Another Chapter in THE SEARCH FOR THE HOLY GRAIL: A MECHANISTIC BASIS FOR HYDRAULIC RELATIONS FOR...

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Another Chapter inTHE SEARCH FOR THE HOLY GRAIL:

A MECHANISTIC BASIS FOR HYDRAULIC RELATIONS FOR BANKFULL FLOW IN GRAVEL-BED RIVERS

Gary Parker

With help from François Metivier and John Pitlick

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What is the physical basis relations for bankfull geometry of gravel-bed streams?

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Where do the following relations come from?

• Bankfull Depth Hbf ~ (Qbf)0.4

• Bankfull Width Bbf ~ (Qbf)0.5

• Bed Slope S ~ (Qbf)-0.3

where Qbf = bankfull discharge

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THE GOAL:

A Mechanistic Description of the Rules Governing Hydraulic Relations at Bankfull Flow in Alluvial Gravel-bed Rivers

The Parameters:

Qbf = bankfull discharge (m3/s)QbT,bf = volume bedload transport rate at bankfull

discharge (m3/s)Bbf = bankfull width (m)Hbf = bankfull depth (m)S = bed slope (1)D = surface geometric mean or median grain size (m)g = gravitational acceleration (m/s2)R = submerged specific gravity of sediment ~ 1.65 (1)

The Forms Sought:bTsbh n

bfbf,bTnbf

nbfbf

nbfbf Q~Q,Q~S,Q~B,Q~H

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DATA SETS1. Alberta streams, Canada1

2. Britain streams (mostly Wales)2

3. Idaho streams, USA3

4. Colorado River, USA (reach averages)

1 Kellerhals, R., Neill, C. R. and Bray, D. I., 1972, Hydraulic and geomorphic characteristics of rivers in Alberta, River Engineering and Surface Hydrology Report, Research Council of Alberta, Canada,No. 72-1.2 Charlton, F. G., Brown, P. M. and Benson, R. W., 1978, The hydraulic geometry of some gravel rivers in Britain, Report INT 180, Hydraulics Research Station, Wallingford, England, 48 p. 3 Parker, G., Toro-Escobar, C. M., Ramey, M. and Beck S., 2003,The effect of floodwater extraction on the morphologyof mountain streams, Journal of Hydraulic Engineering, 129(11), 2003.4 Pitlick, J. and Cress, R., 2002, Downstream changes in the channel of alarge gravel bed river, Water Resources Research 38(10), 1216,doi:10.1029/2001WR000898, 2002.

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NON-DIMENSIONALIZATION

2bf,bT

T2bf

5/2bf

bf5/1

5/2bf

bf5/1

DgDQ

Q,DgD

QQ,Q

BgB~,Q

HgH~

These forms supersede two previous forms, namely

which appear in reference 3 of the previous slide. Note:

DBB,

DHH bfbf

5/25/2 QB~B,QH~H

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WHAT THE DATA SAY

0.0001

0.001

0.01

0.1

1

10

100

1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07

Qhat

Btil

de, H

tilde

, S

Britain widthAlberta widthIdaho widthColorado widthBritain depthAlberta depthIdaho depthColorado depthBritain slopeAlberta slopeIdaho slopeColorado slope

H~

B~

S

The four independent sets of data form a coherent set!

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REGRESSION RELATIONS BASED ON THE DATA

y = 0.3785x4E-05

y = 4.6977x0.0661

y = 0.1003x-0.3438

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Qdim

Bdi

mtil

de, H

dim

tilde

, S

BdimtildeHdimtildeSPower (Hdimtilde)Power (Bdimtilde)Power (S)

344.00661.000004.0 Q100.0S,Q70.4B~,Q379.0H~

To a high degree of approximation,

379.0H~H~ c Remarkable, no?

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WHAT DOES THIS MEAN?

4.0bfbf

4.0bfbf

Qg

379.0H

orQ~H

461.0bf

2/10661.02/550sbf

461.0bfbf

QgDg70.4B

orQ~B

344.0bf

344.02/550s

344.0bf

QDg100.0S

orQ~S

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THE PHYSICAL RELATIONS NECESSARY TO CHARACTERIZE THE PROBLEM

Required: four relations in the four unknownsHbf, Bbf, S, QbT,bf.

1. Resistance relation (Manning-Strickler):

2. Gravel bedload transport relation (Parker 1979 approximation of Einstein 1950):

3. Relation for channel-forming Shields number bf*

(Parker 1978): and

4. Relation for gravel yield from basin (not determined solely by channel mechanics).

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RESISTANCE RELATION

rnbf

rbfbfbf

bf

bf,

bf

DH

SgHHBQ

uUCz

Manning-Strickler form: where Ubf = Qbf/(Bbf Hbf) denotes bankfull flow velocity,

Here we leave r and nr as parameters to be evaluated.

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BEDLOAD TRANSPORT RELATION

Use Parker (1979) approximation of Einstein (1950) relation applied to bankfull flow:

2.11,RD

SHwhere

1DRgDB

Qq

Gbf

bf

5.4

bf

c2/3bfG

bf

bf,bTbf

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RELATION FOR CHANNEL-FORMING SHIELDS NUMBER

Base the form of the relation on Parker (1978):

constrc

bf

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RELATION FOR GRAVEL YIELD FROM BASIN AT BANKFULL FLOW

This relations is external to the channel itself, and instead characterizes how the channels in a watershed interact with the unchannelized hillslopes. The necessary relation should be a dimensionless version of the form

where nbT must be evaluated.

bTnbfbf,bT Q~Q

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WORKING BACKWARD

Rather than working forward from the basic physical relations to the hydraulic relations, let’s work backward and find out what the form the physical relations must be to get the observed hydraulic relations.

SB nS

nBo QS,QB~,H~H~

344.0n,100.0,0661.0n,70.4,379.0H~ SSBBo

Recall that

2bf,bT

T2bf

5/2bf

bf5/1

5/2bf

bf5/1

DgDQ

Q,DgD

QQ,Q

BgB~,Q

HgH~

DBB,

DHH bfbf

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Now using the definition of Cz, the non-dimensionalizations and the relations

it is found that

But so that

RESISTANCE RELATION

SB nS

nBo QS,QB~,H~H~

The desired form is

r

rn

r

nbf

rbfbfbf

bf

bf,

bf HD

HSgHHB

QuUCz

o5/25/2bf H~QH~Q

DHH

]nn)2/1)[(2/5(

o2/1

SB2/3

c

]nn)2/1[(2/1

SB2/3

c2/12/3

BS

BS

H~H

H~1Q

H~1

SH~B~1Cz

]nn)2/1[(2/1

SB2/3

o2/12/3

bfbfbf

bf BSQH~

1SH~B~

1SgHHB

QCz

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RELATION FOR BANKFULL SHIELDS NUMBER

RDSHbf

bf

]n)5/2[(Soccbf

SQrR

H~,r

By definition

Using the relations

it is found that

This can be rewritten as

SB nS

nBo QS,QB~,H~H~

]n)5/2[(So5/2

bfSQ

RH~

RSH~Q

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RELATION FOR GRAVEL TRANSPORT AT BANKFULL FLOW

Recall that

Now from the last relation of the previous slide,

Using the previously-introduced non-dimensionalizations,

Thus

RD

SH,1DRgDB

Qq bf

bf

5.4

bf

c2/3bfG

bf

bf,bTbf

]n)5/2)[(2/3(

2/3

So5.4

2/3G

2/3c

5.42/3

GbfSQ

rRH~

r11r

r11rq

5/2T

bf

bf,bTbf QB~R

QDRgDB

Qq

]}n)5/2[(]n)5/2)[(2/3{(

2/3

So5.4

2/3GBT

BSQrR

H~

r11rRQ

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From the regression relations,

In addition, for natural sediment it is reasonable to assume

In the Parker approximation of the Einstein relation,

The data of the four setsindicate an average valueof bf

* of 0.04870, or thus

EVALUATION OF THE CONSTANTS

344.0n,100.0,0661.0n,70.4,379.0H~ SSBBo

65.1R

03.0c

63.1r

0.001

0.01

0.1

1

100 1000 10000 100000 1000000 10000000

Qhat

taus

bf

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THE RESULTING RELATIONS

rnrHCz

nc Q

ynyT QQ

73.3H~ ]n)2/5(n)4/5()2/3[(o

2/1S

1Br

BS

264.0nn21

25n BSr

0141.0rR

H~ So

0562.0n52n s

00318.0R

H~r11 2/3

S2/3

oB

5.4

G

y

550.0n23n1n SBy

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TEST OF RELATION FOR Czusing all four data sets

1

10

100

1 10 100 1000

Hhat

Cz Cz

Fit

263.0H43.3Cz

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0.001

0.01

0.1

1

1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Qhat

taus

bf tausbfFitQ

TEST OF RELATION FOR bf*using all four data seta

0562.0cbf Q0230.0r

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FINAL RESULTS

If we assume mechanistic relations of the following form:264.0

bf

bfbfbf

bf

bf,

bf

DH73.3

SgHHBQ

uUCz

5.4

bf

c2/3bf

bf

bf,bTbf 12.11

DRgDBQ

q

cbf

bf 63.1RD

SH

0562.0c Q0141.0

344.00661.0 Q100.0S,Q70.4B~,379.0H~

550.0T Q00318.0Q

resistance

bedload transport

channel-forming Shields number

sediment yield relation

The first three of these correspond precisely to the data!

then we obtain the results

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1

10

100

1000

1 10 100 1000

Reported Bbf (m)

Pred

icte

d B

bf (m

)

predicted Albertapredicted Britain Ipredicted Idahopredicted Coloradoequality1/22

Test against the original data set

0661.0Q70.4B~

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0.1

1

10

0.1 1 10

Reported Hbf (m)

Pred

icte

d H

bf (m

)



379.0H~

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0.0001

0.001

0.01

0.1

0.0001 0.001 0.01 0.1

Reported S

Pred

icte

d S


344.0Q100.0S Test against the original data set

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1

10

100

1000

10000

1 10 100 1000 10000

Reported Qbf (m3/s)

Pred

icte

d Q

bf (m

3 /s)

predictedequality1/22

264.0bf

bfbfbf

bf

bf,

bf

DH73.3

SgHHBQ

uUCz


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1

10

100

1 10 100

Reported Bbf (m)

Pred

icte

d B

bf (m

)

predicted Marylandpredicted Britain IIpredicted Tuscanyequality1/22predicted Colo Andr

Test against four new data sets

0661.0Q70.4B~

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0.1

1

10

0.1 1 10

Reported Hbf (m)

Pred

icte

d H

bf

predicted Marylandpredicted Britain IIequality1/22predicted Tuscanypredicted Colo Andr


379.0H~

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0.0001

0.001

0.01

0.1

0.0001 0.001 0.01 0.1

Reported S

Mea

sure

d S

predicted Marylandpredicted Britain IIequality1/22predicted Tuscanypredicted Colo Andr


344.0Q100.0S

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0.1

1

10

100

1000

0.1 1 10 100 1000

Measured Qbf (m3/s)

Pred

icte

d Q

bf (m

3 /s)

predicted Marylandpredicted Britain IIequality1/22predicted ColoAndr

Test against three new data sets

264.0bf

bfbfbf

bf

bf,

bf

DH73.3

SgHHBQ

uUCz

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1

10

100

1 10 100

Reported Bbf (m)

Pred

icte

d B

bf (m

)

Class 1Class 2Class 3Class 4equality1/22

BRITAIN II STREAMS: ROLE OF BANK STRENGTHClass 1 has least vegetation, Class 4 has most vegetation

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1

1.5

2

2.5

3

1 2 3 4

Vegetation Class

r

cbf r

RELATION BETWEEN VEGETATION DENSITY AND BANK STRENGTH, BRITAIN II STREAMS

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HOW WOULD VARIED BANK STRENGTH (r), SEDIMENT SUPPLY (Y) AND RESISTANCE (r) AFFECT HYDRAULIC

GEOMETRY?

rnrHCz

2/35.4

G

yB

)r(r11R

rB n52n

21

51n

Rn11

ry

5.4

G

o

rr11

H~

Rn11

ry

5.4

G

S

rr11

R n

52nS

nc Q

ynyT QQ

SnS QS

oH~H~

BnB QB~

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0.0001

0.001

0.01

0.1

1

10

100

1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Qhat

Bt,

Ht,

S

Britain I widthAlberta widthIdaho widthColorado widthr = 1.1r = 1r = 0.9Britain I depthAlberta depthIdaho depthColorado depthr = 1.1r = 1r = 0.9Britain I slopeAlberta slopeIdaho slopeColorado sloper = 1.1r = 1r = 0.9

S,H~ ,

B~

Q

B~

H~

S

VARIATION IN r (BANK STRENGTH)

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0.0001

0.001

0.01

0.1

1

10

100

1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Qhat

Bt,

Ht,

S

Britain I widthAlberta widthIdaho widthColorado widthar = 1.2ar = 1ar = 0.8Britain I depthAlberta depthIdaho depthColorado depthar = 1.2ar = 1ar = 0.8Britain I slopeAlberta slopeIdaho slopeColorado slopear = 1.2ar = 1ar = 0.8Q

S,H~ ,

B~

rr

r

r

r

r

rrr

B~

H~

S

VARIATION IN y (GRAVEL SUPPLY)

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0.0001

0.001

0.01

0.1

1

10

100

1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Qhat

Bt,

Ht,

SBritain I widthAlberta widthIdaho widthColorado widthay = 1.5ay = 1ay = 0.5Britain I depthAlberta depthIdaho depthColorado depthay = 1.5ay = 1ay = 0.5Britain I slopeAlberta slopeIdaho slopeColorado slopeay = 1.5ay = 1ay = 0.5

Q

S,H~ ,

B~

y

y

y

y

yy

y

y

y

B~

H~

S

VARIATION IN r (CHANNEL RESISTANCE)

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QUESTIONS?

Another Chapter in THE SEARCH FOR THE HOLY GRAIL: A MECHANISTIC BASIS FOR HYDRAULIC RELATIONS FOR...

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