A STUDY OF FLUVIAL GEOMORPHOLOGY ASPECTS OF HYDRAULIC DESIGNiri.ku.edu/.../files/files/pdf/publications/Fluvial_Geomorphology.pdf · ¾A Study of Fluvial Geomorphology Aspects of

11

A STUDY OF FLUVIAL GEOMORPHOLOGY

ASPECTS OF HYDRAULIC DESIGN

A STUDY OF FLUVIAL GEOMORPHOLOGY

ASPECTS OF HYDRAULIC DESIGN

A. David Parr, Ph.D.and John Shelley

CEAE DepartmentUniversity of Kansas(Funded by KDOT)

A. David Parr, Ph.D.and John Shelley

CEAE DepartmentUniversity of Kansas(Funded by KDOT)

22

Jim Richardson

Brad Rognlie

Mike Orth

KDOT Bridge Section

Jim RichardsonJim Richardson

Brad RognlieBrad Rognlie

Mike OrthMike Orth

KDOT Bridge SectionKDOT Bridge Section

AcknowledgmentsAcknowledgments

33

Stable Channel DesignStable Channel DesignStable Channel DesignKDOT is sometimes required to realign short reaches of alluvial channels to facilitate highway improvements or to provide protection for highway structures or roadway embankments.

The new stream reaches should be dynamically stable and should have geomorphic properties that are characteristic of natural streams in similar settings.

They should also be hydraulically and ecologically compatible with the contiguous upstream and downstream stream reaches.

KDOT is sometimes required to realign KDOT is sometimes required to realign short reaches of alluvial channels to short reaches of alluvial channels to facilitate highway improvements or to facilitate highway improvements or to provide protection for highway structures provide protection for highway structures or roadway embankments.or roadway embankments.

The new stream reaches should be dynamically The new stream reaches should be dynamically stable and should have geomorphic properties that stable and should have geomorphic properties that are characteristic of natural streams in similar are characteristic of natural streams in similar settings. settings.

They should also be hydraulically and ecologically They should also be hydraulically and ecologically compatible with the contiguous upstream and compatible with the contiguous upstream and downstream stream reaches.downstream stream reaches.

44

Stream Modification - Road ProjectStream Modification Stream Modification -- Road ProjectRoad Project

Old RoadOld Road

Old StreamOld StreamNew StreamNew Stream

New RoadNew Road

(b)(b)(a)(a)

55

Protection - Meanders on the Kansas River Protection - Meanders on the Kansas River

(a)(b)

66

Approaches to Stable Channel DesignApproaches to Stable Channel DesignApproaches to Stable Channel DesignRegime Methods

Empirical regression equations - not for natural channels

Reference-Reach MethodsRosgen-type methods – The design channel geometry is scaled from a stable reference reach on the same stream network or from a stream of the same type with similar geologic and hydrologic characteristics.

Analytical MethodsUse hydraulic resistance equations and sediment transport equations to design a channel reach that has the same flow and sediment transport capacity as a representative stable upstream supply reach for Bank-full Discharge Conditions.

Regime MethodsRegime MethodsEmpirical regression equations Empirical regression equations -- not for natural not for natural channelschannels

ReferenceReference--Reach MethodsReach MethodsRosgenRosgen--type methods type methods –– The design channel The design channel geometry is scaled from a stable reference reach on geometry is scaled from a stable reference reach on the same stream network or from a stream of the the same stream network or from a stream of the same type with similar geologic and hydrologic same type with similar geologic and hydrologic characteristics.characteristics.

Analytical MethodsAnalytical MethodsUse hydraulic resistance equations and sediment Use hydraulic resistance equations and sediment transport equations to design a channel reach that transport equations to design a channel reach that has the same flow and sediment transport capacity as has the same flow and sediment transport capacity as a representative stable upstream supply reach for a representative stable upstream supply reach for BankBank--full Discharge Conditionsfull Discharge Conditions..

77

Bank-full Discharge ConditionsBankBank--full Discharge Conditionsfull Discharge Conditions

Copeland* states “Bank-full discharge is the maximum discharge that a steam can convey without overflowing into the floodplain.” The water surface elevation for this condition is called the bank-full stage.

Bank-full discharge is also referred to as the channel-forming discharge.

Copeland* states Copeland* states ““BankBank--full discharge is full discharge is the maximum discharge that a steam can the maximum discharge that a steam can convey without overflowing into the convey without overflowing into the floodplainfloodplain..”” The water surface elevation The water surface elevation for this condition is called the bankfor this condition is called the bank--full full stage. stage.

BankBank--full discharge is also referred to as full discharge is also referred to as the the channelchannel--forming dischargeforming discharge..

*http://chl.erdc.usace.army.mil/library/publications/chetn/pdf/chetn-viii-5.pdf*http://chl.erdc.usace.army.mil/library/publications/chetn/pdf/chetn-viii-5.pdf

88

Upstream Supply ReachUpstream Supply ReachUpstream Supply Reach

Project SiteProject Site

Flow

Riffle on Stable UpstreamSupply Reach

Riffle on Stable UpstreamSupply Reach

Supply Reach Cross SectionSupply Reach Cross Section

Sinuosity = Lstream/Lvalley

99

Bankfull Conditions for Supply Reach Cross Section

(1.2 to 2 year recurrence interval)

Bankfull Conditions for Supply Bankfull Conditions for Supply Reach Cross SectionReach Cross Section

((1.2 to 2 year recurrence interval)

Wbf

Abf

dmax

Wbf = bankfull widthAbf = bankfull areadbf = Abf /Wbf = bankfull depth

Bankfull Stage

1010

Determination of Bank-full Stage(http://www.stockton.edu/~epsteinc/rosgen~1.htm)

Determination of Bank-full Stage(http://www.stockton.edu/~epsteinc/rosgen~1.htm)

Involves assessing the elevation where the channel, under bank-full discharge conditions, ends and the floodplain begins. The indicators used to assess this elevation are as follows:

Top of the point barA change in vegetationSlope change in channel cross sectionTop of the undercut slopeChange in particle size (where soils end and sediments begin),Drift lines and water marks

Involves assessing the elevation where the channel, under bank-full discharge conditions, ends and the floodplain begins. The indicators used to assess this elevation are as follows:

Top of the point barA change in vegetationSlope change in channel cross sectionTop of the undercut slopeChange in particle size (where soils end and sediments begin),Drift lines and water marks

1111

University of Kansas StudiesUniversity of Kansas StudiesUniversity of Kansas Studies

Guidelines for Stream Realignment Design – KAM Method

McEnroe, Young and ShelleyReport No. K-TRAN KU-08-2

Stream Realignment Design using a Reference Reach – ARR Method

McEnroe, Young and ShelleyReport No. K-TRAN KU-09-4

A Study of Fluvial Geomorphology Aspects of Hydraulic Design (HEC-RAS applications)

Parr and ShelleyReport No. K-TRAN: KU-08-5

Guidelines for Stream Realignment Design Guidelines for Stream Realignment Design –– KAM KAM MethodMethod

McEnroe, Young and ShelleyMcEnroe, Young and ShelleyReport No. KReport No. K--TRAN KUTRAN KU--0808--22

Stream Realignment Design using a Reference Stream Realignment Design using a Reference Reach Reach –– ARR MethodARR Method

McEnroe, Young and ShelleyMcEnroe, Young and ShelleyReport No. KReport No. K--TRAN KUTRAN KU--0909--44

A Study of Fluvial Geomorphology Aspects of Hydraulic Design (HEC-RAS applications)

Parr and ShelleyParr and ShelleyReport No. KReport No. K--TRAN: KUTRAN: KU--0808--55

This StudyThis Study

1212

KAM and ARR MethodsKAM and ARR MethodsKAM and ARR MethodsConsider alluvial (noncohesive) and threshold (cohesive) channels.

StrengthsInclude planform design for stream meanders and pool spacingDesigns pool depth Uses a simple version of the Meyer-Peter Mueller sediment transport equation for analytical methodsARR incorporates features of both analytical and reference reach methods

Consider alluvial (noncohesive) and threshold Consider alluvial (noncohesive) and threshold (cohesive) channels.(cohesive) channels.

StrengthsStrengthsInclude planform design for stream meanders and Include planform design for stream meanders and pool spacingpool spacingDesigns pool depth Designs pool depth Uses a simple version of the MeyerUses a simple version of the Meyer--Peter Mueller Peter Mueller sediment transport equation for analytical methodssediment transport equation for analytical methodsARR incorporates features of both analytical and ARR incorporates features of both analytical and reference reach methodsreference reach methods

1313

KAM and ARR Methods (Cont.)KAM and ARR Methods (Cont.)KAM and ARR Methods (Cont.)Limitations

Plane bed (no bedforms)Wide channels (Large width to depth ratios)No consideration of grain size distribution other than d50Does not allow for separation of bed and bank hydraulic roughness

LimitationsLimitationsPlane bed (no bedforms)Plane bed (no bedforms)Wide channels (Large width to depth Wide channels (Large width to depth ratios)ratios)No consideration of grain size No consideration of grain size distribution other than ddistribution other than d5050

Does not allow for separation of bed Does not allow for separation of bed and bank hydraulic roughnessand bank hydraulic roughness

1414

Study ObjectivesStudy ObjectivesStudy ObjectivesDevelop procedures to use HEC-RAS 4.0 in the design of stable channel reaches for alluvial streams using the Analytical Approach.

Provide examples for streams with Sand beds Gravel/cobble beds.

Compare HEC-RAS methods with McEnroe’s KAM and ARR Methods.

Develop procedures to use HECDevelop procedures to use HEC--RAS 4.0 in the RAS 4.0 in the design of stable channel reaches for alluvial design of stable channel reaches for alluvial streams using the streams using the Analytical ApproachAnalytical Approach..

Provide examples for streams with Provide examples for streams with Sand beds Sand beds Gravel/cobble beds.Gravel/cobble beds.

Compare HECCompare HEC--RAS methods with McEnroeRAS methods with McEnroe’’s s KAM and ARR Methods.KAM and ARR Methods.

1515

Stable Channel Design in HEC-RASStable Channel Design in HECStable Channel Design in HEC--RASRASUses Steady Flow modeling to determine parameters needed for the sediment transport modeling components.

VelocityDepthArea

Only Manning’s n-values can be used in the HEC-RAS steady flow model for resistance.

Uses Hydraulic Design Functions to perform uniform flow and sediment transport capacity calculations. Brownlie, Strickler, Limerinos and Manning equations can be used to account for channel resistance. Brownlie and Limerinos resistance equations account for bed form resistance as well as resistance due to grains.

Uses Uses Steady FlowSteady Flow modeling to determine parameters modeling to determine parameters needed for the sediment transport modeling components. needed for the sediment transport modeling components.

VelocityVelocityDepthDepthAreaArea

Only ManningOnly Manning’’s ns n--values can be used in the HECvalues can be used in the HEC--RAS RAS steady flow model for resistance.steady flow model for resistance.

Uses Uses Hydraulic Design FunctionsHydraulic Design Functions to perform uniform flow to perform uniform flow and sediment transport capacity calculations. Brownlie, and sediment transport capacity calculations. Brownlie, Strickler, Limerinos and Manning equations can be used Strickler, Limerinos and Manning equations can be used to account for channel resistance. Brownlie and to account for channel resistance. Brownlie and Limerinos resistance equations account for bed form Limerinos resistance equations account for bed form resistance as well as resistance due to grains.resistance as well as resistance due to grains.

1616

HEC-RAS Hydraulic Design Functions used in Analytical Design

HECHEC--RAS Hydraulic Design RAS Hydraulic Design Functions used in Analytical DesignFunctions used in Analytical Design

Stable Channel Design

Uniform Flow

Sediment Transport Capacity

Stable Channel Design Stable Channel Design

Uniform FlowUniform Flow

Sediment Transport CapacitySediment Transport Capacity

Sand BedsSand Beds

Gravel/Cobble BedsGravel/Cobble Beds

1717

Resistance Formulas Used

in

Hydraulic Design Functions

Resistance Formulas Used Resistance Formulas Used

in in

Hydraulic Design FunctionsHydraulic Design Functions

1818

HEC-RAS Resistance Formulas for Alluvial Channels

HECHEC--RAS Resistance Formulas RAS Resistance Formulas for Alluvial Channels for Alluvial Channels

Equation Applicability Strengths LimitationsManning All natural and artificial

streams.Easy to use and to understand. Required for HEC-RAS hydraulic modeling.

Requires a high level of engineering judgment to choose an appropriate value from a table or from a book of reference streams.

Stricker Cobble bed streams dominated by grain size friction.

Quantified the hydraulics losses due to grain size friction based on measurable parameters.

Does not include losses due to bed forms. May be unrealistically low.

Limerinos Stream beds with sediment sizes from coarse sand to cobble under an upper flow regime.

Includes losses due to both grain roughness and bedforms. Based on measurable parameters.

Not applicable to other sediment sizes or to the lower regime flow.

Brownlie Sand bed streams of either an upper or a lower regime.

Includes losses due to both grain roughness and bedforms. Based on measurable parameters. Can be used for either the upper or the lower flow regime. Correlates with the Brownlie sediemnt transport function.

Not applicable to other sediment sizes.

1919

Manning’s Equation ManningManning’’s Equation s Equation

V= Mean Velocity in ft/sec,1.49 = coefficient for English Units (1.0 for Metric),n = Manning’s n value,R = Hydraulic radius, ft. = Area/Wetted Perimeter,S = Slope of the Energy Grade Line.

(Bed slope for uniform flow)

V= Mean Velocity in ft/sec,V= Mean Velocity in ft/sec,1.49 = coefficient for English Units (1.0 for Metric),1.49 = coefficient for English Units (1.0 for Metric),n = Manningn = Manning’’s n value,s n value,R = Hydraulic radius, ft. = Area/Wetted Perimeter,R = Hydraulic radius, ft. = Area/Wetted Perimeter,S = Slope of the Energy Grade Line.S = Slope of the Energy Grade Line.

(Bed slope for uniform flow)(Bed slope for uniform flow)

2 /3 1/ 21.49n R SV

=

2020

Strickler Equation Strickler Equation Strickler Equation 1/ 6s

s

Rn kk

φ⎛ ⎞

= ⎜ ⎟⎝ ⎠

50

90

, ,.

0.0342

0.0342

s

s

wherek Nikuradseequivalet sand roughness ft or m dfor natural channels and d for riprap lined channels

R Strickler function for natrual channelsk

for velocity and stone sizecalculations in riprapchann

φ

= =

−

= =

=0.038

elsfor dischargecalculations in riprapchannels

R hydraulic radius==

2121

Limerinos Equation Limerinos Equation Limerinos Equation

d84 = the particle size, ft, for which 84% of the sediment mixture is finer. Data ranged from 0.00328 to 0.820 ft (1.5 to 250 mm). BIG STUFFn = Manning’s n value. Data ranged from 0.02 to 0.10.R = Hydraulic radius, ft. Data ranged from 1 to 6 ft (0.35 to 1.83 m).

dd8484 = the particle size, ft, for which 84% of the = the particle size, ft, for which 84% of the sediment mixture is finer. Data ranged from sediment mixture is finer. Data ranged from 0.00328 to 0.820 ft (1.5 to 250 mm). 0.00328 to 0.820 ft (1.5 to 250 mm). BIG STUFFBIG STUFFn = Manningn = Manning’’s n value. Data ranged from 0.02 to s n value. Data ranged from 0.02 to 0.10.0.10.R = Hydraulic radius, ft. Data ranged from 1 to 6 R = Hydraulic radius, ft. Data ranged from 1 to 6 ft (0.35 to 1.83 m). ft (0.35 to 1.83 m).

1/ 6

1084

0.0926

1.16 2.0 log

RnR

d

=⎛ ⎞

+ ⎜ ⎟⎝ ⎠

2222

Brownlie Resistance Equations –Sand Only

Brownlie Resistance Equations Brownlie Resistance Equations ––Sand OnlySand Only

d50 = the particle size, ft, for which 50% of the sediment mixture is finer by weight,s = the geometric standard deviation of the

sediment mixture.

dd5050 = the particle size, ft, for which 50% of the = the particle size, ft, for which 50% of the sediment mixture is finer by weight,sediment mixture is finer by weight,s s = the geometric standard deviation of the = the geometric standard deviation of the

sediment mixture. sediment mixture.

( )

( )

0.13740.1670.1112 0.1605

5050

0.0.6620.1670.0395 0.1282

5050

1.6940 0.034

1.0213 0.034

Lower Regime

Rn S dd

Upper Regime

Rn S dd

σ

σ

⎛ ⎞⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

⎛ ⎞⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

2323

Brownlie Resistance Equations (Cont.)

Brownlie Resistance Equations Brownlie Resistance Equations (Cont.)(Cont.)

'

' '

'

'1/3

50

0.8 1.25

0.006

1.74

( 1)

( 2.65 )

g g

g g g

g g

g

gs

s

Lower Regime F F

Transition F F F

Upper Regime S or F F

FS

VF grain Froude numbers gd

wheres sediment specific gravity for sand

<

< <

> >

=

= =−

= ≈

2424

HEC-RAS

Sediment Transport

Equations

HECHEC--RASRAS

Sediment Transport Sediment Transport

EquationsEquations

2525

Sand Beds - Brownlie Sediment Transport Equation

Sand BedsSand Beds -- Brownlie Sediment Brownlie Sediment Transport EquationTransport Equation

Used in the Stable Channel Hydraulic Design Function for sand bed channels only. The method is called the Copeland Method.

Based on dimensional analysis and regression of a very large body of field and laboratory sediment transport data for sand beds.

Only applied to the movable bed – does not consider sediment transport from the main channel banks.

Used in the Stable Channel Hydraulic Design Used in the Stable Channel Hydraulic Design Function for Function for sand bed channels onlysand bed channels only. The . The method is called the Copeland Method.method is called the Copeland Method.

Based on dimensional analysis and regression Based on dimensional analysis and regression of a very large body of field and laboratory of a very large body of field and laboratory sediment transport data for sand beds.sediment transport data for sand beds.

Only applied to the movable bed Only applied to the movable bed –– does not does not consider sediment transport from the main consider sediment transport from the main channel banks. channel banks.

2626

Brownlie Sediment Transport Equation(Sand Bed Natural Channels)

Brownlie Sediment Transport EquationBrownlie Sediment Transport Equation(Sand Bed Natural Channels)(Sand Bed Natural Channels)

( ) 0.33011.978 0.6601509022( ) /g goC F F S r d −= −

( )

*

50

50

0.5293 0.14

/ /

4.596o

g s

go

where C bed material concentration in ppm by weightr representative grain roughness heightd geometric mean grain size of bed materialS slope of energy grade line

F V gd grain Froude number

F S

ρ ρ ρ

τ −

===

=

= − =

=

( )( )

*

05 0.1606

7.7

0.6

350

0.22 0.06(10)

/

/

o

g

Y

g

g s

g

critical grain Froude number

Y critical shear stress

geometric standard deviation of bed material

Y R

R gd grain Reynolds number

σ

τ

σ

ρ ρ ρ

ν

−

−

−

=

= + =

=

= −

= =

2727

Gravel/Cobble Beds – Sediment Transport Potential Functions

Gravel/Cobble Beds Gravel/Cobble Beds –– Sediment Transport Sediment Transport Potential FunctionsPotential Functions

Ackers-White

Engelund-Hansen

Laursen-Copeland

Meyer-Peter Muller

Toffaleti

Yang

AckersAckers--WhiteWhite

EngelundEngelund--HansenHansen

LaursenLaursen--CopelandCopeland

MeyerMeyer--Peter MullerPeter Muller

ToffaletiToffaleti

YangYang

2828

Ranges for Sediment Transport FunctionsRanges for Sediment Transport FunctionsRanges for Sediment Transport Functions

d ra

nge, mm

d range, m

m

Mean d , mm

Mean d , mm

Velocity, fp

s

Velocity, fp

s

Depth, ft

Depth, ft

Energy Grad

Energy Grad

Width ft

Width ft

Temp, o F

Temp, o F

Spec Gravity

Spec Gravity

2929

Sand Beds

HEC-RAS Stable Channel Design

Sand Beds Sand Beds

HECHEC--RAS Stable Channel RAS Stable Channel DesignDesign

3030

HR Stable Channel DesignHR Stable Channel DesignHR Stable Channel Design

Copeland Method using Brownlie Resistance and Sediment Transport Eqs.Sand Bed Channels Only.Resistance Due to Sidewall Roughness, Grains of the Bed Material and Bed Forms.Sediment Transport from Bed Only.Sidewall Roughness Method Applied.Does not specify channel plan form geometry or profile features. (See McEnroe KAM and ARR methods.)

Copeland Method using Brownlie Resistance Copeland Method using Brownlie Resistance and Sediment Transport Eqs.and Sediment Transport Eqs.Sand Bed Channels Only.Sand Bed Channels Only.Resistance Due to Sidewall Roughness, Grains Resistance Due to Sidewall Roughness, Grains of the Bed Material and Bed Forms.of the Bed Material and Bed Forms.Sediment Transport from Bed Only.Sediment Transport from Bed Only.Sidewall Roughness Method Applied.Sidewall Roughness Method Applied.Does not specify channel plan form geometry or Does not specify channel plan form geometry or profile features. (See McEnroe KAM and ARR profile features. (See McEnroe KAM and ARR methods.)methods.)

3131

HR Stable Channel Design Requirements

HR Stable Channel Design HR Stable Channel Design RequirementsRequirements

Upstream Supply Channel: (Trapezoidal channel geometry required.) Bottom width, depth, channel slope, side slopes, discharge, Manning’s n for sidewalls, sediment gradation or sediment conc. Design Channel: Manning’s n for sidewalls, side slopes, sediment gradation ,and either the bottom width, depth, or channel slope.Both: Need d16, d50 and d84

Upstream Supply Channel:Upstream Supply Channel: (Trapezoidal channel (Trapezoidal channel geometry required.) Bottom width, depth, geometry required.) Bottom width, depth, channel slope, side slopes, discharge, channel slope, side slopes, discharge, ManningManning’’s n for sidewalls, sediment gradation or s n for sidewalls, sediment gradation or sediment conc. sediment conc. Design Channel:Design Channel: ManningManning’’s n for sidewalls, side s n for sidewalls, side slopes, sediment gradation ,and either the slopes, sediment gradation ,and either the bottom width, depth, or channel slope.bottom width, depth, or channel slope.Both:Both: Need dNeed d1616, d, d5050 and dand d8484

3232

Procedure for Stable Channel Design of Sand Bed Channels

Procedure for Stable Channel Design of Procedure for Stable Channel Design of Sand Bed ChannelsSand Bed Channels

Establish the bank-full properties of an upstream reference reach riffle cross section. Discharge, cross section geometry via station-elevation data, stage, bed material (d16, d50 and d84), longitudinal energy grade line slope.Open the Uniform Flow function with the Manning for the bank resistance and Brownlie for the movable bed resistance. Input the slope and discharge. By iteration, determine the bank n-values needed to obtain the desired bank-full water surface elevation (stage) for the given bank-full discharge and slope.

Establish the bankEstablish the bank--full properties of an upstream full properties of an upstream reference reach riffle cross section. Discharge, cross reference reach riffle cross section. Discharge, cross section geometry via stationsection geometry via station--elevation data, stage, bed elevation data, stage, bed material material ((dd1616, d, d5050 and dand d8484)), longitudinal energy grade line , longitudinal energy grade line slope.slope.Open the Uniform Flow function with the Manning for the Open the Uniform Flow function with the Manning for the bank resistance and Brownlie for the movable bed bank resistance and Brownlie for the movable bed resistance. Input the slope and discharge. By iteration, resistance. Input the slope and discharge. By iteration, determine the bank ndetermine the bank n--values needed to obtain the values needed to obtain the desired bankdesired bank--full water surface elevation (stage) for the full water surface elevation (stage) for the given bankgiven bank--full discharge and slope. full discharge and slope.

3333

Procedure for Stable Channel Design of Sand Bed Channels (Cont.)

Procedure for Stable Channel Design of Procedure for Stable Channel Design of Sand Bed Channels (Cont.)Sand Bed Channels (Cont.)

Using the bank n-values from the previous step, change the resistance formula for the bed to Manning then by iteration determine the appropriate n-value for the bed to obtain the desired bankfull stage.Create an upstream supply reach that has three of the natural channels using the bank and bed n-values determined above for the bankfull channel.Run the HEC-RAS model.

Using the bank nUsing the bank n--values from the previous step, values from the previous step, change the resistance formula for the bed to change the resistance formula for the bed to Manning then by iteration determine the Manning then by iteration determine the appropriate nappropriate n--value for the bed to obtain the value for the bed to obtain the desired bankfull stage.desired bankfull stage.Create an upstream supply reach that has three Create an upstream supply reach that has three of the natural channels using the bank and bed of the natural channels using the bank and bed nn--values determined above for the bankfull values determined above for the bankfull channel.channel.Run the HECRun the HEC--RAS model.RAS model.

3434

Procedure for Stable Channel Design of Sand Bed Channels (Cont.)

Procedure for Stable Channel Design of Procedure for Stable Channel Design of Sand Bed Channels (Cont.)Sand Bed Channels (Cont.)

Determine an equivalent trapezoidal channel that has the same conveyance as the natural supply reach. Open the Stable Channel Design function.Input the side slopes, base width, bank n-values and the energy grade line slope of the equivalent upstream supply channel.Input the side slopes and bank n-values for the design channel.Run the Stable Channel Design model.

Determine an equivalent trapezoidal channel Determine an equivalent trapezoidal channel that has the same conveyance as the natural that has the same conveyance as the natural supply reach. Open the Stable Channel Design supply reach. Open the Stable Channel Design function.function.Input the side slopes, base width, bank nInput the side slopes, base width, bank n--values values and the energy grade line slope of the and the energy grade line slope of the equivalent upstream supply channel.equivalent upstream supply channel.Input the side slopes and bank nInput the side slopes and bank n--values for the values for the design channel.design channel.Run the Stable Channel Design model.Run the Stable Channel Design model.

3535

Sand Bed ExampleSand Bed ExampleSand Bed Example

49.021.6

0

5

10

15

20

25

0 10 20 30 40 50 60 70 80

Station (ft)

Elev

atio

n (ft

)

Sta-Elev Bankfull Elevation Movable Bed

49.021.6

0

5

10

15

20

25

0 10 20 30 40 50 60 70 80

Station (ft)

Elev

atio

n (ft

)


2828 4444

Bank-full Conditionsd16, d50, d84 = 1.33, 2 and 3 mm, respectivelyQ = 325 cfsStage = 7 ftSlope = 0.00157

BankBank--full Conditionsfull Conditionsdd1616, d, d5050, d, d8484 = 1.33, 2 and 3 mm, respectively= 1.33, 2 and 3 mm, respectivelyQ = 325 cfsQ = 325 cfsStage = 7 ftStage = 7 ftSlope = 0.00157Slope = 0.00157

3636

Uniform Flow with Final Manning’s n values(Initially Brownlie for movable bed, unknown for banks)

Uniform Flow with Final ManningUniform Flow with Final Manning’’s n valuess n values(Initially Brownlie for movable bed, unknown for banks)(Initially Brownlie for movable bed, unknown for banks)

3737

Natural Supply ReachNatural Supply ReachNatural Supply Reach

3838

Equivalent Trapezoidal ChannelEquivalent Trapezoidal ChannelEquivalent Trapezoidal Channel

Abnk Pbnkh

0

5

10

15

20

0 10 20 30 40 50 60 70 80

Station (ft)

Elevation (ft)

Sta‐Elev Points Bankfull Water SurfaceEquivalent Channel Movable Bed

0

5

10

15

20

0 10 20 30 40 50 60 70 80

Station (ft)

Elevation (ft)

Sta‐Elev Points Bankfull Water SurfaceEquivalent Channel Movable Bed

EquivalentTrapezoidal Channel

EquivalentTrapezoidal Channel

3939

Trapezoidal Channel Supply ReachTrapezoidal Channel Supply ReachTrapezoidal Channel Supply Reach

4040

Stable Channel Design FunctionStable Channel Design FunctionStable Channel Design Function

4141

ComputeComputeCompute

4242

Select Design Channel for b = 20 ftSelect Design Channel for b = 20 ftSelect Design Channel for b = 20 ft

20 FT20 FT

4343

Stability Curve, Width vs. SlopeStability Curve, Width vs. SlopeStability Curve, Width vs. Slope

181.42 ppm181.42 ppm

4444

Gravel/Cobble Beds

HEC-RAS Sediment Transport Capacity

Function

Gravel/Cobble BedsGravel/Cobble Beds

HECHEC--RAS Sediment RAS Sediment Transport Capacity Transport Capacity

FunctionFunction

4545

HR Sediment Transport Capacity (STC)HR Sediment Transport Capacity (STC)HR Sediment Transport Capacity (STC)

Grain size classes are input as grain size and percent finer.

Computes STC for each size class, gsi

Total STC is computed by the equationgs,total =∑ pigsi

where pi = fraction of size class i in the bed.

Can compute the total STC for all six Sediment Transport Potential functions.

Grain size classes are input as grain size Grain size classes are input as grain size and percent finer.and percent finer.

Computes STC for each size class, gComputes STC for each size class, gsisi

Total STC is computed by the equationTotal STC is computed by the equationggs,totals,total ==∑∑ ppiiggsisi

where pwhere pii = fraction of size class i in the bed.= fraction of size class i in the bed.

Can compute the total STC for all six Can compute the total STC for all six Sediment Transport Potential functions.Sediment Transport Potential functions.

4646

Gravel/Cobble ExampleGravel/Cobble ExampleGravel/Cobble ExampleThe stream has the following bank-full conditions

Water surface elevation = 11.7 feetDischarge = 3,100 cfs Slope = 0.0015.

The stream has the following bank-full conditions

Water surface elevation = 11.7 feetDischarge = 3,100 cfs Slope = 0.0015.

12027.78

0

10

20

30

0 20 40 60 80 100 120 140 160Station (ft)

Elev

atio

n (ft

)


12027.78

0

10

20

30

0 20 40 60 80 100 120 140 160Station (ft)

Elev

atio

n (ft

)


5252 9696

4747

Pebble Count for Gravel/Cobble Stream Pebble Count for Gravel/Cobble Stream Pebble Count for Gravel/Cobble Stream INCHES PARTICLE MILLIMETER SIZE CLASS COUNT % CUM % Dtop (mm)

Silt/Clay < 0.062 S/C 12 12 12Very Fine .062 - .125 S 7 7 19 0.125

Fine .125 - .25 A 2 2 21 0.25Medium .25 - .50 N 2 2 23 0.5Coarse .50 - 1.0 D 4 4 27 1

.04 - .08 Very Coarse 1.0 - 2 S 3 3 30 2.00

.08 - .16 Very Fine 2 - 4 12 12 42 4.00

.16 - .24 Fine 4 - 5.7 G 2 2 44 5.7

.24 - .31 Fine 5.7 - 8 R 3 3 47 8.00

.31 - .47 Medium 8 - 11.3 A 1 1 48 11.3

.47 - .63 Medium 11.3 - 16 V 0 0 48 16.00

.63 - .94 Coarse 16 - 22.6 E 2 2 50 22.6.94 - 1.26 Coarse 22.6 - 32 L 5 5 55 32.001.26 - 1.9 Very Coarse 32 - 45 S 7 7 62 45.001.9 - 2.5 Very Coarse 45 - 64 6 6 68 64.002.5 - 3.8 Small 64 - 90 C - 6 6 74 90.003.8 - 5.0 Small 90 - 128 O L 6 6 80 1285.0 - 7.6 Large 128 - 180 B E 6 6 86 1807.6 - 10 Large 180 - 256 B S 5 5 91 25610 - 15 Small 256 - 362 B D 1 1 92 36215 - 20 Small 362 - 512 O E 1 1 93 51220 - 40 Medium 512 - 1024 U R 0 0 93 102440 - 160 Lrg to Very Lrg 1024 - 2048 L S 0 0 93 2048

BEDROCK BDRK 7 7 100

NOTENOTE

4848

Log-probability plot of Bed Material LogLog--probability plot of Bed Material probability plot of Bed Material

Sand Gravel Cobble

4949

Make New HEC-RAS Model with one cross section and no discharge

Make New HECMake New HEC--RAS Model with one RAS Model with one cross section and no dischargecross section and no discharge

5050

Uniform Flow – Bed uses Limerinos, banks use Manning (T&E gives nbank = 0.077)

Uniform Flow Uniform Flow –– Bed uses Limerinos, banks Bed uses Limerinos, banks use Manning (T&E gives nuse Manning (T&E gives nbankbank = 0.077) = 0.077)

5151

Uniform Flow – Bed uses Mannings, Banks use n = 0.077 (T&E gives nbed = 0.0363)

Uniform Flow Uniform Flow –– Bed uses Mannings, Banks Bed uses Mannings, Banks use n = 0.077 (T&E gives nuse n = 0.077 (T&E gives nbedbed = 0.0363) = 0.0363)

5252

Create and Run Natural Supply ReachCreate and Run Natural Supply ReachCreate and Run Natural Supply Reach

5353

Sediment Transport Capacity Function Input Grain Sizes (Fake size for banks)

Sediment Transport Capacity Function Sediment Transport Capacity Function Input Grain Sizes (Fake size for banks)Input Grain Sizes (Fake size for banks)

Diam, mm % Finer Diam, mm % Finer Diam, mm % Finer2000 19 0.125 19 2000 192000 21 0.25 21 2000 212000 23 0.5 23 2000 232000 27 1 27 2000 272000 30 2 30 2000 302000 42 4 42 2000 422000 44 5.7 44 2000 442000 47 8 47 2000 472000 48 11.3 48 2000 482000 48 16 48 2000 482000 50 22.6 50 2000 502000 55 32 55 2000 552000 62 45 62 2000 622000 68 64 68 2000 682000 74 90 74 2000 742000 80 128 80 2000 802000 86 180 86 2000 862000 91 256 91 2000 912000 92 362 92 2000 922000 93 512 93 2000 932000 93 1024 93 2000 932000 93 2048 93 2000 93

ROBLOB Main

5454

Compute, Sediment Rating Curve Plot, Generate Report

Compute, Sediment Rating Curve Plot, Compute, Sediment Rating Curve Plot, Generate ReportGenerate Report

5555

Select All Grains Sizes to see a more detailed Report

Select All Grains Sizes to see a more Select All Grains Sizes to see a more detailed Reportdetailed Report

A-W E-H Laur MPM Toff YangClass dm (mm) Left Main Right (tons/day) (tons/day) (tons/day) (tons/day) (tons/day) (tons/day)

All Grains 836000 7991 645200 1083 680.4 272901 0.003 0 0 02 0.006 0 0 03 0.011 0 0 04 0.023 0 0 05 0.045 0 0 06 0.088 0 0.19 0 834600 631600 648.4 254707 0.177 0 0.02 0 985.9 2096 7930 21.51 646.18 0.354 0 0.02 0 152.2 1096 1811 92.51 6.911 313.89 0.707 0 0.04 0 113.9 1195 1443 181.3 2.592 485.9

10 1.41 0 0.03 0 33.98 544.1 548 130.3 0.3168 355.211 2.83 0 0.12 0 37.11 1445 1218 477.3 0.2369 3.44612 5.64 0 0.05 0 8.899 384.8 372.1 163.9 0.02274 2.5613 11.3 0 0.01 0 0.2901 49.26 50.42 20.28 .00586... 0.84614 22.6 0 0.07 0 0 220 185 17.05 0.05728 7.79215 45.1 0 0.13 0 0 262.1 128.6 0 0.1555 5.68116 90.5 0 0.12 0 0 154.1 0 0 0.1482 017 181 0 0.11 0 0 90.2 0 0 0 018 362 0 0.02 0 0 10.58 0 0 .00010... 019 724 0 0 0 0 0.1343 0 0 0 020 1448 1 0.07 1 0 443.2 0 0 0 0

Bed Material Fraction by Standard Grade Size A-W E-H Laur MPM Toff YangClass dm (mm) Left Main Right (tons/day) (tons/day) (tons/day) (tons/day) (tons/day) (tons/day)

All Grains 836000 7991 645200 1083 680.4 272901 0.003 0 0 02 0.006 0 0 03 0.011 0 0 04 0.023 0 0 05 0.045 0 0 06 0.088 0 0.19 0 834600 631600 648.4 254707 0.177 0 0.02 0 985.9 2096 7930 21.51 646.18 0.354 0 0.02 0 152.2 1096 1811 92.51 6.911 313.89 0.707 0 0.04 0 113.9 1195 1443 181.3 2.592 485.9

10 1.41 0 0.03 0 33.98 544.1 548 130.3 0.3168 355.211 2.83 0 0.12 0 37.11 1445 1218 477.3 0.2369 3.44612 5.64 0 0.05 0 8.899 384.8 372.1 163.9 0.02274 2.5613 11.3 0 0.01 0 0.2901 49.26 50.42 20.28 .00586... 0.84614 22.6 0 0.07 0 0 220 185 17.05 0.05728 7.79215 45.1 0 0.13 0 0 262.1 128.6 0 0.1555 5.68116 90.5 0 0.12 0 0 154.1 0 0 0.1482 017 181 0 0.11 0 0 90.2 0 0 0 018 362 0 0.02 0 0 10.58 0 0 .00010... 019 724 0 0 0 0 0.1343 0 0 0 020 1448 1 0.07 1 0 443.2 0 0 0 0


All Grains 836000 7991 645200 1083 680.4 272901 0.003 0 0 02 0.006 0 0 03 0.011 0 0 04 0.023 0 0 05 0.045 0 0 06 0.088 0 0.19 0 834600 631600 648.4 254707 0.177 0 0.02 0 985.9 2096 7930 21.51 646.18 0.354 0 0.02 0 152.2 1096 1811 92.51 6.911 313.89 0.707 0 0.04 0 113.9 1195 1443 181.3 2.592 485.9

10 1.41 0 0.03 0 33.98 544.1 548 130.3 0.3168 355.211 2.83 0 0.12 0 37.11 1445 1218 477.3 0.2369 3.44612 5.64 0 0.05 0 8.899 384.8 372.1 163.9 0.02274 2.5613 11.3 0 0.01 0 0.2901 49.26 50.42 20.28 .00586... 0.84614 22.6 0 0.07 0 0 220 185 17.05 0.05728 7.79215 45.1 0 0.13 0 0 262.1 128.6 0 0.1555 5.68116 90.5 0 0.12 0 0 154.1 0 0 0.1482 017 181 0 0.11 0 0 90.2 0 0 0 018 362 0 0.02 0 0 10.58 0 0 .00010... 019 724 0 0 0 0 0.1343 0 0 0 020 1448 1 0.07 1 0 443.2 0 0 0 0


All Grains 836000 7991 645200 1083 680.4 272901 0.003 0 0 02 0.006 0 0 03 0.011 0 0 04 0.023 0 0 05 0.045 0 0 06 0.088 0 0.19 0 834600 631600 648.4 254707 0.177 0 0.02 0 985.9 2096 7930 21.51 646.18 0.354 0 0.02 0 152.2 1096 1811 92.51 6.911 313.89 0.707 0 0.04 0 113.9 1195 1443 181.3 2.592 485.9

10 1.41 0 0.03 0 33.98 544.1 548 130.3 0.3168 355.211 2.83 0 0.12 0 37.11 1445 1218 477.3 0.2369 3.44612 5.64 0 0.05 0 8.899 384.8 372.1 163.9 0.02274 2.5613 11.3 0 0.01 0 0.2901 49.26 50.42 20.28 .00586... 0.84614 22.6 0 0.07 0 0 220 185 17.05 0.05728 7.79215 45.1 0 0.13 0 0 262.1 128.6 0 0.1555 5.68116 90.5 0 0.12 0 0 154.1 0 0 0.1482 017 181 0 0.11 0 0 90.2 0 0 0 018 362 0 0.02 0 0 10.58 0 0 .00010... 019 724 0 0 0 0 0.1343 0 0 0 020 1448 1 0.07 1 0 443.2 0 0 0 0

Bed Material Fraction by Standard Grade Size

10831083

5656

Meyer-Peter Mueller Function ResultsMeyerMeyer--Peter Mueller Function ResultsPeter Mueller Function Results

1010

77

5757

Design 1 (b = 35 ft, m = 3:1 hor: vert)Design 1 (b = 35 ft, m = 3:1 hor: vert)Design 1 (b = 35 ft, m = 3:1 hor: vert)

Assume slopeCreate 3 cross section model with trapezoidal xsecs with same n’s and Q as supply reachRun steady flow modelRun Sediment Transport Capacity functionSee if STC equals 1083 tons/day – if not back to the top with a new slope

Assume slopeAssume slopeCreate 3 cross section model with Create 3 cross section model with trapezoidal xsecs with same ntrapezoidal xsecs with same n’’s and Q as s and Q as supply reachsupply reachRun steady flow modelRun steady flow modelRun Sediment Transport Capacity functionRun Sediment Transport Capacity functionSee if STC equals 1083 tons/day See if STC equals 1083 tons/day –– if not back if not back to the top with a new slopeto the top with a new slope

5858

Design 1 (b = 35 ft, m = 3:1 hor: vert)Design 1 (b = 35 ft, m = 3:1 hor: vert)Design 1 (b = 35 ft, m = 3:1 hor: vert)

S = 0.003 S = 0.0017b = 35 Natural Design b = 35 Natural Design

Function FunctionA-W 836000 410900... NA A-W 836000 113600... NAE-H 8217 26640 0.31 E-H 8217 9883 0.83Laur 645200 212400... NA Laur 645200 750300 0.86MPM 1083 2427 0.45 MPM 1083 1151 0.94Toff 680.4 627 1.09 Toff 680.4 586.7 1.16

Yang 27290 88360 0.31 Yang 27290 32300 0.84


Function FunctionA-W 836000 992100 0.84 A-W 836000 102100... NAE-H 8217 8908 0.92 E-H 8217 9105 0.90Laur 645200 673200 0.96 Laur 645200 688800 0.94MPM 1083 1063 1.02 MPM 1083 1081 1.00Toff 680.4 582.9 1.17 Toff 680.4 583.7 1.17

Yang 27290 29000 0.94 Yang 27290 29670 0.92

Nat/Des

Nat/Destons/day

Nat/Des

tons/day

tons/day

Nat/Destons/day


Function FunctionA-W 836000 410900... NA A-W 836000 113600... NAE-H 8217 26640 0.31 E-H 8217 9883 0.83Laur 645200 212400... NA Laur 645200 750300 0.86MPM 1083 2427 0.45 MPM 1083 1151 0.94Toff 680.4 627 1.09 Toff 680.4 586.7 1.16

Yang 27290 88360 0.31 Yang 27290 32300 0.84


Function FunctionA-W 836000 992100 0.84 A-W 836000 102100... NAE-H 8217 8908 0.92 E-H 8217 9105 0.90Laur 645200 673200 0.96 Laur 645200 688800 0.94MPM 1083 1063 1.02 MPM 1083 1081 1.00Toff 680.4 582.9 1.17 Toff 680.4 583.7 1.17

Yang 27290 29000 0.94 Yang 27290 29670 0.92

Nat/Des

Nat/Destons/day

Nat/Des

tons/day

tons/day

Nat/Destons/day

T & E gives S = 0.00162T & E gives S = 0.00162

5959

Final DesignFinal DesignFinal Design

6060

Design 2 - Select slope and side slopes, find bDesign 2Design 2 -- Select slope and side Select slope and side slopes, find bslopes, find b

b = 15bnk ht = 15.62 S = 0.0018 m = 2.5 Function All Grains

RS 0 RS 100 RS 200 A-W 192600...Station Elevation Station Elevation Station Elevation E-H 12280-46.550 15.620 -46.550 15.800 -46.550 15.980 Laur 901500-7.500 0.000 -7.500 0.180 -7.500 0.360 MPM 10917.500 0.000 7.500 0.180 7.500 0.360 Toff 438.7

46.550 15.620 46.550 15.800 46.550 15.980 Yang 37320

STC in Tons/Dayb = 15bnk ht = 15.62 S = 0.0018 m = 2.5 Function All Grains

RS 0 RS 100 RS 200 A-W 192600...Station Elevation Station Elevation Station Elevation E-H 12280-46.550 15.620 -46.550 15.800 -46.550 15.980 Laur 901500-7.500 0.000 -7.500 0.180 -7.500 0.360 MPM 10917.500 0.000 7.500 0.180 7.500 0.360 Toff 438.7

46.550 15.620 46.550 15.800 46.550 15.980 Yang 37320

STC in Tons/Day

S = 0.0018, m = 2.5:1 hor: vertS = 0.0018, m = 2.5:1 hor: vertS = 0.0018, m = 2.5:1 hor: vert

6161

S = 0.0018, m = 2.5:1 hor: vertS = 0.0018, m = 2.5:1 hor: vertS = 0.0018, m = 2.5:1 hor: vert

Design Channel, b = 15', hor:vert = 2.5:1, S = 0.0018

0

4

8

12

16

-60 -40 -20 0 20 40 60

Station (ft)

Elev

atio

n (ft

)

Design Channel Bankfull Elevation Movable Bed

Design Channel, b = 15', hor:vert = 2.5:1, S = 0.0018

0

4

8

12

16

-60 -40 -20 0 20 40 60

Station (ft)

Elev

atio

n (ft

)


b = 15 ft, bank ht = 15.62 ftb = 15 ft, bank ht = 15.62 ft

6262

Design 3 - S = 0.0013, m = 2.5:1 hor: vertDesign 3 Design 3 -- S = 0.0013, m = 2.5:1 hor: vertS = 0.0013, m = 2.5:1 hor: vert

0

5

10

-60 -40 -20 0 20 40 60Station (ft)

Ele

vatio

n (ft

)


0

5

10

-60 -40 -20 0 20 40 60Station (ft)

Ele

vatio

n (ft

)


b = 76 ft, bank ht = 8.73 ftb = 76 ft, bank ht = 8.73 ft

4 b = 76.0 1082bnk ht = 8.73 S = 0.0013 m = 2.5 Tons/day

RS 0 RS 100 RS 200 Function All GrainsStation Elevation Station Elevation Station Elevation A-W 388200-59.825 8.730 -59.825 8.860 -59.825 8.990 E-H 4981

-38 0.000 -38 0.130 -38 0.260 Laur 54540038 0.000 38 0.130 38 0.260 MPM 1082

59.825 8.730 59.825 8.860 59.825 8.990 Toff 1078Yang 19240

Final Design 2MPM Gs (Tons/day) =4 b = 76.0 1082bnk ht = 8.73 S = 0.0013 m = 2.5 Tons/day

RS 0 RS 100 RS 200 Function All GrainsStation Elevation Station Elevation Station Elevation A-W 388200-59.825 8.730 -59.825 8.860 -59.825 8.990 E-H 4981

-38 0.000 -38 0.130 -38 0.260 Laur 54540038 0.000 38 0.130 38 0.260 MPM 1082

59.825 8.730 59.825 8.860 59.825 8.990 Toff 1078Yang 19240

Final Design 2MPM Gs (Tons/day) = Final Design 3

6363

Sidewall Correction MethodSidewall Correction MethodSidewall Correction Method

bd

yd

md

1md

1 1md

1md

nwnw

nb

Wd

} 4 /32 /3 1/ 2 2

2 4 /3

3/ 4 3/ 43/ 42 4 /3 2 2

2 4 /3 2 3/ 2

1 1'

1 1 1

square

co

AMetric version of Manning s Equation V R S V Sn n P

Einstein assumed V and S are constant in bank area and sidewall area

V A V A A VS n P S n P n P S

= → =

⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞= → = → =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠

( ) ( ) ( )

3/ 2 3/ 2 3/ 2

3/ 2 3/ 2 3/ 2 3/ 2 3/ 2 3/ 2

2 /3 3/ 2 3/ 23/ 2 3/ 2

3/ 2 3/ 2

1

nstant

Constantw b

w bw b

w bw b w b w w b b

w w b bw w b b w b

P PPn n nA A A

P PPA A A n n n n P n P n P

n P n Pn n P n P also A A and A AP n P n P

β

β β β

= = =

= + → = + → = +

⎡ ⎤= + → = =⎢ ⎥⎣ ⎦

64748

Aw/2Aw/2 AbAbAw/2Aw/2

6464

Sidewall ExampleSidewall ExampleSidewall Example

50’

5’

n = 0.025n = 0.045 n = 0.045

12

12

S = 0.0002

nw = 0.040 nw = 0.040nb = 0.025

( ) ( )

2 2

2 2 2

2/3 2 /33/ 2 3/ 2 3/ 2 3/ 2

50 ; 2 5 10 22.4 72.4(5)(10)50*5 250 ; 2 50 300

2

1 1 (0.040) 22.4 (0.025) 5072.4

0.03001.49 (300

0.0300

b w b w

rect tria rect tria

w w b b

P ft P ft P P P ft

A ft A ft A A A ft

n n P n PP

n

Q

= = + = → = + =

⎡ ⎤= = = = → = + =⎢ ⎥⎣ ⎦

⎡ ⎤ ⎡ ⎤= + = +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦=

=

( ) ( )

5/33

2 /3

3/ 2 3/ 22 2

3/ 2 3/ 2

2 2

) 0.0002 543 /(72.4)

(0.040) (0.025)22.4 50300 143 300 157(0.030) 72.4 (0.030) 72.4

50 250

w b

w b

ft s

A ft and A ft

Geometric values A ft and A ft

=

= = = =

= =

6565

ARR Analytical MethodARR Analytical MethodARR Analytical Method

( )

( )( )

5/3

2 /3

3/ 250

3/2

50

50

50

1.49 . 4 2

8 0.047 . 4 6

0.047 1. 4 8

0.047 1

, , , , 2.65, , .

dd d

d d

s

m smm d d m

d d s

d d d m d s m m

AManning for Design Channel Q S ARR Eqn P

bMPM B yS d ARR Eq

y S G dB B b b ARR Eq

y S G d

Given Q m n S S G d b and y

γ γ γρ

→ = −

→ = − − −⎡ ⎤⎣ ⎦

⎡ ⎤− −= → = −⎢ ⎥

− −⎢ ⎥⎣ ⎦= =

. 4 2 4 8, .

d dUse iteration to solve Eqs and for b and y by iterationSubscripts d and m denote design channel and match reach channels respectively

− −

bd

yd

md

1md

1 1md

1md

nwnw

nb

Wd

nd = Manning’s composite n

6666

ARR vs HEC-RASComposite n – 0.061 from HEC-RAS Supply Reach

ARR vs HECARR vs HEC--RASRASComposite n Composite n –– 0.061 from HEC0.061 from HEC--RAS Supply ReachRAS Supply Reach) ARR solution for Design 1 ( HR solution bd = 35 ft) dm (mm)

md = 3 Sd = 0.001622 nd = 0.061 22.6yd bd Ad Pd Wb Qb ΔQ

10.96 42.9 829.9 112.2 108.6 3100 0.000

) ARR solution for Design 2 ( HR solution bd = 15 ft) dm (mm)md = 2.5 Sd = 0.0018 nd = 0.061 22.6

yd bd Ad Pd Wb Qb ΔQ13.38 22.8 752.1 94.8 89.7 3100 0.000

) ARR solution for Design 3 ( HR solution bd = 76 ft) dm (mm)

md = 2.5 Sd = 0.0013 nd = 0.061 22.6yd bd Ad Pd Wb Qb ΔQ

11.91 58.8 1055.6 123.0 118.4 3897.35 797.350

) ARR solution for Design 1 ( HR solution bd = 35 ft) dm (mm)md = 3 Sd = 0.001622 nd = 0.061 22.6

yd bd Ad Pd Wb Qb ΔQ10.96 42.9 829.9 112.2 108.6 3100 0.000


yd bd Ad Pd Wb Qb ΔQ13.38 22.8 752.1 94.8 89.7 3100 0.000



11.91 58.8 1055.6 123.0 118.4 3897.35 797.350


yd bd Ad Pd Wb Qb ΔQ10.96 42.9 829.9 112.2 108.6 3100 0.000


yd bd Ad Pd Wb Qb ΔQ13.38 22.8 752.1 94.8 89.7 3100 0.000



11.91 58.8 1055.6 123.0 118.4 3897.35 797.350


yd bd Ad Pd Wb Qb ΔQ10.96 42.9 829.9 112.2 108.6 3100 0.000


yd bd Ad Pd Wb Qb ΔQ13.38 22.8 752.1 94.8 89.7 3100 0.000



11.91 58.8 1055.6 123.0 118.4 3897.35 797.350

bHR = 35 ftbARR= 42.9 ftbHR/bARR= 0.82




ARR did not convergebHR = 76 ftbARR= 58.8 ftbHR/bARR= 1.29

ARR did not convergebHR = 76 ftbARR= 58.8 ftbHR/bARR= 1.29

6767

ARR vs HEC-RAS Composite n values from HR Design Reaches

ARR vs HECARR vs HEC--RAS RAS Composite n values from HR Design ReachesComposite n values from HR Design Reaches

(b) ARR solution for Design 1 ( HR solution bd = 35 ft)md = 3 Sd = 0.00162 nd = 0.066 dm (mm) = 22.6

yd bd Ad Pd Wb Qb ΔQ12.35 33.2 867.7 111.3 107.3 3100 0.000

(c) ARR solution for Design 2 ( HR solution bd = 15 ft)md = 2.5 Sd = 0.0018 nd = 0.072 dm (mm) = 22.6

yd bd Ad Pd Wb Qb ΔQ15.19 17.8 847.3 99.6 93.8 3100 0.000

(d) ARR solution for Design 3 ( HR solution bd = 76 ft)md = 2.5 Sd = 0.0013 nd = 0.054 dm (mm) = 22.6

yd bd Ad Pd Wb Qb ΔQ11.91 58.8 1055.6 123.0 118.4 4402.562 1302.562

(b) ARR solution for Design 1 ( HR solution bd = 35 ft)md = 3 Sd = 0.00162 nd = 0.066 dm (mm) = 22.6

yd bd Ad Pd Wb Qb ΔQ12.35 33.2 867.7 111.3 107.3 3100 0.000

(c) ARR solution for Design 2 ( HR solution bd = 15 ft)md = 2.5 Sd = 0.0018 nd = 0.072 dm (mm) = 22.6

yd bd Ad Pd Wb Qb ΔQ15.19 17.8 847.3 99.6 93.8 3100 0.000

(d) ARR solution for Design 3 ( HR solution bd = 76 ft)md = 2.5 Sd = 0.0013 nd = 0.054 dm (mm) = 22.6

yd bd Ad Pd Wb Qb ΔQ11.91 58.8 1055.6 123.0 118.4 4402.562 1302.562

n = 0.066bHR = 35 ftbARR= 33.2 ftbHR/bARR= 1.05




ARR did not convergen = 0.054bHR = 76 ftbARR= 58.8 ftbHR/bARR= 1.29

ARR did not convergen = 0.054bHR = 76 ftbARR= 58.8 ftbHR/bARR= 1.29

6868

ARR’s Simplified MPM EquationARRARR’’s Simplified MPM Equations Simplified MPM Equation

( ) 3/ 23/ 2

'

'

( )8 ( ) 0.047b s m

b

b

b

b

Meyer Peter Mueller MPMbB RKR R S d

nRKR Nikuradse roughness ration

n Manning coefficient for grain sizen total Manning coefficient

γ γ γρ

−

⎡ ⎤= − −⎣ ⎦

⎛ ⎞= =⎜ ⎟⎝ ⎠

==

}( )

}503/ 21

3/ 28 ( ) 0.047dy

b s m

ARR simplified MPM equation

bB RKR R S dγ γ γρ

⎡ ⎤⎢ ⎥= − −⎢ ⎥⎣ ⎦

64748

6969

ConclusionsConclusionsConclusionsMcEnroe’s KAM and ARR methods provide very useful tools and should serve as references for all stable channel design projects.

It is recommended that HEC-RAS be used in lieu of the analytical approaches of KAM and ARR if the grain size distribution is known for an alluvial channel.

If the grain size distribution is unknown or if the channel has a cohesive bed, the KAM and ARR methods should be used in their entirety.

The HEC-RAS methods reported herein only provide for design of the riffle cross section and do not include help for planform design aspects. The methods outlined in McEnroe’s reports should be used for the overall stream design. They provide guidance for the design of meanders, riffle pool spacing and pool dimensions.

McEnroeMcEnroe’’s KAM and ARR methods provide very useful s KAM and ARR methods provide very useful tools and should serve as references for all stable tools and should serve as references for all stable channel design projects.channel design projects.

It is recommended that HECIt is recommended that HEC--RAS be used in lieu of the RAS be used in lieu of the analytical approaches of KAM and ARR if the grain size analytical approaches of KAM and ARR if the grain size distribution is known for an alluvial channel. distribution is known for an alluvial channel.

If the grain size distribution is unknown or if the channel If the grain size distribution is unknown or if the channel has a cohesive bed, the KAM and ARR methods should has a cohesive bed, the KAM and ARR methods should be used in their entirety.be used in their entirety.

The HECThe HEC--RAS methods reported herein only provide for RAS methods reported herein only provide for design of the riffle cross section and do not include help design of the riffle cross section and do not include help for planform design aspects. The methods outlined in for planform design aspects. The methods outlined in McEnroeMcEnroe’’s reports should be used for the overall stream s reports should be used for the overall stream design. They provide guidance for the design of design. They provide guidance for the design of meanders, riffle pool spacing and pool dimensions.meanders, riffle pool spacing and pool dimensions.

7070

Analysis of Bed Grain Size Distribution

Analysis of Bed Grain Size Analysis of Bed Grain Size DistributionDistribution

Sieve AnalysisVisual-Accumulation TubePebble Count

Sieve AnalysisSieve AnalysisVisualVisual--Accumulation TubeAccumulation TubePebble CountPebble Count

7171

Log-normal DistributionLogLog--normal Distributionnormal Distribution

84.1

log 15.9

50

loglog

log84.1 84

15.9 16

log

log log1 1(log ) exp22

10 10

(log ) (log ) (log ) /100

d

dd

dd

g

d

Probability Density Function PDF

d df d

Cumulative Distribution Function CDF

d dd d

F d f d d d P

Standa

σ

σσ π

σ

−∞

⎡ ⎤⎛ ⎞−= ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

= = = ≈

= =∫

84.1

log 15.9

84.1 84.1log 84.1 15.9

15.9 15.9

log84.1 84

15.9 16

1 1(log log ) log log2 2

10 10d

d

dd

g

rdDeviation

d dd dd d

Geometric standard deveiation

d dd d

σ

σ

σ

⎛ ⎞= − = =⎜ ⎟

⎝ ⎠

= = = ≈

7272

Standardized Random VariableMean = 0, standard deviation =1

Standardized Random VariableStandardized Random VariableMean = 0, standard deviation =1Mean = 0, standard deviation =1

250

log

log log 1( ) exp22

( ) ( ) ( ) 1 ( )

d

z

d d zz PDF f z

CDF F z f z dz where F z F z

σ π

−∞

⎡ ⎤−= → → = ⎢ ⎥

⎣ ⎦

→ = − = −∫

F(z) for Standard Normal Random Variable zF(z) for Standard Normal Random Variable zz 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359

0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.57530.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.61410.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.65170.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.68790.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.72240.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.75490.7 0.7580 0.7611 0.7642 0.7673 0.7703 0.7734 0.7764 0.7793 0.7823 0.78520.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.81330.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.83891 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621

1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.88301.2 0.8849 0.8869 0.8888 0.8906 0.8925 0.8943 0.8962 0.8980 0.8997 0.90151.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.91771.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.93191 5 0 9332 0 9345 0 9357 0 9370 0 9382 0 9394 0 9406 0 9418 0 9429 0 9441

z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359

0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.57530.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.61410.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.65170.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.68790.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.72240.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.75490.7 0.7580 0.7611 0.7642 0.7673 0.7703 0.7734 0.7764 0.7793 0.7823 0.78520.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.81330.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.83891 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621

1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.88301.2 0.8849 0.8869 0.8888 0.8906 0.8925 0.8943 0.8962 0.8980 0.8997 0.90151.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.91771.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.93191 5 0 9332 0 9345 0 9357 0 9370 0 9382 0 9394 0 9406 0 9418 0 9429 0 9441

7373

Example - Sand Bed MaterialExample Example -- Sand Bed MaterialSand Bed Material

7474

Log-Probability Plot of Sand Bed DataLogLog--Probability Plot of Sand Bed DataProbability Plot of Sand Bed Data

d (m

m)

Cumulative Distribution Function Expressed as a probability (%)

d84=0.363 mm

d50=0.232 mm

d16=0.158 mm

( ) ( )log 50

84.1log

15.9

0.181

0.386 log 0.386 0.181 log 0.232 .56565

0.363log log 0.1810.158

10 1.52

10 10 10 0.272d

d

g

d

dd

d mmσ

σ

σ+ + −

= = =

= =

= = = =

7575

Log Probability Plot using NORMSINV Function in EXCEL

Log Probability Plot using NORMSINV Log Probability Plot using NORMSINV Function in EXCELFunction in EXCEL

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

-4 -3 -2 -1 0 1 2 3

log 1

0(d

)

d15.9

NORMSINV(P/100)

d50 d84.1

log(d84)log(d84))

log(d50)log(d50))

log(d16)log(d16))

7676

NORMSINV and NORMSDIST EXCEL Functions

NORMSINV and NORMSDIST NORMSINV and NORMSDIST EXCEL FunctionsEXCEL Functions

d50 = 0.48 mm log10(d50) = -0.3188σg = 1.28 mm log10(σg) = 0.1072

P (% finer) F(z) z=NORMSINV(F) di (mm) F(z)=NORMSDIST(z)1 2 3 4 5 6

20 0.2 -0.8416 d20 = 0.390 0.240 0.4 -0.2533 d40 = 0.451 0.460 0.6 0.2533 d60 = 0.511 0.680 0.8 0.8416 d80 = 0.591 0.8

99.99 0.9999 3.7190 d99.99 = 1.202 0.999950 0.5 0.0000 d50 = 0.480 0.5

di (mm) z F(z)=NORMSDIST(z) P (% finer)1 2 3 4

0.41 -0.6385 0.2616 26.20.63 1.1016 0.8647 86.5

50

50

( ) ( )

( ) ( )( )

10 i g

ii

g

log d z logi

log d log dzlog

d σ

σ+

−=

=

50( ) ( )( )

ii

g

log d log dzlog σ

−=

d50 = 0.48 mm log10(d50) = -0.3188σg = 1.28 mm log10(σg) = 0.1072

P (% finer) F(z) z=NORMSINV(F) di (mm) F(z)=NORMSDIST(z)1 2 3 4 5 6

20 0.2 -0.8416 d20 = 0.390 0.240 0.4 -0.2533 d40 = 0.451 0.460 0.6 0.2533 d60 = 0.511 0.680 0.8 0.8416 d80 = 0.591 0.8

99.99 0.9999 3.7190 d99.99 = 1.202 0.999950 0.5 0.0000 d50 = 0.480 0.5

di (mm) z F(z)=NORMSDIST(z) P (% finer)1 2 3 4

0.41 -0.6385 0.2616 26.20.63 1.1016 0.8647 86.5

50

50

( ) ( )

( ) ( )( )

10 i g

ii

g

log d z logi

log d log dzlog

d σ

σ+

−=

=

50( ) ( )( )

ii

g

log d log dzlog σ

−=

7777

Geometric Standard DeviationGeometric Standard DeviationGeometric Standard Deviation

( ) ( )( ) ( ) ( ) ( )( )

( ) ( )

( )

84 50 50 16

84 50 50 16 84 50 16 50

84 16 84 16 84 16

84 16

84 50 84 50

2 log log log log log

2log log / log / log /

12log log / log log / log /2

/

log log log /

logtop top

d d d d

d d d d d d d d

d d d d d d

d d

Alternative Method in HEC RAS Manuald d d d

σ

σ

σ σ

σ

σ σ

= − + −

= + =

= → = =

=

−

= − → =

( )

( )

50 16 50 16

84 50

50 16

log log /

0.5 0.5

bot bot

ave top bot

d d d d

d dd d

σ σ

σ σ σ σ

= − → =

⎛ ⎞= = + = +⎜ ⎟

⎝ ⎠

7878

Geometric Standard Deviation when Pi for smallest sample d is greater than 0.16

Geometric Standard Deviation when Geometric Standard Deviation when PPii for smallest sample d is greater than 0.16for smallest sample d is greater than 0.16

( )

( )

1

1 1

1

1

84

84 1(1 )

84 84

1(1 )

84

(1 ) log log log

loglog log

log log(1 ) (1 )

P

zP P

P

z

P

z d d

dd d d d

z z d

dd

σ

σ

σ

−

−

− = −

⎛ ⎞⎜ ⎟⎜ ⎟− ⎛ ⎞⎝ ⎠= = = ⎜ ⎟⎜ ⎟− − ⎝ ⎠

⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠

Let Pi = the lowest percent finer from the pebble count analysis and let zi = the standardized normal variable that gives F(z) = Pi/100.

Let Pi = the lowest percent finer from the pebble count analysis and let zi = the standardized normal variable that gives F(z) = Pi/100.

7979

Example for Smallest d > d16Example for Smallest d > dExample for Smallest d > d1616

(

( )

1

84 32

1

1 1 1(1 ) (1 0.469 ) 1.469

84

8450 84

8450

8 232 ( ) 0.32 ( ) 1 ( ) 1 0.32 0.680.469 0.469

5.7 5.7 2.042 2

log log log 2.04 log2.04

2.

z

P

Given d mm and d mmP F z F z F z

z z

dd

dd d

dd

σ− − −

= =

= → = → − = − = − =

− = → = −

⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠ ⎝ ⎠⎝ ⎠⎛ ⎞= − = ⎜ ⎟⎝ ⎠

=5.7 2.79

04 2.04mm⎛ ⎞ ⎛ ⎞= =⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

8080

Pebble Count for Gravel/Cobble Stream Pebble Count for Gravel/Cobble Stream Pebble Count for Gravel/Cobble Stream

Sand Gravel Cobble

8181

Theoretical Justification for Pebble Count

Theoretical Justification for Pebble Theoretical Justification for Pebble CountCount

0

, , ,

/ ; ; ( )

(1 )

/

(1 )

L

pores total pores pores pores

s

i

i i s

i s i i ii

s total s s total s total total

i

porosity V V A nA V A x dx

A n A area of A occupied by soilA area of A occupied by particles of a specified size rangef A A

W V V VpW V V n V

p

γγ

= = =

= − ===

= = = =−

∫

( ) [ ]( )

[ ]( )

0 0 0 0

0 0

(1 )

(1 ) (1 ) (1 ) (1 )

(1 ) (1 )(1 )

(1 ) (1 ) (1 )

L L L L

i i i s i

totalL L

i ii

i

i i

Adx Adx f A dx p n A dx

n V n AL n AL n AL

f n A dx f n Adxf n ALp

n AL n AL n ALp f

−= = = =

− − − −

− −−

= = =− − −

=

∫ ∫ ∫ ∫

∫ ∫

L

SOIL PARTICLES

PORES

A

Actual Pebble Count – Shielding, settling, etc.Actual Pebble Count – Shielding, settling, etc.

8282

Mixed Sand and Gravel BedsWatershed Institute, Inc. Pebble Count Data

Mixed Sand and Gravel BedsMixed Sand and Gravel BedsWatershed Institute, Inc. Pebble Count DataWatershed Institute, Inc. Pebble Count Data

MT043442RR01

D16 D50 D84

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

normsinv(P/100)

log1

0(D)

Pebble Count Data D16 D50 D84 Extrapolated

D (mm)D16 0.0763D50 0.344D84 14.8

d (mm)

d16 0.0763

d50 0.344d84 14.8

d16 d50 d84MT043442RR01

D16 D50 D84

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

normsinv(P/100)

log1

0(D)


D (mm)D16 0.0763D50 0.344D84 14.8

d (mm)

d16 0.0763

d50 0.344d84 14.8

d16 d50 d84

SN321115RR

D16 D50 D84

-3

-2

-1

0

1

2

3

-1.5 -1 -0.5 0 0.5 1 1.5 2

normsinv(P/100)

log1

0(D)


D (mm)D16 0.00224D50 0.500D84 20.6

d (mm)

d16 0.00224

d50 0.5d84 20.6

d16 d50 d84SN321115RR

D16 D50 D84

-3

-2

-1

0

1

2

3

-1.5 -1 -0.5 0 0.5 1 1.5 2

normsinv(P/100)

log1

0(D)


D (mm)D16 0.00224D50 0.500D84 20.6

d (mm)

d16 0.00224

d50 0.5d84 20.6

d16 d50 d84

8383

Mixed Sand and Gravel Beds (cont.)Mixed Sand and Gravel Beds (cont.)Mixed Sand and Gravel Beds (cont.)Two Bed Materials

0

10

20

30

40

50

60

-1.5 -1 -0.5 0 0.5 1 1.5 2

log D

W fi

ner (

gm)

Gravel Sand

Sand GravelD50 0.4 10σg 1.5 2

W (gm) 40 50

Sand Gravel

d50 0.4 10

σg 1.5 2W (gm) 40 50

log10(d)

Two Bed Materials

0

10

20

30

40

50

60

-1.5 -1 -0.5 0 0.5 1 1.5 2

log D

W fi

ner (

gm)

Gravel Sand


W (gm) 40 50

Sand Graveld50 0.4 10

σg 1.5 2W (gm) 40 50

log10(d)

Two Bed Materials

-2-1.5

-1-0.5

00.5

1

1.52

2.53

-10 -5 0 5 10norminv(P/100)

log

D

Sand Gravel


W (gm) 40 50

Sand Gravel

d50 0.4 10

σg 1.5 2W (gm) 40 50

log 1

0(d)

NORMINV(P/100)

Two Bed Materials

-2-1.5

-1-0.5

00.5

1

1.52

2.53

-10 -5 0 5 10norminv(P/100)

log

D

Sand Gravel


W (gm) 40 50

Sand Gravel

d50 0.4 10

σg 1.5 2W (gm) 40 50

log 1

0(d)

NORMINV(P/100)

8484

Mixed Sand and Gravel Beds (cont.)Mixed Sand and Gravel Beds (cont.)Mixed Sand and Gravel Beds (cont.)Combined

0

10

20

30

40

50

60

70

80

90

100

-1.5 -1 -0.5 0 0.5 1 1.5 2

log D

W F

iner

(gm

)


W (gm) 40 50

log10(d)

Sand Gravel

d50 0.4 10

σg 1.5 2W (gm) 40 50

Combined

0

10

20

30

40

50

60

70

80

90

100

-1.5 -1 -0.5 0 0.5 1 1.5 2

log D

W F

iner

(gm

)


W (gm) 40 50

log10(d)

Sand Gravel

d50 0.4 10

σg 1.5 2W (gm) 40 50

Combined

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-8 -6 -4 -2 0 2 4

norminv(P/100)

log

D

log D50 D50 (mm)0.537 3.44


W (gm) 40 50

Sand Gravel

d50 0.4 10

σg 1.5 2W (gm) 40 50

log 1

0(d)

log10(d50) d50 (mm)

0.537 3.44

Combined

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-8 -6 -4 -2 0 2 4

norminv(P/100)

log

D

log D50 D50 (mm)0.537 3.44


W (gm) 40 50

Sand Gravel

d50 0.4 10

σg 1.5 2W (gm) 40 50

log 1

0(d)

log10(d50) d50 (mm)

0.537 3.44

Documents

A STUDY OF FLUVIAL GEOMORPHOLOGY ASPECTS OF HYDRAULIC DESIGNiri.ku.edu/.../files/files/pdf/publications/Fluvial_Geomorphology.pdf · ¾A Study of Fluvial Geomorphology Aspects of