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
)
Sta-Elev Bankfull Elevation Movable Bed
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
)
Sta-Elev Bankfull Elevation Movable Bed
12027.78
0
10
20
30
0 20 40 60 80 100 120 140 160Station (ft)
Elev
atio
n (ft
)
Sta-Elev Bankfull Elevation Movable Bed
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
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
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
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
S = 0.0016 S = 0.00162b = 35 Natural Design b = 35 Natural Design
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
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
S = 0.0016 S = 0.00162b = 35 Natural Design b = 35 Natural Design
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
)
Design Channel Bankfull Elevation Movable Bed
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
)
Design Channel Bankfull Elevation Movable Bed
0
5
10
-60 -40 -20 0 20 40 60Station (ft)
Ele
vatio
n (ft
)
Design Channel Bankfull Elevation Movable Bed
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
) 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
) 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
) 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
bHR = 35 ftbARR= 42.9 ftbHR/bARR= 0.82
bHR = 35 ftbARR= 42.9 ftbHR/bARR= 0.82
bHR = 15 ftbARR= 22.8 ftbHR/bARR= 0.66
bHR = 15 ftbARR= 22.8 ftbHR/bARR= 0.66
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
n = 0.066bHR = 35 ftbARR= 33.2 ftbHR/bARR= 1.05
n = 0.072bHR = 15 ftbARR= 17.8 ftbHR/bARR= 0.84
n = 0.072bHR = 15 ftbARR= 17.8 ftbHR/bARR= 0.84
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)
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 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)
Pebble Count Data D16 D50 D84 Extrapolated
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)
Pebble Count Data D16 D50 D84 Extrapolated
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
Sand GravelD50 0.4 10σg 1.5 2
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
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
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
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
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
)
Sand GravelD50 0.4 10σg 1.5 2
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
)
Sand GravelD50 0.4 10σg 1.5 2
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
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
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
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
log 1
0(d)
log10(d50) d50 (mm)
0.537 3.44
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