alessandro cantelli, miguel wong, chris paola and gary parker st. anthony falls laboratory

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