alessandro cantelli, miguel wong, chris paola and gary parker st. anthony falls laboratory
DESCRIPTION
EROSIONAL NARROWING OF A CHANNEL RAPIDLY INCISING INTO A RESERVIOR DEPOSIT IN RESPONSE TO SUDDEN DAM REMOVAL. Some Preliminary Results. Alessandro Cantelli, Miguel Wong, Chris Paola and Gary Parker St. Anthony Falls Laboratory University of Minnesota Mississippi River at 3rd Ave SE - PowerPoint PPT PresentationTRANSCRIPT
EROSIONAL NARROWING OF A CHANNEL RAPIDLY INCISING INTO A RESERVIOR DEPOSIT IN RESPONSE
TO SUDDEN DAM REMOVAL
Alessandro Cantelli, Miguel Wong, Chris Paola and Gary Parker
St. Anthony Falls LaboratoryUniversity of Minnesota
Mississippi River at 3rd Ave SEMinneapolis, MN 55414 USA
Some Preliminary Results
CONSIDER THE CASE OF THE SUDDEN REMOVAL, BY DESIGN OR ACCIDENT,
OF A DAM FILLED WITH SEDIMENT
Before removal
REMOVAL OF THE DAM CAUSES A CHANNEL TO INCISE INTO THE DEPOSIT
After removal
AS THE CHANNEL INCISES, IT ALSO REMOVES SIDEWALL MATERIAL
sidewall sediment eroded as channel incises
top of reservoir deposit
Can we describe the morphodynamics of this process?
A SEEMINGLY UNRELATED PROBLEM:DEGRADATION OF THE LITTLE WEKIVA RIVER, FLORIDA
The Little Wekiva River drains a now heavily-urbanized area in the northern suburbs of Orlando, Florida.
The stream was straightened, and the floodplain filled in in the period 1940 – 1970.
Urbanization of the basin a) reduced infiltration, increasing the severity of floods, and b) cut off most of the supply of sediment to the stream. As a result, the stream degraded severely and produced substantial sidewall erosion.
MORPHODYNAMICS OF DEGRADATION IN THE LITTLE WEKIVA RIVER
Two of us (Cui and Parker, consulting) developed a morphodynamic model of the evolution of the Little Wekiva River as a tool for designing bank
protection and grade control structures, which have since been installed.
EXNER EQUATION OF SEDIMENT CONTINUITY INCLUDING SIDEWALL EROSION
Bb = channel bottom width, here assumed constantb = bed elevationt = elevation of top of bankQb = volume bedload transport rateSs = sidewall slope (constant)p = porosity of the bed deposits = streamwise distancet = time
Bs = width of sidewall zone = volume rate of input per unit length of sediment from sidewalls
ss
bt SB
sQ
tB bb
b
tS2
tB2 b
s
btbs
Ss
Bb
sidewall sediment eroded as channel incises
ttb
Bs
t
b
> 0 for a degrading channel, i.e. b/t < 0
EXNER EQUATION OF SEDIMENT CONTINUITY INCLUDING SIDEWALL EROSION contd.
sQ
tS2B bb
s
btb
Ss
Bb
sidewall sediment eroded as channel incises
ttb
Bs
t
b
Reduce to obtain the relation:
or
sQ
S2B
1t
b
s
btb
b
That is, when sidewall erosion accompanies degradation, the sidewall erosion suppresses (but does not stop) degradation and augments the downstream rate of increase of bed material load.
MODELING OF SEDIMENT PULSES IN MOUNTAIN RIVERS DUE TO LANDSLIDES AND DEBRIS FLOWS
Landslide into the Navarro River, California, USA, 1995
Test of model (Cui, Parker, Lisle, Pizzuto)developed for EPA
ADAPTATION TO THE PROBLEM OF CHANNEL INCISION SUBSEQUENT TO DAM REMOVAL:
THE DREAM MODELS(Cui, Parker and others)
1200 ft
Dam
Saeltzer Dam, California before its removal in 2001.
THE DREAM MODELS
sQ
S2B
1t
b
s
btbm
b
Specify an initial top width Bbt and a minimum bottom width Bbm.
If Bb > Bbm, the channel degrades and narrows without eroding its banks.
If Bb = Bbm the channel degrades and erodes its sidewalls without further narrowing.
)(S2BBs
QB1
t
btsbtb
b
b
b
But Bbm must be user-specified.
Ss
Bb > Bbm
Bbt
Ss
Bb = Bbm
sidewall sediment eroded as channel incises
SUMMARY OF THE DREAM FORMULATION
Ss
increasing time
no narrowing and sidewall erosion when Bb = Bbm
trajectories of left and right bottom
bank position
top of depositnarrowing without sidewall erosion when Bb > Bbm
But how does the process of incision really work?Experiments of Cantelli, Paola and Parker follow.
(Tesi di Laurea, Cantelli)
NOTE THE TRANSIENT PHENOMENON OFEROSIONAL NARROWING!
(Experiments of Cantelli, Paola, Parker)
EVOLUTION OF CENTERLINE PROFILEUPSTREAM (x < 9 m) AND DOWNSTREAM (x > 9 m)
OF THE DAMUpstream degradation Downstream aggradation
Former dam location
CHANNEL WIDTH EVOLUTION UPSTREAM OF THE DAMThe dam is at x = 9 m downstream of sediment feed point.
Note the pattern of rapid channel narrowing and degradation, followed by slow channel widening and degradation. The pattern is strongest near the dam.
-6
-5
-4
-3
-2
-1
0
20 21 22 23 24 25
Water Surface Width (cm)
Wat
er S
urfa
ce E
leva
tion
(cm
)
Progress in time
Subsequent 16.0 minutes of run: period of erosional widening
First 4.3 minutes of run: period of erosional narrowing
REGIMES OF EROSIONAL NARROWING AND EROSIONAL WIDENING
The dam is at x = 9 m downstream of sediment
feed point.
The cross-section is at x = 8.2 m downstream of the
sediment feed point, or 0.8 m upstream of the dam.
SUMMARY OF THE PROCESS OF INCISION INTO A RESERVOIR DEPOSIT
rapid incision with
narrowing
Ss
trajectories of left and right bottom
bank position top of deposit
slow incision with
widening
incisional narrowing suppresses sidewall
erosion
incisional widening enhances sidewall
erosion
CAN WE DESCRIBE THE MORPHODYNAMICS OF RAPID EROSIONAL NARROWING, FOLLOWED BY SLOW
EROSIONAL WIDENING?
The earthflow is caused by the dumping of large amounts of waste rock from the Porgera Gold Mine, Papua New Guinea.
PART OF THE ANSWER COMES FROM ANOTHER SEEMINGLY UNRELATED SOURCE:
AN EARTHFLOW IN PAPUA NEW GUINEA
THE EARTHFLOW CONSTRICTS THE KAIYA RIVER AGAINST A VALLEY WALL
The Kaiya River must somehow “eat” all the sediment delivered to it by the earthflow.
Kaiya River
earthflow
THE DELTA OF THE UPSTREAM KAIYA RIVER DAMMED BY THE EARTHFLOW
The delta captures all of the load from upstream, so downstream the Kaiya River eats only earthflow sediment
earthflow
THE EARTHFLOW ELONGATES ALONG THE KAIYA RIVER, SO MAXIMIZING “DIGESTION” OF ITS SEDIMENT
A downstream constriction (temporarily?) limits the propagation of the earthflow.
THE VIEW FROM THE AIR
Kaiya River
The earthflow encroaches on the river, reducing width, increasing bed shear stress and increasing the ability of the river to eat sediment!
THE BASIS FOR THE SEDIMENT DIGESTER MODEL(Consulting work of Parker)
• The earthflow narrows the channel, so increasing the sidewall shear stress and the ability of the river flow to erode away the delivered material.• The earthflow elongates parallel to the channel until it is of sufficient length to be “digested” completely by the river.
This is a case of depositional narrowing!!!
River
Earthflow Sediment taken sideways into stream
Upstream dam created by earthflow
GEOMETRYH = flow depthn = transverse coordinatenb = Bb = position of bank toeBw = width of wetted banknw = Bb + Bw = position of top
of wetted bankSs = slope of sidewall (const.)b = elevation of bed = volume sediment input
per unit streamwisewidth from earthflow
• The river flow is into the page.• The channel cross-section is assumed to be trapezoidal.• H/Bb << 1.• Streamwise shear stress on the bed region = bsb = constant in n• Streamwise shear stress on the submerged bank region = bss = bsb = constant
in n, < 1.• The flow is approximated using the normal flow assumption.
sw
SBH
Ss
inerodible valley wall
b
b+H
Hn
nb
Bb Bw
river
earthflownw
neq̂
neq̂
EXNER EQUATION OF SEDIMENT BALANCE ON THE BED REGION
Local form of Exner:
where qbs and qbn are the streamwise and transverse volume bedload transport rates per unit width.
Integrate on bed region with qbs = qbss, qbn = 0;
nq
sq
t)1( bnbs
p
bbb n
0bn
n
0bs
n
0p dnn
qdns
qdnt
)1(
Ss
inerodible valley wall
b
b+H
Hn
nb
Bb Bw
river
earthflownw
neq̂
bnbnbnsb
bnsbsbbp qq̂,
Bq̂
sq
t)1(
/t(sediment in bed region)
differential steamwise transport
transverse input from wetted bank region
EXNER EQUATION OF SEDIMENT BALANCE ON THE WETTED BANK REGION
Integrate local form of Exner on wetted bank region with region with:qbs = qbss for nb < n < nb + Bw
qbn = - at n = nt where q denotes the volume rate of supply of sedimentper unit length from the earthflow
Geometric relation:
Result:
w
n
w
b
w
b
n
nbn
n
nbs
n
np dnn
qdns
qdnt
)1(
bebnsb
bssbsss
bs
bwp q̂q̂
sBqHq
sS1
tBS
tB)1(
tBS
tt)Bn(S b
sb
bsb
Ss
inerodible valley wall
b
b+H
Hn
nb
Bb Bw
river
earthflownw
neq̂
neq̂
/t(sediment in wetted bank region)
differential steamwise transport
transverse output to bed region
transverse input from earthflow
EQUATION FOR EVOLUTION OF BOTTOM WIDTH
Eliminate b/t between
Note that there are two evolution equations for two quantities, channel bottom elevation b and channel bottom width Bb. To close the relations we need to have forms for qbsb, qbss and . The parameter is specified by the motion of the earthflow.
bnsq̂
b
bnsbsbbp B
q̂s
qt
)1(
and
to obtain
bewbw
bwbns
b
w
bss
ws
bssbss
ws
bsbbsp q̂
B1
BBBBq̂
sB
Bq
sH
BSq
sq
BSH
sq
tBS)1(
bebnsb
bssbsss
bs
bwp q̂q̂
sBqHq
sS1
tBS
tB)1(
neq̂
Ss
inerodible valley wall
b
b+H
Hn
nb
Bb Bw
river
earthflownw
neq̂
FLOW HYDRAULICS
Flow momentum balance: where S = streamwise slope and Bw = H/Ss,
Flow mass balance
Manning-Strickler resistance relation
bsbb
bs
2ss
bbb BH
S21BgHSB
BH
SS1S
1B
bsbw B
HS211UHBQ
Dnk,kHC,UC ks
6/1
sr
2/1f
2fb
Here ks = roughness height, D = grain size, nk = o(1) constant. Reduce under the condition H/Bs << 1 to get:
10/3
2b
2r
2w
3/1s
SgBQkH
Ss
inerodible valley wall
b
b+H
Hn
nb
Bb Bw
river
earthflownw
neq̂
BEDLOAD TRANSPORT CLOSURE RELATIONSShields number on bed region:
where R = (s/ - 1) 1.65. Shields number on bank region:
Streamwise volume bedload transport rate per unit width on bed and bank regions is qbsb and qbss, respectively: where s = 11.2 and c* denotes a critical Shields stress,
10/7
10/3
2b
2r
2w
3/1sbb
bb SgBQk
D1
RgD
R
10/710/3
2b
2r
2w
3/1s
bbbs
bs SgBQk
DRgD
R
5.4
bb
c5.1bbsbss
5.4
bb
c5.1bbsbsb 1DRgDq,1DRgDq
(Parker, 1979 fit to relation of Einstein, 1950). Transverse volume bedload transport rate per unit width on the sidewall region is qbns, where n is an order-one constant and from Parker and Andrews (1986),
sbb
cn
5.4
bb
c5.1bbss
bb
cnbssBbnbns S1DRgDSqqq̂
b
Ss
inerodible valley wall
b
b+H
Hn
nb
Bb Bw
river
earthflownw
neq̂
SUMMARY OF THE SEDIMENT DIGESTER
10/710/3
2b
2r
2w
3/1s
bb SgBQk
D1
R bbbs
5.4
bb
c5.1bbsbss
5.4
bb
c5.1bbsbsb
1DRgDq
1DRgDq
sbb
cn
5.4
bb
c5.1bbsbns S1DRgDq̂
b
bnsbsbbp B
q̂s
qt
)1(
10/3
2b
2r
2w
3/1s
SgBQkH
Equation for evolution of bed elevation
Equation for evolution of bottom width
bewbw
bwbns
b
w
bss
ws
bssbss
ws
bsbbsp q̂
B1
BBBBq̂
sB
Bq
sH
BSq
sq
BSH
sq
tBS)1(
Hydraulic relations
Sediment transport relations As the channel narrows the Shields number increases
Higher local streamwise and transverse sediment transport rates counteract channel narrowing
A higher Shields number gives higher local streamwise and transverse sediment transport rates.
The earthflow encroaches on the channel
EQUILIBRIUM CHANNELTransports sediment without changing slope or eroding the banks
(flow turned off below: Pitlick and Marr)
EQUILIBRIUM CHANNEL SOLUTION
As long as < 1, the formulation allows for an equilibrium channel without widening or narrowing as a special case (without input from an earthflow).
cbbbsbb
c Choose bed shear stress so that bank shear stress = critical value
01DRgDq5.4
bb
c5.1bbsbss
Streamwise sediment transport on wetted bank
region = 0
0S1DRgDq̂ sbb
cn
5.4
bb
c5.1bbsbns
Transverse sediment transport on
wetted bank region = 0
b5.4
5.1
csb B1DRgDQ
Total bedload transport rate
10/710/3
2b
2r
2w
3/1sbsbc
bb SgBQk
D1
RgD
R
10/3
2b
2r
2w
3/1s
SgBQkH
Three equations; if any two of Qw, S, H, Qb and Bb are specified, the other three can be computed!!
ADAPTATION OF THE SEDIMENT DIGESTER FOR EROSIONAL NARROWING
• As the channel incises, it leaves exposed sidewalls below a top surface t.• Sidewall sediment is eroded freely into the channel, without the
external forcing of the sediment digester.• Bb now denotes channel bottom half-width• Bs denotes the sidewall width of one side from channel bottom to top
surface.• The channel is assumed to be symmetric, as illustrated below.
nt
Bb t
b
Ss
Hn
nb
Bs
rivers
s
bt SB
INTEGRAL SEDIMENT BALANCE FOR THE BED AND SIDEWALL REGIONS
On the bed region, integrate Exner from n = 0 to n = nb = Bb to get
On the sidewall region, integrate Exner from n = nb to n = nt under the conditions that streamwise sediment transport vanishes over any region not covered with water, and transverse sediment transport vanishes at n = nt
nt
Bb t
b
Ss
Hn
nb
Bs
rivers
s
bt SB
b
bnsbsbbp B
q̂s
qt
)1(
INTEGRATION FOR SIDEWALL REGION
Upon integration it is found that
or reducing with sediment balance for the bed region,
nt
Bb t
b
Ss
Hn
nb
Bs
rivers
s
bt SB
bnsb
bssbsss
bs
bsp q̂
sBqHq
sS1
tBS
tB)1(
bs
bsbns
b
s
bss
ss
bssbss
ss
bsbbsp BB
BBq̂s
BBq
sH
BSq
sq
BSH
sq
tBS)1(
INTEGRAL SEDIMENT BALANCE: SIDEWALL REGION
For the minute neglect the indicated terms:
The equation can then be rewritten in the form:
As the channel degrades i.e. b/t < 0, sidewall material is delivered to the channel.
Erosional narrowing, i.e. Bb/t < 0 suppresses the delivery of sidewallmaterial to the channel.
bnsb
bssbsss
bs
bsp q̂
sBqHq
sS1
tBS
tB)1(
t
BSt
B)1(q̂ bs
bspbns
INTEGRAL SEDIMENT BALANCE: SIDEWALL REGION contd.
rapid incision with
narrowing
Ss
trajectories of left and right bottom
bank position top of deposit
slow incision with
widening
incisional narrowing suppresses sidewall
erosion
incisional widening enhances sidewall
erosion
t
BSt
B)1(q̂ bs
bspbns
INTERPRETATION OF TERMS IN RELATION FOR EVOLUTION OF HALF-WIDTH
bs
bsbns
b
s
bss
ss
bssbss
ss
bsbbsp BB
BBq̂s
BBq
sH
BSq
sq
BSH
sq
tBS)1(
This term always causes widening whenever it is
nonzero.
Auxiliary streamwise terms
This term causes narrowing whenever sediment transport
is increasing in the streamwise direction.
But this is exactly what we expect immediately
upstream of a dam just after removal: downward
concave long profile!
REDUCTION FOR CRITICAL CONDITION FOR INCEPTION OF EROSIONAL NARROWING
(on the plane and in the train)
bs
bsbns
b
b
bsbB
bsbS
bsp BB
BBq̂s
BBqN
sS
SqN
tBS)1(
Narrows if slope increases downstream
WidensEither way
Where NS and NB are order-one parameters,
At point of width minimum Bb/s = 0
REDUCTION FOR CRITICAL CONDITION FOR INCEPTION OF EROSIONAL NARROWING contd
(on the plane and in the train)
bs
bsbns
b
b
bsbB
bsbS
bsp BB
BBq̂s
BBqN
sS
SqN
tBS)1(
Where Ns and Nb are order-one parameters,
After some reduction,
where M is another order-one parameter.
That is, erosional narrowing can be expected if the long profile of the river is sufficiently downward concave, precisely the condition to be expected immediately after dam removal!
sbb
c
s
sbb SB
BBMsS
SB
I HOPE THAT MY TALK WAS NOT TOO CONTROVERSIAL
THANK YOU FOR LISTENING