstream of the bridge 

from sediment transport conditions at the bridge crossing. 

• Estimate the cost of methods to solve the aggradation or degradation problem. 
4.35 

Additional issues that need to be addressed include: 

• Is the cost of a sediment transport analysis commensurate with the cost of the bridge and 

the accuracy of the analysis? 

• Are there sediment transport data available from the USGS or other Federal or state 

Agencies for this stream or similar streams? 

4.11.2 Step 2: Determine Scope of Sediment Transport Analysis 

Determine if the need for information on the quantity of sediment being transported is for the 

quantity transported by the occasional high or peak flows or for the amount transported 

annually, i.e., determine the sediment transport design parameters. 

• To check previous scour calculations, only sediment transported by high or peak flows 

may be needed. 

• The design of a check dam or debris basin may require annual or multi-year cumulative 

quantities of sediment transport. 

• Refining estimates of long-term aggradation or degradation will require multi-year 

cumulative quantities. 

4.11.3 Step 3: Determine the Type of Analysis 

The types of analyses available to determine the quantity and gradation of sediment 

transport are field measurements, computer models, and basic methods given in this 

chapter. 

Field Measurement 

Field measurement to determine sediment transport is a specialized activity and generally 

should be contracted for. The USGS and the U.S. Army Corps of Engineers are specialists 

in this type of field measurement. Field measurements used in conjunction with the modified 

Einstein method (Colby and Hubbell 1962, Vanoni 1977, Simons and Senturk 1992) provide 

the most accurate method of determining the total sediment discharge of a stream. 

Basic Methods for Sediment Transport Calculations 

The methods and equations given in this chapter will normally be sufficient for most 

determinations of the bed-material transport for a given discharge or to develop a bedmaterial discharge rating curve to be used with a flow duration curve to determine the annual 

sediment discharge. Only for the most important and costly bridges would it be necessary to 

use field measurements and/or computer models to determine bed-material transport. 

The basic equations and methods given in this chapter and recommended for practical 

applications are: 

• Meyer-Peter and Müller (1948) bed load equation (also USBR 1960) 

• Colby’s (1964) curves and method 
4.36 

• Simons et al. (1981) basic power function method 

• Kodoatie et al. (1999) expanded power function method 

• Yang (1996) sand and gravel equations 

A summary of each method is given below to aid in selecting which method to use. 

Meyer-Peter and Müller Bed Load Equation 

The Meyer-Peter and Müller bed load equation only calculates the bed material moving in 

contact with the bed (contact load). It does not give total bed-material discharge unless there 

is no bed-material in suspension. It is, therefore, best utilized for coarser bed material 

streams (D50 coarser than 2.00 mm). 

Colby's Curves and Method 

Colby’s method is only for sand size bed material (D50 from 0.6 to 2.00 mm). It is very good 

method for rapid determination of total bed-material discharge. It can be used to check or 

compare with the other methods. The method can be used to determine the total bedmaterial transport by size fraction. 

Simons et al. (1981) Power Function Relationship 

The Simons et al. power relationship can be used for steep streams with sand and fine 

gravel-size bed material (D50 from 0.1 to 5.0 mm) that normally exhibit critical or supercritical 

flow. The relationship takes into consideration the size distribution of the bed material (Table 

4.2). The difficulty in the method is the need to interpolate between sand sizes and gradation 

coefficient. 

Kodoatie et al. (1999) Power Function Relationship 

The Kodoatie method (modified Posada (1995)) can be used for streams with bed material 

size ranging from silt to gravel. It provides a coefficient and exponents based on classifying 

streams as silt, fine sand, medium to coarse sand and gravel bed (Table 4.4). The method is 

useful for rapid calculations of bed-material discharge. 

Yang (1996) Sand and Gravel Equations 

The Yang equations can be used for sand and gravel total load estimation. The equations 

can be applied by size fraction or to the median size. The gravel equation should be limited 

to median sizes between 2.0 and 10.0 mm. 

Computer Models 

The use of computer models is beyond the scope of this manual, but programs are available. 

Two of the better known computer models are: 

• BRI-STARS (Molinas 2000) 

• HEC-6 (U.S. Army Corps of Engineers 1993) 

These computer models have the ability for user-selected sediment transport relations 

(including most of those shown in Table 4.1 – see Section 4.6) and power relationships. The 

models can be used to determine the total sediment discharge for a given flow or to 
4.37 

determine the average annual sediment discharge. To determine the average annual total 

bed-material discharge the computer model is used to determine the bed-material discharge 

for a range of discharges to develop a sediment discharge rating curve or table. This rating 

curve is used in combination with a flow duration curve or table to determine the average 

annual bed-material discharge. BRI-STARS can be used for single event or long-term 

sediment transport simulations, whereas HEC-6 is better suited for long-term simulations and 

should be used with caution for single event simulations. 

The U.S. Army Corps of Engineers SAM package (Thomas et al. 2000) includes sediment 

transport calculations using a wide variety of equations. The SAM package uses normal 

depth hydraulics and does not perform sediment routing. It is intended for basic sediment 

transport analysis and for developing sediment rating curves as part of a sediment yield 

analysis. 

4.12 SOLVED PROBLEMS FOR SEDIMENT TRANSPORT (SI) 

4.12.1 Introduction 

The following example problems illustrate the application of concepts and equations for 

sediment transport. Because several of the equations are commonly given in English units 

(for example Einstein’s equation) and the methodology is very complex, some of the 

examples will only be given in English units. Readers who use metric units (SI) are 

encouraged when encountering similar problems to convert the input variables to English 

units, solve the problem, and convert the results back to SI units. 

4.12.2 Problem 1 Suspended Sediment Concentration Profile 

From data observed on the Missouri River, calculate the vertical suspended sediment 

concentration profile for the D50 sediment size in mg/l. 

Given: 

l/mg000,775c

s/m10x00.1

s/m0338.0)D(

m000255.0D

)F5.72(C5.22Temp

000210.0S

s/m27.1V

m30.4y

a

26

50

50

o

=

=υ

=ω

=

°°=

=

=

=

−

From Equation 4.12: 

* V

zand

ay

a

y

yy

c

c

Z

o

o

a βκ

ω  =



 





−

− =
4.38 

Assume: 

β = 1.0 

κ = 0.4 

a = 2D50 = 0.00051 m 

Calculate: 

V* = (g yoS)1/2 = [(9.81) (4.30) (0.00021)]1/2 = 0.0941 m/s 

898.0

)0941.0()4.0()0.1(

0338.0

V

z

*

= = βκ

ω =

Calculate the concentration versus distance profile given in Table 4.5: 

Table 4.5. Concentration vs. Elevation Above the Bed. 

y (m) C (mg/l) 

0.091 7,230 

0.305 2,370 

0.61 1,160 

1.22 531 

2.44 181 

3.05 104 

3.66 48 

4.27 3 

4.12.3 Problem 2 Using the Meyer-Peter and Müller Equation 

Calculate the Bed Sediment Discharge (Bed Load) 

In this example problem the U.S. Bureau of Reclamation formulation of the equation 

(Equation 4.19) will be used. 

Given: 

A rectangular channel and the following data: 

D50 = 1.9 mm 

D90 = 2.8 mm 

G = 1.41 

n = 0.015 

nw = 0.022 

yo = 2.99 m 

Sf = 0.0005 m/m 

Dm = 2.01 mm 

W = 60.96 m 
4.39 

Use Equation 4.19: 

2/3

fo m

2/3

b

6/1

90b

B D627.0Sy

n

D

Q

Q 846.10780.4q 















− 





	

















	 







 =

From Equation 4.21: 

014.0

015.

022. 1

96.60

98.5 1)015(.n

3/2 2/3

b =





















	











 

	


 




 = + −

From Equation 4.23: 

84.0

014.0

022.0

96.60

98.5 1

1

Q

Q

5.1

b =





 



 +

=

2/3 2/3 6/1

B )01.2(627.0)0005(.)99.2( 014.0

8.2 )84.0(846.10780.4q 















− 





	









 =

qB = 137 metric-tons/day/m 

Qs = 137 x 60.96 = 8350 metric-tons/day 

4.12.4 Problem 3 Application of the Einstein Method to Calculate 

Total Bed-Material Discharge 

The application of the Einstein method to calculate total bed material discharge is given in 

English units only. See Example Problem 4.13.4. The example is taken from Einstein’s 

1950 publication. 

4.12.5 Problem 4 Calculation of Total Bed-Material Discharge 

 Using Colby’s Method 

The application of Colby method to calculate total bed-material discharge is given in English 

units only. See Example Problem 4.13.5. The example problem uses the hydraulic and 

sediment data given by Einstein for the purpose of comparison of the two methods. 

The Colby method is easy to use. Organizations using SI units are encouraged to convert the 

input variables to English units, solve the problem, and convert the results back to SI units. 
4.40 

4.12.6 Problem 5 Calculation of Total Bed-Material Discharge Using the 

Basic Power Function Relationship 

Determine the bed-material discharge for the 100-year discharge for a stream with the 

following data. Note that these data are not in the Fr range of Table 4.2. Compare the result 

with the expanded power function application in Problem 6. 

Width W = 60.96 m, Depth y = 1.829 m, Velocity V = 2.438 m/s, Q = 271.83 m3

/s. Sediment 

properties of D50 = 0.25 mm and size distribution factor G =1.5. 

Simons et al. (1981) Equation 4.48 is CC 3s2s = 1ss Vycq

Use Table 4.1 and interpolate to obtain the values of Cs1, Cs2, and Cs3. For Cs1 in SI units 

use the following Equation 

Cs1 (SI) = Cs1 (English units) x )CC2( 3s2s 3048.0 −−

Interpolate in Table 4.1 

Cs1 = 1.50 x 10-5 (English) 

Cs2 = 0.50 

Cs3 = 3.58 

Cs1 = 1.50 x 10-5 x 0.3048(2-0.5-3.58) = 0.000178 (SI) 

qs = 0.000178 (1.829)0.50 (2.438)3.58 = 0.0058 m2

/sec 

Qs = 60.96 x 0.0058 = 0.354 m3

/s 

Using a specific gravity of 2.65 

Qs = 2.65 x 9810/9.81 x 3600 x 24 x 0.354 /1000 = 81,000 metric-tons/day 

Cppm = 81000/(271.83 x 86400 + 81000) x 106

 = 3440 ppm (wt) 

4.12.7 Problem 6 Calculation of Total Bed-Material Discharge Using the 

Expanded Power Function Relationship 

Determine the bed-material discharge for the 100-year discharge for a stream with the data 

given in Problem 5. The data are repeated below. 

Width W = 60.96 m, Depth y = 1.829 m, Velocity V = 2.438 m/s, Q = 271.83 m3

/s, S = 

.000521. Sediment properties of D50 = 0.31 mm and size distribution factor G =1.32. 

Kodoatie et al. (1999) Equation 4.50 is qs = a Vb yc Sd

The sand size places this in a medium sand bed stream in Table 4.3 

From Table 4.3 

a = 2123.4, b = 3.30, c = 0.468, d = 0.613 

qs = 2123.4 (2.438)3.30 (1.829)0.468 (0.000521)0.613 = 518 metric tons/m/day 
4.41 

Qs = 518 x 60.96 = 31,580 metric-tons/day 

Note Qs = 31,580 metric-tons/day = 31,580 x 2204.62/2000 = 34,800 tons/day 

 

4.12.8 Problem 7 Calculate Total Bed-Material Discharge Using Yang's Sand Equation

Yang's sand equation (Equation 4.51) is as follows 

log . . log . log * . . log . log * C log t

DV DV VS VcrS =− − + − − − 



 



 



 

 5 435 0 286  50 0 457 1799 0 409 50 0 314 ω

ν ω

ω

ν ω ωω 

V

V D for cr V D

ω

ν

ν =

−

+ <<

2 5

0 06

0 66 12 70

50

50 .

log .

. . *

* 

and 

V for cr V D

ω ν

= ≥ 2 05 70 50 . * 

Compute the bed-material transport in metric-tons/day. 

Given: 

 Q = 221.39 m3

/sec W = 103.95 m ν = 1.16 x 10-6 m2/s 

 y = 2.31 m D50 = 0.32 mm ω = 0.043 m/sec 

 V = 0.922 m/sec S = 0.00022 m/m 

Calculate: 

τo = 9800 x 2.31 x 0.00022 = 4.98 N/m2

 

V* = (4.98/ 1000)0.5 = 0.0706 m/s 

5.1910x16.1/00032.0x0706.0 DV 50* 6 = = ν

−

69.266.0

06.05.19log

Vcr 5.2

=+ − = ω

00059.000022.0x69.2 crSV = = ω

00472.0043.0/00022.0x922.0 VS = = ω
4.42 

64.1043.0/0706.0 V* = = ω

86.1110x16.1/00032.0x043.0 D50 6 = = ν

ω −

t −= − 64.1log457.086.11log286.0435.5Clog +

− − − )00059.000472.0(log)64.1log314.086.11log409.0799.1(

= 95.1

weightbyppm8910C 95.1

t ==

 Qs = 221.3 x 89 x 3600 x 24/106

 = 1700 metric-tons/day 

4.12.9 Problem 8 Sediment Transport Conversions 

Given: 

Qwater = 100 m3

/s 

 Qs = 0.173 m3

/s 

 Sg = 2.65 

ρw = 1000 kg/m3

 Reach length = 2500 m 

 Channel width = 46 m 

 Porosity of deposited material = 38 percent (0.38) 

 

Compute: 

(1) Sediment Load in metric-tons/day 

(2) Concentration by Volume 

(3) Concentration by Weight 

(4) Concentration in PPM 

(5) Concentration in mg/l 

(6) Using Cppm, use approximate equation to check given Qs in m3

/s 

(7) Volume of material eroded if this entire amount were eroded in one day 

(8) Depth of one day of degradation if this entire amount of material were eroded from 

 the reach. 

 

(1) Qs(metric-tons/day) = Qs(m3/s) x 86.4 x ρs 

 Qs(metric-tons/day) = 0.173 x 86.4 x 2.65 x 1000 = 39,610 metric-tons/day 

(2) Cv = Volume of Sediment / Total Volume of Water and Sediment Mixture 

 Cv = 0.173 / (100 + 0.173) = 0.001727 

(3) Cw = Weight of Sediment / Total Weight of Water and Sediment Mixture 

 Weight of sediment = Qs(m3/s) x ρs = 0.173 x 2650 = 458.45 kg/s 
4.43 

 Weight of water = Qw x ρw = 100 x 1000 = 100,000 kg/s 

 Cw = 458.45 / (100,000 + 458.45) = 0.004564 

(4) Cppm = 106

Cw = 106 x 0.004564 = 4564 ppm (weight) 

(5) Cmg/l = 106

CwSg = 106 x 0.001727 x 2.65 = 4577 mg/l 

(6) Qs ≅ Cppm x 10-6 x Qw / Sg = 4564 x 10-6 x 100 / 2.65 = 0.172 m3

/s 

(7) ∆V = Qs(m3/s) x ∆t / (1 – η) = 0.173 x 86400 / (1 – 0.38) = 24100 m3

(8) ∆Z = ∆V / (W x L) = 24100 / (46 x 2500) = 0.21 m 

4.13 SOLVED PROBLEMS FOR SEDIMENT TRANSPORT (ENGLISH) 

4.13.1 Introduction 

The following example problems illustrate the application of concepts and equations for 

sediment transport. 

4.13.2 Problem 1 Suspended Sediment Concentration Profile 

From data observed on the Missouri River, calculate the vertical suspended sediment 

concentration profile for the D50 sediment size in mg/l. 

Given: 

)ft/lb3.48(l/mg000,775c

s/ft10x017.1

s/ft1109.0)D(

)ft000838.0(mm255.0D

)C5.22(F5.72Temp

000210.0S

s/ft164.4V

ft10.14y

3

a

25

50

50

o

=

=υ

=ω

=

°°=

=

=

=

−

From Equation 4.12: 

* V

zand

ay

a

y

yy

c

c

Z

o

o

a βκ

ω  =



 





−

− =
4.44 

Assume: 

β = 1.0 

κ = 0.4 

a = 2D50 = 0.0017 ft 

Calculate: 

V* = (g yoS)1/2 = [(32.2) (14.1) (0.00021)]1/2 = 0.309 ft/s 

898.0

)309.0()4.0()0.1(

1109.0

V

z

*

= = βκ

ω =

Calculate the concentration versus distance profile given in Table 4.6: 

Table 4.6. Concentration vs. Elevation Above the Bed. 

y (ft) C (mg/l) 

0.3 7,300 

1 2,360 

2 1,180 

4 538 

8 183 

10 105 

12 49 

14 3 

4.13.3 Problem 2 Using the Meyer-Peter and Müller Equation Calculate the 

Bed Sediment Discharge (Bed Load) 

In this example problem the U.S. Bureau of Reclamation formulation of the equation 

(Equation 4.19) will be used. 

Given: 

A rectangular channel and the following data: 

D50 = 1.9 mm 

D90 = 2.8 mm 

G = 1.41 

n = 0.015 

nw = 0.022 

yo = 9.8 ft 

Sf = 0.0005 (ft/ft) 

Dm = 2.01 mm 

W = 200 ft 
4.45 

Use Equation 4.19: 

2/3

fo m

2/3

b

6/1

b 90

B D627.0Sy

n

D

Q

Q 306.3606.1q 















− 





	

















	 







 =

From Equation 4.21: 

014.0

015.0

022. 1

200

6.19 1)015(.n

3/2 2/3

b =





















	











 

	


 




 = −+

From Equation 4.23: 

84.0

014.0

022.0

200

6.19 1

1

Q

Q

5.1

b =





 



 +

=

2/3 2/3 6/1

B )01.2(627.0)0005(.)8.9( 014.0

8.2 )84.0(306.3606.1q 















− 





	









 =

B = ft/day/tons0.46q

S = = day/tons9200200x0.46Q

4.13.4 Problem 3 Application of the Einstein Method to Calculate 

 Total Bed-Material Discharge 

A test reach, representative of the Big Sand Creek near Greenwood, Mississippi was used 

by Einstein (1950) as an illustrative example for applying his bed-load function. His numerical 

example is reproduced here. For simplicity, the effects due to bank friction are neglected. 

The reader can refer to the original example for the construction of the representative cross 

section and the consideration of bank friction. The characteristics of the channel crosssection follow. 

The channel slope was determined to be S = 0.00105. The relations of the cross-sectional 

area, hydraulic radius and wetted perimeter versus stage for the representative cross section 

are given in Figure 4.19. For this wide and shallow channel, the wetted perimeter is 

assumed to equal the surface width. The averaged values of the four bed-material samples 

are given in Table 4.7, and the grain size distribution is presented in Figure 4.20. Note that 

of these composite samples, 95.8% of the bed material falls between 0.589 and 0.147 mm, 

which is divided into four fractions. The sediment transport calculations will be made for 

individual size fractions with selected representative grain sizes equal to the geometric mean 

grain diameter of each fraction. 

The kinematic water viscosity ν is 1.06 x 10-5 ft2

/sec, and the specific gravity of the sediment 

is 2.65. 
4.46 

Table 4.7. Bed Material Information for Sample Problem. 

Grain Size Average Grain Size Settling Velocity 

Distribution, mm mm ft % mm/sec fps 

D > 0.589 -- -- 2.4 -- -- 

0.589 > D > 0.417 0.495 0.00162 17.8 5.20 0.195 

0.417 > D > 0.295 0.351 0.00115 40.2 3.75 0.148 

0.295 > D > 0.208 0.248 0.00081 32.0 2.70 0.100 

0.208 > D > 0.147 0.175 0.00057 5.8 1.70 0.064 

0.147 > D -- -- 1.8 -- -- 

Hydraulic Calculations 

The important hydraulic parameters were calculated, as given in Table 4.8. The table 

headings, their meanings and calculations are explained after the table. In the calculations, 

Equation 4.42 and Figure 4.7 were used to evaluate the resistance to flow. Any similar 

relation discussed in Chapter 2 can be utilized for the same purposes. 

Bed-Material Discharge Calculations 

The bed-material transport is calculated for each representative grain size of the bed material 

at each given flow depth. The procedure and results are given in Table 4.9, see the notes 

section for explanation of symbols, column by column.

Figure 4.19. Description of the average cross section. 
4.47 

Figure 4.20. Grain size distribution of bed material. 
4.48 

 Table 4.8. Hydraulic Calculations for Sample Problem 3, Application of the Einstein 

 Procedure (after Einstein 1950). 

R′b V′* δ′ ks/δ′ X ∆ V Ψ′ V/V"

* V"* R"

b 

1 2 3 4 5 6 7 8 9 10 11 

0.5 0.129 0.00095 1.21 1.59 0.00072 2.92 2.98 16.8 0.17 0.86 

1.0 0.184 0.00067 1.72 1.46 0.00078 4.44 1.49 27.0 0.16 0.76 

2.0 0.259 0.00047 2.44 1.27 0.00091 6.63 0.75 51.0 0.13 0.50 

3.0 0.318 0.00039 2.95 1.18 0.00097 8.40 0.50 87.0 0.10 0.30 

4.0 0.368 0.00033 3.50 1.14 0.00101 9.92 0.37 150.0 0.07 0.14 

5.0 0.412 0.00030 3.84 1.11 0.00104 11.30 0.30 240.0 0.05 0.07 

6.0 0.450 0.00027 4.26 1.08 0.00107 12.58 0.25 370.0 0.03 0.03 

Rb y = Rb Stage A Pb Q X Y β x (β/βx)

2 PE

12 13 14 15 16 17 18 19 20 21 22 

1.36 1.36 150.2 140 103 410 0.00132 0.84 1.29 0.63 10.97 

1.76 1.76 150.9 240 136 1,065 0.00093 0.68 1.19 0.85 11.10 

2.50 2.50 152.1 425 170 2,820 0.00069 0.56 0.91 1.27 11.30 

3.30 3.30 153.3 640 194 5,380 0.00075 0.55 0.91 1.27 11.40 

4.14 4.14 154.9 970 234 9.620 0.00079 0.54 0.91 1.27 11.70 

5.07 5.07 156.9 1,465 289 16,550 0.00084 0.54 0.91 1.27 11.90 

6.03 6.03 159.5 2,400 398 30,200 0.00082 0.54 0.91 1.27 12.04 

Notes: 

(1) R′b ft (bed hydraulic radius due to grain roughness), values are assumed to cover the entire desired discharge 

 range 

(2) V′* = SgR

'

b fps (shear velocity due to grain roughness) 

(3) δ′ = 11.6ν / V′* ft (thickness of laminar sublayer) 

(4) ks = D65, ft (roughness diameter) 

(5) X = f (ks/δ) (correction factor in the logarithmic velocity distribution), given in Figure 4.7 

(6) ∆ = ks/X ft (apparent roughness diameter) 

(7) V = V′* 5.75 log (12.26 R′b /∆ fps (average flow velocity) 

(8) Ψ′ = (ρs - ρ /ρ ) (D35 /R′bS) (intensity of shear on representative particles), given by Equation 4.44 

(9) V/V"

* = f (Ψ′) given in Figure 4.11 

(10) V"

* fps (shear velocity due to form roughness) 

(11) R"b ft (bed hydraulic radius due to form roughness) from V"* = SgR

"

b

(12) Rb = R′b + R"b ft (bed hydraulic radius), with no additional friction from the banks, Rb represents the total 

 hydraulic radius R 

(13) y ft (average flow depth), y ≈ Rb for wide shallow streams 

(14) Stage, ft, from description of cross section, Figure 4.19 for R = Rb. 

(15) A, ft2

, (cross-sectional area), from Figure 4.19 for the given stage 

(16) Pb ft, (bed wetted perimeter), from Figure 4.19 for the given stage 

(17) Q = AV cfs (flow discharge), a stage-discharge relationship can be plotted by relating the 

 computed Q to the stage 

(18) X ft (characteristic distance), from Equation 4.36, X = 0.77∆ for ∆/δ′ > 1.80 and X = 1.39δ′ for ∆/δ′ < 1.8

(19) Y = f (ks/δ′) (pressure correction term), given in Figure 4.8 

(20) β x = log (10.6 X /∆), (logarithmic function) 

(21) β = log 10.6 

(22) PE = 2.303 log (30.2 y /∆), (Einstein’s transport parameter), given by Equation 4.32 
4.49 

Table 4.9. Bed-Material Load Calculations for Sample Problem by Applying the Einstein 

 (1950) Procedure. 

D iB R′b Ψ D/X ξ Ψ* φ * iBqB iBQB 

1 2 3 4 5 6 7 8 9 10 

0.00162 0.178 0.5 

1.0 

2.0 

3.0 

4.0 

5.0 

6.0 

5.08 

2.54 

1.27 

0.85 

0.63 

0.51 

0.42 

1.23 

1.74 

2.35 

2.16 

2.05 

2.03 

1.98 

1.08 

1.00 

1.00 

1.00 

1.00 

1.00 

1.00 

2.90 

1.73 

0.90 

0.60 

0.43 

0.35 

0.29 

1.90 

4.00 

8.20 

12.80 

18.00 

22.50 

27.00 

0.0267 

0.0561 

0.1150 

0.1800 

0.2530 

0.3160 

0.3800 

119.0 

330.0 

845.0 

1,510.0 

2,560.0 

3,950.0 

6,350.0 

0.00115 0.402 0.5 

1.0 

2.0 

3.0 

4.0 

5.0 

6.0 

3.38 

1.69 

0.85 

0.56 

0.42 

0.34 

0.28 

0.82 

1.16 

1.57 

1.44 

1.37 

1.35 

1.32 

1.36 

1.10 

1.01 

1.04 

1.05 

1.05 

1.05 

2.44 

1.27 

0.61 

0.41 

0.30 

0.25 

0.20 

2.45 

5.50 

12.60 

19.00 

26.00 

31.50 

39.00 

0.0471 

0.1060 

0.2420 

0.3640 

0.5000 

0.6040 

0.7490 

210.0 

623.0 

1,780.0 

3,050.0 

5,050.0 

7,540.0 

12,900.0 

0.00081 0.320 0.5 

1.0 

2.0 

3.0 

4.0 

5.0 

6.0 

2.54 

1.27 

0.63 

0.42 

0.32 

0.25 

0.21 

0.61 

0.87 

1.17 

1.08 

1.04 

1.01 

0.99 

2.25 

1.26 

1.10 

1.12 

1.15 

1.17 

1.19 

3.03 

1.09 

0.49 

0.33 

0.25 

0.20 

0.17 

1.75 

6.80 

15.80 

23.50 

31.50 

39.50 

46.00 

0.0155 

0.0600 

0.1390 

0.2070 

0.2790 

0.3490 

0.4060 

69.0 

353.0 

1,020.0 

1,730.0 

2,820.0 

4,360.0 

6,980.0 

0.00057 0.058 0.5 

1.0 

2.0 

3.0 

4.0 

5.0 

6.0 

1.80 

0.90 

0.45 

0.30 

0.22 

0.18 

0.15 

0.43 

0.61 

0.83 

0.76 

0.72 

0.71 

0.70 

5.40 

2.28 

1.37 

1.52 

1.60 

1.65 

1.70 

5.15 

1.39 

0.44 

0.32 

0.25 

0.20 

0.18 

0.58 

5.10 

17.50 

25.00 

31.50 

39.50 

43.50 

0.00056 

0.00500 

0.01710 

0.02460 

0.31000 

0.03870 

0.04260 

2.5 

29.4 

126.0 

206.0 

313.0 

483.0 

732.0 

 iBQB 103

E Z I1 -I2 PEI1+I2+1 iTqT iTQT  iTQT

11 12 13 14 15 16 17 18 19 

400 

1335 

3,771 

6,496 

10,745 

16,333 

27,142 

2.38 

1.84 

1.30 

0.98 

0.78 

0.63 

0.54 

3.78 

2.65 

1.88 

1.53 

1.33 

1.18 

1.08 

0.078 

0.131 

0.240 

0.385 

0.560 

0.810 

1.090 

0.44 

0.74 

1.27 

2.01 

2.80 

3.85 

4.90 

1.42 

1.71 

2.44 

3.44 

4.75 

6.78 

9.20 

0.03800 

0.09580 

0.28100 

0..61700 

1.20000 

2.13000 

3.48000 

168 

561 

2,050 

5,170 

12,100 

26,500 

59,800 

670 

3,928 

30,500 

113,000 

324,000 

800,000 

1,940,000 

 1.69 

1.31 

0.92 

0.70 

0.56 

0.45 

0.38 

2.88 

2.02 

1.44 

1.17 

1.01 

0.90 

0.83 

0.117 

0.210 

0.450 

0.830 

1.370 

2.120 

2.950 

0.68 

1.19 

2.33 

3.85 

5.70 

8.10 

10.50 

1.60 

2.14 

3.76 

6.73 

11.30 

17.20 

26.00 

0.07540 

0.22700 

0.91000 

2.44000 

5.65000 

10.40000 

19.60000 

335 

1,330 

6,660 

20,400 

57,100 

129,000 

335,000 

1.19 

0.92 

0.65 

0.49 

0.39 

0.32 

0.27 

1.94 

1.36 

0.97 

0.79 

0.68 

0.61 

0.55 

0.230 

0.520 

1.530 

3.350 

6.200 

9.800 

15.000 

1.29 

2.60 

6.10 

11.00 

17.50 

25.50 

36.00 

2.23 

4.16 

12.20 

28.70 

56.00 

92.00 

146.00 

0.03450 

0.25000 

1.70000 

5.95000 

15.00000 

32.00000 

59.50000 

153 

1,460 

12,500 

49,700 

157,000 

397,000 

1,020,000 

0.85 

0.65 

0.46 

0.35 

0.28 

0.23 

0.19 

1.12 

0.86 

0.61 

0.49 

0.43 

0.38 

0.35 

0.720 

2.440 

8.400 

19.300 

32.000 

51.000 

70.000 

3.35 

8.10 

21.50 

41.00 

63.00 

91.00 

122.00 

5.55 

20.00 

74.40 

183.00 

312.00 

516.00 

722.00 

0.00312 

0.10000 

1.26000 

4.50000 

9.68000 

20.00000 

30.80000 

14 

587 

9,350 

37,600 

97,800 

248,000 

526,000 
4.50 

Table 4.9 (continued) 

Notes: 

(1) D is the representative grain size, ft, given in Table 4.7 

(2) iB is the fraction of bed material, given in Table 4.7 

(3) R′b is the bed hydraulic radius due to grain roughness, ft, given in Table 4.8 

(4) Ψ′ is the intensity of shear on a particle = ((ρs - ρ)/ρ) (D/(R′bS)) 

(5) D/X for values of X, see Table 4.8, Column 18 

(6) ξ = f (D65 / X) hiding factor, given in Figure 4.6 

(7) Ψ* is the intensity of shear on individual grain size = ξY (β/βx)

2 Ψ, values of Y and (β/βx)2

 are given in Table 4.8 

(8) φ * is the intensity of sediment transport for an individual grain 

(9) iBqB is the bed load discharge per unit width for a size fraction, given in Equation 4.33 

(10) iBQB is the bed load discharge for a size fraction for entire cross section, ton/day = 43.2WiBqB, W = Pb given in Table 4.8 

(11)  iBQB is the total bed load discharge for all size fractions for the entire cross section, tons/day 

(12) E is the ratio of bed layer thickness to water depth = 2D/y. For values of y, see Table 4.8. 

(13) Z is the exponent for concentration distribution = ω/(0.4 V′*) given in Equation 4.41. Values of ω and V* are given in Tables 

 4.7 and 4.8. 

(14) I1 is an integral given by Equation 4.39 and read from Figure 4.9 

(15) -I2 is an integral given by Equation 4.40 and read from Figure 4.10 

(16) PEI1+I2+1 is the factor between bed load and total load given by Equation 4.31 

(17) iTqT is the bed material load per unit width of stream for a size fraction lb/sec-ft = iBqB (PEI1+I2+1) given by Equation 4.31 

(18) iTQT is the bed material load for a size fraction for entire cross section, tons/day = 43.2 WiTqT

(19)  iTQT is the total bed material load for all size fractions, tons/day 
4.51 

4.13.5 Problem 4 Calculation of Total Bed-Material Discharge 

Using Colby’s Method 

In applying the Colby (1964) method, the required data are: the mean velocity, the depth, 

the median size or size distribution of the bed material, the water temperature and the fine 

sediment (wash load) concentration. The mean velocity and the depth of flow may be 

obtained by hydraulic calculations, as in Problem 3. However, to obtain best results from 

calculations, directly measured values of velocity and depth are required. 

For purposes of comparison, the sample problem used to illustrate Einstein’s method is 

solved using the Colby method. The required data are taken from Table 4.8 established by 

hydraulic calculations. In addition, the water temperature and the fine sediment concentration 

are assumed to equal 70°F and 10,000 ppm, respectively. For convenience, the calculations 

are summarized in tables. Two procedures are presented. Table 4.10 gives the calculations 

using the mean diameter of bed material (D50 = 0.31 mm), whereas Table 4.11 gives the 

calculations for individual fractions using the bed material size distribution. Note that depths, 

y, of 5.07 ft and 6.03 ft from Table 4.8 are not shown in Tables 4.10 and 4.11 since the 

corresponding velocities are greater than 10 ft/sec. The Colby curves (Figure 4.13) are 

limited to 10 ft/sec. 

Table 4.10. Bed Material Load Calculations for Sample Problem by Applying the Colby 

 Method (Median Diameter). 

y W V qn k1 k2 k3 qT QT 

1 2 3 4 5 6 7 8 9 

1.36 

1.76 

2.50 

3.30 

4.14 

103 

136 

170 

194 

234 

2.93 

4.44 

6.63 

8.38 

9.92 

14.5 

50.0 

135.0 

220.0 

325.0 

0.92 

0.92 

0.91 

0.90 

0.90 

1.20 

1.21 

1.22 

1.23 

1.25 

0.99 

0.99 

0.99 

0.99 

0.99 

16 

56 

150 

243 

365 

1,648 

7,620 

25,500 

47,142 

85,410 

Notes: 

(1) y, ft (mean depth), taken from Table 4.8 

(2) W, ft (surface width), taken from Table 4.8 

(3) V, fps (average velocity), taken from Table 4.8 

(4) qn tons/day-ft (uncorrected sediment discharge per unit width of channel), taken from Figure 4.13 

 for the given V, y, and D50 

(5) k1 = f (y, T) (correction factor for temperature), given in Figure 4.14 

(6) k2 = f (y, Cf) (correction factor for fine sediment concentration), given in Figure 4.14 

(7) k3 = f (D50) (correction factor for sediment size), given in Figure 4.14 

(8) qT = [1+(k1k2 -1) k3]qn tons/day-ft (true bed material discharge per unit width of stream), given by 

 Equation 4.46 

(9) QT = WqT tons/day (bed material discharge for all size fractions for entire cross section) 
4.52 

Table 4.11. Bed Material Discharge Calculations for Sample Problem by Applying the Colby 

 (Individual Size Fraction). 

D 102ib y W V qn 1 k

1 2 3 4 5 6 7 

0.495 17.8 1.36 

1.76 

2.50 

3.30 

4.14 

103 

136 

170 

194 

234 

2.93 

4.44 

6.63 

8.38 

9.92 

12 

40 

112 

193 

265 

0.92 

0.91 

0.91 

0.90 

0.90 

0.351 40.2 1.36 

1.76 

2.50 

3.30 

4.14 

103 

136 

170 

194 

234 

2.93 

4.44 

6.63 

8.38 

9.92 

15 

45 

120 

210 

290 

0.92 

0.91 

0.91 

0.90 

0.90 

0.248 32.0 1.36 

1.76 

2.50 

3.3 

4.14 

103 

136 

170 

194 

234 

2.93 

4.44 

6.63 

8.38 

9.92 

18 

53 

140 

240 

345 

0.92 

0.91 

0.91 

0.90 

0.90 

0.175 5.8 1.36 

1.76 

2.50 

3.30 

4.14 

103 

136 

170 

194 

234 

2.93 

4.44 

6.63 

8.38 

9.92 

23 

64 

163 

305 

420 

0.92 

0.91 

0.91 

0.90 

0.90 

k2 k3 qT IbqT IbQT ibQT 

8 9 10 11 12 13 

1.20 

1.21 

1.22 

1.23 

1.25 

0.62 

0.62 

0.62 

0.62 

0.62 

13 

43 

119 

205 

288 

2.3 

7.7 

21.0 

37.0 

51.0 

237 

1,050 

3,570 

7,180 

11,900 

1,710 

6,908 

22,780 

45,560 

77,640 

1.20 

1.21 

1.22 

1.23 

1.25 

0.92 

0.92 

0.92 

0.92 

0.92 

16 

49 

132 

230 

323 

6.4 

20.0 

53.0 

93.0 

130.0 

659 

2,720 

9,010 

18,000 

30,420 

1.20 

1.21 

1.22 

1.23 

1.25 

1.00 

1.00 

1.00 

1.00 

1.00 

20 

58 

155 

266 

388 

6.4 

19.0 

50.0 

85.0 

124.0 

659 

2,580 

8,500 

16,500 

29,000 

1.20 

1.21 

1.22 

1.23 

1.25 

0.97 

0.97 

0.97 

0.97 

0.97 

25 

70 

180 

337 

471 

1.5 

4.1 

10.0 

20.0 

27.0 

155 

558 

1,700 

3,880 

6,320 

Notes: 

(1) D, mm (representative grain size; given in Table 4.7 

(2) ib (fraction of bed material), taken from Table 4.7 

(3) y, ft, (average flow depth), taken from Table 4.8, Column 13 

(4) W, ft, (top width), taken from Table 4.8, Column 16 

(5) V, fps (average velocity), taken from Table 4.8, Column 7 

(6) qn, tons/day-ft (incorrect sediment discharge per unit width by assuming the bed is composed entirely of one sand of size 

 D) taken from Figure 4.13 by interpolation on logarithmic paper for the given V, D, and y 

(7) k1 = f (y,T) (correction factor for temperature), given in Figure 4.14 

(8) k2 = f (y,Cf) (correction factor for fine sediment concentration), given in Figure 4.14 

(9) k3 = f (D50) (correction factor for sediment size), given in Figure 4.14 

(10) qT = [1 + (k1K2 -1) k3]qn, tons/day-ft (corrected bed material discharge per unit width by assuming the bed is 

 composed entirely of one sand of size D) 

(11) ibqt, tons/day-ft, (bed material discharge per unit width for a size fraction for entire cross section) 

(12) ibQt = WibqT, tons/day, (bed material discharge for a size fraction for entire cross section) 

(13) ibQT, tons/day, (bed material discharge for all size fractions for entire cross section) 
4.53 

The results of the bed-material discharge calculations for the sample problem using 

Einstein's (1950) and Colby's (1964) methods are shown in Figure 4.21. The curves indicate 

that the sediment discharge increases rapidly with an increase in water discharge. In 

general, the two methods compare relatively well. 

Figure 4.21. Comparison of the Einstein and Colby methods. 

4.13.6 Problem 5 Calculation of Total Bed-Material Discharge Using the 

Basic Power Function Relationship 

Determine the bed-material discharge for the 100-year discharge for a stream with the 

following data. Note that these data are not in the Fr range of Table 4.2. Compare the result 

with the expanded power function application in Problem 6. 

Width W = 200 ft, Depth y = 6.0 ft, Velocity V = 8.0 ft/s, Q = 9600 cfs. Sediment properties of 

D50 = 0.25 mm and size distribution factor G =1.5. 

Simons et al. (1981) Equation 4.48 is 2s CC 3s = 1ss Vycq

Use Table 4.1 and interpolate to obtain the values of Cs1, Cs2, and Cs3. 

 Cs1 = 1.50 x 10-5 

 Cs2 = 0.50 

 Cs3 = 3.58 

 qs = 1.50 x 10-5 (6.0)0.5 (8.00)3.58 = 0.0628 ft2

/sec 

 Qs = 200 x 0.0628 = 12.56 ft3

/sec 

Using a specific gravity of 2.65 

Qs = (2.65 x 62.4 x 3600 x 24 x 12.56) /2000 = 90,000 tons/day 

 Cppm = 90000 (9600 x 62.4 x 86400/2000 + 90000) x 106

 = 3470 ppm (wt) 
4.54 

4.13.7 Problem 6 Calculation of Total Bed-Material Discharge Using the 

Expanded Power Function Relationship 

Determine the bed-material discharge for the 100-year discharge for a stream with the data 

given in Problem 5. The data are repeated below. 

Width W = 200 ft, Depth y = 6 ft, Velocity V = 8 ft/s, Q = 9,600 cfs, S = 0.000521. Sediment 

properties of D50 = 0.31 mm and size distribution factor G =1.32. 

Kodoatie et al. (1999) Equation 4.50 is qs = a Vb yc Sd

The sand size places this in a medium sand bed stream in Table 4.3. 

From Table 4.3 

a = 2123.4, b = 3.30, c = 0.468, d = 0.613 

a (English units) = 1.1 x 0.3048(1+b+c) a = 1.1 x 0.3048(1+3.30+0.468) (2123.4) = 8.095 

qs = 8.095 (8)3.30 (6)0.468 (0.000521)0.613 = 173.78 tons/ft/day 

Qs = 173.78 x 200 = 34,800 tons/day 

4.13.8 Problem 7 Calculate Total Bed-Material Discharge Using Yang's Sand Equation 

Yang's sand equation (Equation 4.51) is as follows: 

log . . log . log * . . log . log * C log t

DV DV VS VcrS =− − + − − − 



 



 



 

 5 435 0 286  50 0 457 1799 0 409 50 0 314 ω

ν ω

ω

ν ω ωω 

V

V D for cr V D

ω

ν

ν =

−

+ <<

2 5

0 06

0 66 12 70

50

50 .

log .

. . *

* 

and 

V for cr V D

ω ν

= ≥ 2 05 70 50 . * 

Compute the bed-material transport in tons per day. 

Given: 

 Q = 7,820 ft3

/sec W = 341 ft ν = 1.21 x 10-5 ft2/s 

 y = 7.58 ft D50 = 0.00105 ft ω = 0.141 ft/sec 

 V = 3.02 ft/sec S = 0.00022 
4.55 

Calculate: 

τo = 62.4 x 7.58 x 0.00022 = 1.04 lb/ft2

 

V* = (0.104/1.94)0.5 = 0.232 ft/s 

1.2010x121.1/00105.0x232.0 DV 50* 5 = = ν

−

67.266.0

06.01.20log

Vcr 5.2

=+ − = ω

00059.000022.0x67.2 cr SV = = ω

00471.0141.0/00022.0x02.3 SV = = ω

64.1

141.0

V* 232.0 == ω

2.1210x21.1/00105.0x141.0 D50 5 = = ν

ω −

96.1

)00059.000471.0(log)64.1log314.02.12log409.0799.1(

t 64.1log457.02.12log286.0435.5Clog

=

− − −

−= − +

weightbyppm9110C 96.1

t ==

day/tons1900)10x2000/(24x3600x4.62x91x7820Q 6

s = =

4.13.9 Problem 8 Sediment Transport Conversions 

Given: 

Qwater = 3531 ft3

/s 

 Qs = 6.11 ft3

/s 

 Sg = 2.65 

γw = 62.4 lb/ft3

 Reach length = 8200 ft 

 Channel width = 151 ft 

 Porosity of deposited material = 38 percent (0.38) 

 

Compute: 

(1) Sediment Load in tons/day 

(2) Concentration by Volume 
4.56 

(3) Concentration by Weight 

(4) Concentration in PPM 

(5) Concentration in mg/l 

(6) Using Cppm, use approximate equation to check given Qs in ft3

/s 

(7) Volume of material eroded if this entire amount were eroded in one day 

(8) Depth of one day of degradation if this entire amount of material were eroded from 

the reach. 

 

(1) Qs(tons/day) = Qs(ft3/s) x 43.2 x γs 

 Qs(tons/day) = 6.11 x 43.2 x 2.65 x 62.4 = 43,647 tons/day 

(2) Cv = Volume of Sediment / Total Volume of Water and Sediment Mixture 

 Cv = 6.11 / (3531 + 6.11) = 0.001727 

(3) Cw = Weight of Sediment / Total Weight of Water and Sediment Mixture 

 Weight of sediment = Qs(ft3/s) x γs = 6.11 x 165.36 = 1010 lb/s 

 Weight of water = Qw x γw = 3531 x 62.4 = 220,334 lb/s 

 Cw = 1010 / (220,334 + 1010) = 0.004563 

(4) Cppm = 106

Cw = 106 x 0.004563 = 4563 ppm (weight) 

(5) Cmg/l = 106

CwSg = 106 x 0.001727 x 2.65 = 4577 mg/l 

(6) Qs ≅ Cppm x 10-6 x Qw / Sg = 4563 x 10-6 x 3531 / 2.65 = 6.08 ft3

/s 

(7) ∆V = Qs(ft3/s) x ∆t / (1 – η) = 6.11 x 86400 / (1 – 0.38) = 851,000 ft3

(8) ∆Z = ∆V / (W x L) = 851,000 / (151 x 8200) = 0.68 ft 
5.1 

CHAPTER 5

RIVER MORPHOLOGY AND RIVER RESPONSE

5.1 INTRODUCTION

Rivers and river systems have served man in many ways. Rivers are passage ways for 

navigation and are essential to agriculture, particularly in the arid and semiarid parts of the 

world. To a large degree, the flooding by rivers and the deposition of sediment on the river 

valleys have been a means of revitalizing the river valleys to keep them productive. Rivers 

have provided a means of traveling inland and developing trade. This has played a significant 

role in the development of all countries wherever rivers of significant size exist. 

Rivers have different alignments and geometry. There are meandering rivers, braided rivers, 

and rivers that are essentially straight. In general, braided rivers are relatively steep and 

meandering rivers have more gentle slopes. Meandering rivers that are not subject to rapid 

movement, are reasonably predictable in behavior; however, meandering rivers are generally 

unstable with eroding banks which may result in destruction of productive land, bridges, bridge 

approaches, control works, buildings, and urban properties during floods. Bank protection 

works are often necessary to stabilize certain reaches of many rivers and to improve them for 

other aspects of flood control and navigation. 

5.2 FLUVIAL CYCLES AND PROCESSES

Fundamental characteristics and processes governing the formation of river systems are 

discussed in this section. A very general classification of rivers which considers their age is 

introduced. The morphology of floodplains, deltas, and alluvial fans is described as well as the 

processes of headcutting and nickpoint migration. The concept of geomorphic threshold 

completes this section on fundamentals leading to a discussion of variability and change in 

large alluvial rivers. Specific aspects of stream form and a simple geomorphic classification of 

streams are presented later in this chapter. 

5.2.1 Youthful, Mature, and Old Streams

One of the early methods to classify rivers was by relative age as youthful, mature, and old 

(Davis 1899, see King and Schumm 1980). As a general concept this has validity with steep 

irregular young streams becoming mature with a narrow valley, a floodplain, and a graded 

condition; that is, the slope and energy of the streams are just sufficient to transport the 

materials delivered to it. As time passes, the valley widens and a fully meandering channel of 

low gradient develops. Unfortunately, for this classification, the lower Mississippi River would 

be designated as old, when, in fact it is one of the most youthful rivers. It developed on the 

alluvium deposited by drainage from the continental ice sheet perhaps 10,000 years ago. 

More recently, rivers have been classified based upon type of sediment load (Schumm 1977), 

and pattern (Brice 1982). Nevertheless, the concept of landform and channel evolution through 

time is a valuable one. Hydraulic Engineering Circular (HEC) No. 20 (Lagasse et al. 2001) 
5.2 

presents several examples of landform evolution, including the long-term cycle of erosion and 

evolution of incised stream channels. 

In the context of a rivers age, the process of channel rejuvenation refers to an increase in 

erosional activities in mature or old channels caused by lowering base level elevation, tectonic 

activities or other causes. Rejuvenated mature or old channels then exhibit some properties of 

youthful channels such as channel incision and erosion processes. 

5.2.2 Floodplain and Delta Formations

Over time, the highlands of an area are worn down. The streams erode their banks. The 

material that is eroded is utilized downstream to build banks and bars to further enhance the 

meandering process. Streams move laterally pushing the highlands back. Low flat valley land 

and floodplains are formed. As the streams transport sediment to areas of flatter slopes, and 

in particular to bodies of water where the velocity and turbulence are too small to sustain the 

transport of the material, the material is deposited forming deltas. As deltas build outward the 

up-river portion of the channel is elevated through deposition and becomes part of the 

floodplain. Also, the stream channel is lengthened and the slope is further reduced. The 

upstream river bed is filled in and average flood elevations are increased. As they work across 

the river valley, these processes cause the total floodplain to raise in elevation. Hence, even 

old streams are far from static. Old rivers meander, are affected by changes in sea level, are 

influenced by movements of the earth's crust, are changed by delta formations or glaciation, 

and are subject to modifications due to climatological changes and as a consequence of 

human development. 

5.2.3 Alluvial Fans

Alluvial fans are very dynamic landforms that can create significant hazards to highways as a 

result of floods, debris flows, deposition, channel incision, and avulsion (Schumm and Lagasse 

1998). They occur whenever there is a change from a steep to a flat gradient. As the bed 

material and water reaches the flatter section of the stream, the coarser bed materials can no 

longer be transported because of the sudden reduction in both slope and velocity. 

Consequently, a cone or fan builds out as the material is dropped. There is considerable 

similarity between a delta and an alluvial fan. Both result from reductions in slope and velocity 

and both tend to reduce upstream slopes. Alluvial fans, like deltas, are characterized by 

unstable channel geometries and rapid lateral movement. An action very similar to the delta 

develops where a steep tributary enters a main channel. The steep channel tends to drop part 

of its sediment load in the main channel building out into the main stream. In some instances, 

the main stream can be forced to make drastic changes at the time of major floods by the 

stream's tributaries. 

Fans can be of two types, dry or mudflow fans formed by ephemeral streamflow, and wet or 

fluvial fans formed by perennial stream flow. Two different conditions of fan morphology are 

observed on modern dry fans. The first situation occurs when deposition is near the mountain 

front and the fan surface is undissected. The second situation occurs when sediment material 

is moved through a fan-head trench and deposition occurs at the toe of the fan. Good 

relationships exist between fan area and drainage basin area (Schumm 1977). These 

relationships among fan slope, area, and drainage basin characteristics are not surprising. The 

presence of fan-head trenches, however, is sometimes attributed to tectonic activity or climate 

change. 
5.3 

The longitudinal profile of fans may be concave. Two types of concavity are recognized. The 

first is due to intermittent uplift of the mountains which gradually steepens the fan head. The 

other case is due to trenching and the building out of a low flatter reach of recent alluvium at 

the toe of the fan. Normally the coarsest material is found at the fan apex, although fan-head 

trenching might result in a slight increase in sediment size with the fan radius. 

Fluvial (wet) fans can become very large, which contrasts with dry fans. The almost random 

distribution of erosion and deposition patterns on the arid fan is often replaced by a 

progressively shifting channel. Lateral migration of streams on fluvial fans can be anticipated 

by the concavity of the contours (i.e., topographic lows or swales). New orientation of a river 

channel is also an equally possible shifting process. 

The potential for avulsion, deposition, and channel blockage and channel incision are 

important for highway design. To minimize these impacts on highways, a reconnaissance of 

the fan and its drainage should be undertaken so that potential changes can be identified 

and countermeasures taken. The ideal result of any study of alluvial fans is a geomorphic 

map delineating active and inactive portions of the fan and the identification of problem sites 

within the active portions of the fan. For example, local aggradation in a channel can lead to 

avulsion because avulsion is likely to occur in places where deposition has raised the floor of 

the channel to a level that is nearly as high as the surrounding fan surface. This condition 

can be identified in the field by observation or by the surveying of cross-fan profiles 

(Schumm and Lagasse 1998). 

Experimental studies show that growth at the fan-head is intermittent, being interrupted by 

periods of incision, sediment reworking and downfan distribution of sediment. The greatest 

variation in sediment yield is related to fan-head trenching and aggradation. Geomorphic 

thresholds controlling fan growth are sketched in Figure 5.1 (see also Section 5.2.5). 

Threshold concepts must be considered when evaluating fan-related hazards to highways. 

For example, identification of relatively recent debris flow deposits, which suggests very high 

sediment delivery from the drainage basin may, in fact, be an indication of future stability. 

That is, stored sediment has been flushed from the drainage basin, and it may be a very 

long time before sufficient sediment accumulates again to produce debris flows even under 

extreme rainfall. This situation has been documented along the Wasatch Mountain front 

north of Salt Lake City (Keaton 1995; Lowe 1993), where drainage basins that produced 

debris flows in 1983 do not contain sufficient stored sediment to produce debris flows at 

present. Therefore, not only the fan itself, but its drainage basin requires investigation. 

5.2.4 Nickpoint Migration and Headcutting

Abrupt changes in the longitudinal profile of the stream are shown in Figure 5.2. This break in 

the profile induces a perturbation moving upstream, especially during floods. Above and below 

the profile break the river may be stable. As the perturbation migrates past a point, a dramatic 

change in channel morphology and stability occurs. These perturbations are of two types: the 

first is a sharp break in profile which forms an in-channel scarp called a headcut (Figure 5.2a), 

and the second, called a nickpoint, has a gradual change in elevation over a greater length of 

channel, but still represents an oversteepened reach with respect to the overall channel slope 

(Figure 5.2b). 
5.4 

Figure 5.1. Changing slope at fan-head leading to fan-head trenching (Schumm 1977). 

Figure 5.2. Headcuts and nickpoints. 

The result of headcut or nickpoint formation and migration is, of course, lowering of the stream 

bed. Erosion of the bed material will be dramatic as a headcut or nickpoint migrates under a 

bridge. During a major flood in Tujunga Wash, California, erosion above the headwall of a 

gravel pit led to the failure of three highway bridges. In cohesive material or erodible rock, 

headcut migration normally lowers the channel abruptly to its new position. In alluvial material, 

however a nickpoint produces degradation that persists for some distance (Figure 5.2b). In 

both cases, scour continues until the gradient has been reduced and bank erosion has 

widened the channel to the point that deposition can begin. As the nickpoint or headcut 

migrates farther upstream, the quantity of sediment delivered to the reach at which a stream 

crossing is located increases greatly due to the erosion of the bed upstream and subsequent 

erosion of the banks of the stream. Therefore, a period of degradation may be followed at a 

site by a period of aggradation. 
5.5 

The most obvious way to identify nickpoints or headcuts is by the use of aerial photographs. 

Particularly in arid and semi-arid regions, headcuts are very easily recognizable because the 

upstream valley floor or channel is essentially undisturbed, whereas the channel below shows 

significant erosion. On topographic maps of large scale the presence of a nickpoint or headcut 

is indicated by closely-spaced contours. This can be verified with field surveys which show the 

break in the longitudinal profile of the stream. A change in the dimension of the channel and a 

change in the character of the bank line may indicate the presence of nickpoint or headcut 

migration. A low width-depth ratio below the break in slope is an indication of scour and 

deepening of the channel. Bank erosion is also a possible consequence and a sharp change 

in the bankline characteristics representing a change from stability to instability may identify the 

presence of a nickpoint or headcut. 

5.2.5 Geomorphic Threshold

Evolution of a drainage network and sediment production from drainage basins are very 

complex processes. Geomorphic history and climatic changes introduce a new degree of 

complexity into the response of watersheds and fluvial systems. Experimental studies 

demonstrate that within a complex natural system, one event can trigger a complex reaction as 

the components of the system respond to change. The magnitude of this complex response is 

likely to appear during early stages of an erosion cycle or during rejuvenation of high sediment 

producing areas. Nevertheless, these areas produce major land management and 

conservation problems, hence their practical interest. 

It is possible to explain variations in sediment yield and channel adjustment by using the 

concept of geomorphic thresholds described by Schumm (1977). For example, in a given 

drainage area, it is possible to define a valley slope above which the valley floor is unstable. 

Stable valley floors above the threshold line are incipiently unstable and any major flood may 

eventually cause erosion and trenching of the alluvium stored in these valleys (Figure 5.1 

illustrates the concept of geomorphic threshold). Another very common example of a 

geomorphic threshold is the progressive increase in channel sinuosity and meander amplitude 

until a cutoff or channel avulsion results on alluvial plains and deltas. This is due to channel 

lengthening and gradient reduction accompanying increases in sinuosity and delta size. Since 

permanent changes result only when a geomorphic threshold has been exceeded, events with 

high magnitude and low frequency may at times have only minor and local effect on the 

landscape, but at other times may produce seemingly "instantaneous" change with potentially 

catastrophic consequences. 

5.3 VARIABILITY AND CHANGE IN ALLUVIAL RIVERS

Those who work with rivers are aware of the great variability that exists among rivers and 

between river reaches. In addition, the impact of human activities can greatly alter the 

behavior, dimensions, and general morphology of a river. Large alluvial rivers have always 

played an important role in human affairs. All of the early great civilizations rose on the 

banks of large alluvial rivers such as the Nile, Indus, Yellow, and Euphrates. River 

engineering began early in human history to minimize the effects of floods and channel 

changes. Today, engineers face the same problems and they have been successful in 

developing flood control, navigation, and channel stabilization programs, but often at great 

cost and with the need to continually maintain and repair structures and channels (Schumm 

and Winkley 1994). In this section, historic change and channel response on the Mississippi 

River and the Nile are surveyed to illustrate the range of variability and change to be 

expected on alluvial rivers. 
5.6 

In order to cope with alluvial rivers, an understanding of their complexity in space and 

through time is necessary. Alluvial rivers differ in three ways: 

1. Rivers differ among themselves depending on hydrology, sediment loads, and geologic 

history (in other words, rivers differ among themselves). 

2. Rivers change naturally through time and as a result of climate and hydrologic change. 

3. Along any one river there can be considerable variability of channel morphology as a 

result of geologic and geomorphic controls. 

Information on these differences, especially the last two, will aid in predicting future river 

behavior and their response to human activities. 

An important consideration in predicting future river behavior and response is the sensitivity 

of the channel. That is, how readily will it respond to change or how close is it to undergoing 

a change without an external influence? For example, individual meanders frequently 

develop progressively to an unstable form, and a chute or neck cutoff results, which leads to 

local and short-term channel adjustments. The cutting off of numerous meanders along the 

Mississippi River caused dramatic changes, as a result of steepening of the gradient, which 

led to serious bank erosion and scour (Winkley 1977). However, Brice (1980) studied the 

effect of cutoffs and channel alignment on numerous smaller streams, and he found that, 

although some responded in a manner similar to the Mississippi, many did not. Those that 

did not were characterized by stable banks and gentle slopes. For example, Stevens (1994) 

found that the Citanduy River in Java has remained relatively stable after a total of 23 cutoffs 

because it has very resistant clayey bed and banks. Therefore, some streams are sensitive 

whereas others are not. Obviously, care must be taken before the behavior and response of 

one stream can be extrapolated to another. 

The differences among rivers is often reflected in their channel patterns. During an 

experimental study of a channel in a laboratory flume, the channel changed from straight, to 

sinuous and finally to braided, as flume slope (and thereby sediment load and stream power) 

increased (Figure 5.3). The relatively straight River Nile and the relatively meandering 

Mississippi River reflect geomorphic history as well as current geologic controls such as 

active tectonics, tributary contributions and other factors that affect valley slope. For 

example, the difference between the Nile and other large rivers may be due to the fact that 

the Mediterranean Sea evaporated during early Tertiary time (Hsü 1983, Said 1981). A 

blockage at the Straights of Gibraltar stopped the inflow of Atlantic Ocean water into the 

Mediterranean basin. This significant lowering of base level caused incision of streams 

draining into the Mediterranean, and a deep canyon was cut which formed the Nile valley. 

When sea level rose again, marine water entered this canyon, and marine sediments are 

found as far up-river as Aswan. The Nile then filled this trough with fluvial sediments at a 

relatively gentle slope that was needed to move water and sediment to the newly established 

higher base level. In contrast to this unique history, the Mississippi River developed a 

relatively steep valley slope as the result of the influx of outwash sediment from melting 

continental glaciers. After retreat of the ice, the valley slope was steeper than was required 

to transport the reduced sediment load, and the Mississippi developed a sinuous course in 

order to reduce its gradient. The difference between these two great alluvial rivers, 

therefore, has much to do with their geologic and geomorphic history. 
5.7 

Figure 5.3. Relationship between flume slope and sinuosity (ratio between channel length 

 and valley length) during flume experiments at constant water discharge (from 

 Schumm and Khan 1972). 

5.3.1 Differences Through Time

During the climate changes of Quaternary time, many rivers completely changed their 

morphology and behavior (Figure 5.4). This metamorphosis reflected great changes of 

discharge and sediment load. In addition to the Mississippi River, rivers on the Polish Plain 

(Kozarski and Retnicki 1977) and the Riverine Plain of Australia (Schumm 1968) underwent 

the same type of changes. 

 Figure 5.4. Sequence of channel changes as water discharge and sediment loads 

 decrease. (A) braided channel; (B) transitional meandering-braided 

 channel with well-defined thalweg; (C) low-sinuosity channel; (D) relatively 

 narrow and deep moderately sinuous channel; and (E) multiphase 

 meandering channel (after Schumm and Brakenridge 1987). 
5.8 

Perhaps of more interest here are the changes during shorter periods of time, as river bends 

form, grow, and cutoff. Most meandering rivers will show significant change of sinuosity 

through time. Figure 5.5 shows the range of sinuosity between 1765 and 1930 for 24 

reaches of the lower Mississippi River. Obviously the Mississippi River is composed of very 

different reaches that behaved differently during the period of record. These are the types of 

changes to be expected with time along any active river, but the changes also can be 

avulsive and of a catastrophic nature with abandonment of one channel, and formation of 

another. This frequently occurs on deltas, and alluvial plains, and it is exemplified by great 

lateral shifts of the Indus River (Holmes 1968). Indeed, avulsion of the Mississippi River 

down the Atachafalaya River channel is only prevented by major flood-control structures. 

If as a result of climatic fluctuations and human activities, the sediment load, flood peaks, 

and water discharge of a river are altered, a river response can be expected. However, the 

type of response will depend upon the nature of the river. For example, sinuous rivers could 

become straight, and braided rivers could become sinuous, or the changes could be very 

minor, depending upon the sensitivity of the river (Schumm and Beathard (1976). 

Figure 5.5. Variability of sinuosity between 1765 and 1915 for 24 Mississippi River reaches. 

 Numbers identify the reaches of Figures 5.6 and 5.7 (after Schumm et al. 1994). 

The greatest modern river changes have been the result of human activity. The Mississippi 

River was straightened and shortened 229 kilometers (142 miles) between 1933 and 1942. 

The goal was to reduce flood peaks and to improve navigation. This great river experiment 

yielded both beneficial and undesirable results. Floods were reduced, but channel stability 

decreased (Winkley 1977, 1994). 

5.3.2 Differences Between Reaches

Perhaps of greater interest to river engineers is the variability along a single river. One could 

assume that large alluvial rivers should have a relatively uniform morphology because the 

controlling factors of water discharge and sediment load should not vary greatly. However, 

other factors intervene to cause considerable variability. For example, a glance at even a 

coarse-scale map of the pre-cutoff Mississippi River reveals great variability. In fact, 24 

distinct reaches were identified between Cairo, Illinois and Old River, a distance of 768 valley 

kilometers (477 miles) (Figure 5.6). The reaches were identified by changes of valley slope, 
5.9 

channel variability through time and the pre-cutoff 1930 river pattern. Figure 5.5 shows great 

differences in the variability of sinuosity among reaches and for each of the reaches through 

time. The differences among the reaches are for the most part the result of tectonic 

deformation of the valley floor and the presence of more resistant materials in the bed and or 

banks, such as clay plugs, Pleistocene-age alluvium, and Tertiary-age bedrock. A plot of the 

valley (floodplain) slope, based upon the 1880 bankfull elevations (Figure 5.7) shows 

considerable variability, as a result of these controls. The Mississippi is characterized by a 

lack of uniformity in its morphology and dynamics, which has made the job of the river 

engineer difficult. The great variability is largely the result of geomorphic and geologic 

controls (Schumm et al. 1994). 

Figure 5.6. Mississippi River reaches between Cairo, Illinois and Old River, Louisiana (after 

 Schumm et al. 1994). 

Figure 5.7. Mississippi River valley profile (after Schumm et al. 1994). 
5.10 

Another great alluvial river, the Nile, when studied using the same geomorphic techniques 

(Schumm and Galay 1994) is found to be much less variable and dynamic (Figure 5.8), but 

because of the great population density along its course even minor changes are of 

importance. There appears to have been little change of the pattern of the River Nile since 

mapping in the 18th century, except in the reach between El Saff and Cairo. 

A plot of the Nile valley slope (Figure 5.9) shows differences similar to that of the Mississippi 

(Figure 5.7), but they are much less. Sinuosity does increase with increased valley slope, 

and valley slope actually increases downvalley (Figure 5.10). The change of valley slope 

and sinuosity below Qena is undoubtedly the result of the large Wadi Quena, which in the 

past has introduced coarse sediments into the Nile Valley. The other changes of slope are 

probably related to tectonics, such as faulting at Assiut. 

The anticipated impact of the High Aswam Dam on hydrology and sediment loads was a 

matter of great concern for engineers concerned with bank stability and potential channel 

degradation. Perhaps of greatest concern was the potential for major degradation of the 

Nile following construction of the dam. Because degradation has been controversial, it was 

studied by many researchers prior to, and after, construction of the dam (Fathy 1956, 

Mostafa 1957, Shalash 1980, Shalash 1983, Richardson and Clyma 1980, Gasser et al. 

1978, and El-Moatassem and El-Mottaleb 1979). The construction of the High Aswam Dam 

commenced in 1963 and proceeded to 1968. Pre-dam estimates of degradation ranged 

from 2.0 to 8.5 m (6.5 to 28 ft), but 18 years after the dam was in operation, maximum 

degradation was 0.70 m (2.3 ft). 

Several factors account for the fact that degradation has been minimal after closure of the 

dam. For example, it is important to recognize that during past humid periods in Egypt, 

wadis delivered coarse sediments to the valley, which could act as a control of degradation 

depending upon their location in the valley and the depth at which they are encountered. At 

present, many wadis appear to contribute only relatively fine sediment that can be readily 

transported downstream, and the bed of the Nile is sand. However, during wetter periods of 

the past, the wadis probably contributed abundant coarse sediment to the river, and even 

today wadi flooding must introduce coarse sediments into the river. For example, there is an 

abundant supply of sand, gravel, and cobbles in Wadi Qena, which drains a large area to the 

north of Qena. A deep trench was excavated in a small wadi that enters the Nile valley at 

Khuzam about 30 km (18.5 mi) downstream from Armant. Boulders and cobbles are 

abundant in the trench, and such sediments undoubtedly were moved into the Nile during 

wetter periods. All of the wadis contain much stored sediment of gravel, cobble, and boulder 

size. 

Based upon an analysis of borings in the Nile valley, Attia (1954) concluded that within the 

valley "coarse deposits composed of coarse sand, sand and gravel, or gravel lie beneath the 

fine alluvial deposits." It is well known that only a small percentage of coarse bed material 

can armor a bed and it appears that the minimal degradation by the River Nile in response to 

the High Aswam Dam is the result of coarse sediment beneath a veneer of sand. 
5.11 

Figure 5.8. River Nile between Qena and Cairo showing six reaches of steep valley slope. 

Figure 5.9. River Nile valley profile. Numbers identify reaches of Figure 5.8 (after Schumm 

 and Galay 1994). 
5.12 

Figure 5.10. Plot of River Nile sinuosity and valley slope. Numbers represent reaches of 

 Figure 5.9. The vertical line joining two points for Reach 12 shows the 

 difference between 18th century and present low sinuosity (after Schumm and 

 Galay 1994). 

5.3.3 Summary

The contrast between the lower Mississippi River and the Nile is striking. The Nile is much 

less dynamic, but it too responds to geologic controls and tributary influences. Both the 

Mississippi and Nile Rivers have sand-beds, yet what lies beneath the sand may be of critical 

importance to the character and response of these great rivers. It is probable that the 

Mississippi is controlled totally by Tertiary-age clay and Pleistocene-age gravels, and the 

Nile has apparently developed an armor of gravel and cobbles beneath its sand bed. 

Anomalous river behavior can often be explained by factors that are related to river history. 

Both the Mississippi and the Nile provide support of this fact. Hence, factors that are 

normally beyond the engineer's range of expertise may dominate river morphology and 

behavior. 

In order to understand alluvial rivers, one cannot limit a study to modern conditions only. In 

fact, past channel morphology and behavior can provide valuable information about modern 

river behavior. For example, although both the Mississippi and the Nile are large and 

transport similar sediments, they differ greatly morphologically and dynamically as a result of 

quite different histories. Therefore, in order to anticipate and to predict river response, both 

the past and the present must be combined in what could be termed a geomorphicengineering approach to river maintenance and/or restoration. 

5.4 STREAM FORM AND GEOMETRY OF ALLUVIAL CHANNELS

A study of the plan and profile of a stream is very useful in understanding stream morphology. 

Planview appearances of streams are varied and result from many interacting variables. Small 
5.13 

changes in a variable can change the planview and profile of a river, adversely affecting a 

highway crossing or encroachment. Conversely, the highway crossing or encroachment can 

inadvertently change the planview or profile, adversely affecting the river environment. In this 

section, stream form is classified and channel processes are discussed. 

5.4.1 Classification of River Channels

Brice and Blodgett (1978) developed a simple classification scheme oriented primarily toward 

lateral stability of rivers. The common geomorphic terms for the various types of streams (e.g., 

meandering, braided) are shown in Figure 5.11. Each term is defined on the small sketches. 

This classification is also used in HEC-20 (Lagasse et al. 2001) as a basis for identifying 

geomorphic factors important to stream stability analyses. 

Additional information on specific channel features is illustrated in Figure 5.12. Classification 

based on oxbow lakes is illustrated in Figure 5.12a. In Figure 5.12b, types of meander scroll 

formations are shown. By studying scroll formations in terms of age of vegetation the rate and 

direction of channel migration can be quantified. The sinuosity index is the ratio of the length of 

the watercourse over the valley length between the same points. Classification based on 

natural levees is illustrated in Figure 5.12c. Well developed levees are associated with older 

rivers. The floodplain that is broad in relation to the channel width is indicative of an older river. 

Conversely, when the river valley is narrow and confined by terraces or valley walls, the river is 

usually mature. In general, the growth of vegetation (tree cover) is indicative of the presence of 

silts and clays in the river banks and the floodplain. This is particularly true if the floodplain is 

well drained. With good drainage, the silt and clay are essential to the growth of vegetation 

because of their water holding capability. 

Natural levees are a characteristic of alluvial river systems. Levees form during floods as the 

river stage exceeds bankfull conditions. Sediment is then deposited on the floodplain due to 

the reduced velocity and transporting capacity. The natural levees near the river are rather 

steep because coarse material drops out quickly. Farther from the river the gradients are 

flatter and the finer materials drop out. Beyond the levees are the swamp areas. On the lower 

Mississippi River, natural levees on the order of 10 ft in height are common. The rate of growth 

of natural levees is slower after they reach a height equal to the average annual flood stage. 

A detailed knowledge of the hydraulic characteristics of different types of streams is of great 

value when dealing with the location of highway crossings and encroachment, training works, 

flood control works and other river structures. A channel classification in Figure 5.13 shows the 

relative stability and types of hazards encountered. Figure 5.13 is also useful in making a 

qualitative assessment of stream stability based on stream characteristics. It shows that 

straight channels are relatively stable only where flow velocities and sediment load are low. As 

these variables increase, flow meanders in the channel causing the formation of alternate bars 

and the initiation of a meandering channel pattern. Similarly, meandering channels are 

progressively less stable with increasing velocity and bed load. At high values of these 

variables, the channel becomes braided. The presence and size of point bars and middle bars 

are indications of the relative lateral stability of a stream channel. Bed material transport is 

directly related to stream power, and relative stability decreases as stream power increases as 

shown by Figure 5.13. An example is given in Section 5.9 (Problem 2) on how to use this 

classification. 
5.14 

Figure 5.11. Stream properties for classification (after Brice and Blodgett 1978). 
5.15 

Figure 5.12. Classification of river channels (after Culbertson et al. 1967). 
5.16 

Figure 5.13. Channel classification showing relative stability and types of hazards encountered 

 with each pattern (after Shen et al. 1981). 

Additional approaches to stream channel classification including those by Brice (1975), 

Schumm (1977 and 1981), Montgomery and Buffington (1997), and Rosgen (1994 and 1996) 

are introduced in HEC-20 (Lagasse et al. 2001). Further discussion of the opportunities and 

limitations related to using stream classification for river analysis, engineering and 

management can be found in Thorne (1997). 

The next sections provide a brief discussion concerning the nature and stability of straight, 

braided, and meandering channels. Each behaves in a slightly different way when subject to 

human-related or natural impacts. A knowledge of this behavior is important in anticipating and 

understanding stability problems. 
5.17 

5.4.2 Straight River Channels

Straight river channels can be of two types. The first forms on a low-gradient valley slope, has 

a low width-depth ratio channel, and is relatively stable. The second type is a steep gradient, 

high width-depth ratio, high energy river that has many bars, and at low flow is braided. It is 

relatively active. The first type of straight channel may contain alternate bars (Figure 5.14), that 

result in a sinuous thalweg (flow path connecting deepest points in successive cross sections) 

within the straight channel. The braided channel, as discussed in detail later, has numerous 

bars and multiple thalwegs. 

Figure 5.14. Planview and cross section of a meandering stream. 

5.4.3 Meandering River Channels 

Alluvial channels of all types deviate from a straight alignment. The thalweg oscillates 

transversely and initiates the formation of bends. In general, the river engineer concerned with 

channel stabilization should not attempt to develop straight channels fully protected with riprap. 

In a straight channel the alternate bars and the thalweg are continually changing; thus, the 

current is not uniformly distributed through the cross-section but is deflected toward one bank 

and then the other. Sloughing of the banks, nonuniform deposition of bed load caused by 

debris such as trees, and the Coriolis force due to the earth's rotation have been cited as 

causes for meandering of streams. When the current is directed toward a bank, the bank is 

eroded in the area of impingement and the current is deflected and impinges upon the opposite 

bank further downstream. The angle of deflection of the thalweg is affected by the curvature 

formed in the eroding bank and the lateral depth of erosion. 
5.18 

The meandering river consists of pools and crossings (Figure 5.14). The thalweg, or main 

current of the channel, flows from the pool through the crossing to the next pool forming the 

typical S-curve. In the pools, the channel cross-section is somewhat triangular. Point bars 

form on the inside of the bends. In the crossings, the channel cross-section is more 

rectangular and depths are smaller. At low flows, local slope is steeper and velocities are 

larger in the crossing than in the pool. At low stages, thalweg is located very close to the 

outside of the bend. At higher stages, thalweg tends to straighten, that is, thalweg moves away 

from the outside of the bend encroaching on the point bar to some degree. In the extreme 

case, the shifting of the current causes chute channels to develop across the point bar at high 

stages. In Figure 5.14, one can observe the position of the thalweg, the location of the point 

bars, alternate bars and the location of the pools and crossings. Note that in the crossing the 

channel is shallow compared to pools and the banks may be more subject to erosion. 

Figure 5.14 illustrates the change in water surface profile from low to high water discharge. At 

low flow the water surface slope is steep in the crossing and flatter in the pool. The reverse is 

true at higher discharges. At higher discharges the thalweg straightens, shortening the path of 

travel and increasing the local friction slope. In the extreme case, river slope approaches the 

valley slope at flood stage. It is during high floods that the flow often cuts across the point 

bars, developing chute channels and a steeper channel prevails under this condition. 

In general, bends are formed by the process of erosion and deposition. Erosion without 

deposition to assist in bend formation would result only in scalloped banks. Under these 

conditions the channel would simply widen until it becomes so large that the erosion would 

terminate. The material eroded from the bank is normally deposited over a period of time on 

the point bars that are formed downstream. The point bars constrict the bend and enable 

erosion in the bend to continue, accounting for the lateral and longitudinal migration of the 

meandering stream. Erosion is greatest across the channel from the point bar. As the point 

bars build out from the downstream sides of the bars, the bends gradually migrate down the 

valley. The point bars formed in the bendways clearly define the direction of flow. The bar is 

generally streamlined and its largest portion is oriented downstream. If there is very rapid 

caving in the bendways upstream, the sediment load may be sufficiently large to cause middle 

bars to form in the crossing. 

As a meandering river system moves laterally and longitudinally, the meander limbs move at an 

unequal rate because of the unequal erodibility of the banks. This causes the channel to 

appear as a bulb form, generally skewed in a downvalley direction. The channel geometry 

depends upon the local slope, the bank material, and the geometry of the adjacent bends. 

Over time the local steep slope caused by the cutoff is distributed both upstream and 

downstream. Years may be required before a configuration characteristic of average 

conditions in the river is attained. 

When a cutoff occurs, an oxbow lake is formed (Figure 5.12a). Oxbow lakes may persist for 

long periods of time before filling. Usually the upstream end of the lake fills quickly to bank 

height. Overflow during floods carries fine materials into the oxbow lake area. The lower end 

of the oxbow remains open and the drainage and overland flow entering the system can flow 

out from the lower end. The oxbow gradually fills with fine silts and clays. Fine material that 

ultimately fills the bendway is plastic and cohesive. As the river channel meanders it 

encounters old bendways filled with cohesive materials (referred to as clay plugs). These plugs 

are sufficiently resistant to erosion to serve as semipermanent geologic controls and can 

drastically affect river geometry. The variability of bank materials, and the fact that the river 

encounters such features as clay plugs, cause a wide variety of river forms in a meandering 

river. 
5.19 

In summary, a meandering river has regular inflections that are sinuous in plan. It consists of a 

series of bends connected by crossings. In the bends, deep pools are carved adjacent to the 

concave bank by the relatively high velocities. Because velocities are lower on the inside of the 

bend, sediments are deposited in this region, forming a point bar. The centrifugal force in the 

bend causes a transverse water surface slope, and in many cases, helicoidal flow occurs in 

the bend. Point bar building is enhanced when large transverse velocities occur. In so doing, 

they sweep the heavier concentrations of bed load toward the convex bank where they are 

deposited to form the point bar. Some transverse currents have a magnitude of about 15 

percent of the average channel velocity. The bends are connected by crossings (short straight 

reaches) which are quite shallow compared to the pools in the bendways. Much of the 

sediment eroded from the outside bank is deposited in the crossing and on the point bar in the 

next bend downstream. At low flow, large sandbars form in the crossings if the channel is not 

well confined. 

The scour in the bend causes the bend to migrate laterally and sometimes downstream. 

Lateral movements as large as 760 m (2,500 ft) per year have been observed in alluvial rivers. 

The meander belt formed by a meandering river is often fifteen to twenty times the channel 

width. When compared to most braided rivers, meandering rivers have relatively flat slopes. 

The geometry of meandering rivers is measured quantitatively in terms of: (1) meander 

wavelength λ, (2) meander width Wm, (3) mean radius of curvature rc, (4) meander amplitude 

A, and (5) bend deflection angle φ. These variables are shown in Figure 5.15. 

The actual meanders in natural rivers are obviously not as regular as indicated in Figure 5.15. 

The precise measurement of meander dimensions is therefore difficult in natural channels and 

tends to be subjective. An example on how to measure these characteristics is presented in 

Section 5.9 (Problem 1). The analysis of the mean meander dimension in nature shows that 

the meander length and meander width are both related to the width of the channels. The 

empirical relationships for the meander length λ and the bank-full channel width as well as the 

meander amplitude, A, and the bank-full channel width are shown in Figure 5.16 and Table 5.1. 

Figure 5.15. Definition sketch for meanders. 
5.20 

Figure 5.16. Empirical relations for meander characteristics (Leopold et al. 1964). 

Table 5.1. Empirical Relations for Meanders in Alluvial Valleys. 

Meander Length to 

Channel Width 

Amplitude to 

Channel Width 

Meander Length to 

Radius of Curvature Source 

λ = 6.6W0.99 A = 18.6W0.99 - Inglis (1949) 

- A = 10.9W1.04 - Inglis (1949) 

λ = 10.9W1.01 A = 2.7W1.10 λ = 4.7rc

0.98 Leopold and 

Wolman (1960) 

5.4.4 Braided River Channels 

A braided stream is one that consists of multiple and interlacing channels (Figures 5.11 and 

5.13). One cause of braiding is the large quantity of bed load. Generally, the magnitude of the 

bed load is more important than its size. If the channel is overloaded with sediment, deposition 

occurs, the bed aggrades, and the slope of the channel increases in an effort to obtain a 

graded state. As the channel steepens, the velocity increases, and multiple channels develop. 

These interlaced multiple channels cause the overall channel system to widen. Multiple 

channels are generally formed as bars of sediment are deposited within the main channel. 

Another cause of braiding is easily eroded banks. If the banks are easily eroded, the stream 

widens at high flow and forms bars at low flow which become stabilized, thus forming islands. 

In general, a braided channel has a relatively steep slope, a large bed-material load in 

comparison with its suspended load, and relatively small amounts of silts and clays in the bed 

and banks. Figure 5.17 will assist in defining the various conditions for multiple channel 

streams. 
5.21 

Figure 5.17. Types of multi-channel streams. 

The braided stream may present difficulties for highway construction because it is unstable, 

changes its alignment rapidly, carries large quantities of sediment, is very wide and shallow 

even at flood flow and is, in general, unpredictable. 

5.4.5 River Conditions for Meandering and Braiding

It can be shown that changes in water discharge, sediment discharge or both can cause 

significant changes in channel slope (see Section 5.5). The changes in sediment discharge 

can be in quantity, Qs, or sediment size, D50, or both. Often, such changes can alter the 

planview in addition to the profile of a river. 

According to Lane (1957), Figures 5.18a and b illustrate the dependence of sand bed river form 

on channel slope and discharge. They show that when: 

SQ k 0 25 . ≤ (5.1) 

where: 

 k = 0.0007 SI 

 k = 0.0017 English 

a sandbed channel meanders. Similarly, when: 

SQ k 0 25 . ≥ (5.2) 
5.22 

0.00001

0.0001

0.001

0.01

1 10 100 1000 10000 100000

MEAN ANNUAL DISCHARGE (m3/s)

SLOPE (m/m)

MEANDERING

TRANSITIONAL

BRAIDED

LANE

LANE

LEOPOLD & WOLMAN

SQ0.25 = 0.00070

SQ0.25 = 0.0041

SQ0.44 = 0.0125

0.00001

0.0001

0.001

0.01

100 1000 10000 100000 1000000

MEAN ANNUAL DISCHARGE (ft3/s)

SLOPE (ft/ft)

MEANDERING

TRANSITIONAL

BRAIDED

LANE

LANE

LEOPOLD & WOLMAN

SQ0.25 = 0.0017

SQ0.25

S

= 0.010

Q0.44 = 0.060

(a)

(b)

Figure 5.18a,b. Slope-discharge relationship for braiding or meandering in sandbed streams 

 (after Lane 1957) (a = SI units b = English units). 
5.23 

where: 

 k = 0.0041 SI 

 k = 0.010 English 

the sandbed river is braided. In these equations, S is the channel slope in m/m (ft/ft) and Q is 

the mean discharge in m3

/s (ft3/s). Between these values of S Q0.025 is the transitional range. 

Many of the U.S. rivers, classified as intermediate sandbed streams, plot in this zone. If a river 

is meandering but its discharge and slope border on the transitional zone a relatively small 

increase in channel slope may cause it to change to a transitional or braided river. 

Leopold and Wolman (1960) plotted slope and discharge for a variety of natural streams. They 

observed that a line could separate meandering from braided streams (Figures 18a and b). 

The equation of this line is: 

SQ k 0 44 . = (5.3) 

where: 

 S = Slope in m/m (ft/ft) 

 Q = Bank-full discharge is m3

/s (ft3/s) 

 k = 0.0125 SI 

 k = 0.06 English 

Streams classified as meandering by Leopold and Wolman are those whose sinuosity is 

greater than 1.5. Braided streams are those which have relatively stable alluvial islands and, 

therefore, two or more channels. Leopold and Wolman note that sediment size is related to 

slope and channel pattern, but they do not try to account for the effect of sediment size on the 

morphology of the streams. They further note that braided and meandering streams can be 

differentiated based on combinations of slope, discharge, and width/depth ratio, but regard the 

width as a variable dependent on mainly discharge. 

Leopold and Wolman recognize that their analysis treats only two of the many variables 

affecting morphology, therefore, they do not expect this method to apply in every condition. 

However, because the data were all taken from natural streams, and because the analysis 

obviously does indicate a significant relation between slope and discharge, the analysis should 

give a reasonably effective prediction of channel pattern if slope and discharge are known. 

Problem 1 in Section 5.9 gives an example of this type of prediction and Section 5.5.3 uses 

these concepts in an engineering geomorphic analysis. 

5.4.6 Hydraulic Geometry of Alluvial Channels

Hydraulic geometry is a general term applied to alluvial channels to denote relationships 

between discharge Q and the channel morphology, hydraulics and sediment transport. 

Channels forming in their own sediments are called alluvial channels. In alluvial channels, the 

morphologic, hydraulic and sedimentation characteristics of the channel are determined by a 

large variety of factors. The mechanics of such factors are not fully understood. However, 

alluvial streams do exhibit some quantitative hydraulic geometry relations. In general, these 

relations apply to channels within a physiographic region and can be obtained from data 

available on gaged rivers. It is understood that hydraulic geometry relations express the 

integral effect of all the hydrologic, meteorologic, and geologic variables in a drainage basin for 

in-bank flows. 
5.24 

The hydraulic geometry relations of alluvial streams are useful in river engineering. The 

forerunner of these relations are the regime theory equations of stable alluvial canals (see for 

example, Kennedy 1895, Lacy 1930, and Leliavsky 1955). Hydraulic geometry relations were 

developed by Leopold and Maddock (1953) for different regions in the United States and for 

different types of rivers. In general the hydraulic geometry relations are stated as power 

functions of the discharge: 

W aQb = (5.4) 

y cQ o

f = (5.5) 

V kQm = (5.6) 

Q pQ T

j = (5.7) 

S tQ f

z = (5.8) 

n rQy = (5.9) 

where: 

 W = Channel width 

 yo = Channel depth 

 V = Average velocity of flow 

 QT = Total bed sediment load 

 Sf = Friction slope 

 n = Manning's roughness coefficient 

 Q = Discharge as defined in the following paragraphs 

The coefficients a, c, k, p, t, r and exponents b, f, m, j, z, y in these equations are determined 

from analysis of available data on one or more streams. From the continuity equation (Q = 

WyoV), it is seen that 

axcxk =1 (5.10) 

and 

bfm ++ =1 (5.11) 

Leopold and Maddock (1953) have shown that in a drainage basin, two types of hydraulic 

geometry relations can be defined: (1) relating W, yo, V and Qs to the variation of discharge 

at-a-station; and (2) relating these variables to the discharges of a given frequency of 

occurrence at various stations in a drainage basin. Because QT is not readily available, they 

used Qs, the suspended sediment transport rate. The former are called at-a-station

relationships and the latter downstream relationships. The distinction between at-a-station 

and downstream hydraulic geometry relations is illustrated in Figures 5.19 and 5.20. 

Figures 5.19 and 5.20 illustrate how the hydraulic relations at-a-station and in the downstream 

direction may be different from one basin to another. For example, the width and depth at-astation do not change very much in Basin A. The width to depth ratio is almost constant but the 

velocity increases, as it must, as the discharge increases at-a-station. In Basin B the width to 

depth ratio decreases with an increase in discharge. That is, the width changes very little but 

the depth increases significantly with discharge at-a-station. Note, that in both basins, width 
5.25 

increases with discharge in the downstream direction, but in Basin B, depth changes very little 

in the downstream direction. Leopold and Maddock (1953) show that the type of relations 

illustrated in Figure 5.20 exist in natural rivers. 

Figure 5.19. Variation of discharge at a given river cross section and at points downstream 

 (after Leopold and Maddock 1953). At-a-station relations pertain to individual 

 sites such as A or B. Downstream relations pertain to a channel (segment A-B) 

 or drainage network for discharge of a given frequency of occurrence. 

Figure 5.20. Schematic variation of width, depth, and velocity with at-a-station and downstream 

 discharge variation (after Leopold and Maddock 1953). 
5.26 

The mean values of exponents b, f, m, j, z, and y as reported by Leopold et al. (1964) are 

given in Table 5.2. These values are based on an extensive analysis of stream data in the 

United States and are stream specific. 

Table 5.2. At-A-Station and Downstream Hydraulic Geometry Relationships. 

Average At-A-Station Relations 

 b f m j z y 

Average values Midwestern United States .26 .40 .34 2.5 

Brandywine Creek, PA .04 .41 .55 2.2 .05 -.2 

Ephemeral Streams in Semiarid U.S. .29 .36 .34 

Average of 158 Gaging Stations in U.S. .12 .45 .43 

Ten Gaging Stations on Rhine River .13 .41 .43 

Average Downstream Relations 

(bank-full or mean annual flow) 

 b f m j z y 

Average values Midwestern United States .5 .4 .1 .8 -.49 

Brandywine Creek, PA .42 .45 .05 -1.07 -.28 

Ephemeral Streams in Semiarid U.S. .5 .3 .2 1.3 -.95 -.3 

Appalachian Streams .55 .36 .09 

In Table 5.3, derived hydraulic geometry relations for conditions at-a-station (variable discharge 

frequency) and in the downstream direction on the same stream at bankfull discharge (or 

constant frequency) are given. Note: the term "downstream" implies any other location along 

the channel, either upstream or downstream from a selected station. Bray (1982) and Julien 

and Simons (1984) determined that bed material size is an important variable in hydraulic 

geometry relations in alluvial gravel streams. Their relations for non-cohesive gravel-bed rivers 

in the downstream direction at bankfull discharge (or constant frequency) are also given in 

Table 5.3. Applications of these relationships are illustrated in Section 5.9 (Problems 5 and 6). 

Table 5.3. Derived At-A-Station and Downstream Geometry Relationships. 

At-A-Station(1) Downstream(1) Downstream(2) 

40.0

o ≈ Qy 46.0 ≈ Qy bo

40.0 ≈ Qy bo

26.0 ≈ QW 46.0 ≈ QW 33.0

50

53.0

b DQW − ≈

34.0 ≈ QV 08.0 ≈ QV b33.0

50

07.0 ≈ b DQV

00.0

f ≈ QS 46.0 QS bf

− ≈ 00.1

50

4.0

bf DQS +− ≈

(1) Sand bed 

(2) Gravel bed 
5.27 

Work by Julien and Wargadalam (1995) has updated the theory and applications of hydraulic 

geometry relationships for alluvial channels. In their study, downstream hydraulic geometry of 

alluvial channels, in terms of bankfull width, average flow depth, mean flow velocity, and friction 

slope, is examined from a 3-dimensional stability analysis of noncohesive particles under 2-

dimensional flows. Four exponent diagrams illustrate good agreement with several empirical 

regime equations found in the literature. The analytical formulations were tested with a 

comprehensive data set consisting of 835 field channels and 45 laboratory channels. The data 

set covers a wide range of flow conditions from meandering to braided sand-bed, and gravelbed rivers. 

5.4.7 Dominant Discharge in Alluvial Rivers

The hydraulic geometry relations discussed in Section 5.4.6 indicate how the channel 

morphology and other characteristics vary with discharge at-a-station or in the downstream 

direction in a drainage network. In the hydraulic design of river crossings and encroachments, 

the relations need to be defined to determine the downstream hydraulic geometry of the 

channel at a site between two gaged sites. The question then arises about the frequency of 

discharge to be used in the hydraulic geometry relations. The downstream hydraulic geometry 

relations expressed in Section 5.4.6 relate to the bank-full stage, which for many humid region 

U.S. rivers has a frequency of occurrence of one in 1.5 years. For arid region streams, the 

bankfull return period may be on the order of five to ten years. 

Analysis of bed sediment load estimations indicates that on most rivers, up to 90 percent of the 

total transport is caused by flows that are equaled or exceeded about ten percent of the time 

only. Thus, the average bed sediment load in a river may be described in terms of a formative 

discharge much larger than the mean annual flow. Also, the average channel width, depth and 

meander geometry may be defined in terms of different formative discharges rather than an 

arbitrarily chosen dominant discharge. 

Research by the U.S. Army Corps of Engineers (Watson et al. 1999, Copeland and Hall 1998) 

suggests that the effective discharge is the best representation of the channel forming 

discharge. The effective discharge is the increment of discharge that transports the most 

sediment on an annual basis. This discharge may be determined by integrating a sediment 

transport rating curve with the annual flow-duration curve. Where possible, it is important to 

attempt to verify this channel-forming discharge with field indicators of bankfull discharge. 

Appendix A to the 1999 USACE reference cited above provides a practical guide to effective 

discharge calculations. 

The concept of the frequency of occurrence of flows is important in the hydraulic design of 

highway crossings and encroachments. Both the at-a-station and downstream hydraulic 

geometry relations are especially useful when the hydraulic design is based on the frequency 

of occurrence of flows. The concept of bank-full condition corresponding to a discharge with a 

period of return of 1.5 years for perennial streams and five to ten years for ephemeral streams 

is recommended for practical use when detailed analysis of formative discharge is not possible, 

or feasible. 
5.28 

5.4.8 River Profiles and Bed Material

The slope of a river channel or a river system is usually steepest in the headwater regions. 

The river profile is concave upward and the slope of the river profile can be represented by the 

equation: 

S Se x o

x = − α

 (5.12) 

where: 

 Sx = Slope at any station a distance x downstream of the reference station 

 So = Slope at the reference station 

α = Coefficient 

Similarly, the bed sediment size is coarser in the upper reaches where the channel slopes are 

steep and the bed sediment size becomes finer with distance downstream. Generally, the size 

of the bed material reduces with distance according to the relationship 

D De x o

x

50 50 = − 

β (5.13) 

where: 

50x D = Median size of bed material at distance x downstream of reference station 

50o D = Median size of bed material at the reference station 



β = Coefficient 

The hydraulic geometry relations are applicable to continuous channel behavior. In some 

cases, this behavior (in this case the slope) may become discontinuous as the channel pattern 

changes from meandering to braided by the formation of cutoffs. Application of Equation 5.13 

is illustrated with field data in Section 5.9 (Problem 7). 

5.5 QUALITATIVE RESPONSE OF RIVER SYSTEMS

Many rivers have achieved a state of practical equilibrium throughout long reaches. For 

practical engineering purposes, these stable reaches can be also called "graded" streams by 

geologists and "poised" streams by engineers. However, this does not preclude significant 

changes over a short period of time or over a period of years. Conversely, many streams 

contain long reaches that are actively aggrading or degrading. These aggrading and 

degrading channels pose a definite hazard to any highway crossing or encroachment, as 

compared to poised streams. 

Regardless of the degree of channel stability, local human activities may produce major 

changes in river characteristics locally and throughout the entire reach. All too frequently the 

net result of a river improvement is a greater departure from equilibrium than that which 

originally prevailed. Good engineering design must invariably seek to enhance the natural 

tendency of the stream toward stable conditions. To do so, an understanding of the direction 

and magnitude of change in channel characteristics caused by human activity and natural 

processes is required. This understanding can be obtained by: (1) studying the river in a 

natural condition; (2) having knowledge of the sediment and water discharge; (3) being able to 
5.29 

predict the effects and magnitude of future human activities; and (4) applying a knowledge of 

geology, soils, hydrology, and hydraulics of alluvial rivers. 

To predict river response to channel modifications is a very complex task. A large number of 

variables are involved in the analysis. These variables are interrelated and can respond to 

changes in a river system in the continual evolution of river form. The channel geometry, bars, 

and forms of bed roughness all change with changing water and sediment discharges. 

Because such a prediction is necessary, methods have been developed to predict the 

response of channel systems to changes both qualitatively and quantitatively. 

5.5.1 General River Response to Change

Quantitative prediction of response can be made if all of the required data are known with 

sufficient accuracy. Usually, however, the data are not sufficient for quantitative estimates, and 

only qualitative estimates are possible. Examples of studies that have been undertaken by 

various investigators for qualitative estimates follow. Lane (1955) studied the changes in river 

morphology caused by modifications of water and sediment discharges. Similar but more 

comprehensive treatments of channel response to changing conditions in rivers have been 

presented by Leopold and Maddock (1953), Schumm (1971, 1977), Santos-Cayado (1972), 

Richards (1982), ASCE (1983), Thorne et al. (1997), Thorne (1998), and Knighton (1998). 

Research results support the following general statements: 

1. Depth of flow is directly proportional to water discharge and inversely proportional to bed 

material discharge, 

2. Width of channel is directly proportional to water discharge and to bed material discharge, 

3. Shape of channel expressed as width-depth ratio is directly related to bed material 

discharge, 

4. Meander wavelength is directly proportional to water discharge and to bed material 

discharge, 

5. Slope of stream channel is inversely proportional to water discharge and directly 

proportional to bed material discharge and grain size, and 

6. Sinuosity of stream channel is proportional to valley slope and inversely proportional to bed 

material discharge. 

It is important to remember that these statements pertain to natural rivers and not necessarily 

to artificial channels with bank materials that are not representative of sediment load. In any 

event, the relations will help to determine the response of water conveying channels to change. 

Bed material sediment transport (Qs) can be directly related to stream power (τoV) and 

inversely related to the fall diameter of bed material (D50). 

Q VWC

D s o f ~ τ

50

 (5.14) 

Here τo is the bed shear stress, V is the cross-sectional average velocity, W is the width of 

the stream and Cf is the volumetric concentration of fine sediments. Equation 5.14 can be 

written as: 
5.30 

Q y SWVC

D

QS C

D s of f ~ γ γ

50 50

= (5.15) 

If the specific weight, γ, is considered constant and if the concentration of wash load Cf can be 

incorporated in the fall diameter, D50, then the relationship can be expressed as: 

QS Q D ~ s 50 (5.16) 

which is the relationship originally proposed by Lane (1955), except Lane used the median 

diameter of the bed material as defined by sieving instead of the fall diameter. The fall 

diameter includes the effect of temperature on the transportability of the bed material, but use 

of physical diameter is sufficiently precise for most qualitative analyses. 

The proportionality represented by Equation 5.16 is very useful to qualitatively predict channel 

response to climatological changes, river modifications, or both. Two simple example problems 

are analyzed using Equation 5.16. 

 

In a first example, consider a tributary entering the main river at point C that is relatively small 

but carries a large sediment load (Figure 5.21). This increases the sediment discharge in the 

main stream from Qs to + Qs . It is seen from Equation 5.16 that, for a significant increase in the 

sediment discharge ( + Qs ) the channel gradient (S) below C must increase if Q and D50 remain 

constant. The line CA (indicating the original channel gradient) therefore changes with time to 

position C'A. Upstream of the confluence the slope will adjust over a long period of time to the 

original channel slope. The river bed will aggrade from C to C'. This change may induce a 

change in the channel morphology downstream of point C as the downstream reach may tend 

toward braiding. This possible change in planform geometry must be considered if any 

structure is to be built between points A and C'. 

Figure 5.21. Changes in channel slope in response to an increase in sediment load at Point C. 

In a second example, the construction of a dam on a river usually causes a decrease in 

sediment discharge downstream. Referring to Figure 5.22, and using Equation 5.16 and the 

earlier discussion, it can be concluded that for a decrease in bed material discharge from Qs to − Qs , the slope S decreases downstream of the dam (assuming discharge and sediment size 

remain unchanged). In Figure 5.22, the line CA, representing the original channel gradient, 
5.31 

changes to C'A, indicating a decrease in bed elevation and slope in the downstream channel 

with time. Note, however, if the dam fills with sediment so that the incoming sediment 

discharge passes through, that, except for local scour at the dam, the grade line C'A would 

return to the line CA. Also upstream of the dam the grade would return to the original 

equilibrium grade but would be offset vertically by the height of the dam. Thus small reservoirs 

(storage capacity small in relation to annual discharge) may cause degradation below the dam 

and then aggradation over a relatively short period of time. 

Figure 5.22. Changes in channel slope in response to a dam at Point C. 

The engineer is also interested in quantities in addition to qualitative trends. The geomorphic 

relation QS ~ QsD50 is only an initial step in analyzing long-term channel response problems. 

However, this initial step is useful because it warns of possible future difficulties related to 

channel modifications and flood protection works. The prediction of the magnitude of possible 

errors in flood protection design, because of changes in stage with time, requires the 

quantification of changes in stage. To quantify these changes, it is necessary to be able to 

quantify future changes in the variables that affect the stage. In this respect, knowledge of the 

future flow conditions is necessary. 

In many instances it is important to assess the effects of changes in water and sediment 

discharge on specific variables such as depth of flow, channel width, characteristics of bed 

materials and velocity. For this type of analysis, we can use Equation 5.15 as follows: 

Q D C SV y W S fo 50 / ~ (5.17) 

Using this form, the response of a river system to changes in specific variables is given in 

Table 5.4. A plus (+) sign signifies an increase in the value of the variable and a minus (-) sign 

signifies a decrease in the value of the variable. The letter B indicates an increase in the 

product SQ1/4 and a shift toward a braided condition and the letter M indicates a reduction in 

SQ1/4 and a shift toward the meandering condition (see Section 5.4.5). No attempt is made 

here to determine whether or not the channel actually braids or meanders. 
5.32 

Table 5.4. Change of Variables Induced by Changes in Sediment 

 Discharge, Size of Bed Sediment and Wash Load. 

Relationship Tendency to Braid or Meander 

QD C SVyW sf o

+ + + − +

50 / ~ B 

QD C SVyW sf o

− − − + −

50 / ~ M 

QD C SV y W sf o 50

+ + + − + / ~ B 

QD C SV y W sf o 50

− − ± ± ± / ~ M 

QD C SV y W sf o 50 / ~+ − ±± ± M 

QD C SV y W sf o 50 / ~− + +− + B 

QD C SVyW sf o

+ + + +− +

50 / ~ B 

QD C SVyW sf o

− − − ±± −

50 / ~ M 

QD C SVyW sf o

+ + − + +− +

50 / ~ B 

QD C SVyW sf o

− − + − ±− ±

50 / ~ M 

QD C SVyW sf o

+ + + + +− +

50 / ~ B 

Note: An increase in the value of the variable is denoted by a +; 

 and a decrease is denoted by a -. As an example, in the 

 first line, if the value of Qs increases, the slope, velocity, 

 and width will increase, the depth of flow will decrease and 

 the channel may tend toward a braided form. 

5.5.2 Prediction of Channel Response to Change

In Section 5.5.1, it was illustrated that the proportionality of Equation 5.16 could be used to 

predict changes in channel profiles caused by changes in water and sediment discharge. It is 

now possible to talk qualitatively about changes in channel profile, changes in river form and 

changes in river cross section both at-a-station and along the river channel using the other 

relations presented above. 

This can be best illustrated by application. Referring to Table 5.5, consider the effect of an 

increase in discharge indicated by a plus sign on line (a) opposite discharge. The increase in 

discharge may affect the river form, energy slope, stability of the channel, cross-sectional area 

and river regime. Equations 5.1 and 5.2 or Figure 5.18 (which illustrates SQ0.25) show that an 

increase in discharge could change the channel form in the direction of a braided form. 

Whether or not the channel form changes would depend on the river form prior to the increase 

in discharge. With the increase in discharge the stability of the channel would be reduced, 

which indicates an increase in velocity. On the other hand, this prediction could be affected by 

changes in form of bed roughness that dictate resistance to flow. 

The information presented in Chapter 4 can be used to determine the direction of change of 

hydraulic variables when sediment characteristics or discharges are varied. It is important to 

notice that Einstein, Colby, and Manning's equations apply to a cross section or reach and 

differ from some of the available geomorphic equations that have been derived by considering 
5.33 

a reach or total length of river. Einstein, Colby, and Manning's equations deal with depth of 

flow, width of flow and energy slope whereas most geomorphic equations deal with channel 

depth, channel width and channel slope. 

The interdependency of top width, depth of flow, energy slope, bed-material size, and 

kinematic viscosity on the water and sediment discharge allows the establishment of the 

relative influence of those variables on stage-discharge relationships. This information can be 

used to establish the direction of variation of hydraulic variables as a consequence of changes 

imposed on the bed-material and sediment discharge as shown in Table 5.5. 

Table 5.5. Qualitative Response of Alluvial Channels. 

Effect On 

Variable 

Change in 

Magnitude of 

Variable Regime 

of Flow 

River 

Form 

Resistance

to Flow 

Energy 

Slope 

Stability of 

Channel Area Stage 

Discharge (a) 

 (b) 

+ 

- 

+ 

- 

M→B 

B→M 

±

±

- 

+ 

- 

+ 

+ 

- 

+ 

- 

Bed Material (a) 

Size (b) 

+ 

- 

- 

+ 

M→B 

B→M 

+ 

- 

+ 

- 

±

±

+ 

- 

+ 

- 

Bed Material (a) 

Load (b) 

+ 

- 

+ 

- 

B→M 

M→B 

- 

+ 

- 

+ 

+ 

- 

- 

+ 

- 

+ 

Washload (a) 

 (b) 

+ 

- 

+ 

- 

 - 

+ 

- 

+ 

±

±

- 

+ 

- 

+ 

Viscosity (a) 

 (b) 

+ 

- 

+ 

- 

 - 

+ 

- 

+ 

±

±

- 

+ 

- 

+ 

Seepage Force (a) 

 (b) 

Outflow 

Inflow 

- 

+ 

B→M 

M→B 

+ 

- 

- 

+ 

+ 

- 

+ 

- 

+ 

- 

Vegetation (a) 

 (b) 

+ 

- 

- 

+ 

B→M 

M→B 

+ 

- 

- 

+ 

+ 

- 

+ 

- 

+ 

- 

Wind (a) 

 (b) 

Downstream 

Upstream 

+ 

- 

M→B 

B→M 

- 

+ 

+ 

- 

- 

- 

- 

+ 

- 

+ 

Chapter 4 indicates that the wash load increases the apparent viscosity of the water and 

sediment mixture. This makes the bed material behave as if it were smaller. In fact, the fall 

diameter of the bed material is made smaller by significant concentrations of wash load. With 

more wash load, the bed material is more susceptible to transport and any river carrying 

significant wash load will change from lower to upper regime at a lower Froude number than 

otherwise. Also, the viscosity is affected by changes in temperature. 

 Seepage forces resulting from seepage losses help stabilize the channel bed and banks. With 

seepage inflow, the reverse is true. Vegetation adds to bank stability and increases resistance 

to flow, reducing the velocity. Wind can retard flow, increasing roughness and depth, when 

blowing upstream. The reverse is true with the wind blowing downstream. Wind generated 

waves and their adverse influence on channel stability are the most significant effects of wind. 
5.34 

In terms of channel stability, it is important to recognize that three geomorphic responses or 

processes can result from changes in dominant channel flow and sediment conditions. These 

are channel widening, channel deepening, and changing planform (a change in sinuosity or 

meander pattern). All of these responses will cause some level of stream bank erosion. 

Channel widening is evidenced through an increase in channel width, with or without an 

increase in channel depth. An increase in flow or sediment discharge results in a tendency 

toward channel widening. When both sediment discharge and flow increase, however, the 

channel section can be expected to increase its depth as well as its width. When only 

sediment load increases, width increases but the depth may decrease. In this case the 

channel is said to be aggrading, implying that the channel has filled in because of an excess of 

sediments. 

Channel deepening is a process of channel degradation that increases the depth of the 

channel. Channel degradation can cause bank instability by producing a steeper bank angle. 

Whether or not instability actually occurs is a function of the properties of the bank materials 

and the original bank geometry. Channel deepening results from increased flow without an 

appreciable increase in sediment discharge. Increased flow rates can result from an overall 

increase in the volume of water moving through the channel or an increase in channel slope. 

Changing channel planform includes changes in channel pattern and position as viewed from 

above. Changes in planform are most often exhibited through the downstream migration of 

meandering bends and changes in the sinuosity of meander bends. Other examples include 

the shifting of channels and the cutting off of meander bends. Generally, these changes are 

manifested by an adjustment of channel slope to conform with changes in flow or sediment 

discharge. 

A reduction in sediment discharge or an increase in water discharge will result in a reduction of 

the channel slope. These slope reductions result from increased channel sinuosity and/or 

channel-bed degradation; both of which lead to a tendency toward increased bank erosion. 

Also, a reduction in sediment discharge will result in an increase in channel sinuosity, again, 

leading to increased bank erosion. 

It is also important to recognize that the three geomorphic processes just discussed (channel 

widening, channel deepening, and changing planform) are often interrelated and can occur 

simultaneously or in sequence. For example, adjustments in channel slope through 

degradation often are accompanied by increases in channel sinuosity and bank caving or 

channel widening. Also, the initiation of a given process at a particular site may initiate another 

process either upstream or downstream. For example, an aggrading channel reach can cause 

an increase in sinuosity in a downstream reach. 

5.5.3 River Pattern Thresholds and Response 

 

The work of Lane (1957) and Leopold and Wolman (1960) as summarized in Section 5.4.5 

indicates that there is a gradient or discharge threshold above which rivers tend to be 

braided (Figure 5.18). The experimental work reported by Schumm and Khan (1972) shows 

that for a given discharge, as valley-floor slope is progressively increased, a straight river 

becomes sinuous and then eventually braided at high values of stream power and sediment 

transport (Figure 5.3). Rivers that are situated close to the meandering-braided threshold 

should have a history which is characterized by transitions in morphology from braided to 

meandering and vice versa. 
5.35 

If the natural range of patterns along a river can be identified, then within that range it should 

be possible to identify the most appropriate channel pattern and sinuosity. If so, the 

engineer can work with the river to produce its most efficient or most stable channel. 

Obviously, a river can be forced into a straight configuration or it can be made more sinuous, 

but there is a limit to the changes that can be induced beyond which the channel cannot 

function without a radical morphologic adjustment as suggested by Figure 5.4. Identification 

of rivers that are near the pattern threshold would be useful, because a braided river near 

the threshold might be converted to a more stable, meandering, single-thalweg stream. On 

the other hand, a meandering stream near the threshold should be identified in order that 

steps could be taken to prevent braiding due, perhaps, to changes of land use. 

Perhaps the best qualitative guide to river stability is a comparison of the morphology of 

numerous reaches, and the determination of whether or not there has been a change in the 

position and morphology of the channel in the past. Another approach might be to 

determine the position of the river on the Leopold-Wolman (1957) or Lane (1957) gradientdischarge graphs (Figure 5.18). If a braided river plots among the meandering channels or 

vice versa, it is a likely candidate for change because it is incipiently unstable. 

An example of the way that this could be done is provided by the Chippewa River of 

Wisconsin, a major tributary to the Mississippi River (Schumm and Beathard 1976). The 

Chippewa River rises in northern Wisconsin and flows 320 km (200 mi) to the Mississippi 

River, entering it 120 km (75 mi) below St. Paul (Figure 5.23). It is the second largest river in 

Wisconsin, with a drainage basin area of 24 600 km2 (9,500 sq mi). 

From its confluence with the Mississippi to the town of Durand 26.5 km (16.5 mi) up the 

valley, the Chippewa is braided (Table 5.6). The main channel is characteristically broad 

and shallow, and it contains shifting sand bars. The bankfull width as measured from U.S. 

Geological Survey topographic maps is 333 m (1,092 ft). The sinuosity of this reach is very 

low, being only 1.06. However, in the 68 km (42 mi) reach from Durand to Eau Claire, the 

Chippewa River has a meandering configuration with a bankfull width of 194 m (636 ft) and a 

sinuosity of 1.49. The valley slope and channel gradient are different for each reach of the 

river. The braided section has a gentler valley slope than the meandering reach upstream, 

0.00035 as opposed to 0.00040, contrary to what is expected from Figure 5.18, but the 

situation is reversed for channel slope. The braided reach has a channel gradient of 

0.00033, whereas the meandering reach has a gradient of 0.00028. 

Table 5.6. Chippewa River Morphology. 

Location 

Channel 

Pattern 

Channel Width 

m (ft) Sinuosity 

Valley 

Slope 

Channel 

Slope 

Below Durand Braided 333 (1,092) 1.06 0.00035 0.00033 

Above Durand Meandering 194 (636) 1.49 0.0004 0.00028 

Buffalo Slough Meandering 212 (695) 1.28 0.00035 0.00027 
5.36 

Figure 5.23. Map of lower Chippewa River. The river is braided below Durand, Wisconsin. 

 Buffalo Slough is an old course of Chippewa River (Schumm and Beathard 

 1976). 

There is no evidence to suggest that the Chippewa River is either progressively eroding or 

aggrading its channel at present. In fact, the river below Durand has remained braided 

during historic time. It has maintained its channel position and its pattern, but a significant 

narrowing as the result of the attachment of islands and the filling of chute channels has 

occurred downstream of Durand, which resulted in a recent decrease in channel width of 

over 40 percent. The only other change in the lower river is a noticeable growth of the 

Chippewa delta into the Mississippi valley. The delta deposits show increased vegetational 

cover as well as extension into the Mississippi River valley. 

The relations described by Leopold and Wolman (1957) and Lane (1957), which are 

combined in Figure 5.18, provide a means of evaluating the relative stability of the modern 

channel patterns of the Chippewa River. The bankfull discharge was plotted against channel 

slope on Figure 5.24 (the Leopold and Wolman relationship) for both the braided and the 

meandering reaches of the Chippewa. The value used for the bankfull discharge is 1,503 

m3

/s (53,082 cfs), which is the flood discharge having a return period of 2.33 years. The 

braided reach plots higher than the meandering reach, but both are well within the 

meandering zone, as defined by Leopold and Wolman. This suggests that the braided reach 

is anomalous; that is, according to this relation the lower Chippewa would be expected to 

display a meandering pattern rather than a braided one. Even when the 25-year flood of 

2,787 m3

/s (98,416 cfs) is used, the braided reach still plots within the meandering region of 

Figure 5.24. 

When the Chippewa data are plotted on Lane's graph (Figure 5.25) the same relation exists. 

The Chippewa River falls in the intermediate region, but within the range of scatter about the 

regression line for meandering streams. Again the braided reach is seen to be anomalous 

because it should plot much closer to or above the braided stream regression line. The 

position of the braided reach as plotted on both figures indicates that this reach should be 

meandering. 
5.37 

Figure 5.24. Leopold and Wolman's (1957) relation between channel patterns, channel 

 gradient, and bankfull discharge. Letters B and M identify braided and 

 meandering reaches of Chippewa River. Letters K and W refer to Kowhai 

 and Wairau Rivers (Schumm and Beathard 1976). 

Figure 5.25. Lane's (1957) relation between channel patterns, channel gradient, and mean 

 discharge. The regression lines were fitted to data from streams that Lane 

 classified as highly meandering to braided. Letters B and M identify position of 

 the braided and meandering reaches of the Chippewa River (Schumm and 

 Beathard 1976). 
5.38 

This conclusion requires an explanation that can be based on the geomorphic history of 

Chippewa River. For example, there is a significant morphologic feature on the Chippewa 

River floodplain, Buffalo Slough, which occupies the southeastern edge of the floodplain 

(Figure 5.23). It is a sinuous remnant of the Chippewa River that was abandoned, and it is 

evidence of a major channel change in the Chippewa River valley (Table 5.6). 

Flow through Buffalo Slough has decreased during historic time, and indeed flow was 

completely eliminated in 1876, when the upstream end of Buffalo Slough was permanently 

blocked. The abandonment of the former Buffalo Slough channel by the Chippewa River is 

the result of an avulsion, but one that took many years to complete. The channel shifted 

from Buffalo Slough to a straighter, steeper course along the northwestern edge of the 

floodplain. This more efficient route gradually captured more and more of the total 

discharge. The new Chippewa channel produced a braided configuration due to a higher 

flow velocity and the resulting bank erosion. As the new channel grew, the old course 

deteriorated and eventually its discharge was so reduced that the Mississippi was able to 

effectively dam the old channel mouth with natural levees and plug its outlet. 

The sinuosity of Buffalo Slough is approximately 1.28. Although sinuosity is usually defined 

as the ratio of stream length to valley length, it is also the ratio of valley slope to channel 

slope. A sinuosity of 1.28 is, therefore, the ratio of the present valley slope (0.00035) to a 

channel slope of about 0.0027. This channel slope value is very similar to the channel slope 

of the meandering reach upstream of Durand (0.00028) therefore, the meandering pattern of 

the Buffalo Slough channel was appropriate. Although delta construction at the Mississippi 

River confluence was responsible for the lower valley slope of the lower Chippewa River 

valley, the Buffalo Slough channel had a gradient that was not appreciably different from that 

in the upstream reach. Channel sinuosity decreased from 1.49 to 1.28 in a downstream 

direction, thereby maintaining a channel gradient of about 0.00027. 

This sinuous channel could not have transported the large amounts of sediment that the 

present braided channel carries to the Mississippi River, or it too would have followed a 

straight braided course. Therefore, the present sediment load carried by the Chippewa 

River is greater than that conveyed by the Buffalo Slough channel, but this is due almost 

entirely to the formation of the new straight, steep, braided channel, which is 121 m (397 ft) 

wider than the old sinuous Buffalo Slough channel. It appears that the lower Chippewa has 

not been able to adjust as yet to its new position and steeper gradient, and the resulting bed 

and bank erosion has supplied large amounts of sediment to the Mississippi. The normal 

configuration of the lower Chippewa is sinuous, and if it could be induced to assume such a 

pattern, the high sediment delivery from the Chippewa might be controlled. An appropriate 

means of channel stabilization and sediment load reduction in this case is the development 

of a sinuous channel. 

Since the above suggestions were made (Schumm and Beathard 1976), more detailed 

studies of the Chippewa River basin indicate that upstream sediment production must be 

controlled, especially where the upper Chippewa River is cutting into the Pleistocene 

outwash terraces. If the contribution of sediment from these sources were reduced, the lower 

Chippewa could resume its sinuous course. 

An indication that the pattern conversion of the Chippewa could be successful if the 

upstream sediment sources were controlled is provided by the Rangitata River of New 

Zealand. The Rangitata River is the southern most of the major rivers which traverse the 
5.39 

Canterbury Plains of South Island. It leaves the mountains through a bedrock gorge. Above 

the gorge, the valley of the Rangitata is braided, and it appears that the Rangitata should be 

a braided stream below the gorge, as are all the other rivers which cross the Canterbury 

Plain. However, below the gorge, the Rangitata is meandering. A few miles farther 

downstream, the river cuts into high Pleistocene outwash terraces, and it abruptly converts 

from a meandering to a braided stream. The braided pattern persists to the sea. If the 

Rangitata could be isolated from the gravel terraces, it probably could be converted to a 

single-thalweg sinuous channel, because the Rangitata is obviously a river near the pattern 

threshold. 

There are other New Zealand rivers that are near the pattern threshold. and therefore, they 

are susceptible to pattern change. In fact, New Zealand engineers are attempting to 

accomplish this pattern change in order to produce 'single-thread' channels which will cause 

less flood damage, have greater stability at bridge crossings, and be less likely to acquire 

large sediment loads from their banks and terraces. For example, the engineers have had 

success in converting the Wairau River, a major braided stream, from its uncontrolled 

braided mode to that of a slightly sinuous, single-thalweg, relatively more stable channel. 

The increase in sinuosity is only from 1.0 to 1.05, and this was accomplished by the 

construction of curved training banks. On Figure 5.24 the Wairau River plots close to the 

threshold line, and with the reduction of sediment load produced by bank stabilization, it 

appears that the pattern threshold can be crossed successfully. 

Farther to the south near Kaikoura, the Kowhai River is being modified in the same manner 

as was the Wairau. Whereas much of the sediment load in the Wairau River is derived from 

bank and terrace erosion which can be controlled, high sediment loads are delivered to the 

Kowhai River directly from steep and unstable mountain slopes. On Figure 5.24 the Kowhai 

River plots well above the threshold line, and without a major reduction in upstream 

sediment, it may be difficult to maintain a single-thalweg channel at this location. 

The variability of the Rangitata River pattern indicates that braided to single-thalweg 

conversions should be possible for the Chippewa and Wairau Rivers. However, not all 

braided rivers can be so readily modified, as this depends on their position with regard to the 

line defining the pattern thresholds on Figures 5.24 and 5.25. 

Perhaps the simplest way to determine the most likely pattern for a river is to determine its 

pattern from the earliest maps and aerial photographs. If the river was braiding in the past, it 

seems unlikely that an attempt to convert to a sinuous channel will be successful. However, 

if the historic river was meandering and it is now braided, although there have been no major 

erosional or hydrologic changes in the drainage basin, then a conversion back to a 

meandering pattern is appropriate. Another approach is to determine the characteristics of 

other nearby rivers. If the subject river is very different it may be converted to the local 

character if neither the sediment load or hydrologic character are significantly different. 

5.6 MODELING OF RIVER SYSTEMS

The necessity for quantitative prediction of river channel response is increasing. The accuracy 

of such a prediction depends on the quality of the data. There are generally two ways of 

predicting response. One is the mathematical model and the other is the physical model. 

Mathematical models utilize a number of mathematical equations governing the motion of 

water and sediments in a channel. Regardless of the potential of mathematical models, to date 

they have been best used to study channel response using 1-dimensional, or at most 2-

dimensional, approximations. For complex three-dimensional channel processes, it is very 

difficult to accurately formulate mathematically what happens in a river. Studies of channel 
5.40 

response to development for complex situations are usually made using a physical model. The 

physical model is designed to achieve similar behavior as that of the prototype. The 

relationship of the governing physical processes and parameters must then be the same for 

the model as for the prototype. 

5.6.1 Physical Modeling

Physical models are used to test the performance of a design or to study the details of a 

phenomenon. The performance tests of proposed structures can be made at moderate costs 

and small risks on small-scale (physical) models. Similarly, the interaction of a structure and 

the river environment can be studied in detail. In this regard, HEC-23 (Lagasse et al. 2001) 

discusses the use of physical modeling in bridge scour and stream instability countermeasure 

design. 

The natural phenomena are governed by appropriate sets of governing equations. If these 

equations can be integrated, the prediction of a given phenomena in time and space domains 

can be made mathematically. In many cases related to river engineering, all the governing 

equations are not known. Also, the known equations cannot be directly treated mathematically 

for the geometries involved. In such cases, physical models are used to physically represent 

solutions to the governing equations. 

The similitude required between a prototype and a model implies two conditions: 

1. To each point, time and process in the prototype, a uniquely coordinated point, time and 

process exists in the model; and 

2. The ratios of corresponding physical magnitudes between prototype and model are 

constant for each type of physical quantity. 

Rigid Boundary Models. To satisfy the preceding conditions in clear water, geometric, 

kinematic, and dynamic similarities must exist between the prototype and the model. 

Geometric similarity refers to the similarity of form between the prototype and its model. 

Kinematic similarity refers to similarity of motion, while dynamic similarity is a scaling of masses 

and forces. For kinematic similarity, patterns or paths of motion between the model and the 

prototype should be geometrically similar. If similarity of flow is maintained between the model 

and prototype, mathematical equations of motion will be identical for the two. Considering the 

equations of motion, the dimensionless ratios of V gy / (Froude number) and Vy/ν (Reynolds 

number) are both significant parameters in models of rigid boundary clear water open channel 

flow. The Froude number represents the ratio of inertial to gravity forces in the system being 

modeled, while the Reynolds number represents the ratio of inertial to friction (viscosity) forces. 

It is seldom possible to achieve kinematic, dynamic and geometric similarity all at the same 

time in a model. For instance, in open channel flow, gravitational forces predominate, and 

hence, the effects of the Froude number are more important than those of the Reynolds 

number. Therefore, the Froude criterion is used to determine the geometric scales, but only 

with the knowledge that some scale effects, that is, departure from strict similarity, exists in the 

model. 

Ratios (or scales) of velocity, time, force and other characteristics of flow for two systems are 

determined by equating the appropriate dimensionless number which applies to a dominant 

force. If the two systems are denoted by the subscript m for model and p for prototype, then 
5.41 

the ratio of corresponding quantities in the two systems can be defined. The subscript r is 

used to designate the ratio of the model quantity to the prototype quantity. For example, the 

length ratio is given by: 

Lxx yy zz r mp mp mp === /// (5.18) 

for the coordinate directions, x, y, and z. Equation 5.18 assumes a condition of exact 

geometric similarity in all coordinate directions. 

Frequently, open channel models are distorted. A model is said to be distorted if there are 

variables that have the same dimension but are modeled by different scale ratios. Thus, 

geometrically distorted models can have different scales in horizontal (x, y) and vertical (z) 

directions, and two equations are necessary to define the length ratios in this case. 

Lxx yy r mp mp = = / / (5.19) 

and 

zzz r mp = / (5.20) 

If perfect similitude is to be obtained the relationships that must exist between the properties of 

the fluids used in the model and in the prototype are given in Table 5.7 for the Froude (gravity), 

Reynolds (viscosity), and Weber (surface tension) criteria. The use of this table is presented in 

Section 5.9, Problems 5 and 6. 

In free surface flow, the length ratio is often selected arbitrarily, but with certain limitations kept 

in mind. The Froude number is used as a scaling criterion because gravity has a predominant 

effect. However, if a small length ratio is used (very shallow water depths) then surface tension 

forces, which are included in the Weber number (/ / ) V L σ ρ , may become important and 

complicate the interpretations of results of the model. It is desirable that the length scale be 

made as large as possible so that the Reynolds number is sufficiently large that friction 

becomes a function of the boundary roughness (and essentially independent of the Reynolds 

number). A large length scale also ensures that the flow is as turbulent in the model as it is in 

the prototype. 

The boundary roughness is characterized by Manning's roughness coefficient, n, in free 

surface flow. Analysis of Manning's equation and substitution of the appropriate length ratios, 

based upon the Froude criterion, results in an expression for the ratio of the roughness which is 

given by 

n L r r = 1 6/

 (5.21) 

It is not always possible to achieve boundary roughness in a model and prototype that 

corresponds to that required by Equation 5.21 and additional measures such as adjustment 

of the slope, may be necessary to offset disproportionately high resistance in the model. 
5.42 

Table 5.7. Scale Ratios for Similitude. 

Characteristic Dimension Re Fr We 

Length L L L L 

Area L2 L2 L2 L2 

Volume L3 L3 L3 L3 

Time T ρL2/μ (Lρ/γ)

1/2 (L3

ρ/σ)

1/2 

Velocity L/T μ/Lρ (Lγ/ρ)

1/2 (σ/Lρ)1/2 

Acceleration L/T2 

μ

2

/ρ

2

L3

γ/ρ σ/L2

ρ

Discharge L3/T Lμ/ρ L5/2 (γ/ρ)

1/2 L3/2 (σ/ρ)1/2 

Mass M L3

ρ L3ρ L3ρ

Force ML/T2

μ

2

/ρ L3

γ Lσ

Density M/L3

ρ ρ ρ

Specific Weight M/L2T2

μ

2

/L2 

ρ γ σ/L2 

Pressure ML/T μ

2

/L2 

ρ Lγ σ/L 

Impulse & Momentum ML2/T2 L2

μ L7/2(ργ)

1/2 L5/2(ρσ)1/2

Energy and Work ML2/T2 Lμ

2

/ρ L4 

γ L2σ

Power ML2/T3

μ

3

/Lρ

2

 L7/2

γ

3/2

ρ

-1/2

σ

3/2 (Lρ)1/2 

Mobile Bed Models. In modeling highway crossings and encroachments in the river 

environment, two- and three-dimensional mobile bed models are often used. These models 

have the bed and sides molded of materials that can be moved by the model flow. Similitude 

in mobile bed models implies that the model reproduces the fluvial processes such as bed 

scour, bed deposition, lateral channel migration, and varying boundary roughness. It is 

generally not possible to faithfully simulate all of these processes simultaneously on scale 

models. Distortions of various parameters are often made in such models. 

Two approaches are available to design mobile bed models. One approach is the analytical 

derivation of distortions explained by Einstein and Chien (1956) and the other is based on 

hydraulic geometry relationships given by Lacey, Blench, and others (Mahmood and Shen 

1971). In both of these approaches, a first approximation of the model scales and distortions 

can be obtained by numerical computations. The model is built to these scales and then 

verified for past information obtained from the prototype. In general, the model scales need 

adjusting during the verification stage. 

Model verification consists of the reproduction of observed prototype behavior under given 

conditions on the model. This is specifically directed to one or more alluvial processes of 

interest. For example, a model may be verified for bed-level changes over a certain reach of 

the river. The predictive use of the model should be restricted to the aspects for which the 

model has been verified. The use is based on the premise that if the model has successfully 
5.43 

reproduced the phenomenon of interest over a given hydrograph as observed on the prototype, 

it will also reproduce the future response of the river over a similar range of conditions. 

The mobile bed models are more difficult to design and their theory is significantly more 

complicated as compared to clear water rigid bed models. However, many successful 

examples of their use are available. In general, important river training and control works are 

invariably studied on physical models. The interpretation of results from a mobile bed model 

requires a basic understanding of the fluvial processes and some experience with such 

models. Even in the many cases where it is only possible to obtain qualitative information from 

mobile bed models, this information is of great help in comparing the performance of different 

designs. 

Adoption of a particular method for estimating river response depends on quality and 

availability of data as well as the engineer's experience. More information on physical modeling 

can be found in HEC-23 and reports by Gessler (1971), Yalin (1971), Shen (1979), and 

Richardson et al. (1987, 1989). Additional applications of physical model studies of alluvial 

channel flow at highway crossings can be found in: "Hydraulic, Erosion, and Channel Stability 

Analysis of the Schoharie Creek Bridge Failure, New York," (Richardson et al. 1987); "Flume 

Modeling Experimental Plan for the Replacement of the Herbert C. Bonner Bridge," (Parsons 

Brinkerhoff Quade and Douglas, Inc. 1996); "Laboratory Report of the Acosta Bridge Scour 

Study," (Stein, S.M. et al. 1990); "FHWA Hydraulics Laboratory and Partners Perform Scour 

Evaluation for Woodrow Wilson Bridge" (Jones, J.S. 1999); and "Local Pier Scour Model Tests 

for Jensen Beach Bridge," (Sheppard, D.M. 1999). 

5.6.2 Computer Modeling

The design engineer's interest in alluvial river response is generally focused on anticipating 

how the river bed and water-surface elevations will change if an existing stable or equilibrium 

situation is perturbed. This perturbation may be the occurrence of an unusually large annual 

flood that temporarily scours the bed and banks to accommodate the higher flow before 

returning to normal conditions. The perturbation may also be a permanent change in river 

discharge and sediment supply caused by upstream regulation of flows, or a change in channel 

geometry resulting from bank stabilization or channelization. The first type of perturbation can 

often be simulated using a physical scale model. Although problems arise with interpretation of 

the results, physical models in the hands of experience modelers can yield valuable information 

on local scour and deposition around structures. However, the sheer expense and space 

requirements of physical scale models generally disqualify them for simulation of long-term, 

large-scale river bed response to the second type of perturbation. This is where numerical, 

computer-based models, which can simulate both short- and long-term response, find their 

natural area of application. Another area where prediction of long-term response is desired is 

river stabilization and river restoration design. 

Numerical models of alluvial river response are the natural outgrowth of rigid-boundary, 

unsteady flood propagation models that have proven so useful in engineering design. These 

unsteady flow models have succeeded because they are based on mathematical descriptions 

that incorporate all the important physical processes involved and use reliable, carefully 

implemented numerical methods to obtain approximate solutions to the appropriate 

partial-differential equations. However, alluvial river-response models have not enjoyed the 

success of their rigid-boundary cousins, precisely because of the weaknesses in our 
5.44 

understanding and mathematical formulation of the relevant physical processes. 

Notwithstanding this fundamental difficulty, design engineers have an immediate need for 

reliable numerical simulations, and hydraulic research engineers have targeted alluvial river 

hydraulics as a prime area for continuing fundamental and applied research. Out of this 

fortunate confluence of interest have arisen a variety of simulation techniques and software 

systems, as well as many apparently successful simulations of prototype situations. 

The most basic one-dimensional description of water and sediment flow in an alluvial river 

consists of four relations: conservation of water; conservation of water momentum; 

conservation of sediment; and sediment-transport relationships. These equations form a 

nonlinear partial-differential system that in general cannot be solved analytically. 

When the water wave propagation effects are of secondary importance for sediment-transport 

phenomena, the system of equations can be simplified by assuming that the water flow 

remains quasi-steady during a certain interval of time. 

Most published 1-dimensional software systems for the solution of the water- and 

sediment-flow equations use one form or another of the finite-difference method, in which time 

and space derivatives are approximated by differences of nodal values of grid functions that 

replace the continuous functions, leading to a system of algebraic equations. Some authors 

have used the finite-element method, but in one dimension there does not appear to be any 

strong reason for doing so. In any case, the quality and reliability of numerical models for bed 

evolution are determined primarily by the sediment-transport formulation and mechanisms 

adopted for sorting, armoring, and so forth. The particular numerical method used, as long as 

it is consistent with the partial-differential equations and is stable, has only a secondary effect 

on simulation quality. 

Whether the full unsteady set of equations or the quasi-steady set of equations is solved 

numerically, two basic approaches are possible: coupled or uncoupled. In the coupled case, a 

simultaneous solution of both water and sediment equations is sought. This is evidently the 

physically proper way to proceed, because the water-flow and sediment-transport processes 

occur simultaneously. However, the simultaneous solution may involve certain computational 

complications, especially when the sediment-transport flow resistance equation involves not 

just an analytic mathematical expression, but a whole series of procedures and computations 

to simulate alluvial channel processes such as armoring, sorting, and bed forms. 

The uncoupled procedure has arisen essentially to circumvent the computational difficulties of 

the coupled approach. The uncoupling of the liquid and solid transport occurs during a short 

computational time step. First the water-flow equations are solved to yield new values of depth 

and velocity throughout the reach of interest, assuming that neither the bed elevation nor the 

bed-sediment characteristics change during the time step. Then the depths and velocities are 

taken as constant, known inputs to the sediment continuity and transport equations; these 

equations then become relatively easy to solve numerically, yielding the new bed elevations. 

When the overall model includes bed-sediment sorting or armoring, these processes are 

simulated in a third uncoupled computational phase using new depths, velocities, and bed 

elevations as known inputs. Although it is difficult to quantify the error associated with this 

artificial uncoupling of simultaneous, mutually dependent processes, it is intuitively obvious that 

the uncoupling is justified only if bed elevations and bed-material characteristics change very 

little during one time step. Experience in the use of uncoupled models, with both the unsteady 
5.45 

and quasi-steady water-flow equations, has shown that the uncoupling is not a serious obstacle 

to successful simulation. 

Another distinguishing feature of numerical bed-evolution models is the representation of 

sediment sorting and bed-surface armoring. Alluvial sediments are rarely of uniform grain size. 

A broad range of sizes are represented, from gravels and coarse sands down to fine silt and 

clay in varying proportions. Finer particles are preferentially entrained into the flow as erosion 

occurs, so that the material remaining on the bed contains a progressively higher proportion of 

coarser material. This so-called sorting process tends to increase the mean bed-sediment size 

as degradation occurs, thus affecting the sediment-transport rate, river regime (existence of 

ripples and dunes), and flow resistance through both particle roughness and bed-form effects. 

If the original bed material contains a high enough proportion of large, non-moveable materials 

(coarse gravel, cobbles, and small boulders), an interlocking armor layer may form on the 

surface, arresting further degradation. These processes are qualitatively reversed during 

deposition, but become even more difficult to quantify. Some models attempt to simulate 

sorting effects on bed evolution; others ignore them completely. Thus another important 

distinguishing feature of computer-based models is the degree to which they incorporate 

sorting and armoring effects. 

Numerical modeling of alluvial river flows has become very popular in the 1990s because of the 

advancement of computer technology. However, the number of computer-based, alluvial 

riverbed prediction models that are readily available for application to prototype cases seems to 

be quite small. Many of the available models have been developed for specific rivers under 

particular flow and alluvial river bed conditions, and many of them are, to some extent, well 

tuned or calibrated only for those particular rivers. 

The following assessment of selected models is made for two different groups: short-term 

models and long-term models. The short-term models are best suited to compute changes in 

alluvial river bed level during a relatively short time period. Long-term models employ simpler 

implementations of steady-state flow equations, and thus are suited for long-term prediction of 

river bed level for multiple-flood events over multiple years. However, it should be recognized 

that the short-term models can also be applied for long-term prediction if variable time steps 

are employed. In that case, a shorter time step is used for highly unsteady flows and a longer 

time step is used otherwise. 

BRI-STARS MODEL (The Bridge Stream Tube Model for Alluvial River Simulation). The 

BRI-STAR (Molinas 2000) model developed for the Federal Highway Administration is a 

semi-two-dimensional model capable of computing alluvial scour/deposition through subcritical, 

supercritical, and a combination of both flow conditions involving hydraulic jumps. Both energy 

and momentum equations are used so the water surface profile computation can be carried 

out through combinations of subcritical and supercritical flows without interruption. The stream 

tube concept is used in a semi-two-dimensional way, which allows the lateral and longitudinal 

variation of hydraulic conditions as well as sediment activity at various cross sections along the 

study reach. The sediment continuity equation and sediment transport capacity equations are 

used for sediment routing computations to simulate the general scour in the river bed elevation. 

The sediment routing is performed for each size fraction to account for the bed composition 

changes and the bed armoring processes. The minimum rate of energy dissipation theory is 

used for decisions as to whether channel adjustments taking place at a given cross section. 

BRI-STARS is capable of simulating channel widening/narrowing phenomenon. 
5.46 

The channel widening/narrowing is accomplished by coupling a stream tube computer model 

with a decision-making algorithm using rate of energy dissipation or total stream power 

minimization. The first component, the fixed-width streamtube computer model, simulates the 

scouring/deposition process taking place in the vertical direction across the channel. The user 

can select from one of seven sediment transport functions including one general function 

including velocity depth, discharge, energy slope, and grain size. The second component, the 

total stream power minimization algorithm, determines what takes place in the lateral or vertical 

direction. It is this component that allows the lateral changes in channel geometries. Finally, 

the bridge component allows the computation of the hydraulic flow variables and the resulting 

scour due to highway encroachments. 

HEC-6. The HEC-6 program (Scour and Deposition in Rivers and Reservoirs) was developed 

at the Hydrologic Engineering Center of the U.S. Army Corps of Engineers (HEC 1993). The 

quasi-steady backwater equation is used to compute water-flow conditions uncoupled from the 

sediment-continuity equation, and is intended for long-term river response simulation. 

The sediment-continuity equation is solved using an explicit finite-difference scheme, with 

sediment-transport capacities determined from water-flow conditions previously determined in 

the uncoupled backwater computation. The entire movable bed portion of the channel is 

assumed to aggrade or degrade uniformly. Sediments are routed by individual size fraction, 

which makes possible a detailed accounting of hydraulic sorting and development of an 

armored layer. The user can select from thirteen sediment transport functions including one 

general function including depth and slope. Bank lines are assumed to be stable and fixed in 

the HEC-6 computation. 

HEC-6 is strictly a one-dimensional model with no provision for simulating the development of 

meanders or specifying a lateral distribution of sediment transport rate across the section. The 

model is not suitable for rapidly changing flow conditions but can be applied to predict reservoir 

sedimentation, degradation of the streambed downstream from a dam, and long-term trends of 

scour or deposition in a stream channel, including the effects of dredging. It is anticipated that 

future versions of HEC-RAS will incorporate sediment transport and replace HEC-6. 

FLUVIAL-12. This uncoupled model was developed at San Diego State University by Chang 

(1998) to simulate one dimensional, unsteady, gradually varied, water and sediment flows for 

channels. The model also includes the capacity to simulate bank erosion and is described as 

an erodible-boundary model as opposed to an erodible bed model by Chang. FLUVIAL-12 first 

solves the unsteady, flow-continuity and flow-momentum equations in one time step by 

neglecting storage effects due to unsteady flow. The model uses an implicit, 

central-difference, numerical scheme in solving for the two unknown variables of water 

discharge and cross-sectional area. The flow information is then used to compute the bed 

sediment discharge at each section using one of six user selected sediment transport 

functions. 

Next, the net change in cross-sectional area is obtained by solving the sediment-continuity 

equation using a backward-difference scheme for space and a forward-difference scheme for 

time. The computed cross-sectional area change is then adjusted for the effects of channel 

migration. Width adjustments are made in such a manner that the spatial variation in power 

expenditure per unit channel length is reduced along the reach by a trial and error technique. 

Further adjustment of cross-sectional area is made to reduce the spatial variation in power 

expenditure along the channel. The effect of lateral channel migration is determined by solving 

the sediment-continuity equation in the transverse direction, which incorporates the effect of 

radius of curvature of the river bend into the transverse component of the sediment transport 

rate. 
5.47 

Chang (1998) states that the actual sediment rate is obtained by considering sediment 

material of all size fractions already in the flow as well as the exchange of sediment load with 

the bed using the method by Borah et al. (1982). If the stream carries a load in excess of its 

capacity, it will deposit the excess material on the bed. In the case of erosion, any size 

fraction available for entrainment at the bed surface will be removed by the flow and added 

to the sediment already in transport. During sediment removal, the exchange between the 

flow and the bed is assumed to take place in the active layer at the surface. Thickness of 

the active layer is based upon the relation defined by Borah, et al. This thickness is a 

function of the material size and composition, but also reflects the flow condition. During 

degradation, several of these layers may be scoured away, resulting in the coarsening of the 

bed material and formation of an armor coat. However, new active layers may be deposited 

on the bed in the process of aggradation. Materials eroded from the channel banks, 

excluding that portion in the wash load size range, are included in the accounting. Bed 

armoring develops if bed shear stress is too low to transport any available size. 

Two-Dimensional Sediment Transport Models are also available. Both finite-element and 

finite-difference numerical schemes have been applied. Examples of finite-element models 

are the SED2D (Letter et al. 1998) and Flo2DH (Froehlich 1996) models. SED2D is an 

uncoupled sediment transport model that generally uses RMA-2V (USACE 1997) hydraulic 

simulation results to compute sediment transport, scour and fill, then updates the model 

geometry and cycles back to RMA-2V to update the flow field. Other sources of hydraulic 

computations can be used as input to SED2D. Also under development are Flo2DH Version 

3 and Flo1D (Arneson 2001). FESWMS Version 3 will incorporate sediment transport 

computations. These models are best suited for short-term sediment transport simulations 

although SED2D has features that allow for long-term simulations. 

5.6.3 Data Needs 

Common to all alluvial river-flow models are requirements for the following input information: (1) 

accurate initial conditions, including a cross-sectional profile and bed-material size distribution 

at each computational cross section; (2) accurate boundary conditions such as water and 

sediment inflows along the boundaries, quantitative expressions of bed-load and 

suspended-load discharges, size distributions of boundary-sediment input, and stage 

hydrographs at the upstream and downstream boundaries; and (3) bed-roughness 

characteristics at each computational point. It is clear that a computer simulation would be 

meaningless without the first and second requirements, and the lack of the third requirement 

would yield an erroneous estimation of flow characteristics, resulting in erroneous feedback of 

flow information to the riverbed. Included in the first set of data are depths of alluvium and 

controls such as structures and bedrock outcrops. 

The exclusion of even one of these three requirements may lead to serious errors in computer 

simulations. However, one can hardly be provided with a complete set of input data in any 

prototype numerical application. Therefore, a great number of assumptions often have to be 

made to fill the gap in the input data. Even if adequate data are provided for a study river, 

there still remains a need to calibrate and verify the model by means of field data. In most 

natural rivers, only extremely limited field data are available for high flood stages at which major 

riverbed changes occur, and, consequently, adequate calibration or verification of the models 

normally cannot be obtained. In this sense, the capability of the alluvial river-flow models can 

best be assessed according to how accurately they can predict riverbed changes with limited 

sources of input data. A numerical modeler should be aware of which input information is most 

important to the final result of predicting riverbed changes. The success of the numeral model 

depends on the capabilities of the model, the quality of the input data and the abilities of the 

modeler. 
5.48 

5.7 HIGHWAY PROBLEMS RELATED TO GRADATION CHANGES

Gradation problems at highway crossings include changes in the vertical dimension such as 

aggradation and degradation; however, lateral erosion problems often occur as a consequence 

of these changes. The highway problem most associated with aggradation is reduction of flow 

area, which increases backwater effects upstream of bridges and culverts. Problems 

associated with degradation are undermining of footings, pile bents, abutments, cutoff walls, 

and other flow-control or crossing structures. Degradation has also been found to undermine 

bank protection resulting in the instability of channel banks and increasing debris problems. A 

common problem associated with lateral erosion is bank slumping, which undermines 

abutments and piers located near the bank line. Another very common problem arises when 

meandering streams, migrate laterally and encroach upon roadways. 

Causes of gradation changes that have an impact on highway crossings can be grouped into 

two basic categories: (1) the result of human activities; and (2) natural causes or factors. An 

analysis of case histories indicates that very few gradation changes were due to natural factors. 

 Some gradation changes should perhaps be classified as being caused by a combination of 

both natural and human-induced factors. However, their number is so small that a separate 

category is not warranted. Because human activities dominate the causes for gradation 

problems, they will be discussed first. 

5.7.1 Changes Due to Human Activities

Human activities are literally changing the face of the Earth and generate accelerated erosion 

from watersheds. Some activities have had far-reaching consequences on streams and have 

caused, or contributed to aggradation and degradation problems at bridges. Construction of a 

bridge and approach embankments may also have consequences, but they are unlikely to be 

far-reaching. Human activities were found to be the major cause of streambed elevation 

changes. Because accelerated erosion is associated with human activities, it is often possible 

to anticipate many impacts on bank stability and provide adequate bank protection in advance. 

From an analysis of case histories, human activities resulting in gradation problems can be 

grouped into the following categories: (1) channel alterations; (2) land use changes; (3) 

streambed mining/excavation; and (4) dams and reservoirs. 

Channel Alterations. Straightening, dredging, clearing and snagging, artificial constrictions, and 

other alterations of natural channels are the major causes of streambed elevation changes. 

Channel straightening is the dominant activity. Examples of channel response to straightening 

are presented in Keefer et al. (1980). 

Many of the straightened channels have degraded, and degradation is usually accompanied by 

widening of the channel, unstable banks and serious debris problems. The degradation is 

attributed to an increase in channel slope that results from shortening of channel length. The 

increase in channel slope increases the velocity and the shear stress on the bed. As a result, 

the channel bed degrades until the bed becomes armored or the channel widens and begins to 

meander to reduce the channel slope back to an equilibrium, or stable condition. There is some 

evidence that degradation, if it is to occur as a consequence of channel alteration, will be most 

rapid during a period shortly following the alteration and will thereafter occur at a decreasing 

rate. 
5.49 

Land Use Changes. Urbanization, agriculture, strip mining, and unregulated logging are 

other activities that cause gradation problems. Natural vegetation is extremely important in 

maintaining channel stability. The lateral stability of most streams in the United States, 

particularly in regions where agriculture or lumbering is practiced, has very probably been 

affected by the clearing of natural vegetation. Because this clearing has occurred more or less 

gradually over the past hundred years, the magnitude of the effect at a particular crossing site 

is sometimes difficult to assess. 

The response to deforestation and agricultural activities is generally toward increased peak 

flows and increased sediment yield. Channel widening and reduced sinuosity are common. 

Grazing along the stream banks may have significant effects on bank stability. 

Urbanization normally causes significant increase in the magnitude of runoff events while 

reducing their duration. Urban areas are also low sediment producers because of the large 

percentage of land covered by impervious surfaces. The combination of increased peak runoff 

rates and reduced sediment loads results in channel degradation, channel widening, and a 

reduction in channel sinuosity. 

Improper construction activities, on the other hand, are known to increase both discharge and 

sediment load. The removal of the vegetative cover accelerates the erosion process. The 

response of the system to the increased discharge is to increase channel width and reduce the 

radius of curvature. In response to increased sediment load, the stream will increase its 

tendency for bank erosion. 

Mining in an upland area may cause aggradation of channels, which are then subject to 

degradation after the mining ceases. 

Streambed Mining/Excavation. If sand or gravel is removed from an alluvial channel in 

quantities that represent a substantial percentage of the annual bedload in transport, the 

channel will probably degrade. In addition, removal of gravel from pits or trenches in or along 

the stream may result in a change in flow alignment at a bridge. 

Downstream mining can also produce headcutting through a bridge waterway, undermining the 

structure. Mining operations upstream of the bridge can also produce degradation at the 

bridge site and endanger the structure. Equation 5.28 provides a subjective tool for analysis of 

gradation changes. 

Highway engineers should, as a minimum, consider up- and downstream factors that might 

cause gradation problems during scheduled bi-annual inspections of bridges. 

Dams And Reservoirs. The effects of dams and reservoirs on a stream are complex. The 

consequences include clear-water releases; high sustained, regulated flows; backwater; low 

sustained, regulated flows; dam breach or removal; and high, controlled irrigation canal 

releases. 

Downstream from a reservoir, channel degradation is to be expected because of removal of 

sediment. This effect has been documented for many streams (see for example, Williams and 

Wolman 1984 and Lagasse 1994). The total amount of degradation is difficult to predict; if a 

sand-bed channel becomes armored with gravel, the amount may be small. On gravel-bed 

streams, aggradation may occur downstream from the dam because the flow releases are 
5.50 

insufficient to transport gravel brought in by tributary streams. Channel avulsions, which can 

present a serious threat to many engineering structures, are associated with most aggrading 

situations. Rapid lowering of river stage may result in severe bank slumping from pore-water 

pressures in the banks. However, the more general effect of reservoirs is probably to reduce 

hydraulic problems at highway crossing bridges, both by reduction of flood peaks and a 

reduction of lateral erosion rates. 

Interbasin transfers of flow and diversions result in periods of channel instability and bank 

erosion until the new channel regime is established. 

5.7.2 Natural Causes

Although problems resulting from natural causes are not as frequent as those resulting from 

human activities, it is important to recognize natural causes in both design and maintenance of 

highway crossings. 

Natural causes and complications from gradation problems include: alluvial fans, natural 

armoring, braiding, meandering/migration (natural cutoffs), recurrent flooding, high stream 

velocity, channel bed and bank material erodibility, fire, floating debris, mud and debris flows, 

earthquakes, tectonic activity, volcanic activity, and landslides. 

Floating Debris. Floating debris causes hydraulic problems at highway crossings nationwide. 

The problems are the greatest in the Pacific Northwest and the upper and lower Mississippi 

River Valley. Debris hazards are generally a local phenomena often associated with large 

floods. Most bridge destruction from debris is due to accumulation of debris against bridge 

components. Debris may partially or totally block waterways, create adverse hydraulic 

conditions that erode pier foundations and bridge abutments or may overtop roadways and 

cause structural damage. 

Many debris problems exist in forested areas with active logging operations. Highway 

crossings on streams where stream slopes are mild or moderate, in contrast to headwater 

streams, are more vulnerable to debris related hazards. Debris hazards occur more frequently 

in unstable streams where bank erosion is active. Countermeasures presently used by 

highway agencies include: (1) sufficient freeboard, (2) proper pier spacing, (3) solid piers, (4) 

debris deflectors, (5) special superstructure designs, (6) flood relief structures, and (7) routine 

and emergency removal of debris at bridge crossings. Most debris transported in floods does 

not travel a great distance and often is observable locally along the stream banks upstream 

from the bridge prior to the flood. Debris usually moves as individual logs in a non-random 

path concentrating in the thalweg of the stream. Therefore, methods for evaluating its 

abundance and for mitigating its hazard are deemed feasible. HEC-20 (Lagasse et al. 2001) 

summarizes the results of studies by the U.S. Geological Survey to develop methods to 

estimate potential debris accumulations at bridges (Diehl 1997 and Diehl and Bryan 1993). 

Examples of debris control structures for culverts are given in Reihsen (1964). 

Mud Flows And Debris Flows. Fast melting snowpack and overabundance of soil moisture 

on steep slopes throughout the Western United States causes mudflows, debris flows and 

landslides, threatening bridges and highway structures. There is considerable evidence of 

damages to highway structures in the literature. For example Hungr et al., (1984) documented 

a bridge for which a concrete bridge beam was demolished by point impact during a debris flow 

event. 
5.51 

In another example, the volcanic eruption from the magmatic blast of Mount St. Helens 

triggered a major slope failure on the north flank of the mountain. Mudflows and debris flows 

were generated and swept down the Toutle and Cowlitz Rivers destroying bridges, inundating 

buildings, and eventually blocking the navigation channel of the Columbia River. Bradley 

(1984) reported the Cowlitz has aggraded markedly as a result of the post eruption 

hyperconcentrated flows. The upper Cowlitz and the lower Toutle have shifted from 

meandering to braided streams, thereby causing difficulties in preventing the failure of some 

remaining bridges. 

Examination of typical watershed behavior and response provides information on the impact of 

changes on the fluvial system. Channel stability assessments and possible gradation changes 

are indicated in Table 5.8. Those findings reflect the observations of Keefer et al. (1980). An 

example relating to the use of this table is presented in Section 5.9, Problem 3. 

5.7.3 Resulting Problems at Highway Crossings

Brown et al. (1980) reviewed design practices to evaluate crossing design procedures and the 

effect of grade changes on these procedures. The parameters most influenced by grade 

changes are those used as input to the hydraulic design procedures currently in use. These 

input parameters include: design discharge; channel roughness; energy slope; bed slope; 

velocities; shear stresses; cross-sectional geometry; base level; flow depth; and flow alignment. 

Other components of crossing design affected by grade changes include foundation depth, 

bridge deck clearance, and flow opening size. 

Problems encountered at bridge crossings include bridge capacity, backwater, pier and 

abutment alignment, footing depth at piers and abutments, and construction depth for 

flow-control and debris-control structures. With respect to bridge capacity and backwater, 

aggradation produces the most severe problems. However, debris problems associated with 

degradation can also have a significant impact on flow capacity and scour. Foundation depths 

for piers, abutments, and flow-control structures can be influenced in two ways by grade 

changes: the normal streambed base level will be altered; and the "normal" hydraulic 

conditions at a site used as input to local scour computations will be changed. The important 

components of bank protection design adversely affected by grade changes are key depths 

and the vertical extent of bank protection above and below the streambed. 

Problems encountered at culvert crossings can be the result of general grade changes 

produced by long-term changes in stream morphology or inadequate design and/or 

construction of culvert systems. The design components most often influenced are culvert 

capacity and structural stability. The greatest danger produced by aggradation is partial 

plugging of the culvert opening resulting in a damming effect and increasing the magnitude 

and frequency of flooding upstream of the structure. Degrading stream reaches affect culvert 

systems by reducing their structural stability. General streambed degradation has undermined 

the foundations of culverts resulting in their complete failure. 
5.52 

Table 5.8. Channel Response to Changes in Watershed and River Condition 

 (after Keefer et al. 1980). 

Observed Channel Response 

Condition Stable Unstable Degrading Aggrading 

Alluvial Fan 

 Upstream X X 

 Downstream X X 

Dam and Reservoir 

 Upstream X X 

 Downstream X X 

River Form 

 Meandering X X Unknown Unknown 

 Straight X Unknown Unknown 

 Braided X Unknown Unknown 

Bank Erosion Unknown Unknown 

Vegetated Banks X Unknown Unknown 

Headcuts X X 

Diversion 

 Clear Water Diversion X X 

 Overloaded with Sediment X 

Channel Straightened X X 

Deforested Watershed X X 

Drought Period X X 

Wet Period X X 

Bed Material Size 

 Increase X X 

 Decrease X Unknown X 

5.8 STREAM STABILITY PROBLEMS AT HIGHWAY CROSSINGS

In the United States, the annual damage related to hydraulic problems at bridges and highways 

has been estimated as high as $60 million during years of extreme floods. Damages by 

streams can be reduced by considering channel stability in site selection, bridge design, and 

countermeasure placement. Ideally, a stable channel is one that does not change in size, 

form, or position through time. However, all alluvial channels change to some degree and 

therefore have some degree of instability. For engineering purposes, an unstable channel is 

one whose rate or magnitude of change is great enough to be a significant factor in the 
5.53 

planning or maintenance of a highway crossing during the service life of the structure. The 

kinds of changes considered here are: (1) lateral bank erosion; (2) degradation or aggradation 

of the streambed that continues progressively over a period of years; and (3) natural short-term 

fluctuations of streambed elevation that are usually associated with the passage of a flood 

(scour and fill). Stability is inferred mainly from the nature of point bars, the presence or 

absence of cut banks, and the variability of stream width (see also HEC-20, Lagasse et al. 

2001). 

5.8.1 Bank Stability

On a laterally unstable channel, or at actively migrating bends on an otherwise stable channel, 

the point bars are usually wide and unvegetated and the bank opposite to a point bar is cut and 

often scalloped by erosion. The crescentic scars of slumping may be visible from place to 

place along the bankline. The presence of a cut bank opposite of a point bar is evidence of 

instability, even if the point bar is vegetated. Sand or gravel on the bar appears as a light tone 

on airphotos. The unvegetated condition of the point bar is attributed to a rate of growth that is 

too rapid for vegetation to become established. However, the establishment of vegetation on a 

point bar is dependent on other factors besides rate of growth, such as climate and the timing 

of floods. If the width of an unvegetated point bar is considered as part of the channel width, 

the channel tends to be wider at bends. Streams whose width at bends is about twice or more 

the width at straight reaches are called wide-bend streams. 

Oxbow lakes are formed by the cutoff of meander loops, which occurs either by gradual 

closure of the neck (neck cutoffs) or by a chute that cuts across the neck (chute cutoffs). Neck 

cutoffs are associated with relatively stable channels, and chute cutoffs with relatively unstable 

channels. Recently formed oxbow lakes along a channel are evidence of recent lateral 

migration. A recently formed lake is usually immediately adjacent to the channel and it 

transmits flow at high river stages. Commonly, a new meander loop soon forms at the point of 

cutoff and grows in the same direction as the previous meander. Cutoffs tend to induce rapid 

bank erosion at adjacent meander loops. The presence of abundant oxbow lakes on a 

floodplain does not necessarily indicate a rapid channel migration rate, because an oxbow lake 

may persist for hundreds of years. 

Along an unstable channel, bank erosion tends to be localized at bends, and straight reaches 

tend to be relatively stable. However, meandering of the thalweg in a straight reach is likely to 

be a precursor of instability. Bars that occur alternately from one side to the other of a straight 

reach are somewhat analogous to point bars and are indicative of a meandering thalweg. 

The following paragraphs summarize the characteristics of unstable and stable banks. For a 

more detailed discussion of bank stability and the mechanics of bank failure see HEC-20 

(Lagasse et al. 2001). 

Unstable Banks With Moderate To High Erosion Rate. The slope angle of unstable banks 

usually exceeds 30 percent, and a cover of woody vegetation is rarely present. At a bend, the 

point bar opposite of an unstable cut bank is likely to be bare at normal stage, but it may be 

covered with annual vegetation and low woody vegetation, especially willows. Where very rapid 

erosion is occurring, the bankline may have irregular indentations. Fissures, which represent 

the boundaries of actual or potential slump blocks along the bankline indicate the potential for 

very rapid bank erosion. 
5.54 

Unstable Banks With Slow To Moderate Erosion Rate. If a bank is partly graded (smooth 

slope) the degree of instability is difficult to assess and reliance is placed mainly on vegetation. 

The grading of a bank typically begins with the accumulation of slumped material at the base 

such that a slope is formed, and progresses by smoothing of the slope and the establishment 

of vegetation. 

Stable Banks With Very Slow Erosion Rate. Stable banks tend to be graded to a smooth slope 

and the slope angle is usually less than about 30 percent. In most regions of the United 

States, the upper parts of stable banks are vegetated, but the lower part may be bare at 

normal stage, depending on bank height and flow regime of the stream. Where banks are low, 

dense vegetation may extend to the water's edge at normal stage. Mature trees on a graded 

bank slope are particularly convincing evidence for bank stability. Where banks are high, 

occasional slumps may occur on even the most stable graded banks. Shallow mountain 

streams that transport coarse bed sediment tend to have stable banks. 

Field information on lateral migration rates for channels of different sizes has been compiled by 

Brice (1982). Bank erosion rates tend to increase with increasing stream size. In Figure 5.26 

channel width is taken as a measure of stream size. The dashed line is drawn arbitrarily to 

have a slope of 1 and a position (intercept) to separate most equiwidth streams from most 

wide-bend and braided point-bar streams. For a given channel width, equiwidth streams tend 

to have the lowest erosion rates, and braided point-bar streams the highest. Braided streams 

without point bars (diamond symbol, Figure 5.26) plot well below the arbitrary curve because 

their channels are very wide relative to their discharges. Channel width is an imperfect 

measure of stream size, as are drainage area and discharge, particularly for the comparison of 

streams in arid and semiarid regions with streams in humid regions. If braided streams and 

braided point-bar streams (which are uncommon in most parts of the United States) are 

excluded, the dashed curve in Figure 5.26 provides a preliminary estimate of erosion rates that 

may be encountered at a particular site. An example on the use of these results is presented 

in Section 5.9, Problem 4. 

5.8.2 Stability Problems Associated With Channel Relocation

For some highway encroachments, a change in the river channel alignment is advantageous. 

When a river crossing site is so constrained by non-hydraulic factors that consideration of 

alternative sites is not possible, the engineer must attempt to improve the local situation to 

meet specific needs. Also, the engineer may be forced to make channel improvements in 

order to maintain and protect existing highway structures in or adjacent to the river. 

Suppose a meandering river is to be crossed with a highway, as shown in Figure 5.27a. 

Assume that the alignment is fixed by constraints in the acquisition of the right-of-way. To 

create better flow alignment with the bridge, consideration is given to channel improvement as 

shown in Figure 5.27b. Similarly, consideration for improvement to the channel would also be 

advisable for a hypothetical lateral encroachment of a highway as depicted in Figure 5.27c. In 

either case, the designer's questions are how to realign the channel, and what criteria to use to 

establish the cross-sectional dimensions. 
5.55 

Figure 5.26. Median bank erosion rate in relation to channel width for different types of 

 streams (after Brice 1982). 
5.56 

Figure 5.27. Encroachment on a meandering river. 

Prior to realigning a river channel the stability of the existing channel must be examined using 

the methods outlined earlier. A stream classification, recent and past aerial photographs and 

field surveys are generally necessary. The realigned channel may be made straight without 

curves, or may include one or more curves. If curves are included, the radii of curvature, the 

number of bends, the limits of rechannelization (hence the length or slope of the channel) and 

the cross-sectional area are decisions which have to be made by the designer. Different rivers 

have different characteristics and historical background with regard to channel migration, 

discharge, stage, geometry and sediment transport. As indicated in the previous chapters, it is 

important for the designer to understand and appreciate river hydraulics and geomorphology 

when making decisions concerning channel relocation. It is difficult to state generalized criteria 

for channel relocation applicable to every river. Knowledge about river systems has not yet 

advanced to such a state as to make this possible. Nevertheless, it is possible to provide some 

principles and guidelines for the design engineer. 

As the general rule, the radii of bends should be made about equal to the mean radii of bends 

in extended reaches of the river. When the angle φ defined in Figure 5.15 exceeds about 40 

degrees, this provides a sufficient crossing length for the thalweg to shift from one side of the 

channel to the other. Generally, it is necessary to stabilize the outside banks of the curves in 

order to hold the new alignment, and depending upon crossing length some amount of 

maintenance may be necessary to remove sandbars after large floods so that the channel 

does not develop new meander patterns in the crossings during normal flows. 

The sinuosity and channel bed slope are related in the following way. The bed elevations at 

the ends of the reach being rechannelized, (designated 1 and 2, in Figure 5.27) are established 

by existing boundary conditions. Hence, the total drop in bed elevation for the new channel 

(subscript 2) and the old channel (subscript 1) are the same. 

∆∆ ∆ zzz 1 2 = = (5.22) 
5.57 

The length of channel measured along the thalweg is labeled Lc. Thus, the mean slope of the 

channel bed before relocation is: 

S z

Lc1

1 = ∆ (5.23) 

and after relocation is 

S z

Lc

2

2

= ∆ (5.24) 

Sinuosity is defined as the ratio of the length of channel, Lc to the length of the valley, or 

S L

L n c

v

= ≥1 (5.25) 

Clearly, 

S

L

L n

c

v

1 1

1

= (5.26) 

S

L

L n

c

v

2 2

2

= (5.27) 

but 

LLL VVv 1 2 = = (5.28) 

and 

∆∆ ∆ z

L

z

L

z

vvv L

1 2

1 2

= = (5.29) 

Thus, 

S S

L

L

z

L

L

L

z

L n s S c

v c1

c

v c

1 1 n

2

2 2 1 2 = 



 



 



 



 = 



 



 



 



 = ∆ ∆

 (5.30) 

The new channel slope and channel sinuosity are inversely related. If Sn2 < Sn1 then S2 > S1. 

The new channel alignment, hence Sn2, can be chosen by the designer with due consideration 

given to the radii of curvature, deflection angles and tangent lengths between reversing curves. 

As indicated before, consideration should also be given to prevailing average conditions in the 

extended reach. The new slope S2 can be calculated from Equation 5.30, and the relationship 

for meandering (Equation 5.1) should be satisfied. 
5.58 

0007.QS 4/1

2 ≤ (SI) 

or 

S Q2

1 4 0 0017 / ≤ . (English) 

If S1 is of such magnitude that Equation 5.1 cannot be satisfied with still larger S2, the possibility 

of the river changing to a braided channel because of steeper slope should be carefully 

evaluated. With steeper slope, there could be an increase in sediment transport which could 

cause degradation, and the effect would be extended both upstream and downstream of the 

relocated reach. The meander patterns could change. Considerable bank protection might be 

necessary to contain lateral migration which is characteristic of a braided channel, and if the 

slope is too steep, head cuts could develop which migrate upstream with attendant effects on 

the geometry of the channel. Even when changes in slope are not very large, a short-term 

adjustment of the average river slope occurs, consistent with the sediment transport rate, flow 

velocities and roughnesses, beyond the upstream and downstream limits of channel 

improvement. For small changes in slope, the proportionality (Equation 5.16), QS ~ Qs D50

tends toward equilibrium with slight increases in bed sediment size, D50, and adjustment in the 

sediment transport rate, Qs. 

A small increase in the new channel width could be considered which tends to maintain the 

same stream power, τo V, in the old and new channels. That is, 

( )( ) τ τ o o V V 1 2 = (5.31) 

With substitution of τo = γRS, V = Q/A and R = A/P  A/W, Equation 5.42 leads to 

W SW S 2 21 1 = / (5.32) 

Any designed increase in width should be limited to about 10 to 15 percent. Wider channels 

would be ineffective. Deposition would occur along one bank and the effort of extra excavation 

would be wasted. Furthermore, bar formation would be encouraged, with resultant tendencies 

for changes in the meander pattern leading to greater maintenance costs for bank stabilization 

and removal of the bars to hold the desired river alignment. 

The depth of flow in the channel is dependent on discharge, effective channel width, sediment 

transport rate (because it affects bed form and channel roughness) and channel slope. 

Methods for evaluating flow depth were discussed earlier in this chapter. 

The foregoing discussion pertains to alluvial channels with silt and sand sized bed materials. 

For streams with gravel and cobble beds, the usual concern is to provide adequate channel 

cross-sectional dimensions to convey flood flows. If the realigned channels are made too 

steep, there is an increased stream power with a consequent increase in transport rate of the 

bed material. The deposition of material in the reaches downstream of the crossing tends to 

form gravel bars and encourages changes in the planform of the channel. Short-term changes 

in channel slope can be expected until equilibrium is reestablished over extended reaches both 

upstream and downstream of the rechannelized reach. Bank stabilization may be necessary to 

prevent lateral migration, and periodic removal of gravel bars may also be necessary. 
5.59 

5.8.3 Assessment of Stability for Relocated Streams

Brice (1980) reported case histories for channel stability of relocated streams in different 

regions of the United States. Based on his study, the recommendations and conclusions 

presented here apply to specific aspects of the planning and construction of channel relocation. 

They are intended for assessment of the risk of instability and for reduction of the degree of 

instability connected with relocation. Serious instability resulting from relocation can be 

observed either when the prior natural channel is unstable or when floods of high recurrence 

interval occur during or soon after construction. Although there is an element of uncertainty in 

channel stability, the experience represented by Brice's study sites provides useful guidelines 

for improvement in the performance of channels relocated by highway agencies. 

Consideration of the following aspects of the channel relocation is recommended. 

Channel Stability Prior To Relocation. Assessment of the stability of a channel prior to 

relocation is needed to assess erosion-control measures and risk of instability. An unstable 

channel is likely to respond unfavorably to relocation. Bank stability is assessed by field study 

and the stereoscopic examination of aerial photographs. The most useful indicators of bank 

instability are cut or slumped banks, fallen trees along the bankline, and wide, unvegetated, 

exposed point bars. Bank recession rates are measured by comparison of time-sequential 

aerial photographs. Vertical instability is equally important but more difficult to determine. It is 

indicated by changes in channel elevation at bridges and gaging stations. Serious degradation 

is usually accompanied by generally cut or slumped banks along a channel. 

Erosion Resistance Of Channel Boundary Materials. The stability of a channel, whether 

natural or relocated, is partly determined by the erosion resistance of materials that form the 

wetted perimeter of the channel. Resistant bedrock outcrops, which extend out into the 

channel bottom, or that lie at shallow depths, will provide protection against degradation. Not 

all bedrock is resistant. Erosion of shale, or of other sedimentary rock types interbedded with 

shale, has been observed. Degradation was slight or undetected at most sites where bed 

sediment was of cobble and boulder size. However, serious degradation may result from 

relocation. Degradation may result from the relocation of any alluvial channel, whatever the 

size of bed material, but the incidence of serious degradation of channels relocated by highway 

agencies is slight. 

The cohesion and erosion resistance of banks tend to increase with clay content. Banks of 

weakly coherent sand or silt are clearly subject to rapid erosion, unless protected with 

vegetation. No consistent relation was found between channel stability and the cohesion of 

bank materials, probably because of the effects of vegetation. 

Length Of Relocation. The length of relocation contributes significantly to channel instability at 

sites where its value exceeded 250 channel widths. When the value is below 100 channel 

widths, the effects of length of relocation are dominated by other factors. The probability of 

local bank erosion at some point along a channel increases with the length of the channel. The 

importance of vegetation, both in appearance and in erosion control, would seem to justify a 

serious and possibly sustained effort to establish it as soon as possible on the graded banks. 

Bank Revetment. Revetment makes a critical contribution to stability at many sites where it is 

placed at bends and along roadway embankments. Rock riprap is by far the most commonly 

used and effective revetment. Concrete slope paving is prone to failure. Articulated concrete 

block can be effective especially when vegetation can establish in the interstices between 

blocks. Bank revetment is discussed in detail in the next chapter. 
5.60 

Check Dams (drop structures). In general, check dams are effective in preventing channel 

degradation. The potential for erosion at a check dam depends on its design and construction, 

its height and the use of revetment on adjoining banks. A series of low check dams, less than 

about 0.5 m (2 ft) in height, is probably preferable to a single higher structure, because the 

potential for erosion and failure is reduced. By simulating rapids, low check dams may add 

visual interest to the flow in a channel. One critical problem arising with check dams relates to 

improper design for large flows. Higher flows have worked around the ends of many 

installations to produce failure. 

Maintenance. The following problems that can be controlled by maintenance were observed 

along relocated channels: (1) growth of annual vegetation in channel; (2) reduction of channel 

conveyance by overhanging trees; (3) local bank cutting; and (4) bank slumping. The expense 

of routine maintenance or inspection of relocated channels beyond the highway right-of-way is 

probably prohibitive. However, most of the serious problems could be detected by periodic 

inspection, perhaps by aerial photography, during the first 5 or 10 years after construction. 

Relationship Between Sinuosity And Stability. This relationship is summarized as follows: 

(1) Meandering does not necessarily indicate instability; an unstable stream will not remain 

highly sinuous for very long, because the sinuosity will be reduced by frequent meander 

cutoffs; (2) Where instability is present along a reach, it occurs mainly at bends; straight 

segments may remain stable for decades; and (3) The highest instability is for reaches whose 

sinuosity is in the range of 1.2 to 2 and whose type is either wide bend or braided point bar. 

5.8.4 Estimation of Future Channel Stability and Behavior

One objective of stability assessment is to anticipate the migration of bends and the 

development of new bends. Lateral erosion is probably more frequently involved in hydraulic 

problems at bridges than any other stream process. Problems caused by general scour, local 

scour, channel degradation and accumulation of debris are somewhat less common. 

The lateral stability of a stream can be measured from records of its position at two or more 

different times where the available records are usually maps or aerial photographs. Historic 

surveyed cross-sections are extremely useful and can frequently be located in bridge 

inspection files. It is recognized that some progress is being made on the numerical prediction 

of loop deformation and bend migration (see Section 5.8.5). At present, however, the best 

available estimates are based on past rates of lateral migration at a particular reach. However, 

erosion rates may fluctuate substantially from one period of years to the next. 

Measurements of bank erosion on two time-sequential aerial photographs (or maps) require 

the identification of reference points which are common to both. Useful reference points 

include roads, buildings, irrigation canals, bridges and fence corners. This analysis of lateral 

stability is greatly facilitated by a drawing of time changes in bankline position. To prepare 

such a drawing, aerial photographs are matched in scale and the photographs are 

superimposed holding the reference points fixed. For further discussion see HEC-20 (Lagasse 

et al. 2001). 

Bank erosion rates increase with the stream size as shown in Figure 5.26. Sinuous canaliform 

streams can then be expected to have the lowest erosion rates and the sinuous braided 

streams, the highest. 
5.61 

The lateral stability of different stream reaches can be compared by means of a dimensionless 

erosion index. The erosion index is the product of its median bank erosion rate expressed in 

channel widths per year, multiplied by the percent of reach along which erosion occurred, 

multiplied by 1,000. Erosion indexes for 41 streams in the United States are plotted against 

sinuosity in Figure 5.28. The length of most of these reaches is 25 to 100 times the channel 

width. The highest erosion index values are for reaches with sinuosity ranging between 1.2 

and 2. Erosion indexes are large for sinuous braided and sinuous point bar (wide bend) 

streams. Equiwidth streams tended to be relatively stable. The erosion index value of 5, in 

Figure 5.28, is suggested as a boundary between stable and unstable reaches. Brice (1984) 

considers that reaches having erosion indexes values less than 5 are unlikely to cause lateral 

erosion problems at bridges. An example on the use of Figure 5.28 is presented in Section 

5.9, Problem 4. 

A general assessment of bank stability can be made considering the following aspects. 

Bank Erosion Rates. It is theoretically possible to determine bank erosion rates from factors 

such as water velocity and resistance of the banks to erosion. See HEC-18 (Richardson and 

Davis 2001) for a discussion of recent advances in measuring erosion rates. The results in 

Figure 5.26 provide a first approximation of migration rate of a bend regardless of the hydraulic 

conditions and sediment characteristics. Past rates of erosion at a particular site provide the 

best estimate of future rates. In projecting past rates into the future, consideration must be 

given to the following factors: (1) the past flow history of the site during the period of 

measurement, in comparison with the probable future flow history during the life span of the 

highway crossing. The duration of floods, or of flows near bankfull stages, is probably more 

important than the magnitude of floods; and (2) human-induced factors that are likely to affect 

bank erosion rates. Among the most important of these are urbanization and the clearing of 

floodplain forests. 

Figure 5.28. Erosion index in relation to sinuosity (Brice 1984). 
5.62 

Behavior Of Meander Loops. If the proposed bridge or roadway is located near a meander 

loop, it is useful to have some insight into the probable way in which the loop will migrate or 

develop, as well as its rate of growth. No two meanders will behave in exactly the same way, 

but the meanders on a particular stream reach tend to conform to one of the several modes of 

behavior illustrated in Figure 5.29. 

Mode A (Figure 5.29) represents the typical development of a loop of low amplitude, which 

decreases in radius as it extends slightly in a downstream direction. Mode B rarely occurs 

unless meanders are confined by valley sides on a narrow floodplain, or are confined by 

artificial levees. Well developed meanders on streams that have moderately unstable banks 

are likely to follow Mode C. Mode D applies mainly to large loops on meandering or highly 

meandering streams. The meander has become too large in relation to stream size and flow, 

and secondary meanders develop along it, converting it to a compound loop. Mode E also 

applies to meandering or highly meandering streams, usually of the equiwidth point-bar type. 

The banks have been sufficiently stable for an elongated loop to form (without being cut off), 

but the neck of the loop is gradually being closed and cutoff will eventually occur at the neck. 

Modes F and G apply mainly to locally braided sinuous or meandering streams having unstable 

banks. Loops are cut off by chutes that break diagonally or directly across the neck. 

Effects Of Meander Cutoff. If cutoffs seem imminent at any meanders in the vicinity of a 

proposed bridge crossing the probable effects of cutoff need to be considered. The local 

increase in channel slope due to cutoff usually results in an increase in the growth rate of 

adjoining meanders, and an increase in channel width at the point of cutoff. On a typical 

wide-bend point-bar stream the effects of cutoff do not extend very far upstream or 

downstream. 

Figure 5.29. Modes of meander loop development. A, Extension. B, Translation. C, Rotation. 

 D, Conversion to a compound loop. E, Neck cutoff by closure. F, Diagonal cutoff 

 by chute. G, Neck cutoff by chute. 
5.63 

Assessment Of Degradation. Field sites having degradation problems are more numerous 

than sites having aggradation problems. Annual rates of degradation averaged from past 

records such as the closure of a dam give poor estimates of future rates of degradation. 

Typical situations exhibit an exponential decay function of the rate of channel degradation. 

Recent evidence of degradation can be detected from field surveys or by stereo viewing of 

aerial photographs. Indicators of degradation are: (1) channel scarps, headcuts and 

nickpoints; (2) gullying of minor side tributaries; (3) high and steep unvegetated banks; (4) 

measurements of streambed elevation from a bridge deck; (5) changes in stream discharge 

relationships; and (6) measurements of longitudinal profiles. 

Assessment Of Scour And Fill. Natural scour and fill refer to fluctuations of streambed 

elevation about an equilibrium condition. These fluctuations are associated mainly with floods 

and occur by three different mechanisms operating jointly or independently: (1) bed form 

migration; (2) convergence and divergence of flow; and (3) lateral shift of thalweg or braids. 

The maximum scour induced by the migration of a dune is almost one-half the dune height, 

and dune heights are roughly estimated as one-third of the mean flow depth. In gravel bed 

streams, most migrating bed forms can be regarded as bars, the height of which is related to 

flow depth. The migration of a bar through a bridge waterway is mainly of concern because of 

the deflection and concentration of flow. Bar migration tends to be a random process and its 

motion can best be tracked from time-sequential aerial photographs. 

Gravel bars tend to migrate on braided streams and to remain fixed at riffles on unbraided pool 

and riffle streams. 

Flow convergence in natural streams is associated with scour, whereas divergent currents are 

associated with deposition. Persistent pools have the strongest convergence of flow and the 

greatest potential for scour. Such pools are best identified by a continuous bed profile along 

the thalweg. In braided streams, scour holes are found at the confluence of braids. Field 

measurement of cross-sectional area and flow velocity at an incised reach near bankfull stage 

provides a good basis for calculation of scour by extrapolation to the design flood. 

Instability of the streambed that results from shift of thalweg is related to stream type and can 

be assessed from study of aerial photographs. On sinuous canal form streams, shift of the 

thalweg during flood is minimal. A greater shift of the thalweg can be expected on sinuous 

point-bar streams. In straight reaches, alternate bars visible on aerial photographs taken at low 

stage are commonly present. These alternate bars indicate the potential for thalweg shifting 

and also for bank erosion when the current is deflected against the bankline. Shift of the 

thalweg with increase in stage must be considered when determining the location of the point 

of maximum bed scour, bank erosion, and the alignment of piers with flood flow. 

Site Selection For Highway Crossings. For most streams the magnitude of scour is 

substantially greater at specific places along the channel. Bends and narrow sections may 

scour at high stages regardless of the effect of bridge structures. Straight or gently curved 

reaches with stable banks are preferred. 

Considerations for the selection of a crossing site on a non-sinuous reach include: (1) is the 

site at a pool, riffle or transition section; (2) are alternate bars visible at low stage; and (3) what 

is the effect of migration of mid-channel bars, if any? With respect to meandering reaches, 

questions requiring solution include: (1) what has been the rate and mode of migration of the 

meander; (2) what is the probable future behavior, as based on the past; (3) is the site at a 

pool, riffle or transition section; and (4) is meander cutoff probable? 
5.64 

5.8.5 Advances in Predicting Meander Migration 

Meander Behavior. According to Hooke (1991), it has long been assumed that, after initial 

development, meanders tend towards an equilibrium of form, given stability of external 

conditions. These ideas were based on early experiments in flumes (Friedkin 1945) and on 

the analysis of process-form relations in the 1960s in which statistical relationships were 

established, e.g., between meander wavelengths and discharge (Carlston 1965). Much 

evidence has now been accumulated that many meanders exhibit no such equilibrium 

(Carson and Lapointe 1983, Hooke 1984). 

If the behavior of individual bends on active rivers is examined (using for example, historical 

evidence or meander scrolls) then a continuous evolution with an increasing complexity of 

pattern is frequently found. Meanders may migrate at first, but then they begin to grow in 

amplitude, then to become compound in form. Based on analysis of historical sequences of 

meander change on rivers using maps and aerial photographs, models of meander 

development have been produced (Hooke 1991; Harvey 1989; Keller 1972). 

These show a sequence from low sinuosity bends to bends of symmetrical form which tend 

to migrate; then bends enter a phase of rapid growth with maximum erosion at the apex 

leading to extension in the cross-valley direction. Beyond a certain increase in path length 

(Hooke and Harvey 1983) the bends start to become compound in form by development of 

lobes on one or both parts of the apex. These lobes may go on to develop into separate 

bends if there is space. This increasing asymmetry and complexity is similar to other models 

such as those of Brice (1974) and Hickin and Nanson (1975). 

This process of growth cannot, of course, go on indefinitely. Eventually the bends are likely 

to intersect or spatial limits of the floodplain are reached so that cut-offs take place. 

Therefore, following a phase of growth, cutoffs are likely. Rivers show an early phase of 

increasing sinuosity then oscillating sinuosity thereafter. 

Of course, not all bends exhibit complexity or rapid growth. Some simply progressively 

migrate or tend to stabilize at various stages. This could be for a number of reasons, e.g., 

slope and overall energy in the system, floodplain form, or it could simply be that changes 

are progressing at a very slow rate. Different domains of meander behavior can therefore 

be visualized (Figure 5.30). Theoretically, it should be possible to identify domains of 

behavior with thresholds between them. The thresholds will vary with the actual river and 

may be rather 'fuzzy.' Further work is needed to substantiate whether they do indeed exist, 

but the indication is that any single mathematical model of meander behavior will be 

inadequate. This is supported by problems with existing models. 

The mathematical modeling of helical and cross channel flows and their related channel 

characteristics has proved to be difficult. Most such models are based on simplifications of 

the equations of continuity for water and sediment and on the Reynold's (or St. Venant's) 

equations of motion (Engelund 1974; Smith and McLean 1984; Odgaard 1986). As noted by 

several researchers (Odgaard and Bergs 1987; Yen and Ho 1990), most of these models 

simulate bend flow and channel topography only in the "fully developed" portions of the 

bend, i.e., where velocity and thalweg depth do not vary longitudinally. Other models predict 

these characteristics throughout the bend, even when velocity and depth change 

downstream (Dietrich and Smith 1983; Engelund 1974; Odgaard 1986; Yen and Ho 1990). 

In addition, considerable disagreement exists over the simplifications and eliminations that 

have been used to solve these equations (Dietrich and Smith 1983). 
5.65 

Figure 5.30. Domains of meander behavior (from Hooke 1991). 

Most of the mathematical models have been tested or verified in flumes, some of which had 

fixed beds or constant radius of curvature segments connected to straight segments 

(Odgaard and Bergs 1987). Even when model testing is done on rivers with constantlyvarying radii of curvature and mobile bed materials, problems with measurement accuracy of 

the various model components often dominate the final result (Dietrich and Smith 1983). 

One difficulty is the assumption that all meanders are simple, whereas as Hooke (1991) 

demonstrates they are not. In addition to other controls on meander migration, the mode of 

bank retreat varies from grain by grain removal to mass failure (Hasegawa 1989; Knighton 

1984). 

There are many authors who believe that it is combinations of processes that are important 

(e.g., Hooke 1979; Thorne 1982; Lawler 1986). In view of the wide range of alluvial 

materials, riverine forms and hydroclimatic environments, the limitations of mathematical 

models are many and the range of validity of the models is limited. Therefore a reasonable 

forecast of the future planform of the river has not yet been realized successfully and 

remains a challenging topic (Thein 1994). 

Mosselman (1995) states that the non-periodicity of meanders, which develop from a straight 

alignment in the numerical models of Howard and Knutson (1984) and others "suggest 

chaotic behavior, that is, a sensitive dependence on initial and boundary conditions." He 

concludes that this "chaotic behavior bears on the predictability of river meandering." 

It is only by considering the great variety of meander configurations and how they influence 

future patterns that reasonable predictions of meander shift can be accomplished. Indeed, 

Howard (1996) concludes that the chaotic behavior suggests that "even the most detailed 

model has limited predictive power." This is perhaps because "small differences in initial 

geometry or boundary conditions between two identical streams will cause different meander 

patterns" (Howard 1996). In addition, he concludes that "predictability of future meander 

patterns decrease with time." 
5.66 

The problem appears to lie in the lack of recognition of different types of rivers with different 

meander behavior (Figure 5.30) and the assumption that all meanders are represented by 

idealized bends. 

Analysis Options. A study by Johns Hopkins University (Cherry et al. 1996) for the U.S. 

Army Corps of Engineers Waterways Experiment Station investigated the use of both 

empirical and analytical approaches to provide solutions to the problem of predicting 

meander migration. This study evaluated empirical and analytical (computational) methods 

for forecasting planform change and bankline migration in flood control channels, using data 

originally assembled and analyzed by Brice (1982) to assess stream channel instability 

problems at bridges for the FHWA. Twenty-six sites were used to evaluate the predictive 

capabilities of bend-flow meander migration computer models. The computational bend-flow 

meander migration model used by Cherry et al. was developed by Garcia et al. (1994). 

The Johns Hopkins study recognized that a meandering river is a complex system involving 

relations among many variables. The erosion rate for a meander bend is determined by the 

balance between the erosive forces applied to the channel bank and the resistance to 

erosion provided by the bank material and bank vegetation. Erosive force is a complex 

function of discharge, channel cross section geometry, sediment load, bed roughness, 

presence of bedforms and bars, and the planform geometry. Resistance to erosion is 

related to the properties of the bank material, the bank geometry (slope, height, shape), the 

presence of vegetation, and the state of the pore water in the bank (Cherry et al. 1996). 

Although simplified, single valued correlations between a number of variables were 

established empirically and expressed as power functions, Johns Hopkins concluded that 

they did not adequately describe meander behavior. 

In regard to computer modeling, a number of authors have developed versions of the bend 

flow model (e.g., Parker et al. 1982; Beck and Melfi 1984; Hasegawa 1989; Odgaard and 

Bergs 1987; Garcia et al. 1994) and although the models have typically been tested with 

plots of predicted vs. observed channel form for a limited number of channels, there has 

been little general testing of these models over a range of hydrologic and geologic conditions 

(Cherry et al. 1996). 

After testing the bend flow model for 26 of the meandering sites in the Brice data set, the 

Johns Hopkins study concluded that both the accuracy and applicability of the bend-flow 

meander migration model are limited by a number of simplifying assumptions. Among the 

most important of these are the use of a single discharge and the assumption of constant 

channel width, both of which prevent the model from successfully forecasting the spatial and 

temporal variability that appears to be inherent in the process of bend migration. 

It was also concluded that much of the discrepancy between the predicted and observed 

distributions of erosion can be accounted for by the fact that meander migration is modeled 

as a smooth, continuous process. In reality, erosion occurs predominantly in discrete 

events, and varies greatly both temporally and spatially along the channel from bend to bend 

(Nanson and Hickin 1983). The Johns Hopkins study noted that the identification of local 

factors that influence the amount of bank erosion that occurs is a subject "that will require 

further investigation." 
5.67 

In addition to channel and bank characteristics, floodplain characteristics must also be 

incorporated into an analysis procedure. The floodplain characteristics that should affect 

meander migration include geologic controls, alluvial deposits and topographic variability. 

Geologic controls include bedrock outcrops and erosion resistant features along the valley 

sides. Alluvial deposits frequently include oxbows, meander scrolls and scars, and clay 

plugs, each with different erodibility characteristics. Topographic variability that should be 

considered include the cross valley slope of the adjacent floodplain and valley slope. These 

factors could be incorporated into regression equations but would be difficult to include in 

computer modeling of bendway migration. 

5.9 SOLVED PROBLEMS RIVER MORPHOLOGY AND RESPONSE

5.9.1 PROBLEM 1 Meandering and Braiding 

(a) Consider the sinuous point bar stream in Figure 5.31. Determine the following 

characteristics: meander wavelength λ; meander width Wm; mean radius of curvature rc; 

meander amplitude A; and the bend deflection angle φ. 

Figure 5.31. Sinuous point bar stream. 

Meander wavelength λ is approximately 11,000 ft (3,353 m) 

Radius of curvature rc is approximately 2,400 ft (732 m) 

Channel width ranges from 250 ft to 850 ft (76 to 260 m) at high stage 

Meander amplitude A is 2,200 ft (670 m) 

Meander width Wm is 2,700 ft (823 m) 

Bend deflection angle φ is 105°

Sinuosity is 1.2 

(b) Given the sand size D50 = 0.5mm, the bankfull discharge Q = 10,000 cfs (283 m3

/s) and 

the slope S = 2x10-4, determine the effect of increasing slope, discharge, sediment size and 

sediment discharge on the planform geometry. 
5.68 

The result of increasing any or several of these variables (Figure 5.18 and Table 5.4) is to 

promote braiding of this channel. When locating this stream on Lane's diagram (Figure 5.18) it 

is shown that with SQ0.25 = 0.002 (SI = .00082), this river is very close to the line SQ0.25 = 

0.0017 (.0007 for SI) for meandering streams. Hence an increase in discharge or slope would 

be required to change the planform to a braided stream. 

(c) Determine the effect of increased discharge, bed sediment size, bed sediment load, or 

washload on channel stability, resistance to flow, energy slope and stage of the same river. 

Table 5.5 can be used to provide a qualitative response to these changes. 

• Increase in discharge results in an increase in stage and a decrease in energy slope and 

channel stability. 

• Increase in sediment size results in an increase in stage, energy slope and resistance to 

flow. The channel stability might not be changed. 

• Increase in bed sediment load should increase the channel stability through a decreased 

resistance to flow, slope and stage. 

• Effect of increasing washload is similar to that of increasing bed sediment load except for 

channel stability, which is uncertain. 

5.9.2 PROBLEM 2 Classification of Alluvial Reaches 

Identify the three types of alluvial river reaches sketched below. Discuss the relative stability of 

each channel. 

Figure 5.32. River channels (After Petersen 1986). 
5.69 

Based on the classification presented in Figures 5.11, 5.12, and 5.13, the first stream (a) is a 

straight channel with alternate bars and sinuous thalweg. This stream has a relatively low 

slope and low width-depth ratio. Figure 5.13 also indicates that it has an intermediate sediment 

size, sediment load and moderate ratio of bedload to total load (between 3 - 10 percent). This 

stream can be classified as relatively stable. 

The second stream (b) is a sinuous point bar stream which is somewhat wider at bends. The 

meander bends are expected to shift gradually with possible neck cutoff. The stream is then 

relatively stable. 

The third stream (c) is a braided sand-bed stream with multiple bars and channels. Bars are 

likely to be comprised of coarse sand and the large flow velocities and stream power will 

generate large sediment load with a large proportion transported as contact load. The overall 

stability of this braided channel is very low, channel shifting, and avulsions are certainly 

common. 

5.9.3 PROBLEM 3 Channel Response to Changes in Watershed Conditions 

Determine the effect of watershed deforestation, bank erosion and headcutting on channel 

stability and gradation changes in an alluvial stream. 

Referring to Table 5.8, deforestation generally causes aggradation problems and therefore 

channel instability. The reason for this is that deforestation increases runoff and peak runoff 

discharge as well as sediment transport from upland areas. 

Headcutting is a degradation process. Upstream migration of headcuts induces bank failure 

and channel stability problems. Bank erosion has basically the same consequence on channel 

stability as that of headcutting. The gradation changes, however, are more difficult to assess 

because bank erosion changes the width-depth ratio. 

5.9.4 PROBLEM 4 Channel Migration Rate 

(a) Determine the bank erosion rate and the erosion index of the sinuous point bar stream 

sketched in Figure 5.33. 

The best method to estimate the rate of bank erosion is to compare two sets of aerial 

photographs. However, a first assessment can be obtained from Figure 5.26. Entering the 

figure with an average width around 150 m (492 ft), the median erosion rate should be around 

2 meters (6.5 ft) per year. Wide bend streams have slightly larger erosion rates than given by 

the dashed line. Entering Figure 5.28 with a sinuosity of 1.2, the erosion index might be as 

large as 18 indicating channel instability. Erosion is to be expected on the concave side of the 

bends. Other means for assessing lateral migration rates such as study of past aerial 

photographs, scroll formation (Figure 5.12(b)) and field studies should be undertaken if any 

bridge crossing was to be built across this river. 

(b) Sketch the likely future changes in the meandering river shown in the sketch below (Figure 

5.34). 
5.70 

Figure 5.33. Meandering river sketch (after Petersen 1986). 

Figure 5.34. Estimated future location of a meandering river (after Petersen 1986). 
5.71 

The river (left bank) has moved downvalley approximately 6,000 feet (1,829 m) in 50 years or 

an average of 120 ft (36 m) per year. The narrowest point of the meander neck is about 2,500 

ft (762 m). At its historical rate, this meandering channel will most likely lead to a cutoff in the 

next 15 to 20 years. The estimated position of the channel is sketched in Figure 5.34. 

5.9.5 PROBLEM 5 At-A-Station and Downstream Hydraulic Geometry 

 Relationships (SI) 

At bankfull discharge conditions Q1 = 227 m3

/s and the width of a sand-bed stream (Ds1 = 0.6 

mm) is W1 = 76 m, the maximum flow depth is yo1 = 2.4 m, the slope is Sf1 = 2.5 x 10 -4, and the 

maximum velocity is V1 = 1.5 m/s. 

(a) Estimate the width, W2, depth yo2, slope Sf2 and velocity V2 at the same station when the 

discharge Q2 is 5.7 m3

/s if the cross-sectional geometry is unknown. 

The at-a-station hydraulic geometry relationships (Table 5.3) can be used when no specific 

field data is available. 

For width: 

m2.29

227

7.5 76

Q

Q WW

26.0 26.0

1

2

12  = 

 



 = 













 =

For depth: 

m55.0

227

7.5 4.2

Q

Q

yy

40.0 40.0

1

2

2 oo 1  = 

 



 = 













 =

Slope is unchanged: 

4

ff 10x5.2SS 1 2

− ==

For velocity: 

s/m43.0

227

7.5 5.1

Q

Q VV

34.0 34.0

1

2

12  = 

 



 = 













 =

(b) Using the same station as for part (a), estimate the width, W2. depth yo2, slope Sf2 and 

velocity V2 in an upstream section of this stream if the bankfull discharge is 14 m3

/s and the 

bed material is gravel (D50 = 8 mm). How would the hydraulic geometry change if the bed 

material upstream is sand (D50 = 0.6 mm)? 
5.72 

The "downstream" geometry relationships can be used in this case. Two types of relationships 

are given in Table 5.3. The sand bed relationships are a function of discharge only, whereas 

the gravel bed relationships are a function of both discharge and sediment size. Both methods 

are compared in the following. 

Flow Depth

Sand bed: m67.0

227

14 4.2

Q

Q

yy

46.0 46.0

1b

2b

2 oo 1  = 

 



 = 













 =

Gravel bed: m79.0

227

14 4.2

Q

Q

yy

40.0 4.0

1b

2b

2 oo 1  = 

 



 = 













 =

 

Channel width

Sand bed: m1.21

227

14 76

Q

Q WW

46.0 46.0

1b

2b

12  = 

 



 = 













 =

Gravel bed: m4.7

6.0

8

227

14 76

D

D

Q

Q WW

53.0 33.0 33.0

s

s

53.0

1b

2b

12

1

2

 = 

 



 



 



 = 





























 =

− −

Friction slope

Sand bed: 4

46.0

4

46.0

1b

2b

ff 10x9

227

14 10x5.2

Q

Q SS 2 1

−

−

−

−

 = 

 



 = 













 =

Gravel bed: 01.0

6.0

8

227

14 10x5.2

D

D

Q

Q SS

4.0

4

1

s

s

4.0

1b

2b

ff

1

2

2 1  =



 



 



 



 = 





























 =

−

−

−

Velocity

Sand bed: s/m2.1

227

14 5.1

Q

Q VV

08.0 08.0

1b

2b

12  = 

 



 = 













 =

Gravel bed: s/m9.2

6.0

8

227

14 5.1

D

D

Q

Q VV

07.0 33. 33.

s

s

07.0

1b

2b

12

1

2

 = 

 



 



 



 = 





























 =

The gravel bed relationships give a steeper, faster flowing and narrower channel as compared 

to the sand bed relationships. Unless the sediment size is markedly different for two streams, 

the resulting hydraulic geometry calculated from both sets of equations will be similar. 
5.73 

5.9.6 PROBLEM 6 At-A-Station and Downstream Hydraulic Geometry 

 Relationships (English) 

At bankfull discharge conditions Q1 = 8000 cfs and the width of a sand-bed stream (Ds1 = 0.6 

mm) is W1 = 250 ft, the maximum flow depth is yo1 = 8 ft m, the slope is Sf1 = 2.5 x 10 -4, and 

the maximum velocity is V1 = 5 ft/s. 

(a) Estimate the width, W2, depth yo2, slope Sf2 and velocity V2 at the same station when the 

discharge Q2 is 200 cfs if the cross-sectional geometry is unknown. 

The at-a-station hydraulic geometry relationships (Table 5.3) can be used when no specific 

field data is available. 

For width: 

ft96

8000

200 250

Q

Q WW

26.0 26.0

1

2

12  = 

 



 = 













 =

For depth: 

ft8.1

40

1 8

Q

Q

yy

40.0 40.0

1

2

2 oo 1  = 

 



 = 













 =

Slope is unchanged: 

4

ff 10x5.2SS 1 2

− ==

For velocity: 

s/ft4.1

40

1 5

Q

Q VV

34.0 34.0

1

2

12  = 

 



 = 













 =

(b) Using the same station as for part (a), estimate the width, W2. depth yo2, slope Sf2 and 

velocity V2 in an upstream section of this stream if the bankfull discharge is 500 cfs and the 

bed material is gravel (D50 = 8 mm). How would the hydraulic geometry change if the bed 

material upstream is sand (D50 = 0.6 mm)? 

The "downstream" geometry relationships can be used in this case. Two types of relationships 

are given in Table 5.3. The sand bed relationships are a function of discharge only, whereas 

the gravel bed relationships are a function of both discharge and sediment size. Both methods 

are compared in the following. 
5.74 

Flow Depth

Sand bed: ft2.2

8000

500 8

Q

Q

yy

46.0 46.0

1b

2b

2 oo 1  = 

 



 = 













 =

Gravel bed: ft6.2

16

1 8

Q

Q

yy

40.0 4.0

1b

2b

2 oo 1  = 

 



 = 













 =

 

Channel width

Sand bed: ft70

16

1 250

Q

Q WW

46.0 46.0

1b

2b

12  = 

 



 = 













 =

Gravel bed: ft25

6.0

8

16

1 250

D

D

Q

Q WW

53.0 33.0 33.0

s

s

53.0

1b

2b

12

1

2

 = 

 



 



 



 = 





























 =

− −

Friction slope

Sand bed: 4

46.0

4

46.0

1b

2b

ff 10x9

16

1 10x5.2

Q

Q SS 2 1

−

−

−

−

 = 

 



 = 













 =

Gravel bed: 01.0

6.0

8

16

1 10x5.2

D

D

Q

Q SS

4.0

4

1

s

s

4.0

1b

2b

ff

1

2

2 1  =



 



 



 



 = 





























 =

−

−

−

Velocity

Sand bed: s/ft4

16

1 5

Q

Q VV

08.0 08.0

1b

2b

12  = 

 



 = 













 =

Gravel bed: s/ft7.9

6.0

8

16

1 5

D

D

Q

Q VV

07.0 33. 33.

s

s

07.0

1b

2b

12

1

2

 = 

 



 



 



 = 





























 =

The gravel bed relationships give a steeper, faster flowing and narrower channel as compared 

to the sand bed relationships. Unless the sediment size is markedly different for two streams, 

the resulting hydraulic geometry calculated from both sets of equations will be similar. 

5.9.7 PROBLEM 7 Downstream Sediment Size Distribution 

Measurements of sediment size in the St.-Lawrence Seaway between Cornwall and Valleyfield 

are given in the following table. 
5.75 

Table 5.9. Sediment Size Distribution in the St. Lawrence Seaway. 

Distance Downstream of Cornwall 

(km) (mi) 

D50 

(mm) 

0 0 28 

24.1 15 0.25 

40.2 25 0.018 

48.3 30 0.003 

56.3 35 0.001 

Estimate the mean sediment size D50 10 miles (16.1 km) and 20 miles (32.2 km) downstream 

of Cornwall. 

The gradual decrease in sediment size with downstream distance can be approximated by the 

following equation: 

(SI) x082.0

50 10x28D − =

(English) x132.0

50 10x28D − =

This equation was obtained by regression analysis based on Equation 5.13. At distances of 10 

and 20 miles (16.1 and 32.2 km), the expected mean sediment sizes D50 obtained by this 

relationship are, respectively, 1.34 mm and 0.064 mm. 

5.9.8 PROBLEM 8 Scale Ratios for Physical Models (SI) 

A physical model is to be built in the Hydraulics Laboratory to simulate the flow pattern around 

a structure in a complex multiple channel stream. About 334 m2

 of space (with a maximum 

length of 18.3 m) is available in the laboratory to model a 800 m reach. Knowing that the same 

fluid (water) will be used for both the model and the prototype, determine the appropriate scale 

ratios for time, discharge and force. Also, required is the flow depth in the model at the 

location where flow depth reaches 6 m in the prototype. 

A fixed boundary model will be used and open channel flow modeling is scaled by similarity in 

Froude number. The scale ratios for γ and ρ equal unity and thus, scaling depends uniquely on 

the length scale L = 18.3/(800) = 2.3 x 10-2. The following scale ratios for time, discharge and 

force are calculated from the expressions shown in Table 5.7. 

Parameter Scale Ratio

Time (Lρ/γ)

1/2 = L1/2 = 0.15 

Discharge L 5/2 (γ/ρ)

1/2 = L 5/2 = 8.0 x 10 -5

Force L3

γ = L 3

 = 1.2 x 10 -5

The flow depth of the model at h = 6.0 m is given by the product hL = 0.14 m. 
5.76 

5.9.9 PROBLEM 9 Scale Ratios for Physical Models (English) 

A physical model is to be built in the Hydraulics Laboratory to simulate the flow pattern around 

a structure in a complex multiple channel stream. About 3,600 ft2

 of space (with a maximum 

length of 60 ft) is available in the laboratory to model a 1/2-mile reach. Knowing that the same 

fluid (water) will be used for both the model and the prototype, determine the appropriate scale 

ratios for time, discharge and force. Also, required is the flow depth in the model at the 

location where flow depth reaches 20 ft in the prototype. 

A fixed boundary model will be used and open channel flow modeling is scaled by similarity in 

Froude number. The scale ratios for γ and ρ equal unity and thus, scaling depends uniquely on 

the length scale L = 60/(0.5 x 5280) = 2.3 x 10-2. The following scale ratios for time, discharge 

and force are calculated from the expressions shown in Table 5.7. 

Parameter Scale Ratio

Time (Lρ/γ)

1/2 = L1/2 = 0.15 

Discharge L5/2 (γ/ρ)

1/2 = L5/2 = 8.0 x 10 -5

Force L3

γ = L3

 = 1.2 x 10 -5

The flow depth of the model at h = 20 ft is given by the product hL = 0.45 ft. 
6.1 

CHAPTER 6 

RIVER STABILIZATION AND BANK PROTECTION 

6.1 OVERVIEW 

From a study of river morphology and river response (Chapter 5), it should be clear that both 

short-term and long-term changes can be expected on river systems as a result of natural and 

man-made influences. Recommended structures and design methods for river control are 

presented in this chapter. The integrated and interactive effects of these structures with the 

river are discussed in Chapter 9. 

Numerous types of river control and bank stabilization devices have evolved through past 

experience. Concrete, brick, willow, rock, and asphalt mattresses; sacked concrete and sand; 

riprap grouted slope protection; sheet and timber piles; steel jack and brush jetties; angled and 

sloped rock-filled, earth-filled, and timber dikes; automobile bodies; and concrete armor units 

have all been used in the practice of training, restoring, and stabilizing rivers. 

An early treatise on the subject of bank and shore protection was prepared by the California 

Division of Highways (1959). A large number of publications on river training and stabilization 

have been prepared by the U.S. Army Corps of Engineers (1981, 1994a,b) and the U.S. 

Bureau of Reclamation. Many more publications on the subject exist in the open literature. It 

is not intended that an exhaustive coverage of the various types of river control structures and 

methods of design be made in this manual; rather, the purpose of this manual is to recommend 

methods and devices which provide useful alternatives to the highway engineer for the majority 

of circumstances which are likely to be encountered in highway practice. A treatise of great 

interest in relation to highway crossings is the Report FHWA-RD-78-162 and 163 on 

countermeasures for hydraulic problems at bridges by Brice and Blodgett (1978). The 

interested reader is referred to these two volumes for an analysis and assessment (Vol. 1) and 

283 case histories (Vol. 2). 

The Federal Highway Administration has prepared Hydraulic Engineering Circular (HEC) No. 

23 (Lagasse et al. 2001) to provide experience, selection, and design guidelines for a wide 

range of stream instability and bridge scour countermeasures. When used in combination with 

HEC-20 (Lagasse et al. 2001) and HEC-18 (Richardson and Davis 2001), these three 

publications provide a comprehensive integrated approach to analyzing stream instability and 

bridge scour problems and selecting and designing countermeasures for specific problems. 

Generally, changes to river alignment, river cross section, training, and bank stabilization of 

rivers associated with highway projects are confined to short reaches of the river. While the 

methods for river training and bank stabilization discussed herein are applicable to short and 

long reaches of the river, they are not a panacea to all problems associated with highway 

encroachments on rivers. An understanding of river system dynamics is essential to selection, 

design and successful installation of river stabilization, restoration, and bank protection works. 

It must also be recognized that the solution to a particular problem may generate problems 

elsewhere in the river system. 
6.2 

6.2 STREAM BANK EROSION 

The erosion, instability, and/or retreat of a stream bank is dependent on the processes 

responsible for the erosion of material from the bank and the mechanisms of failure resulting 

from the instability created by those processes. Bank retreat is often a combination of these 

processes and mechanisms varying at seasonal and sub-seasonal time scales. Changes in 

channel geometry with time are particularly significant during periods when alluvial channels 

are subjected to high flows. The converse situation exists during relatively dry periods. Erosive 

forces during high flow periods may have a capacity approximately 100 times greater than 

those forces acting during periods of intermediate and low flow. In most instances when 

considering the instability of alluvial rivers, it can be shown that approximately 90 percent of all 

river changes occur during the small percentage of the time when the discharge exceeds the 

dominant discharge. 

Regardless of the fact that the majority of bank changes occur during comparatively short time 

periods, there may also be regions within a river in which some degree of instability is exhibited 

for all flow conditions. Raw banks may develop on the outside of bends as a consequence of 

direct impingement of the flowing water. Sloughing banks may occur as a result of seepage 

and other secondary forces created by water draining back through the banks into the river. 

Continuous wave action, generated either naturally or by human activities, may also precipitate 

erosion problems. 

6.2.1 Causes of Stream Bank Failure 

Bank retreat processes may be grouped into three categories: weakening and weathering 

processes, direct fluvial entrainment, and mass failure. HEC-20 (Lagasse et al. 2001) provides 

more detail on these processes. The impact of these processes on bank retreat is dependent 

on site characteristics, especially near-bank hydraulic fields, bank height, and the geotechnical 

properties of the bank material. Table 1 lists factors affecting bank erosion. 

Table 6.1. Factors Affecting Erosion of River Banks. 

A. Hydraulic 

 Factors 

B. Geomorphic 

 Factors 

C. Human 

 Factors 

D. Biological 

 Factors 

E. Climatic 

 Factors 

F. Other 

Fluid Properties River Planform Agricultural Activities Vegetation Freezing Subsurface Flows 

 Specific Weight Meandering Mining Trees Ice Thickness Seepage Forces 

 Viscosity (Temp) Straight Dams Shrubs Duration Piping 

Flow Characteristics Braided Navigation Grass Frequency Waves 

 Discharge Anabranched Transportation Animals Thawing Wind 

 Magnitude Bed and Bank Material Urbanization Domestic Permafrost Boats 

 Duration Size Drainage Wild Precipitation 

 Frequency Gradation Floodplain Develop. 

 Velocity Shape Recreational Boating 

 Velocity Distrib. Specific Weight Commercial Boating 

 Turbulence Fall Velocity 

 Shear Stress Bank Characteristics 

 Drag/Lift Forces Cohesive 

 Momentum Noncohesive 

 Energy Stratified 

 Height 
6.3 

6.2.2 Bed and Bank Material 

Resistance of a river bank to erosion is closely related to several characteristics of the bank 

material. Bank material deposited in the river can be broadly classified as cohesive, 

noncohesive, and composite. Failure of banks for various situations is shown in Figure 6.1. 

Cohesive bank material is more resistant to surface erosion and has low permeability which 

reduces the effects of seepage, piping, frost heaving, and subsurface flow on the stability of 

the banks. However, such banks when undercut and/or saturated are more likely to fail due to 

mass wasting processes such as sliding. 

Noncohesive bank material tends to be removed grain by grain from the bank line. The rate of 

particle removal, and hence the rate of bank erosion, is affected by factors such as the 

direction and magnitude of the velocity adjacent to the bank, the turbulent fluctuations, the 

magnitude and fluctuations in the shear stress exerted on the banks, seepage force, piping 

and wave forces, many of which may act concurrently. 

Composite or stratified banks are very common on alluvial rivers and generally are the product 

of past transport and deposition of sediment by the river. More specifically, these types of 

banks consist of layers of materials of various sizes, permeability, and cohesion. The layers of 

noncohesive material are subject to surface erosion, but may be partly protected by adjacent 

layers of cohesive material. This type of bank is also vulnerable to erosion and sliding as a 

consequence of subsurface flows and piping. 

6.2.3 Subsurface Flow 

With flow of water from the river into the adjacent banks, a stabilizing seepage force is 

generated. Rivers that continuously seep water into the banks tend to have smaller widths and 

larger depths for a particular discharge. The reverse is true of the rivers that continuously gain 

water by an inflow through their banks. The inflowing water creates a seepage force that 

makes the banks less stable. The movement of water through the bank material can be 

attributed to various factors. 

If the water table is higher than river stage, flow will be from the banks into the river. The high 

water table may result from: (1) a wet period during which water draining from tributary 

watersheds saturates the floodplain to a higher level; (2) poor drainage conditions resulting 

from deterioration or failure of surface drainage systems; (3) increased infiltration resulting 

from changes in land use causing an increase in water level; (4) irrigated floodplains; and (5) 

development of the adjacent floodplain for homes and businesses that utilize septic tanks and 

leach fields to dispose of waste water and sewage. 

With a rise in river stage an outward gradient is developed that induces flow into the banks. 

This can be caused by: (1) the storage and release of water for pumped storage hydropower 

generation which causes numerous fluctuations in river stage; (2) boat and wind waves which 

cause local variations in stage that introduce inflow and outflow of water from the banks; (3) 

predominately dry and semi-arid channels subject to intermittent floods. However, because the 

duration of the change in stage is small, the inflow and outflow phenomena are usually 

concentrated locally in the surface of the banks; and (3) the formation and loss of backwater 

caused by ice flows and ice jams which lead to both seepage into and out of the banks. 

Frequent stage fluctuations, such as may occur with hydropower operations, may exacerbate 

the bank erosion process. 
6.4 

Figure 6.1. Typical bank failure surfaces (a) noncohesive, (b) cohesive, (c) composite 

 (after Brown 1985a,b,c). 
6.5 

The presence of water in the banks of rivers and its movement toward or away from the river 

affect bank stability and bank erosion in various ways. The related erosion of banks is a 

consequence of seepage forces, piping, and mass wasting. 

6.2.4 Piping of River Banks 

Piping is another phenomenon common to the alluvial banks of rivers. With stratified banks, 

i.e., lenses of sand and coarser material sandwiched between a layer of finer cohesive 

materials, flow is induced in more permeable layers by changes in river stage and by wind-and 

boat-generated waves. If the flow through the permeable lenses is capable of dislodging and 

transporting particles from the permeable lenses, the material is slowly removed, undermining 

portions of the bank. Without this foundation material to support the overlying layers, a block 

of bank material drops down and results in the development of tension cracks sketched in 

Figure 6.1c. These cracks allow surface flows to enter, further reducing the stability of the 

affected block of bank material. Bank erosion may continue on a grain-by-grain basis or the 

block of bank material may ultimately slide downward and outward into the channel, causing 

bank failure as a result of a combination of seepage forces, piping, and mass wasting. 

6.2.5 Mass Wasting 

An alternative form of bank erosion is caused by local mass wasting. If the bank becomes 

saturated and possibly undercut by flowing water, blocks of the bank may slump or slide into 

the channel. Mass wasting may be further aggravated by construction of homes on river 

banks, operation of equipment on the floodplain adjacent to the banks, added gravitational 

force resulting from tree growth, location of roads that cause unfavorable drainage conditions, 

saturation of banks by leach fields from septic tanks, and increased infiltration of water into the 

floodplain as a result of changing land-use practices. 

Landslides, the downslope movement of earth and organic materials, result from an imbalance 

of forces. Various forces are involved in mass wasting. These forces are associated with the 

downslope gravity component of the slope mass. Resisting these downslope forces are the 

shear strength of the earth's materials and any additional contributions from vegetation via root 

strength or engineered slope reinforcement activities. When a slope is acted upon by a stream 

or river, an additional set of forces is added. These forces are associated with removal of 

material from the toe of the slope, fluctuations in groundwater levels, and vibration of the slope. 

 A slope may fail if stable material is removed from the toe. When the toe of a slope is 

removed, the slope loses more resistance by buttressing than it does by downslope 

gravitational forces. The slope materials may then tend to move downward into the void in 

order to establish a new balance of forces or equilibrium. Often this equilibrium is a slope 

configuration with less than original surface gradient. The toe of the failed mass can provide a 

new buttress against further movements. However, if this buttress is removed by stream 

erosion, the force equilibrium may again be upset. For slope toes acted upon by erosive 

stream water, the continual removal of toe material can upset the force balance. 
6.6 

6.3 RIVER TRAINING AND STABILIZATION 

Various devices and structures have been developed to control river flow along a preselected 

path and to stabilize the banks. Most have been developed through trial and error applications, 

aided in some instances by hydraulic model studies. Specific functions of bank protection and 

training works in relation to bridges and their approaches include: (1) stabilize eroding river 

banks and channel location in the case of shifting streams; (2) economize on bridge lengths by 

constricting the natural waterway; (3) direct flow parallel to piers and thereby minimize local 

scour; (4) improve the hydraulic efficiency of a waterway opening, thereby reducing backwater 

and scour and facilitating passage of ice and debris; (5) protect road approaches from stream 

attack and prevent meanders from migrating into the approaches; (6) permit construction of a 

well-aligned bridge crossing by diverting the channel from a skewed alignment; (7) reduce the 

overall cost of a road project by diverting the channel away from the base of a valley slope, 

thereby allowing a reduction in bridge length and height; (8) secure existing works, or to repair 

damage and improve initial designs; and (9) protect longitudinal encroachments. 

A comprehensive bank stabilization and channel rectification program to control a river reach 

completely normally requires extensive work on concave banks in bends, minor work on 

convex bars, and control work on both banks through bridge crossings. 

To minimize attack by the stream on stabilization and rectification structures, the river is 

shaped to an alignment consisting of a series of easy bends, with the flow directed from one 

bend into the next bend downstream in such a way as to maintain a direction essentially 

parallel to the channel flow line (see Section 5.8.4). Straight reaches and reaches of very small 

curvature should be avoided, insofar as practicable, because there is a tendency for flows to 

shift from side to side in such reaches. The optimum bend radius approximates that of 

relatively stable bends in the general river reach. 

6.3.1 Fixed Points 

One of the essential requirements in designing a system of stabilization works is that 

construction starts at a stable, fixed point on the bank and continues downstream to another 

stable location or to some point below which the river can safely be left uncontrolled. 

Construction of relatively short isolated stabilization work has often proved unsuccessful 

because eventual changes in the direction of flow inherent in bank caving in the upstream 

uncontrolled reach either will set up a direct attack against the isolated protective work and 

severely damage or destroy it, or will shift the attack to some other nearby reach of bank, 

requiring additional work and possible abandonment of the original work. 

Revetments should be constructed on a smooth alignment, with no irregularities, in order to 

avoid eddies set up by disturbances to the flow that can lead to local scour and subsequent 

undermining of the revetment. 

6.3.2 Radius of Curvature

The most appropriate radius of curvature for rectification and stabilization varies from river to 

river and from reach to reach for a given river. It must be determined on the basis of relatively 

stable natural bends for each stream (see Section 5.8.3). 
6.7 

The shorter the radius of curvature of a bend, the deeper the channel will be adjacent to the 

concave bank. The deeper the channel is, the greater the possibility of undermining bank 

protection work in the bend and the greater the cost of maintaining the structure. Therefore, 

sharp curvature of bends should be avoided to obtain the most economical control of the river. 

6.3.3 Countermeasures for Channel Instability

Countermeasures can be used to control both lateral and vertical channel instability and 

include river training structures and revetment armoring. River training structures are those 

which modify the flow (flow control). River training structures are distinctive in that they alter 

hydraulics to mitigate undesirable erosional and/or depositional conditions at a particular 

location or in a river reach. River training structures can be constructed of various material 

types and are not distinguished by their construction material, but rather, but their orientation to 

flow. River training structures are described as transverse, longitudinal or areal depending on 

their orientation to the stream flow. 

To protect against lateral channel instability, flow control structures are used to: 

• Direct flow from one bend into the next bend downstream 

• Flair out sharp bends to a larger radius of curvature to provide a more desirable channel 

alignment 

• Close off secondary channels and old bendways 

• Concentrate flow on a limited width within a wider channel 

To protect against vertical channel instability, flow control structures are used for: 

• Limiting or halting long-term degradation 

• Establishing a desired channel bed elevation in a bridge reach 

• Arresting the migration of a head cut or nickpoint through a bridge reach 

 

Revetments are structures parallel to the current and are used to armor the channel bank from 

erosive/hydraulic forces. They are usually applied in a blanket type fashion for areal coverage. 

Revetments can be classified as either rigid or flexible/articulating. Rigid revetments are 

typically impermeable and do not have the ability to conform to changes in the supporting 

surface. These countermeasures often fail due to undermining. Flexible/ articulating 

revetments can conform to changes in the supporting surface and adjust to settlement. These 

countermeasures often fail by removal and displacement of the armor material. 

Rock riprap is probably the most widely used revetment material to stabilize river banks and 

protect the side slopes of embankments and river training devices. In the final report to 

Congress, the U.S. Army Corps of Engineers (1981) concluded that rock will likely continue to 

be the first choice of bank protection materials where material of sufficient size is available and 

affordable, because of durability, and other advantages. Because of its wide use and 
6.8 

importance in highway practice, a separate section (Section 6.6) is devoted to design and 

placement of rock riprap. 

6.3.4 Countermeasure Design Guidelines

Hydraulic Engineering Circular (HEC) No. 23 (Lagasse et al. 2001) provides experience, 

selection, and design guidance for a wide range of stream instability and bridge scour 

countermeasures, including river training devices and revetment armoring. 

HEC-23 is organized to: 

• Highlight the various groups of countermeasures and identify their individual 

characteristics. 

• For specific countermeasures, list information on their functional applicability to a particular 

problem, their suitability to specific river environments, the general level of maintenance 

resources required, and which State Highway Agencies (SHAs) have experience with 

specific countermeasures. 

• Provide general criteria for selection of countermeasures for bridge scour and stream 

instability problems. 

• Discuss countermeasure design concepts including design approach, hydraulic analysis, 

environmental permitting, special design considerations related to riprap, filters, and edge 

treatment, and biotechnical engineering approaches. 

• Provide detailed design guidelines for specific countermeasures. 

Specific design guidelines are provided in HEC-23 for the following river training devices: 

• Bendway Weirs/Stream Banks 

• Spurs 

• Guide Banks 

• Check Dams/Drop Structures 

Design guidelines are also provided for the following armoring (revetment) countermeasures. 

• Revetment (summary of HEC-11 guidance) 

• Soil Cement 

• Wire Enclosed Riprap Mattress 

• Articulating Concrete Block Systems 

• Articulating Grout Filled Mattress 

Hydraulic Engineering Circular (HEC) No. 11 (Brown and Clyde 1989) is a comprehensive 

design manual for riprap revetment. HEC-11 includes discussions on erosion potential, 

erosion mechanisms and riprap failure modes, as well as riprap types including rock riprap, 

rubble riprap, gabions, preformed blocks, grouted rock, and paved linings. Design concepts 

included in HEC-11 are: design discharge, flow types, channel geometry, flow resistance, 

extent of protection, and toe depth. Detailed design guidelines are presented for rock riprap, 
6.9 

and design procedures are summarized in charts and examples. Design guidance is also 

presented for: 

• Wire-enclosed rock (gabions) 

• Precast concrete blocks 

• Concrete paved linings 

The following section (6.4) summarizes design concepts for selected flow control (river training) 

structures. Section 6.5 deals with riprap design and placement, and Section 6.6 describes 

design for revetment types other than riprap. General filter design concepts are discussed in 

Section 6.7 and issues related to overtopping flows on embankments are summarized in 

Section 6.8. Environmental considerations for stream bank protection are presented in Section 

6.9. 

6.3.5 Protection of Training Works 

Granular or geosynthetic filters are essential to the performance of hydraulic countermeasures, especially armoring countermeasures such as bankline revetment or armoring 

used as protection for river training works such as spurs and guide banks. Filters prevent 

soil erosion beneath the armoring material, prevent migration of fine soil particles through 

voids in the armoring material, distribute the weight of the armor units to provide a more 

uniform settlement, and permit relief of hydrostatic pressure within the soils. Experience has 

indicated that the proper design of filters is critical to the stability of revetments. If openings 

in the filter material are too large, excessive piping through the filter can result in erosion of 

the subgrade beneath the armor. Conversely, if openings in the filter are too small, 

hydrostatic pressures can build up in the underlying soil and result in failure of the 

countermeasure. Guidelines for the selection, design, and specifications of filter material 

can be found in HEC-11 (Brown and Clyde 1989), HEC-23 (Lagasse et al. 2001), and 

Section 6.7. 

Undermining of the edges of armoring countermeasures is another of the primary 

mechanisms of failure. The edges of the armoring material (head, toe, and flanks) should 

be designed so that undermining will not occur. For revetment slope protection, this is 

achieved by trenching the toe of the revetment below the channel bed to a depth which 

extends below the combined expected contraction scour and long-term degradation depth. 

When excavation to the contraction scour and degradation depth is impractical, a launching 

apron can be used to provide enough volume of rock to launch into the channel while 

maintaining sufficient protection of the exposed portion of the bank. Continuous systems, 

such as articulating concrete block systems and grout filled mattresses applied on side 

slopes, should be designed with an apron or toe trench so that the system provides 

protection below the combined expected contraction scour and long-term degradation depth. 

Tension anchors may be used to increase stability at the edges of these continuous 

systems. Additional guidelines on edge treatment for armoring countermeasures can be 

found in HEC-11 and HEC-23. 

Variations in bed elevation during flow events or after bank hardening can result in the 

undermining of bank protection structures including longitudinal structures. Deep sections at 

the toe of the outer bank of a bendway are the result of scour. As discussed in Section 

5.4.3, high velocity along the outer bank is caused by secondary currents and greater outerbank depths, and together with the resultant shear stress, produce scour and cause a 
6.10 

difference between the sediment load entering and exiting the outer-bank zone. Since 

secondary currents transport sediment supplied, in large part, from outer bank erosion 

toward the inner bank of a bend, hardening of the outer bank by longitudinal bank protection 

structures may cause the channel cross section to narrow and deepen by preventing the 

recruitment of eroded outer bank sediments. 

Experience is usually the most reliable means of estimating scour depth when designing a 

bank protection project for a particular stream. Lacking experience on a particular stream, 

scour depths may be estimated using physically based analytical models or empirical 

methods. Although scour-depth can be estimated analytically or empirically, empirical 

methods were generally found to provide better agreement with observed data. 

Maynord (1996) provides an empirical method for determining scour depths on a typical 

bendway bank protection project. Although his studies are restricted to sand bed streams, 

the Maynord method agrees reasonably well with the limited number of gravel-bed data 

points obtained by Thorne and Abt (1993). Nonetheless, the techniques presented by 

Maynord are restricted to meandering channels having naturally developed widths and 

depths, and cannot be applied to channels that have been confined to widths significantly 

less than a natural system. 

HEC-23 provides application guidelines for Maynord's approach to estimating toe protection 

requirements on stabilized bendways. HEC-23 also contains guidance for estimating scour 

at vertical wall structures (e.g., retards and bulkheads) from flow parallel to and impinging on 

the wall. 

6.4 FLOW CONTROL STRUCTURES 

A flow control structure is defined here as a structure, either within or outside a channel that 

acts as a countermeasure by controlling the direction, velocity, or depth of flowing water. 

Structures within this category are sometimes called "river training works". Among the most 

important properties of a flow control structure is its degree of permeability. An impermeable 

structure may deflect a current entirely, whereas a permeable structure may serve mainly to 

reduce water velocity. As used here, the term "permeable" means that a structure has definite 

openings through which water is intended to pass, such as openings between adjacent boards 

or pilings, or the meshes of wire. Structures made of riprap, or filled with riprap, have some 

degree of permeability, but these are classed as impermeable because they act essentially as 

impermeable barriers to a rapidly moving current of water. 

Types of flow control structures are distinguished on Figure 6.2. 

6.4.1 Spurs 

A spur is a structure or embankment projected into a stream from the bank at some angle and 

for a short distance to deflect flowing water away from critical zones, to prevent erosion of the 

bank, and to establish a more desirable channel alignment or width. By deflecting the current 

from the bank and causing sediment deposits behind them, a spur or a series of spurs may 

protect the stream bank more effectively and at less cost than revetment riprap applied directly 

on the bank. Also, by moving the location of any scour away from the bank, failure of the 

riprap on the spur can often be repaired before damage is done to structures along and across 

the river. Conversely, failure of riprap on the bank may immediately endanger adjacent 

structures. 
6.11 

Figure 6.2. Placement of flow control structures relative to channel banks, crossing, and 

 floodplain. Spurs, retards, dikes, and jack fields may be either upstream or 

 downstream from the bridge (from Brice and Blodgett 1978). 

Spurs are also used to protect highway embankments that form the approaches to a bridge 

crossing. Often these highway embankments cut off the overbank flood flows causing these 

flows to run parallel to the embankment enroute to the bridge opening. Spurs constructed 

perpendicular to the highway embankment keep the potentially erosive current away from the 

embankment, thus protecting it. Spurs as used in this report encompass the terms dikes, 

jetties, and groins, which are also used to describe these structures. 

Spurs are also used to channelize a wide, poorly defined stream into a well-defined channel 

that neither aggrades nor degrades, thus maintaining its location from year to year. Spurs on 

streams with suspended sediment discharge can cause deposition to establish and maintain 

the new alignment. The use of spurs in this instance may decrease the length necessary for 

the bridge opening and may make a more suitable, stable channel approach to the bridge. 

This decreases the cost of the bridge structure. 

Recommendations for spur design from Brown (1985) are summarized in HEC-23 (Lagasse et 

al. 2001). The major considerations are: 

• Extent of channel bank protection 

• Spur length 

• Spur spacing 

• Spur angle/orientation 

• Spur height 

• Spur crest profile 

• Channel bed and channel bank contact 

• Spur head form 
6.12 

6.4.2 Bendway Weirs 

Bendway weirs, also referred to as stream barbs, bank barbs, and reverse sills, are low 

elevation stone sills used to improve lateral stream stability and flow alignment problems at 

river bends and highway crossings. Bendway weirs are used for improving inadequate 

navigation channel width at bends on large navigable rivers. They are used more often for 

bankline protection on streams and smaller rivers. Design guidelines for bendway weirs are 

presented in HEC-23 (Lagasse et al. 2001). 

6.4.3 Hardpoints 

Hardpoints are an erosion control technique consisting of stone fills spaced along an eroding 

bank line (Figure 6.3). The structures protrude only short distances into the river channel and 

are supplemented with a root section extending landward into the bank to preclude flanking, 

should excessive erosion persist. The majority of the structure cannot be seen as the lower 

part consists of rock placed underwater, and the upper part is covered with topsoil and seeded 

with native vegetation. The structures are especially adaptable in long, straight reaches not 

subject to direct attack. 

Figure 6.3. Perspective of hard point with section detail (after Brown 1985a,b,c). 

6.4.4 Retards 

Retards are devices placed parallel to embankments and river banks to decrease the stream 

velocities and prevent erosion (Figures 6.2 and 6.4). 

Pile retards can be made of concrete, steel or timber. The design of timber pile retards is 

essentially the same as timber pile dikes shown in Figure 6.6. They may be used in 

combination with bank protection works such as riprap. The retard then serves to reduce the 

velocities sufficiently so that either smaller riprap can be used, or riprap can be eliminated. 
6.13 

Figure 6.4. Retard (after Brown 1985a, b, c). 

Timber and concrete cribs are sometimes used for bulkheads and retaining walls to hold 

highway embankments, particularly where lateral encroachment into the river must be limited. 

Cribs are made up by interlocking pieces together in the manner shown in Figure 6.5. The crib 

may be slanted or vertical depending on height and the crib is filled with rock or earth. 

Reinforced concrete retaining walls are alternatives to timber cribs which can be considered. 

However, concrete retaining walls are expensive and are generally only used in special 

confined locations where space precludes other methods of bank protection. In constructing 

concrete retaining walls, drainage holes (weep holes) must be provided. The foundation of 

these walls should be placed below expected scour depths. 

Figure 6.5. Concrete or timber cribs (after Brown 1985a, b, c). 
6.14 

6.4.5 Dikes (Floodplain) 

A floodplain dike is an impermeable linear structure for the control or containment of overbank 

flow. Most dikes are on floodplains (Figure 6.2) but in some situations (as on wide braided 

rivers or on alluvial fans) they may be within channels (see Section 6.4.6). Floodplain dikes are 

used to prevent flood water from bypassing a bridge, or to confine channel width and maintain 

channel alinement. Some dikes extend upstream from one or both sides of the bridge 

opening. These are similar in function to guide banks, but are usually much longer and may 

extend to the valley side. Such dikes are commonly called "training dikes." The use of fencetype structures for training dikes is mostly restricted to arid and semi-arid regions (Brice et al. 

1978). 

6.4.6 Dikes (Channel) 

Both permeable and impermeable dikes are installed in channels. Permeable dikes are those 

which permit flow through the dike but at reduced velocities, thereby preventing further erosion 

of the banks and causing deposition of suspended sediment from the flow. 

Timber or steel pile dikes (also retards) may consist of closely-spaced single, double, or 

multiple rows. There are a number of variations to this scheme. For example, wire fence may 

be used in conjunction with pile dikes to collect debris and thereby cause effective reduction of 

velocity. Double rows of piles can be placed together to form cribs, and rocks may be used to 

fill the space between the piles. Pile dikes are vulnerable to failure through scour. This can be 

overcome if the piles can be driven to a large depth to achieve safety from scour, or the base 

of the piles can be protected from scour with dumped rock in sufficient quantities. The various 

forms of pile dikes are illustrated in Figure 6.6. 

The arrangement of piles depends upon the velocity of flow, quantity of suspended sediment 

transport, and depth and width of the river. If the velocity of flow is large, pile dikes are not 

likely to be very effective. Stabilization of the bank by other methods should be considered. 

On the other hand, in moderate flow velocities with high concentrations of suspended 

sediments, these dikes can be quite effective. Deposition of suspended sediments in the pile 

dike field is a necessary consequence of reduced velocities. If there is not sufficient 

concentration of suspended sediment in the flow, or the velocities in the dike fields are too 

large for deposition, the permeable pile dikes will only partially be effective in training the river 

and protecting the bends. 

The length of each dike depends on channel width, position relative to other dikes, flow depth 

and available pile lengths. Generally, pile dikes are not used in large rivers where depths are 

great, although timber pile dikes have been used in the Columbia River. On the other hand, 

banks of wide shallow rivers can be successfully protected with dikes. The spacing between 

dikes varies from 3 to 20 times the length of the upstream dike, with closer spacing favored for 

best results. 

Vane dikes are low-elevation structures designed to guide the flow away from an eroding bank 

line (Figure 6.7). The structures can be constructed of rock or other erosion-resistant material, 

the tops of which are constructed below the design water surface elevation and would not 

connect to the high bank. Water would be free to pass over or around the structure with the 

main thread of flow directed away from the eroding bank. The structures will discourage high 

erosive velocities next to an unprotected bank line, encourage diversity of various channel 

depths, and protect existing natural bottomland characteristics. The findings from a model 

investigation of these structures include the effects of various vane dike orientation, vane dike 

length, and gap length U.S. Army Corps of Engineers, 1981. 
6.15 

Figure 6.6. Pile dikes (retards would be similar) (after Brown 1985a, b, c). 

Figure 6.7. Vane dike model, ground walnut shell bed, during low stage portion of test run 

 (U.S. Army Corps of Engineers 1981). 
6.16 

6.4.7 Jetties 

The purpose of a jetty field is to add roughness to a channel or overbank area to train the main 

stream along a selected path. The added roughness along the bank reduces the velocity and 

protects the bank from erosion. Jetty fields are usually made up of steel jacks tied together 

with cables. Both lateral and longitudinal rows of jacks are used to make up the jetty field as 

shown in Figure 6.8. 

Figure 6.8. Typical jetty-field layout (after Brown 1985a, b, c). 

The lateral rows are usually angled about 45 to 70 degrees downstream from the bank. The 

spacing varies, depending upon the debris and sediment content in the stream, and may be 15 

to 75 m (50 to 250 ft) apart. Jetty fields are effective only if there is a significant amount of 

debris carried by the stream and the suspended sediment concentration is high. 

When jetty fields are used to stabilize meandering rivers, it may be necessary to use jetty fields 

on both sides of the river channel because in flood stage the river may otherwise develop a 

chute channel across the point bar. A typical layout is shown in Figure 6.8. 

Steel jacks are devices with basic triangular frames tied together to form a stable unit. The 

resulting framework is called a tetrahedron. The tetrahedrons are placed parallel to the 

embankment and cabled together with the ends of the cables anchored to the bank. Wire 

fencing may be placed along the row of tetrahedrons. In order to function well, there must be 

considerable debris in the stream to collect on the fence and the suspended sediment 

concentration must be large so that there will be deposition behind the retard. Various forms of 

steel jacks may be assembled. Two types are shown in Figure 6.9. Tiebacks should be 

spaced every 30 m (l00 ft) and space between jacks should not be greater than their width. 
6.17 

Figure 6.9. Steel jacks (after Brown 1985a, b, c). 

6.4.8 Bulkheads 

Bulkheads can be used to prevent stream bank erosion or failure. As an additional benefit, a 

bulkhead may provide a substantial increase in waterfront area and an improvement in 

water/land access. Concrete, steel, timber and more recently, aluminum, corrugated asbestos, 

and used tires have been used to construct bulkheads. Concrete and steel bulkheads 

generally cost at least four time as much as a comparable bulkhead of another material; 

however, the service life is longer and less maintenance is required. Timber is the most 

commonly available material for economical bulkhead construction. 

Timber bulkhead construction is similar to common fence construction except that a few 

precautions should be observed: 

• All wood should be treated with preservative to minimize deterioration due to repetitive 

wetting and drying or insect activity. 

• The toe of the bulkhead should always be protected with riprap. The most common cause 

of bulkhead failure is scour around the pilings, followed by the structure tipping over due to 

the pressure of the bank behind the bulkhead. 

• Piles should be anchored to deadmen buried in the bank. 

• Fill material placed between the bulkhead and natural bank should be free draining so that 

the soil behind the bulkhead will not become saturated and push the structure over. 

• If there are no cracks between the planks, weepholes should be drilled in the fence at 

regular intervals to allow the bank to drain. Filter fabric or gravel can be placed as a filter 

behind openings in the fence to prevent fine soils from leaching through. A filter must be 

properly designed to match the filter with the soil. 

• The bulkhead should be tied into the bank at the upstream and downstream end of the 

structure to prevent flow behind the bulkhead. 
6.18 

6.4.9 Fencing 

Fencing can be used as a low-cost bank protection technique on small to medium size 

streams. Special structural design considerations are required in areas subject to ice and 

floating debris. Both longitudinal (parallel to stream) fence retards and transverse 

(perpendicular to stream) fences have been used in the prototype with varying degrees of 

success. A model investigation and literature review of longitudinal fence retards with tiebacks 

were conducted to identify the following important design considerations: 

• Channel gradient must be stable and not be steep (tranquil flow) 

• Toe scour protection can be provided by extending the support posts well below the 

maximum scour expected or by placing loose rock at the base of the fence to launch 

downward if scour occurs at the toe 

• Tiebacks to the bank are important to prevent flanking of the fence and to promote 

deposition behind the fence 

• Fence retards generally reduce attack on the bank so that vegetation can establish 

• Metal or concrete fences are preferred due to ice damage and fire loss of wooden fences 

6.4.10 Guide Banks 

Guide banks are placed at or near the ends of approach embankments to guide the stream 

through the bridge opening. Constructed properly, flow disturbances, such as eddies and 

cross-flow, will be minimized to make a more efficient waterway under the bridge. They are 

also used to protect the highway embankment and reduce or eliminate local scour at the 

embankment and adjacent piers. The effectiveness of guide banks is a function of river 

geometry, quantity of flow on the floodplain, and size of bridge opening. A typical guide bank 

at the end of an embankment is shown in Figure 6.10. 

Figure 6.10. Typical guide bank. 
6.19 

The recommended shape of a guide bank is a quarter ellipse with a major to minor axis ratio of 

2.5. The major axis should be approximately parallel to the main flow direction. For bridge 

crossings normal to the river, the major axis would be normal to the highway embankment. 

However, for skewed crossings, the guide bank should be placed at an angle with respect to 

the embankment with the view of streamlining the flow through the bridge opening. Design 

guidelines and a design chart for guide banks are provided in HEC-23 (Lagasse et al. 2001). 

The length of the guide bank, Ls, required depends upon quantity of flow on the floodplain, 

width of bridge opening and skewness of the highway crossing. Shorter guide banks may be 

used where floodplain flow is small, scour potential at piers and embankment ends is small, or 

where trees or brush are intersected by the guide bank. 

The crest elevation of guide banks should be higher than the elevation of the design flood 

taking into consideration the effect of the contraction of the flow; this is because the design 

flow should not overtop the guide bank. 

6.4.11 Drop Structures 

Check dams or channel drop structures are used downstream of highway crossings to arrest 

head cutting and maintain a stable streambed elevation in the vicinity of the bridge. Check 

dams are usually built of rock riprap, concrete, sheet piles, gabions, or treated timber piles. 

The material used to construct the structure depends on the availability of materials, the height 

of drop required, and the width of the channel. Definition sketches for a vertical wall and a 

sloping sill drop structure are shown in Figures 6.11 and 6.12. Design considerations for 

vertical wall or sloping sill structures are given in texts by Rouse (1950), Chow (1959), 

Peterson (1986), and Simons and Senturk (1992). Design guidelines for a vertical drop 

structure and stilling basins for drop structures are given in HEC-23 (Lagasse et al. 2001). 

Figure 6.11. Definition sketch for a vertical drop (Lagasse et al. 2001). 
6.20 

Figure 6.12. Flow and scour patterns at a sloping sill (after Laursen and Flick 1983). 

6.5 RIPRAP DESIGN AND PLACEMENT 

6.5.1 Factors to Consider 

When available in sufficient size, rock riprap is usually the most economical material for bank 

protection. Rock riprap has many other advantages over other types of protection. A riprap 

blanket is flexible and is neither impaired nor weakened by slight movement of the bank 

resulting from settlement or other minor adjustments. Local damage or loss is easily repaired 

by the placement of more rock. Construction is not complicated so for many applications 

special equipment or construction practice is not necessary. Riprap is usually durable and 

recoverable and may be stockpiled for future use. The cost-effectiveness of locally available 

riprap provides a viable alternative to many other types of bank protection. Riprap stability 

increases with increasing thickness as more material is available to move to damaged areas 

and more energy is dissipated before it reaches the filter and stream bank. Although riprap 

must be placed to the proper level below the bed, there are no special foundation 

requirements. The appearance of rock riprap is natural and after a period of time vegetation 

will grow between the rocks. Wave runup on rock slopes is usually less than on other types of 

bank protection. 

The important factors to be considered in designing rock riprap bank protection are: 

• Durability of the rock 

• Density of the rock 

• Velocity (both magnitude and direction) of the flow in the vicinity of the rock 

• Slope of the bed and bankline being protected 

• Angle of repose for the rock 

• Shape and angularity of the rock 

• What shape and weight of stones will be stable in the streamflow 

• What blanket thickness is required 

• Is a filter needed between the bank and the blanket to allow seepage but to prevent 

erosion of bank soil through the blanket 
6.21 

• How will the blanket be stabilized at the toe of the bank 

• How will the blanket be tied into the bank at its upstream and downstream ends 

6.5.2 Stability Factor Design Methods 

Stability Factors For Riprap. In the absence of waves and seepage, the stability of rock riprap 

particles on a side slope is a function of: (1) the magnitude and direction of the stream velocity 

in the vicinity of the particles; (2) the angle of the side slope; and (3) the characteristics of the 

rock including the geometry, angularity and density. The functional relations between the 

variables are developed below. This development closely follows that given by Stevens and 

Simons (1971). 

Consider flow along an embankment as shown in Figure 6.13. The fluid forces on a rock 

particle identified as P in Figure 6.13a result primarily from fluid pressure around the surface of 

the particles. The lift force F is defined herein as the fluid force normal to the plane of the 

embankment. The lift force is zero when the fluid velocity is zero. The drag force Fd is defined 

as the fluid force acting on the particle in the direction of the velocity field in the vicinity of the 

particle. The drag force is normal to the lift force and is zero when the fluid velocity is zero. 

The remaining force is the submerged weight of the rock particle Ws. 

Figure 6.13. Diagram for riprap stability conditions. 

Rock particles on side slopes tend to roll rather than slide, so it is appropriate to consider the 

stability of rock particles in terms of moments about the point of rotation. In Figure 6.13b the 

direction of movement is defined by the vector R . The point of contact about which rotation in 

the R direction occurs is identified as point "0" in Figure 6.13c. 
6.22 

The forces acting in the plane of the side slope are Fd and Ws sinθ as shown in Figure 6.13b. 

The angle θ is the side slope angle. The lift force acts normal to the side slope and the 

component of submerged weight Ws sinθ acts normal to the side slope as shown in Figure 

6.13c. 

At incipient motion, there is a balance of moments about the point of rotation such that 

eW eW 2 1 34 ss d cos sin cos cos θ θβ δ = ++ eF eF (6.1) 

The moment arms e1, e2, e3 and e4 are defined in Figure 6.13c and the angles δ and β are 

defined in Figure 6.13b. 

The stability factor S.F. against rotation of the particle is defined as the ratio of the moments 

resisting particle rotation out of the bank to the submerged weight and fluid force moments 

tending to rotate the particle out of its resting position. Accordingly, 

S F e W

eW eF eF

s

s d

. . cos

sin cos cos = + +

2

1 34

θ

θβ δ 

 (6.2) 

The following particle stability analysis was first derived by Stevens (1968). The analysis is also 

presented in Simons and Senturk (1992). This analysis shows that the stability factor for rock 

riprap on side slopes where the flow has a non-horizontal velocity vector is related to properties 

of the rock, side slope and flow by the following equations: 

S F. . cos tan

' tan sin cos = +

θ φ

ηφ θβ (6.3) 

in which 

β λ

θ

η φ

λ

=

+





























− tan cos

sin

tan

sin

1

2 (6.4) 

s s

o

D)1S(

21

γ−

τ

=η (6.5) 

and 











 β+λ+ ′ η=η 2

)(sin1 (6.6) 

Given a rock size Ds, of specific weight Ss and angle of repose φ and given a velocity field at an 

angle λ to the horizontal producing a tractive force τo on the side slope of angle θ, the set of 

four equations (Equations 6.3, 6.4, 6.5, and 6.6) can be solved to obtain the stability factor S.F. 

 If S.F. is greater than unity, the riprap is stable; if S.F. is unity, the rock is at the condition of 
6.23 

incipient motion; if S.F. is less than unity, the riprap is unstable. Problem F.1 in Appendix F 

illustrates how to determine the stability of riprap. 

Simplified Design Aid For Side Slope Riprap. When the velocity along a side slope has no 

downslope component (i.e., the velocity factor is along the horizontal), some simple design aids 

can be developed. 

For horizontal flow along a side slope, the equations relating the stability factor, the stability 

number, the side slope angle, and the angle of repose for the rock are obtained from 

Equations 6.4 and 6.6 with λ = 0. 

















θ

φη =β −

sin2

tan tan 1 (6.7) 

and 





 



 β+ ′ η=η 2

sin1

 (6.8) 

When Equations 6.7 and 6.8 are substituted into Equation 6.3, the expression for the stability 

factor for horizontal flow on a side slope is: 

{ } 4 ζ−+ζ= 2

S .F.S m 2 (6.9) 

in which 

m secS θη=ζ (6.10) 

and 

θ

φ = tan

tan Sm (6.11) 

If we solve Equations 6.9 and 6.10 for η, then: 

θ − =η cos

S.)F.S(

.)F.S(S

2

m

2 2

m (6.12) 

The interrelation of the variables in these two equations is represented in Figure 6.14. Here, 

the specific weight of the rock is taken as 2.65 and a stability factor of 1.5 is employed. This 

recommended stability factor for the design of riprap (S.F. = 1.5) is the result of studies of the 

riprap embankment model data obtained by Lewis. These studies were reported by Simons 

and Lewis (1971). 
6.24 

Figure 6.14. Stability numbers for a 1.5 stability factor for horizontal flow along a side slope. 

The curves in Figure 6.14 are computed in the following manner: (1) Select an angle of repose 

φ. For example φ = 45°; (2) Select a side slope angle. For example θ = 25°; (3) Compute Sm

from Equation 6.11: 

;14.2 25tan

45tan Sm =

°

° =

(4) Compute η from Equation 6.12 with S. F. = 1.5 

31.025cos

)14.2()5.1(

)5.1()14.2(

2

2 2

=° 













 − =η

(5) Repeat the above steps for the full range of interest for φ and θ. 

If the shear stress τo on the side slope is known, then, from Equation 6.5, the riprap size 

required is obtained from: 

ηγ−

τ = )1S(

21 D

s

o

m (6.13) 

where the stability number η is obtained from Figure 6.14. 

6.5.3 Velocity Profile and Tractive Force 

In riprap design, it is often desirable to relate the tractive force (shear stress) acting on the 

riprapped bed or bank to the fluid velocity in the vicinity of the riprap. For fully turbulent flow, 

the reference velocity Vr at a distance D50 above the bed is determined from Equation 6.14. 
6.25 

ρ

τ ==













 = o

*

50

50

r * 52.8V52.8

D

D2.30lnV5.2V (6.14) 

Thus, the relation between Vr and τo is: 

72

V2

r

o

ρ =τ (6.15) 

The relation is valid only for uniform flow in wide prismatic channels in which the flow is fully 

turbulent. For the purpose of riprap design, Equation 6.15 can be used when the flow is 

accelerating such as on the tip of spur dikes or abutments. The equation should not be used in 

areas where the flow is decelerating or below energy dissipating structures. In these areas, 

the shear stress is larger than calculated for Equation 6.15. 

One can also demonstrate (Richardson et al. 1975) that the reference velocity Vr is related to 

the velocity against the stone Vs (Vr  1.4 Vs). 

In summary the following expressions for η are equivalent: 

s s

o

D)1S(

21

γ−

τ

=η (6.5) 

s s

2

r

Dg)1S(

V30.0

− =η (6.16) 

s s

2

s

Dg)1S(

V60.0

− =η (6.17a) 

s s

2

2

s

o gD)1S(

V

) D

y 3.12(ln

4.3 30.0 −

































	











 =η (6.17b) 

6.5.4 U.S. Army Corps of Engineers Design Equation 

U.S. Army Corps of Engineers design equations are based on local depth-averaged velocity 

(V), local depth of flow (y), and coefficients for a factor of safety (Sf ), stability (Cs), vertical 

velocity distribution (Cv ), blanket thickness (CT ), and side slope (K1). The principal equation 

determines the riprap size of which 30 percent is finer by weight (D30) (U.S. Army Corps of 

Engineers EM 1110-2-1601, 1991 and 1994a and Maynord 1988). The equation is: 
6.26 

5.2

2/1

1

2/1

s

30 Tvsf )gyK(

V yCCCSD 





















	











γ−γ

γ = (6.18) 

where: 

 D30 = Riprap size, m, ft 

 Sf = Safety factor, minimum 1.1 

 Cs = Stability coefficient for incipient failure, (D85/ D15 = 1.7 to 5.2) 

 = 0.30 for angular rock 

 = 0.36 for rounded rock 

 D85/D15 = Gradation uniformity coefficient (D50 = D30 (D85/D15)

1/3

 Cv = Vertical velocity distribution coefficient 

 = 1.0 for straight channels, inside of bends 

 = 1.283 – 0.2 log (R/W) for outside of bends, 1 for (R/W) > 26 

 = 1.25 downstream of concrete channels and at ends of dikes 

 R = Centerline radius of curvature of bend (main channel only) m, ft 

 W = Water-surface width at upstream end of bend (main channel only) m, ft 

 CT = Blanket thickness coefficient 

 = 1.0 for thickness = 1 D100 (max) or 1.5 or D50 (max), whichever is 

greater 

 y = Local depth of flow, m, ft 

γ = Unit weight of water, Kg/m3

, lb/ft3

γs = Unit weight of rock, Kg/m3

, lb/ft3

 V = Local depth-average velocity, m/s, ft/s 

 Vss = Local depth average velocity on side slopes, m/s, ft/s 

 Vavg = Average velocity in the main channel, m/s, ft/s 

 K1 = Side slope correction factor 

 g = Acceleration of gravity, 9.81 m/s2

, 32.2 ft/s2

The use of D30 for the sizing of the riprap is somewhat controversial. Its use probably results 

in an overall increase in the sizes of the riprap in the blanket compared to using the 

traditional D50 approach. Maynord (1988) states ”Stability tests conducted at a thickness of 1 

X D100, which is the most commonly used thickness for bank protection, showed that 

gradations ranging from uniform, to highly nonuniform exhibited the same stability if they had 

the same D30.” The Corps manual gives an approximate relationship between D50 and D30. It 

is D50 = D30 (D85 / D15 )1/3. 

The U.S. Army Corps of Engineers manual (1994a) states that the minimum safety factor 

may have to be increased for the following conditions: 

• Impact forces from logs, uprooted trees, vessels, ice etc. 

• Natural variations in the quality of the rock used as riprap 

• Vandalism 

• Placement quality 

• Freeze-thaw of the riprap is anticipated 

• Precision of the determination of the hydraulic variables 
6.27 

These factors for determining the value of the safety factor coefficient must be considered 

and the largest value used as the safety factor. The Corps manual (1994) provide the 

following additional guidance. 

If the riprap design is for a channel bottom, use the local depth-averaged velocity and local 

depth in Equation 6.18. Determining the depth-averaged velocity to design riprap to protect 

side slopes is more difficult because the velocity varies greatly form the toe to the top of the 

bank. The Corps manual EM 1110-2-1601 uses the depth-averaged velocity at a point 20 

percent up slope from the toe. This velocity can be determined by physical or computer 

models, empirical equations or methods, and prototype data. The Corps manual EM 1110-

2-1601 gives the following equation for estimating the side slope riprap design velocity. 





 



 −= W

R Log52.074.1 V

V

avg

ss (6.19) 

Note that the value of Vss / Vavg rarely exceeds 1.6 in alluvial or man made channels for large 

discharges for which riprap is designed. 

K1, the side slope correction factor in Equation 6.18, is taken from the Carter et al. (1953) 

relation in the Corps manual EM 1110-2-1601. The relationship is: 

2/1

2

2

1

Sin

Sin 1K 



 





φ

θ −= (6.20) 

where: 

θ = Angle of side slope with horizontal 

φ = Angle of repose of the riprap (normally 40o

) 

The Corps manual ME 1110-2-1601 gives an equation to determine riprap size for steep 

slopes (2 to 20 percent) where the unit discharge is low. A typical application is a rock-lined 

chute. The equation is: 

3/1

3/2555.0

30

g

qS95.1 D = (6.21) 

EM 1110-2-1601 states Equation 6.21 is applicable for thickness = 1.5 D100, angular rock, 

unit weight of 167 lb/ft3

, D85/ D15 from 1.7 to 2.7, slopes from 2 to 20 percent and uniform flow 

with no tailwater. Also, q is determined using the bottom width of the chute. 

The Corps manual 1110-2-1601 gives charts and figures as design guides and guidance for 

toe scour, toe scour protection, quality control, revetment top protection and revetment end 

protection (see also HEC-23). Termination of riprap bank protection often leads to the 

formation of an eddy (wake vortex) that erodes the bank. This erosion is undesirable and 

can initiate failure of the riprap. For this reason, end protection is needed. Maynord (1996) 

also gives equations and methods for toe scour estimation and toe scour and end section 

protection. He describes a computer program “CHANLPRO” available from the U.S. Army 

Engineer Waterways Experiment Station that incorporates the above riprap design and 

scour depth procedures as well as sizing gabion mattresses. 
6.28 

6.5.5 Riprap Gradation and Thickness 

The concept of a representative grain size for riprap is simple. A uniformly graded riprap with a 

median size D50 scours to a greater depth than a well-graded mixture with the same median 

size. The uniformly distributed riprap scours to a depth at which the velocity is less than that 

required for the transportation of D50 size rock. The well-graded riprap, on the other hand, 

develops an armor plate. That is, some of the finer materials, including sizes up to D50 and 

larger, are transported by the high velocities, leaving a layer of large rock sizes which cannot 

be transported under the given flow conditions. Thus, the size of rock representative of the 

stability of the riprap is determined by the larger sizes of rock. The representative grain size 

Dm for riprap is larger than the median rock size D50. 

The recommended gradation for riprap is illustrated in Figure 6.15 in terms of D50. The 

computations of the representative grain size Dm for the recommended gradation are given in 

Table 6.2. 

Figure 6.15. Suggested gradation for riprap. 

Table 6.2. Data for Suggested Gradation. 

Percent Finer Sieve Diameter DI

 0 0.25 D50 -- 

 10 0.35 D50 0.28 D50

 20 0.5 D50 0.43 D50

 30 0.65 D50 0.57 D50

 40 0.8 D50 0.72 D50

 50 1.0 D50 0.90 D50

 60 1.2 D50 1.10 D50

 70 1.6 D50 1.50 D50

 90 1.8 D50 1.70 D50

100 2.0 D50 1.90 D50
6.29 

The rock sizes in the last column in Table 6.2 are used in the following equation (Stevens 

1968) to find the representative grain size Dm. This effective grain size, Dm, of the mixture 

corresponds to the D65 size of the riprap. 

50

3/1 10

i

3

i

m D25.1

10

D

D ≅

















=  (6.22) 

When the bed material has a log-normal distribution, the representative size of the bed 

material based on the weight of the particles is given by Mahmood (1973) as a function of the 

gradation coefficient G: 











 = 2

50m )G(ln 2

3 expDD (6.23) 

For gradation coefficients of 2 and 3, Dm = 0.72 D50 and 1.81 D50 respectively. G is 

determined by Equation 3.9. 

With a distributed size range, the interstices formed by the larger stones are filled with the 

smaller sizes in an interlocking fashion, preventing formation of open pockets. Riprap 

consisting of angular stones is more suitable than that consisting of rounded stones. Control of 

the gradation of the riprap is almost always made by visual inspection. If it is necessary, poor 

gradations of rock can be employed as riprap provided the proper filter is placed between the 

riprap and the bank or bed material. Where available rock size is inadequate, wire enclosed 

(gabion) riprap can be used. 

Considering the practical problems of quarry production, a gradation band is usually specified 

by the U.S. Army Corps of Engineers (1981) rather than a single gradation curve, and any 

stone gradation within the limits is acceptable. The Corps criteria for establishing gradation 

limits in terms of stone weight (W) for riprap are as follows: 

• Lower limit of W50 stone should not be less than the weight of stone required to withstand 

the design shear forces. 

• Upper limit of W50 stone should not exceed five times the lower limit of W50 stone, the size 

which can be obtained economically from the quarry, or the size that satisfies layer 

thickness requirements. 

• Lower limit of W100 stone should not be less than two times the lower limit of W50 stone. 

• Upper limit of W100 stone should not exceed five times the lower limit of W50 stone, the size 

which can be obtained economically from the quarry, or the size that satisfies layer 

thickness requirements. 

• Lower limit of W15 stone should not be less than one-sixteenth the upper limit of W100 stone. 

• Upper limit of W15 stone should be less than the upper limit of the filter material. 
6.30 

• Bulk volume of stone lighter than the W15 stone should not exceed the volume of voids in 

the structure without this lighter stone. 

The riprap thickness should not be less than 300 mm (12 in.) for practical placement, less than 

the diameter of the upper limit of W100 stone, or less than 1.5 times the diameter of the upper 

limit W50 stone, whichever is greater. If riprap is placed under water, the thickness should be 

increased by 50 percent, and if it is subject to attack by large floating debris or wave action it 

should be increased 150 to 300 mm (6 to 12 in.). 

6.5.6 Riprap Placement 

General. Riprap placement is usually accomplished by dumping directly from trucks. If riprap 

is placed during construction of the embankment, rocks can be dumped directly from trucks 

from the top of the embankment. Rock should never be placed by dropping down the slope in 

a chute or pushed downhill with a bulldozer. These methods result in segregation of sizes. 

With dumped riprap there is a minimum of expensive hand work. Poorly graded riprap with 

slab-like rocks requires more work to form a compact protective blanket without large holes or 

pockets. Draglines, backhoes, and other power equipment can also be used to place the 

riprap. 

Hand placed rock riprap is another method of riprap placement. Stones are laid out in more or 

less definite patterns, usually resulting in a relatively smooth top surface. This form of 

placement is used rarely in modern practice because it is usually more expensive than 

placement with power machinery, and it is more likely to fail than dumped riprap. 

Dumped riprap that is keyed (or plated) by tamping has proved to be effective. Guidelines for 

placement of keyed riprap have been developed by the Oregon Department of Transportation 

and distributed by the Federal Highway Administration. In the keying of a riprap, a 1,818 kg 

(4,000 lb) or larger piece of steel plate is used to compact the rock into a tight mass and to 

smooth the revetment surface. Keyed riprap is more stable than loose riprap revetment 

because of reduced drag on individual stones, its angle of repose is higher, and its cost is less 

because a lesser volume of rock per unit area is required. 

Riprap should not be used at slopes steeper than IV:1.5H. This criterion is widely followed for 

abutment fill-slopes, but it is sometimes disregarded for stream banks. 

Broken concrete is used for riprap in many states where rock riprap is unavailable or unusually 

expensive. Broken concrete riprap has proven to be unsatisfactory if the material is not broken 

into riprap size particles. 

Rounded stones less than 150 mm (6 in.) in diameter have a significantly lower angle of repose 

than angular stones (Figure 3.4). Although they are less desirable than angular stones, 

rounded stones are nevertheless effective for larger diameters. 

The blanket should be stabilized at its base with a key trench or apron to prevent the stone 

from sliding down the bank. The upstream and downstream ends of the blanket should be tied 

back into the bank to prevent stream currents from unraveling the blanket. The most common 

method to tie into the bank is to dig a trench at the ends of the blanket. (Figure 6.16). The 

depth of a trench should be twice the blanket thickness and the bottom width of the trench 

three times the thickness. 
6.31 

Figure 6.16. Tie-in trench to prevent riprap blanket from unraveling (after Keown 1983). 

Rock-fill Trenches and Launching Aprons. Rock-fill trenches are structures used to protect 

banks from caving caused by erosion at the toe. A trench is excavated along the toe of the 

bank and filled with riprap as shown in Figure 6.17. 

Figure 6.17. Rock-fill trench. 

As the stream bed adjacent to the toe is eroded, the toe trench is undermined and the rock fill 

slides downward to pave the bank. The size of trench to hold the rock fill depends on expected 

depths of scour. It is advantageous to grade the banks before paving the slope with riprap and 

placing rock in the toe trench. The slope should be at such an angle that the saturated bank is 

stable while the river stage is falling. 

The rock-fill trench need not be at the toe of the bank. An alternative method is to excavate a 

trench above the water line along the top of the river bank and backfill with rocks. Then as the 

bank erodes toward the trench, the rocks in the trench slide down and pave the bank. This 

method is applicable in areas of rapidly eroding banks of medium to large size rivers. 

Launching aprons can provide similar protection against undermining for channel banks and 

revetment. A flexible launching apron is placed horizontally on the bed at the foot of the 

revetment, so that when scour occurs the materials will settle and cover the side of the scour 

hole on a natural slope. This method is generally the most economical for cohesionless 

channel beds where deep scour is expected. Materials used for launching aprons include 
6.32 

stone riprap, articulated concrete matting, concrete blocks, gabions, and wire mesh mattresses 

filled with stone. Stone riprap is most commonly used. 

In cohesionless channel beds, the design of stone aprons should be based on the stone 

launching to a slope of up to lV:2H. Model tests have indicated that these slopes are realistic 

for sand beds, but little field confirmation seems to have been reported. 

Stone sizes for launching should be the same as for slope revetment. The volume of stone 

should be sufficient to cover the final scoured slope to a thickness of 1.25 times the size of the 

largest stones in the specified grading. At the nose of a guide bank or spur, there should be 

sufficient stone to cover the final conical surface of the scoured slope. Piers should not be 

located within the launching apron slope unless it is unavoidable. 

Launching aprons do not perform well on cohesive channel beds where scour occurs in the 

form of slumps with steep slip faces. In such cases, bank revetment should be continued 

down to the expected worst scour level, and the excavation then refilled. 

A variation of these methods of toe protection is to pile the rocks in a "windrow" along the bank 

line instead of excavating a trench. Then as the bank is scoured, the rocks in the windrow drop 

down to pave the bank. 

Windrow Revetment. Windrow revetment is an erosion control technique (Figure 6.18) 

consisting of the depositing of a fixed amount of erosion-resistant material (riprap) landward 

from the existing bank line at a predetermined location, beyond which additional erosion is to 

be prevented. The technique consists of burying or piling a sufficient supply of 

erosion-resistant material in a windrow below or on the existing land surface along the bank, 

then permitting the area between the natural river bank and the windrow to erode though 

natural processes until the erosion reaches and undercuts the supply of rock. As the rock 

supply is undercut, it falls onto the eroding area, thus giving protection against further 

undercutting, and eventually halting further landward movement. The resulting bank line 

remains in a near natural state, with an irregular appearance due to intermittent lateral erosion 

in the windrow location. The treatment particularly lends itself to the protection of adjacent 

wooded areas, or placement along stretches of presently eroding, irregular bank line. The 

following observations and conclusions were obtained from model investigations on windrow 

revetments. 

(1) The "application rate" is the weight of stone applied per meter (foot) of bank line. The 

amount of stone in the windrow indicates the degree to which lateral erosion will be permitted 

to occur; 

(2) Various windrow shapes were investigated in the model investigations, and a rectangular 

cross section was the best windrow configuration. This type of windrow is most easily placed 

in an excavated trench of the desired width. The second best windrow shape was found to be 

a trapezoidal shape. This shape provides a steady supply of stone to produce a uniform 

blanket of stone on the eroding bank line. A triangular shape was found to be the least 

desirable; 

(3) Studies indicated that varying the bank height did not significantly affect the final 

revetment; however, high banks tended to produce a nonuniform revetment alignment. 

Studies showed that the high banks had a tendency for large segments of the bank to break 

loose and rotate slightly, whereas the low banks simply "melted" or sloughed into the stream. 

The slight rotation of the high bank segment probably induced a tendency for ragged 

alignment; and 
6.33 

Figure 6.18. Windrow revetment, definition sketch (after U.S. Army Corps of Engineers 1981). 

(4) The velocity and characteristics of the stream dictate the size of stone that must be used to 

form a windrow revetment. The size of stone used in the windrow was not significant as long 

as it was large enough to resist being transported by the stream. An important design 

parameter is the ratio of the relative thickness of the final revetment to the stone diameter. It 

was found that large stone sizes will require more material than smaller stone sizes to produce 

the same relative thickness. Since a filter cannot be used, a well-graded stone is important to 

ensure that the revetment does not fail from leaching of the underlying bank material. The 

stream velocity was found to have strong influence on the ultimate side slope of the revetment. 

It was determined that the initial bank slope was on the average approximately 15 percent 

steeper than the final revetment slope. In general, the greater the velocity, the steeper the side 

slope of the final revetment. 

Filters. Filters are used under riprap to allow water to drain easily from the bank without 

carrying out soil particles. Filters must meet two basic requirements: stability and permeability. 

The filter material must be fine enough to prevent the base material from escaping through the 

filter, but it must be more permeable than the base material. Section 6.7 provides general 

specifications for granular and geosynthetic filters. HEC-11 (Brown and Clyde 1989) contains 

more detailed design guidance. 

6.5.7 Riprap Failure Modes 

In a preliminary evaluation of various riprap design techniques, Blodgett and McConaughy 

(1985) concluded that the procedures based on velocity as a means of estimating stresses on 

the boundary provide the most reliable and consistent results. The following procedures were 

investigated: The 1967 version of FHWA HEC-11, FHWA HEC-15, USACE (EM-1601), 
6.34 

Caltrans Bank and Shore Manual, Simons and Senturk, and Oregon Department of 

Transportation. A major shortcoming of all present design techniques is their assumption that 

failures of riprap revetment are due only to particle erosion. Procedures for the design of riprap 

protection need to consider all the various causes of failures. 

Classic riprap failure modes are identified as follows: (1) particle erosion; (2) translational slide: 

(3) modified slump; and (4) slump. These modes of failure are illustrated in Figure 6.19. 

Particle erosion is the most commonly considered erosion mechanism (Figure 6.19a). Particle 

erosion occurs when individual particles are dislodged by the hydraulic forces generated by the 

flowing water. Particle erosion can be initiated by abrasion, impingement of flowing water, 

eddy action/reverse flow, local flow acceleration, freeze/thaw action, ice, or toe erosion. 

Probable causes of particle erosion include: (1) stone size not large enough; (2) individual 

stones removed by impact or abrasion; (3) side slope of the bank so steep that the angle of 

repose of the riprap material is easily exceeded; and (4) gradation of riprap too uniform. 

A translational slide is a failure of riprap caused by the downslope movement of a mass of 

stones, with the fault line on a horizontal plane (Figure 6.19b). The initial phases of a 

translational slide are indicated by cracks in the upper part of the riprap bank that extend 

parallel to the channel. This type of riprap failure is usually initiated when the channel bed 

scours and undermines the toe of the riprap blanket; this could be caused by particle erosion of 

the toe material, or some other mechanism which causes displacement of toe material. Any 

other mechanism which would cause the shear resistance along the interface between the 

riprap blanket and base material to be reduced to less than the gravitational force could also 

cause a translational slide. It has been suggested that the presence of a filter blanket may 

provide a potential failure plane for translational slides. Probable causes of translational slides 

are as follows: (1) bank side slope too steep; (2) presence of excess hydrostatic (pore) 

pressure; and (3) loss of foundation support at the toe of the riprap blanket caused by erosion 

of the lower part of the riprap blanket. 

Modified slump failure of riprap (Figure 6.19c) is the mass movement of material along an 

internal slip surface within the riprap blanket; the underlying material supporting the riprap does 

not fail. This type of failure is similar in many respects to the translational slide, but the 

geometry of the damaged riprap is similar in shape to initial stages of failure caused by particle 

erosion. Probable causes of modified slump are: (1) bank side slope is so steep that the riprap 

is resting very near the angle of repose, and any imbalance or movement of individual stones 

creates a situation of instability for other stones in the blanket; and (2) material critical to the 

support of upslope riprap is dislodged by settlement of the submerged riprap, impact, abrasion, 

particle erosion, or some other cause. 

Slump failure is a rotational-gravitational movement of material along a surface of rupture that 

has a concave upward curve (Figure 6.19d). The cause of slump failures is related to shear 

failure of the underlying base material that supports the riprap. The primary feature of a slump 

failure is the localized displacement of base material along a slip surface, which is usually 

caused by excess pore pressure that reduces friction along a fault line in the base material. 

Probable causes of slump failures are: (1) nonhomogeneous base material with layers of 

impermeable material that act as a fault line when subject to excess pore pressure; and (2) 

side slopes too steep and gravitational forces exceeding the inertia forces of the riprap and 

base material along a friction plane. 
6.35 

Figure 6.19. Riprap failure models (after Blodgett and McConaughy 1985). 
6.36 

Because of the general effectiveness of dumped riprap, a more detailed analysis of the 

relatively small number of cases in which it failed has been presented by Brice and Blodgett 

(1978). The principal causes of failure and methods of mitigation are given in Table 6.3. 

Table 6.3. Causes of Riprap Failure and Solutions (Brice et al. 1978) 

Cause Solution 

Inadequate size of riprap 

Impingement of current directly upon riprap 

rather than having flow parallel to riprap 

Channel degradation 

Internal slope failure (slump) 

Riprap with high percentage of fines causes 

washing out of the fines 

Larger riprap 

Heavier stones, flatten riprap slopes, redirect 

flow 

Provide a volume of reserve riprap at the 

revetment toe 

Reduce the riprap slope angle 

Follow gradation specifications 

 

6.6 BANK PROTECTION OTHER THAN RIPRAP 

There are many methods other than riprap that can be used for bank protection, including: 

vegetation, mattresses, baskets, and blocks. This section provides an overview of several of 

these methods. HEC-23 (Lagasse et al. 2001) and HEC-11 (Brown and Clyde 1989) provide 

detailed design guidance for several of these methods (see Section 6.3.4). 

6.6.1 Bioengineering Erosion Control 

Vegetation is probably the most natural method for protecting stream banks because it is 

relatively easy to establish and maintain, is visually attractive and environmentally more 

desirable. 

Below a stream's waterline, vegetation can effectively protect a bank in two ways. First, the 

root system helps to hold the soil together and increases overall bank stability by forming a 

binding network. Second, the exposed stalks, stems, branches and foliage provide resistance 

to the streamflow, causing the flow to lose energy by deforming the plants rather than by 

removing soil particles. Above the waterline, vegetation prevents surface erosion by absorbing 

the impact of falling raindrops and reducing the velocity of overbank drainage flow and rainfall 

runoff. Further, vegetation takes water from the soil providing additional capacity for infiltration 

and may improve bank stability by water withdrawal. 

Vegetation is generally divided into two broad categories: grasses and woody plants (trees 

and shrubs). The grasses are less costly to plant on an eroding bank above the toe and 

require a shorter period of time to become established. Woody plants offer greater protection 

against erosion because of their more extensive root systems; however, under some 

conditions the weight of the plant will offset the advantage of the root system. On very high 

banks, tree root systems do not always penetrate to the toe of the bank. If the toe becomes 

eroded, the weight of the tree and its root mass may cause a bank failure. 
6.37 

The major factor affecting species selection is the length of time required for the plant to 

become established on the slope. 

Water-tolerant grasses such as canarygrass (Phalaris), reedgrass (Calamagrostis), cordgrass 

(Spartina), and fescue (Festuca) are effective in prevent erosion on upper banks which are 

inundated from time to time and are primarily subject to erosion due to rainfall, overland flow, 

and minor wave action. Along the lower bank, where erosive forces are high, grasses are 

generally not effective as a protective measure; however, cattails (Typha), bulrushes (Scripus), 

reeds (Phragmites), knotweed and smartweed (Plygonum), rushes (Juncus), and mannagrass 

(Glyceria) are helpful in inducing deposition and reducing velocities in shallow water or wet 

areas at the bank toe and in protecting the bank in some locations. Willows (Salix) are among 

the most effective woody plants in protecting low banks because they are resilient, are 

sufficiently dense to promote deposition of sediment, can withstand inundation, and become 

established easily. 

Grass can be planted by hand seeding, sodding, sprigging, or by mechanical broadcasting of 

mulches consisting of seed, fertilizer, and other organic mixtures. Several commercial 

manufacturers market economical erosion control matting that will hold the seed and soil in 

place until new vegetation can become established. The matting is generally installed by hand 

and secured to the bank where plantings have been made to prevent erosion, then a fence 

should be placed along the top of bank. If livestock require access to the stream for watering 

or crossing, gates should be placed in the fence at locations where the cattle will do the least 

amount of damage to the planted bank; additionally, crossings should be fenced. 

6.6.2 Bioengineering Countermeasures 

The past few decades have seen increasing use of vegetation as a bank stabilizer. It has 

been used primarily in stream restoration and rehabilitation projects and can be applied 

independently or in combination with structural countermeasures. The term bioengineering 

and is generally used to describe stream bank erosion countermeasures and bank 

stabilization methods that incorporate vegetation. 

Stabilization of eroding stream banks using vegetative countermeasures has proven 

effective in many documented cases in Europe and the United States. However, the use of 

bioengineering with respect to scour and stream instability at highway bridges is a relatively 

new field. There is research being conducted in these fields, but these techniques have 

generally not been tested specifically as a countermeasure to protect bridges in the river 

environment. 

Bioengineering erosion control is not suitable where flow velocities exceed the strength of 

the bank material or where pore water pressure causes failures in the lower bank. In 

contrast, bioengineering maybe suitable where some sort of engineered structural solution is 

required, but the risk associated with using just vegetation is considered too high. 

Nonetheless, this group of countermeasures is not as well accepted as the classical 

engineering approaches to bridge stability. 

Design of bioengineered countermeasures to minimize rates of stream bank erosion requires 

accounting for hydrologic, hydraulic, geomorphic, geotechnical, vegetative, and construction 

factors. Although most of the literature dealing with biotechnical engineering on rivers is 
6.38 

associated with stream bank stabilization relative to channel restoration and rehabilitation 

projects, it is also generally applicable to bank stabilization associated with bridge crossings. 

HEC-23 (Lagasse et al. 2001) provides an overview of bioengineering approaches, a 

summary of design considerations, and reference to recent reports, studies, and handbooks 

containing design guidelines. 

6.6.3 Rock-and-Wire Mattresses 

When adequate riprap sizes are not available, rocks of cobble sizes may be placed in wire 

mesh mats made of galvanized fencing and placed along the bank forming a mattress. HEC23 (Lagasse et al. 2001) contains specifications developed by the New Mexico State Highway 

and Transportation Department for wire enclosed riprap mattress. The individual wire units are 

called baskets if the thickness is greater than 300 mm (12 in.). The term mattress implies a 

thickness no greater than 300 mm (12 in.). Toe protection is offered by either extending the 

mattresses on to the channel bed as shown in Figure 6.20 or embedding the mattress to some 

predetermined scour depth. As the bed along the toe is scoured, the extended mattress drops 

into the scour hole. Special wire baskets and mattresses are manufactured and sold 

throughout the United States. It should be noted that when rock-and-wire mattresses are used 

in streams transporting cobble and rocks, the wires of the basket can be cut by abrasion rather 

rapidly, which will destroy the intended protection along the base of the bank. Corrosion of the 

wire mesh and vandalism may also be a problem. 

Mattresses and baskets can be made up in large sizes in the field. These are flexible and can 

conform to scour holes which threaten the stability of the banks. They should be linked 

together to prevent separation as subsidence takes place. 

Figure 6.20. Rock and wire mattress. 
6.39 

The most economical combination of rock and wire for stream bank protection is simply laying 

wire mesh over stone. The major problem with this approach is keeping the mesh in place. 

One successful solution has been to bend pipe or rebar into the shape of a staple and then 

drive it through the mesh into the bank. The major drawback is that a rock and wire mattress 

generally costs more to place than a comparable riprap blanket. 

6.6.4 Gabions 

Gabions are patented rectangular wire boxes (or baskets) filled with relatively small-size stone, 

usually less than 200 mm (8 in.) in diameter. Where flow velocities are such that small stone 

would not be stable if used in a riprap blanket, the wire boxes provide an effective restraint. 

Limiting recommended maximum velocity for use of gabions ranges from 2.4 to 4.6 m/s (8 to 

15 ft/s), depending on the manufacturer. Gabions are used primarily for revetment-type 

structures, but have also been used for dikes and sills. HEC-11 (Brown and Clyde 1989) 

provide design guidelines for gabion revetment. 

Gabions act as a large heavy porous mass having some flexibility. The baskets are 

commercially available in a range of standard sizes and are made of heavy galvanized wire 

(coated when used in a corrosive environment). They are supplied at a job site folded flat and 

are assembled manually, using noncorrosive wire. The baskets are normally 0.5 m deep by 1 

m by 2 m (1.6 x 3.3 x 6.5 ft) and are set on a graded bank for revetments. A filter blanket or 

synthetic filter fabric is used, where required, to prevent leaching of base material and 

undermining of the baskets. 

6.6.5 Sacks 

Burlap sacks filled with soil or sand-cement mixtures have long been used for emergency work 

along levees and stream banks during floods (Figure 6.21). In recent years commercially 

manufactured sacks (burlap, paper, plastics, etc.) have been used to protect stream banks in 

areas where riprap of suitable size and quality is not available at a reasonable cost. Although 

most types of sacks are easily damaged and will eventually deteriorate, those sacks filled with 

sand-cement mixtures can provide long-term protection if the mixture has set up properly. 

Sand-cement sack revetment construction is not economically competitive in areas where good 

stone is available. However, if quality riprap must be transported over long distances, this type 

of sack revetment can often be placed on an eroding stream bank at a lesser cost than riprap. 

If a permanent revetment is to be constructed, the sacks should be filled with a mixture of 15 

percent cement (minimum) and 85 percent dry sand (by weight). The filled sacks should be 

placed in horizontal rows like common house brick beginning at an elevation below any toe 

scour (alternatively, riprap can be placed at the toe to prevent undermining of the bank slope). 

The successive rows should be stepped back approximately 1/2-bag width to a height on the 

bank above which no protection is needed. The slope steepness of the completed revetment 

should be no more than 1:1. After the sacks have been placed on the bank, they can be hosed 

down for a quick set or the sand-cement mixture can be allowed to set up naturally through 

rainfall, seepage or condensation. If cement leaches through the sack material, a bond will 

form between the sacks and prevent free drainage. For this reason weepholes should be 

included in the revetment design. The installation of weepholes will allow drainage of 

groundwater from behind the revetment thus helping to prevent pressure buildup that could 

cause revetment failure. Detailed design guidelines for commercially available articulating 

grout filled mattresses are provided in HEC-23 (Lagasse et al. 2001). 
6.40 

Figure 6.21. Typical sand-cement bag revetment (after Keown 1983). 

6.6.6 Articulated Concrete Block Systems (ACBs) 

Precast cellular blocks can be manufactured using locally available sand, cement, and 

aggregate or can be obtained from commercial sources. Cellular blocks are cast with 

openings to provide for drainage and to allow vegetation to grow through the blocks thus 

permitting the root structure to strengthen the bank. Fabric or a gravel blanket can be used as 

a filter under the blocks if there is any danger that the bank soil will be eroded through the 

block openings by streamflow or seepage. Although specialized equipment can be used to 

install large sections of blocks, hand placement is frequently used when mechanized apparatus 

is not available, access to the bank is limited, or costs need to be minimized. After the blocks 

have been placed, the revetment has sufficient flexibility to conform to minor changes in bank 

shape. Solid blocks should not be used because the bank may not be able to drain freely and 

failure could occur. 

Small precast concrete blocks held together by steel rods or cables can be used to form a 

flexible mat as shown in Figure 6.22. Design guidelines for ACBs are provided in HEC-23 

(Lagasse et al. 2001) and HEC-11 (Brown and Clyde 1989). 

The sizes of blocks may vary to suit the contour of the bank. It is particularly difficult to make a 

continuous mattress of uniform sized blocks to fit sharp curves. The open spacing between 

blocks permits removal of bank material unless a filter blanket of gravel or geosynthetic 

material is placed underneath. For embankments that are subjected only to occasional flood 

flows, the spaces between blocks may be filled with earth and vegetation can be established. 
6.41 

Figure 6.22 Articulated concrete block system. 

6.6.7 Used Tires 

Tires have been placed both as a mattress and stacked back against the bank. Both methods 

appear to have some potential as an economical approach to protect a stream bank, but 

environmental issues and aesthetics must be considered. 

During construction of a tire mattress on an eroding bank, two precautions should be 

considered to ensure that the mattress will stay in place. 

• The tires must be banded together; alternatively, cables running the length and width of the 

mattress can be woven through the tires. 

• The top, toe and the upstream and downstream ends of the mattress must be tied to the 

bank (Figure 6.23). If scour is anticipated, riprap should be placed at the toe of the 

mattress for additional protection. 

 

Figure 6.23. Used tire mattress (after Keown 1983). 
6.42 

While the precautions listed above are essential for successful construction of a stable 

mattress, other considerations can further improve the chances that the revetment will provide 

long-term bank protection. 

• Holes can be cut, drilled, or burned in the tire sidewalls to prevent flotation. 

• Presorting the tires by size may help to fit them together. 

• Earth screw anchors (or some other type of anchor) fastened to the mattress can be 

placed in the bank at various points on the face of the revetment. 

• The tires can be packed with stone or rubble. 

• Willows can be planted inside the tires preferably at the beginning of the growing season. 

Once established the root system will further strengthen the bank and obscure the 

unsightly mattress. If willows are not readily available, other species should be planted. 

Possible species for use are discussed later in this text. 

If the mattress effectively controls the stream bank erosion and remains intact, sediment may 

gradually cover the revetment. If willows have not been planted, volunteer vegetation will may 

become established in areas with a temperate climate. 

 

Prior to constructing a stacked-tire revetment, the bank face should be shaped so that the tires 

can be laid in horizontal rows on a geosynthetic filter material. The revetment should be 

started at the toe of the bank and stepped back 150 to 300 mm (6 to 12 in.) per row. Each tire 

should overlap the two tires under it. The stacked tires should be packed tightly with stone or 

rubble. Any space behind the tires should be filled with free-draining soil so that the soil mass 

will not become saturated and cause the revetment to fail. In addition, the upstream and 

downstream ends of the revetment should be tied into the bank so that there is no flow behind 

the revetment. 

6.6.8 Soil Cement 

In areas where riprap is scarce, use of in-place soil can sometimes be combined with cement 

to provide a practical alternative. HEC-23 (Lagasse et al. 2001) provides design guidelines 

and specifications for soil cement revetments. Figure 6.24 shows a detail of a typical 

soil-cement construction for bank protection. For use in soil-cement, soils should be easily 

pulverized and contain at least five percent, but not more than 35 percent, silt and clay 

(material passing the No. 200 sieve). Finer textured soils usually are difficult to pulverize and 

require more cement as do 100 percent granular soils which have no material passing the No. 

200 sieve. Soil cement can be placed and compacted on slopes as steep as one horizontal to 

one vertical. 

A stairstep construction is recommended on channel embankments with relatively steep 

slopes. Placement of small quantities of soil-cement for each layer 150 mm (6-inch) layers can 

progress more rapidly than a large quantity of fill material. Special care should be exercised to 

prevent raw soil seams between successive layers of soil-cement. If uncompleted 

embankments are left at the end of the day, a sheepsfoot roller should be used on the last 

layer to provide an interlock for the next layer. The completed soil-cement installation must be 

protected from drying out for a seven day hydration period. After completion, the material has 

sufficient strength to serve as a roadway along the embankment. Procedures for constructing 

soil-cement slope protection by the stairstep method can be found in "Suggested 

Specifications for Soil-Cement Slope Protection for Embankments (Central-Plant Mixing 

Method)," Portland Cement Association Publication IS052W and numerous other PCA 

publications. 
6.43 

Figure 6.24. Typical soil-cement bank protection. 

When velocities exceed 1.8 to 2.4 m/s (6 to 8 ft/s) and the flow carries sufficient bed load to be 

abrasive, special precautions are advisable. The aggregates in this case should contain at 

least 30 percent gravel particles retained on a No. 4 (4.75 mm) sieve. It should be emphasized 

that soil-cement provides a rigid bank protection. The depth of the bank protection should be 

sufficient to protect the installation from the anticipated total scour. 

A soil cement blanket with 8 to 15 percent cement may be an economical and effective stream 

bank protection method for use in areas where vegetation is difficult to establish and the bank 

material is predominately sand. The sand can be mixed with cement by hand or mechanically 

to a depth of at least 4 inches. The mixture should then be wet down and allowed to set up. 

This method has the advantage of low cost. However, there are three major disadvantages: 

impermeability, low strength, and susceptibility to temperature variations. If the bank behind 

the blanket becomes saturated and cannot drain, failure may occur. Also, because a 

sand-cement blanket is relatively brittle, very little if any traffic (vehicular, pedestrian, or 

livestock) can be sustained without cracking the thin protective veneer. In northern climates 

the blanket can break up during freeze-thaw cycles. 

6.7 FILTERS 

Filters are used under riprap to allow water to drain easily from the bank without carrying out 

soil particles. Filters must meet two basic requirements: stability and permeability. The filter 

material must be fine enough to prevent the base material from escaping through the filter, but 

it must be more permeable than the base material. There is no standard filter that can be used 

in all cases, see HEC-11 (Brown and Clyde 1989) for additional guidelines on filter design. 

Two types of filters are commonly used: granular (gravel) filters and geosynthetic filters. 

Granular Filters. A layer or blanket of well-graded gravel should be placed over the 

embankment or river bank prior to riprap placement. Sizes of gravel in the filter blanket should 

be from 5 mm (3/l6 in.) to an upper limit depending on the gradation of the riprap with 

maximum sizes of about 76 to 89 mm (3 to 3-l/2 in.). Thickness of the filter may vary 

depending upon the riprap thickness but should not be less than 152 to 228 mm (6 to 9 in.). 

Filters that are one-half the thickness of the riprap are quite satisfactory. Suggested 

specifications for gradation are as follows: 
6.44 

40

)Base(D

)Filter(D )1(

50

50 < (6.24) 

40

)Base(D

)Filter(D 5)2(

15

15 < < (6.25) 

5

)Base(D

)Filter(D )3(

85

15 < (6.26) 

If the base material is a fine-grained cohesive soil, such as fat or lean clay, these requirements 

are not applicable, and the stability criterion is that the D15 size of the filter cannot exceed 0.4 

mm. 

When the base material is very fine, the required filter material may also be quite fine, and 

more than one layer of filter (a graded filter) may be needed. In such a case, each layer must 

satisfy the stability and permeability requirements relative to the underlying layer. 

If the filter is designed for protection against the upward flow of water, the graded filter is 

constructed so that each layer is coarser than the one beneath (a "reverse" or "inverted" filter). 

Geosynthetic Filters. Geosynthetic materials are also used as filters, replacing a component of 

a graded filter. Detailed information on the use of geosynthetic filters can be found in Holtz et 

al. 1995 (FHWA HI-95-038) who define permeable geosynthetics as "geotextiles." Numerous 

geotextiles are on the market, with a wide variation in size and number of openings and in 

strength and durability. Geotextiles which provide opening areas of 25 to 30 percent are 

desirable to minimize the possibility of clogging and to reduce head loss. 

When geotextiles are used, care must be taken not to puncture the material during 

construction. If the filter fabric is placed on top of the base material, gravel can sometimes be 

placed directly on the fabric, eliminating the need for filter sand. If the paving materials is 

dumped or cast stone, however, it is desirable to place a protective blanket of sand or gravel 

on the filter, or to take care in placing the rock, so that the filter fabric is not punctured. Stones 

weighing as much as 1,360 kg (3,000 lbs) have been placed on synthetic filters with no 

apparent damage. If a protective covering is not used, the size and drop of the rock should be 

limited. The sides and toe of the filter fabric must be sealed or trenched so that base material 

does not leach out around the filter fabric. Care is also required in joining adjacent sections of 

filter fabric together; sewn, overlapped, and welted seams are used. See Holtz et al. 1995 for 

AASHTO specifications and construction guidelines. 

6.8 OVERTOPPING FLOW ON EMBANKMENTS

6.8.1 Introduction 

Floodwaters which exceed the crest elevation of roadways, approach embankments, levees, 

and similar earth embankment structures result in a hydraulic condition referred to as 

overtopping flow. Overtopping flow can be broadly characterized into two categories 

according to the level of tailwater on the downstream side of the structure: (1) "submerged 

flow," where tailwater is higher than the embankment crest and presents a backwater 

condition sufficient to affect the discharge over the crest; and (2) "free flow," where tailwater 

may be present against the downstream embankment slope but at an elevation low enough 

such that interference with flow over the crest does not occur. 
6.45 

Since the mid–1980s, extensive research has been conducted on overtopping flow over 

embankments and on the effectiveness of alternative methods for protecting the 

downstream slope of embankments against erosion during overtopping events. A number of 

different aspects of overtopping flow have been investigated, including the hydraulic 

conditions of flow, the mechanics of overflow erosion, and the prevention of erosion under 

varying conditions of soil types, hydraulic conditions, and embankment geometry. The 

ASCE Task Committee on the Mechanics of Overflow Erosion on Embankments (Powledge 

et al. 1989b) summarizes more than 30 dam, levee and roadway embankments that 

experienced overtopping flows during the 1980s. Field studies and laboratory investigations 

conducted both at full scale and at model scales during the last two decades have 

contributed to the understanding of analysis and design issues associated with overtopping 

flow. Many of these research activities are also summarized by the Task Committee 

(Powledge 1989a). 

 

6.8.2 Hydraulics of Overtopping Flow 

Flow Zones. In the case where there is minimal tailwater during an overtopping event, flow 

over an embankment structure transitions through three zones. Zone 1 is characterized by 

subcritical flow over the dam crest. In this zone the total energy is essentially equal to the 

elevation of the low-velocity pool upstream of the embankment, and erosive energy in this 

flow zone is low. In Zone 1, the embankment will experience erosion only when its crest is 

comprised of highly erodible material. Zone 2 consists of a transition zone near the 

downstream shoulder where the flow transitions from subcritical to supercritical. While the 

total energy remains similar to that in Zone 1, an increase in velocity creates an increase in 

local shear stress, and the erosive energy begins to build. Zone 3 consists of supercritical 

flow on the downstream slope of the embankment. In this zone, the erosive energy 

increases significantly as the flow accelerates down the slope. The energy slope increases 

as well, causing high erosion potential. Figure 6.25 illustrates these flow regimes (Powledge 

et al. 1989b). 

 

 Figure 6.25. Hydraulic flow zones on an embankment during overtopping 

 flow (Powledge et al. 1989b). 

Estimating Discharge. Flow over an embankment can exhibit two types of behavior. 

Kindsvater (1964) was one of the first to classify these conditions. The first type of flow 

condition is known as free flow, and is characterized by tailwater conditions that are low 
6.46 

enough such that backwater conditions on the crest do not occur. Free flow is further 

subdivided into either free-plunging or free-surface flow. The term plunging flow is used 

when the high-velocity jet plunges under the tailwater surface, creating a hydraulic jump. 

Surface flow occurs when the jet separates from the downstream slope of the embankment 

and "rides" over the tailwater surface. The generally accepted equation for discharge over 

an embankment is 

3/2 = H Cq 1 (6.27) 

where: 

 q = Discharge per unit width, m2

/s (ft2/s) 

 C = Experimentally determined coefficient (for level-crested embankments, 

C ≈ 0.51 for SI units or 3.0 for English units) 

 H1 = Total head above the embankment crest, m (ft) 

The second flow behavior is known as submerged flow. In this case, the tailwater is high 

enough to affect the discharge over the embankment. For the case of submerged flow, the 

equation becomes 





 



 = C

C H Cq 3/2 s (6.28)

 

where Cs is the submergence coefficient and the term Cs/C is dimensionless. Figure 6.26 

illustrates the variables needed to describe overtopping flow (Kindsvater 1964). 

Figure 6.26. Definition sketch of variables involved in overtopping flow (Kindsvater 1964). 

Hydraulic Analysis. In 1983, FHWA initiated research for which a full-scale testing facility 

was constructed in Fort Collins, Colorado. This facility was used to carry out a two-phase 

research project. Phase I consisted of an extensive literature review, field studies, and 

prototype-scale embankment tests involving different types of protection systems. Analysis 

of these studies developed embankment erosion equations and a computer simulation 

program called EMBANK (Chen and Anderson 1987). This program was used to develop 
6.47 

design charts that could be applied to estimate embankment erosion caused by a range of 

overtopping depths and tailwater conditions. 

When a bare soil embankment fails, it is often directly related to the bed shear stress that is 

applied as the flow accelerates over and down the embankment. Clopper and Chen (1988) 

describe a computational approach to calculating the bed shear stress. Using the principle 

of the conservation of momentum, the following equation was developed for non-uniform 

flow on the downstream embankment slope. 

( ) ( ) 





















	









 −ρ−θ− γ +θ+ γ =τ

22

2 2

2

2

o 21 1 d

1

d

1 qcosdd 2L

1 sin dd

2 (6.29) 

where γ is the unit weight of water, ρ is the unit mass of water, q is the volumetric discharge 

per unit width, and the remaining parameters are presented in Figure 6.27. 

 Figure 6.27. Definition sketch for application of the momentum equation 

 for embankment overtopping flow. 

If uniform flow is assumed, Manning’s equation can be used to estimate the depth of flow, 

and shear stress calculated by 

o S d f γ=τ (6.30) 

It should be noted that when using Manning’s equation, the assumption is made that the 

friction slope Sf is equal to the bed slope S0. This assumption usually yields a conservatively 

high estimate of bed shear stress, unless the combination of slope length and unit discharge 

result in conditions which approach uniform flow. 
6.48 

In practice, it is recommended that a range of Manning’s n values of approximately +/- 10 

percent of the base value be used to calculate the range of anticipated flow conditions. The 

minimum value of Manning’s n is used to determine the peak anticipated velocity, and the 

maximum value of Manning’s n is used to determine peak anticipated flow depth and shear 

stress. 

6.8.3 Mechanics of Overflow Erosion 

 

Erosion Processes. When flow overtops an embankment, locally high velocities and shear 

stresses will create strong erosion forces, typically at the downstream shoulder and on the 

embankment slope, that are too great for the soil of the embankment to withstand. There 

are two primary processes of erosion that occur during an overtopping event. 

When the overtopping flow is submerged, erosion typically begins at the downstream 

shoulder. This is the condition often experienced by roadways and approach embankments. 

Figure 6.28 (Chen and Anderson 1987) shows the progression of this type of failure at times 

t1, t2, and t3. As flow accelerates over the embankment, a surging hydraulic jump is formed 

which causes a nick point between the shoulder and downstream slope. This nick point will 

begin to migrate upstream due to the high velocities and erosion will begin to move 

downstream. The downstream migration of the erosion is caused by the turbulence 

associated with the hydraulic jump. It should be noted that, depending on the tailwater 

conditions and embankment geometry and soil type, even events of long duration may not 

ultimately result in a full breach of the embankment section. This is because a balance of 

forces can develop once the initial embankment erosion has produced a geometry that 

achieves equilibrium between hydraulic forces and the erosion resistance of the remaining 

embankment material. 

 Figure 6.28. Typical embankment erosion pattern with submerged flow 

 (Chen and Anderson 1987). 
6.49 

The second general erosion pattern results from the case of free flow. With low tailwater, 

the flow will accelerate down the slope with high velocity and shear stress associated with 

supercritical flow. Erosion typically initiates near the toe of the embankment, whether or not 

a hydraulic jump is present. Erosion progresses in the upslope and upstream direction 

through the embankment. Figure 6.29 (Chen and Anderson 1987) illustrates this 

progression. Given sufficient duration, this type of erosion pattern typically will result in a full 

breach of the embankment section. 

 Figure 6.29. Typical embankment erosion pattern with free flow 

 (Chen and Anderson 1987). 

Erodibility Rate Equation. Many analytical equations have been presented to estimate the 

rate of erosion under conditions of overtopping flow. Most of these equations use the 

effective shear stress and/or velocity along with some measure of resistance of the 

embankment material to estimate the rate of soil loss. These equations were assessed 

(Chen and Anderson 1987) by relating the erosion rates measured during tests to those 

calculated by the various equations. 

The erodibility rate relation developed by the Agricultural Research Laboratory was found to 

be the most applicable to embankment overtopping flow conditions based on these studies. 

This general form of the equation is: 

( )a

c -K E ττ= (6.31) 

where: 

 E = Detachment rate per unit area 

τ = Local effective shear stress based on hydraulic conditions 

τc = Critical shear stress of the soil 

 K&a = Empirical coefficients dependent on soil properties 

Local Shear Stress Calculation. For an embankment undergoing erosion, another 

parameter of importance is local shear stress. Chen and Anderson (1987) present the 

following equation (in English units) for calculating the shear stress in the immediate vicinity 

of the erosion location. 
6.50 

2 V f 

8

1

ρ=τ (6.32) 

 

where: 

1/3

2

d

n 116 f = (6.33) 

and V is the local velocity, f is the Darcy-Weisbach coefficient, and ρ is the water density, n 

is Manning's resistance coefficient, and d is the local flow depth. The use of the local shear 

stress equation can assist in accommodating the irregular pattern of erosion and its 

nonuniform progression through an embankment section at various times during an 

overtopping event. This form of the shear stress equation is utilized in the program 

EMBANK, described previously in this section. 

6.8.4 Erosion Protection in Overtopping Flow 

Materials or manufactured systems designed to protect against overtopping erosion can be 

selected and designed using methods based on permissible velocity, permissible shear 

stress, or both. Because velocity in and of itself is not a force, procedures based on 

permissible velocity are generally derived from extensive testing and field experience using a 

particular material under a variety of flow conditions, and are empirical in nature. 

Procedures utilizing shear stress as a fundamental variable tend to be more theoretical in 

nature, and are typically developed using a force-balance or moment-balance concept. Both 

types of methods are described in this section. 

Permissible Velocity Approach. Based on a battery of tests of vegetated trapezoidal 

channels using various types of reinforcement materials on steep waterways, Hewlett et al 

(1987) present permissible velocities for reinforced grass channels. These curves are 

presented in Figure 6.30 as a function of duration of the overtopping event. Note that the 

permissible velocity of concrete-based systems are independent of duration, as shown in the 

figure. 

Permissible Shear Stress Approach. Permissible shear stress based approaches are 

usually related to force-balance (sliding) or moment-balance (overturning) representations of 

the stability paradigm. Typically, a discrete-particle approach to the stability analysis is 

developed using the force- or moment-balance equations for the particular system under 

consideration. These approachs are presented in Hydraulic Engineering Circular No. 23 

(Lagasse et al. 2001) for selected erosion control systems, e.g., articulating concrete blocks, 

concrete armor units, and grout-filled fabric mats. 

Laboratory or field tests are recommended in order to develop the calibration parameters 

needed to fully describe the performance of the system. The permissible shear stress from 

the controlled testing program is then extended to various conditions of bed slope and side 

slope. Figure 6.31 shows a typical full-scale test of an articulated concrete block system in 

progress under steep-slope, high-velocity flow. 
6.51 

 Figure 6.30. Permissible velocities of various protection materials as 

 a function of flow duration (Hewlett et al. 1987). 

Notes to Figure: 

1. Minimum superficial mass 135 kg/m2

 (28 lb/ft2) 

2. Minimum nominal thickness 20 mm (0.8 inch) 

3. Installed within 20 mm (0.8 inch) of soil surface, or in conjunction with a surface mesh 

4. These graphs should only be used for erosion resistance to unidirectional flow. Values 

are based on available experience and information at the date of this report. 

5. All reinforced grass values assume well established, good grass cover. 

6. Other criteria (such as short-term protection, ease of installation and management, 

susceptibility to vandalism) must be considered in choice of reinforcement. 

6.8.5 Overtopping Protection Systems 

A number of material types and manufactured systems have been identified for use in 

minimizing or preventing erosion of embankments subjected to overtopping flow. Listed 

below are protection systems which have been the subject of various investigation over the 

last several decades. These systems are described in detail in a summary report issued by 

the ASCE Task Committee on Overtopping Protection (Oswalt et al. 1994): 

1. Reinforced Vegetation 

2. Roller Compacted Concrete 

3. Soil Cement 

4. Cast in Place Concrete 

5. Articulating Concrete Blocks 

6. Geotextile 

7. Riprap 

8. Gabions 
6.52 

 Figure 6.31. Full-scale test of an embankment overtopping protection system under 

 steep-slope, high-velocity flow conditions. 
6.53 

Each of these protection systems has its own unique mode of failure and threshold capability 

for erosion resistance. When using these systems, careful attention should be placed on the 

design of termination details at the crest, sides, and toe of the embankment. Powledge, et 

al. (1989a, 1989b) note that many of these systems were originally designed for uses other 

than the protection of embankments during overtopping flow, and have been adapted to this 

application as a result of a recognized need. For application of the Factor of Safety method 

(Section 6.5) to the design of riprap for the complex geometries often associated with 

overtopping flows, reference to Julien (1995) is suggested. 

6.9 ENVIRONMENTAL CONSIDERATIONS 

Stream bank protection projects should be planned, designed and constructed with 

consideration being given to environmental quality factors and project objectives. 

Environmental quality should address preservation or restoration of environmental resources 

within the project boundaries and avoidance of adverse impacts associated with the project. 

Some environmental factors are project specific and are necessarily defined during the 

planning phase while others are mandated by existing regulations. Stream bank protection 

projects should strive to preserve or restore existing environmental quality to the extent 

possible. 

6.9.1 Environmental Impacts 

Impacts of stream bank protection projects are dependent on project location and regional 

characteristics. For example, in arid regions of the western United States, forested habitat may 

be restricted to riparian areas and be directly and extensively impacted by project construction; 

whereas in the southeastern United States, forested habitat may be more extensive, but 

changes in stream hydraulics caused by the project can result in adverse impacts throughout 

the floodplain ecosystem. While general categories of impacts may be stated, site-specific and 

regional considerations and individual project features are critical in determining the magnitude 

and type of environmental impact. In some cases stream bank protection is performed in 

conjunction with other projects having different purposes, and it is difficult to isolate impacts 

due to stream bank protection alone. Categories of environmental impacts associated with 

stream bank protection projects include aesthetic, physical, water quality, and biological. 

Asthetic impacts most often occur because the natural appearance of the project area is 

disturbed or changed and replaced by an artificial appearance due to structures or channel 

alignment. The physical impacts of stream bank protection can affect channel morphology, 

sediment-carrying capacity of the stream, and stream hydraulics. These physical effects tend 

to manifest themselves as changes in landscape diversity and associated aquatic habitat 

diversity or quality; for example, loss of side channels or shallow areas or replacement of 

natural bank with revetment. Losses or changes in habitat will affect wildlife and aquatic life 

either by a reduction or change in community structure; however, changes in habitat 

composition for a specific project can be either detrimental or beneficial depending on 

circumstances. 

Water quality impacts from changes in turbidity together with alteration of riparian habitat (e.g., 

shading) affect stream temperature and photosynthetic activities that in turn may affect algae 

or aquatic plant populations, dissolved oxygen, and other water quality parameters. 

Temporary changes in water quality may occur as a result of construction activities. 
6.54 

Biological impacts can be broadly categorized into either terrestrial or aquatic. The major 

terrestrial impact involves alteration or elimination of riparian zone vegetation due to 

construction or project features. The riparian zone provides and supports a wide variety of 

plant and animal life and often provides critical habitat for certain species. Riparian vegetation 

also supports aquatic species by providing habitat for these species and input to the food 

chain. Channel stabilization can affect succession of riparian vegetation and decrease 

diversity. Aquatic organisms, including benthos and fish, may also be affected due to changes 

or reductions in required habitats as a result of project features. 

Other impacts that may occur due to stream bank protection projects include loss of wetlands 

and historic sites, changes in land use, increased recreational pressure, and economic or 

social impacts. 

6.9.2 Effects of Channelization 

Patrick (1973) assessed that the stream and its floodplain constitutes an integrated system that 

is well designed for moderating the effects of flooding waters and for maintaining high 

productivity in the stream. Disturbing the system inevitably results in a reduction in diversity of 

species and productivity. Because the functioning of the aquatic ecosystems is impaired, the 

ability of the stream to cleanse itself and to assimilate wastes is lessened, and the 

improvement of water quality is slower. The stream, instead of being one that is aesthetically 

pleasing and highly productive, becomes degraded and its recreational use is minimized. The 

chief effects of channelization are as follows: 

• Removes the natural diverse substrate materials that allow the development of many types 

of habitats for aquatic organisms 

• Increases sediment load that decreases light penetration and primary production 

• Creates a shifting bed load that is inimical to bottom-dwelling organisms 

• Simplifies the current pattern and eliminates habitats of diverse currents 

• Lowers the stream channel and often drains adjacent swamp areas and aquifers that help 

to maintain stream flow during time of low precipitation 

• Destroys floodplain ponds that are the breeding ground for aquatic life and that act as a 

reservoir for species of the river proper 

• Reduces the stability of the banks and causes cave-in of trees and other overhanging 

vegetation that are an important food source for stream life and whose shade reduces high 

stream temperatures during the summer months 

Kellerhals et al. (1985) concluded that most existing engineering design criteria for river training 

works are in direct conflict with general concepts of fish habitat maintenance since their aims 

are well aligned, uniform and stable channels, with minimum local scour and no opportunity for 

debris jamming. The objectives of fisheries mitigation works are often quite the opposite; a 

degree of instability, rough and irregular banks, deep local scour holes, debris jams and 

overhanging vegetation. Well founded guidelines for designing such diverse and irregular 
6.55 

channels (particularly in larger rivers) are needed and can only be developed on the basis of 

studies with a far broader scope than the normal project-oriented work funded by developers. 

6.9.3 Channel Restoration and Rehabilitation

Over the last several years, numerous agencies and practitioners have published guidelines for 

stream corridor restoration and channel rehabilitation design. For example, in 1998, fifteen 

Federal agencies and partners published a manual, Stream Corridor Restoration - Principles, 

Processes and Practices (Federal Interagency Stream Restoration Working Group 1998). This 

document represents a cooperative effort by the participating agencies to produce a common 

technical reference on stream corridor restoration. Recognizing that no two stream corridors 

and no two restoration initiatives are identical, this technical document broadly addresses the 

elements of restoration that apply in the majority of situations encountered. Reference is also 

suggested to Rosgen (1996) and U.S. Army Corps of Engineers publications such as Watson 

et al. 1999. HEC-20 (Lagasse et al. 2001) provides an introduction to stream restoration 

concepts and HEC-23 (Lagasse et al. 2001) presents guidelines and references for 

bioengineering bank protection treatments. 

6.10 SOLVED PROBLEMS FOR STABILITY OF RIPRAP (SI) 

6.10.1 PROBLEM 1 Stability of Particles Under Downslope Flow

Downslope flow over a plane bed inclined at an angle θˆ shown in Figure 6.32 is equivalent to 

an oblique flow on a side slope with θ = θˆ and λ = 90°. Then, according to Equation 6.4, β = 0, 

and from Equation 6.6, 

 η=

	 




 +°+ ′ η=η 2

090sin1

It follows from Equation 6.3 that the stability factor is 

θ+φη

φθ = sintan ˆ

cos ˆ tan .F.S

Alternatively, solving for η yields 





 





φ

θ −θ=η tan

ˆ tan

.F.S

1 ˆ cos

Using this information we wish to calculate (if the angle of repose, φ = 40°), what is the 

maximum bed angle θˆ at which η will be 5 percent different from that of a horizontal bed. 

Solving for θˆ with η = 0.95, S.F. = 1 and φ = 40° yields θˆ = 2.35° or 4.1 percent. 
6.56 

Figure. 6.32. Definition sketch for riprap on a channel bed. 

6.10.2 PROBLEM 2 Riprap Design on Embankment Slopes 

In a design situation water flows parallel to an embankment built of crushed rock riprap, (Ss = 

2.65), at a sloping angle θ = 20°. 

(a) If the design shear stress is τo = 95.8 N/m2

, calculate the riprap size that gives a stability 

factor equal to 1.5. 

Assuming angle of repose φ = 40°, the stability number η is obtained from Equation 6.3 or 

Figure 6.14 for θ = 20° (η = 0.36). The stone size is then calculated from Equation 6.13. 

 

m35.0

36.0x9800)165.2(

8.95x21

)1S(

21 D

s

o

m = − = − γη

τ =

(b) For the same design shear stress τo = 95.8 N/m2

, determine the stability factor of particle 

sizes Dm = 0.15 m. From Equation 6.5 

83.0

15.0x9800)165.2(

8.95x21

D)1S(

21

s m

o = − = γ−

τ

=η

Then, Equations 6.11, 6.10, and 6.9 are used to calculate the stability factor 

31.2

20tan

40tan

tan

tan Sm =

°

° = θφ =

m =θη=ζ = 04.220sec83.0x31.2secS
6.57 

( ) 194.0)04.2)404.2(( 2

31.2 ))4( 2

S .F.S 2/12 2/1 m 2 =ζ−+ζ= <=−+

This size fraction (Dm = 0.15 m) is unstable. 

(c) Determine riprap size Dm for a side slope. The side slope angle θ = 20°; very angular rock 

with angle of repose φ = 40°; Vss = 3.66 m/s; y = 3.05 m; S.F. = 1.1 and D85 / D15 = 2.0. From 

Equations 6.11, 6.12, and 6.16: 

31.2

20tan

40tan

tan

tan Sm == θφ =

66.020cos

31.2x1.1

1.131.2

cos

Sx.)F.S(

.)F.S(S

2

22

2

m

2 2

m = − =θ − =η

m38.0

66.0x81.9x65.1

66.3x3.0

g)1S(

V3.0 D

2

s

2

m = = η− =

(d) Compare the size calculated in (c) with a riprap size calculated using the U.S. Army Corps 

of Engineers equation (Section 6.5.4). From Equations 6.18 and 6.20: 

5.2

5.0

1

5.0

s

Tvsf30 )ygK(

V yCCCSD 





















	











γ−γ

γ =

85.0

40Sin

20Sin 1

Sin

Sin 1K

2/1

2

2 2/1

2

2

1  =



 



  −=



 





φ

θ −=

m24.0

)05.3x81.9x85.0

66.3

175,16

9800 05.3x0.1x0.1x3.0x1.1D

5.2

5.0

5.0

30 = 













 



	 




 =

m30.0)2(x24.0)D/D(xDD 3/1 3/1

50 = 30 1585 = =

6.10.3 PROBLEM 3 Stability Factors for Riprap Design

For flow around spill through abutments the angle between the horizontal and the velocity 

vector can be large (Figure 6.13). The draw down as the flow goes around the upstream 

end of the abutment can be very large. The draw down angle can range from 0 to 35 

degrees and the reference velocity Vr in the vicinity of the riprap can be very large (Lewis 

1972, Richardson et al. 1975, 1990). The following problem addresses the design of riprap 

for the protection of the spill through embankment. 
6.58 

The reference velocity Vr = 1.83 m/s, and the embankment side slope angle θ = 18.4° which 

corresponds to a 3:1 side slope. The velocity vector angle with the horizontal λ = 20°. If the 

embankment is covered with dumped rock having a specific weight Ss = 2.65 and an effective 

rock size Dm = 0.305 m, determine the stability factor. 

From Equation 6.16 

 

( ) 203.0

)305.0()81.9()165.2(

)83.1)30.0(

gD)1S(

V30.0 2

s m

2

r = − = − =η

This dumped rock has an angle of repose of approximately 35° according to Figure 3.4. 

Therefore, from Equation 6.4. 

 

°=





























°+ °

°

° =





























λ+

φη

θ

λ =β − − 11

20sin

35tan203.0

4.18sin2

20cos tan

sin

tan

sin2

cos tan 1 1

and from Equation 6.6 

154.0

2

)1120(sin1 203.0

2

)(sin1 =











 °+°+ =











 β+λ+ ′ η=η

The stability factor for the rock is given by Equation 6.3 

59.1

11cos4.18sin35tan154.0

35tan4.18cos

cossintan

tancos .F.S =

°°+°

°° = ′ βθ+φηφθ =

Thus, with a stability factor of 1.59, this rock is more than adequate to withstand the flow 

velocity. 

By repeating the above calculations over the range of interest for Dm (with φ = 35°), the curve 

given in Figure 6.33 is obtained. This curve shows that the incipient motion rock size is 

approximately 0.107 m and the maximum stability factor is less than 2.0 on the 3:1 side slope. 

The stability factor of a particular side slope riprap design can be increased by decreasing the 

side slope angle θ. If the side slope angle is decreased to zero degrees, then Equation 6.3 is 

applicable and 

93.4

203.0

11 .F.S == η =

The curve in Figure 6.34 relates the stability factor and side slope angle of the embankment 

(for λ = 20°, Dm = .0305 m ft and Vr = 1.83 m/s). The curve is obtained by employing 

Equations 6.3, 6.4, 6.16, and 6.6 for various values of θ. 
6.59 

Figure 6.33. Stability factors for various rock sizes on a side slope. 

Figure 6.34. Safety factors for various side slopes. 
6.60 

6.10.4 PROBLEM 4 Riprap Design on an Abutment 

(a) Consider a spill through abutment with side slope θ = 18.5o (3:1 side slope). The flow 

near the surface, for the 100-year flood, has a velocity vector Vr with a magnitude of 1.83 

m/s and angle with the horizontal λ = 20o

. The specific gravity of the available rock is 2.65. 

Determine the size of riprap in this area required to resist the erosive force of water. 

The spill slope riprap rock size is obtained by iteration from assuming Dm and calculating 

successively Equations 6.16, 6.4, 6.6, and 6.3 until the stability factor, S.F. equals 1.5. As a 

first approximation, a stone size of 0.091 m is used. From Figure 3.4, φ ≅ 40°. 

From Equation 6.16, 

68.0

091.x81.9x)165.2(

83.1x3.0 2

= − =η

From Equation 6.4, 

8.32

20sin

40tan68.0

5.18sin2

20cos tan 1 =





























+

=β −

From Equation 6.6, 

61.0

2

)8.3220(sin1 68.0 =












 ++ ′ =η

From Equation 6.3, 

02.1

8.32cos5.18sin40tan61.0

40tan5.18cos .F.S = + =

The procedure is repeated with increasing stone size until S.F. = 1.5. A riprap size of 0.23 m 

has a S.F. slightly over 1.5. 

(b) At the toe of the spill through abutment the velocity vector Vr has a magnitude of 2.88 

m/s and an angle with the horizontal λ of 0 degrees. Determine the size of riprap to protect 

the toe of the abutment. 

The riprap size is determined either by the same iterative procedure used in (a) or by using 

Equations 6.11, 6.12, and 6.16. 

Using the latter method: 

Equation 6.11 
6.61 

Sm = tanφ/tanθ = tan 40/tan 18.5 = 2.51 

Equation 6.12 

 

40.05.18cos

5.2x5.1

5.1)5.2( cos

S.F.S

.F.SS

2

2 2

2

m

2 2

m = 













 − =θ 













 − =η

Riprap size is obtained from Equation 6.16 

m38.0

4.0x81.9x)165.2(

)88.2(x3.0

g)1S(

V3.0 D

2

s

2

t

m = − = η− =

This is the size recommended. 

6.11 SOLVED PROBLEMS FOR FILTER DESIGN (SI)

The requirements for a gravel filter are given in Section 6.7. The gradation of a filter should be 

such that 

40

)Base(D

)Filter(D

50

50 < (6.24) 

40

)Base(D

)Filter(D 5

15

15 < < (6.25) 

5

)Base(D

)Filter(D

85

15 < (6.26) 

6.11.1 PROBLEM 1 Filter Design 

The properties of the riprap and base material are given in Table 6.4. Determine if filter is 

needed between the riprap and the base material. 

Table 6.4. Sizes of Materials. 

Base Material (Sand) Riprap (Gravel) 

D85 = 1.50 mm D85 = 24 mm 

D50 = 0.75 mm D50 = 12 mm 

D15 = 0.38 mm D15 = 6 mm 

In accordance with the recommended sizes for filters: 

16

75.0

12

)Base(D

)Riprap(D

50

50 ==
6.62 

which satisfies requirement 6.24 

16

38.0

6

)Base(D

)Riprap(D

15

15 ==

which satisfies the requirement 6.25 

4

5.1

6

)Base(D

)Riprap(D

85

15 ==

which satisfies the requirement 6.26. 

The riprap itself satisfies the requirements for the filter so no filter is needed. 

6.11.2 PROBLEM 2 Filter Design 

The following filter design is taken from Anderson et al. (1968). The properties of the base 

material and the riprap are given in Table 6.5. Determine if a filter is needed. 

Table 6.5. Sizes of Materials. 

Base Material (Sand) Riprap (Rock) 

D85 = 1.5 mm D85 = 400 mm 

D50 = 0.5 mm D50 = 200 mm 

D15 = 0.17 mm D15 = 100 mm 

The riprap does not contain sufficient fines to act as the filter because 

600

17.0

100

)Base(D

)Riprap(D

15

15 ==

which is much greater than 40, the recommended upper limit (requirement 6.25). Also 

67

5.1

100

)Base(D

)Riprap(D

85

15 ==

which is much greater than 5, the recommended upper limit (requirement 6.26). 

The properties of the filter to be placed adjacent to the base are as follows: 

(1) 40

)Base(D

)Filter(D

50

50 < 

so D50 (Filter) < (40) (0.5) = 20 mm 
6.63 

(2) 40

)Base(D

)Filter(D

15

15 < 

so D15 (Filter) < (40) (0.17) = 6.8 mm 

(3) 5

)Base(D

)Filter(D

85

15 < 

so D15 (Filter) < (5) (1.5) = 7.5 mm 

(4) 5

)Base(D

)Filter(D

15

15 > 

so D15 (Filter) > (5) (0.17) = 0.85 mm 

Thus, with respect to the base 

0.85 mm < D15 (Filter) < 6.8 mm 

and 

D50 (Filter) < 20 mm 

The properties of the filter to be placed adjacent to the riprap are as follows: 

(1) 40

)Filter(D

)Riprap(D

50

50 < 

so mm5

40

200

50 )Filter(D =>

(2) 5

)Filter(D

)Riprap(D

15

15 > 

so mm20

5

100

15 )Filter(D =<

(3) 40

)Filter(D

)Riprap(D

15

15 < 

so mm5.2

40

100

15 )Filter(D =>

(4) 5

)Filter(D

)Riprap(D

85

15 < 
6.64 

so mm20

5

100

85 )Filter(D =>

Therefore, with respect to the riprap, the filter must satisfy these requirements 

 2.5mm < D15 (Filter) < 20 mm 

 D50 (Filter) > 5 mm 

 D85 (Filter) > 20 mm 

These riprap filter requirements along with those for the base material are shown in Figure 

6.35. Any filter having sizes represented by the double cross-hatched area is satisfactory. For 

example, a good filter could have these sizes: 

 D85 = 40 mm 

 D50 = 10 mm 

 D15 = 4 mm 

Figure 6.35. Gradations of filter blanket for Problem 2 (after Anderson et al. 1968). 

6.12 SOLVED PROBLEMS FOR STABILITY OF RIPRAP (ENGLISH) 

6.12.1 PROBLEM 1 Stability of Particles Under Downslope Flow 

Downslope flow over a plane bed inclined at an angle θˆ shown in Figure 6.36 is equivalent to 

an oblique flow on a side slope with θ = θˆ and λ = 90°. Then, according to Equation 6.4, β = 0, 

and from Equation 6.6, 
6.65 

 η=

	 




 +°+ ′ η=η 2

090sin1

It follows from Equation 6.3 that the stability factor is 

θ+φη

φθ = sintan ˆcos ˆ tan .F.S

Alternatively, solving for η yields 





 





φ

θ −θ=η tan

ˆ tan

.F.S

1 ˆ cos

Figure. 6.36. Definition sketch for riprap on a channel bed. 

Using this information we wish to calculate (if the angle of repose, φ = 40°), what is the 

maximum bed angle θˆ at which η will be 5 percent different from that of a horizontal bed. 

Solving for θˆ with η = 0.95, S.F. = 1 and φ = 40° yields θˆ = 2.35° or 4.1 percent. 

6.12.2 PROBLEM 2 Riprap Design on Embankment Slopes 

In a design situation water flows parallel to an embankment built of crushed rock riprap, (Ss = 

2.65), at a sloping angle θ = 20°. 

(a) If the design shear stress is τo = 2.0 psf, calculate the riprap size that gives a stability factor 

equal to 1.5. 
6.66 

Assuming angle of repose φ = 40°, the stability number η is obtained from Equation 6.3 or 

Figure 6.14 for θ = 20° (η = 0.36). The stone size is then calculated from Equation 6.13. 

 

ft13.1

36.0x4.62)165.2(

0.2x21

)1S(

21 D

s

o

m = − = − γη

τ =

(b) For the same design shear stress τo = 2.0 psf, determine the stability factor of particle sizes 

Dm = 6 inches. From Equation 6.5 

82.0

5.0x4.62)165.2(

2x21

D)1S(

21

s m

o = − = γ−

τ

=η

Then, Equations 6.11, 6.10, and 6.9 are used to calculate the stability factor 

31.2

20tan

40tan

tan

tan Sm =

°

° = θφ =

m =θη=ζ = 02.220sec82.0x31.2secS

( ) 195.0)02.2)402.2(( 2

31.2 ))4( 2

S .F.S 2/12 2/1 m 2 =ζ−+ζ= <=−+

This size fraction (Dm = 6 inches) is unstable. 

(c) Determine riprap size Dm for a side slope. The side slope angle θ = 20°; very angular rock 

with angle of repose φ = 40°; Vss = 12 fps; y = 10 ft; S.F. = 1.1 and D85 / D15 = 2.0. From 

Equations 6.11, 6.12, and 6.16 

31.2

20tan

40tan

tan

tan Sm == θφ =

66.020cos

31.2x1.1

1.131.2

cos

Sx.)F.S(

.)F.S(S

2

2 2

2

m

2 2

m = = =θ − =η

ft25.1

66.0x2.32x65.1

12x3.0

g)1S(

V3.0 D

2

s

2

m = = η− =

(d) Compare the size calculated in (c) with a riprap size calculated using the U.S. Army Corps 

of Engineers equation (Section 6.5.4). From Equations 6.18 and 6.20: 

5.2

5.0

1

5.0

s

Tvsf30 )ygK(

V yCCCSD 





















	











γ−γ

γ =
6.67 

85.0

40Sin

20Sin 1

Sin

Sin 1K

2/1

2

2 2/1

2

2

1  =



 



  −=



 





φ

θ −=

ft79.0

)10x2.32x85.0

12

96.102

4.62 10x0.1x0.1x3.0x1.1D

5.2

5.0

5.0

30 = 













 



	 




 =

ft00.1)2(x79.0)D/D(xDD 3/1 3/1

50 = 30 1585 = =

6.12.3 PROBLEM 3 Stability Factors for Riprap Design

For flow around spill through abutments the angle between the horizontal and the velocity 

vector can be large (Figure 6.13). The draw down as the flow goes around the upstream 

end of the abutment can be very large. The draw down angle can range from 0 to 35 

degrees and the reference velocity Vr in the vicinity of the riprap can be very large (Lewis 

1972, Richardson et al. 1975, 1990). The following problem addresses the design of riprap 

for the protection of the spill through embankment. 

The reference velocity Vr = 6 fps, and the embankment side slope angle θ = 18.4° which 

corresponds to a 3:1 side slope. The velocity vector angle with the horizontal λ = 20°. If the 

embankment is covered with dumped rock having a specific weight Ss = 2.65 and an effective 

rock size Dm = 1.0 ft, determine the stability factor. 

From Equation 6.16 

 

( ) 203.0

)0.1()2.32()165.2(

)6()30.0(

gD)1S(

V30.0 2

s m

2

r = − = − =η

This dumped rock has an angle of repose of approximately 35° according to Figure 3.4. 

Therefore, from Equation 6.4. 

 

°=





























°+ °

°

° =





























λ+

φη

θ

λ =β − − 11

20sin

35tan203.0

4.18sin2

20cos tan

sin

tan

sin2

cos tan 1 1

and from Equation 6.6 

154.0

2

)1120(sin1 203.0

2

)(sin1 =











 °+°+ =











 β+λ+ ′ η=η

The stability factor for the rock is given by Equation 6.3 

59.1

11cos4.18sin35tan154.0

35tan4.18cos

cossintan

tancos .F.S =

°°+°

°° = ′ βθ+φηφθ =
6.68 

Thus, with a stability factor of 1.59, this rock is more than adequate to withstand the flow 

velocity. 

By repeating the above calculations over the range of interest for Dm (with φ = 35°), the curve 

given in Figure 6.37 is obtained. This curve shows that the incipient motion rock size is 

approximately 0.107 m and the maximum stability factor is less than 2.0 on the 3:1 side slope. 

Figure 6.37. Stability factors for various rock sizes on a side slope. 

The stability factor of a particular side slope riprap design can be increased by decreasing the 

side slope angle θ. If the side slope angle is decreased to zero degrees, then Equation 6.3 is 

applicable and 

93.4

203.0

11 .F.S == η =

The curve in Figure 6.38 relates the stability factor and side slope angle of the embankment 

(for λ = 20°, Dm = 1.0 ft and Vr = 6.0 fps). The curve is obtained by employing Equations 6.3, 

6.4, 6.16, and 6.6 for various values of θ. 

Figure 6.38. Safety factors for various side slopes. 
6.69 

6.12.4 PROBLEM 4 Riprap Design on an Abutment 

(a) Consider a spill through abutment with side slope θ = 18.5 o (3:1 side slope). The flow 

near the surface, for the 100-year flood, has a velocity vector Vr with a magnitude of 6.0 

fps and angle with the horizontal λ = 20o

. The specific gravity of the available rock is 2.65. 

Determine the size of riprap in this area required to resist the erosive force of water. 

The spill slope riprap rock size is obtained by iteration from assuming Dm and calculating 

successively Equations 6.16, 6.4, 6.6, and 6.3 until the stability factor, S.F. equals 1.5. As a 

first approximation, a stone size of 0.3 ft is used. From Figure 3.4, φ ≅ 40°. 

From Equation 6.16, 

68.0

)3.0(x)2.32(x)165.2(

6x3.0 2

= − =η

From Equation 6.4, 

8.32

20sin

40tan68.0

5.18sin2

20cos tan 1 =





























+

=β −

From Equation 6.6, 

61.0

2

)8.3220(sin1 68.0 =












 ++ ′ =η

From Equation 6.3, 

02.1

8.32cos5.18sin40tan61.0

40tan5.18cos .F.S = + =

The procedure is repeated with increasing stone size until S.F. = 1.5. A riprap size of 9 inches 

would give a S.F. slightly over 1.5. 

(b) At the toe of the spill through abutment the velocity vector Vr has a magnitude of 9.45 

fps and an angle with the horizontal λ of 0 degrees. Determine the size of riprap to protect 

the toe of the abutment. 

The riprap size is determined either by the same iterative procedure used in a or by using 

Equations 6.11, 6.12, and 6.16. 

Using the latter method: 

Equation 6.11 
6.70 

Sm = tanφ/tanθ = tan 40/tan 18.5 = 2.51 

Equation 6.12 

 

40.05.18cos

5.2x5.1

5.1)5.2( cos

S.F.S

.F.SS

2

2 2

2

m

2 2

m = 













 − =θ 













 − =η

Riprap size is obtained from Equation 6.16 

ft26.1

4.0x2.32x)165.2(

)45.9(x3.0

g)1S(

V3.0 D

2

s

2

r

m = − = η− =

This is the size recommended. 

6.13 SOLVED PROBLEMS FOR FILTER DESIGN (ENGLISH)

The requirements for a gravel filter are given in Section 6.7. The gradation of a filter should be 

such that 

40

)Base(D

)Filter(D

50

50 < (6.24) 

40

)Base(D

)Filter(D 5

15

15 < < (6.25) 

5

)Base(D

)Filter(D

85

15 < (6.26) 

6.13.1 PROBLEM 1 Filter Design 

The properties of the riprap and base material are given in Table 6.6. Determine if filter is 

needed between the riprap and the base material. Note: Generally, sediment sizes are given 

in mm for the English system. 

Table 6.6. Sizes of Materials. 

Base Material (Sand) Riprap (Gravel) 

D85 = 1.50 mm D85 = 24 mm 

D50 = 0.75 mm D50 = 12 mm 

D15 = 0.38 mm D15 = 6 mm 
6.71 

In accordance with the recommended sizes for filters: 

16

75.0

12

)Base(D

)Riprap(D

50

50 ==

which satisfies requirement 6.24 

16

38.0

6

)Base(D

)Riprap(D

15

15 ==

which satisfies the requirement 6.25 

4

5.1

6

)Base(D

)Riprap(D

85

15 ==

which satisfies the requirement 6.26. 

The riprap itself satisfies the requirements for the filter so no filter is needed. 

6.13.2 PROBLEM 2 Filter Design 

The following filter design is taken from Anderson et al. (1968). The properties of the base 

material and the riprap are given in Table 6.7. Determine if a filter is needed. 

Table 6.7. Sizes of Materials. 

Base Material (Sand) Riprap (Rock) 

D85 = 1.5 mm D85 = 400 mm 

D50 = 0.5 mm D50 = 200 mm 

D15 = 0.17 mm D15 = 100 mm 

The riprap does not contain sufficient fines to act as the filter because 

600

17.0

100

)Base(D

)Riprap(D

15

15 ==

which is much greater than 40, the recommended upper limit (requirement 6.25). Also 

67

5.1

100

)Base(D

)Riprap(D

85

15 ==

which is much greater than 5, the recommended upper limit (requirement 6.26). 
6.72 

The properties of the filter to be placed adjacent to the base are as follows: 

(1) 40

)Base(D

)Filter(D

50

50 < 

so D50 (Filter) < (40) (0.5) = 20 mm 

(2) 40

)Base(D

)Filter(D

15

15 < 

so D15 (Filter) < (40) (0.17) = 6.8 mm 

(3) 5

)Base(D

)Filter(D

85

15 < 

so D15 (Filter) < (5) (1.5) = 7.5 mm 

(4) 5

)Base(D

)Filter(D

15

15 > 

so D15 (Filter) > (5) (.17) = 0.85 mm 

Thus, with respect to the base 

0.85 mm < D15 (Filter) < 6.8 mm 

and 

D50 (Filter) < 20 mm 

The properties of the filter to be placed adjacent to the riprap are as follows: 

(1) 40

)Filter(D

)Riprap(D

50

50 < 

so mm5

40

200

50 )Filter(D =>

(2) 5

)Filter(D

)Riprap(D

15

15 > 

so mm20

5

100

15 )Filter(D =<
6.73 

(3) 40

)Filter(D

)Riprap(D

15

15 < 

so mm5.2

40

100

15 )Filter(D =>

(4) 5

)Filter(D

)Riprap(D

85

15 < 

so mm20

5

100

85 )Filter(D =>

Therefore, with respect to the riprap, the filter must satisfy these requirements 

 2.5mm < D15 (Filter) < 20 mm 

 D50 (Filter) > 5 mm 

 D85 (Filter) > 20 mm 

These riprap filter requirements along with those for the base material are shown in Figure 

6.39. Any filter having sizes represented by the double cross-hatched area is satisfactory. For 

example, a good filter could have these sizes: 

 D85 = 40 mm 

 D50 = 10 mm 

 D15 = 4 mm 

Figure 6.39. Gradations of filter blanket for Problem 2 (after Anderson et al. 1968). 
6.74 

(page intentionally left blank) 
7.1 

CHAPTER 7 

SCOUR AT BRIDGES 

 

7.1 INTRODUCTION 

7.1.1 General 

Scour at highway structures is the result of the erosive action of flowing water removing bed 

material from around the abutments and piers which support the bridge and bed and bank 

material of the stream the structure crosses. Both scour at highway structures and stream 

migration (instability) can cause a bridge failure.

All material in a streambed will erode. It is just a matter of time. However, some material such 

as granite may take hundred's of years to erode. Whereas, sandbed streams will erode to the 

maximum depth of scour in hours. Sandstone, shales, and other sedimentary bedrock 

materials, although they will not erode in hours or even days will, over time, if subjected to the 

erosive forces of water, erode to the extent that a bridge will be in danger unless the 

substructures are founded deep enough. Cohesive bed and bank material such as clays, silty 

clays, silts and silty sands or even coarser bed material such as glacial tills, which are 

cemented by chemical action or compression, will erode if subjected to the forces of flowing 

water. The erosion of cohesive and other cemented material is slower than sand bed material, 

but their ultimate scour will be as deep if not deeper than the scour depth in a non-cohesive 

sandbed stream (Briaud et al. 1999). It might take the erosive action of several major floods 

but ultimately the scour hole will be equal to or greater in depth than with a sand bed material. 

This does not mean that every bridge foundation must be buried below the calculated scour 

depth determined for non-bedrock streams. It does mean that so-called bedrock streams must 

be carefully evaluated. 

Scour at bridge crossings is a sediment transport process. Long-term degradation, general 

scour, and local scour at piers and abutments result from the fact that more sediment is 

removed from these areas than is transported into them. If there is no transport of bed 

material into the bridge crossing, clear-water scour exists. Transport of appreciable bed 

material into the crossing results in live-bed scour. In this latter case the transport of the bed 

material limits the scour depth. Whereas, with clear-water scour the scour depths are limited 

by the critical velocity or critical shear stress of a dominant size in the bed material at the 

crossing. 

7.1.2 Costs of Bridge Failure from Scour 

Hydraulic factors (scour/ice/debris) cause 60 percent of bridge failures in the United States 

(Shirole and Holt 1991). In the United States there are over 580,000 bridges in the National 

Bridge Inventory. These numbers include federal highway system, state, county and city 

bridges. Approximately 84 percent of these bridges are over water. Bridge failures cost 

millions of dollars each year as a result of both direct cost necessary to replace and restore 

bridges, and indirect costs related to disruption of transportation facilities. However, of even 

greater consequence is loss of life from bridge failures. Chang, in a 1973 study for the 
7.2 

Federal Highway Administration, indicated that about $75 million was expended annually up 

to 1973 to repair roads and bridges that were damaged by floods. This cost does not include 

the additional indirect costs to highway users for fuel and operating costs resulting from 

temporary closure and detours and to the public for costs associated with higher tariffs, 

freight rates, additional labor costs and time. 

Rhodes and Trent (1999) document that $1.2 billion was expended for restoration of flood 

damaged highway facilities during the 1980s. They state that this amount is conservative 

because it (1) only includes the amount funded by the U.S. Government which ranges from 

75 to 100 percent of the total restoration costs, and (2) the funds were only for disasters 

related to very large floods and do not include the hundreds of smaller events that occur 

every year. They also demonstrate that the added cost of operating a vehicle over a detour 

and time lost traveling when a bridge failed (only part of the indirect costs) exceed by several 

times the direct cost of bridge replacement or repair. 

7.1.3 Bridge Scour Evaluation Program 

The U.S. Department of Transportation, Federal Highway Administration (FHWA) in 1988 

issued Technical Advisory T5140.20 (superseded by T5140.23 in 1991) requiring the States 

to conduct a scour evaluation program. This program resulted from the failure of the I-90 

bridge over Schoharie Creek in upstate New York which killed ten people (NTSB 1988 and 

Richardson et al. 1987). The evaluation is to be conducted by an interdisciplinary team of 

hydraulic, geotechnical and structural engineers who can make the necessary engineering 

judgments to determine the vulnerability of a bridge to scour. As of February 2001, the 

481,313 bridges over water in the National Bridge Inventory have been screened as to their 

scour vulnerability (100 percent). 

The evaluation program in the U.S. is on schedule and scour countermeasures are being 

implemented on bridges that have been identified as scour susceptible or scour critical. 

Replacement bridges are being constructed as rapidly as funds can be provided. Scour 

countermeasures include riprap protection, scour monitoring before, during and after a flood 

and the inspection program (Richardson and Davis 2001, Lagasse et al. 1997, 2001a, and 

2001b, Schall et al. 1997a and 1997b). 

7.1.4 Comprehensive Scour Analysis 

This chapter presents background knowledge on scour analysis. Earlier chapters give a 

comprehensive overview of fluvial geomorphology, sediment transport, and flow in alluvial 

channels and stream stability fundamentals needed to understand streams, stream instability 

and scour. The comprehensive procedure for scour and stream instability analyses and 

countermeasures to control them are given in the FHWA publications HEC-18, HEC-20, and 

HEC-23 (Richardson and Davis 2001; Lagasse et al. 2001a, 2001b). The interrelationships 

and procedures recommended in the three documents are shown in the Figure 7.1. 
7.3 

Office Data Collection & Site Visit

Field Data Collection

Define / Classify Stream

Evaluate Stream Stability

Assess Stream Response

Sufficient

Data

HEC-20

Stream Stability and

Geomorphic Assessment

HEC-18

Hydrologic, Hydraulic

and Scour Analysis

HEC-23

 Bridge Scour and Stream 

Instability Countermeasures

Hydrologic Analysis

Hydraulic Analysis

Tidal Riverine

Scour Analysis

Plot Scour Prism

Multi Disciplinary Evaluation

Hydraulics/Structures/Geotechnical

Low Risk

Scour Critical: Plan of Action Required

Replace

Bridge

Structure

Stable

Evaluate CM

Options with Matrix

Design CM / 

Monitoring Plan

Environmental

Considerations / Permitting

Install CM /

Implement Monitoring Plan

Inspection & Maintenance

Select CM 

Type

CM

Viable

Yes

No

Yes

Yes

No (Existing Bridge)

No

Hydraulic Structural

Monitoring

Structural / 

Geotechnical

Input

(New Bridge,

revise design) Bridge

Type

New

Existing

Scour

Susceptible

Yes

Low Risk

No

Establish Level of Analysis

Evaluate CM Impact

Acceptable

CM Impact

Yes

No

No

Develop Plan of Action

Figure 7.1. Flow chart for scour and stream stability analysis and countermeasures. 
7.4 

7.2 TOTAL SCOUR 

Total scour at a highway crossing is composed of three components: 

 

(1) Long-Term Aggradation or Degradation

The change in river bed elevation (aggradation or degradation) over long lengths and time due 

to changes in controls, such as dams, changes in sediment discharge, head cuts, daily tidal 

flow, and changes in river geomorphology, such as changing from a meandering to a braided 

stream. These processes may be natural or human induced. 

(2) General Scour 

The scour that results from the acceleration of the flow due to either a natural or bridge 

contraction or both (contraction scour). General scour may also result from the location of the 

bridge on the stream. For example, its location with respect to a stream bend or its location 

upstream from the confluence with another stream. In this latter case, the elevation of the 

downstream water surface will affect the backwater on the bridge, hence, the velocity and 

scour. General scour may occur during the passage of a flood and the stream may fill in on the 

falling stage. This type of scour involves the removal of material from the bed and banks 

across all or most of the width of a channel. 

(3) Local Scour

The scour that occurs at a pier or abutment as the result of the pier or abutment obstructing the 

flow. These obstructions accelerate the flow and create vortices that remove the material 

around them. 

 

Generally, scour depths from local scour are much larger than long-term degradation or 

general scour, often by a factor of ten. But, if there are major changes in the stream 

conditions, such as a large dam built upstream or downstream of the bridge or severe 

straightening of the stream, long-term bed elevation changes can be the larger element in the 

total scour. Also, scour depths from severe contraction of the flow, (often causing ponding 

upstream of the bridge) can be larger than local scour. 

(4) Lateral Shifting of the Stream

In addition to the above, lateral shifting of the stream may also erode the approach roadway 

to the bridge and, by changing the angle of the flow in the waterway at the bridge crossing, 

change the total scour. 

7.3 CLEAR-WATER AND LIVE-BED SCOUR 

The two conditions of general and local scour are (1) clear-water scour and (2) live-bed scour. 

 

Clear-water scour occurs when there is no movement of the bed material in the main channel 

of the stream upstream of the crossing, or the sediment transport in the upstream reach or 

floodplain is transported through the bridge opening or local scour holes in suspension. The 

increase in velocity by contraction of the flow by the bridge or the acceleration of the flow and 

vortices created by the piers or abutments causes the bed material in the bridge opening or at 

their base to move. 
7.5 

Live-bed scour occurs when the bed material upstream of the crossing is moving and moves 

as contact sediment discharge in the contracted bridge opening and/or into the local scour 

holes. 

 

Typical clear-water scour situations include (1) coarse bed material streams, (2) flat gradient 

streams during low flow, (3) local deposits of larger bed materials that are larger than the 

biggest fraction being transported by the flow (rock riprap is a special case of this situation), 

(4) armored stream beds where the only locations that tractive forces are adequate to 

penetrate the armor layer are at piers and/or abutments, (5) vegetated channels where, 

again, the only locations that the cover is penetrated is at piers and/or abutments, and (6) 

streams with fine bed material and large velocity or shear stress whereby the bed material in 

transport washes through a contraction or local scour hole 

Clear-water scour reaches its maximum over a longer period of time than live-bed scour 

(Figure 7.2). This is because clear-water scour occurs mainly on coarse bed material streams. 

In fact, clear-water scour may not reach its maximum until after several floods. Also, maximum 

clear-water scour is about 10 percent greater than the equilibrium live-bed scour. Bridges over 

coarse bed material streams often have clear-water scour at the lower part of a hydrograph, 

live-bed scour at the higher discharges, and then clear-water scour on the falling stages. 

Figure 7.2. Local scour depth at a pier as a function of time. 

Live-bed scour in sand bed streams with a dune bed configuration fluctuates about an 

equilibrium scour depth (Figure 7.2). The reason for this is the fluctuating nature of the 

sediment transport of the bed material in the approaching flow when the bed configuration of 

the stream is dunes. In this case (dune bed configuration in the channel upstream of the 

bridge), maximum depth of scour is about 30 percent larger than equilibrium depth of scour. 
7.6 

The maximum depth of scour is the same as the equilibrium depth of scour for live-bed scour 

with a plane bed configuration. With antidunes occurring upstream and in the bridge crossing; 

the maximum depth of scour, from the limited research of Jain and Fisher (1979), is from 10 to 

20 percent greater than the equilibrium depth of scour. In general, with sand bed streams a 

dune bed changes to plane bed or antidune flow during flood flow. 

7.4 ARMORING 

 

Armoring occurs on a stream or in a scour hole when the forces of the water during a particular 

flood are unable to move the larger sizes of the bed material. This protects the underlying 

material from movement. Contraction scour or local scour around an abutment or pier may 

initially occur, but as the scour depth increases the coarser bed material moves down in the 

bridge cross-section or local scour hole and protects the bed so that the full scour potential is 

not reached. 

When armoring occurs, the coarser bed material will tend to remain in place or quickly 

redeposit so as to form a layer of riprap-like armor in the cross-section or local scour hole and, 

thus, limit further scour for a particular discharge. When larger flows occur the armor layer can 

be broken and the scour depth deepened either until a new armor layer is developed or the 

maximum scour is reached. 

7.5 LONG-TERM BED ELEVATION CHANGES 

Long-term bed elevation changes (aggradation or degradation) may be the natural trend of the 

stream or may be the result of some modification to the stream or watershed condition. In 

scour analyses, only long-term degradation is considered. Aggradation, if it occurs, is not used 

to decrease total scour estimate. 

The streambed may be aggrading, degrading or not changing (equilibrium) in the bridge 

crossing reach. When the bed of the stream is neither aggrading or degrading, it is in 

equilibrium with the sediment discharge supplied to the bridge reach and the elevation of the 

bed does not change. In this section, we consider long-term trends, not the cutting and filling 

of the bed of the stream that might occur during a runoff event. A stream may cut and fill 

during a runoff event and also have a long-term trend of an increase or decrease in bed 

elevation. The problem for the engineer is to determine what the long-term bed elevation 

changes will be during the lifetime of the structure. What is the current rate of change in the 

stream bed elevation? Is the stream bed elevation in equilibrium? Is the streambed 

degrading? Is it aggrading? Is there a head cut or nickpoint moving upstream? What is the 

future trend in the stream bed elevation? 

During the life of the bridge the present trend may change. These long-term changes are the 

result of modifications of the state of the stream or watershed. Such changes may be the 

result of natural processes of the result of human activities. The engineer must assess the 

present state of the stream and watershed and determine future changes in the river system, 

and from this assessment determine the long-term stream bed elevation. 

Factors that affect long-term bed elevation changes are: dams and reservoirs (up or 

downstream of the bridge), changes in watershed land use (urbanization, deforestation, etc.), 

channelization, cutoff of a meander bend (natural or human induced), changes in the 
7.7 

downstream base level (control) of the bridge reach, gravel mining from the stream bed, 

diversion of water into or out of the stream, natural lowering of the total system, movement of a 

bend, bridge location in reference to stream planform, and stream movement in relation to the 

crossing. 

Analysis of long-term streambed elevation changes must be made using the principles of river 

mechanics in the context of a fluvial system analysis. Such analysis of a fluvial system 

requires the consideration of all influences upon the bridge crossing, i.e., runoff from the 

watershed to the channel (hydrology), the sediment delivery to the channel (erosion), the 

sediment transport capacity of the channel (hydraulics) and the response of the channel to 

these factors (geomorphology and river mechanics). Many of the largest impacts are from 

human activities, either in the past, the present or the future. The analysis requires a study of 

the past history of the river and human activities on it; a study of present water and land use 

and stream control activities; and, finally, contacting all agencies involved with the river to 

determine future changes in the river. 

A method to organize such an analysis is to use a three level fluvial system approach (see 

Chapter 9 and HEC-20). This method provides three levels of detail in an analysis: (1) a 

qualitative determination based on general geomorphic and river mechanics relationships; (2) 

engineering geomorphic analysis using established qualitative and quantitative relationships to 

establish the probable behavior of the stream system to various scenarios of future conditions; 

and (3) quantifying the changes in bed elevation using available physical process mathematical 

models such as BRI-STARS (Molinas 2000) or HEC-6, (U.S. Army Corps of Engineers 1993). 

Methods to be used in the three levels of analysis are given in this manual, FHWA’s HEC-18 

(Richardson and Davis 2001) and HEC-20 (Lagasse et al. 2001). 

7.6 GENERAL SCOUR

General scour at a bridge can be caused by a decrease in channel width, either naturally or by 

the bridge, which decreases flow area and increases velocity. This is contraction scour. 

General scour can also be caused by short-term (daily, weekly, yearly or seasonally) changes 

in the downstream water surface elevation that controls the backwater and hence the velocity 

through the bridge opening. Because this scour is reversible it is included in general scour 

rather than in long-term scour. General scour can result from the location of the bridge with 

regard to a bend. If the bridge is located on or close to a bend, the concentration of the flow on 

the outer part of the channel can erode the bed. 

General scour can be cyclic. That is, during a runoff event the bed scours during the rise in 

stage (increasing discharge) and fills on the falling stage (deposition). 

General scour from a contraction occurs when the flow area of a stream is decreased from the 

normal either by a natural constriction or by a bridge. With the decrease in flow area there is 

an increase in average velocity and bed shear stress. Hence, there is an increase in stream 

power at the contraction and more bed material is transported through the contracted reach 

than is transported into the reach. The increase in transport of bed material lowers the bed 

elevation. As the bed elevation is lowered, the flow area increases and the velocity and shear 

stress decreases until equilibrium is reached, that is the bed material transported into the reach 

is equal to that which is transported out of the reach. 
7.8 

The contraction of the flow by the bridge can be caused by a decrease in flow area of the 

stream channel by the abutments projecting into the channel and/or the piers taking up a large 

portion of the flow area. Also, the contraction can be caused by the approaches to the bridge 

cutting off the overland flow that normally goes across the floodplain during high flow. This 

latter case causes clear-water scour at the bridge section because the overland flow normally 

does not transport any bed material sediments. This clear-water flow picks up additional 

sediment from the bed when it returns to the bridge crossing. In addition, if it returns to the 

stream channel at an abutment can increase the local scour there. A guide bank at that 

abutment decreases the risk from scour from this returning overbank flow. Also, relief bridges 

in the approaches decrease the scour problem at the bridge cross section by decreasing the 

amount of flow returning to the natural channel. 

Other Factors that can cause contraction scour are: 

(1) a natural stream constriction 

(2) long approaches over the floodplain to the bridge 

(3) ice formation or jams 

(4) berm forming along the banks by sediment deposits 

(5) island or bar formations upstream or downstream of the bridge opening 

(6) debris 

(7) growth of vegetation in the channel or floodplain 

To determine the magnitude of general scour from a variable backwater requires a study of the 

stream system to (1) determine if this condition exists and (2) determine the magnitude of 

general scour for this condition. Of particular value in determining if backwater effects exist 

and the magnitude of the effects on the velocity and depth is the WSPRO computer model. 

The difference in depth between the highest expected bed elevation and the lowest expected 

bed elevation for the design discharge is the value of the general scour. 

General scour of the bridge opening may be concentrated in one area. If the bridge is located 

on or close to a bend the scour will be concentrated on the outer part of the bend. In fact, 

there may be deposition on the inner portion of the bend, further concentrating the flow, which 

increases the scour at the outer part of the bend. Also at bends, the thalweg (the part of the 

stream where the flow or velocity is largest) will shift toward the center of the stream as the flow 

increases. This can increase scour and the non-uniform distribution of the scour in the bridge 

opening. 

Often the magnitude of general scour cannot be predicted and inspection is the solution for 

general scour problems. Also, a physical model study can be used to determine general scour. 

7.6.1 Contraction Scour 

Contraction Scour Conditions. Contraction scour equations are based on the principle of 

conservation of sediment transport (continuity). In the case of live-bed scour, the fully 

developed scour in the bridge cross section reaches equilibrium when sediment transported 

into the contracted section equals sediment transported out. As scour develops, the shear 

stress in the contracted section decreases as a result of a larger flow area and decreasing 

average velocity. For live-bed scour, maximum scour occurs when the shear stress reaches 

the point that bed-material transported in equals the bed-material transported out and the 

conditions for sediment continuity are in balance. For clear-water scour, the bed-material 
7.9 

transported into the contracted section is essentially zero and maximum scour occurs when 

the shear stress or velocity reaches the critical shear stress or critical velocity of the bed 

material in the section. Chapter 3 and HEC-18 give equations and methods for calculating 

critical shear stress or critical velocity. 

Live-bed contraction scour occurs at a bridge when there is transport of bed material in the 

upstream reach into the bridge cross section. With live-bed contraction scour, the area of the 

contracted section increases until, in the limit, the transport of bed material out of the 

contracted section equals the bed material transported in. Normally, the width of the 

contracted section is constrained and depth increases until the limiting conditions are 

reached. 

Clear-water contraction scour occurs in a long contraction when (1) there is no bed material 

transport from the upstream reach into the downstream reach or (2) the material being 

transported in the upstream reach is transported through the downstream reach mostly in 

suspension and at less than capacity of the flow. With clear-water contraction scour, the 

area of the contracted section increases until, in the limit, the velocity (V) of the flow or the 

shear stress (τo) on the bed is equal to the critical velocity (Vc) or the critical shear stress (τc) 

of a certain particle size (D) in the bed material. Normally, the width (W) of the contracted 

section is constrained and the depth (y) increases until the limiting conditions are reached. 

Live-bed scour depths may be limited if there are appreciable amounts of large-sized 

particles in the bed material. It is appropriate, then, to use the clear-water scour 

equation in addition to the live-bed scour equation and use the smaller of the two 

depths. Also, it is appropriate to use the clear-water scour equation if the transport of 

bed material from upstream of the contraction is small in quantity or composed of fine 

material that washes through the contraction in suspension. 

There are four conditions (cases) of contraction scour at bridge sites depending on the type 

of contraction, and whether there is overbank flow or relief bridges. Regardless of the case, 

contraction scour can be evaluated using two basic equations: (1) live-bed scour, and (2) 

clear-water scour. For any case or condition, it is only necessary to determine if the flow in 

the main channel or overbank area upstream of the bridge, or approaching a relief bridge, is 

transporting bed material (live-bed) or is not (clear-water), and then apply the appropriate 

equation with the variables defined according to the location of contraction scour (channel or 

overbank). 

Critical Velocity. To determine if the flow upstream of the bridge is transporting bed material, 

calculate the critical velocity for beginning of motion Vc of the D50 size of the bed material and 

compare it with the mean velocity V of the flow in the main channel or overbank area 

upstream of the bridge opening. If the critical velocity of the bed material is larger than the 

mean velocity (Vc > V), then clear-water contraction scour will exist. If the critical velocity is 

less than the mean velocity (Vc < V), then live-bed contraction scour will exist. To calculate 

the critical velocity use the equation derived in Chapter 3. This equation is: 

3/16/1 = uc DyKV (7.1) 

 

where: 

 Vc = Critical velocity above which bed material of size D and smaller will be 

transported, m/s (ft/s) 

 y = Depth of flow, m (ft) 
7.10 

 D = Particle size for Vc, m (ft) 

 D50 = Particle size in a mixture of which 50 percent are smaller, m (ft) 

 Ku = Coefficient derived in Chapter 3 

 Ku = 6.19 SI units 

 Ku = 11.25 English units 

Live-Bed Contraction Scour. A modified version of Laursen's 1960 equation for live-bed 

scour at a long contraction is recommended to predict the depth of scour in a contracted 

section. The modification is to eliminate the ratio of Manning's n (see the following Note #3). 

k1

2

1

7/6

1

2

1

2

W

W

Q

Q

y

y 



 



 



 



 = (7.2) 

yyy o2s =−= average scour depth (7.3) 

where: 

 y1 = Average depth in the upstream main channel, m (ft) 

 y2 = Average depth in the contracted section, m (ft) 

 yo = Existing depth in the contracted section before scour, m (ft) (see Note 7) 

 Q1 = Flow in the upstream channel transporting sediment, m3

/s (ft3/s) 

 Q2 = Flow in the contracted channel, m3

/s (ft3/s) 

 W1 = Bottom width of the upstream main channel, m (ft) 

 W2 = Bottom width of the main channel in the contracted section less pier 

width(s), m (ft) 

 k1 = Exponent determined below 

V*/ω k1 Mode of Bed Material Transport 

<0.50 0.59 Mostly contact bed material discharge 

0.50 to 2.0 0.64 Some suspended bed material discharge 

>2.0 0.69 Mostly suspended bed material discharge 

V* = (τo/ρ)

1⁄2 = (gy1 S1)1⁄2, shear velocity in the upstream section, m/s (ft/s) 

ω = Fall velocity of bed material based on the D50, m/s (Figure 3.1) 

For fall velocity in English units (ft/s) multiply fall velocity in m/s by 3.28 

 g = Acceleration of gravity (9.81 m/s2

) (32.2 ft/s2) 

 S1 = Slope of energy grade line of main channel, m/m (ft/ft) 

τo = Shear stress on the bed, Pa (N/m2

) (lb/ft2) 

ρ = Density of water (1000 kg/m3

) (1.94 slugs/ft3) 

Notes:

1. Q2 is the total flow going through the bridge opening. 

2. Q1 is the flow in the main channel upstream of the bridge, not including overbank flows. 

3. The Manning’s n ratio is eliminated in Laursen's live-bed equation to obtain Equation 7.2. 

This was done for the following reasons. The ratio can be significant for a condition of 
7.11 

dune bed in the main channel and a corresponding plane bed, washed out dunes or 

antidunes in the contracted channel. However, Laursen's equation does not correctly 

account for the increase in transport that will occur as the result of the bed planing out 

(which decreases resistance to flow, increases the velocity and the transport of bed 

material at the bridge). That is, Laursen's equation indicates a decrease in scour for this 

case, whereas in reality, there would be an increase in scour depth. In addition, at flood 

flows, a plane bedform will usually exist upstream and through the bridge waterway, and 

the values of Manning’s n will be equal. 

4. W1 and W2 are not always easily defined. In some cases, it is acceptable to use the 

topwidth of the main channel to define these widths. Whether topwidth or bottom width is 

used, it is important to be consistent so that W1 and W2 refer to either bottom widths or 

topwidths. 

5. The average width of the bridge opening (W2) is normally taken as the bottom width, with 

the width of the piers subtracted. 

6. Laursen's equation will overestimate the depth of scour at the bridge if the bridge is 

located at the upstream end of a natural contraction or if the contraction is the result of 

the bridge abutments and piers. At this time, however, it is the best equation available. 

7. In sand channel streams where the contraction scour hole is filled in on the falling stage, 

the y0 depth may be approximated by y1. Sketches or surveys through the bridge can 

help in determining the existing bed elevation. 

8. Coarse sediments in the bed material which armor the bed may limit scour depths with 

live-bed contraction scour. Where coarse sediments are present, it is recommended that 

scour depths be calculated for live-bed scour conditions using the clear-water scour 

equation (given in the next section) in addition to the live-bed equation, and that the 

smaller calculated scour depth be used. 

Clear-water Contraction Scour. The recommended clear-water contraction scour equation is 

based on a development suggested by Laursen (1963). Its development is presented in 

Chapter 3 as part of the development of the critical velocity equation. The equation is: 

7/3

23/2

m

2

u

2 WD

QK

y 













 = (7.4) 

o2s =−= average(yyy scour )m,depth (7.5) 

where: 

 y2 = Average depth in the contracted section after contraction scour, m (ft) 

 Q = Discharge through the bridge or on the set-back overbank area at the 

bridge associated with the width W, m3

/s (ft3/s) 

 Dm = Diameter of the smallest nontransportable particle in the bed material 

(1.25 D50) in the contracted section, m (ft) 

 D50 = Median diameter of bed material, m (ft) 

 W = Bottom width of the contracted section less pier widths, m (ft) 

 yo = Existing depth in the contracted section before scour, m (ft) 

 K = Coefficient derived in Chapter 3 as an extension of critical velocity 
7.12 

 Ku = 0.025 SI units 

 Ku = 0.0077 English units 

For stratified bed material the depth of scour can be determined by using the clear-water 

scour equation sequentially with successive Dm of the bed material layers. 

7.6.2 Computer Models for General Scour 

The above equations give satisfactory but conservative contraction scour depths. Computer 

models, if carefully used by competent hydraulic engineers experienced in their use, will give 

a more precise determination of contraction scour. These models can also be used for other 

general scour depth determinations. The one-dimensional models presently being used and 

maintained are BRI-STARS (Molinas 1990, 2000) and HEC-6 (U.S. Army Corps of Engineers 

1993). Also, the one-dimensional water surface profile models HEC-RAS (U.S. Army Corps 

of Engineers 2001) and WSPRO (Arneson and Sherman 1998) and two-dimensional models 

FESWMS-2D (Froehlich 1996) and RMA-2V (Thomas and McAnally 1985; U.S. Army Corps 

of Engineers 2001) can be used to obtain the input variables for the contraction scour 

equations. 

7.7 LOCAL SCOUR AT PIERS 

7.7.1 Introduction 

Local scour at piers is a function of bed material size, flow characteristics, fluid properties and 

the geometry of the pier. The subject has been studied extensively in the laboratory since the 

research of Dr. Laursen in the late 1940s and 1950s (Laursen 1958, 1960, 1963; Laursen and 

Toch 1956; Richardson and Lagasse 1999). Richardson (1999) gives a brief listing of scour 

investigations in the United States. As a result of the many studies there are many equations. 

In general, the equations are for live-bed scour in cohesionless sand bed streams, and they 

give widely varying results. Since 1988, through the efforts of the USGS and FHWA, a 

considerable number of field measurements of local pier scour depths have been collected 

(Landers and Mueller 1999). The data is given by Richardson and Lagasse (1999), page 585. 

In this section, we give two equations for determining the ultimate local pier scour and an 

equation to determine the topwidth of a local pier scour hole. These equations are as follows: 

1. Colorado State University's (CSU) equation. (Richardson et al. 1975) 

2. FHWA’s HEC-18 equation (Richardson and Davis 2001) 

3. Topwidth equation (Richardson and Abed 1999; Richardson and Davis 2001) 

This discussion of the equations is only for simpler flow conditions and pier geometry. 

Equations and methods to determine local scour depths for piers in more complex flow 

conditions (for example tidal flow) and pier geometry (for example a pier on piles with the pile 

cap at the water surface) are given in HEC-18 (Richardson and Davis 2001). The HEC-18 

equation is a modification of the CSU equation resulting from additional research and field 

measurements that have occurred since 1975. 

 

As explained in Section 7.7.2, FHWA’s HEC-18 equation is recommended for determining the 

ultimate scour depth for both live-bed and clear-water scour. Briaud et al. (1999) present a 

method for determining local pier scour depth in cohesive bed material is given for those 

special occasions when it is assumed that the ultimate scour depth is not needed. For 
7.13 

example, a bridge with foundations in clay that is to be replaced in a few years. However, as 

Briaud et al. (1999) point out the ultimate scour in cohesive material is as deep as in sand bed 

material. 

7.7.2 Comparison of Pier Scour Equations 

Jones (1983) compared many of the more common equations. His comparison of these 

equations is given in Figures 7.3 and 7.4. Some of the equations have velocity as a variable 

(normally in the form of a Froude number). However, some equations, such as Laursen's, do 

not include velocity. A Froude number of 0.3 was used (Fr = 0.3) in Figure 7.3 for purposes of 

comparing commonly used scour equations. In Figure 7.4, the equations are compared with 

some field data measurements. As can be seen from Figure 7.4 the CSU equation encloses 

all the data points, but gives lower values of scour than the other equations. The CSU 

equation includes the velocity of the flow just upstream of the pier by including the Froude 

Number in the equation. 

The equations illustrated in Figures 7.3 and 7.4 do not take into account the possibility that 

larger sizes in the bed material could armor the scour hole. That is, the large sizes in the bed 

material will at some depth of scour limit the scour depth. FHWA’s HEC-18 scour depth 

equation has a coefficient, which is applied to the CSU equation, that decreases the scour 

depth when the bed material has large particles. 

Mueller (1996) compared 22 scour equations using field data collected by the USGS

(Landers and Mueller 1996; Landers, Mueller, and Richardson 1999). He concluded that the 

HEC-18 equation was good for design because it rarely under predicted measured scour 

depth. However, it frequently over predicted the observed scour. The data contained 384 

measurements of scour at 56 bridges. Figure 7.5 gives six of his 22 comparisons. The six 

equations in Figure 7.5 are Shen’s, Froehlich’s, Laursen’s, Melville and Sutherland’s, HEC18, and Mueller’s modified HEC-18 equation (HEC-18BM). 

7.7.3 Colorado State University's Equation 

The Colorado State University's Equation (Richardson et al. 1975) is as follows: 

43.0

1

65.0

1

21

1

s Fr

y

a KK0.2

y

y 



 



 = (7.6) 

where: 

ys = Scour depth, m (ft) 

y1 = Flow depth just upstream of the pier, m (ft) 

K1 = Correction for pier shape from Table 7.1 and Figure 7.6 

K2 = Correction for flow angle of attack of flow from Table 7.2 and Equation 7.8 

a = Pier width, m (ft) 

Fr1 = Froude number = V1 / (gy1) 0.5

V1 = Velocity upstream of pier, m/s (ft/s) 
7.14 

Figure 7.3 Comparison of scour formulas for variable depth ratios (y/a) (Jones 1983). 

Figure 7.4. Comparison of scour formulas with field scour measurements (Jones 1983). 
7.15 

Figure 7.5. Comparison of pier scour equations with field measurements (after Mueller 1996). 
7.16 

7.7.4 FHWA HEC-18 Equation

The FHWA HEC-18 equation (Richardson and Davis 2001) to predict local scour depths at a 

pier, based on the CSU equation, is recommended for both live-bed and clear-water scour. 

The equation predicts maximum pier scour depths. The equation is: 

43.0

1

65.0

1

54321

1

s Fr

y

a KKKKK0.2

y

y 



 



 = (7.7) 

where: 

ys = Scour depth, m (ft) 

y1 = Flow depth just upstream of the pier, m (ft) 

K1 = Correction for pier shape from Table 7.1 and Figure 7.6 

K2 = Correction for flow angle of attack of flow from Table 7.2 and Equation 7.8 

K3 = Correction factor for bed condition from Table 7.3 

K4 = Correction factor for armoring by bed material size from Equation 7.9 

K5 = Correction factor for pier width from Equation 7.13 or 7.14 

L = Length of pier, m (ft) 

Fr1 = Froude Number directly upstream of the pier = V1/(gy1)

1/2

V1 = Mean velocity of flow directly upstream of the pier, m/s (ft/s) 

g = Acceleration of gravity (9.81 m/s2) (32.2 ft/s2) 

For round nose piers aligned with the flow the depth of scour has the following limits. 

ys ≤ 2.4 times the pier width (a) for Fr ≤ 0.8 

ys ≤ 3.0 times the pier width (a) for Fr > 0.8 

The correction factor for angle of attack of the flow K2 given in Table 7.2 can be calculated 

using the following equation: 

65.

2 Sin

a

L CosK 



 



 θ+θ= (7.8) 

If L/a is larger than 12, use L/a = 12 as a maximum in Equation 7.8 and Table 7.2. 
7.17 

Figure 7.6. Common pier shapes. 

Table 7.1. Correction Factor, K1,for

 Pier Nose Shape. 

 Table 7.2. Correction Factor, K2, for Angle of 

 Attack, θ, of the Flow. 

Shape of Pier Nose K1 Angle L/a=4 L/a=8 L/a=12 

(a) Square nose 1.1 0 1.0 1.0 1.0 

(b) Round nose 1.0 15 1.5 2.0 2.5 

(c) Circular cylinder 1.0 30 2.0 2.75 3.5 

(d) Group of cylinders 1.0 45 2.3 3.3 4.3 

(e) Sharp nose 0.9 90 2.5 3.9 5.0 

 Angle = skew angle of flow 

L = length of pier, m 

Table 7.3. Increase in Equilibrium Pier Scour Depths, K3, for Bed Condition. 

Bed Condition Dune Height m K3

Clear-Water Scour N/A 1.1 

Plane bed and Antidune flow N/A 1.1 

Small Dunes 3 > H > 0.6 1.1 

Medium Dunes 9 > H > 3 1.2 to 1.1 

Large Dunes H > 9 1.3 

Notes:

1. The correction factor K1 for pier nose shape should be determined using Table 7.1 

for angles of attack up to 5 degrees. For greater angles, K2 dominates and K1

should be considered as 1.0. If L/a is larger than 12, use the values for L/a = 12 as 

a maximum in Table 7.2 and Equation 7.8. 
7.18 

2. The values of the correction factor K2 should be applied only when the field conditions 

are such that the entire length of the pier is subjected to the angle of attack of the 

flow. Use of this factor will result in a significant over-prediction of scour if (1) a 

portion of the pier is shielded from the direct impingement of the flow by an abutment 

or another pier; or (2) an abutment or another pier redirects the flow in a direction 

parallel to the pier. For such cases, judgment must be exercised to reduce the value 

of the K2 factor by selecting the effective length of the pier actually subjected to the 

angle of attack of the flow. 

3. The correction factor K3 results from the fact that for plane-bed conditions, which is 

typical of most bridge sites for the flood frequencies employed in scour design, the 

maximum scour may be 10 percent greater than computed with Equation 7.6. In the 

unusual situation where a dune bed configuration, with large dunes, exists at a site 

during flood flow, the maximum pier scour may be 30 percent greater than the 

predicted equation value. This may occur on very large rivers, such as the Mississippi. For smaller streams that have a dune bed configuration at flood flow, the dunes 

will be smaller and the maximum scour may be only 10 to 20 percent larger than. 

Correction Factor for Bed Material Size

The correction factor K4 decreases scour depths for armoring of the scour hole for bed 

materials that have a D50 equal to or larger than 2.0 mm and D95 equal to or larger than 20 

mm. The correction factor results from research by Molinas et al. (1998) and Mueller (1996). 

Molinas’ research for FHWA showed that when the approach velocity (V1) is less than the 

critical velocity (Vc90) of the D90 size of the bed material and there is a gradation in sizes in 

the bed material, the D90 will limit the scour depth. Mueller and Jones (1999) developed a K4

correction coefficient from a study of 384 field measurements of scour at 56 bridges. 

if D50 > = 2 mm and D95 > = 20 mm 

then 

15.0

4 R = )V(4.0K (7.9)

 

where: 

0

VV

VV

V

50 95

50

cD icD

icD1

R > − =

−

 (7.10) 

and 

VicDx = Approach velocity corresponding to critical velocity (m/s or ft/sec) for incipient scour in 

the accelerated flow region at the pier for the grain size Dx (m or ft) 
7.19 

x x cD

053.0

x

icD V

a

D 645.0V 



 



 = (7.11) 

 VcDx = Critical velocity (m/s or ft/s) for incipient motion for the grain size Dx (m or ft) 

3/1

x

6/1

cD 1u DyKV x = (7.12) 

y1 = Depth of flow just upstream of the pier, excluding local scour, m (ft) 

V1 = Velocity of the approach flow just upstream of the pier, m/s (ft/s) 

Dx = Grain size for which x percent of the bed material is finer, m (ft) 

Ku = 11.25 SI units 

Ku = 6.19 English units 

Although this K4 provides a good fit with the field data, the velocity ratio terms are so formed 

that if D50 is held constant and D95 increases, the value of K4 increases rather than 

decreases. For field data an increase in D95 was always accompanied with an increase in 

D50. 

The minimum value of K4 is 0.4 and VR must be greater than 0. The bed material size must 

have D50 > 2.0 mm and D95 > 20.0 mm. 

Correction Factor for Wide Piers

Flume studies on scour depths at wide piers in shallow flows and field observations of scour 

depths at bascule piers in shallow flows indicate that existing equations, including the CSU 

equation, overestimate scour depths. Johnson and Torrico (1994) suggest the following 

equations for a K factor to be used to correct Equation 7.7 for wide piers in shallow flow. 

The correction factor to be applied when the ratio of depth of flow (y) to pier width (a) is less 

than 0.8 (y/a < 0.8); the ratio of pier width (a) to the median diameter of the bed material (D50) 

is greater than 50 (a/D50 > 50); and the Froude Number of the flow is subcritical. 

1V/VforF

a

y 58.2K c

65.0

34.0

5  <



 



 = (7.13) 

1V/VforF

a

y 0.1K c

25.0

13.0

5  >



 



 = (7.14) 

 

Engineering judgment should be used in applying K5 because it is based on limited 

data from flume experiments. Engineering judgment should take into consideration 

the volume of traffic, the importance of the highway, cost of a failure (potential loss of 

lives and dollars) and the change in cost that would occur if the K5 factor is used.

7.7.5 Pier Scour in Cohesive Bed Material 

The rate of scour in cohesive silt and clay bed materials is many times slower than for noncohesive materials. The maximum depth of scour at a pier will often be reached during one 
7.20 

runoff event in a sand bed material stream. Whereas, it will take many runoff events to get 

the maximum depth scour at a pier in a stream with clay bed material. However, as Briaud et 

al. (1999) point out, "the maximum depth of scour in sand and in clay appears to be the 

same; this is confirmed by the fact that the HEC-18 equation developed from sand 

experiments fits this data on clay quite well." This data refers to the Briaud et al. flume 

studies of pier scour in clay bed material. Also, in their flume studies of scour at a circular 

cylinder in clay bed material, the maximum scour depth occurred behind the pier. 

Using an Erosion Function Apparatus (EFA) to measure scour in cohesive soils Briaud et al. 

(1999) demonstrated that the scour rate for sand was approximately 1,000 times faster than 

clay. Other researchers and field experience have also demonstrated that the scour rate in 

clay is many times slower than in sand. In addition, Briaud et al. describe the phenomena, 

bonds, and factors that give and affect cohesion in clays. They note "because of the 

number and complexity of these bonds, it is very difficult to predict the critical shear stress for 

clays empirically on the basis of a few index properties." They propose that the critical shear 

stress be measured for the clay bed material at the bridge site directly. The EFA was 

developed to measure the erosion rate directly. Briaud et al. propose a method and 

equations to use this measurement to predict depth of scour corresponding to the duration of 

the flood or the design life of the bridge. 

7.7.6 Pier Scour for Other Pier Geometry, Flow Conditions, and Debris 

 

HEC-18 (Richardson and Davis 2001) gives procedures and equations to determine local 

pier scour depths for piers with exposed footings and piles groups. multiple columns skewed 

to the flow, composite pier configurations (pier on pile cap on piles exposed to the flow), 

debris on piers, and piers subjected to pressure flow. 

7.7.7 Topwidth of Scour Holes 

The topwidth of a scour hole in cohesionless bed material from one side of a pier or footing 

can be estimated from the following equation (Richardson and Abed 1999). 

s θ+= )cotK(yW (7.15) 

where: 

 W = Topwidth of the scour hole from each side of the pier or footing, m (ft)

 ys = Scour depth, m (ft)

 K = Bottom width of the scour hole as a fraction of scour depth

θ = Angle of repose of the bed material ranging from about 30° to 44°

The angle of response of cohesiveness material in air ranges from about 30° to 44°. 

Therefore, if the bottom width of the scour hole is equal to the depth of scour ys (K = 1), the 

topwidth in cohesionless sand would vary from 2.07 to 2.80 ys. At the other extreme, if K = 0, 

the topwidth would vary from 1.07 to 1.8 ys. Thus, the topwidth could range from 1.0 to 2.8 ys

and will depend on the bottom width of the scour hole and composition of the bed material. 

In general, the deeper the scour hole, the smaller the bottom width. In water, the angle of 

repose of cohesionless material is less than the values given for air; therefore, a topwidth of 

2.0 ys is suggested for practical applications (Figure 7.7). 
7.21 

Figure 7.7. Topwidth of scour hole (HEC-18). 

7.7.8 Physical Model Studies 

For unusual or complex pier foundation configurations a physical model study should be 

made. The scale between model and prototype is based on the Froude criteria, that is, the 

Froude number for the model should be the same as for the prototype. In general, it is not 

possible to scale the bed material size. Also, at flood flows in sand bed streams the 

sediment transport conditions will be live-bed and the bed configuration will be plane bed. 

Whereas, in the model live-bed transport conditions may be ripples or dunes. These are 

incomparable pier scour conditions. Therefore, it is recommended that a bed material be 

used that has a critical velocity just below the model velocity (i.e., clear-water scour 

conditions). This will usually give the maximum scour depth; but a careful study of the 

results need to be made by persons with field and model scour experience. For additional 

discussion of the use of physical modeling in hydraulic design, see Chapter 5 and HEC-23 

(Lagasse et al. 2001). 

7.8 LOCAL SCOUR AT ABUTMENTS 

7.8.1 Introduction 

The two causes of scour when the flow is obstructed by the abutment and approach highway 

embankment are: (1) A horizontal vortex starting at the upstream end of the abutment and 

running along the toe of the abutment, and (2) a vertical wake vortex at the downstream end 

of the abutment (Figure 7.8). 
7.22 

Figure 7.8. Schematic representation of abutment scour. 

The horizontal vortex at the toe of the abutment is very similar to the horseshoe vortex that 

forms at piers, and the vertical vortex that forms at the downstream end is similar to the wake 

vortex that forms downstream of a pier, or that forms downstream of any flow separation. A 

large amount of laboratory research has been conducted to determine the depth and location 

of the scour hole that develops for the horizontal (so called horseshoe) vortex that occurs at 

the upstream end of the abutment. From this research, numerous abutment scour equations 

have been developed to predict this scour depth. However, very little field data exist to verify 

the equations. In fact, many of the equations, as explained in Section 7.8.2, are spurious 

correlations for the field case. 

Only since 1992 has research been conducted to develop equations for the field case. 

Maryland SHA has developed an abutment scour program titled ABSCOUR (Chang and 

Davis 1999) based on Laursen’s long contraction scour equations. Sturm (1999a, b) 

developed equations based on his research at Georgia Institute of Technology. 

Kouchakzadeh and Townsend (1999) and Trivino and Richardson (2000) present equations 

based on momentum transfer for the determination of abutment scour depths. These 

equations have had limited testing of computed vs. measured abutment scour and will not be 

given. 

The wake vortex at the downstream end of the abutment also causes abutment failures. 

Sometimes the abutment does not fail but only the approach embankment is eroded. 

Research and the development of methods to determine the erosion from the wake vortex 

has not been conducted. Methods to protect abutments and the embankments against wake 

vortex erosion are given in HEC-23 (Lagasse et al. 2001b). 

In summary, engineering judgment is required in designing foundations for 

abutments. As a minimum, abutment foundations should be designed assuming no 

ground support (lateral or vertical) as a result of soil loss from long-term degradation 

and contraction scour. The abutment should be protected from local scour from both 
7.23 

the upstream and downstream vortexes using riprap and/or guide banks. Guidelines 

for the design of riprap and guide banks are given in HEC-23 (Lagasse et al. 2001b). 

To protect the abutment and approach roadway from scour by the wake vortex several 

SHAs use a 15-meter (50-ft) guide bank extending from the downstream corner of the 

abutment. Otherwise, the downstream abutment and approach should be protected 

with riprap.

 

In the following sections, two equations from HEC-18 are presented for use in estimating 

scour depths as a guide in designing abutment foundations. The methods can be used for 

either clear-water or live-bed scour. 

 

7.8.2 Commentary on Abutment Scour Equations 

Until recently, the equations in the literature were developed using the abutment and 

roadway approach length as one of the variables. This approach results in excessively 

conservative estimates of scour depth. Richardson and Richardson (1993) pointed this out 

in a discussion of Melville's (1992) paper and in a 1999 paper (Richardson and Lagasse 

1999, p. 457). They stated. 

"The reason the equations in the literature predict excessively conservative 

abutment scour depths for the field situation is that, in the laboratory flume, 

the discharge intercepted by the abutment is directly related to the abutment 

length; whereas, in the field, this is rarely the case." 

Figure 7.9 illustrates the difference. Thus, equations for predicting abutment scour would be 

more applicable to field condition if they included the discharge intercepted by the 

embankment rather than embankment length. 

(a) (b)

 Figure 7.9. Comparison of laboratory flow characteristics to field flow conditions. 
7.24 

Abutment scour depends on the interaction of the flow obstructed by the abutment and 

roadway approach and the flow in the main channel at the abutment. The discharge 

returned to the main channel at the abutment is not simply a function of the abutment and 

roadway length in the field case. Richardson and Richardson (1993) noted that abutment 

scour depth depends on abutment shape, discharge in the main channel at the abutment, 

discharge intercepted by the abutment and returned to the main channel at the abutment, 

sediment characteristics, cross-sectional shape of the main channel at the abutment 

(especially the depth of flow in the main channel and depth of the overbank flow at the 

abutment), and alignment. In addition, field conditions may have tree-lined or vegetated 

banks, low velocities, and shallow depths upstream of the abutment. Most of the early 

laboratory research failed to replicate these field conditions. 

7.8.3 Abutment Site Conditions 

Abutments can be set back from the natural stream bank, placed at the bankline or, in some 

cases, actually set into the channel itself. Common designs include stub abutments placed 

on spill through slopes, and vertical wall abutments, with or without wingwalls. Scour at 

abutments can be live-bed or clear-water scour. The bridge and approach road can cross 

the stream and floodplain at a skew angle and this will have an effect on flow conditions at 

the abutment. Finally, there can be varying amounts of overbank flow intercepted by the 

approaches to the bridge and returned to the stream at the abutment. More severe abutment 

scour will occur when the majority of overbank flow returns to the bridge opening directly 

upstream of the bridge crossing. Less severe abutment scour will occur when overbank 

flows gradually return to the main channel upstream of the bridge crossing. 

The skew angle for an abutment (embankment) is depicted in Figure 7.10. For an abutment 

angled downstream, the scour depth is decreased whereas the scour depth is increased for 

an abutment angled upstream. An equation for adjusting abutment scour depth for 

embankment skew is given in Section 7.8.5. 

θ

Figure 7.10. Orientation of embankment angle θ to the flow. 
7.25 

7.8.4 Abutment Shape 

The three general shapes for abutments are: (1) spill-through abutments, (2) vertical walls 

without wing walls, and (3) vertical-wall abutments with wing walls (Figure 7.11). These 

shapes can all have varying angles to the flow. As shown in Table 7.4, depth of scour is 

approximately double for vertical-wall abutments as compared with spill-through abutments 

for very short sections of the abutment and approach road. As the length of the abutment 

and approach road in the floodplain increase, the effect of the spill-through slope is 

decreased. For long approach road sections on the floodplain, this coefficient will approach 

a value of 1.0. Similarly, scour for vertical wall abutments with wingwalls on short abutment 

sections is reduced to 82 percent of the scour of vertical wall abutments without wingwalls. 

As the length of the abutment and approach road in the floodplain increase, the effect of the 

wingwall is decreased. For long approach road sections in the floodplain, this coefficient will 

approach a value of 1.0. 

Figure 7.11. Abutment shape. 

Table 7.4. Abutment Shape Coefficients for Short Abutment Sections. 

Description K1

Vertical-wall abutment 1.00 

Vertical-wall abutment with wing walls 0.82 

Spill-through abutment 0.55 

7.8.5 Froehlich’s Live-Bed Scour Equation 

To determine the potential depth of scour at existing bridges and to aid in the design of 

foundations and placement of rock riprap or guide banks at new bridges, Froehlich's (1989) 

live-bed scour equation or an equation from HIRE (Richardson et al. 1995) can be used. 

Froehlich analyzed by regression analysis, 170 live-bed scour measurements in laboratory 

flumes to obtain the following equation: 
7.26 

1Fr

y

L KK27.2

y

y 61.0

43.0

a

21

a

s

 +



 



 ′ = (7.16) 

where: 

K1 = Coefficient for abutment shape (Table 7.4) 

K2 = Coefficient for angle of embankment to flow 

K2 = (θ/90)0.13 (Figure 7.10 for definition of θ) 

θ<90° if embankment points downstream 

θ>90° if embankment points upstream 

L ́ = Length of abutment projected normal to flow, m, ft 

Ae = Flow area of the approach cross section obstructed by the 

embankment, m2

, ft2

Fr = Froude Number of approach flow upstream of the abutment = Ve/(gya)

1/2

Ve = Qe/Ae, m/s, ft/s 

Qe = Flow obstructed by the abutment and approach embankment, 

m3

/s, ft3/s 

ya = Average depth of flow on the floodplain, m, ft 

ys = Scour depth, m, ft 

Note: That as L ́ tends to 0, ys also tends to 0. In a regression equation, 50 percent of the 

data are above or below the regression line. The 1 was added to the equation so as to 

encompass 98 percent of the data. 

7.8.6 1975 and 1990 HIRE Equation 

An equation in HIRE was developed from Corps of Engineers field data of scour at the end of 

spurs in the Mississippi River (Richardson et al. 1975, 1990). This field situation closely 

resembles the laboratory experiments for abutment scour in that the discharge intercepted by 

the spurs was a function of the spur length. The HIRE equation is applicable when the ratio 

of projected abutment length (a) to the flow depth (y1) is greater than 25. This equation can 

be used to estimate scour depth (y1) at an abutment where conditions are similar to the field 

conditions from which the equation was derived: the equation is: 

55.0

k Fr4

y

y 33.0 1

1

S = (7.17) 

where: 

ys = Scour depth, m, ft 

y1 = Depth of flow at the abutment on the overbank or in the main channel, m, ft 

Fr = Froude Number based on the velocity and depth adjacent to and upstream 

of the abutment 

K1 = Coefficient for abutment shape (Table 7.4) 
7.27 

To correct Equation 7.17 for abutments skewed to the stream, use K2 for Equation 7.16. 

Clear-Water Scour at an Abutment. Use Equations 7.16 or 7.17 for live-bed scour because 

clear-water scour equations potentially decreases scour at abutments due to the presence of 

coarser material. This decrease is unsubstantiated by field data. 

7.9 SCOUR PROBLEMS

Solved problems for contraction, pier, and abutment scour in metric (SI) and English units 

are given in HEC-18 (Richardson and Davis 2001). Example problems for stream instability 

problems are given in HEC-20 (Lagasse et al. 2001a). In addition, guidelines for the design 

of scour and stream instability countermeasures are given in HEC-23 (Lagasse et al. 2001b). 

In Chapter 10 of this manual, three design examples are given which demonstrate the use of 

some of the equations and methods given in this chapter. 
7.28 

(page intentionally left blank) 
8.1 

CHAPTER 8 

DATA NEEDS AND DATA SOURCES

The purpose of this chapter is to identify data needed for calculations and analyses which will 

lead to recommendations for highway crossings and encroachments of rivers. The types and 

amounts of data needed for planning and designing river crossings and lateral encroachments 

can vary from project to project depending upon the class of the proposed highway, the type of 

river and the geographic area. 

8.1 BASIC DATA NEEDS

The data, preliminary calculations, alternative route selections and analyses of these routes 

should be documented in a report. Such a report serves to guide the detailed designs, and 

provides reference background for environmental impact analysis and other needs such as 

application for permits and historical documentation for any litigation which may arise. 

8.1.1 Area Maps

An area map is needed to identify the location of the entire highway project and all streams and 

river crossings and encroachments involved. The purpose of the map is to orient the highway 

project geographically with other area features. The map may be very small scale showing 

towns, cities, mountain ranges, railroads and other highways and roads. The area map should 

be large enough to identify river systems and tributaries. 

8.1.2 Vicinity Maps

Vicinity maps for each river crossing or lateral encroachment are needed to layout the 

proposed highway alignment and alternate routes. There should be sufficient length of river 

included on the vicinity map to enable identification of stream type and to locate river 

meanders, sand bars, and braided channels. Other highways and railroads should be 

identified. The maps should show contours and relief. Intakes for municipal and industrial 

water, diversions for irrigation and power, and navigation channels should be clearly identified. 

Recreational areas such as camping, picnic grounds, beaches, and recreational boat docks 

should be identified. Cultivated areas and urban and industrial areas, in the vicinity of towns 

and cities should be noted on the map. The direction of river flow should of course be clearly 

identified. 

8.1.3 Site Maps

Site maps are needed to determine details for hydraulic, roadway, and structural designs. The 

site map should show detailed contours (1- or 2-foot intervals), vegetation distribution and type, 

and other structures. The site map is used to locate highway approach embankments, piers 

and alignments of piers, channel changes, and protection works. High water lines should be 
8.2 

indicated on the site maps for the purpose of estimating flood flows and distributions across the 

river cross section. 

8.1.4 Aerial and Other Photographs

It is highly desirable in preparing vicinity and site maps that aerial photographs be obtained. 

Multi-image cameras use different ranges of the light spectrum to assist in identifying various 

features such as sewer outfalls, groundwater inflows, types of vegetation, sizes and heights of 

sandbars, river thalwegs, river controls and geologic formations, existing bank protection 

works, old meander channels, and other features. Topographic information can also be 

developed from aerial photographs for vicinity and site maps where such information is not 

readily available. Historic aerial photographs are a valuable resource to assess river planform 

stability. 

Land photographs (as opposed to aerial photos) of existing structures near the crossing are 

always helpful in documentation and evaluation of potential effects of highway construction. 

Photographs of water intake works likely to be affected by the highway project should be 

obtained, and specific data should be noted and briefly discussed. High water marks recorded 

photographically along with dates of occurrence are useful. Photographs aid the designer, who 

may not have the opportunity to visit the site, to visualize crossings and encroachments, and 

they aid documentation. 

Conditions of the river channel in the river reach of concern are easy to record 

photographically, and such pictures can be very helpful in analysis of the river reach. 

Vegetation on floodplains and seasonal variations of vegetation should be recorded 

photographically. Notable geologic formations should be photographed as well and 

supplemented with adequate notes. All photographs should be referenced on the site or on 

vicinity maps. 

8.1.5 Field Inspection

A field inspection of potential highway encroachment sites of rivers should be made prior to the 

analysis. This has been implied in the foregoing paragraphs but is emphasized again because 

of the underlying importance of making first hand appraisals of specific sites before 

conclusions and recommendations are advanced for possible highway routes. Of course, they 

are important in making detailed designs as well, but it is not always feasible to provide 

opportunities for site inspection by the entire design staff. Forms and checklists for a detailed 

geomorphic reconnaissance of the stream corridor are provided in HEC-20 (Lagasse et al. 

2001). 

8.1.6 Geologic Map

A geologic vicinity map, on which geophysical features are indicated, is a basic need. The rock 

formations, outcroppings, and glacial and river deposits which form control points on rivers are 

valuable in analysis of rivers. Soil type determines the size of sediment in transport, infiltration 

rates, and groundwater flows. Channel geometry and roughness are important factors in river 

mechanics. 
8.3 

Soil survey maps with engineering interpretations are available for a significant proportion of 

the United States. They may be helpful in selecting layouts and assessing the suitability of fill 

materials. 

8.1.7 Climatologic Data

Stream gaging stations have been established on many streams throughout the United States. 

However, there are some streams where either a gaging station does not exist near the project 

site or a gaging station does not exist at all. In such cases, it is necessary to estimate flood 

flows. These estimates may be based on regionalized estimating procedures or other 

prediction models using meteorological and watershed data inputs. These meteorological data 

are available from the National Weather Service (NWS) Data Center of the National Oceanic 

and Atmospheric Administration (NOAA), and estimates of average conditions can be made 

from rainfall data published by the NWS. Temperature records are helpful in making snowmelt 

estimates, and wind data are helpful in making wave height estimates on rivers, lakes and 

reservoirs as well as for coastal areas. 

8.1.8 Hydraulic Data

Whenever possible, sediment load data should be provided as auxiliary data for river analyses. 

Bed-material load, suspended load and wash load data may be obtained for some rivers in the 

water supply papers published by the U.S. Geological Survey, state engineers' reports, flood 

control and other water resources investigation reports. Information may also be obtained by 

direct sampling of the river. 

Riverbed cross sections and profiles may be obtained with an ultrasonic depth sounder and are 

helpful in sediment transport and backwater studies. It is also helpful to know water 

temperatures. Direct measurement of flood flows should be made when historical records may 

be deficient. Depth and velocity measurements need to be made at a sufficient number of 

subsections in a cross section to determine total flow rate. Discharge measurements made at 

various stages at a gaging site can provide data for developing a stage-discharge rating curve. 

Observations of high water marks along the river reach should be made. Each high water 

mark and relevant profile should be established. These are helpful in calculating historical 

flood discharges. Also, stages achieved by ice jams at specific locations should be noted. 

Records of the performance of existing bridges and other drainage structures should be 

obtained. Data on scour at piers of existing bridges (or at bridges which have failed) in the 

vicinity should be obtained. For bridges which have failed, as much information as possible 

should be obtained relative to direction of flow (angle of attack) at the piers or embankment 

ends. Flood duration, debris in the river, distribution of flows, and magnitudes of scour are 

useful information. Historical records of damage to adjacent property and results of legal 

actions brought about because of damage are useful information also. 
8.4 

8.1.9 Hydrologic Data

The purpose of hydrologic data is to determine the stream discharge, flood magnitudes, and 

duration and frequencies of flood prior to analysis of river behavior and design of the river 

encroachments and crossings. Hydrologic data and hydraulic analyses should be documented 

in report form for project development. After construction, the documentation would be helpful 

in evaluating any damage from floods and failures, in the event they occur, and providing 

background for any litigation which may arise as a consequence. 

Sometimes a highway crossing and/or encroachment may have a significant effect on flood 

hydrographs and a hydrologic analysis should be made to determine the level of significance. 

This analysis would involve hydrograph development and flow routing within the zone of 

influence of such highway structures. 

The basic data needed are stream discharge data at the nearest gaging station, historical 

floods and highwater marks. It is also desirable to prepare a drainage map for the region 

upstream of the proposed highway project, with delineation of size, shape, slope, land use, and 

water resource facilities such as storage reservoirs for irrigation and power and flood control 

projects. It is desirable whenever possible to obtain flood histories of the river from residents 

and accounts by the news media, particularly for events prior to stream gaging records. 

Estimates of flood discharge can be made from these accounts which are valuable in 

flood-frequency analysis. 

A flood-frequency curve is prepared from recorded stream flow data and augmented by 

estimated discharges (using Manning's equation or equivalent) from high water marks. 

Several methods ranging from sophisticated stochastic analysis to simple methods have been 

developed. The greatest difficulty in constructing a flood-frequency curve is lack of sufficient 

and reliable data. Approximate methods for extrapolating the range of flood-frequency curves 

are available but are not discussed in detail here (see HDS-2, FHWA 1961). 

A simple graphical method based on extreme value theory is reasonably satisfactory. The 

method consists of ordering the annual peak flood discharges of record from the largest to 

smallest, irrespective of chronological order. The annual (flood) discharge is plotted against its 

recurrence interval on special probability (Gumbel, or other) paper. The recurrence interval, RI 

is calculated from 

m

1n RI + = (8.1) 

in which n is the number of years of records, and m is the order (largest flood is ranked 1) of 

the flood magnitude. Thus, the highest flood discharge would have a recurrence interval of n + 

1 years and lowest would have a recurrence interval of (1+1/n) years. The U.S. Water 

Resources Council (1981) has adopted the log-Pearson III distribution for use as a base 

method for determining flood flow frequencies. Details of the method and plotting paper may 

be obtained from the U.S. Geological Survey in Bulletin 17B. 

When adjusting discharge records from a nearby gaging station to the project site, the flood 

peaks are often prorated on the basis of drainage area ratios. Depending on drainage basin 

characteristics, the exponent of the ratio varies from 0.5 to 0.8. Slope-area calculations for 
8.5 

peak discharges can also be used. In using this method, the conveyance of the channel is 

calculated using the Manning equation in which the roughness coefficient, n, needs to be 

estimated from the discussion presented in Chapters 2 and 3. By referring to a catalog of 

(color) photographs, similar channel situations to the specific site can be identified and a 

relatively inexperienced engineer may make a reliable estimate for n. 

Whatever approach is used, the reader is cautioned not to blindly accept computer printout as 

the final answer in estimating a flood frequency relationship. The data should be plotted on 

probability paper as analyzed by several commonly used methods. Sometimes paleo (ancient) 

hydrology techniques need to be employed to resolve historic outliers at very sensitive sites. 

8.1.10 Environmental Data

In making environmental impact analyses of highway projects on streams and rivers, it is 

necessary to obtain water quality and biological data for the streams. Such data are not readily 

available for many rivers. Municipal water and sewage treatment facilities and industrial plants 

utilizing river water should have recent records regarding river water quality which will be 

helpful in making comprehensive environmental analyses. Water quality data for certain rivers 

can be obtained from the U.S. Geological Survey. Wildlife information such as migration 

patterns of deer and elk should be determined and local game refuges should be located. 

Information regarding fishes and their river habitat should be obtainable from the state fish and 

game agencies. Species of trees and other vegetation should be determined, and some 

information regarding sensitivity of the flora to auto emissions should be obtained. Data should 

also be obtained in order to enable assessment of stream turbidity during and after highway 

construction. Information on soil type to be used in construction of embankments would be 

helpful in this regard. 

8.2 CHECKLIST OF DATA NEEDS

As an aid in collecting data preparatory to analysis of rivers and highway encroachment of 

rivers, the relevant types of data have been listed in Table 8.1. There may be more data items 

included in this table than are needed for a given project site, and some judgment is required. 

For data which are not available, the checklist should be helpful for planning a field 

investigation or other data acquisition program. 

8.3 DATA SOURCES

The best data sources are national data centers where the principal function is to disseminate 

data. But it might be necessary to collect data from a variety of other sources such as from a 

field investigation, interviews with local residents, and a search through library material. 

Detailed information on the location of these federal agencies across the U.S. is available in 

Appendix A of the manual HEC-19, (FHWA 1984). The list of sources in Table 8.2 is provided 

to serve as a guide to the data collection task.
8.6 

Table 8.1. Checklist of Data Needs. 

Maps and Charts: 

(1) Geographic 

(2) Topographic 

(3) Geologic 

(4) Navigation Charts 

(5) Potamology Surveys 

(6) County and City Plats

Aerial and Other Photos: 

(1) Large Scale Photos for Working Plans 

(2) Small Scale Stereo Pairs of River and Surrounding Terrain 

(3) Color Infrared Photos for Flow Patterns, Scour Zones, and Vegetation 

(4) Ground Photos 

(5) Underwater Photos 

Information on Existing Structures, Bridges, Dams, Diversion, or Outfalls: 

(1) Plans and Details 

(2) Construction Details 

(3) Alterations and Repairs 

(4) Foundations 

(5) Piers and Abutments 

(6) Scour 

(7) Dikes 

(8) Field Investigations: 

• Bridge structure and repairs to bridge and approach 

• Damage due to ice or debris 

Hydraulic, Hydrology, and Soils: 

(1) Discharge Records 

(2) Stage-Discharge Records 

(3) Flood Frequency Curves for Stations Near Site 

(4) Flow Duration Curves (hydrographs) 

(5) Newspaper, Radio, Television, Accounts of Large Floods 

(6) Channel Geometry: 

• Main channel 

• Side channel 

• Navigation channel 

• Floodplain 

• Slopes 

• Backwater calculation 

• Bars 

• Sinuosity 

• Type (braided, meandering, straight) 

• Controls (falls, rapids, restriction, rock outcropping dams, diversions) 

(7) Sediment Discharge: 

• Size distribution 

• Bed and Bank Material Sizes 

• Roughness Coefficient n 

(8) Ice: 

• Recorded thickness 

• Dates of freeze up and break up 

• Flow patterns and jams 
8.7 

Table 8.1. Checklist of Data Needs. 

Hydraulic, Hydrology, and Soils (continued): 

(9) Regulating Structures: 

• Dams, diversions 

• Intake, outfalls 

• Scour survey around existing piers, abutments, spur dikes 

• Inspect and photograph stabilization works, riprap sizes, filter blankets 

• Check wells for groundwater levels in areas 

• Install gaging stations 

(10) Soils Information: 

• Excavation data 

• Borrow pits 

• Gravel pits 

• Cuts 

• Tunnels 

• Core boring logs 

• Well drilling logs 

• Soil tests 

• Permeability 

• Rock for riprap 

(11) Planned and Anticipated Water Resources Projects 

(12) Lakes, Tributaries, Reservoirs or Side Channel Impoundments 

(13) Field Surveys: 

• Onsite inspections and photographs 

• Samples of sediments 

• Measure water and sediment discharge 

• Observe channel changes or realignment since last maps or photos 

• Identify high water lines or debris deposits due to recent floods 

• Check magnitude of velocities and direction of flow in vicinity of proposed structure 

• Outcroppings 

• Subsurface Exploration 

Climatological Data: 

(1) National Weather Service Records for Precipitation 

(2) Wind 

(3) Temperatures 

Land Use: 

(1) Zoning Maps 

(2) Recent Aerial Photographs 

(3) Planning Committee Records 

(4) Urban Areas 

(5) Industrial Areas 

(6) Recreational Areas 

(7) Primitive Areas 

(8) Forests 

(9) Vegetation 
8.8 

Table 8.2. List of Data Sources. 

Topographic Maps: 

(1) Quadrangle maps -- U.S. Department of the Interior, Geological Survey, Topographic Division; and 

U.S. Department of The Army, Army Map Service 

(2) River plans and profiles -- U.S. Department of the Interior, Geological Survey, Conservation Division 

(3) National parks and monuments -- U.S. Department of the Interior, National Park Service 

(4) Federal reclamation project maps -- U.S. Department of the Interior, Bureau of Reclamation 

(5) Local areas -- commercial aerial mapping firms 

(6) American Society of Photogrammetry

Planimetric Maps: 

(1) Plans of public land surveys -- U.S. Department of the Interior, Bureau of Land Management 

(2) National forest maps -- U.S. Department of Agriculture, Forest Service 

(3) County maps -- State Highway Agency 

(4) City plans -- city or county recorder 

(5) Federal reclamation project maps -- U.S. Department of the Interior, Bureau of Reclamation 

(6) American Society of Photogrammetry 

(7) ASCE Journal -- Surveying and Mapping Division

Aerial Photographs: 

(1) The following agencies have aerial photographs of portions of the United States: U.S. Department of 

the Interior, Geological Survey, Topographic Division; U.S. Department of Agriculture, Commodity 

Stabilization Service, Soil Conservation Service and Forest Service; U.S. Air Force; various State 

agencies; commercial aerial survey and mapping firms; National Oceanic and Atmospheric 

Administration 

(2) American Society of Photogrammetry 

(3) Photogrammetric Engineering 

(4) Earth Resources Observation System (EROS); Photographs from Gemini, Apollo, Earth Resources, 

Technology Satellite (ERTS) and Skylab 

(5) City or County Records 

(6) State Highway Agency

Transportation Maps: 

(1) State Highway Agency. 

(2) Large Cities

Triangulation and Benchmarks: 

(1) State Engineer. 

(2) State Highway Agency. 

(3) Cities 

Geologic Maps: 

(1) U.S. Department of the Interior, Geologic Survey, Geologic Division; and State Geological Surveys 

 Departments. (Note - some regular quadrangle maps show geological data also.)

Soil Data: 

(1) County soil survey reports -- U.S. Department of Agriculture, Soil Conservation Service. 

(2) Land use capability surveys -- U.S. Department of Agriculture, Soil Conservation Service. 

(3) Land classification reports -- U.S. Department of the Interior, Bureau of Reclamation. 

(4) Hydraulic laboratory reports -- U.S. Department of the Interior, Bureau of Reclamation. 

(5) State Universities and State Agricultural and Conservation Agencies.
8.9 

Table 8.2. List of Data Sources. 

Climatological Data: 

(1) National Weather Service Data Center. 

(2) Hydrologic bulletin -- U.S. Department of Commerce, National Oceanic and Atmospheric 

Administration. 

(3) Technical papers -- U.S. Department of Commerce, National Oceanic and Atmospheric 

Administration. 

(4) Hydrometeorological reports -- U.S. Department of Commerce, National Oceanic and Atmospheric 

Administration, and U.S. Department of the Army, Corps of Engineers. 

(5) Cooperative study reports -- U.S. Department of Commerce, National Oceanic and Atmospheric 

 Administration and U.S. Department of the Interior, Bureau of Reclamation.

Stream Flow Data: 

(1) Water supply papers -- U.S. Department of the Interior, Geological Survey, Water Resources 

Division. 

(2) Reports of State Engineers. 

(3) Annual reports -- International Boundary and Water Commission, United States and Mexico. 

(4) Annual reports -- various interstate compact commissions. 

(5) Hydraulic laboratory reports -- U.S. Department of the Interior, Bureau of Reclamation. 

(6) Corps of Engineers, U.S. Army, Flood control studies. 

(7) Tennessee Valley Authority. 

(8) State Highway Agency. 

(9) USGS, FEMA Flood Studies. 

(10) University Studies

Sedimentation Data: 

(1) Water supply papers -- U.S. Department of the Interior, Geological Survey, Quality of Water branch. 

(2) Reports -- U.S. Department of the Interior, Bureau of Reclamation; and U.S. Department of the 

Agriculture, Soil Conservation Service. 

(3) Geological Survey Circulars -- U.S. Department of the Interior, Geological Survey.

Quality of Water Reports: 

(1) Water supply papers -- U.S. Department of the Interior, Geological Survey, Quality of Water Branch. 

(2) Reports -- U.S. Department of Health, Education, and Welfare, Public Health Service. 

(3) Reports -- State Public Health Departments. 

(4) Water Resources Publications -- U.S. Department of the Interior, Bureau of Reclamation. 

(5) Environmental Protection Agency, regional offices. 

(6) State Water Quality Agency. 

Irrigation and Drainage Data: 

(1) Agricultural census reports -- U.S. Department of Commerce, Bureau of the Census. 

(2) Agricultural Statistics -- U.S. Department of Agriculture, Agricultural Marketing Service. 

(3) Federal Reclamation Projects -- U.S. Department of the Interior, Bureau of Reclamation. 

(4) Reports and Progress Reports -- U.S. Department of the Interior, Bureau of Reclamation. 

Power Data: 

(1) Directory of Electric Utilities -- McGraw Hill Publishing Co. 

(2) Directory of Electric and Gas Utilities in the United States -- Federal Power Commission. 

(3) Reports -- various power companies, public utilities, State power commissions, etc. 
8.10 

Table 8.2. List of Data Sources. 

Basin and Project Reports and Special Reports: 

(1) U.S. Department of the Army, Corps of Engineers. 

(2) U.S. Department of the Interior, Bureau of Land Management, Bureau of Mines, Bureau of 

Reclamation, Fish and Wildlife Service, and National Park Service. 

(3) U.S. Department of Agriculture, Soil Conservation Service. 

(4) U.S. Department of Health, Education, and Welfare, Public Health Service. 

(5) State Departments of Water Resources, Departments of Public Works, power authorities, and 

planning commissions. 

Environmental Data: 

(1) Sanitation and public health -- U.S. Department of Health, Education, and Welfare, Public Health 

Service; State Departments of Public Health. 

(2) Fish and Wildlife -- U.S. Department of the Interior, Fish and Wildlife Service; State Game and Fish 

Departments. 

(3) Municipal and Industrial Water Supplies -- City Water Departments; State Universities; Bureau of 

Business Research; State Water Conservation Boards or State Public Works Departments; State 

Health Agencies; Environmental Protection Agency, Public Health Service. 

(4) Watershed Management -- U.S. Department of Agriculture, Soil Conservation Service, Forest 

Service; U.S. Department of the Interior, Bureau of Land Management, Bureau of Indian Affairs. 

(5) State Highway Administration. 

8.4 COMPUTERIZED LITERATURE AND DATA SEARCH

Recent literature information can be retrieved from computerized databases of technical 

information. The principal databases related to highway and river environment are: 

COMPENDEX, ENVIRONMENTAL BIBLIOGRAPHY, CIVIL ENGINEER DATABASE, WATER 

RESOURCES ABSTRACTS, TRIS, and GeoRef. 

8.4.1 COMPENDEX

1970 - Present, monthly updates (Engineering Information, Inc., Hoboken, New Jersey). The 

COMPENDEX database is the electronic version of the Engineering Index (Monthly/Annual), 

which provides abstracted information from the world's significant engineering and 

technological literature. The COMPENDEX database provides worldwide coverage of 

approximately 3,500 journals and selected government reports and books. 

8.4.2 ENVIRONMENTAL BIBLIOGRAPHY 

1971 - Present, bimonthly updates (Environmental Studies Institute of the International 

Academy, Santa Barbara, California). The ENVIRONMENTAL BIBLIOGRAPHY covers the 

fields of general human ecology, atmospheric studies, energy, land resources, water 

resources, and nutrition and health. More than 400 periodicals are currently indexed in 

ENVIRONMENTAL BIBLIOGRAPHY and the database includes more than 1,000 journals. 
8.11 

8.4.3 CIVIL ENGINEERING DATABASE 

1973 - Present, The Civil Engineering Database is designed to provide easy bibliographic 

access to all American Society of Civil Engineers ASCE publications. The database covers 

ASCE documents published since 1973. It provides access to all the journals, conference 

proceedings, books, standards, manuals, magazines, and newsletters. Engineering subject 

areas include: Aerospace Engineering, Architectural Engineering, Bridges, Cold Regions, 

Computer Practices, Construction, Earthquake Engineering, Education, Engineering 

Mechanics, Environmental Engineering, Forensic Engineering, Geotechnical Engineering, 

Highways, Hydrology, Hydraulics, Irrigation and Drainage, Management, Materials 

Engineering, Structural Engineering, Transportation, Urban Planning, Water Resources, 

Waterway, Port, Coastal, and Ocean Engineering. 

8.4.4 WATER RESOURCES ABSTRACTS

1967 - Present, monthly updates (U.S. Dept. of the Interior, Washington, D.C.) WATER 

RESOURCES ABSTRACTS is prepared from materials collected by over 50 water research 

centers and institutes in the United States. The file covers a wide range of water resource 

topics including water resource economics, ground and surface water hydrology, metropolitan 

water resources planning and management and water-related aspects of nuclear radiation and 

safety. The collection is particularly strong in the literature on water planning (demand, 

economics, cost allocations), water cycle (precipitation, snow, groundwater, lakes, erosion, 

etc.), and water quality (pollution, waste treatment). WRA covers predominantly 

English-language materials and includes monographs, journal articles, reports patents, and 

conference proceedings. 

8.4.5 TRIS

TRIS is the Transportation Research Information Services data base, a computerized 

information file maintained and operated by the Transportation Research Board (TRB) of the 

National Research Council. It is sponsored by the U.S. Department of Transportation's 

Federal Highway Administration, Federal Transit Administration, and National Highway Traffic 

Safety Administration; the highway and transportation departments of the fifty states, the 

District of Columbia and Puerto Rico; the U.S. Army Corps of engineers; the American 

Automobile Manufacturers Association; the National Asphalt Pavement Association; and the 

Association of American Railroads. 

TRIS contains information on transportation modes and practices, including planning, design, 

finance, construction, maintenance, equipment, traffic, operations, management, and 

marketing. TRIS contains more than 350,000 abstracts of completed research and summaries 

of research projects in progress. More than 1,000 journals are scanned for TRIS. Primary 

domestic information sources include the U.S. Department of Transportation's Federal 

Highway Administration and Federal Transit Administration; the U.S. Congress; the U.S. 

General Accounting Office; trade and professional associations; universities; research 

institutes; and other regional and state organizations. International information sources include 

the International Union of Public Transport, the European Conference of Ministers of 

Transport, and the Organization for Economic Cooperation and Development. 
8.12 

8.4.6 GeoRef 

GeoRef is an international resource for bibliographic information about geology and 

geosciences. It is a comprehensive database of citations established by the American 

Geological Institute (AGI) in 1966. The purpose of GeoRef is to facilitate research and 

development and to meet the needs of the geoscience community. GeoRef provides access 

to over 1.8 million references to articles, books, maps, conference papers, reports, and theses, 

covering the geology of North America from 1785 to the present. It includes more than 3,500 

journals in 40 languages and references to all publications of the U.S. Geological Survey. 
9.1

CHAPTER 9

DESIGN CONSIDERATIONS FOR HIGHWAY ENCROACHMENTS

AND RIVER CROSSINGS

9.1 INTRODUCTION

The objective of this chapter is to present applications of the fundamentals of hydraulics, 

hydrology, fluvial geomorphology, and river mechanics to the hydraulic and environmental 

design of river crossings and highway encroachments. The principal factors to be

considered in design are presented, followed by a discussion of the procedures

recommended for the evaluation, analysis and design of river crossings and encroachments. 

The design of most complex problems in river engineering can be facilitated by a qualitative 

evaluation combined with a quantitative analysis. In most cases, the systematic approach of 

a qualitative assessment of channel response, followed by a quantitative estimate, is 

necessary for a meaningful analysis of complex river response problems.

This chapter contains several hypothetical cases of river environments and their response to 

crossings and encroachments based upon geomorphic principles given in Chapter 5. These 

cases indicate the trend of change in river morphology for given initial conditions. The 

hypothetical cases are followed by actual case histories for river crossings in the United 

States. These histories document river response to highway crossings and encroachments 

and illustrate river response qualitatively.

This chapter uses two types of examples (conceptual and actual examples) related to river 

crossings and highway encroachments. Applications of the basic principles developed in 

Chapters 1 through 8 are illustrated by these conceptual examples and specific case

histories related to the subject matter of this manual.

9.2 PRINCIPAL FACTORS TO BE CONSIDERED IN DESIGN

Identification of the principal factors to be considered in design of river crossings and

encroachments is useful. These factors are generally interrelated, and fundamental

mechanisms and relationships of the governing physical processes must be clearly

understood prior to design.

9.2.1 Types of Rivers

In selecting the site for a crossing or an encroachment on a river it is necessary to give 

detailed consideration and study to the type of river or rivers involved. A sandbed river may be 

meandering, it may be essentially straight, or it may be braided. In addition, a meandering 

river may be small, medium, or large. The same channel can be classified as youthful, 

mature, or old. Each of these different river types requires different design procedures. For 

example, in designing training works for large sandbed channels (braided or meandering) it is 

unlikely that Kellner jetties alone will be useful to stabilize the bank alignment (see Chapter 6). 

It may be necessary to stabilize the banks with rock riprap and to control the overbank flows 

using jetties to achieve a set of specific purposes. Gravel and cobble bed channels are 
9.2

normally considerably steeper than sandbed channels and in general have narrower river 

valleys. In the extreme are torrential rivers, the beds of which are comprised of large rocks. 

This type of river usually exists in a youthful or canyon type environment near the upper end of 

large river systems where the slopes are relatively steep.

9.2.2 Location of the Crossing or Encroachment

In selecting the site of a crossing or a longitudinal encroachment, several considerations are 

necessary. First, the crossing or encroachment must mesh with the transportation system in 

the area. Second, environmental factors must be considered. In fact, unless appropriate 

weight is given to the environmental impacts it may not be possible to obtain permission to 

proceed with the project at all. Economic considerations are equally important. Depending 

upon the characteristics of the river and the environmental considerations, the cost of a 

particular crossing or encroachment can be significantly affected by its location. The length 

of the approaches versus the length of the crossing, the cost of real estate that must be 

acquired to accomplish the crossing, the maintenance cost required to keep the crossing 

functional over its estimated life, and the method of construction are some of the specific 

aspects that should be considered in locating the crossing. The cost of protective measures 

should also be considered in locating an encroachment.

9.2.3 River Characteristics

The subclassifications of river form can be utilized to identify the range of conditions within 

which the particular river operates. It is necessary to determine if a river is relatively stable in 

form or is likely to be unstable. In Figure 1.3 of Chapter 1, it was pointed out that rivers can be 

essentially poised so that a small change in discharge characteristics can change a river 

from meandering to braided or vice versa. It is important to know the sensitivity of any river 

system to change. Criteria given in Chapter 5, for example Figure 5.18, or Chapter 3, Figure 

3.13, can be used to predict this sensitivity. A meandering stream whose slope and

discharge plot close to the braided river line in Figure 5.18 may change to a braided stream 

with a small increase in discharge or slope (see also Section 5.5.3).

In addition to river form, it is important to determine other characteristics of the channel: that 

is, the channel may have a sand bed and cohesive banks; it may be formed in cobbles or it 

may be formed in other combinations of these materials. Each of these river systems 

behaves differently depending upon the characteristics of the floodplain material, the bank 

material, and the bed material of the river both over the short term and the long term. Hence, 

a detailed survey of the characteristics of the bed and bank material coupled with river form 

plus other pertinent information are essential to design.

9.2.4 River Geometry

For planning a river crossing or an encroachment it is important to know the river geometry 

and its variation with discharge and time. It is essential to know the slope of the channel and, 

preferably, the energy gradient through the reach. In Chapter 5, relations were presented that 

illustrate how width and depth vary with stage at-a-section as well as along the length of a
9.3

channel. For most rivers, if the appropriate hydraulic and hydrologic data are available, it is 

possible to develop simple relations showing how width and depth vary with discharge.

9.2.5 Hydrologic Data

It is necessary to gather all of the hydrologic data pertinent to the behavior of the river and to 

the design of the river crossing or encroachment. As pointed out in Chapter 8, records of the 

flood flows are essential. From such information, flow duration curves can be developed, 

seasonal variations in the river system can be considered, and design discharge values can 

be established depending upon the discharge frequency criteria used in the design. 

Also, it is important to consider the low flows that the river channel will be subjected to and the 

possible changes in flow conditions that may be imposed on the river system as a

consequence of water resources development in the area. Sometimes low flows may lead to 

a more severe local scour situation at bridge piers and footings. Finally, in terms of

hydrologic data it is usually necessary to synthesize some of the required data. Conventional 

techniques may be used to fill in missing records or it may be essential to synthesize records 

where limited hydrologic data exist. In synthesizing data it is very important to compare the 

particular watershed with other watersheds having similar characteristics. With this

information, reasonably good estimates of what can be anticipated at the site can be

established.

9.2.6 Hydraulic Data

At the site of a crossing or a longitudinal encroachment, it is essential to know the discharge 

and its variation over time. Coupled with this, it is necessary to know the velocity distribution 

across the river cross-section and its variation in the river system. This involves determining 

the type of velocity distribution across the channel as well as in the vertical. Knowledge of the 

distribution of velocities should be coupled with a study of changes in position of the thalweg 

to estimate the severity of attack that may occur along the river banks and in the vicinity of the 

crossing. Furthermore, it is essential to develop stage-discharge relations since these 

relations fix key elevations of the structure in design and serve as bench-mark data when 

considering channel protection measures that may alter the stage of the river. Large changes 

in velocity can occur in a river system with changing discharge and stage. In a sandbed river, 

as flow conditions bring about a transition from lower regime to upper regime, the average 

velocity in the cross section may actually double. From another viewpoint, changes induced 

in the river system, such as those due to artificial cutoffs or channelization, may sufficiently 

steepen the gradient so the river operates in upper regime over its whole range of discharge. 

These possibilities must be considered in the detailed design.

9.2.7 Characteristics of the Watershed 

The water flowing in the river system and the sediment transported with the flow are usually 

intimately related to the watershed feeding the river system. Consequently, one needs to 

study the watershed considering its geology, geometry and land use. In the case of 

development, land uses include recreation, industrial development, urbanization, flood control, 

agriculture, and grazing. Similarly, one needs to consider the vegetation cover on the

watershed and the watershed response to changes in vegetation cover by human activities or 

by climatic changes. 
9.4

Significant changes in vegetation cover affect the amount of sediment delivered from the 

watershed to the river system. One of the most common techniques to study the sources of 

sediment in a watershed is to employ aerial photography and remote sensing techniques 

coupled with ground investigations. The utilization of remote sensing techniques enables the 

skilled observer to determine which areas of the watershed are stable and which are

unstable. Viewing the total watershed from this perspective and using water and sediment 

routing techniques, it is possible to evaluate the sediment yield as a function of time. 

9.2.8 Flow Alignment

In order to design a safe crossing or longitudinal encroachment, it is necessary to consider 

the flow alignment in detail. The direction of flow must be considered as a function of time. 

The position of the thalweg will vary with low, intermediate and high stages. The changing 

characteristics of the river with stage, such as the change in velocity distribution, the position 

of the thalweg, and the river form can have a significant effect on the intensity of attack on the 

approaches, the abutments, the piers and embankments. The detailed study of the behavior 

of the river over time and with varying discharge is necessary for proper design of training 

works. Two-dimensional computer modeling, or in some cases physical modeling, as

discussed in Chapter 5 can provide the detailed hydraulic data to support design. Only with 

this type of information can one adequately consider the intensity of attack, the duration of 

attack and the necessity for training works to make the river system operate within a range of 

conditions acceptable at the crossing or encroachment. Certainly, changes over time at a 

particular crossing affect the channel geometry, the geometry of the crossing itself, general 

scour and local scour. If the characteristics of the flow and how they vary with time are 

known, then the information in Chapter 6 can be utilized to design against excessive

contraction and local scour in order to make the highway functional with minimum

maintenance over the life of the project.

9.2.9 Flow on the Floodplain

Up to this point the concern has been principally with flow in the main channel. However, 

design floods usually flow in both the main channel and on the floodplain. Only by studying 

the characteristics and geometry of the river and the floodplain can one determine the type of 

flows that are apt to occur on the floodplain. This particular topic should be studied in 

adequate detail so that the magnitude and intensity of the flows on the floodplain can be 

approximated. The characteristics of flow on the floodplain are especially relevant to the 

design of longitudinal encroachments. As an example, consider a sinuous channel. At flood 

stage there is a tendency for the water to flow in the main channel in such a way as to 

develop chute channels across the point bars. Often, the water spills over the outside of the 

bends onto the floodplain. Flow conditions on the floodplain and in the main channel can be 

greatly different at flood stage than at low flow, and this must be taken into consideration. 

Again, 2-dimensional computer modeling, as discussed in Chapter 5, can provide detailed 

hydraulic data on overbank and floodplain flows.
9.5

9.2.10 Site Selection

Most of the factors cited in the preceding sections have a bearing on the final site selection. 

In summary, such factors as the form of the river, the alignment of the river, variations of the 

river form over time, the type of bed and bank material, the hydrologic and hydraulic 

characteristics of the river, and past, present and future watershed conditions are all

important inputs to the site selection. In addition, it is necessary to consider the requirements 

of the area to be served and the economic and environmental factors that relate to the 

crossing. Having made a detailed study of possible alternate sites, and having determined 

the best site considering these important factors, one can then proceed with the

determination of the geometry and length of the approaches to the crossing, the type and 

location of the abutments, the number and location of the piers, and the depth of structural 

support for the piers to insure against danger from local scour. The location of the

longitudinal encroachment in the floodplain, the amount of allowable longitudinal

encroachment into the main channel, and the requirement for river training works all need to 

be considered.

9.2.11 Channel Stability Investigations

In conjunction with the background information discussed in the preceding paragraphs, it is 

essential to determine the need for bank stabilization. The location, selection, and design of 

various types of river training works must be considered. The selection of training works is 

significantly affected by the characteristics of the river and the river system itself. The 

magnitude of local scour at the training structure must be considered. The possible

necessity of holding the river in a selected alignment must also be adequately explored. With 

regard to these particular issues, the principles of Chapter 6 can be applied to develop 

suitable designs for stabilizing the approaches and banks of the main channel, and the design 

of training works that assist in controlling the alignment of the river relative to the crossing or 

longitudinal encroachment. Hydraulic Engineering Circular (HEC) No. 23 (Lagasse et al. 

2001) also provides experience, selection, and design guidance for a variety of river training 

works and their applicability under a range of characteristics of the river environment.

9.2.12 Short-Term Response

Having completed the tentative design of the crossing or the encroachment based on river 

form, channel geometry, hydrologic and hydraulic data; it is essential to consider the shortterm response of the river system to construction. Similarly, existing or proposed

developments up- and downstream of the site and at the site itself should also be considered. 

 The techniques that may be utilized to investigate the short-term response at the site or in 

the vicinity of the crossing or encroachment involve the utilization of qualitative geomorphic 

relationships followed by the application of more sophisticated analyses using the principles 

presented in the chapters on open channel flow, sediment transport and river mechanics. As 

discussed in Chapter 5, it is possible to establish a mathematical model designed to route 

both water and sediment through the system. If this model is appropriately designed and 

utilized, it is possible to evaluate the response of the river system to both the construction of 

the crossing or encroachment and to other river development projects in the immediate area. 

For example, it may be important to establish the pattern of clear-water releases from a dam 

upstream of a crossing. Knowing the type of flow the channel would be subjected to and that 

the water being released is clear, one can make an estimate of the extent of degradation in
9.6

the channel, the amount of sediment derived from the bed and bank, the instability of the 

banks and even the types of lateral shifting that may be induced in the river system as it 

affects the crossing or encroachment.

9.2.13 Long-Term Response

The long-term river response at a crossing or a longitudinal encroachment and in the river 

system itself should be considered based on all river development projects including the 

highway. This type of treatment is, in general, beyond the scope of this particular manual. 

Nevertheless, significant advances have been made pertaining to the mathematical modeling 

of river systems, considering both their short and long-term response. Mathematical

modeling can be time consuming and expensive, requiring a substantial amount of additional 

data for calibration of the model. However, this approach is worth considering on important 

projects where determining long-term response may be critical to project success (see 

Section 5.6).

9.3 PROCEDURE FOR EVALUATION AND DESIGN OF RIVER CROSSINGS AND 

 ENCROACHMENTS

This section presents a summary of a general procedure to evaluate and design river

crossings and encroachments. Due to the multi-disciplined complexity of these problems, it 

is difficult to develop a procedure which is applicable to all situations that may be

encountered. A generalized approach can be described; however, modification to this 

procedure must be made to tailor the procedure to an individual project.

9.3.1 Approach to River Engineering Projects

The evaluation and design of river crossings should proceed from a broad evaluation of the 

characteristics of the river and the principles to be considered in design (described in Section 

9.2) to detailed computations and analysis. The evaluation should begin with a qualitative 

assessment of the river. As the analysis progresses, the analysis becomes more and more 

detailed and subsequently more quantitative. At all stages of the investigation and design, 

qualitative evaluation is important to determine, if possible, the interrelationship of all aspects 

of the project.

The three-level procedure outlined in HEC-20 (Lagasse et al. 2001) and illustrated in Figure 

9.1 is the recommended approach for evaluating projects in the river environment. The 

method begins with broad considerations and proceeds through a series of steps of

increasing complexity to narrow down to the finer points of the project. Additionally, this 

approach provides for back checking or "feedback" loops to insure that the interdependence 

of all the variables is continually adjusted (see HEC-20, Chapters 1 and 3) (see also Figure 

7.1). 

In Level 1, the analysis consists of: (1) identifying the goals of the project; (2) developing 

several options to achieve those goals; (3) determining the problems and possible solutions 

to problems associated with each option; and (4) performing a qualitative assessment of all 

aspects of the project.
9.7

Level 1

Simple Geomorphic Concepts and 

Qualitative Analysis

 ̄

Level 2

Advanced Geomorphic Concepts and 

Quantitative Engineering Analysis

 ̄

Level 3

Mathematical or Physical 

Model Studies

Figure 9.1. Three-level analysis procedure for river engineering studies.

A Level 2 analysis involves a more detailed qualitative analysis combined with a quantitative 

evaluation. Computation of water surface profiles using 1-dimensional computer models can 

be included in this level of analysis. In many cases, the evaluation and analysis can be 

considered adequate at this level if the goals are met, the interrelationship between different 

aspects of the project and river system are adequately explained and all of the problems 

resolved.

A Level 3 analysis involves mathematical or physical modeling of water and sediment. These 

procedures are not always necessary. Water can be modeled using a steady or unsteady 

flow rigid boundary flow model. In some cases, a movable boundary flow model can be 

employed, routing both water and sediment through the study reach. Sediment routing 

models will require a substantial historic data set and analysis time to calibrate and verify the 

model. Experienced modelers should be employed if sediment routing is to be performed.

9.3.2 Project Initiation 

The success of a project can be dramatically influenced by careful planning in the initial 

stages of the project. The following guidelines will help insure the success of the project.
9.8

Develop a Project Concept. When the project is conceived the goals of the project should be 

carefully defined and several options to meet these goals identified. The factors discussed in 

Section 9.2. should be considered when identifying design options. These options will be 

refined as the project progresses, eventually focusing on one or two options.

Assemble Available Data. All available data should be compiled and checked. The data 

checklist presented in Chapter 8 should be used as a guide. Data which is unavailable or has 

periods of missing data should also be listed on the checklist. Missing data can be ranked 

according to need (i.e., essential, nonessential and optional). Field programs designed to 

collect the essential data could be implemented at this time, however it is recommended that 

a field reconnaissance and evaluation be completed prior to implementing field programs. 

The field reconnaissance will provide a clearer definition of the project and will influence the 

types and quantity of additional data requirements.

Conduct a Field Reconnaissance. An initial field reconnaissance should be performed by a 

small group of technical personnel. It is advisable that the group be multi-disciplined so that 

geologic, geomorphic, hydrologic, hydraulic, alignment and highway constraints can be

identified. They should define the problems for each option and identify possible solutions to 

each problem. Options which are least feasible should be eliminated. Detailed procedures 

and check lists for a field reconnaissance that considers most geomorphic factors important 

to river engineering analyses are provided in HEC-20 (Lagasse et al. 2001). The field

reconnaissance team should identify the most favorable options, recommend the types of 

analyses which will be needed, and design field programs to collect specific data which will 

be required by the analysis.

Collect Additional Field Data. Field programs should be designed to collect only data that will 

be required to analyze and design the project. The field reconnaissance discussed previously 

is an important tool for the design and implementation of efficient field programs. By 

designing and implementing field programs after the field reconnaissance, the collection of 

unnecessary data can be avoided, providing more time and funding for collection of essential 

data.

It is also advisable that the field crews be supervised in the field by personnel who will be 

directly involved in the analysis and design. These personnel should be completely familiar 

with the types of data and the methods used to collect the data, providing an interface 

between the field and the analysis in the office. In this way, the field work, analysis, and 

design can be closely coordinated.

9.4 CONCEPTUAL EXAMPLES OF RIVER ENCROACHMENTS

This section discusses conceptual examples of river response to highway encroachments. 

Sixteen hypothetical cases are tabulated in Table 9.1. Each individual case is identified in the 

first column to show the physical situation that exists prior to the construction of the highway 

crossing. In the following three columns some of the major effects (local, upstream, and 

downstream) resulting from construction of a particular crossing are given. Only the gross 

local, upstream and downstream effects are identified in this table. In an actual design 

situation, it is worthwhile first of all to consider the gross effects as listed in Table 9.1 (Level 1 

analysis).
9.9

The Lane relation QS ~ Qs D50 (Equation 5.28) is one method for determining qualitative river 

response. Having identified the qualitative response that can be anticipated, other river 

mechanics techniques can be used to predict the possibility of change in river form and to 

estimate the magnitude of local, upstream and downstream river response. 

The initial river conditions in Table 9.1 include storage dams or water diversions. These 

examples are used as illustrations relating to common experience. In general, the effect of a 

storage reservoir is to cause a sudden increase of base level for the upstream section of the 

river. The result is aggradation of the channel upstream, degradation downstream and a 

modification of the flow hydrograph. Similar changes in the channel result if the base level is 

raised by some other mechanism, say a tectonic uplift. The effect of diversions is to

decrease the river discharge downstream of the diversion with or without an overall reduction 

of the sediment concentration. Similarly, changes in water and sediment input to a river often 

occur due to river development projects upstream from the proposed crossings or due to 

natural causes.

Case (1) of Figure 9.1 involves the construction of a bridge across a tributary stream 

downstream of where the steeper tributary stream has reached the floodplain of the parent 

stream. In most cases, the change in gradient of the tributary stream causes significant 

deposition. In the case illustrated for Case (1), an alluvial fan develops which in time can 

divert the river around the bridge, or, if the water continues to flow under the bridge, the 

waterway is significantly reduced, endangering the usefulness and stability of the structure. 

Usually, streams on alluvial fans shift laterally so that the future direction of the approach flow 

to the bridge is uncertain.

Case (2) sketched in Table 9.1 illustrates a situation where a bridge is constructed across a 

tributary stream. The average water surface elevation in the main channel acts as the base 

level for the tributary. It is assumed that at some point in time after the construction of the 

bridge the base level in the main channel has been lowered. Under the new condition, the 

local gradient of the tributary stream is significantly increased. This increased energy

gradient induces head cutting and causes a significant increase in water velocities in the 

tributary stream. The result is bank instability, possible major changes in the geomorphic 

characteristics of the tributary stream and increased local scour. When the base level is 

raised the gradient in the tributary is decreased, resulting in deposition, lateral channel

instability, increased flood levels, and a decrease in flow area under the bridge, similar to 

Case (1). 

Case (3) illustrates a situation where a bridge supported by piers and footings is constructed 

across a channel that is subjected to long periods of low stage. When a river is subject to 

long periods of low flows, there is a tendency for the low flow to develop a new low-water 

thalweg in the main channel. If the low-water channel aligns itself with a given set of piers, it 

is possible that the depth of local scour resulting from this flow condition may be greater than 

the depth of local scour at high stage. There are several documented failures where bridges 

have been safe in terms of local scour at high stage, but have failed as a consequence of the 

development of greater local scour during low-flow periods.
9.10

Table 9.1. River Response to Highway Encroachments and to River Development.

Bridge

Location

Local

Effects

Upstream

Effects

Downstream

Effects

(1) Crossing downstream of an

 alluvial fan

1 - Fan reduces

 waterway

2 - Direction of flow 

 at bridge site is 

 uncertain

3 - Channel location

 is uncertain

1 - Erosion of banks

2 - Unstable 

channel

3 - Large transport

 rate

1 - Aggradation

2 - Flooding

3 - Development of

 tributary bar in

 the main 

 channel

(2) Lowering of base level for the

 channel

1 - Headcutting

2 - General scour

3 - Local scour

4 - Bank instability

5 - High velocities

1 - Increased 

velocity

2 - Increased bed

 material transport

3 - Unstable 

channel

4 - Possible change

 of form of river 

1 - Increased

 transport to 

main

 channel

2 - Aggradation

3 - Increased flood

 stage

Case (4) illustrates a situation where artificial cutoffs have straightened the channel

downstream of a particular crossing. Straightening the channel downstream of the crossing 

significantly increases the channel slope. This causes higher velocities, increased bed

material transport, degradation and possible head cutting in the vicinity of the structure. This 

can result in unstable river banks and a braided streamform. The straightening of the main 

channel can drop the base level, adversely affecting tributary streams flowing into the

straightened reach of the main channel, which was discussed in Case (2).

Table 9.1. River Response to Highway Encroachments and to River Development 

(continued).

Bridge

Location

Local

Effects

Upstream

Effects

Downstream

Effects

(3) Channel characterized by 

 prolonged low flows 

1 - At low flow a low

 water channel

 develops in river

 bed

2 - Increased danger to

 piers due to

 channelization and 

 local scour

3 - Bank caving

----- -----

(4) Cutoffs downstream of 

 crossing 

1 - Steeper slope

2 - Higher velocity

3 - Increased transport

4 - Degradation and

 possible headcutting

5 - Banks unstable

6 - River may braid

7 - Danger to bridge

 foundation from

 degradation and

 local scour

See local effects 1 - Deposition 

down-

 stream of 

 straightened 

 channel

2 - Increase in flood 

 stage

3 - Loss of channel

 capacity

4 - Degradation in

 tributary
9.11

Case (5) illustrates the situation where a bridge is constructed across a river immediately 

downstream of the confluence with a steep tributary. The tributary introduces relatively large 

quantities of bed material into the main channel. As a result, an island has formed in the main 

channel and divided flow exists. In order to reduce the cost of the bridge structure, the bridge 

is built across one subchannel to the island or bar formed by deposition, closing the

secondary channel. Such a procedure forces all of the water and sediment to pass through a 

reduced width. This contraction of the river in general increases the local velocity, increases 

general and local scour, and may increase bank instability. In addition, the contraction can 

change the alignment of the flow in the vicinity of the bridge and affect the downstream 

channel for a considerable distance. A chute channel can develop across the second point 

bar downstream, adversely affecting several meander loops downstream. Upstream of the 

bridge, there is aggradation. The amount depends on the magnitude of water and sediment 

being introduced from the tributary. Also, there is an increase in the backwater upstream of 

the bridge at high flows which in turn affects other tributaries farther upstream of the crossing, 

see Case (2).

Case (6) illustrates a situation where the main channel is realigned in the vicinity of the bridge 

crossing. A cutoff is made to straighten the main channel through the selected bridge site. 

As discussed in Case (4), increased local gradient, local velocities, local bed material

transport, and possible changes in the characteristics of the channel are expected due to the 

new conditions. As a result the channel may braid. A short cut off section (1 or 2 bends) can 

be designed to transport the same sediment loads that the river is capable of carrying 

upstream and downstream of the straightened reach; however, it may be difficult to achieve 

stability when multiple bends in a long reach are cut off. 

It is possible to build modified reaches of main channels that do not introduce major adverse 

responses due to local steepening of the main channel. In order to design a straightened 

channel so that it behaves essentially as the natural channel in terms of velocities and 

magnitude of bed material transport, it is usually necessary to build a wider, shallower

section.

Case (7) illustrates a bridge constructed across a main channel. Subsequently, the base level 

for the channel is raised by the construction of a dam downstream. Whenever the base level 

of a channel is raised, a pool is created extending a considerable distance upstream 

depending on the size of dam and slope of the channel. As the water and sediment being 

transported by the river encounter this pool, most of the sediments drop out, forming a 

delta-like structure at the mouth. If the bridge lies within the effects imposed by the new base 

level, the following response at the crossing will be expected: aggradation of the bed, a loss of 

waterway at the bridge site, significant changes in river geometry, and increased flood stages 

and lateral channel instability. This is similar to Case (1).

Case (8) considers the situation where the sediment load is reduced in the channel after a 

bridge has been constructed. This may happen due to the construction of a storage dam 

upstream of the crossing. As stated in the preceding case, the raising of the base level of a 

river, as in the development of storage by constructing a dam on a river, provides a

sedimentation basin for the water flowing in the system. In most instances, all of the

sediment coming into a reservoir drops out within the reservoir. Water released from the 

reservoir is mostly clear. With sediment-free flow, the channel is too steep and bed

sediments are entrained from the bed and the banks causing significant degradation. If the 

bridge is sufficiently close to the reservoir to be affected by the degradation in the channel, the 

depth due to general and local scour at the bridge may be significantly increased. Also, the 

channel banks may become unstable due to degradation, and there is a possibility that the 

river, as its profile flattens, may change its form. In the extreme case, it is possible that the 

degradation may cause failure of the dam and the release of a flood wave.
9.12

Table 9.1. River Response to Highway Encroachments and to River Development (continued).

Bridge

Location

Local

Effects

Upstream

Effects

Downstream

Effects

(5) Excess of sediment at bridge

 site due to upstream tributary

1 - Contraction of the

 river

2 - Increased velocity

3 - General and local

 scour

4 - Bank instability

1 - Aggradation

2 - Backwater at flood 

 stage

3 - Changed response

 of tributaries

1 - Deposition of 

 excess sediment 

 eroded at down-

 stream of the 

 bridge

2 - More severe attack

 at first bend 

 downstream

3 - Possible develop-

 ment of a chute 

 channel across

 the second point 

 bar downstream of 

 the bridge

(6) River channel relocation at

 crossing site

1 - None if straight 

 section is designed 

 to transport the 

 sediment load of 

 the river and if it is

 designed to be

 stable when sub-

 jected to anticipated 

 flow. Otherwise 

 same as in Case (4)

1 - Similar to local

 effects

1 - Similar to local

 effects

Table 9.1. River Response to Highway Encroachments and to River Development 

(continued).

Bridge

Location

Local

Effects

Upstream

Effects

Downstream

Effects

(7) Raising of river base level

1 - Aggradation of bed

2 - Loss of waterway

3 - Change in river

 geometry

4 - Increased flood 

 stage

5 - Lateral channel

 instability

1 – See local effects

2 - Change in base

 level for tributaries

3 - Deposition in

 tributaries near 

 confluences

4 - Aggradation caus-

 ing a perched river

 channel to develop

 or changing align-

 ment of main 

 channel

1 - See upstream 

 effects

(8) Reduction of sediment load

 upstream

1 - Channel 

 degradation

2 - Possible change in 

 river form 

3 - Local scour

4 - Possible bank 

 instability

5 - Possible destruction

 of structure due to

 dam failure

1 - Degradation

2 - Reduced flood 

 stage

3 - Reduced base level

 for tributaries,

 increased velocity

 and reduced 

 channel stability

 causing increased

 sediment transport

 to main channel

1 - Degradation

2 - Increased 

 velocity

 and transport in

 tributaries
9.13

Case (9) through Case (13) illustrate a more complicated set of circumstances. These 

cases involve the interrelationship of Cases (1) through (8). In Case (9) the river crossing is 

affected by Dam A constructed upstream as well as Dam B constructed downstream. As 

documented in Case 8, Dam A causes significant degradation in the main channel. Dam B 

causes aggradation in the main channel (Case 7). The final condition at the bridge site is 

estimated by summing the effects of both dams on the main channel and the tributary flows. 

Normally, this analysis requires water and sediment routing techniques studying both longand short-term effects of the construction of these dams, and it is necessary to consider the 

extreme possibilities to develop a safe design.

Table 9.1. River Response to Highway Encroachments and to River Development 

(continued).

Bridge Location

(9) Combined increase of base level and reduction of sediment load upstream

Local Effects Upstream Effects Downstream Effects

1 - Dam A causes degradation

2 - Dam B causes aggradation

3 - Final condition at bridge site is the 

 combined effect of (1) and (2).

 Situation is complex and combined

 interaction of dams, main channel

 and tributaries must be analyzed 

 using water and sediment routing 

 techniques and geomorphic factors

1 - Channel could aggrade or 

 degrade with effects similar 

 to cases (7) and (8)

1 - See upstream effects

In Case (10) Bridges A and B cross two major tributaries a considerable distance upstream 

of their confluence. Upstream of Bridge A, a diversion structure is built to divert essentially 

clear water by canal to the adjacent tributary on which Bridge B has been constructed. 

Upstream of Bridge B the clear water diverted from the other channel enters the storage 

reservoir and the water from the tributary plus the transfer water is released through a 

hydro-power plant. It is anticipated that a larger storage reservoir may be constructed

downstream of the confluence on the main stem at C. These changes in normal river flows 

give rise to several possible complex responses at bridge sites A and B, in the tributary 

systems as well as on the main stem. Bridge Site A may aggrade due to the excess of 

sediment left in that tributary when clear water is diverted. However, initially there may be 
9.14

lowering of the channel bed in the vicinity of the diversion structure because of the deposition 

upstream of the diversion dam and the release of essentially clear water for a relatively short 

period of time until the sediment storage capacity of the reservoir is satisfied. Bridge site B 

is subjected to degradation due to the increased discharge and an essentially clear water 

release. However, the degradation of the channel could induce degradation in the tributaries 

causing them to provide additional sediment to the main channel, see Case (8). This 

response would to some degree counteract the degrading situation in this reach of river. 

Such changes in river systems are not uncommon and introduce complex responses 

throughout the system. Complete analysis must consider the individual effects and sum 

them over time to determine a safe design for the crossings.

Table 9.1. River Response to Highway Encroachments and to River Development (continued).

Bridge Location

(10) Change in water discharge, no change in sediment load

Local Effects Upstream Effects Downstream Effects

1 - Bridge A may be subjected to 

 aggradation due to excess 

 sediment left in the channel by 

 diversion of clear water 

2 - Bridge B may be subjected to

 degradation due to increased

 discharge in the channel

3 - If a storage reservoir was 

 constructed at C, it could induce 

 aggradation in both main

 tributaries

1 - Upstream of Bridge A - 

 aggradation and possible 

 change of river form

2 - Upstream of Bridge B - 

 degradation and change of

 river form

3 - Channel instabilities

4 - Significant effects on flood

 stage

1 - See upstream effects

2 - Construction of reservoir C could

 induce aggradation in the main

 channel and in the tributaries.

 Effects same as in Case 7

Case (11) shows a highway that crosses the main channel at Bridge A and its tributary at 

Bridge B. The confluence of the main channel and its tributary is downstream of both 

bridges. The alignment of the main channel is continually changing. The rate of change in the 

river system should be evaluated as part of the geomorphic and hydraulic analysis of the site. 

 If the main channel shifts to the alternate position shown and moves the confluence closer to 

Bridge B, the gradient in the tributary is significantly increased causing degradation as well as 

channel instabilities and possible changes in river form. Excess sediment from the tributary 

causes aggradation in the main channel and possibly significant changes in channel

alignment. 
9.15

Table 9.1. River Response to Highway Encroachments and to River Development 

(continued).

Bridge

Location

Local

Effects

Upstream

Effects

Downstream

Effects

(11) Naturally shifting river channel

1 - Rivers are dynamic

 (ever changing) and

 the rate of change

 with time should be

 evaluated as part of

 the geomorphic and

 hydraulic analysis

2 - Alignment of main

 channel continually

 changes affecting

 alignment of flow

 with respect to 

 Bridge A.

3 - If the main channel

 shifts to the alter-

 nate position, the

 confluence shifts 

 and the tributary

 gradient is signi-

 ficantly increased

 causing degrada-

 tion in the tributary.

 Local effects on 

 Bridge B same as 

 1, 2, 3, and 4 in

 Case (8)

4 - Excess sediment 

 from the tributary,

 assuming (3) 

 causes aggrada-

 tion in the main 

 channel and 

 possible significant

 changes in channel

 alignment 

1 - The river could 

 abandon its present 

 channel. Changing 

 position of the main 

 channel may 

 require realignment 

 with training works.

1 - See upstream 

 effects

2 - Shifts in the posi-

 tion of the main

 channel relative

 to the position of

 the confluence

 with the tributary

 alternatively flat-

 tens or steepens 

 the gradient of 

 the tributary

 causing corres-

 ponding aggra-

 dation and de-

 gradation

3 - Shifts in the posi-

 tion of the main

 channel causes

 aggradation, de-

 gradation and

 instabilities de-

 pending upon 

 direction and

 magnitude of

 channel change

 

Considering the possible changes in the position of the main channel, training works may be 

required at and upstream of Bridge A to assure a satisfactory approach of the flow to the 

bridge crossing. Otherwise, the river could abandon its present channel. A shift in the 

position of the main channel relative to the position of the confluence with the tributary also 

alternately flattens or steepens the gradient of the tributary causing corresponding

aggradation or degradation in the tributary. This type of problem is difficult because of the 

continuously changing characteristics of such river systems. Rivers of this type are usually 

stable for several years at a time or at least between major flows. Consequently, if crossing 

locations are properly selected and appropriate stabilization techniques are taken, it may be 

possible to maintain the usefulness of the crossings for the life of the structures. However, 

the disadvantages associated with such locations will often require expensive solutions and 

these locations should be avoided if possible.
9.16

Case (12) illustrates a meandering channel with several tributaries and a major storage 

reservoir constructed on the main channel. Two crossings are shown on the main channel 

upstream of the reservoir. It is assumed that complete channelizing of the meandering river 

has been authorized. This shortens the path of travel of the water by an appreciable distance. 

Considering local effects at the bridges, bridge site A is first subjected to possibly severe 

degradation and then aggradation. Bridge site B is primarily subjected to degradation. The 

magnitude of this degradation can be large. With the degree of straightening indicated in the 

sketch, severe head cutting may be initiated up the main channel as well as the tributaries. 

The whole system may be subjected to passage of sediment waves and the river form can 

dramatically change over time. The flood level in the system and the local and general scour 

in the vicinity of the bridges is greatly affected by the channelization.

As a result of channelization, the river reach at bridge site B could become braided. Also, in 

this reach the rate of sediment transport is increased, head cutting is induced and flood 

stages are reduced. The tributaries in the upper reach are subjected to severe degradation. 

For the bridge at position A, the channel would probably degrade and then significantly

aggrade. Significant reactions are possible when channelization is undertaken in a river

system. A detailed analysis of all of the responses is necessary before it is possible to safely 

design crossings such as those at location A and B.

Table 9.1. River Response to Highway Encroachments and to River Development 

(continued).

Bridge

Location

Local

Effects

Upstream

Effects

Downstream

Effects

(12) Human-induced reduction of

 channel length 

1 - Bridge A is first 

 subjected to 

 degradation and 

 then aggradation.

 Action can be very 

 severe

2 - Bridge B is 

primarily

 subjected to 

 degradation. The

 magnitude can be

 large

3 - The whole system 

is

 subjected to pass-

 age of sediment 

 waves

4 - River form could

 change to braided

5 - Flood levels are

 reduced at B and

 increased at A

6 - Local and general 

 scour is 

significantly

 affected

1 - A change of river

 form from 

meander-

 ing to braiding is

 possible

2 - Rate of sediment

 transport is

 increased

3 - Headcutting is

 induced in the

 whole system

 4 - Upstream of B

 flood stage is

 reduced

5 - Velocity increases

6 - Tributaries 

respond

 to main channel

 changes

1 - For Bridge B see 

 upstream effects

2 - For Bridge A the

 channel first 

 degrades and 

 then significantly

 aggrades

3 - Large quantities 

 of bed material

 and wash load

 are carried to the

 reservoir

4 - Delta forms in 

the

 reservoir

5 - Wash load may

 affect water

 quality in the

 entire reservoir

6 - Tributaries 

 respond to main

 channel 

changes
9.17

Case (13) is a series of situations, unrelated in some instances and combined in others, 

which can affect bridge crossings. Tidal flows, seiches and bores can have significant 

effects on scour and depth in the channel system. The tidal flows, seiches and bores, as well 

as wind waves, can rapidly and violently destroy existing bank lines. 

When considering earthquakes, it is of importance to examine a seismic probability map of 

the United States. Large portions of the United States are subjected to at least infrequent 

earthquakes. Associated with earthquake activity are severe landslides, mud flows, uplifts in 

the terrain, and liquefaction of otherwise semi-stable materials, all of which can have a 

profound effect upon channels and structures located within the earthquake area. Historically, 

several rivers have completely changed their course as a consequence of earthquakes. For 

example, the Brahmaputra River in Bangladesh and India shifted its course laterally a 

distance of some 320 km (200 miles) as a result of earthquakes that occurred approximately 

200 years ago. Although it may not be possible to design for this type of natural disaster, 

knowledge of the probability of its occurrence is important so that certain aspects of the 

induced effects from earthquakes can be taken into consideration when designing the

crossings and affiliated structures.

Table 9.1. River Response to Highway Encroachments and to River Development 

(continued).

Bridge

Location

Local

Effects

Upstream

Effects

Downstream

Effects

(13) Tectonics and other natural causes

1 - Scour or

 aggradation

2 - Bank erosion

3 - Channel change

4 - Bed form 

 change

1 - Bank erosion

2 - Inundated 

 highway

3 - Increase in 

 velocity

4 - Wave action

1 - Channel 

 changes

2 - Scour or 

 deposition

3 - Decrease in 

 bank stability

4 - Landslides

5 - Rockslides

6 - Mudflows

1 - See local effects

2 - Channel erosion

3 - Changes in 

 channel slope

1 - See local effects

1 - See local effects

2 - Slide lakes

1 - See local effects

2 - Beach erosion

1 - See local effects

1 - See local effects

2 - Slide lakes
9.18

Case (14), (15), and (16) illustrate three examples of longitudinal encroachment. In Case 

(14), a few bends of a meandering stream have been realigned to accommodate a highway 

(see Case 6). There are two problems involved in channel realignment. First, the length of 

realigned channel is generally shorter than the original channel which results in a steeper 

energy gradient in the reach (Case 4). Second, the new channel bank material in the 

realigned reaches may have a smaller resistance to erosion. As a result of these two 

problems, the channel may suffer instability by the formation of a headcut from the

downstream end and increased bank erosion. The realigned channel may also exhibit a 

tendency to regain the lost sinuosity and may approach and scour the highway embankment. 

 To counter these local effects one could design the realignment to maintain the original 

channel characteristics (length, sinuosity). Another way would be to control the slope by a 

series of low check dams. In any case, bank protection by riprap, jacks or spurs will be 

needed. The up- and downstream effects of the channel realignment will be the same as 

discussed for channel length reduction in Case (12). For example, as the degradation travels 

through the realigned reach, sediment load generation in the river by bed and bank erosion will 

cause aggradation downstream.

Table 9.1. River Response to Highway Encroachments and to River Development (continued).

Bridge

Location

Local

Effects

Upstream

Effects

Downstream

Effects

(14) Longitudinal encroachment

1 - Increased energy 

 gradient and 

 potential bank 

 and bed scour

2 - Highway fill is 

 subject to scour 

 as channel tends 

 to shift to old

 alignment

3 - Reach is subject 

 to bed degra-

 dation as headcut 

 develops at the 

 downstream

 end and travels

 upstream

4 - Lateral drainage 

 into the river is 

 interrupted and 

 may cause 

 flooding and

 erosion 

1 - Energy gradient 

 also increased in 

 the reach upstream

 and may cause 

 change of river form

 from meandering to

 braided

2 - Rate of sediment 

 transport is 

 increased. As the

 headcut travels

 upstream severe 

 bank and bed 

 erosion is possible

3 - If tributaries in the

 zone of influence 

 exist they will 

 respond to lowering

 of base level

1 - Channel will 

 aggrade as the

 sediment load

 coming from 

 bed and bank 

 erosion

 is received

2 - Channel may 

 deteriorate from

 meandering to

 braided

Case (15) illustrates encroachment on the waterway of an incised stream flowing through a 

narrow gorge. Locally, the effect is to reduce the waterway and to increase the velocities and 

bank and bed erosion potential. The erosion protection of the highway slope exposed to the 

flow, and possibly, the opposite bank are important problems. The backwater induced by this 

obstruction may cause upstream aggradation and higher flood levels. On the downstream 

side, channel aggradation may be experienced if bed erosion occurs locally in the encroached 

reach.
9.19

Table 9.1. River Response to Highway Encroachments and to River Development (continued).

Bridge

Location

Local

Effects

Upstream

Effects

Downstream

Effects

(15) Longitudinal encroachment

1 - Reduced 

 waterway

 causes a local

 obstruction to flow

 and higher

 velocities

2 - Significant erosion

 problem on the

 highway fill and

 induced bed 

 degradation

3 - Lateral drainage 

 into the river is 

 interrupted and 

 may cause flooding

 and erosion

1 - Backwater 

 generated by the

 obstruction 

 increases flood

 stage

2 - Deposition induced

 by the backwater

1 - Large sediment

 load may cause

 aggradation

2 - Local scour at 

 end of con- 

 tracted section

Case (16) is a case of floodplain encroachment. It is assumed that during bankfull and lower 

stages the highway does not interact with the flow. However, during high stages, the total 

flow area is decreased by the encroachment. Locally, the highway must be protected against 

inundation and erosion during a flood. The effect on the river channel depends on the extent 

of encroachment on the waterway. If the highway significantly reduces the floodplain, it may 

increase river stages for a given flood. If the river channel tends to shift, the highway 

encroachment may alter the interaction of floodplain flow and channel flow, affecting the 

channel floodplain flow pattern. Very often this type of encroachment has little or no effect on 

flood stages or on the stream upstream or downstream, however, possible adverse effects 

should be investigated.

In all cases of longitudinal encroachment, the lateral drainage into the river will be intercepted. 

An important consideration in the design of encroachments will be to provide for this drainage.

Table 9.1. River Response to Highway Encroachments and to River Development (continued).

Bridge

Location

Local

Effects

Upstream

Effects

Downstream

Effects

(16) Longitudinal encroachment

1 - Erosion of highway

 fill and submer-

 gence possible

 during floods

2 - Pattern of overbank

 spill are affected

 by the encroach-

 ment and in highly

 shifting channels

 may change river

 course downstream

3 - Lateral drainage into

 the river is interrupted

 and may cause flood-

 ing and erosion

1 - If significant

 encroachment on 

 the floodplain

 waterway, back-

 water may be

 induced

1 - If the river 

 channel is highly

 shifting, the 

 channel align-

 ment may 

 change

2 - If significant 

 erosion

 experienced

 upstream, 

 aggradation will

 occur
9.20

9.5 PRACTICAL EXAMPLES OF RIVER ENCROACHMENTS

In the preceding paragraphs, a series of conceptual cases were discussed. Each case 

considers the interactions between the river and the encroachment over a period of time. In 

general, the particular practical examples presented in this section are not as complex as 

some of the earlier conceptual cases. For example, there is no consideration of water 

resources development throughout the basins, including construction of reservoirs

transmountain diversions, and so forth.

9.5.1 Cimarron River, East of Okeene, Oklahoma (Example 1)

The bridge and a rock dike on its north abutment were built in 1934 (Figure 9.2a). A high 

discharge year in 1938 caused the south bank to erode. A Kellner jetty field was installed to 

prevent further erosion of the bank. The jetty field was ineffective due to the lack of debris and 

suspended sediment load. A large flood in 1957 spread out over the floodplain in several 

places. After the flood, seven timber pile diversion units and a riprapped dike were installed in 

the old jetty field location to prevent future damage to the highway (Figure 9.2b). As of 1968, 

the south bank had been held in line by the timber pile diversion structures and the dike 

(Figure 9.2c).

9.5.2 Arkansas River, North of Bixby, Oklahoma (Example 2)

The bridge was built in 1938. A Kellner jetty field was installed on the north bank in 1939 to 

protect the north bridge abutment (Figure 9.3a). In 1948 minor floods eroded the south bank. 

A Kellner jetty field was installed to prevent further erosion (Figure 9.3b). Some time after, 

riprap was put on the south bank up- and downstream of the jetty field (Figure 9.3c). In 1959, 

a 50-year frequency flood eroded the north bank and washed out a section of the north 

approach to the bridge. The flood also washed out two sections of roadway further north on 

the floodplain. The approach was rebuilt and riprap was installed on the embankment. A

riprapped spur dike (guide bank) was also constructed just south of the north abutment. Five 

pile diversion structures were built to prevent further erosion of the north bank (Figure 9.3c). 

As of 1968, the south bank has remained stationary, and the north bank has filled (Figure 

9.3d).

9.5.3 Washita River, North of Maysville, Oklahoma (Example 3)

In 1949, floods washed out the north span of the bridge. Also, both banks upstream from the 

bridge were damaged. A temporary structure was installed in place of the north span of the 

bridge. In October of 1949, two Kellner jetty fields were completed upstream from the bridge 

to provide bank protection (Figure 9.4a). In 1950, a new bridge was constructed just

downstream from the old bridge. State Highway 74 was realigned to conform to the new 

bridge. In eight months of operation, the Kellner jetty field on the northeast bank had

completely silted in. This was largely due to the clay content in the suspended sediment and 

the large amount of drift in the stream (Figure 9.4b). The floods of 1957 did very little damage 

to the new bridge site or the banks. Floods in 1968 and 1969 have caused bank erosion on 

the north bank upstream of the jetty field which could eventually cut in behind the jettyfield 

(Figure 9.4c).
9.21

Figure 9.2. Cimarron River, east of Okeene, Oklahoma (Example 1).
9.22

Figure 9.3. Arkansas River, north of Bixby, Oklahoma (Example 2).
9.23

Figure 9.4. Washita River, north of Maysville, Oklahoma (Example 3).

9.5.4 Beaver River, North of Laverne, Oklahoma (Example 4)

During high flows of 1938, the river washed over the south bank and damaged the approach 

roadway south of the bridge. The south end of the bridge was also damaged. Jetty fields 

were constructed in several locations upstream of the bridge in an attempt to reduce bank 

erosion. Two jetty lines were constructed in a side channel downstream of the bridge to 

discourage flow in that channel to prevent eddy currents from eroding the north embankment 

(Figure 9.5a). A new longer bridge was constructed in 194l. High flows in 1946 caused 

severe erosion on the south bank upstream from the bridge. In 1949, an earth dike and jetty 

field were constructed on the south bank to prevent further erosion. In 1969, the river cut 

through a portion of the 1949 jetty field and eroded the earth dike (Figure. 9.5b). Car bodies 

were used as bank protection. However, car bodies are not environmentally acceptable and 

are difficult to hold in place unless anchored with cable or weighted down with concrete or 

rocks.

9.5.5 Powder River, 40 Miles East of Buffalo, Wyoming (Example 5)

The Powder River has very fine bed material and a high sinuosity. The river contains dunes 

and antidunes at low flows. At the bridge site, there was a grove of cottonwood trees on the 

upstream east bank of the bridge and a dry draw coming in from the upstream left bank 

(Figure 9.6a). Upon completion of the bridge, a large flood occurred. The river attempted to 

straighten out its meanders possibly as a result of instream or floodplain mining in conjunction 

with the road construction. At the same time, the draw on the upstream left bank was 

bringing in a large amount of sediment, forcing the stream toward the upstream side of the 

(east) abutment. 

The flood flow uprooted the grove of cottonwoods which impacted on the sacked concrete 

riprapped spur dike (guide bank) at the east abutment and destroyed the dike (Figure 9.6b). 

To restore the channel to its original alignment, a training dike was constructed from the 

bridge upstream to a nearby bluff (Figure 9.6c). A jack jetty field was also constructed on the 

upstream meander to prevent the river from flowing across the point bar. The jetty field has 

begun to fail as a result of the streamside piles being undercut. The streamside anchors 

were rebuilt by driving the piles deeper in the center of small riprapped mounds.
9.24

Figure 9.5. Beaver River, north of Laverne, Oklahoma (Example 4).
9.25

Figure 9.6. Powder River, 40 miles east of Buffalo, Wyoming (Example 5).

9.5.6 North Platte River, Near Guernsey, Wyoming (Example 6)

The North Platte River is a fairly stable river in this reach as a result of reservoir control 

upstream (Figure 9.7). The bed is coarse granular material with cobbles in evidence. It was

decided at this crossing to build the bridge over the main channel and part of the island and to 

block off the active overflow channel on the opposite side of the island (Figure 9.7a). Two 

situations can occur due to this choice for the bridge crossing. One situation is that the 

concave bank erodes due to the high velocities resulting from the decreased area of flow 

under the bridge (Figure 9.7b). In fact, with extreme flows the river could erode a chute 

across the point bar on the first bend downstream. The other situation which may occur is 

that the high velocity flow carries increased sediment load and deposits this material in the 

eddies downstream (Figure 9.7c), which in this case would not cause any problems. The 

county insisted on low construction costs so the training dike had minimal riprapping as 

shown in the figure. As a result the dike is now eroding badly. The high bank began eroding 

so the property owner dumped in some broken concrete, rock and debris. The channel 

degraded through the opening but as expected the coarse bed has armored quite well. 

Except for the eroding of training dike, everything has worked well for over 30 years.
9.26

Figure 9.7. North Platte River near Guernsey, Wyoming (Example 6).

9.5.7 Coal Creek, Tributary of Powder River, Wyoming (Example 7)

A small bridge was constructed over Coal Creek over an intermittent tributary of the Powder 

River. Coal Creek is a dry draw that is incised. Head cuts existed downstream from the 

bridge at the time of its construction (Figure 9.8). Several headcuts were over a mile 

downstream. During a subsequent flood, at least one head cut moved upstream through the 

bridge site. This headcut almost undercut the midstream piles and exposed some of the 

abutment piles. To prevent further degradation under the bridge when the remaining

headcuts move through, chainlink enclosed riprap was placed as shown. The lower portion 

was placed as articulated riprap. The migration of the downstream headcuts has since 

resulted in some downward articulation. A heavy steel girder fence-like device to allow the 

lower portion of the replaced riprap to be stable at a steeper slope was necessary to establish 

a larger waterway opening. The original waterway was significantly smaller than the final 

configuration. Other alternatives would have been to excavate the head cuts in the channel 

through the bridge site before the bridge construction, allowing them to move naturally

upstream from there; to set the piles deeper in anticipation of the lowering of the bed 

elevation; or to construct a weir (check dam) control structure at the location of the headcut to 

prevent upstream migration.

9.5.8 South Fork of Deer River at US-51 Near Halls, Tennessee (Example 8)

Examples 8 through 10 were taken from Federal Highway Report Number FHWA/RD-80/038. 

In general, these examples are presented as in the original form. They have, however, been 

edited slightly and modified to fit the format of this text.
9.27

Figure 9.8. Coal Creek, tributary of Powder River, Wyoming (Example 7).

The location for Example 8 is near the Dyer and Lauderdale County line in western

Tennessee (Figure 9.9a). Dual bridges were built in 1963, with a 16 m (52.5 ft) main span 

supported by wall-type piers in the main channel, and thirty 8 m (26 ft) approach spans 

supported by concrete pile bents. In 1975, both bridges over the main channel were rebuilt, 

with a main span of 22.5 m (73.8 ft) supported by hammerhead piers. Spill-through 

abutments, set back from the main channel, were protected with sacked concrete in 1963 

and have remained stable.

 Figure 9.9a. Map showing South Fork of Deer River at U.S. Highway 51 Crossing 

 (Example 8).
9.28

The drainage area is 2,688 km2 (1,038 mi2); the bankfull discharge is 28 m3/s (989 ft3/s), and 

the river width (where bordered by natural vegetation) is 24 m (79 ft). The stream is 

perennial, alluvial, sand bed, in a valley of moderate relief and in a wide floodplain. The natural 

channel has a sinuosity of about 2.5 but the channel has been straightened; it is equiwidth, 

not incised, cut banks are rare, with silt-sand banks.

The channel was first straightened and enlarged in the 1920s by local drainage districts; but, 

probably because the natural floodplain forest was not cleared, the banks remained stable. In 

1969, the Corps of Engineers straightened and enlarged a reach about 4.8 km (3 mi) in length 

downstream from the bridge, reducing the length about 20 percent. During the 1960s, and 

particularly early 1970s, the floodplain was cleared of trees for agricultural purposes.

Figure 9.9b illustrates the Corps of Engineers' channel modifications that reduced the channel 

length and increased the channel slope. Figure 9.9b also provides a profile of the channel 

before any modifications were made in 1975. 

Between 1970 and 1971 the left bank (near bent 7) at the bridge receded an average distance 

of 4 m (13 ft). The peak discharge during this period was 215 m3/s (7,592 ft3/s) (1.5 year 

recurrence interval). Timber pile retards were built at the left bank near bent 7 and a single 

row of pile with wood face planks extending from the downstream end of bent 7 for a distance 

of 37.5 m (123 ft) upstream (Figure 9.9c).

Figure 9.9b. Channel modifications to South Fork of Deer River at U.S. Highway 51 near

 Halls, Tennessee (Example 8).
9.29

Figure 9.9c. Elevation sketch of U.S. Highway 51 Bridge (Example 8).

Between 1971 and 1973 the peak discharge was 751 m3/s (26,518 ft3/s) (17 years 

recurrence interval). Bankfull stage occurred several times with high flows sustained for 

periods of weeks. The left bank continued to erode behind the retard where the average 

distance of recession was about 2 m (6.5 ft) for the 3-year period. Bent 7 became exposed 

below the ground line. Concrete was poured at the base to prevent further erosion. Slumping 

from the left bank deflected flow toward the right bank, causing rapid erosion and failure of 

bent 8. The south lane of the bridge was closed. A detailed inspection was made in 1973, 

but little field data was collected. Data collected indicated a large local scour problem at the 

bridge. No profile data was available to evaluate the gradation problem.

In 1975 both lanes of the bridge were rebuilt, with new piers having deeper footings and less 

area normal to the flow (Figure 9.9c). Single-row timber pile retards were built along both 

banks in the vicinity of the bridge. A large scour hole in the center of the channel downstream 

from the bridge, attributed to flow constriction during bridge construction, was filled with 

gravel.

In 1977 a detailed inspection was made and cross-section data at the bridge indicated that 

the local scour problem was somewhat corrected and the gradation problem was fairly 

stable. Effectiveness of the timber pile retard had not yet been tested, and the area between 

the retard and the bank was not accumulating sediment. The lowermost face plank on the 

retard was about 1.5 m (5 ft) above the streambed. From experience face planks should be 

extended to, or below, streambed elevation. In addition, the upstream end of the retard 

seemed to be keyed into the bank for an insufficient distance. Vegetation was becoming 

re-established on the banks, which appeared more stable in 1977 than in the recent past.

The original bridge failed in 1974 because of channel degradation and concurrent bank

recession, which are directly attributable to straightening of the channel for drainage purposes 

and clearing of the banks and floodplain for agricultural purposes. Channel width increased 

by a factor of approximately 2, between 1969 and 1976. The clearing of vegetation is 

apparently one of the most critical factors, because channel straightening in the 1920s which 

was not accompanied by extensive clearing, did not result in significant bank instability. The 

timber pile retard installed in 1971 was of inadequate design, in view of the seriousness of the 

problem. 
9.30

Bank recession might have been controlled by an adequate retard or other countermeasure, 

but channel degradation is more difficult to control.

The site was inspected in 1973, in 1974, four times in 1975 and in 1976 by numerous federal 

and state agency employees. It should be pointed out that little actual field data were 

collected to document the progressive channel change. All inspections concluded that the 

channel changes were responsible for the gradation problems and other related hydraulic 

problems.

9.5.9 Elk Creek at SR-15 Near Jackson, Nebraska (Example 9)

Elk Creek is located in Dakota County and is a tributary of the Missouri River. It flows into the 

Missouri River just upstream of Sioux City, Iowa. The State Road 15 Bridge just west of 

Jackson is of interest. The stream is perennial but flashy, alluvial, sand-silt bed and in a valley 

of low relief with a wide floodplain. The channel is sinuous, incised by degradation, and has 

silt-clay banks.

The stream bed has degraded at least 2 m (6.6 ft) since 1955. There are two primary 

reasons for this degradation. First, channel modifications have been made to improve and 

maximize agricultural production. As a result, the channel has been straightened and

changed at isolated locations. Second, and probably more important, is the general

degradation below Gavins Point Dam. Missouri river stage trends, for almost 50 years for 

eight of the key main stream gaging stations below Sioux City, indicated at least 2 m (6.6 ft) of 

degradation at Sioux City, as indicated in Figure 9.10. This degradation is probably due to 

three main reasons as follows:

1. Between 1890 and 1960 the Missouri River length from Sioux City to Omaha has been 

reduced 21 percent by the Corps of Engineers. As a result the stream bed slope was 

increased.

2. The sediment-free water released at Gavins' Point Dam is transporting the bed sediment 

that is available.

3. The rather high sustained flows of the regulated Missouri River system do not allow for 

any aggradation or filling.

 

The degradation is primarily responsible for lateral instability as the channel has almost 

doubled in width, and degradation has exposed bridge pier footings.

Tributary degradation, resulting from degradation on the mainstream of the Missouri River is 

to be expected. The Missouri River has historically degraded, as indicated in the Missouri 

River stage trends. This condition should be evaluated on an annual basis and bridges 

inspected that are subject to this headcutting. This degradation should be expected on each 

tributary that is not protected by a grade control structure. Failure of this particular bridge due 

to degradation is not likely because of the great depth to which bridge foundations have been 

placed.
9.31

Figure 9.10. Stage trends at Sioux City, Iowa, on the Missouri River (Example 9).
9.32

9.5.10 Big Elk Creek at I-90 Near Piedmont, South Dakota (Example 10)

Big Elk Creek is located in Meade and Lawrence counties and is a tributary of the Cheyenne 

River. The headwaters of Big Elk Creek originate in the Black Hills National Forest. I-90 

crosses Big Elk Creek on an alluvial fan just outside of the National Forest, where there is a 

significant reduction in channel slope. The bridge, built in 1964, is 54 m (177 ft) long, has pile 

bents with square piles, spillthrough abutments, and a concrete deck. The creek is

intermittent, flashy and alluvial with cobble and gravel bed. The drainage area above I-90 is 

1,300 km2 (502 mi2) and the design discharge was 85 m3/s (3,000 ft3/s).

The highway crossing is located on an alluvial fan. At this location there is insufficient slope 

(energy) to transport the cobble and gravel material. Since 1964, it has been necessary to 

excavate about 20,000 m3 (26,160 yd3) of deposited bed material on three occasions at an 

expense of hundreds of thousands of dollars. The excavation was necessary to pass the 

flow from the spring snowmelt runoff. The primary aggradation problem is insufficient flow 

area and is aggravated by too many piers in the channel as well as a bad alignment with a 67 

degree skew.

In 1966 several rock and wire basket flow deflectors were installed for several hundred 

meters upstream of the bridge to constrict the flow and increase the transport characteristics. 

Figure 9.11 illustrates the deflector arrangement as well as the alignment problem. The 

deflectors were not very effective. They did constrict the flow and increase the velocity to 

transport the gravel sizes, but the cobble bed material still deposited upstream of the bridge. 

The constriction was not enough compensation for the reduction in slope as the creek comes 

out of the Black Hills.

Figure 9.11. Deflector arrangement and alignment problem on I-90 Bridge across Big Elk

 Creek near Piedmont, North Dakota (Example 10).
9.33

The channel is not well defined as it flows onto the alluvial fan. As a result, it is difficult to 

locate a bridge to take into account both significant lateral channel migration and bed elevation 

changes. A possible solution to the problem would be to build a debris basin upstream of the 

bridge to trap the large-sized bed material. This would provide a source of gravel which is in 

demand in that area. This might be a site where a bedload transport model may be helpful in 

determining the solution to the aggradation problem.

9.5.11 Outlet Creek at US-101 Near Longvale, California (Example 11)

Examples 11 through 13 were taken from Federal Highway Administration Report No.

FHWA/RD-80/158. These examples illustrate situations where the river channel was

relocated to accommodate highway encroachments and crossings. These examples are 

reprinted from the original publication except for minor editing to conform with the format of 

this text.

In Example 11, the Outlet Creek channel was shortened from 435 m (1,427 ft) in length to 335 

m (1,100 ft) and relocated to avoid two crossings on the realigned highway curve (Figure 

9.12). The stream is semi-alluvial, and resistant bedrock crops out in the bottom of the 

relocated channel. The highway embankment, which forms the right bank of the relocated 

channel, is heavily riprapped, and the main potential for instability is at the left bank, which is a 

steep (3/4:1) slope cut into colluvial material. However, no erosion or slumping was

observed.

Figure 9.12. Plan sketch of channel relocation, Outlet Creek (Example 11).

The site location is on US-101, 3.2 km (2 mi) south of Longvale, California. Outlet Creek is 

perennial with a drainage area of about 360 km2 (140 mi2) at the site and with an average 

discharge of 12 m3/s (424 ft3/s). The channel width is 9-15 m (30-50 ft), with a channel slope 

of 0.0036. The bed material is gravel and cobble, bank material is gravel and sand where 

alluvial.

The length of a bend in the natural channel was shortened by a factor of 0.77, and room for 

the relocated channel was made by grading back a steep valley side-slope. The relocated 

channel lies between this graded slope and the riprapped highway embankment. Riprap on 

the embankment includes rocks weighing several tons, and erosion of the embankment is 

unlikely.
9.34

Following an extreme flood in 1964, no maintenance work was needed. Small trees have 

become established on the riprapped highway embankment and along the base of the cut 

slope.

Landslides, the major potential for instability along the relocated channel, are very common in

some California terrains, but are not evident here along the valley of the Outlet Creek. The 

lack of naturally occurring landslides, which is attributed to the resistance of the underlying 

bedrock, was an indication that the cut slope would not be particularly susceptible to failure by 

mass movement. Except for the bare upper part of the cut slope, the appearance of the 

relocated channel is not unnatural for a mountain stream in a narrow valley.

9.5.12 Nojoqui Creek at US-101 At Buellton, California (Example 12)

The lowermost reach of this creek was relocated to enter Santa Ynez River upstream from 

the US-101 bridge, for purpose of avoiding a stream crossing at an interchange (Figure 9.13). 

The performance period of 16 years (1964-1979), during which major floods occurred in 1969 

and 1978, showed no evidence of degradation or lateral erosion in the relocated channel, but 

a sinuous low-water channel had developed in the wide bottom of the relocated channel. 

Severe bank erosion occurred in the natural channel at the bend upstream from the relocation 

during flood of 1978, but this is not attributed to the relocation.

Figure 9.13. Plan sketch of Nojoqui Creek channel relocation (Example 12).

The site is located on US-101 about 1.3 km (0.8 mi) south of Buellton. Nojoqui Creek is 

intermittent, with a drainage area about 39 km2 (15 mi2). The stream is ungaged, but an 

adjacent gaged stream of similar drainage area (Alisal Creek) has an average discharge of 

0.15 m3/s (5.3 ft3/s), with no flow 64 percent of the time. This point-bar braided stream is 

generally incised into terraces. Tree cover along the channel is less than 50 percent. Bed 

material is gravel, cobbles, and small boulders; bank material is moderately cohesive silt, clay 

and gravel.

The lowermost 760 m (2,500 ft) of natural channel was relocated into a straight artificial 

channel 640 m (2,100 ft) in length, resulting in a length change factor of 0.84. The width of the 

natural channel was in the range of 15-20 m (50-65 ft). The relocated channel has a top width 

of 42 m (138 ft), a bottom width of 31 m (102 ft), and is bounded by riprapped dikes that rise 

about 1.5 m (5 ft) above the flat bottom. Slope of the natural channel was 0.0044; of the 

relocated channel, 0.0053, decreasing to 0.0025 at the lower end. The dikes bounding the 
9.35

relocated channel are riprapped in part with large 1 m (3.2 ft) rock and in part, with 0.6 m (2 ft) 

rock, with the toe of riprap extending to a depth of 1.8 m (6 ft) below the channel bottom.

Floods having an estimated recurrence interval greater than 25 years occurred in 1969 and 

1978. The banks of the natural channel at a bend upstream from the relocation were severely 

eroded in 1978, and the channel was subsequently realigned by the bulldozing of bed material 

against the banks. The flood apparently disrupted the riprap facing of dikes along the

relocated channel, but there is no evidence that the dikes were broken. Young willows and 

other vegetation have become established along the dikes and, locally, in the channel bottom.

Because the bottom width of the relocated channel is more than twice that of the natural 

channel, a sinuous low-water channel has developed, which may eventually erode laterally 

against the bounding dikes. In addition, the wide bottom may become overgrown with 

willows, which will impair its transmission of floods.

9.5.13 Turkey Creek at I-10 Near Newton, Mississippi (Example 13)

A reach, 625 m (2,050 ft) in length, was relocated and thereby shortened to 270 m (886 ft), for 

the purpose of improving the channel alignment at the bridges and to accommodate the 

planned roadway location (Figure 9.14). The performance period was 15 years (1964-1979). 

At the U.S. Geological Survey gage on nearby Chunky River, major floods occurred in April 

1974 and January 1975. As specified in the plans, the relocated channel was trapezoidal in 

cross section, with a bottom width of 9 m (29.5 ft), and a side slope of 2:1. Channel slope as 

measured on the topographic map is about 0.0016. No bank protection measures were 

applied. In 1979, the bottom width of the relocated channel between the interstate bridges 

was in the range of 3-4 m (10-13 ft), resistant coherent clay was exposed in the channel 

bottom, and the banks were stable. The natural channel was generally stable (although 

choked with debris) upstream from the relocation; but downstream the bank was eroded and 

unstable at the outside of banks where, bordered by a pasture, the bottom width was in the 

range of 9-11 m (30-36 ft). Bank instability at this point was more severe than in 1955, but 

causes other than the relocation may have contributed to this.

Figure 9.14. Plan sketch of Turkey Creek channel relocation (Example 13).
9.36

Stability of the relocated channel, which is substantially more narrow than specified in the 

plans and also more narrow than the natural channel, is attributed to the resistant clay in the 

bed and lower banks. Evidently because of mowing of the median area, no trees have 

become established along the bankline.

9.5.14 Gravel Mining on the Russian River, California (Example 14)

It is essential to monitor and manage sand and gravel mining so as not to induce undesirable 

instabilities in the river system. In most cases, removal of sand and gravel has caused 

deepening and widening of the channel. These wider, deeper reaches act as sinks for the 

sediment loads and they may trap the finer clays and silts, altering the river environment. An 

interesting example of a river where excessive removal of sands and gravels has caused 

significant changes is in the vicinity of the confluence of Dry Creek with the Russian River. 

The location map is shown in Figure 9.15a. The tributary (Dry Creek) enters the river just 

downstream of a small dam on the Russian River. Previous sand and gravel extraction in the 

tributary has not exceeded the calculated safe yield, however, extraction rates in the middle 

reach of the Russian River significantly exceeded the safe yield for the period 1951 to 1964. 

This excessive extraction of sand and gravel induced a headcut that progressed upstream 

along the tributary, as shown in Figures 9.15b and 9.15c. This headcut was curtailed by a 

rock outcrop acting as a control point near a bridge approximately seven miles upstream of 

the confluence and essentially stabilized around 1972-1973. During the years 1946 to about 

1955, the tributary channel widened (Figure 9.15d). Figure 9.15e shows the corresponding 

stage discharge relationships of the Russian River at the mouth of Dry Creek. Although in 

this example both rivers are bar-braided systems, they indicate the magnitude of possible 

adverse consequences from excessive sand and gravel mining on meandering streams.

Figure 9.15. Case study of sand and gravel mining (Example 14).
9.37

9.5.15 Nowood River and Ten Sleep Creek Confluence, Wyoming (Example 15)

During a site visit a unique situation was uncovered at the confluence of Ten Sleep Creek and 

the Nowood River in the Big Horn Basin of Wyoming. While investigating the site it was noted 

that the Nowood River had become unstable and considerable meandering activity was 

posing a threat to a rancher's numerous hay meadows. There was considerable evidence 

that the rancher had constructed and armored several cutoffs as well as armored incipient 

bendway activity to try and protect his meadows. The river had unstable banks and displayed 

degradation of the bed - obviously the river was in an unstable regime.

The rancher was contacted and could offer no explanation for the unusual bendway activity. 

He did say that prior to 1935 the Nowood River had been stable. Notably, that was the same 

year a highway agency had constructed the new bridge across the Nowood River. The 

rancher, in passing, said the 1935 bridge had replaced two bridges; one on the Nowood River 

and one on Ten Sleep Creek (Figure 9.16). The indiscriminate channel change employed to 

economize by constructing only one bridge had pushed the Nowood River past a stability 

threshold. The result was an unstable reach.

9.5.16 Middle Fork Powder River, Wyoming (Example 16)

Significant and rapid erosion occurred immediately downstream from the new bridge across 

the Middle Fork Powder River at Kaycee, Wyoming. The erosion is primarily bank migration 

in the bendways and threatens to cause a meander cutoff (Figure 9.17). This cutoff would 

destroy the community's rodeo grounds.

The community blamed the highway agency's new bridge, claiming the bridge piers

improperly directed flows into the downstream bendway. The highway agency's hydraulic 

engineers did not agree, as the new bridge was larger and more efficient than the previous 

structure. Aerial photos of the river reach taken several years apart through this period were 

obtained. New photos were obtained showing the river's present planform. Together these 

photos displayed a planform history. 

The river was noted to be relatively stable until such time as when a downstream rancher had 

constructed a major cutoff to gain additional pasture land. Unfortunately the bridge was 

constructed about the same time as the cutoff was completed. However, the evidence was 

overwhelming in attributing the sudden instability to the rancher's channel change. These 

findings were presented to the community of Kaycee and the complaints ceased.

The erosion problem immediately downstream from the bridge is expected to continue.

Should the community fail to forestall the cutoff, the rodeo grounds will be destroyed and the 

bridge will be in jeopardy from potential headcutting, as will an upstream trailer park.
9.38

Figure 9.16. Nowood River near Ten Sleep, Wyoming (Example 15).
9.39

Figure 9.17. Middle Fork Powder River at Kaycee, Wyoming (Example 16).

9.6 CONCLUDING REMARKS ON DESIGN CONSIDERATIONS

The dynamic features of rivers and river systems and the natural beauty of the river scenery 

make the design of highways in the river environment one of the most challenging and 

stimulating of all engineering designs.

This chapter illustrates the many interactions between rivers and highway structures. Rivers 

are dynamic, at times migrating rapidly across floodplains, at others lying dormant through 

years of low flows, only to break out of their banks during the next flood to recarve the form of 

the immediate landscape and, potentially, seriously impact highway structures. The design of 

bridges, bridge countermeasures, and river restoration must consider low, intermediate, 100-

year, and super floods. The next chapter presents three design examples illustrating the 

application of the principles and methods given in this manual.
9.40

(page intentionally left blank)
10.1 

CHAPTER 10 

OVERVIEW EXAMPLES OF DESIGN FOR 

 HIGHWAYS IN THE RIVER ENVIRONMENT 

In this chapter, three examples are given illustrating the application of the principles, 

methods and concepts of previous chapters to the design of highway encroachments and 

crossings in the river environment. The designs encompass the use of geomorphic, 

hydrologic and hydraulic principles to design safe and economical crossings that protect, 

maintain and restore the river environment. The three-level design procedure discussed in 

Chapter 9 is emphasized in these designs. 

In the examples, the designs are determined by well-established numerical procedures; 

however, they also depend heavily on the judgment of the engineer. The examples should 

be read and studied as an illustrative unit and not as designs from which one can choose the 

correct prescription for a problem at hand. River problems are much too complex for a 

cookbook approach, as is evident from these examples. 

10.1 OVERVIEW EXAMPLE 1 - BIJOU CREEK 

This example presents a geomorphic, hydrologic, and hydraulic analysis of Bijou Creek, a 

tributary to the Narrows Unit near Fort Morgan, Colorado. The proposed location of the Union 

Pacific Railroad is approximately 1,500 ft (457 m) north of the creek (Figure 10.1). The 

purpose of the analysis is to ensure the safety of the proposed railroad location and to evaluate 

river engineering design alternatives. Since the data for this case study are available in 

English units, the figures and tables retain English unit notation. SI (metric) units are given 

parenthetically in the text for reference. 

Figure 10.1. Topographic map of Bijou Creek study area. 
10.2 

10.1.1 Level 1 - Reconnaissance and Geomorphic Analysis 

A Level 1 field study was conducted to obtain morphologic data for the Level 2 hydraulic 

analysis. The information utilized in the analysis includes the representative cross sections of 

the channel, the representative bed material size, and the energy slope. With these data, the 

flow resistance coefficient was estimated. 

The representative cross section in the vicinity of the study site was developed by considering 

five selected cross sections established during the field study. Of these, two cross sections 

were similar in shape and are considered representative of the cross sections of the study 

reach. The dimensions of a representative cross section were established by averaging these 

two cross sections. The results of this analysis are shown in Figures 10.2 and 10.3. 

Figure 10.2 shows the relationship between cross sectional areas and the flow depth as 

follows: 

52.1

n = y6.242A

in which A is the cross sectional area and yn is the normal depth of flow measured from the 

thalweg level. 

Figure 10.3 gives the following relationship between the wetted perimeter and the flow depth. 

31.0

n = y0.682P

The average top width of channel as estimated by field survey and topographic maps is 1,700 

ft (518 m). 

The results of sieve analyses of the bed material samples taken at three cross sections are 

shown in Figure 10.4. The average median bed material size, D50, is 0.45 mm. The average 

D16 size is 0.22 mm, and the average D84 size is 0.91 mm. The bank material has nearly the 

same distribution as that of the bed material, but contains lenses of silt and clay. The bank is 

highly stratified and can be easily eroded. 

The energy slope may be less than the channel bed slope. However, for a safer design, it is 

assumed that the energy slope is equal to the channel bed slope. The average channel bed 

slope from field surveys is 0.00252 ft/ft (m/m). This slope is adopted as the design energy 

slope. 

The floods of June 1965 in South Platte River Basin, Bijou Creek were in upper regime, having 

antidunes with breaking waves. Similarly, computations show that the bed forms of Bijou 

Creek (D50 = 0.45 mm) will be antidunes or standing waves during floods. Resistance to flow 

associated with antidunes depends on how often the antidunes form, the area of the reach 

which they occupy, and the violence and frequency of their breaking. If the antidunes do not 

break, resistance to flow is about the same as for a plane sand bed, and the discharge 

coefficient, C/(g)1/2, (where C is the Chezy resistance coefficient and g is the gravitational 

acceleration) ranges from 14 to 23 (Manning's coefficient, n, is about 0.017 to 0.027 for the 

flow depths being considered). The acceleration and deceleration of the flow through the 

nonbreaking antidunes (frequently called standing waves) causes resistance to flow to be 

slightly more than that for flow over a plane bed. If many antidunes break, resistance to flow 

can be very large because the breaking waves dissipate a considerable amount of energy. 

With breaking waves, C/(g)1/2 may range from 10 to 20, and Manning's coefficient n ranges 

from about 0.019 to 0.038 for the flow depths being considered. 
10.3 

Figure 10.2. Cross sectional area versus flow depth relation. 
10.4 

Figure 10.3. Wetted perimeter versus flow depth relation. 

Figure 10.4. Analysis of bed material size of Bijou Creek. 
10.5 

From available data on the stage-discharge relation at the stream gaging station (near 

Wiggins, Colorado) it is estimated that the Manning roughness coefficient, n, during the design 

floods is about 0.023. This value of Manning's n gives a computed mean flow velocity of 18.7 

ft/s (5.7 m/s) in Bijou Creek for the extreme flood of June 1965 [discharge 466,000 ft3

/s (13,200 

m3

/s)]. 

10.1.2 Level 2 - Quantitative Engineering Analysis 

Hydrologic Analysis. The hydrologic analysis identified three design floods and corresponding 

water surface elevations with return periods of 50, 100, and 200 years. Bank protection will be 

specified for each of the design floods for five channel design alternatives. The design floods 

were estimated using the Gumbel Method of frequency analysis and all of the available 

hydrologic data. The design of riprap bank protection will consider the design flood discharge, 

superelevation in the bend, bedform height, local scour, bed material size, and stability of 

riprap materials. 

The historical records show that there were two extreme floods 282,900 cfs (8,012 m3

/s) and 

466,000 cfs (13,200 m3

/s) observed at the streamflow gaging station of Bijou Creek near 

Wiggins, Colorado. Historic stream flow data were plotted in Figure 10.5 using Gumbel 

probability paper. From this figure, the floods with various return periods can be interpolated or 

extrapolated. The estimated design floods with return periods of 100 and 200 years are 

respectively 62,000 cfs (1,756 m3

/s) and 72,500 cfs (2,053 m3/s) . 

Moveable Bed Hydraulic Analysis. The information on hydraulic conditions needed for 

designing bank protection includes the normal depth of flow, the cross sectional area of flow, 

the mean flow velocity, the Froude number, the bedform height, the local scour depth, the 

superelevation of the flow in the bend, the local depth, and the local velocity. Moreover, 

different design alternatives will result in different hydraulic conditions. In this study, five design 

alternatives were proposed; the first alternative was to protect the existing outer bank with 

riprap (Figure 10.6), the second alternative was to realign the bend to its plan geometry before 

1965 and to protect the outer bank with riprap (Figure 10.7), and the third alternative was to 

realign the bend to that of a mild bend relative to the existing channel alignment and to protect 

the outer bank with riprap (Figure 10.8). The fourth alternative was to determine the necessary 

buffer strip distance between the railroad and the present north river bank if bank protection is 

not utilized The fifth alternative was to use rock riprap spurs to protect the existing river bank 

by realigning the channel (Figure 10.9). 

As shown in Figure 10.9, the fifth alternative, to construct a series of rock riprapped spurs to 

guide the flow away from the existing north bank, has the same flow alignment as the second 

alternative (Figure 10.7). 

A summary of computed hydraulic conditions in the first three basic alternatives is given in 

Table 10.1. The data necessary to determine riprap size are included for three values of radius 

of curvature, rc, of the bend. 

The following normal depth-discharge relation is determined by using Manning's equation, 

using the slope of energy gradient S = 0.00252, and Manning's roughness coefficient n = 

0.023. 

32.2

n = y02.395Q

In this relation Q is the design discharge and yn is the normal depth of flow measured from the 

thalweg level. 
10.6 

Figure 10.5. Gumbel's method of frequency analysis. Annual floods on Bijou Creek near 

 Wiggins, Colorado. 

Figure 10.6. Proposed first alternative. 
10.7 

Figure 10.7. Proposed second alternative. 

Figure 10.8. Proposed third alternative. 
10.8 

Figure 10.9. Proposed fifth alternative. 

 

Table 10.1. Summary of Computer Hydraulic Conditions. 

T 

yrs 

Q 

cfs 

Yn 

ft 

A 

ft 

V 

fps 

F H 

ft 

Ys

ft 

A rc

ft 

∆z

ft 

yo

ft 

Vo

fps 

50 51,600 8.17 5,904 8.74 0.72 1.05 8.82 

1 

2 

3 

950 

1,900 

2,650 

4.24 

2.12 

1.52 

12.40 

10.28 

 9.69 

17.37 

15.30 

14.70 

100 62,000 8.84 6,660 9.31 0.74 1.18 9.63 

1 

2 

3 

950 

1,900 

2,650 

4.82 

2.41 

1.73 

13.65 

11.20 

10.57 

18.50 

16.20 

15.60 

200 72,500 9.46 7,379 9.82 0.75 1.32 10.33 

1 

2 

3 

950 

1,900 

2,650 

5.36 

2.68 

1.92 

14.82 

12.14 

11.38 

19.50 

17.11 

16.40 

Note: T is the return period, Q is the design discharge, yn is the normal depth from the thalweg level, A is the cross-sectional 

area of flow, V is the mean flow velocity, F is the Froude number, H is the antidune height, ys is the scour depth at the 

leading portion of riprap bank protection, A is the alternative of designs, rc is the radius at the center of a bend, ∆z is the 

superelevation, and yo and Vo are the depth and velocity, respectively, for designing riprap sizes. 

For design floods with return periods of 50, 100 and 200 years, the normal depths from thalweg 

level are determined. Then the cross sectional area and the wetted perimeter can be 

computed. Finally, the mean flow velocity, the hydraulic radius, and the Froude number are 

calculated. 
10.9 

As shown in Table 10.1, the average Froude number is approximately 0.74, which implies the 

maximum local Froude number in the center of the flow could be on the order of 1.0 to 1.2. 

According to Simons and Richardson (1963, 1966), these flow conditions should be in the 

upper regime with antidunes with breaking or nonbreaking waves. The estimated Manning's 

coefficient of 0.023 is correct for these flow conditions. In addition, the backwater effects are 

negligible because the flows are in the upper regime. The normal depth as computed should 

be satisfactory for design purposes. 

There are many methods for determining riprap size (Simons and Senturk 1977). Among 

them, the method developed by Stevens et al. (1984) may be the most comprehensive and 

appropriate method to use. In applying their method, the depth of flow, the flow velocity and 

the angle between the horizontal and the velocity vector in the plane of the side slope are 

necessary as discussed in Chapter 6. 

As mentioned in Chapter 3, the bed forms of antidunes or standing waves constitute a series of 

inphase symmetrical sand and water waves. These waves are in the center of the channel. 

Thus, the depth of flow at the bank for designing riprap sizes is taken to be the sum of the 

normal depth from thalweg level and the superelevation (yo = yn + ∆z). The flow velocity for 

designing riprap size is computed by utilizing Manning's equation: 

2/1

f

3/2

o o Sy

n

486.1 V =

in which Sf is the energy slope. The angle between the horizontal and the velocity vector in the 

plane of the side slope is assumed to be negligible. 

Three design alternatives based on different radii of curvature are considered in this phase of 

the study. Different hydraulic designs result for each of the three design alternatives and each 

of three different design floods. The hydraulic design includes the determination of the total 

length of bank protection, the minimum buffer strip distance between the railroad and the river 

bank, the estimated volume of earthwork, the side slope of riprapped bank, the sizes of riprap 

material, the thickness of riprap, the size of the gravel filters, the thickness of the gravel filters, 

the height of riprap protection above the existing bed level, and the depth the riprap should 

extend below thalweg level. Using the hydraulic data, all these values can be computed. 

Table 10.2 provides a summary of the hydraulic designs for different design conditions. The 

methods of design are described in the following sections. 

From Figures 10.6, 10.7, and 10.8 the length of bank protection, minimum buffer strip distance 

between the railroad and the river bank, and the excavation volumes for three different design 

alternatives can be estimated. The minimum buffer strip distance is a measure of how far the 

river bank must migrate due to bank erosion to endanger the railroad. A wider minimum buffer 

strip distance between Bijou Creek and the railroad will provide a larger factor of safety for the 

railroad. However, the length of bank protection and excavation volumes increase accordingly, 

which in turn increase the cost of construction. 
10.10 
10.11 

10.1.3 Riprap Design for Alternatives 1, 2, and 3

Side Slope of Riprap Bank. The side slope of the riprapped bank should be less than the angle 

of repose of the bank material. Analysis shows that the angle of side slope should be at least 

5°

 less than the angle of repose of the bank material. The bank material is medium sand with 

some clay and silt with average D50 of about 0.35-0.45 mm. Its angle of repose is about 29°

(see Chapter 6). A side slope 2.5:1 (horizontal to vertical distance) is utilized and the 

corresponding angle of side slope is 21.75°

 (Figure 10.10). 

Riprap Design. The size of riprap is determined using the method presented in Chapter 6. The 

method developed by the Bureau of Reclamation (1958) to determine the maximum rock size 

in a riprap mixture is used for comparison. The flow conditions required for designing riprap 

sizes were discussed earlier and given in Table 10.1. 

Using a stability factor of 1.3 and an angle of repose of riprap material of 41°

, the required 

riprap sizes with a uniform gradation are determined. The results are shown in Table 10.3. 

Table 10.3. Design of Riprap Sizes. 

Riprap Mixture by 

Return Period Stevens et al. (1974) 

(yrs) Discharge 

(cfs) 

Design 

Alternatives 

Uniform Size 

by Stevens 

et al. (1974) 

Median 

Diameter (ft) 

Maximum 

Size (ft) 

Maximum 

Size by 

USBR (1958) 

(ft) 

 50 51,600 

1 

2 

3 

2.6 

2.0 

1.8 

1.5 

1.1 

1.0 

3.3 

2.4 

2.2 

2.6 

2.0 

1.8 

100 62,000 

1 

2 

3 

3.0 

2.2 

2.0 

1.7 

1.3 

1.2 

3.7 

2.9 

2.6 

3.1 

2.2 

2.0 

200 72,500 

1 

2 

3 

3.4 

2.5 

2.2 

2.0 

1.4 

1.3 

4.4 

3.1 

2.9 

3.5 

2.5 

2.3 

It is not acceptable to riprap a bank with uniform size rock, especially when large rock is 

required. Therefore, it is necessary to design a riprap mixture that includes an adequate range 

of sizes. In addition, the stability factor for determining the D50 of the riprap mixture is assumed 

to be 1.1. According to Chapter 6, riprap gradation should follow a smooth size distribution 

curve with gradation coefficient of about 1.3. Riprap sizes are given in Table 10.3. 

The Bureau of Reclamation (1958) developed a figure to determine the maximum rock size in 

a riprap mixture downstream from stilling basins. If the bottom velocity is assumed equal to the 

reference velocity on the top of the rock, the maximum rock sizes in the riprap mixtures for 

different design conditions can be estimated. The results utilizing both methods are given in 

Table 10.3 for comparison. The sizes obtained by use of the Stevens method given in Chapter 

6 are recommended. 
10.12 

Figure 10.10. The sketch of proposed riprap design. 
10.13 

Gravel Filter Design. A filter should be placed under the riprap unless the material forming the 

core of the structure is coarse gravel or of such a mixture that it forms a natural filter. Two 

types of filters are commonly used: gravel and geotextile filter. 

The sizes of gravel in the filter layer are calculated as explained in Chapter 6 and the results 

are given in Table 10.2. Two layers of filter material are required because the riprap sizes are 

large. 

Thickness of the filters may vary depending on the riprap thickness, but should not be less than 

6 to 9 in. (15 - 23 cm). Filters that are one-half the thickness of the riprap are satisfactory and 

provide a great degree of safety. 

Height and Depth of Riprap. The design height of riprap protection above the existing bed 

level (Figure 10.10) must provide for freeboard, water depth, superelevation and wave height. 

The normal depth from the thalweg level and the superelevation were determined earlier. 

Because the forms of antidunes are a series of inphase symmetrical sand and water waves, 

the wave height at the water surface can be assumed equal to the antidune height. The 

minimum height required for riprap protection above the existing bed level can be determined 

by considering normal depth, superelevation, and antidune height. In order to provide 

additional protection against the breaking waves, an extra foot of freeboard is added to the 

required minimum height of riprap above bed level. 

The riprap (Figure 10.10) must extend some distance below the thalweg level to provide safety 

against possible local scour, general scour, and troughs of passing sand waves. The general 

scour is assumed to be negligible because the proposed structures do not significantly 

constrict the flow. Some local scour is expected at the leading portion of the riprap revetment. 

Hence, a larger depth of riprap is required at this location. The minimum depth of riprap below 

thalweg level can be determined by considering the potential scour and the antidune height. 

In general, the riprap should extend a minimum of 5 ft (1.5 m) below thalweg level in order to 

protect against possible long-term degradation of the river reach. If the computed depth of 

riprap protection below thalweg level is less than 5 ft (1.5 m), the design depth is set at 5 ft (1.5 

m). 

10.1.4 Design for Alternatives 4 and 5

Alternative 4. The fourth alternative is simply to provide sufficient buffer strip distance between 

the railroad and the existing river bank so that bank protection may not be required. The 

distances the north bank will migrate under different design flood conditions can be evaluated 

by utilizing sediment transport rates and the migration history of the 1965 flood. The estimated 

bank migration distances for the design floods are given in Table 10.4. In order to provide 

extra protection against slope failures and long-term bank migration due to smaller sizes of 

floods, the design buffer strip distances should be at least twice the computed migration 

distances (factor of safety - 2.0) because the accumulated bank migration distance due to 

smaller sizes of floods can be very significant. 
10.14 

Table 10.4. Bank Migration for Alternative 4. 

Return Period 

(yrs) 

Discharge 

(cfs) 

Bank Migration 

Distance (ft) 

Design Buffer 

Strip Distance* (ft) 

 50 

100 

200 

51,600 

62,000 

72,500 

562 

629 

715 

1,124 

1,258 

1,430 

*The eroded bank should be restored to its present alignment after each flood to maintain an 

 adequate buffer zone. 

Alternative 5. The fifth alternative is to construct a series of rock riprap spurs to guide the flow 

away from the existing north bank (Figure 10.9). The design of the spurs includes: the form of 

spurs, the angle of spurs to the bank, the length of spurs, the spacing between spurs, the 

height or elevation of spurs, the construction materials, the crest width and slopes, and the 

local scour. The design of the spurs is summarized in Tables 10.5, 10.6, and 10.7. Table 10.5 

gives a summary of spur designs for different design floods including buffer strip distance, 

number of spurs, spacing of spurs, height of spur above the existing thalweg level, riprap size 

at the spur nose, and riprap size in the spur shank. Table 10.6 provides the suggested design 

dimension of spurs including spur form, spur length, angle of spur to bank, the portion requiring 

riprap the same size as used on the nose, and length requiring shank protection. Table 10.7 

summarizes the sizes of riprap and filter design at the nose of the spurs and for the spur 

shanks. HEC-23 (Lagasse et al. 2001) also provides design guidelines for a typical round nose 

spur recommended by FHWA. 

Table 10.5. Summary of Spur Design. 

T 

yrs 

Q 

cfs 

Bm 

ft 

Ns Sd 

ft 

ha

ft 



b h

ft 

n

50 K

ft 

s

50 K

ft 

 50 

100 

200 

51,600 

62,000 

72,500 

1,250 

1,250 

1,250 

7 

7 

7 

600 

600 

600 

12.5 

13.5 

14.5 

10.0 

11.0 

12.0 

1.1 

1.3 

1.4 

0.6 

0.7 

0.7 

Note: T is the flood return period in years, Q is the design discharge, Bm is the minimum buffer strip distance between the 

railroad and the spur nose, Ns is the number of spurs, Sd is the spur spacing, ha is the design height of spur above the 

existing thalweg level, 

b h is the design depth of spur at the spur nose below the existing thalweg level, n50 K is the 

median size of riprap at the spur nose, and s

50 K is the size of riprap for the spur shank. 

Table 10.6. Dimension of Spurs. 

Spur 

No. 

Spur 

Form 

Spur 

Length 

(ft) 

T-Nose 

Length 

(ft) 

Angle of 

Spur to Bank

(°) 

Length 

Requiring 

Nose Riprap 

Size (ft) 

Length 

Requiring 

Shank 

Protection (ft) 

1 

2 

3 

4 

5 

6 

7 

T-nose 

T-nose 

Round nose 

Round nose 

Round nose 

Round nose 

Round nose 

200 

200 

500 

750 

750 

400 

200 

400 

400 

N/A 

N/A 

N/A 

N/A 

N/A 

90 

90 

70 

70 

70 

70 

70 

 20 

 20 

200 

200 

200 

200 

200 

180 

180 

300 

550 

550 

200 

 0 
10.15 

Table 10.7. Size and Filter Design for Riprap at Spur Noses and Spur Shanks. 

T (yrs) Q (cfs) K50 (ft) GK tr (ft) 1

50 f (mm) 1 Gf .)in(t1

f

2

50 f (mm) 2 GF ( .)int 2

f

(a) At Spur Noses 

 50 

100 

200 

51,600 

62,000 

72,500 

1.1 

1.3 

1.4 

1.3 

1.3 

1.3 

2.4 

2.9 

3.1 

6.0 

6.0 

6.0 

3.0 

3.0 

3.0 

 8.0 

 9.0 

10.0 

60.0 

60.0 

60.0 

2.0 

2.0 

2.0 

7.0 

9.0 

9.0 

(b) At Spur Shank 

 50 

100 

200 

51,600 

62,000 

72,500 

0.6 

0.7 

0.7 

1.3 

1.3 

1.3 

1.3 

1.5 

1.5 

6.0 

6.0 

6.0 

3.0 

3.0 

3.0 

6.0 

6.0 

6.0 

20.0 

20.0 

20.0 

2.0 

2.0 

2.0 

6.0 

6.0 

6.0 

Note: T is the return period, Q is the design discharge, K50 is the design riprap size for which 50 percent is finer by weight, GK is 

the gradation coefficient of riprap, tr is the thickness of riprap, 1

50 f , 1 Gf , and 1f t are, respectively, the gravel size for which 

50 percent is finer by weight, the gradation coefficient, and the thickness of the first layer of gravel filter, and 2

50 f , 2 Gf , 

and 2

f t are, respectively, the gravel size for which 50 percent is finer by weight, the gradation coefficient, and the 

thickness of the second layer of gravel filter. 

 

Figure 10.11 shows the suggested spur design. The methods of design are briefly described 

below. 

In Figure 10.9, Spurs No. 1 and 2 are T-nose spurs, and the others are round-nose spurs. The 

angle of spur to the bank is usually 60°

 to 120°. The available literature shows that the angle 

for T-nose spurs is normally 90°

 but the angles for round spurs varies. Mamak (1964) states 

that the best results for deflecting flow and trapping sediment load are obtained with spurs 

inclined upstream from 100°

 to 110°. However, the study by Franco (1967) showed that for 

channelization the normal or angled downstream spurs (60°

) performed better than the angled 

upstream spurs. HEC-23 (Lagasse et al. 2001) suggests that for most applications, spurs can 

be oriented at 90°

 to the bank line, except for the first spur in the spur field. In this case, with a 

short radius bend, a moderate downstream angle is justified. Thus, judging from the purpose 

of spurs and flow conditions being considered, it is determined that the angles of round spurs 

should be constructed about 70°

 to the bank, angled downstream as shown. 

The length of a spur depends on its location, amount of contraction of stream width, and 

purpose of the spur. The purpose of the spurs considered in this study is to guide the flow 

away from the bank and to provide a flow alignment similar to that of 1965. The lengths of 

spurs are determined to serve this purpose (Figure 10.9 and Table 10.7). 

The spacing between spurs is primarily related to the length of the spur. In general, the 

recommended spacing is from one and one-half to six times that of the upstream projected 

spur length into the flow (see HEC-23 for additional guidance). For bank protection in a sharp 

bend, a smaller spacing should be used. For the design conditions, the spacing of spurs is 

taken as 600 ft (183 m). 

The height or elevation of a spur is determined by considering the maximum flow depth above 

thalweg level. In order to provide additional protection against the breaking waves, an extra 

foot of freeboard was added to the design height of each spur above the existing thalweg level. 

The local scour depth at the spur nose is the same as that at the leading portion of the 

continuous riprap revetment. The computed local scour depths for different design floods are 

given in Table 10.2. The minimum depth of riprap below thalweg level is determined by 

considering the local scour depth and the antidune heights, which also gives the design depth 

of spur at the spur nose below the existing thalweg level. 
10.16 

Figure 10.11. Sketch of spur design. 
10.17 

The construction layout is shown in Figure 10.11. The riprap design and the gravel filter design 

at the spur nose are the same as those used in the continuous revetment. The length requiring 

nose riprap size can be estimated by considering the flow separation zone. The riprap size in 

the shank inside this zone may use a smaller rock size. A reduction of fifty percent in rock size 

is determined by considering the decrease in flow velocity. The riprap at the downstream side 

of shank may be eliminated if a larger risk is accepted. This is because the downstream side 

of the shank is not expected to be subjected to a strong velocity. However, it may be subject to 

scour due to overtopping. 

The crest width of rock riprap spurs usually ranges from 3 to 20 ft (1 to 6 m) and the side slope 

from 1.25:1 to 5:1. Considering the convenience in hauling and placing rock riprap, the crest 

width of spurs is determined to be 15 ft (5 m). The side slope was determined to be 2.5:1. 

A second set of alternatives would result from replacing the rock riprap with soil-cement 

revetment in design alternatives 1, 2, and 3. This alternative may be of interest because large 

rocks can be very difficult to obtain and soil-cement revetment may be manufactured at the site 

without much difficulty (see Chapter 6 and HEC-23). 

10.2 OVERVIEW EXAMPLE 2 - RILLITO RIVER

10.2.1 Background 

The Rillito River System in Tucson, Arizona provides an example of the problems encountered 

in bridge crossing design. The objective of this example is to illustrate the methodologies used 

in the analysis and design of a bridge crossing, including a three-level analysis and the 

evaluation of conceptual alternatives. Since the data for this case study are available in 

English units, the figures and tables retain English unit notation. SI (metric) units are given 

parenthetically in the text for reference. 

Two bridge sites are reviewed which provide insight into several outstanding problems 

characteristic of the Rillito system. These are the Sabino Canyon Road site with an existing 

bridge crossing (constructed 1936) and the Craycroft Road site with a dip crossing (where the 

roadway is at the same elevation as the channel bed). The study reach of Rillito River includes 

approximately 11.5 miles (18.5 km) of channel extending from Dodge Boulevard to Agua 

Caliente Wash (Figure 10.12). This includes 2 miles (3.2 km) on the Rillito River below 

Craycroft Road, 6-1/2 miles (10.5 km) upstream of Craycroft road on Tanque Verde Creek, 2 

miles (3.2 km) upstream of Craycroft Road on Pantano Wash and 1 mile (1.7 km) on Sabino 

Creek upstream of the confluence with Tanque Verde Creek. 

The history of flood events and the recent geomorphology of the Rillito system has shown that 

it is very dynamic and illustrates the characteristics of a braided river. The channel is steep, 

dropping at the rate of 21 ft per mile (3.98 m per kilometer). The bed material is predominantly 

in the medium to coarse sand sizes. The natural sinuosity of the river is low. Additionally, the 

river is generally unstable, changes alignment rapidly, and carries large quantities of sediment. 

A large portion of the river system is in the metropolitan area of Tucson where human activities 

in, and adjacent to, the river environment have induced a number of changes in the system. 

Primary impacts on the system have occurred due to encroachment by urban development 

and channelization of segments of the river. Uncontrolled sand and gravel extraction has also 

led to even more rapid and significant changes in the river system. A secondary effect of 

urbanization is a reduction in sediment supply from tributaries draining urban areas. 
10.18 

Figure 10.12. Rillito River system vicinity map. 

Undeveloped land in the study area generally has little protective cover and supplies large 

quantities of sediment to the river system. However, extensive erosion control measures 

established for urban development and the creation of impermeable areas in these tributaries 

has reduced the sediment supply to the river system. As urbanization continues there will be a 

long-term decrease in sediment supply that will have a significant influence on the 

geomorphology of the river system. 

The four bridge sites in the study area are at Dodge Boulevard, Swan Road, Sabino Canyon 

Road, and Tanque Verde Road (Figure 10.12). Measures have been taken at Craycroft Road 

to stabilize the crossing during the low flows and, as a result, the crossing is acting as a grade 

control on Pantano Wash. Stabilization measures have not been successful on the north side 

of Craycroft and no grade control has formed. Complex hydraulic conditions exist at the 

confluence of Tanque Verde Creek and Pantano Wash during the 100-year flood. Divided flow 

occurs with flood water spilling laterally into Pantano Wash from Tanque Verde Creek during a 

flood from that watershed, or flood water spills to Tanque Verde Creek during a flood from the 

Pantano watershed. 

The bridges across the Rillito River and Tanque Verde Creek span a variety of channel 

conditions. The sedimentation and erosion processes due to the proposed bridges will depend 

on the extent to which the bridge influences the hydraulic conditions in the river (primarily the 

velocity and depth). Conversely, the changing form of the channel due to lateral migration or 

long-term changes in the channel profile can alter the hydraulic conditions at the bridge. The 

Dodge Boulevard bridge is assumed to be the downstream control for the study reach. This 

assumption is valid because the bridge crosses a channelized section of the Rillito River which 

has little influence on the water surface elevation in the channel for the 100-year flood. The 

downstream boundary hydraulic condition is assumed to be uniform flow. 

Swan Road also crosses the Rillito River at a channelized section and has little effect on 

upstream water surface. Craycroft Road is one of two crossings discussed in detail later. The 

Sabino Canyon bridge crosses the defined channel on the Tanque Verde Creek. At the bridge 
10.19 

site, Tanque Verde Creek has also formed a sharp bend which will be unstable at high flow. 

The last bridge in the study area is at Tanque Verde Road. This bridge is on an undisturbed 

portion of the river system and crosses only the defined portion of the channel. Because the 

capacity of this channel is much less than the 100-year flood, the bridge creates a significant 

backwater for the 100-year flood. The roadway approaches would sustain heavy damage 

under these conditions. 

A three-level analysis approach was applied to the river system to identify potential erosion and 

sedimentation problems associated with the bridge sites (see Chapter 9). First, a qualitative 

geomorphic analysis was performed documenting the history of the river system, the type of 

river form, the qualitative response of the system, and potential local problems at the bridge 

sites. The second level was an engineering geomorphic analysis which assesses the general 

quantitative response of the system, including determination of sediment supply, sediment 

transport rates, equilibrium slopes in the system for selected conditions, and lateral migration 

tendencies. The third level applied a detailed water and sediment routing procedure to 

evaluate the as-is conditions and various design alternatives. This three-level analysis 

provides the necessary information to: 

1. Evaluate the stability of existing and proposed bridge structures 

2. Determine the lateral migration tendencies of the channel 

3. Estimate the extent of expected general channel scour 

4. Determine the potential local scour around bridge piers and abutments 

5. Estimate the long-term effects of sediment degradation or aggradation on the bed and 

water surface profiles 

6. Determine the effects of debris on scour and water depth at the bridge sites 

Applications of the principles of qualitative geomorphic analysis (plus basic engineering 

relations), and quantitative analysis (sediment routing) are demonstrated for the Craycroft site 

and the Sabino Canyon road bridge. Use of this three-level analysis gives a realistic bridge 

design for moveable bed and bank conditions resulting from the 100-year flood. 

10.2.2 Level 1 - Qualitative Geomorphic Analysis 

General System Response. The purpose of qualitative geomorphic analysis is to identify the 

important physical processes which have been acting on the river system. General 

geomorphic relationships are used to classify the river system. Aerial photographs are 

compiled over a series of years as a means of constructing the recent history of the river 

system. Human activities including gravel mining and river training are documented and the 

river's response noted. A qualitative prediction of river response is developed, based on 

general geomorphic relationships such as the Lane relationship (see Section 5.5). The 

information gained by this level of analysis greatly aids in applying more rigorous methods of 

analysis. 

The data base necessary for this type of analysis includes aerial photographs, topographic 

maps (1" = 100', 2' contour intervals), and site observations. With this approach, the basic 

characteristics of the river can be understood quickly with limited information. 

Much of the system has been significantly disturbed by human activities. Observed activities 

include channelization, sand and gravel mining, construction of bridges, construction of grade 

controls, road crossings, and encroachment by urbanization. Much of the system's shape and 

form, then, is dictated by human activities rather than natural processes. This is especially true 

for the portions of the Rillito River and Pantano Wash within the study area. The Rillito River 

has been subjected to major channelization up- and downstream of Swan Road (Figure 10.12). 
10.20 

There is also a large instream gravel pit below Swan Road. The Pantano Wash system has a 

large instream gravel pit below, and bank stabilization works in the vicinity of Tanque Verde 

Road. The results of these activities have been to change these systems from their natural 

braided forms to defined channels. Pantano Wash still possesses a stretch of over 3,000 ft 

(900 m) which is braided. Tanque Verde Creek, however, has experienced less impact from 

human activities than the other two systems. The islands, bends and natural channel 

alignment observed in 1941 aerial photographs of Tanque Verde Wash are still intact. 

The Tanque Verde Creek system should be classified as a braided system (see Section 5.4.5). 

The evidence of multiple channels, islands, and shifting alignment support this conclusion. A 

braided river can be identified by Equation 5.2. 

01.0QS 25.0 ≥

in which S is the average bed slope and Q is the dominate discharge (cfs). The mean annual 

flood of 5,000 cfs (142 m3

/s) is assumed to represent dominant conditions in the system. The 

average slope for 13.1 miles (21 km) of the Tanque Verde Creek and Rillito River is 0.0044. 

This slope and discharge give a value of 0.037, which is well within the braided range (Figure 

10.13). Pantano Wash has a slightly steeper grade, which would place it even further into the 

braided range. Even though much of the river has been channelized, it should be recognized 

that the river is in the braided range and, hence, is very dynamic. 

Often, the general response of a river system to a flood event can be assessed qualitatively by 

studying its profile and plan view. This is especially true of a system which has been altered by 

human activity. This type of analysis is based on estimating the relative velocity along the 

system. In locations where a channel is constricted or the profile steepens, the velocity would 

be expected to increase. Since velocity is the dominant factor in determining sediment 

transport rate (when the sediment size does not change greatly), areas with large increases in 

velocity should degrade; areas where velocities are slowed considerably should experience 

aggradation. This is expressed in Lane's relationship (Equation 5.28), which can be written: 

50s α SQDQ

In this relationship, Qs is the sediment transport rate, D50 is the median sediment size, Q is the 

flow rate of water and S is the slope of the bed. 

For example, the instream gravel pit below Swan Road will trap sediment and reduce sediment 

supply to the downstream reach. This can be expressed using the Lane relationship as: 

− −

50s α SQDQ

From this, one would expect an overall response of possible degradation in the reach of the 

Rillito River below the gravel pit to Dodge Boulevard. A similar application of the Lane 

relationship to various reaches in the study area indicates that over half of the channel reaches 

are well balanced for sediment transport. None of the reaches has a great potential for either 

aggradation or degradation. In all, the system should not experience large bed elevation 

changes except for those related to increased development and localized flow conditions. This 

qualitative assessment is confirmed by more detailed Level 2 analyses in the next section. 
10.21 

Figure 10.13. Slope-discharge relationship for Rillito River. 

Each proposed or existing bridge site has possible problems associated with local erosion and 

sedimentation processes. These problems are identified below. 

Potential Local Problems at the Craycroft Road Site. Location of a bridge at the confluence of 

Tanque Verde and Pantano Wash (Figure 10.14) could cause several problems. First, the 

confluence of two sand-bed rivers is usually very dynamic and can shift upstream or 

downstream and laterally quite quickly. This is especially true when an abnormal sequence of 

events results in a shift in the relative balance of flows between the two rivers. To compound 

this problem, the grade control structure (dip crossing) has created a situation in which 

Pantano Wash has a bed elevation several feet higher than Tanque Verde Creek at the same 

location. This provides an additional tendency for flows from Pantano Wash to migrate toward 

Tanque Verde Creek, where the flow could attack bridge piers and abutments at angles other 

than designed. As a result, local scour around piers and abutments of a bridge at this site 

could be significantly increased. 

Neither Pantano Wash nor Tanque Verde Creek can contain a 100-year flood within its own 

channel. Since the two usually do not reach peak flows at the same time, the flow spills out of 

the flooding channel across the floodplain area between the two channels and into the opposite 

channel. In the process, the overflow deposits most of its sediment in the floodplain and the 

clear water entering the opposite channel causes degradation. There is also the problem of 

poor flow alignment past piers and abutments under these conditions. These problems are 

analyzed further when engineering alternatives to eliminate the problems are presented. 
10.22 

Figure 10.14. Sketch of Craycroft Road crossing. 
10.23 

Potential Local Problems of the Sabino Canyon Road Bridge Site. The Sabino Canyon Road 

bridge site (Figure 10.15) has several potential erosion and sedimentation problems that 

should be considered in the bridge design. The existing bridge has already experienced 

several such problems. The flow area of the bridge appears to be inadequate for the 100-year 

flood event. Over 4 ft (1.2 m) of scour has occurred around the bridge piers and abutments. In 

addition, the channel is located on a reach that is migrating to the left (looking downstream). 

This is causing the left abutment to be attacked. The migration tendency of Tanque Verde 

Creek is largely due to its braided nature and its lack of confinement by bank stabilization or 

channelization works. The lateral migration tendency is studied in more detail in the Level 2 

quantitative engineering geomorphic analysis. 

Considerable scour is occurring on the left side of the channel under the bridge since it is 

located on the outside of a bend. This is to be expected since high-velocity flow and 

secondary currents can scour sediment from the outside of a bend. 

The final consideration is the gravel mining from the river. Currently, there is a mine 

approximately 3,000 ft (900 m) upstream of the bridge. The pit could act as a sediment trap 

and cause scour downstream of the pit near the bridge site as the water removes sediment 

from the bed to regain an equilibrium sediment transport rate. Because of the distance, the 

threat is not large from the present activity, considering the passage of the 100-year flood; 

however, gravel mining operations located closer to the bridge site could cause problems if not 

properly managed. In addition, over a long period of time the overextraction of sand and gravel 

can cause significant degradation for the entire reach downstream of the operating site, and 

possible headcuts upstream of the mining. 

10.2.3 Level 2 - Engineering Geomorphic Analysis 

Hydrology. The Rillito River is formed by the confluence of Pantano Wash and Tanque Verde 

Creek northeast of Tucson and flows west-northwest about 12 miles (19.3 km) to its confluence 

with the Santa Cruz River. Precipitation in the Rillito River watershed is produced by three 

types of storms: general winter storms, general summer storms, and local thunderstorms. The 

general winter storms usually last for several days and result in widespread precipitation. 

General summer storms are often accompanied by relatively heavy precipitation over large 

areas for periods of up to 24 hours. Local thunderstorms can occur at any time of the year; 

however, they cover comparatively small areas and cause high-intensity precipitation for a few 

hours. 

The flow in the Rillito River is intermittent and Tanque Verde Creek is almost always dry, other 

than during or immediately after rain. The USGS gaging station on the Rillito River near 

Tucson kept daily discharge records from October, 1908 to September, 1975, after which it 

was converted to a crest-stage partial-record station. The gage is located 4.75 mi (7.2 km) 

upstream of the confluence of Rillito River with the Santa Cruz River and about 4 mi (6.4 km) 

downstream of Dodge Boulevard (Figure 10.12). 

Utilizing the USGS records at Rillito Station, all of the extreme events since 1915 are plotted in 

Figure 10.16. Based on these flood data the flood frequency curves are plotted on log-normal 

paper (Figure 10.17). The USGS log-Pearson Type III analysis is shown in Table 10.8. The 

hydrograph of the 1965 flood observed at the Rillito River gage near Tucson (Figure 10.18) 

was used to establish the 100-year flood hydrographs for Tanque Verde Creek and Pantano 

Wash. The design hydrographs for the 100-year flood for Tanque Verde Creek, Sabino Creek, 

Pantano Wash, Ventana Wash and Alamo Wash are given in Figure 10.19. 
10.24 

Figure 10.15. Sabino Canyon Road crossing site. 
10.25 

Table 10.8. Rillito River Near Tucson, Arizona Log-Pearson Type III Frequency Analysis 

 by USGS. 

95 Percent Confidence Limit 

Exceedance (One-Sided Test) 

Probability 

Return 

Period (yr) 

Expected 

Discharge (cfs) Lower (cfs) Upper (cfs) 

0.5000 2 5,000 4,240 5,800 

0.2000 5 9,300 7,670 11,100 

0.1000 10 12,500 10,100 15,600 

0.0400 25 17,200 13,300 22,000 

0.0200 50 21,100 15,800 27,300 

0.0100 100 25,200 18,400 33,000 

0.0050 200 29,800 20,900 39,200 

0.0020 500 35,700 24,400 47,800 

 Figure 10.16. Flood events at Rillito River near Tucson, Arizona (drainage area 915 

 square miles). 

Hydraulics. A complete rigid boundary hydraulic analysis was conducted prior to the moveable 

bed analysis. Water surface profile calculation from Dodge Boulevard to Agua Caliente Wash 

(Figure 10.12) was conducted using the Corps of Engineers HEC-2 program (today the Corps' 

HEC-RAS program would be used). The cross sectional data were modified to reduce the 

number of cross-sections and make the spatial resolution of the water surface computation 

compatible with the sediment routing model. The main channel roughness was reduced to 

near the lower limit of the river flow regime expected during the 100-year flood. The hydraulic 

conditions are predominantly subcritical up to Sabino Creek. The reach from Sabino Creek to 

Agua Caliente Wash increases in gradient and a mix of subcritical and supercritical hydraulic 

conditions is possible. 
10.26 

Figure 10.17. Log-normal frequency analysis for Rillito River near Tucson. 

Figure 10.18. December 1965 flood in Rillito River and Tanque Verde Creek. 
10.27 

Figure 10.19. 100-year flood design hydrographs. 

The stage-discharge plot for the USGS gaging station on Rillito River near Tucson is shown in 

Figure 10.20. In this plot, the gage height values from the flood observations have been 

converted to equivalent stages at the present sites, based on the gage datum information given 

below. The shifting of the stage-discharge relationships from early years to 1970 is also 

indicated in this plot. For a given discharge the water-surface elevation drops about 3 ft from 

the 1956-65 curve to the 1966 curve and drops another 2 ft to the 1974-1978 curve. The 

decrease of the water surface elevation at this station is a result of the channel degradation 

since 1956. 

Figure 10.20. Stage-discharge plot for Rillito River near Tucson. 
10.28 

Spatial Design and Channel Geometry. In the quantitative geomorphic analysis, groups of 

cross sections were aggregated to form a typical cross section for a study reach. The 

geometric properties (area, wetted perimeter, top width, and hydraulic radius) of this section 

are determined for a series of depths. Normal depth for a given discharge, channel roughness 

and slope are then calculated and the hydraulic parameters (velocity and hydraulic depth) are 

used in the sediment transport relationship to determine the transport capacity in the reach. An 

equilibrium slope analysis can then be made. 

A water surface profile calculation is made based on the sections utilizing HEC-2 (HEC-RAS). 

Hydraulic conditions in the main channel and overbank areas are based on output from HEC-2 

(HEC-RAS). 

Bed Material Size Distribution. Sediment size is one of the most important parameters used in 

evaluating sediment transport. A thorough sediment sampling survey was conducted on the 

river system, consisting of 41 bed material samples. Variation of the size distribution within 

these segments of the river did not follow an identifiable trend, and therefore an average size 

distribution was used and the variation from the average size distribution was assumed to be 

sampling error. Three size distributions were used to cover the river segments from Dodge 

Boulevard to Pantano Wash (including Pantano Wash), Pantano Wash to Sabino Creek and 

Sabino Creek to Agua Caliente Wash. Figure 10.21 shows the size distributions used for 

design on various segments of the river system. The size distribution of Pantano Wash is 

included to illustrate its similarity to the Rillito River size distribution. 

Figure 10.21. Rillito-Pantano-Tanque Verde bed sediment distribution. 

 

A large percentage of sediment falls in the coarse sand and fine gravel range with less than 10 

percent classified as medium gravel. Very coarse gravels are not present. From an analysis 

based on Shields' criteria, all sizes present can be easily transported by the mean annual flood. 

Formation of an armoring layer on the bed is unlikely since coarse, nontransportable particle 

sizes are missing from the distribution. 
10.29 

Subsurface bed material samples and bank material samples were also taken. The 

subsurface bed material is slightly coarser in most cases, but still lacked sizes in the 

nontransporting range. Bed material samples had more fine material and these distributions 

varied substantially from one location to the next. 

Riparian Vegetation. The yield of debris to the bridge sites was determined by visual 

inspection of aerial photographs for the riparian zone of the system. A review of these photos 

indicated that large trees along the bank present the greatest problem. Trees along the banks 

were counted and the root zone size estimated. The root zone is the area of the tree capable 

of supporting the weight of the tree. This diameter is estimated as 5 to 6 ft (1.5 - 1.8 m) for 

trees on the Tanque Verde. 

Tree yield will be from the banks of the river for large floods. The accumulation of smaller 

debris at a bridge is assumed to occur only in conjunction with the trapping of larger debris. 

Actual debris yield and trapping at a bridge were analyzed by a qualitative approach. For 

additional information on estimating debris accumulation, see HEC-20, Chapter 4 (Lagasse et 

al. 2001). 

Resistance to Flow. During the December 1965 flood, the Rillito River was in upper regime, 

having antidunes with breaking waves. Similarly, the geomorphic analysis shows that the bed 

forms of the channels in the study system will be antidunes or standing waves during floods. 

Resistance to flow associated with antidunes depends on how often the antidunes form, the 

area of the reach they occupy, and the violence and frequency of their breaking. If many 

antidunes break, resistance to flow can be large because breaking waves dissipate a 

considerable amount of energy. With breaking waves, g/C may range from 10 to 20, and 

Manning's coefficient n ranges from about 0.019 to 0.038 for the flow depths being considered. 

The existing channels will not contain all of the 100-year flood flows. Some overbank flow will 

occur. Sparse vegetation, brush, trees and houses are in the floodplain. These elements 

increase the resistance to flow. For a conservative erosion and sedimentation analysis (high 

channel velocity), a Manning's roughness of 0.025 for the main channels was assumed for this 

study. For overbank flows, a higher Manning's n value of 0.05 was used from Dodge 

Boulevard to Sabino Creek and an n value of 0.06 was used from Sabino Creek to Agua 

Caliente Wash. 

Sediment Transport Rates. The rate of sediment transport is the most important factor in 

conducting a quantitative determination of aggradation and degradation in the channel. Since 

very little actual data were available to calibrate the sediment transport rate determinations, 

there is some uncertainty inherent in the procedure used to compute sediment supply rates. 

Fortunately, the uncertainty in the results was reduced by several factors. An indirect check of 

the sediment transport rate determinations was available on Tanque Verde Creek. This area 

has undergone the least change in river form of all the locations in the study area. The 1941 

and present aerial photographs show this portion of the system to have remained nearly 

unchanged. Therefore, it is expected that these reaches must have sediment transporting 

capacities near equilibrium. The engineering geomorphic analysis is in agreement with this 

conclusion. 

Another factor that helped provide a reliable determination of sediment supply was the grade 

control structure on Pantano Wash at the Craycroft Road bridge site. A channel will quickly 

come to equilibrium behind such a structure since the results of an excess or imbalance in 
10.30 

sediment transport rate to the structure are corrected by removal or storage of material behind 

the structure. This process allows the channel to quickly reach equilibrium behind the structure 

by producing a channel bed slope that will result in a sediment transport rate equal to the 

incoming supply. 

The method used to compute the sediment transport rate was the Meyer-Peter, Muller 

bed-load equation and the Einstein method for suspended bed material discharge (see 

Chapter 4). The shear stress on the bed of the channel was calibrated based on the grain 

resistance of the bed material. The method produced a total bed-material concentration which 

matched available data on the Rillito River and was consistent with similar sand-bed arid 

region rivers. 

Engineering Geomorphology. To determine equilibrium channel slope and possible changes in 

channel alignment quantitatively, an engineering geomorphic analysis was performed. The 

analysis considered various river sections in the study area for a series of water discharges 

ranging from 5,000 cfs (mean annual flow) to 34,000 cfs (100-year flood). A multiple 

regression was developed for sediment transport as a function of velocity and depth. This 

equation was combined with Manning's equation and the water and sediment continuity 

equations to form a computational procedure for determining the equilibrium slope (for details 

see HEC-20, Chapter 6). Computation of equilibrium slopes was performed for the present 

sediment supply and reductions of 25 to 50 percent. Computations with the reduced sediment 

supplies were carried out to determine the effects of increased urbanization on the stability of 

the present channel system. 

The calculation of equilibrium slopes was accomplished by trial and error. The results show 

(Table 10.9) that for the present supply condition most of the study area is close to equilibrium. 

The exceptions occur at the bridge sites, all of which should degrade according to the 

equilibrium analysis. The accuracy of the calculations at the bridge sites is less than the other 

locations, since the normal depth assumption may not be accurate because of local hydraulic 

effects caused by the bridges. In all, the calculations reflect the fact that the system is near 

equilibrium and should not experience significant bed elevation changes if no further 

disturbances are introduced. This agrees with the results of the Level 1 channel morphology 

analysis. 

The sediment supply reduction cases result in degradation at all locations along the system. 

This is a crucial fact. It illustrates the severe consequences which would arise from a reduction 

in sediment supply by urbanization, sand and gravel mining, or other activities. The equilibrium 

conditions which presently exist would be destroyed and significant bed elevation changes 

would result. For example, a 0.0005 decrease in equilibrium slope resulting from a 25 percent 

reduction in supply would cause a degradation of 2.6 ft/mi (.492 m/km). 

The possible extent of lateral migration was determined for Tanque Verde Creek by studying 

the channel alignment characteristics for 13 miles (21 km) of Tanque Verde Wash above 

Craycroft Road. A potentially severe lateral migration potential was identified for Tanque 

Verde Wash at the Sabino Canyon Road Bridge during the Level 1 analysis. In order to 

understand the migration process more thoroughly, a 13-mile (21 km) reach of Tanque Verde 

Wash above the confluence with Pantano Wash was studied closely to determine the range of 

channel plan geometry. Meander amplitudes, wave lengths, and radii of curvature were 

measured. 
10.31 

Table 10.9. Equilibrium Slope Calculations, Dominant Discharge. 

Slope in ft/ft 

Location Actual Present 

Supply 

25 Percent 

Reduction 

50 Percent 

Reduction 

Cross-Sections 32-47 (Pantano) 0.0060 Too much gravel mining to accurately estimate 

Cross-Sections 31-24 (Pantano) 0.0035 Reach assumed in equilibrium at present slope 

(supply reach) 

Cross-Sections 47-54 

(Tanque Verde) 

0.0033 Reach assumed in equilibrium at present slope 

(supply reach) 

Cross-Sections 46-45 

(Tanque Verde-Sabino) 

0.0048 0.0032 0.0026 0.0020 

Cross-Sections 44-27 

(Tanque Verde) 

0.0036 0.0036 0.0030 0.0023 

Cross-Sections 26-23 

(Rillito Craycroft) 

0.0043 0.0033 0.0027 0.0023 

Cross-Sections 22-18 (Rillito) 0.0037 0.0032 0.0026 0.0020 

Cross-Sections 17-14 

(Rillito Swan Road) 

0.0043 0.0032 0.0026 0.0020 

Cross-Sections 13-1 (Rillito) 0.0039 0.0031 0.0025 0.0020 

Meander wave length, channel width and radius of curvature ( λ, B, and rc) pairs were plotted 

for each meander loop. These points were plotted on logarithmic scales and on linear scales. 

Straight lines were fitted and the following equations were obtained and adopted for the lower 

13 miles (21 km) of the Tanque Verde Creek: 

λ = 2.6 rc

λ = 2.75 B1.35

rc = 1.06 B1.35

These relationships are used to determine appropriate channel widths and bend shapes at the 

Sabino Canyon Road Bridge site. 

Sabino Canyon Road Bridge (Figure 10.15) is currently located on a bend with a curvature that 

creates several problems. At low flows, scour occurs at the outside of the bend (south side) 

because of high velocity and secondary currents. This phenomena is evident in the present 

channel cross section under the bridge. At flood flows, the problem is nearly the opposite. The 

north side of the bridge is attacked because of the tendency of the thalweg to straighten in the 

bend. The amplitude of the meander bend is 300 ft (91 m). Therefore, the lateral migration 

tendency is on the order of 300 ft (91 m). 

These facts point to the necessity for engineering control measures to be taken at Sabino 

Canyon Road Bridge in order to prevent future failure of the structure from lateral migration. 

Possible measures are bank stabilization and channelization. 

10.2.4 Level 3 - Sediment Routing for the 100-Year Flood 

The long-term trends of the Tanque Verde, Pantano, Rillito River system were determined 

using a sediment routing procedure. This analysis determined the aggradation and 

degradation for subreaches in the respective river systems, based on sediment supply from the 

upstream reaches and local hydraulic conditions throughout the system. Subcritical hydraulic 

conditions were assumed for all backwater computations. 
10.32 

River and watershed information consists of upstream sediment loading information and the 

discretized hydrographs for the mainstem and tributaries. The mainstem hydrograph was 

discretized into 12 time steps with 5 time steps on the rising limb of the hydrograph. Time 

steps vary in length from 0.5 hours at the peak to 4.0 hours for the end of the recession limb. 

Time steps average typically 1.5 hours. The sediment output from the Pantano Wash was 

used as input to Tanque Verde at the Craycroft Road section. Two cases of sediment supply 

from Pantano were considered. The first case assumed stable conditions existed in the 

Pantano reach. This case corresponds to a no-fail grade control condition. The second case 

assumed the roadway grade control would fail during the 100-year flood. The sediment load 

resulting from the failure of the roadway was calculated independently of the routing procedure. 

Four Tanque Verde sediment routing analyses were conducted for the as-is condition. The 

cases analyzed are: 

Case I. Tanque Verde floods with overflow to the Pantano Wash and no grade control failure. 

Case II. Tanque Verde floods with overflow to the Pantano Wash and the grade control fails. 

Case III. Pantano Wash floods with overflow to the Tanque Verde and no grade control failure. 

Case IV. Pantano Wash floods with overflow to the Tanque Verde and the grade control fails. 

Figures 10.22 and 10.23 show the bed level changes over the duration of the 100-year flood 

for both Tanque Verde flooding and Pantano flooding. The bridge sites show only slight 

aggradation/degradation for the 100-year flood if the Pantano grade control structure does not 

fail. This agrees with the conclusion drawn from the geomorphic analysis. The major potential 

cause of aggradation is the failure of the existing Pantano roadway grade control. If the 

existing Pantano Wash roadway grade control structure fails during Pantano flooding, there is 

a maximum aggradation downstream of Craycroft Road of 4.6 ft (1.4 m). The erosion and/or 

headcut that may occur during Pantano flooding can supply significant amounts of sediment in 

a short period of time (Figure 10.23). 

10.2.5 Results of Analysis

Each of the sites evaluated has its own unique problems that must be considered in the 

formulation of alternative designs. This section presents several conceptual alternatives based 

on the analysis of the as-is conditions. Analysis of the design alternative is broken into three 

areas, (1) low chord criteria, (2) total scour criteria and (3) other additional considerations. 

The criteria for an acceptable low chord elevation for this alternative is that the bridge can pass 

the 100-year flood peak with sufficient freeboard between the water surface elevation from a 

rigid-bed hydraulic analysis, plus additional components resulting from sand-wave movement, 

general aggradation, and superelevation caused by flow curvature. In addition, an increment 

of height is added to provide freeboard for debris passage. 

When designing a bridge foundation, proper consideration of scour must be made to determine 

the required safe depth of piles or other supports. A design which gives adequate support for 

the structure when the channel bed is at its initial elevation may be inadequate after scour 

occurs and lowers the channel bed. The physical processes that must be considered are 

long-term changes in bed elevation, local scour, contraction (or general) scour, and passage of 

sand waves. The total scour is the sum of these, and must be subtracted from the initial design 

elevation to establish the design depth for all supports. The supports must have a depth of 

burial below this elevation sufficient to support the structure. 
10.33 

Figure 10.22. Bed elevation change of Rillito-Tanque Verde System (Tanque Verde flooding). 

Figure 10.23. Bed elevation change of Rillito-Tanque Verde System (Pantano flooding). 
10.34 

Craycroft Road. Three conceptual alternatives are presented which offer various channel 

alignment or bridge design configurations at this site. Figure 10.24 shows Alternative I. This 

alternative utilizes bank protection works with the central embankment designed to guide the 

flow through the bridge. The pier elevations are designed according to the worst scour 

potential of the two branches and are the same for both tributaries. The design should also 

consider the condition resulting from grade control failure on Pantano and should utilize circular 

piers to minimize problems due to adverse flow alignment. 

Figure 10.24. Alternative I for Craycroft Road Bridge crossing. 
10.35 

Alternative II conceptual design avoids some of the difficulties resulting from locating the bridge 

in the vicinity of the confluence. This alternative suggests relocating the bridge and road 

approximately 850 ft (260 m) downstream of the current right-of-way (Figure 10.25). This 

alternative would require only one bridge, and the piers could be properly aligned with the flow 

or circular piers could be used. However, the acquisition of the new right-of-way for this site 

may not be economically feasible. 

Figure 10.25. Alternative II for Craycroft Road Bridge crossing. 
10.36 

Alternative III suggests physically moving the confluence approximately 700 ft (213 m) 

upstream by modifying both branches of the tributaries with excavation, fill, and bank 

protection. The grade control and the deposited material behind the control would be removed. 

This conceptual design is shown in Figure 10.26. This alternative has most of the advantages 

that are provided by Alternative II and does not require relocation of the road right-of-way. 

However, the cost of modifying the channels can be significant and may be prohibitive. 

The results of low chord and total scour analyses are presented in Table 10.10. The freeboard 

requirement for debris was determined qualitatively. Pier scour was estimated with and without 

a vegetative debris accumulation. A maximum debris width of 5 ft (1.5 m) was assumed for 

this analysis. Long-term degradation assuming a 25 percent reduction of sediment supply due 

to urbanization and gravel mining activities was added to the degradation calculated for the 

100-year flood. 

Table 10.10. Low Chord and Total Scour Requirements at Craycroft Road. 

Degradation (ft) 

Alternative 

Water 

Surface 

(ft) 

Aggradation 

(ft) 100-Year Flood 

25% Reduction in 

Sediment Supply 

I 2435.6 1.7 0.5 2.6 

II 2431.0 4.6 0.8 2.6 

III 2434.4 0 0.8 2.6 

Vegetative Pier Local Scour (ft) 

Debris 

Freeboard (ft) 

Minimum 

Low Chord 

(ft) 

Abutment 

Local Scour 

(ft) 

Without 

Debris 

With 

Debris 

Total Abutment 

Scour 

(ft) 

Total Pier 

Scour 

(ft) 

3.0 2441.7 11.1 5.0 7.8 14.2 10.9 

3.0 2437.7 13.2 7.0 10.9 16.6 14.3 

3.0 2439.5 13.2 7.0 10.9 16.6 14.3 

As mentioned, the grade control on Pantano Wash near the confluence with Tanque Verde 

Creek is temporary in nature and is likely to fail during a major flood. Sudden failure of the 

grade control can cause instability of the banks and the new bridge. It is recommended that 

the grade control be removed, or at least the bridge design should consider the consequences 

of its sudden failure. 

The alignment of the flow is a major concern when building a bridge in the vicinity of a 

confluence. Poor flow alignment can cause serious local scour problems. Circular piers are 

strongly recommended in this situation to minimize local scour. 

Although the probability of simultaneous occurrence of the two peak flows at the bridge site 

may be small, the impact on the proposed bridge due to such an event should be considered. 

One way to account for such an occurrence is to design the Craycroft Road Bridge with an 

extra margin of safety, both hydraulically and structurally. 

In selecting the best alternatives for the Craycroft Road Bridge, economic, social and 

environmental constraints must be considered. Based upon this preliminary evaluation, both 

Alternatives I and III are more attractive than Alternative II. Alternative II has the disadvantage 

of a potential for both significant aggradation and degradation, which increases the cost of 

design. The acquisition of the right-of-way to realign the bridge in Alternative II may be 

prohibitive. 
10.37 

Figure 10.26. Alternative III for Craycroft Road Bridge crossing. 

 
10.38 

Sabino Canyon Road Bridge. Four alternatives for the Sabino Canyon road bridge are 

analyzed. Alternative I is the present condition (Figure 10.15). The other three alternatives 

consider variations on a similar scheme. The obvious options involve channelization 

downstream of the bridge to reduce water surface elevations, widening of the bridge opening, 

protection of the south side of the bridge by a guide bank, and bank protection at key locations. 

Alternatives II, III and IV consider a channel bottom width of 300, 350, and 400 ft (91, 107, and 

122 m), respectively. A sketch of the basic design is presented in Figure 10.27. The drawing 

shows a 300-foot (91 m) channel for Alternative II. 

A low chord and total scour analysis similar to the Craycroft Road site presented in Table 10.11 

could be used for the analysis of the Sabino Canyon bridge. Because of the problems 

associated with a bridge located on a bend, Alternatives II, III, and IV include bank protection 

and stabilization 200 ft (60 m) up- and downstream of the bridge site. Also, additional bank 

protection should by provided along the south bank above Sabino Canyon Road. This latter 

protection will prevent the upstream bend from further migration that would cause flow 

alignment difficulties at the bridge. A guide bank should be constructed on the south side to 

protect the bridge from southward channel migration and to assist in controlling overbank flow 

and guiding the flow through the bridge opening (Figure 10.27). 

By using the radius of bend curvature as a function of channel width determined earlier, 

rc = 1.06 B1.35

the appropriate radius of curvature for each alternative can be determined. The results show 

that for the channelization alternatives (II, III, IV), the radius of curvature is within the stable 

range. However, for the present condition (Alternative I), the radius of curvature is too small. 

This is one cause of the present migration problem on the south bank. A very preliminary 

assessment concluded that Alternative II [300 ft (91 m) bottom width channelization] is 

probably the most feasible and practical solution for the new bridge. Very little is gained in 

terms of low chord and scour reduction by the wider channels; however, they would require a 

large amount of additional earthwork. The narrower channel would also cause less conflict 

with private property ownership. 

10.3 BRI-STARS SEDIMENT TRANSPORT MODELING EXAMPLE 

10.3.1 Background

Overview examples 1 and 2 illustrate the application of the three-level analysis procedure. 

Example 1 applied the principles and methods introduced in this manual to the design of 

countermeasure alternatives on a migrating alluvial channel bendway. Techniques used 

included a Level 1 reconnaissance and geomorphic analysis and a Level 2 quantitative 

engineering analysis. Example 2 developed conceptual design alternatives for two bridge 

crossings on a dynamic sand bed river system in the southwest using, primarily, Level 1 and 

2 analysis procedures; however, the results of a Level 3 sediment routing analysis were 

demonstrated. 

This final overview example illustrates in more detail the application of Level 3 sediment 

transport analysis techniques using FHWA's BRI-STARS model. This example problem is 

taken from a paper by Arneson et al. (1991) which is presented as the total scour 

comprehensive example in HEC-18 (Richardson and Davis 2001). For this problem in HEC18, FHWA's WSPRO computer program was used to obtain the hydraulic variables. The 

program uses 20 stream tubes to give a quasi 2-dimensional analysis. Each stream tube 

has the same discharge (1/20 of the total discharge). The stream tubes provide the velocity 

distribution across the flow and the program has excellent bridge routines. Since the data for 

this case study are available in English units, the figures and tables retain English unit 

notation. SI (metric) units are given parenthetically in the text for reference. 
10.39 

 Figure 10.27. General plan view of Sabino Canyon Road crossing for alternative II, 

 III, and IV (II shown). 
10.40 

The Level 1 and 2 analysis results and total scour computations are summarized briefly in 

the following sections. Then the BRI-STARS model is applied to refine the initial qualitative 

analysis of long-term bed elevation changes. 

10.3.2 Level 1 - Reconnaissance and Geomorphic Analysis 

A 650-foot (198 m) long bridge (Figure 10.28) is to be constructed over a channel with 

spill-through abutments (slope of 1V:2H). The left abutment is set approximately 200 ft (60.5 

m) back from the channel bank. The right abutment is set at the channel bank. The bridge 

deck is set at elevation 22 ft (6.71 m) and has a girder depth of 4 ft (1.22 m). Six round-nose 

piers are evenly spaced in the bridge opening. The piers are 5 ft (1.52 m) thick, 40 ft (12.19 

m) long, and are aligned with the flow. 

Figure 10.28. Cross section of proposed bridge. 

A site investigation of the crossing was conducted to identify potential stream stability 

problems at this crossing. Evaluation of the site indicates that the river has a relatively wide 

floodplain. The floodplain is well vegetated with grass and trees; however, the presence of 

remnant channels indicates that there is a potential for lateral shifting of the channel. The 

river and crossing are located in a rural area with the primary land use consisting of 

agriculture and forest. 

The bridge crossing is located on a relatively straight reach of channel. The channel 

geometry is essentially the same for approximately 1,000 ft (300 m) upstream and 

downstream of the bridge crossing. The D50 of the bed material and overbank material is 

approximately 2 mm. The maximum grain size of the bed material is approximately 8 mm. 

The specific gravity of the bed material was determined to be equal to 2.65. Since this is a 

sand-bed channel, no armoring potential is expected. Furthermore, the bed for this channel 

at low flow consists of dunes which are approximately 1 to 1.5 ft (0.3 to 0.5 m) high. At 

higher flows, above the Q5, the bed will be either plane bed or antidunes. 
10.41 

The left and right banks are relatively well vegetated and stable; however, there are isolated 

portions of the bank which appear to have been undercut and are eroding. Brush and trees 

grow to the edge of the banks. Banks will require riprap protection if disturbed. Riprap will 

be required upstream of the bridge and extend downstream of the bridge. 

Evaluation of stage discharge relationships and cross sectional data obtained from other 

agencies does not indicate progressive aggradation or degradation. Review of bridge 

inspection reports for bridges located upstream and downstream of the proposed crossing 

indicates no long-term aggradation or degradation in this reach. Based on these 

observations, the channel is relatively stable vertically, at present. Furthermore, there are no 

plans to change the local land use in the watershed. The forested areas of the watershed 

are government-owned and regulated to prevent wide spread fire damage, and instream 

gravel mining is prohibited. These observations indicate that future aggradation or 

degradation of the channel, due to changes in sediment delivery from the watershed, are 

minimal. 

Based on these observations, and due to the lack of other possible impacts to the river 

reach, it is determined that the channel will be relatively stable vertically at the bridge 

crossing and long-term aggradation or degradation potential is considered to be minimal. 

However, there is evidence that the channel is unstable laterally. 

10.3.3 Level 2 - Hydraulic Engineering Analysis

Hydraulic characteristics at the bridge were determined using FHWA's WSPRO model. 

Three cross sections were used for this analysis and are denoted as "EXIT" for the section 

downstream of the bridge, "FULLV" for the full-valley section at the bridge, and "APPR" for 

the approach section located one bridge length upstream of the bridge. The bridge geometry 

was superimposed on the full-valley section and is denoted "BRDG." 

The WSPRO HP2 option was used to provide hydraulic characteristics at both the bridge and 

approach sections. This option subdivides the cross section into 20 equal conveyance 

tubes. Figures 10.29 and 10.30 illustrate the location of these conveyance tubes for the 

approach and bridge cross sections. Figure 10.30 illustrates the average velocities in each 

conveyance tube and the contraction of the flow from the approach section through the 

bridge. Figure 10.30 also identifies the equal conveyance tubes of the approach section 

which are cut off by the abutments. 

Contraction scour will occur both in the main channel and on the left overbank of the bridge 

opening. For the main channel, contraction scour could be either clear-water or live-bed 

depending on the magnitude of the channel velocity and the critical velocity for sediment 

movement. In the overbank area adjacent to the left abutment, clear-water scour will occur. 

This is because the overbank areas upstream of the bridge are vegetated and the velocities 

in these areas will be low. Thus, returning overbank flow which will pass under the bridge 

adjacent to the left abutment will not be transporting significant amounts of material to 

replenish the scour on the left overbank adjacent to the left abutment. 
10.42 

Figure 10.29. Equal conveyance tubes of approach section. 

Figure 10.30. Plan view of equal conveyance tubes showing velocity distribution at approach 

 and bridge sections. 
10.43 

The hydraulic variables used to estimate the local scour at the piers were determined from a 

plot of the velocity distribution derived from the WSPRO output (Figure 10.31). For this 

example the highest velocities and flow depths in the bridge cross section were used (at 

conveyance tube number 12). Only one pier scour computation was completed because the 

possibility of thalweg shifting and lateral migration requires that all of the piers be set 

assuming that any pier could be subjected to the maximum scour producing variables. 

Figure 10.31. Velocity distribution at bridge crossing. 

Local scour volumes at the left abutment and right abutments were estimated using the 

equations presented in Section 7.8. 

10.3.4 Level 2 - Scour Analysis Results

Use of the hydraulic data from the WSPRO model in the appropriate equations of Chapter 7 

resulted in the following scour estimates: 

• Live-bed contraction scour in main channel (Equation 7.2) - 9.2 ft (2.8 m) 

• Clear-water contraction scour on left overbank (Equation 7.4) - 1.7 ft (0.52 m) 

• Local scour at piers (Equation 7.7, 0° angle of attack) - 11.8 ft (3.6 m) 

• Local scour at piers (Equation 7.7, 10° angle of attack) - 19.3 ft (5.9 m) 

• Local scour at left abutment (Equation 7.16) - 8.4 ft (2.6 m) 

• Local scour at right abutment (Equation 7.16) - 13.3 ft (4.1 m) 

The results of the scour computations should be plotted on the bridge cross section and 

carefully evaluated (Figure 10.32). For this example, only the computations for pier scour 

with piers aligned with the flow were plotted. The topwidth of the local scour holes is 

estimated as 2.0 times the depth of scour. 
10.44 

Figure 10.32. Plot of total scour for example problem. 

It is important to evaluate the results of the scour computations carefully. For example, 

although the total scour plot indicates that the total scour at the overbank piers is less than 

for the channel piers, this does not indicate that the foundations for the overbank piers can 

be set at a higher elevation. Due to the possibility of channel and thalweg shifting identified 

with the Level 1 analysis, all of the piers should be set to account for the maximum total 

scour. Also, the computed contraction scour is distributed uniformly across the channel in 

Figure 10.32. However, in reality this may not occur. With the flow from the overbank area 

returning to the channel, the contraction scour could be deeper at both abutments. The use 

of guide banks would distribute the contraction scour more uniformly across the channel. 

This would make a strong case for guide banks in addition to the protection they would 

provide to the abutments. The WSPRO stream tube velocities could be used to distribute the 

scour depths across this section. 

The plot of the total scour also indicates that there is a possibility of overlapping scour holes 

between the sixth pier and right abutment. Also, it is not clear from where the right abutment 

scour should be measured, since the abutment is located at the channel bank. Both of these 

uncertainties should be avoided for replacement and new bridges whenever possible. 

Consequently, it would be advisable to set the right abutment back from the main channel. 

This would also tend to reduce the magnitude of contraction scour in the main channel. 

The possibility of lateral migration of the channel could also have an adverse effect on the 

magnitude of the pier scour. This is because lateral migration will most likely skew the flow 

to the piers. This problem can be minimized by using circular piers. An alternative approach 

would be to install guide banks to align the flow through the bridge opening. 

The plot of the scour prism in Figure 10.32 should be replotted to show the potential for the 

scour to occur at any location in the bridge opening. This is shown in Figure 10.33. 
10.45 

Figure 10.33. Revised plot of total scour for example problem. 

10.3.5 Level 3 - BRI-STARS Analysis 

The contraction scour computed in Section 10.3.4 using the HEC-18 (Richardson and Davis 

2001) live-bed and clear-water scour equations could also be determined using the Level 3 

approach of applying a sediment transport model such as BRI-STARS (see discussion in 

Section 5.6.2). Contraction scour occurs in a bridge when the sediment supplied from 

upstream is less than the sediment transport capacity in the bridge opening. The HEC-18 

equations compute the ultimate amount of contraction scour that would occur if the hydraulic 

condition persists until the transport capacity in the bridge opening matches the upstream 

sediment supply. 

Two BRI-STARS models were developed for the comprehensive scour example presented in 

HEC-18 and summarized above. The first BRI-STARS model simulated a constant flow of 

30,000 cfs (850 m3

/s), the 100-year flow, for a 3-day period. The second model was run for 

a 3-day hydrograph that peaks at 30,000 cfs (850 m3

/s). The hydrograph is presented in 

Figure 10.34. The models used 1.2 hour computation intervals for a total of 60 time steps for 

each of the 3-day simulations (Arneson 2001). 

Figure 10.35 shows the comparison thalweg profiles in the vicinity of the of the bridge, which 

is located between stations 1400 and 1450. Water surface and bed profiles are shown for 

the steady state (SS) and hydrograph (Hyd.) runs. The starting profile is a uniform slope of 

0.002 ft/ft (m/m). For the steady state run, scour increases for the entire simulation and 

would continue well beyond the 3-day simulation time. At the end of the steady state 

simulation the maximum scour is approximately 4.5 ft (1.4 m) as compared with the HEC-18 

(see Section 10.3.4) result of 9.2 ft (2.8 m). For the steady state run, the starting water 

surface profile shows approximately 1.4 ft (0.43 m) of backwater. At the maximum scour 

condition, however, there is virtually no backwater because the scour enlarges the bridge 

opening and reduces the head loss caused by the structure. 
10.46 

0

5000

10000

15000

20000

25000

30000

0 0.5 1 1.5 2 2.5 3

Time (days)

Discharge (csf)

Figure 10.34. Three-day hydrograph for BRI-STARS analysis. 

0

2

4

6

8

10

12

14

16

18

500 1000 1500 2000 2500 3000

Distance (ft)

Elevation (ft)

Starting Bed

Starting WS, SS

Max. Scour, SS

WS at Max Scour, SS

Max. Scour, Hyd.

Max. WS, Hyd.

Final Scour, Hyd.

Figure 10.35 Comparison longitudinal profiles from BRI-STARS. 
10.47 

The hydrograph run reaches maximum scour of 2.8 ft (0.85 m) shortly after the peak flow. A 

different hydrograph with the same peak discharge but different shape would result in 

different amounts of predicted scour. This is because the sediment transport model 

computes scour based on the difference between rates of sediment supply and transport 

capacity, which vary with discharge. Also shown in Figure 10.35 is the scour at the end of 

the hydrograph. Some infilling of the scour hole is predicted to occur during the recession 

limb of the hydrograph, and the channel would be expected to return to the original condition 

after continued base flow. Because the hydrograph produces less scour than the steady 

state analysis, some backwater is predicted for peak flow conditions in this simulation. 

Figure10.36 shows a comparison of bridge cross sections from the BRI-STARS simulations. 

As expected, the deepest scour is in the center of the main channel. Clear-water contraction 

scour also occurs in the left overbank under the bridge (see Figure 10.33 for comparison). 

The HEC-18 clear-water contraction scour equation predicted 1.7 ft (0.52 m). The steady 

state model reached 0.81 ft (0.25 m) at the end of the 3-day simulation and the hydrograph 

model predicted a maximum scour on the left overbank of 0.17 ft (0.05 m). Figure 10.35 also 

shows how the scoured areas refill during the recession limb of the hydrograph. 

-5

0

5

10

15

20

800 900 1000 1100 1200 1300 1400 1500 1600

Distance (ft)

Elevation (ft)

Starting Bed

Max. Scour SS

Max. Scour, Hyd.

Final Scour, Hyd.

Max. Water Surface

Figure 10.36 Comparison cross-sections from BRI-STARS. 

Although the HEC-18 scour results are larger than those predicted by these BRI-STARS 

runs, the HEC-18 equation computes ultimate scour conditions using simplified relationships 

for sediment transport. If the BRI-STARS run had been extended, greater scour would have 

occurred, and since BRI-STARS does not scour the channel uniformly, the maximum scour 

could exceed that predicted by the HEC-18 equations. 

The original WSPRO model of this bridge included one cross section downstream (EXIT) and 

one cross section upstream (APPROACH) of the bridge (Figure 10.29). These are the 

minimum number of sections required for a hydraulic analysis. Sediment transport models 

typically require additional sections. In addition to cross sections at the upstream and 

downstream bridge faces, the BRI-STARS model included two downstream cross sections 

and 18 upstream cross sections spaced at 650 ft (200 m) intervals along the channel. 
10.48 

Figure10.37 shows the entire BRI-STARS model profile for starting and final conditions from 

the steady state run. By placing the upstream boundary well upstream of the bridge, no 

sediment inflow needed to be specified. The model eroded material from the upstream 

sections and maintained a stable channel upstream of the bridge. This provided the 

necessary sediment supply to the bridge. The model could have been set up with less length 

upstream of the bridge, but an appropriate amount of sediment supply would have to be 

entered in the input file. 

0

5

10

15

20

25

30

35

40

0 2000 4000 6000 8000 10000 12000

Distance (ft)

Elevation (ft)

Starting Bed

Starting WS, SS

Max. Scour, SS

WS at Max Scour, SS

Figure 10.37. BRI-STARS model profiles for steady-state run. 
11.1

CHAPTER 11 

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Sedimentation Section, Hydrology Branch, Division of Project Investigations, Denver, CO. 

U.S. Department of the Interior, Bureau of Reclamation (USBR), 1958, "Total Sediment 

Transport Program, Lower Colorado River Basin, Interim Report," Denver, CO, 175 p. 

U.S. Department of Transportation, Federal Highway Administration, 1988, "Scour at 

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11.26

USGS, 1973b, "Fluvial Sediment Concepts," Techniques of Water-Resources Investigations 

of the USGS, Book 3, Chapt. C1, Reston, VA. 

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Water-Resources Investigations of the USGS, Book 5, Chapt. C1, Reston, VA. 

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Sediment." 

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Report No. 3, 1941a, "Analytical Study of Methods of Sampling Suspended 

Sediment." 

Report No. 4, 1941b, "Methods of Analyzing Sediment Samples." 

Report No. 5, 1941c, "Laboratory Investigation of Suspended Sediment Samplers." 

Report No. 6, 1952, "The Design of Improved Types of Suspended Sediment 

Samplers." 

Report No. 7, 1943, "A Study of New Methods for Size Analysis of Suspended 

Sediment Samples." 

Report No. 8, 1948, "Measurement of the Sediment Discharge of Streams." 

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Report No. 12, 1957b, "Some Fundamentals of Particle-Size Analysis." 

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Engineering Practice – No. 54, Reston, VA. 

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Load of Alluvial Streams," California Institute of Technology Sedimentation Laboratory, 

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ASCE Hydr. Div., Jour., Vol. 93, No. HY3, pp. 121-144. 
11.27

von Karman, T., 1930, "Mechanische Aehnlichkeit und Turbulenz," Proceedings, Third 

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of London, Series A, No. 958, Vol. 174, pp. 322-338. 

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Proc. Inst. Civil Eng. Part 2, Vol. 59, pp. 265-92. 

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11.28

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A.1 

APPENDIX A

Metric System, Conversion Factors, and Water Properties

The following information is summarized from the Federal Highway Administration, National 

Highway Institute (NHI) Course No. 12301, "Metric (SI) Training for Highway Agencies." For 

additional information, refer to the Participant Notebook for NHI Course No. 12301. 

In SI there are seven base units, many derived units and two supplemental units (Table A.1). 

Base units uniquely describe a property requiring measurement. One of the most common 

units in civil engineering is length, with a base unit of meters in SI. Decimal multiples of 

meter include the kilometer (1000m), the centimeter (1m/100) and the millimeter (1 m/1000). 

The second base unit relevant to highway applications is the kilogram, a measure of mass 

which is the inertial of an object. There is a subtle difference between mass and weight. In 

SI, mass is a base unit, while weight is a derived quantity related to mass and the 

acceleration of gravity, sometimes referred to as the force of gravity. In SI the unit of mass is 

the kilogram and the unit of weight/force is the newton. Table A.2 illustrates the relationship 

of mass and weight. The unit of time is the same in SI as in the English system (seconds). 

The measurement of temperature is Centigrade. The following equation converts Fahrenheit 

temperatures to Centigrade, C = 5/9 (F - 32). 

Derived units are formed by combining base units to express other characteristics. Common 

derived units in highway drainage engineering include area, volume, velocity, and density. 

Some derived units have special names (Table A.3). 

Table A.4 provides useful conversion factors from English to SI units. The symbols used in 

this table for metric units, including the use of upper and lower case (e.g., kilometer is "km" 

and a newton is "N") are the standards that should be followed. Table A.5 provides the 

standard SI prefixes and their definitions. 

Table A.6 provides physical properties of water at atmospheric pressure in SI system of 

units. Table A.7 gives the sediment grade scale and Table A.8 gives some common 

equivalent hydraulic units. 
A.2 

Table A.1. Overview of SI Units. 

Units Symbol 

Base units 

length 

mass 

time 

temperature* 

electrical current 

luminous intensity 

amount of material 

meter 

kilogram 

second 

kelvin 

ampere 

candela 

mole 

m 

kg 

s 

K 

A 

cd 

mol 

Derived units 

Supplementary units 

angles in the plane 

solid angles 

radian 

steradian 

rad 

sr 

*Use degrees Celsius (C), which has a more common usage than kelvin. 

Table A.2. Relationship of Mass and Weight. 

Mass 

Weight or 

Force of 

Gravity 

Force 

English slug 

pound-mass 

pound 

pound-force 

pound 

pound-force 

metric kilogram newton newton 
A.3 

Table A.3. Derived Units With Special Names. 

Quantity Name Symbol Expression 

Frequency hertz Hz s-1

Force newton N kg m/s2

Pressure, stress pascal Pa N/m2

Energy, work, quantity of heat joule J N m 

Power, radiant flux watt W J/s 

Electric charge, quantity coulomb C A s 

Electric potential volt V W/A 

Capacitance farad F C/V 

Electric resistance ohm Ω V/A 

Electric conductance siemens S A/V 

Magnetic flux weber Wb V s 

Magnetic flux density tesla T Wb/m2

Inductance henry H Wb/A 

Luminous flux lumen lm cd sr 

Illuminance lux lx lm/m2
A.4 

Table A.4. Useful Conversion Factors. 

Quantity 

From English 

Units 

To Metric 

Units 

Multiplied 

By* 

Length mile 

yard 

foot 

inch 

km 

m 

m 

mm 

1.609 

0.9144 

0.3048

25.40

Area square mile 

acre 

acre 

square yard 

square foot 

square inch 

km2

m2

hectare 

m2

m2

mm2

2.590 

4047 

0.4047 

0.8361 

0.09290 

645.2 

Volume acre foot 

cubic yard 

cubic foot 

cubic foot 

100 board feet 

gallon 

cubic inch 

m3

m3

m3

L (1000 cm3

) 

m3

L (1000 cm3) 

cm3

1233 

0.7646 

0.02832 

28.32 

0.2360 

3.785 

16.39 

Mass lb 

kip (1000 lb) 

kg 

metric ton (1000 

kg) 

0.4536 

0.4536 

Mass/unit length plf kg/m 1.488 

Mass/unit area 

psf kg/m2 4.882 

Mass density pcf kg/m3 16.02 

Force lb 

kip 

N 

kN 

4.448 

4.448 

Force/unit length plf 

klf 

N/m 

kN/m 

14.59 

14.59 

Pressure, stress, 

modulus of elasticity 

psf 

ksf 

psi 

ksi 

Pa 

kPa 

kPa 

MPa 

47.88 

47.88 

6.895 

6.895 

Bending moment, 

torque, moment of 

force 

ft-lb 

ft-kip 

N m 

kN m 

1.356 

1.356 

Moment of mass lb ft m 0.1383 

Moment of inertia lb ft2 kg m2 0.04214 

Second moment of 

area 

in4 mm4 416200 

Section modulus in3 mm3 16390 

Power ton (refrig) 

Btu/s 

hp (electric) 

Btu/h 

kW 

kW 

W 

W 

3.517 

1.054 

745.7 

0.2931 

*4 significant figures; underline denotes exact conversion 
A.5 

Table A.4. Useful Conversion Factors (continued). 

Quantity From English 

Units 

To Metric Units Multiplied by* 

Volume rate of flow ft3

/s 

cfm 

cfm 

mgd 

m3

/s 

m3

/s 

L/s 

m3

/s 

0.02832 

0.0004719 

0.4719 

0.0438 

Velocity, speed ft/s m/s 0.3048

Acceleration f/s2 m/s2 0.3048

Momentum lb ft/sec kg m/s 0.1383 

Angular momentum lb ft2

/s kg m2/s 0.04214 

Plane angle degree rad 

mrad 

0.01745 

17.45 

*4 significant figures; underline denotes exact conversion 

Table A.5. Prefixes. 

Submultiples Multiples 

deci 10-1 d deka 101 da 

centi 10-2 c hecto 102 h 

milli 10-3 m kilo 103 k 

micro 10-6 μ mega 106 M 

nano 10-9 n giga 109 G 

pica 10-12 p tera 1012 T 

femto 10-15 f peta 1015 P 

atto 10-18 a exa 1018 E 

zepto 10-21 z zetta 1021 Z 

yocto 10-24 y yotto 1024 Y 
A.6 
A.7 
A.8 
A.9 
A.10 

(page intentionally left blank) 
B.1 

APPENDIX B 

Analysis of Selected Sediment Transport Relationships 

B.1 BACKGROUND 

This appendix provides a review of several bed material load equations and provides the 

background for several modifications to selected equations. The equations are tested, 

developed, and verified using a large group of field and laboratory data for a wide range of 

river and bed material sizes. 

Sediment transport in rivers has been studied and applied for centuries. Research continues 

because the analysis of bed material transport involves complex interactions among many 

interrelated variables (Vanoni 1977; Ackers and White 1973 and 1980; Simons and Sentürk 

1992). Several sediment transport relationships for alluvial rivers have been proposed during 

the last 50 years. These are based upon simplified and idealized assumptions and limited 

field data. Due to the complexity of the problem, most existing equations are empirical and 

semi-empirical and/or heavily based upon laboratory data and assumptions that are not 

totally justified (Cao et al. 1997; Paceco-Ceballos 1989; Tywoniuk 1972). These methods 

have been developed based upon theoretical considerations and/or statistical interpretations 

of data, and some have been based upon the physics of particle motion. Other methods 

have been developed from experimental work, some were derived empirically, and some 

represent a combination of theories, experiments and empirical methods, but no 

comprehensive study has ever been presented for the total range of alluvial rivers for which 

data exist. 

In practice, engineers have dealt with a variety of granular materials in applying the 

governing laws of transport and movement of sediments (Bogardi 1974). However, 3-

dimensional, time-dependent phenomena still need to be studied, and available knowledge 

to conduct these analyses is still insufficient (Overbeek 1979; and Borah et al. 1982a and 

1982b). In addition, field verification data have not been utilized to validate the various 

relationships to a significant degree (Simons and Sentürk 1992). 

Rivers transporting large sediment loads are found throughout the world. Most sediment 

transport equations can predict with acceptable accuracy sediment transport in small rivers 

since the measurement of pertinent data to develop empirical relations is relatively easy. 

However, due to the difficulty of gathering information and required data for large rivers, 

extrapolation of methods to estimate sediment loads usually do not give accurate results 

(Posada 1995). Hence, the subject of total bed-material transport in large rivers is a 

challenge and needs further investigation. 

Most investigators have recognized these facts. However, the sediment problem remains 

complex and warrants further investigation (Williams and Julien 1989). To date, accepted 

methodologies are not able to accurately calculate sediment transport in order to give 

acceptable results encompassing all alluvial channel conditions. In other words, there is no 

single equation which can calculate sediment transport for the entire range of conditions 

found in the field (Alonso et al. 1982; Shen and Hung 1983; Steven and Yang 1989; Simons 

and Sentürk 1992; Julien 1995). As a result, calculations of sediment transport using 

existing methods for specific rivers with the same input data produce a wide range of 

estimates of sediment transport. Because of the complexity of the problem, identification of 
B.2 

the best methodology is often academic. Specific equations are based upon limited data and 

they may not fit the whole spectrum of data from flumes, canals and rivers (Yang and Wan 

1991). Therefore, engineering judgment must be used when selecting and applying the 

available methods. This Appendix provides additional guidance on the applicability of 

selected sediment transport equations based on a comparison of predicted transport rates 

with a large data set. 

B.2 EVALUATION OF SELECTED SEDIMENT TRANSPORT EQUATIONS 

B.2.1 Introduction 

A large compilation of field data was used to test ten widely used sediment transport (bed 

material load) equations (Kodoatie 1999, Kodoatie et al. 1999). The data encompass a wide 

range of bed materials, from silts to gravels, and a wide range of river sizes, from less than 1 

m width to several thousand meters in width. 

The ten sediment transport equations that were evaluated are: Einstein (1950), Laursen 

(1958), Bagnold (1966), Toffaletti (1969), Shen and Hung (1972), Ackers and White (1973), 

Yang (1973) and (1984) for gravel-bed rivers, Brownlie (1981), Karim and Kennedy (1981), 

and Karim (1998). Field data include a total of 2,946 sets from 33 alluvial systems. 

Additionally, 919 sets of laboratory data from 19 sources were selected to verify the 

proposed methods. Tables B.1a and B.1b identify the field data and laboratory data used in 

the Kodoatie (1999) study. 

B.2.2 Scope of Study 

Because no single equation can encompass all alluvial channel conditions, four subdivisions 

of river data were analyzed based upon bed material size. These included: gravel-bed, 

medium to very coarse sand-bed, very fine to fine sand-bed and silt-bed rivers. Also, the 

data were subdivided based upon size of river, i.e., 

• Small rivers with widths equal to or less than 10 m (33 ft) and depths equal to or less than 

1 m (3.3 ft), 

• Intermediate rivers with widths greater than 10 m (33 ft) and equal to or less than 50 m 

(164 ft) and depths greater than 1 m (3.3 ft) and equal to or less than 3 m (9.8 ft), 

• Large rivers with widths greater than 50 m (164 ft) and depths of greater than 3 m (9.8 ft). 

The range of field and laboratory data are identified in Table B.2. The sediment transport 

relations tested are summarized in Table B.3. 
B.3 

Table B.1a. Field Data. 

Name of River 

Data 

Sets 

No. 

Used Source Also Found In 

ACOP Canal (Pakistan Canal) 151 142 Mahmood (1979) Brownlie (1981b) 

Chops Canal, West Pakistan 33 33 Chaudhry, et al. (1970) Brownlie (1981b) 

American Canal 13 12 Simons (1957) Brownlie (1981b) 

Atchafalaya River 72 72 Toffaletti (1968) Brownlie (1981b) 

Amazon and Orinoco Rivers 114 85 Posada (1995) 

Black Canal 17 7 Williams and Rosgen (1989) 

India Canal 32 32 Chitale (1966) Brownlie (1981b) 

Chippewa River 66 47 Williams and Rosgen (1989) 

Chulitna River 43 4 Williams and Rosgen (1989) 

Colorado River 131 100 USBR (1958) Brownlie (1981b) 

Hii River 38 38 Shinohara and Tsubaki (1979) Brownlie (1981b) 

Middle Loup River 38 15 Hubbel and Matajka (1959) Brownlie (1981b) 

Mississippi River 164 164 Toffaletti (1968) Brownlie (1981b) 

Mississippi River 85 85 Posada (1995) 

Mountain Creek 100 100 Einstein (1944) Brownlie (1981b) 

Niobrara River near Cody 51 19 Colby and Hembree (1955) Brownlie (1981b) 

North Fork Toutle River 10 2 Williams and Rosgen (1989) 

North Saskatchewan and Elbow Rivers 55 55 Samide (1971) Brownlie (1981b) 

Oak Creek 17 17 Milhous (1973) Brownlie (1981b) 

Red River 30 29 Toffaletti (1968) Brownlie (1981b) 

Rio Grande River 293 289 Nordin and Beverage (1965) Brownlie (1981b) 

Rio Grande Conveyance Channel 33 9 Culbertson, et al. (1972) Brownlie (1981b) 

Rio Grande River, Columbia 38 38 Toffaletti (1968) Brownlie (1981b) 

Rio Magdalena and Canal del Dique 113 75 NEDECO (1973) Brownlie (1981b) 

River data of Leopold 72 55 Leopold (1969) Brownlie (1981b) 

Portugal Rivers 219 219 Da Cunha (1969) Brownlie (1981b) 

Snake and Clearwater Rivers 21 17 Seitz (1976) Brownlie (1981b) 

Susitna River 38 2 Williams and Rosgen (1989) 

Toutle River 31 9 Williams and Rosgen (1989) 

Trinity River 4 3 Knott (1974) Brownlie (1981b) 

Wisconsin River 20 9 Williams and Rosgen (1989) 

Yampa River 24 11 Williams and Rosgen (1989) 

Yangtze River 40 40 Long and Liang (1994) 

Yellow River 2,326 1,112 Long and Liang (1995) 

TOTAL DATA 4,532 2,946 
B.4 

Table B.1b. Laboratory Data. 

Number Source Data Sets 

1 Barton and Lin (1955) 30 

2 Brooks (Vanoni and Brooks 1957) 21 

3 Guy (Simons and Richardson 1966) 290 

4 Franco (1968) 19 

5 Kalinske and Hsia (1945) 9 

6 Kennedy and Brooks (1963) 9 

7 Laursen (1958) 24 

8 Meyer-Peter and Müller (1948) 139 

9 Nomicos 1 (Toffaletti 1968) (Vanoni and Brooks 1957) 12 

10 Nomicos 2 (Vanoni and Brooks 1957) 26 

11 Onishi, Jain and Kennedy (1976) 14 

12 Stein (1965) 57 

13 Straub (1954 and 1958) 24 

14 Taylor and Vanoni (1972) 6 

15 Vanoni and Brooks (1957) 15 

16 Vanoni and Hwang (1967) 16 

17 Williams (1970) 83 

18 Willis (Willis, Coleman, and Ellis 1972) 96 

19 Wilcock and Southard (1988) 29 

 TOTAL DATA 919 

Table B.2. Field and Laboratory Data Ranges. 

Hydraulic Geometry Field Data Laboratory Data 

Flow discharge (m3/s) 0.0009 – 235,000 0.001 – 4.614 

Width (m) 0.8 – 3,338 0.267 – 2.438 

Depth (m) 0.02 – 68.00 0.008 – 1.092 

Slope 0.0000021 – 0.0126 0.00015 – 0.0331 
B.5 
B.6 

B.2.3 Analysis of Sediment Transport Relations 

The field data were divided into two categories: Group 1 for analysis of the selected 

sediment transport relations and proposed equations, and Group 2 for verification and 

validation of the proposed methods. The river data sets were divided into two parts in 

random order. 

A comparison between computed results and field data was conducted and examined. 

Statistical approaches were used including the mean discrepancy ratio RD (Bechteler and 

Vetter 1989; Wu 1999; Nakato 1990; Yang and Wan 1991; and Hydrau-Tech, Inc. 1998), and 

the correlation coefficient CC (Hydrau-Tech, Inc. 1998). The equations for each parameter 

follow: 

=  N

R R i

D , 

i

i

i Y

X R = (B.1) 

( )( )

  ( ) ( )

 = 2

i

2

i

ii

c

Y-YX-X

Y-YX-X C (B.2) 

For perfect fit, the values in Equations B.1 and B.2 are D = 1R and Cc = 1. 

The sediment relations were used to calculate the transport of sediment using data from 

Group 1. The sediment load was calculated using mean diameter of riverbed material 

(uniform sediment size). The results of comparing computed sediment concentration (Cppm) 

and measured sediment concentration measured for four ranges of particle sizes and for 

three sizes of rivers are shown in Figure B.1a through d, and Figure B.2a through c. 

B.2.4 Summary of Applicability of Ten Sediment Relations Analyzed 

The applicability of these relationships as reported by Kodoatie et al. (1999) is illustrated in 

Table B.4. While certain equations appear to be more applicable to particular bed 

characteristics or river size, each equation could, potentially, be applied with reasonable 

results for a specific river, if the river characteristics are compatible with those used in the 

equation's development. For any specific river, it is recommended that the results be 

compared with actual measurements. The following conclusions can be drawn from the 

results of this analysis: 

1. Gravel-bed rivers (2 mm < D50 < 64 mm) 

Compared to the measured values, none of the selected sediment relations can accurately 

predict the sediment discharge. The closest values based upon the discrepancy ratio are 

Ackers and White with RD = 0.33 and Brownlie withRD = 4.25. However, based upon the 

Pearson correlation coefficient for comparison of computed to measured Cppm, the best 

equations are Bagnold and Shen and Hung, both with Cc of 0.70. Considering gravel-bed 

rivers, Brownlie’s equations, although developed for sand-bed rivers, are the most 

acceptable of the ten equations. 
B.7 

Cppm (measured)

Cppm (computed)

(a) gravel-bed rivers

Figure B.1. Predicted versus measured total bed-material transport considering four 

 classifications of size of bed materials.
B.8 

Cppm (measured)

Cppm (computed)

(b) medium to very coarse sand-bed rivers

Figure B.1. Predicted versus measured total bed-material transport considering four 

 classifications of size of bed materials (continued).
B.9 

Cppm (measured)

Cppm (computed)

( c ) very fine to fine sand-bed rivers

Figure B.1. Predicted versus measured total bed-material transport considering four 

 classifications of size of bed materials (continued).
B.10 

Cppm (measured)

Cppm (computed)

(d) silt-bed rivers (Ackers and White not plotted because values are too high) 

 Figure B.1. Predicted versus measured total bed-material transport considering four 

 classifications of size of bed materials (continued). 
B.11 

Cppm (measured)

Cppm (computed)

(a) small rivers

Figure B.2. Comparison Cppm computed and measured for three river sets. 
B.12 

Cppm (measured)

Cppm (computed)

(b) intermediate rivers

Figure B.2. Comparison Cppm computed and measured for three river sets. 
B.13 

Cppm (measured)

Cppm (computed)

( c ) large rivers

Figure B.2. Comparison Cppm computed and measured for three river sets. 
B.14 

2. Medium to very coarse sand-bed rivers (0.250 mm < D50 < 2.00 mm) 

For this bed material, Toffaletti with RD = 0.93, Laursen with RD = 0.60, and Bagnold with 

RD = 1.68 compute Cppm closest to measured values. On the other hand, based upon 

Pearson correlation coefficient, Karim with Cc = 0.66 followed by Brownlie with Cc = 0.60 best 

correlate measured concentrations of sediment as compared to measured values. 

3. Very fine to fine sand-bed rivers (0.062 mm < D50 < 0.250 mm) 

The most suitable equations are Karim and Kennedy with RD = 0.90, Karim with RD = 1.28, 

Brownlie with Cc = 0.58, and Toffaletti with Cc = 0.52. 

4. Silt-bed rivers (0.004 mm < D50 < 0.062 mm) 

For silt-bed rivers, Einstein with RD = 1.06, Bagnold with RD = 0.56, Toffaletti with Cc = 

0.48, and Brownlie with Cc = 0.38 are the most acceptable relationships. 

5. Small rivers (width ≤ 10 m and depth ≤ 1 m) 

The closest results obtained in small rivers is Brownlie with RD = 1.18, Karim with RD = 

1.19, Yang with Cc = 0.85 and Toffaletti with Cc = 0.79. 

6. Intermediate rivers (10 m < width ≤ 50 m and 1 m < depth ≤ 3 m) 

For intermediate rivers, Toffaletti with RD = 0.94, Brownlie with RD = 0.94, Karim with Cc = 

0.76, and Yang with Cc = 0.70 are the most acceptable relationships. 

7. Large rivers (width > 50 m and depth > 3 m) 

For large rivers, Bagnold with RD = 1.04, Laursen with RD = 1.13, Brownlie with Cc = 0.80, 

and Shen and Hung with Cc = 0.76 are the most acceptable relationships. It should be noted 

that for silt-bed rivers and for very fine to fine sand-bed rivers, the Yellow River contributes 

about 77 and 63 percent of the data, respectively. As reported by many investigators, this 

river is an extremely heavily sediment-laden river and floods experience hyperconcentrations 

of sediment. Out of all rivers, this river system is unique and therefore should not be 

categorized as a common alluvial river. 

From the analysis, it can be seen that both Ackers and White and Toffaletti have a tendency 

to increase the computed concentration of suspended sediment, as the median diameter of 

bed material becomes finer. This tendency also occurs with these relationships when the 

river size increases. As reported by Kodoatie et al. (1999), the applicability of the ten 

selected sediment transport relations based upon comparison between measured and 

computed sediment transport rates and results from other studies are summarized in Table 

B.4. 
B.15 

Table B.4. Summary of Applicability of Selected Sediment Transport Relations 

 (Kodoatie et al. 1999).

 

Method Gravel 

Med – Very 

Coarse Sand 

Very Fine to 

Fine Sand Silt 

Small 

Rivers 

Intermed 

Rivers 

Large 

Rivers 

1 Ackers and White X X 

2 Bagnold X X X X 

3 Brownlie X X X X 

4 Einstein X X 

5 Karim X X X 

6 Karim and Kennedy X X X 

7 Laursen X X X X 

8 Shen and Hung X X X 

9 Toffaletti X X X X 

10 Yang ’73 and ‘84 X X X 

B.3 POWER FUNCTION RELATIONSHIPS 

B.3.1 Introduction 

Power relationships empirically relate sediment transport with hydraulic conditions and 

sediment characteristics. They can be developed by fitting the coefficients to computed 

sediment transport from more sophisticated equations or to measured data. Their utility is in 

their ease of use and, when developed from measured data, their site-specific accuracy. 

B.3.2 Basic Power Function Relationship 

In 1981, Simons et al. proposed an efficient method of evaluating sediment discharge. The 

method is based on variables flow depth, velocity, and particle diameter, and gradation 

coefficient. It can be easily applied in steep sand and fine gravel-bed creeks and rivers that 

normally exhibit critical or supercritical flow. It will also be shown that a modification of this 

relationship can be applied to subcritical rivers. This is the only transport relationship 

specifically developed for upper flow regime conditions. These power relationships were 

developed by Simons et al., from a computer solution of the Meyer-Peter and Müller bed load 

transport equation and the integration of the Einstein method for suspended bed sediment 

discharge (Julien 1995) and expressed as 

cc 3s2s = 1ss Vycq (B.3) 

where: 

 qs = Unit sediment transport rate ft2

/s (m2/s) 

 cs2, cs3 = Exponents based on mean particle diameter (D50) ranging from sand to 

fine gravel 

 cs1 = Coefficient based on mean particle diameter (note that Cs1 

must be adjusted for SI units - see Section 4.8.2) 

 y = Mean flow depth, ft (m) 

 V = Mean velocity, ft/s (m/s) 
B.16 

Table B.5 provides the coefficient and exponents for Equation B.3 (for English units) for 

different gradation coefficients and sizes of bed material. The term Gr in Table B.5 is defined 

as the gradation coefficient of the bed material and is 





 





+=

50

84

16

50

r D

D

D

D

2

1 G (B.4) 

where: 

Dn = Size of the bed material for which n percent is finer

Equation B.3 was obtained for steep, sand- and fine gravel-bed channels experiencing 

critical and super-critical flows [Simons, et al. (1981), Julien (1995)]. The range of 

parameters utilized to develop this equation is shown in Table B.6. 

The results for the total bed sediment discharge are presented in Table B.5. The high values 

of Cs3 (3.3 < Cs3 < 3.9) show the high level of dependence that sediment transport rates have 

with respect to velocity. Comparatively, the influence of depth is less important (-0.34 < Cs2

< 0.7). For smaller sizes the exponent Cs2 is positive since the smaller material is more 

easily suspended and resulting sediment concentration profiles are more uniform. Thus, the 

larger the depth, the more sediment will be suspended at a given velocity. For the larger 

sediment sizes the sediment is more difficult to suspend and keep in suspension. 

Table B.5. Coefficient and Exponents of Equation B.3 (Simons et al.). 

 D50 (mm) 

 0.1 0.25 0.5 1.0 2.0 3.0 4.0 5.0 

Gr = 1 cs1 3.30x10-5 1.42 x10-5 7.60 x10-6 5.62 x10-6 5.64 x10-6 6.32 x10-6 7.10 x10-6 7.78 x10-6

 cs2 0.715 0.495 0.28 0.06 -0.14 -0.24 -0.3 -0.34 

 cs3 3.3 3.61 3.82 3.93 3.95 3.92 3.89 3.87 

Gr = 2 cs1 1.59 x10-5 9.80 x10-6 6.94 x10-6 6.32 x10-6 6.62 x10-6 6.94 x10-6

 cs2 0.51 0.33 0.12 -0.09 -0.196 -0.27 

 cs3 3.55 3.73 3.86 3.91 3.91 3.9 

Gr = 3 cs1 1.21 x10-5 9.14 x10-6 7.44 x10-6 

 cs2 0.36 0.18 -0.02 

 cs3 3.66 3.76 3.86 

Gr = 4 cs1 1.05 x10-5 

 cs2 0.21 

 Cs3 3.71 

Table B.6. Range of Parameters Equation B.3 Developed by Simons et al. 

Parameter Value Range SI Units 

Froude Number 1 – 4 -- 

Velocity 1.98 – 7.92 m/s 

Bed Slope 0.005 – 0.040 m/m 

Unit Discharge, q 3.05 – 60.96 m2/s 

Particle Size, D50 > 0.062 mm 
B.17 

B.3.3 Power Relationships Presented by Posada (1995) 

Expanding earlier work by Nordin, Posada developed a sediment transport relation in 1995. 

In the simplest form, the sediment transport in sand-bed rivers varies as a function of the 

velocity to about the fifth power (Simons and Simons 1987). Using data from the Mississippi 

River (85 sets of data) and the Amazon and Orinoco River Systems (114 sets of data), 

Posada (1995) proposed a sediment discharge relation for sand-bed rivers as a function of 

velocity, 

5

s = V30q (B.5) 

where: 

 qs = Unit sand discharge (metric tons/m/day) 

 V = Mean velocity (m/s) 

This equation has an advantage for large, flat rivers in that velocity is easily measured and 

conversely slope is very difficult to measure accurately. This is particularly true when dealing 

with large, flat rivers such as the Mississippi, Padma, Amazon, and Orinoco Rivers. The 

application of Equation B.5 can be improved by subdividing the cross sections in the reach of 

channel being analyzed into similar parts, i.e., thalweg, etc. Also, better results are obtained 

for coarse sand and gravel if the exponent is reduced to four (4), and for silt-bed rivers if the 

exponent is increased to six (6). 

Using Equation B.5, Posada computed the unit sand discharge for the data of the 

Mississippi, Amazon, and Orinoco River Systems and compared the results to the measured 

values. About 70 percent of the data for these river systems have a discrepancy ratio RD

between 0.5 and 2.0. However, the quality of this relation is limited when additional sets of 

data from 23 river systems (2503 data points for sand-bed rivers) are included in the 

analysis. The results are shown in Figure B.3. 

The Pearson correlation coefficient of qs computed using Equation B.5 and qs measured is 

0.66. 

 Figure B.3. Relation of unit sediment discharge qs from measured and 

 computed results using Equation B.5 for 2503 sets of field data. 
B.18 

About 60 percent of the computed unit sediment discharge is 50 percent less than the 

measured unit sediment discharge. Only 33 percent of computed qs are between 70 and 200 

percent of measured qs and 7 percent of computed qs are greater than 200 percent of the 

measured qs. Accordingly, the application of Equation B.5 for a wide variety of river data is 

questionable, but not to the degree of several other widely utilized equations. The range of 

hydraulic data and particle diameters included in this analysis is shown in Table B.7. 

Table B.7. Range of Data Utilized by Posada to Develop Equation B.5. 

Width (w) Depth (d) 

< 20 m 20<w<102m 102

≤w<103m >103

m < 1 m 1 m < d < 3 m > 3 m 

2.51% 6.03% 55.28% 36.18% 0.50% 10.05% 89.45% 

Mean Bed Material Diameter 

< 0.125 mm 0.125 – 0.250 mm 0.250 – 2.000 mm 

5.88% 31.93% 77.39% 

Table B.7 illustrates that Posada developed Equation B.5 for large rivers with mostly medium 

to very coarse mean bed material (77.39 percent). Accordingly, the unit sediment discharges 

computed by Equation B.5 fit quite well to measured values for those conditions. For 

medium to very coarse sand bed rivers the discrepancy ratio between measured qs and 

computed qs is 1.12. However, RD is only 0.33 for very fine to fine sand-bed rivers, it is 0.18 

for silt-bed rivers and it is 312.46 for gravel-bed rivers. 

B.3.4 Expanded Power Function Relationship 

Considering the correlation coefficient for each variable of hydraulic geometry and the 

sediment characteristics, Kodoatie (1999) modified the Posada (1995) equation (Section 

B.3.3) using nonlinear optimization and the field data for different sizes of riverbed sediment. 

The resulting equation is (variables are defined in Section 4.8.3): 

dcb

t = SyaVq (B.6) 

A summary of coefficients and exponents is presented in Table B.8 and depend on size of 

bed material. The equation was developed with one group of data (Group 1) and validated 

with another group of data (Group 2). Graphical comparisons for four bed material 

categories are shown in Figures B.4 and B.5 for the two groups of data. Discrepancy ratios 

and correlation coefficients based upon data from Groups 1 and 2 are shown in Tables B.9 

and B.10. Comparisons between the Posada and Kodoatie et al. equations are also shown 

on these tables. Note that the values of "a" must be adjusted for input and results in English 

units (see Section 4.8.3). 

Table B.8. Coefficients and Exponents for Equation B.6. 

 a b c d 

Silt-bed rivers 281.40 2.622 0.182 0

Very fine to fine-bed rivers 2,829.60 3.646 0.406 0.412

Medium to very coarse sand-bed rivers 2,123.40 3.300 0.468 0.613

Gravel-bed rivers 431,884.80 1.000 1.000 2.000
B.19 

Table B.9. Discrepancy Ratios and Correlation Coefficients for Posada and Kodoatie 

 Equations, Data from Group 1. 

 Posada Kodoatie 

RD Cc RD Cc

Silt-bed rivers 0.12 0.8393 0.88 0.8018

Very fine to fine sand-bed rivers 0.33 0.6843 1.00 0.7242

Medium to very coarse sand-bed rivers 1.12 0.7274 1.00 0.8146

Gravel-bed rivers 312.46 0.2175 1.00 0.7625

Table B.10. Discrepancy Ratios and Correlation Coefficients for Posada and Kodoatie 

 Equations, Data from Group 2. 

 Posada Kodoatie 

RD Cc RD Cc

Silt-bed rivers 0.18 0.8393 0.88 0.8018

Very fine to fine sand-bed rivers 0.33 0.7845 1.01 0.8239

Medium to very coarse sand-bed rivers 1.12 0.7511 1.07 0.8006

Gravel-bed rivers 13.516 0.8357 0.47 0.9591

B.4 LAURSEN AND MODIFIED LAURSEN EQUATIONS

B.4.1 Laursen Equation (1958) 

Laursen (1958) working with Rouse developed the following equation. This relationship was 

modified by several scientists in an attempt to improve it by several scientists. 

















ω 













 − τ

τ′ 



 





γ=  ci i

o

6/7

i

i t i

* V

f1

d

d p01.0C (B.7) 

B.4.2 Modified Laursen Equation by Madden (1985) and Copeland and Thomas (1989) 

Madden (1985) modified Laursen’s concepts by utilizing Arkansas River data. The 

modification by Madden permitted the analysis of bed material transport by size fraction. The 

U.S. Corps of Engineers’ (1988) adopted this methodology for computing the transport in 

rivers with a mixture of sand and gravel forming the bed. Madden’s modification to Laursen’s 

methodology was based on three sets of sediment measurements made in the Arkansas 

River. The first two sets were gathered near Dardanelle in June-July 1957 and in April 1958. 

The third set was gathered near Morrilton in April 1958. Madden utilized the Missouri River 

data collected by Bondurant (1958) to validate the rating curves for the Arkansas River. The 

data sets resulted in best-fit curves that were parallel, but the two data sets did not overlap. 

The reader is reminded that the two sets of data were from two different rivers and even 

though they were sand-bed rivers, the methodologies produced by Bondurant and Madden 

were developed for specific rivers and are not generally applicable to other river systems. 
B.20 

 (a) silt-bed rivers (b) very fine to fine sand bed rivers 

( c ) medium to very coarse sand-bed rivers (d) gravel-bed rivers 

Figure B.4. Comparison of measured qs and computed qs using Posada and Kodoatie 

 equations for various river-bed materials, data from Group 1. 
B.21 

qs computed (metric tons/m/day)

qs measured (metric tons/m/day)

 

qs computed (metric tons/m/day)

qs measured (metric tons/m/day)

 (a) silt-bed rivers (b) very fine to fine sand-bed rivers qs computed (metric tons/m/day)

qs measured (metric tons/m/day)

 

qs computed (metric tons/m/day)

qs measured (metric tons/m/day)

 (c) medium to very coarse (d) gravel-bed rivers 

 sand-bed rivers 

Figure B.5. Comparison of measured qs and computed qs using Posada and 

 Kodoatie equations for various river-bed materials, data from 

 Group 2. 
B.22 

An adjustment factor related to the Froude number was utilized to modify the Laursen 

equation by Madden as follows: 

































ω 













 − τ

τ′ 



 



 =  904.0

r ci i

o

67

i

i t b

F

V 1616.0 f1

d

d pC * (B.8) 

In addition, Copeland and Thomas (1989) proposed a modification of Laursen’s 1958 

concept. This methodology was formulated to be utilized in both sand-bed rivers and, to a 

lesser degree, in larger gravel-bed rivers. The Laursen equation as modified by Copeland 

and Thomas is: 

















ω

′















 − τ

τ′ 



 





γ=  ci i

o

67

i

i t i

* V

f1

d

d p01.0C (B.9) 

The results of Copeland and Thomas’ modification to the Laursen concepts are presented in 

Figure B.6. The relation between V*/ωi and f(V*/ωi) for four sizes of sediment sizes and three 

sizes of channels were plotted on the Laursen Graph and are presented on Figure B.7. 

Many researchers have observed that more than one value of bed material transport can be 

obtained from the same values of Q, V, Sw, Sf, and τo, see for example, Brooks (1958), 

Simons and Sentürk (1992) and Yang (1996). Gilbert (1914) proposed the fact that different 

values of transport can occur for the same hydraulic parameters. 

Figure B.6. Modifications of Laursen’s (1958) Graph. 

med to very coarse sand-bed 

gravel-bed 

v. fine to fine sand-bed 

silt-bed 
B.23 

(a) four sizes of riverbed material 

(b) three sizes of rivers 

Figure B.7. Plotting V*/ωi and f(V*/ωi) for data from Group 1 with Laursen’s Equation. 

Accordingly, the validity of the assumption that total discharge for a given particle size can be 

determined by the proposed parameters is questionable. Lane (1955) and Bagnold (1966) 

first proposed the concept that stream power has a strong correlation which bed material 

discharge. Yang (1996) identified stream power, τV, as an independent variable with a strong 

correlation with bed material discharge. Furthermore, Yang suggested that unit stream 

power, VS, has a stronger relationship to bed-material discharge than stream power. Other 

researchers suggested that the stream power relations could be utilized in straight channels 

as well as channels that are in the process of changing their patterns from straight to 

meandering or braiding (Yang 1977 and 1996) (Vanoni 1978) (Yang and Molinas 1982). It is 

important to note that dimensionless unit stream power, VS/ω, further improved the 

correlation with bed material discharge (Yang and Kong 1991). Kodoatie et al. 1999 verified 

that VS/ω has a stronger correlation with bed-material discharge than other stream power 

parameters. 
B.24 

B.4.3. Modified Laursen Equation by Kodoatie et al. (1999) 

Kodoatie adapted the Laursen methodology for analysis because this methodology was 

expressed in terms which are generally recognized as important variables related to bedmaterial transport. Dimensionless unit stream power was used with regression analysis and 

nonlinear optimization techniques to improve the Laursen Equation. The modified Laursen 

equation resulting is (variables are defined in Section 4.9): 

V a flog

50c

o

67

50

t

VS 101 '

d

D 01.0C 50

*





 





ω 













 − τ

τ







 







γ=

















ω (B.10) 

where the coefficient a is a variable related to mean bed material diameter as shown in Table 

B.11. 

Table B.11. Value of "a" in Equation B.10 for Various Sizes of Bed Material. 

Bed Material "a" 

Gravel 0.0 

Medium to very coarse sand -0.2 

Very fine to fine sand 0.078 

Silt 0.06 

In equations B.7 and B.10 

 Ct = Sediment concentration, N/m3

 (lb/ft3) 

γ = Unit weight of water, N/m3

 (lb/ft3) 

 pi = Fraction by weight of bed sediment mean size, Di

 Di = Bed sediment size with I percent finer, m (ft) 

 y = Mean flow depth, m (ft) 

 V = Mean veolcity, m/s (ft/s) 

V* = gRS )s/ft(s/m,/ = ′

o ρτ

ω = Particle fall velocity, m/s (ft/s) 

 S = Slope 

 a = Exponent given in Table B.11 

τo

′ = 

)ft/lb(m/N,

y

D

58

V 2 2

3/1

50

2















ρ 

τci = Particle critical shear stress = ks (γs - γ) Di

, N/m2 (lb/ft2) 

 ks = Shields parameter 

γs = Unit weight of sediment, N/m3

 (lb/ft3) 

Note that in the modified Laursen equation an exponent equal to log f(V*/ω50) is a significant 

variable. This parameter can be determined referring to Figure B.8. Comparison of 

modifications to the Laursen equation by Madden, Copeland, and Kodoatie et al. are 

presented in Table B.12. 
B.25 

Figure B.8. Proposed graph by Kodoatie using Equation B.10. 

Table B.12. Comparison of Modified Laursen Equations. 

 Madden Copeland Kodoatie et al. 

1 Used same equation, but 

modified using Froude No. 

1 Used same equation, 

modified by 0.018 

1 Used same equation but added 

dimensionless streampower as the 

adjustment factor 

2 Used modified graph 2 Used modified graph 2 Used modified graph. More specific 

in particle size from silt to gravel 

3 Used size fraction 3 Used size fraction 3 Used median particle diameter 

4 Used Arkansas River data 4 Used both river and 

flume data (not 

specified) 

4 Used 33 river systems and 18 

sources of flume data (total data 

more than 5300 sets) 

5 Graph is higher than 

original for sand bed; not 

specified for gravel and silt 

5 Graph is higher than 

original for sand bed; 

for silt not specified; 

graph for gravel is 

proposed 

5 Graph is higher than the original for 

sand bed (sand bed is more specific 

for very fine to fine sand and 

medium to very coarse sand); 

smaller for silt compared to original; 

graph for gravel is proposed 

The modified Laursen equation (Kodoatie, 1999) was developed using Group 1 data and 

validated using Group 2 data. The following paragraphs show that this modified Laursen 

equation provides reasonable estimates of sediment transport for sediment sizes ranging 

from silt to gravel bed and rivers ranging in size from small to large. 

B.4.3.1 Gravel-Bed Rivers 

The relationship between V*/ωi and f(V*/ωi) of data from Group 2 (Meyer-Peter and Müller 

laboratory data) for gravel beds is within the range of the relationship between V*/ωi and log 

f(V*/ωi) utilizing data from Group 1 (Figure B.9). However, since few field data sets are 

reported for gravel-bed rivers, which are usually bimodal, further research should be 

conducted in this area. The Meyer-Peter and Müller (MPM) formula may be utilized to check 

the results of this analysis. The MPM method is based upon almost 20 years of experimental 

work and is probably the most widely utilized relationship for coarse bed material (Simons 

and Sentürk 1992). 

med to very coarse sand-bed 

gravel-bed 

v. fine to fine sand-bed

silt-bed 
B.26 

 Figure B.9. Relationships between V*/ωi and log [f(V*/ωi

)] for gravel–bed rivers 

 and gravel-bed Meyer-Peter and Müller laboratory data. 

B.4.3.2 Medium to Very Coarse Sand-Bed Rivers 

Comparisons of Group 2 data between measured Cppm and computed Cppm by the Laursen 

and modified Laursen equation are shown in Figure B.10 and Table B.13. 

 Figure B.10. Comparison of measured and computed concentration using Laursen 

 and modified Laursen for medium to very coarse sand bed-rivers. 

Table B.13. Discrepancy Ratio RD Between Cppm Computed and Cppm Measured for 

 Medium to Very Coarse Sand-Bed Rivers. 

 Laursen Modified Laursen 

Discrepancy Ratio RD 0.24 0.98 
B.27 

B.4.3.3 Very Fine to Fine Sand-Bed Rivers 

Comparison of Group 2 data between measured and computed concentrations by the 

Laursen and modified Laursen equation are shown in Figure B.11 and Table B.14. 

 Figure B.11. Comparison of measured Cppm and computed Cppm using Laursen and 

 modified Laursen equation for very fine sand bed-rivers. 

Table B.14. Discrepancy Ratio RD Between Cppm Computed and Cppm Measured for 

 Very Fine to Fine Sand-Bed Rivers. 

 Laursen Modified Laursen 

Discrepancy Ratio RD 0.36 1.01 

 

B.4.3.4 Silt-Bed Rivers 

Comparison of Group 2 data between measured and computed concentrations by the 

modified Laursen equation are shown in Figure B.12 and Table B.15. 

B.4.3.5 Channel Size 

As stated in previous paragraphs, the analysis of sediment transport relations was based 

upon channel size subdivided into small rivers, intermediate rivers, and large rivers. From 

Figure B.7(b), the relationships between V*/ωi and f(V*/ωi) for these three channel sizes using 

data from Group 1 follow the proposed modified Laursen graph. Validation for this 

consistency is examined using data from Group 2 for three channel sizes. The relationships 

between V*/ωi and f(V*/ωi) for data from Group 2 for these three channel sizes are plotted 

with the Laursen graph in Figure B.13. From Figure B.13, it can be concluded that 

consistency between V*/ωi and logf(V*/ωi) is good for several sizes of rivers based upon data 

from Group 2. This figure also shows that for different sizes of bed sediment (from silt to 

gravel) the proposed modified Laursen relation gives good results in comparison with other 

accepted methods. 
B.28 

Figure B.12. Comparison of Cppm measured and Cppm computed using Laursen and 

 modified Laursen equations for silt-bed rivers. 

Table B.15. Discrepancy Ratio RD Between Computed Cppm and Measured Cppm for 

 Silt-Bed Rivers. 

 Laursen Modified Laursen 

Discrepancy Ratio RD 1.45 0.991 

Figure B.13. Relationships between V*/ωi

 and logf(V*/ωi) for river size. 

B.5 RANGE OF DATA FOR VALIDATION 

To validate the Kodoatie et al. power relationship and modified Laursen equation, data from 

Group 2 were used. The computed Cppm from the proposed methods were compared to the 

measured Cppm. Note that because sets of data for gravel-bed rivers were relatively small, 

the laboratory data of Meyer-Peter and Müller (1948) containing 139 sets of data were used 

gravel-bed 

medium to very 

coarse sand-bed 

very fine to fine 

sand-bed 

silt-bed 
B.29 

for validation and verification. The ranges of hydraulic variables of the Group 2 data are 

shown below. 

B.5.1 Bed Material Size 

Gravel-Bed Rivers. The mean diameter of particle size has the range of 2.69 mm (very fine 

gravel) to 28.65 mm (coarse gravel). The wide range of geometric data is, 

discharge, Q : 0.001 – 4.614 m3

/s 

width, W : 0.1 – 2.0 m 

depth, d : 0.01 – 1.092 m 

slope, S : 0.0004 – 0.0227 

Sand-Bed Rivers. The sand-bed rivers were divided into two groups as follows: very fine to 

fine sand-bed rivers (0.062 mm – 0.250 mm), and medium to very coarse sand-bed rivers 

(0.251 mm – 2.00 mm). 

Very Fine to Fine Sand-Bed Rivers (0.062 – 0.250 mm). A total of 692 data sets from Group 

2 including 23 river systems were used. The range of data is: 

discharge, Q : 0.003673 – 235,000 m3

/s 

width, W : 0.346 – 3,090 m 

depth, d : 0.0341 – 68.00 m 

slope, S : 0.0000027 – 0.00769 

Medium to Very Coarse Sand-Bed Rivers (0.251 mm – 2.00 mm). A total of 546 data sets 

including 23 river systems outside of the United States and an additional 68 data sets from 

rivers in the United States includes: seven sets of data from the Black River near Galesville, 

Wisconsin; three sets data from the Chippewa River near Caryville, Wisconsin; 14 sets of 

data from the Chippewa River of at Durand, Wisconsin; 18 sets of data from the Chippewa 

River near Pepin, Wisconsin; six sets of data from the Toutle River at Tower Road near 

Silver Lake, Washington; nine sets of data from the Wisconsin River at Muscoda, Wisconsin; 

and 11 sets of data from the Yampa River at Deerlodge Park, Colorado. The range of data is 

as follows: 

discharge, Q : 0.0075 – 151,000 m3

/s 

width, W : 1.00 – 2,608 m 

depth, d : 0.061 – 65.00 m 

slope, S : 0.0000138 – 0.0029 

Silt-Bed Rivers. A total of 141 data sets from Group 2 including three river systems were 

used. The range of data is: 

discharge, Q : 1.15 – 41,200 m3

/s 

width, W : 4.3 – 3,110 m 

depth, d : 0.55 – 13.10 m 

slope, S : 0.0000120 – 0.00087 
B.30 

B.5.2 River Size 

Small Rivers (width < 10 m and depth < 1.0 m). A total of 81 data sets from Group 2 

including six river systems were used. The range of data is: 

discharge, Q : 0.2268 – 7.35 m3

/s 

mean bed diameter, D50 : 0.042 mm (coarse silt) – 57.51 mm (very coarse gravel) 

slope, S : 0.000115 – 0.00745 

Intermediate Rivers (10 m < width < 50 m and 1.0 m < depth < 3 m). A total of 61 data sets 

from Group 2 including nine river systems were used. The range of data is: 

discharge, Q : 17.50 – 268.72 m3

/s 

mean bed diameter, D50 : 0.021 (medium silt) – 1.91 mm (very coarse sand) 

slope, S : 0.000058 – 0.0024 

Large Rivers (width > 50 m and depth > 3 m). A total of 374 data sets from Group 2 

including 13 river systems were used. The range of data is: 

discharge, Q : 107 – 235,000 m3

/s 

mean bed diameter, D50 : 0.02 mm (medium silt) – 1.05 mm (coarse sand) 

slope, S : 0.0000027 – 0.000121 
 C.1 

 APPENDIX C 

 Index

TERM SECTION 

alluvial channels 3.1 

alluvial channel flow 1.7.2, 3.1 

alluvial fans 5.2.3 

alluvial river 3.1, 5.3 

anabranch 5.4.1 

angle of repose 3.2.8, 3.7.2 

antidunes 3.3.1, 3.3.7 

average boundary shear stress 2.4.4 

average velocity 2.2.1 

backwater curve classification 2.8.2 

bank protection 6.5, 6.6 

bars 3.3.10 

bends 2.7 

bendway weirs 6.4.2 

Bernoulli Equation 2.2.3 

block revetment 6.6.6 

braided river channel 5.4.4 

bulkhead 6.4.8 

causes of stream bank failure 6.2.1 

celerity 2.5.1 

channel bends 2.7, 5.4 

channel forms 5.4 

channel relocation 6.9.3 

Chenzy's equation 2.4.3 

choking flow 2.6.3, 2.10.4 

chutes and pools 3.3.8 

clay plug 5.4.3 

cohesion 3.2.7, 7.7.5 

Colby Method 4.5.4 

computer modeling 2.11, 5.6.2 

computerized data bases 8.4 

concentration conversion factors 4.8 

concrete mattresses 6.6.6 

conservation of energy 2.2.3 

conservation of mass 2.2.1 

conservation of momentum 2.2.2 

contraction scour 7.2, 7.6.1 
 C.2 

 APPENDIX C 

 Index

TERM SECTION 

control volume 2.2.1 

convective acceleration 2.2.1 

countermeasures for hydraulic problems 6.1, 7.1 

critical shear stress 3.5 

critical velocity 3.5.5, 3.5.6 

culvert analysis programs 2.12.1 

culvert design 2.12 

culvert flow 2.12 

culvert hydraulics 2.12 

data checklist 8.2 

data needs 8.1 

data sources 8.3, 8.4 

debris 5.7.2, 7.7.6 

dikes 6.4.5, 6.4.6 

discharge diagram 2.6.3 

distorted scales 5.6.1 

dominant discharge 5.4.7 

drag coefficient 3.5.3 

drag force 3.5.3 

drop structures 2.6.5, 6.4.11 

dunes 3.3.1, 3.3.5 

effective diameter 3.2.4 

effects of human activities 1.4, 5.7.1 

effects of natural changes 1.2, 5.7.2 

Einstein Method 4.5.2 

encroachment 1.1, 1.4 

environmental impacts 6.9, 8.1.10 

equation of hydrostatistics 2.12 

equation of motion 2.2 

fall velocity 3.2.1, 3.2.3 

fencing 6.4.9 

filter design for riprap 6.7, 6.11.2 

floodplain 5.4.1 

flow classifications 2.1.1, 3.3.1 

flow control structures 6.4 

flow regimes 3.3.1 

flow resistance 2.4, 3.4 
 C.3 

 APPENDIX C 

 Index

TERM SECTION 

forces on particles 3.5 

Froude number 2.1.1, 2.5.3, 5.6.1 

gabion mattresses 6.6.4 

general scour 7.2, 7.6 

geomorphic threshold 5.2.5 

gradation coefficient 3.2.4 

gradually varied flow 2.8 

groins 6.4.1 

guide banks 6.4.10 

hardpoints 6.4.3 

headcut 5.2.4 

hydraulic geometry 5.4.6 

hydraulic jump 2.5.3 

hydraulically rough boundary 2.4.2 

hydraulically smooth boundary 2.4.2 

jetties 6.4.7 

kinetic energy correction factor 2.2.3, 2.4.6 

laminar flow 2.1.1 

limiting shear stress 3.5, 7.6.1 

local acceleration 2.2.1 

local scour at abutments 7.2, 7.8 

local scour at drop structures 2.6.5, 6.4.11 

local scour at embankments 6.8 

local scour at piers 7.2, 7.7 

local shear stress 2.4.5 

lower flow regime 3.3.1 

Manning's equation 2.4.3 

mature river 5.2.1 

meandering river channel 5.4.3 

Meyer-Peter and Müller 4.5.1 

mixing length 2.4.2 

mobile bed model 5.6.1 

momentum coefficient 2.2.2, 2.4.6 

n-value estimate for sand bed streams 2.4.3, 3.4.11 

n-value estimation 2.4.3 

neotectonics 1.2.1 

nickpoint 5.2.4 
 C.4 

 APPENDIX C 

 Index

TERM SECTION 

nominal diameter 3.2.1 

nonuniform flow 2.2.1 

open channel flow 2.1 

overtopping, embankment 2.13, 6.8 

pebble count 3.2.4, 3.7.1 

physical modeling 5.6.1 

pile retard 6.4.6 

pipette method 3.2.4 

plane bed 3.3.1, 3.3.6 

plane bed with sediment movement 3.3.6 

porosity 3.2.6 

power function 4.7 

pressure distribution 2.3 

qualitative river response 5.5 

rapid flow 2.1.1 

rapidly varying flow 2.6 

retards 6.4.4 

revetment 6.3.3, 6.3.4 

Reynolds number 5.6.1 

ripples 3.3.1, 3.3.4 

ripples on dunes 3.3.1 

riprap 6.5 

riprap design 6.5 

riprap failure 6.5.7 

riprap filter 6.7, 6.11.2 

riprap gradation 6.5.5 

riprap placement 6.5.6 

riprap stability 6.5 

river channel classification 5.2, 5.4.1 

river channel profiles 2.8.2, 5.4.8 

river morphology 5.1 

river response to change 1.2, 5.5 

river stabilization 6.4, 6.5 

river variables 1.5.1 

roadway overtopping 2.13, 6.8 

rock fill trench revetment 6.5.6 

roll waves 2.5.4 
 C.5 

 APPENDIX C 

 Index

TERM SECTION 

sack revetment 6.6.5 

sampling bed material 3.2.4 

scour due to overbank flow 2.13, 6.8 

scour, general 7.2, 7.6 

scour, local 7.2, 7.7, 7.8 

sediment concentration conversion 4.8 

sediment discharge 3.6, 4.2 

sediment discharge measurement 3.6 

sediment shape 3.2.2 

sediment terminology 3.6.2, 4.3 

sediment transport 3.6.7, 4.1 

sedimentation diameter 3.2.1 

shear stress relations 2.4.5 

Shields' criteria 3.5.4 

sieve analysis 3.2.4 

sieve diameter 3.2.1, 3.2.4 

similitude 5.6.1 

sinuosity 5.3, 5.5.1 

size classification of sediments 3.2.1 

size distribution 3.2.4 

soil cement revetment 6.6.8 

specific discharge 2.6.3 

specific energy (specific head) 2.6.2 

specific weight 3.2.5 

spur 6.4.1 

spur angle 6.4.1 

spur crest 6.4.1 

spur height 6.4.1 

spur length 6.4.1 

spur spacing 6.4.1 

stability of relocated channels 1.2, 1.3.2 

stabilization measures 6.1, 6.3 

standard fall diameter 3.2.1 

standard fall velocity 3.2.1 

standard step method 2.8.3 

steady flow 2.2.1 

steel jacks 6.4.7 
 C.6 

 APPENDIX C 

 Index

TERM SECTION 

stone fill dike 6.4.6, 6.5.6 

straight river channel 5.4.2 

stream gaging 2.9 

stream power 3.4.11, 4.7, 5.5.1 

stream stability 1.2, 5.1 

superelevation, in rapid flow 2.7.4, 2.10.4 

superelevation, in subcritical flow 2.7.3 

surges 2.5.2 

suspended sediment discharge 3.6, 4.4 

timber crib 6.4.4 

timber pile dike 6.4.6 

tranquil flow 2.1.1 

transition zone flow regime 2.4.2, 3.3.1 

transitions 2.6, 3.3.1 

transitions, in rapid flow 2.6.4 

transverse velocity distribution 2.7.2 

turbulent (Reynolds) shear stress 2.4.2 

turbulent flow 2.1.1, 2.4.2 

turbulent velocities 2.4.2 

uniform flow 2.2.1 

unsteady flow 2.2.1 

upper flow regime 3.3.1 

vane dike 6.4.6 

vegetation revetment 6.6 

velocity of flow on riprap 6.5.3, 6.5.4 

viscous drag 3.5.3 

viscous sublayer 2.4.2 

visual accumulation tube 3.2.4 

von Karman universal velocity coefficient 2.4.2 

windrow revetment 6.5.6