another chapter in the search for the holy grail:
DESCRIPTION
Another Chapter in THE 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. What is the physical basis relations for bankfull geometry of gravel-bed streams?. - PowerPoint PPT PresentationTRANSCRIPT
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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
2
What is the physical basis relations for bankfull geometry of gravel-bed streams?
3
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
4
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
5
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
2
bf,bTT2
bf5/2
bf
bf5/1
5/2bf
bf5/1
DgD
QQ,
DgD
QQ,
Q
BgB~
,Q
HgH~
These forms supersede two previous forms, namely
which appear in reference 3 of the previous slide. Note:
D
BB,
D
HH 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
Bti
lde,
Hti
lde,
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
Bd
imti
lde,
Hd
imti
lde,
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
rn
bfr
bfbfbf
bf
bf,
bf
D
H
SgHHB
Q
u
UCz
Manning-Strickler form: where Ubf = Qbf/(Bbf Hbf) denotes bankfull flow velocity,
Here we leave r and nr as parameters to be evaluated.
12
BEDLOAD TRANSPORT RELATION
Use Parker (1979) approximation of Einstein (1950) relation applied to bankfull flow:
2.11,RD
SH
where
1DRgDB
Gbf
bf
5.4
bf
c2/3
bfG
bf
bf,bTbf
13
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
15
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
2
bf,bTT2
bf5/2
bf
bf5/1
5/2bf
bf5/1
DgD
QQ,
DgD
QQ,
Q
BgB~
,Q
HgH~
D
BB,
D
HH bfbf
16
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
r
nr
n
bfr
bfbfbf
bf
bf,
bf HD
H
SgHHB
Q
u
UCz
o5/25/2bf H~
QH~
QD
HH
]nn)2/1)[(2/5(
o2/1
SB2/3
c
]nn)2/1[(
2/1SB
2/3c
2/12/3
BS
BS
H~H
H~
1Q
H~
1
SH~
B~
1Cz
]nn)2/1[(
2/1SB
2/3o
2/12/3bfbfbf
bf BSQH~
1
SH~
B~
1
SgHHB
QCz
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RELATION FOR BANKFULL SHIELDS NUMBER
RD
SHbfbf
]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
R
H~
R
SH~
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,1
DRgDB
Qq bf
bf
5.4
bf
c2/3
bfG
bf
bf,bTbf
]n)5/2)[(2/3(
2/3
So
5.42/3
G
2/3
c
5.42/3
GbfSQ
rR
H~
r
11r
r
11rq
5/2
T
bf
bf,bTbf
QB~
R
Q
DRgDB
]}n)5/2[(]n)5/2)[(2/3{(
2/3
So
5.42/3
GBTBSQ
rR
H~
r
11rRQ
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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
tau
sb
f
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THE RESULTING RELATIONS
rnrHCz
nc Q
ynyT QQ
73.3H~ ]n)2/5(n)4/5()2/3[(
o2/1
S1
BrBS
264.0nn2
1
2
5n BSr
0141.0rR
H~
So
0562.0n5
2n s
00318.0R
H~
r1
1 2/3S
2/3oB
5.4
G
y
550.0n2
3n1n 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
tau
sb
f
tausbf
FitQ
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
D
H73.3
SgHHB
Q
u
UCz
5.4
bf
c2/3
bf
bf
bf,bTbf 12.11
DRgDB
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)
Pre
dic
ted
Bb
f (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)
Pre
dic
ted
Hb
f (m
)
predicted Albertapredicted Britain Ipredicted Idahopredicted Coloradoequality1/22
Test against the original data set
379.0H~
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0.0001
0.001
0.01
0.1
0.0001 0.001 0.01 0.1
Reported S
Pre
dic
ted
S
predicted Albertapredicted Britain Ipredicted Idahopredicted Coloradoequality1/22
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)
Pre
dic
ted
Qb
f (m
3 /s)
predictedequality1/22
264.0
bf
bfbfbf
bf
bf,
bf
D
H73.3
SgHHB
Q
u
UCz
Test against the original data set
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1
10
100
1 10 100
Reported Bbf (m)
Pre
dic
ted
Bb
f (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)
Pre
dic
ted
Hb
f predicted Marylandpredicted Britain IIequality1/22predicted Tuscanypredicted Colo Andr
Test against four new data sets
379.0H~
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0.0001
0.001
0.01
0.1
0.0001 0.001 0.01 0.1
Reported S
Me
as
ure
d S
predicted Marylandpredicted Britain IIequality1/22predicted Tuscanypredicted Colo Andr
Test against four new data sets
344.0Q100.0S
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0.1
1
10
100
1000
0.1 1 10 100 1000
Measured Qbf (m3/s)
Pre
dic
ted
Qb
f (m
3 /s)
predicted Marylandpredicted Britain IIequality1/22predicted ColoAndr
Test against three new data sets
264.0
bf
bfbfbf
bf
bf,
bf
D
H73.3
SgHHB
Q
u
UCz
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1
10
100
1 10 100
Reported Bbf (m)
Pre
dic
ted
Bb
f (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
34
HOW WOULD VARIED BANK STRENGTH (r), SEDIMENT SUPPLY (Y) AND RESISTANCE (r) AFFECT HYDRAULIC
GEOMETRY?
rnrHCz
2/35.4
G
yB
)r(r1
1R
rB n5
2n
2
1
5
1n
Rn1
1
ry
5.4
G
o
rr1
1H~
Rn11
ry
5.4
G
S
rr1
1R
n5
2nS
nc Q
ynyT QQ
SnS QS
oH~
H~
BnB QB
~
35
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 width
Alberta width
Idaho width
Colorado width
r = 1.1
r = 1
r = 0.9
Britain I depth
Alberta depth
Idaho depth
Colorado depth
r = 1.1
r = 1
r = 0.9
Britain I slope
Alberta slope
Idaho slope
Colorado slope
r = 1.1
r = 1
r = 0.9
S,H~ ,
B~
Q
B~
H~
S
VARIATION IN r (BANK STRENGTH)
36
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 width
Alberta width
Idaho width
Colorado width
ar = 1.2
ar = 1
ar = 0.8
Britain I depth
Alberta depth
Idaho depth
Colorado depth
ar = 1.2
ar = 1
ar = 0.8
Britain I slope
Alberta slope
Idaho slope
Colorado slope
ar = 1.2
ar = 1
ar = 0.8Q
S,H~ ,
B~
rr
r
r
r
r
rrr
B~
H~
S
VARIATION IN y (GRAVEL SUPPLY)
37
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 width
Alberta width
Idaho width
Colorado width
ay = 1.5
ay = 1
ay = 0.5
Britain I depth
Alberta depth
Idaho depth
Colorado depth
ay = 1.5
ay = 1
ay = 0.5
Britain I slope
Alberta slope
Idaho slope
Colorado slope
ay = 1.5
ay = 1
ay = 0.5Q
S,H~ ,
B~
y
y
y
y
yy
y
y
y
B~
H~
S
VARIATION IN r (CHANNEL RESISTANCE)
38
QUESTIONS?