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
Minneapolis, MN 55414 USA
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 constant
b = bed elevation
t = elevation of top of bank
Qb = volume bedload transport rate
Ss = sidewall slope (constant)
p = porosity of the bed deposit
s = streamwise distance
t = time
Bs = width of sidewall zone
 = volume rate of input per unit length
of sediment from sidewalls
sidewall sediment eroded as
channel incises
t  b
 Ss
Bs

Q
Bb b   b  
t
s
  2 Bs
b
  b b
 2 t
t
Ss
t
Ss
t
b
t
t
 > 0 for a degrading channel, i.e. b/t < 0
b
Bb
Bs
EXNER EQUATION OF SEDIMENT CONTINUITY
INCLUDING SIDEWALL EROSION contd.
Reduce to obtain the relation:

  b  b
Q
 Bb  2 t

 b
Ss  t
s

or
b
Qb
1

t

t  b  s
 Bb  2

Ss 

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.
sidewall sediment eroded as
channel incises
Ss
t
b
t
t
b
Bb
Bs
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)
Dam
1200 ft
Saeltzer Dam, California before its removal in 2001.
THE DREAM MODELS
Specify an initial top width Bbt and
a minimum bottom width Bbm.
If Bb > Bbm, the channel degrades
and narrows without eroding its
banks.
b
1 Qb

t
Bb s
Bbt
Ss
Bb > Bbm
Bb  Bbt  2S s (t  b )
If Bb = Bbm the channel degrades
and erodes its sidewalls without
further narrowing.
sidewall sediment eroded as
channel incises
b
Qb
1

t

t  b  s
 Bbm  2

Ss 

But Bbm must be user-specified.
Ss
Bb = Bbm
SUMMARY OF THE DREAM FORMULATION
narrowing without
sidewall erosion
when Bb > Bbm
trajectories of left
and right bottom
bank position
top of deposit
Ss
no narrowing and
sidewall erosion
when Bb = Bbm
increasing
time
But how does the process of incision really work?
Experiments of Cantelli, Paola and Parker follow.
(Tesi di Laurea, Cantelli)
NOTE THE TRANSIENT PHENOMENON OF
EROSIONAL NARROWING!
(Experiments of Cantelli, Paola, Parker)
EVOLUTION OF CENTERLINE PROFILE
UPSTREAM (x < 9 m) AND DOWNSTREAM (x > 9 m)
OF THE DAM
Upstream degradation
Former dam location
Downstream aggradation
CHANNEL WIDTH EVOLUTION UPSTREAM OF THE DAM
The 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.
REGIMES OF EROSIONAL NARROWING AND EROSIONAL
WIDENING
0
Progress in time
Water Surface Elevation (cm)
-1
The dam is at x = 9 m
downstream of sediment
feed point.
-2
First 4.3 minutes of run: period
of erosional narrowing
-3
The cross-section is at x =
8.2 m downstream of the
sediment feed point, or 0.8
m upstream of the dam.
-4
-5
Subsequent 16.0 minutes of run:
period of erosional widening
-6
20
21
22
23
Water Surface Width (cm)
24
25
SUMMARY OF THE PROCESS OF INCISION INTO A
RESERVOIR DEPOSIT
trajectories of left
and right bottom
bank position
incisional narrowing
suppresses sidewall
erosion
incisional widening
enhances sidewall
erosion
top of deposit
Ss
rapid
incision with
narrowing
slow
incision with
widening
CAN WE DESCRIBE THE MORPHODYNAMICS OF RAPID
EROSIONAL NARROWING, FOLLOWED BY SLOW
EROSIONAL WIDENING?
PART OF THE ANSWER COMES FROM ANOTHER
SEEMINGLY UNRELATED SOURCE:
AN EARTHFLOW IN PAPUA NEW GUINEA
The earthflow is caused by the dumping of large amounts of
waste rock from the Porgera Gold Mine, 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)
Upstream dam created by earthflow
Earthflow
River
Sediment taken sideways into stream
• 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!!!
GEOMETRY
H = flow depth
n = transverse coordinate
nb = Bb = position of bank toe
Bw = width of wetted bank
nw = Bb + Bw = position of top
of wetted bank
Ss = slope of sidewall (const.)
b = elevation of bed
q̂ne = volume sediment input
per unit streamwise
width from earthflow
H
 Ss
Bw
•
•
•
•
•
inerodible valley wall
nw
river
earthflow
q̂ne
H
n
Ss
nb
Bb
b
Bw
b+H
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.
EXNER EQUATION OF SEDIMENT BALANCE ON THE BED
REGION
Local form of Exner:
(1   p )
q
q

  bs  bn
t
s
n
inerodible valley wall
where qbs and qbn are the streamwise and
transverse volume bedload transport rates
per unit width.
nw
river
nb
0
/t(sediment in bed region)
q̂ne
H
n
Ss
nb
Integrate on bed region with qbs = qbss, qbn
= 0;
(1   p )
earthflow
Bb
nb q
nb q

bs
bn
dn   
dn  
dn
0
0
t
s
n
b
Bw
b+H

transverse input from wetted bank region
differential steamwise transport
b
qbsb q̂bns
(1  p )


t
s
Bb
,
q̂bns  qbn n
b
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 = - q̂ne at n = nt where q denotes the
volume rate of supply of sediment
per unit length from the
earthflow
inerodible valley wall
nw
river
earthflow
q̂ne
H
n
Ss
nb
Geometric relation:
  b  Ss (n  Bb )
Result:
(1   p )
nw
nb
/t(sediment in wetted
bank region)

Bb
B
 b

 Ss b
t
t
t
n w q
n w q

bs
bn
dn   
dn  
dn
nb
nn
t
s
n
differential
steamwise transport
b
Bw
b+H

transverse input
from earthflow
transverse output
to bed region
B 
1 
 
qbssH  qbss Bb  q̂bns  q̂be
(1  p )Bw  b  Ss b   
t 
Ss s
s
 t
EQUATION FOR EVOLUTION OF BOTTOM WIDTH
inerodible valley wall
Eliminate b/t between
b
qbsb q̂bns
(1  p )


t
s
Bb
nw
river
earthflow
q̂ne
H
n
Ss
nb
Bb
and
b
Bw
b+H
B 
1 
 
qbssH  qbss Bb  q̂bns  q̂be
(1  p )Bw  b  Ss b   
t 
Ss s
s
 t
to obtain
Bb
qbsb
qbss H qbss Bb
B w  Bb
H qbss
1
(1   p )Ss




 q̂bns

q̂be
t
s
SsB w s
SsB w s B w s
B w Bb
Bw
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 q̂bns . The parameter q̂ne is
specified by the motion of the earthflow.
FLOW HYDRAULICS
Flow momentum balance: where S =
streamwise slope and Bw = H/Ss,
inerodible valley wall

Ss  1  S2s H 

1 H


bbBb 1  
 gHSBb  Bb 

Ss
Bb 
2
S
B
s
b 



nw
river
earthflow
q̂ne
H
n
Ss
Flow mass balance
nb

1 H

Qw  UHB b 1 
2
S
B
s
b 

Bb
b
Bw
b+H
Manning-Strickler resistance relation
1/ 6
H
2
1/ 2
b  C f U , C f   r  
, k s  nk D
k
 s
Here ks = roughness height, D = grain size, nk = o(1) constant. Reduce under the
condition H/Bs << 1 to get:
 k 1s/ 3Q 2w 
H   2 2 
  r gBb S 
3 / 10
BEDLOAD TRANSPORT CLOSURE RELATIONS
Shields number on bed region:
bb
bb
1  k1s/ 3Q2w



RgD RD   r2gBb2



3 / 10
S7 / 10
inerodible valley wall
where R = (s/ - 1)  1.65. Shields
number on bank region:
bs 
bs
 k Q 


 bb 
RgD
RD   gB 
1/ 3
s
2
r
2
w
2
b
nw
river
3 / 10
S7 / 10
 
qbsb  RgD D s bb

c 
1   
 bb 
q̂ne
H
n
Ss
nb
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,
1. 5
earthflow
Bb
4.5
,

b
qbss  RgD D s bb

1 .5
Bw
b+H

c 
1   
 bb 
4.5
(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 orderone constant and from Parker and Andrews (1986),
q̂bns  qbn B  qbss n
b
c
Ss 
bb

RgD D s bb

1.5
4. 5

c 
c
1     n
Ss



bb 
bb

SUMMARY OF THE SEDIMENT DIGESTER
Equation for evolution
of bed elevation
(1  p )
b
q
q̂
  bsb  bns
t
s
Bb
The earthflow encroaches on the channel
Equation for evolution of bottom width
(1   p )Ss
Bb
q
q
B  Bb
H qbss
H qbss Bb
1
  bsb 
 bss

 q̂bns w

q̂be
t
s
SsB w s
SsB w s B w s
B w Bb
Bw
Hydraulic relations
 k 1s/ 3Q 2w 
H   2 2 
  r gBb S 
3 / 10
1  k 1s/ 3Q 2w


RD   r2gBb2
bb



3 / 10
S7 / 10
As the channel narrows the Shields
number increases
Sediment transport relations
 
qbsb  RgD D s bb
1 .5

qbss  RgD D s bb

q̂bns  RgD D s bb

c 
1   
 bb 
4.5


c 
1   
 bb 



1 
 
1 .5
1. 5

c

bb
A higher Shields number gives higher
local streamwise and transverse
sediment transport rates.
4.5
4. 5

  n

bs  bb
Higher local streamwise and transverse

S s sediment transport rates counteract

channel narrowing

c

bb
EQUILIBRIUM CHANNEL
Transports 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).
c
 
bb

qbss  RgD D s bb

q̂bns  RgD D s bb


1.5
1.5

c 
1   
 bb 
4.5
0
Streamwise sediment transport on wetted bank
region = 0
4. 5

c 
c
1     n
Ss  0

bb
 bb 
 c
Qb  RgD D s 

1 .5



1  4.5 Bb
 k 1s/ 3Q 2w 
H   2 2 
  r gBb S 
bb
Choose bed shear stress so that bank shear stress
= critical value
 bs  bb  c
Transverse sediment transport on
wetted bank region = 0
Total bedload transport rate
3 / 10
c
bsb
1  k1s/ 3Q2w
 2 2



 RgD RD   r gBb



Three equations; if any two of Qw, S, H,
Qb and Bb are specified, the other three
can be computed!!
3 / 10
S7 / 10
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
t  b
 Ss
Bs
Ss
river
H
n
nb
Bb
b
Bs
t
INTEGRAL SEDIMENT BALANCE FOR THE BED AND
SIDEWALL REGIONS
On the bed region, integrate Exner from n = 0 to n = nb = Bb to get
(1  p )
b
q
q̂
  bsb  bns
t
s
Bb
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
t  b
 Ss
Bs
Ss
river
H
n
nb
Bb
b
Bs
t
INTEGRATION FOR SIDEWALL REGION
Upon integration it is found that
B 
1 
 
qbssH  qbss Bb  q̂bns
(1  p )Bs  b  Ss b   
t 
Ss s
s
 t
or reducing with sediment balance for the bed region,
Bb
qbsb
B s  Bb
H qbss qbss H qbss Bb
(1   p )Ss




 q̂bns
t
s
SsBs s
SsBs s Bs s
BsBb
nt
t  b
 Ss
Bs
Ss
river
H
n
nb
Bb
b
Bs
t
INTEGRAL SEDIMENT BALANCE: SIDEWALL REGION
For the minute neglect the indicated terms:
B 
1 
 
qbssH  qbss Bb  q̂bns
(1  p )Bs  b  Ss b   
t 
Ss s
s
 t
The equation can then be rewritten in the form:
B 
 
q̂bns  (1   p )Bs   b  Ss b 
t 
 t
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 sidewall
material to the channel.
INTEGRAL SEDIMENT BALANCE: SIDEWALL REGION
contd.
trajectories of left
and right bottom
bank position
incisional narrowing
suppresses sidewall
erosion
top of deposit
Ss
incisional widening
enhances sidewall
erosion
rapid
incision with
narrowing
slow
incision with
widening
q̂bns
Bb 
 b
 (1   p )Bs  
 Ss

t 
 t
INTERPRETATION OF TERMS IN RELATION FOR
EVOLUTION OF HALF-WIDTH
Bb
qbsb
B s  Bb
H qbss qbss H qbss Bb
(1   p )Ss




 q̂bns
t
s
SsBs s
SsBs s Bs s
BsBb
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!
This term always causes
widening whenever it is
nonzero.
REDUCTION FOR CRITICAL CONDITION FOR INCEPTION
OF EROSIONAL NARROWING
(on the plane and in the train)
Where NS and NB are order-one parameters,
Bb
qbsb S
qbsb Bb
B s  Bb
(1   p )Ss
 NS
 NB
 q̂bns
t
S s
Bb s
BsBb
Narrows if
slope
increases
downstream
At point of width minimum Bb/s = 0
Either way
Widens
REDUCTION FOR CRITICAL CONDITION FOR INCEPTION
OF EROSIONAL NARROWING contd
(on the plane and in the train)
Where Ns and Nb are order-one parameters,
Bb
qbsb S
qbsb Bb
B s  Bb
(1   p )Ss
 NS
 NB
 q̂bns
t
S s
Bb s
BsBb
After some reduction,
Bb S
Bb  B s
M
S s
Bs
c
Ss

bb
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!
I HOPE THAT MY TALK WAS NOT TOO CONTROVERSIAL
THANK YOU FOR LISTENING