Transcript Slide 1

1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CHAPTER 21
RESPONSE OF A SAND-BED RIVER TO A DREDGE SLOT
Grey Cloud Island
Lock and Dam no. 2
Mississippi River near Grey Cloud Island south of St. Paul, Minnesota
Image from NASA
https://zulu.ssc.nasa.gov/mrsid/mrsid.pl
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
FILLING OF A DREDGE SLOT IN A SAND-BED RIVER
The Mississippi River must be dredged in order to maintain a depth sufficient for
navigation. In addition, gravel and sand are mined for industrial purposes on Grey
Cloud Island adjacent to the river. Now suppose that a) gravel and sand mining is
extended to a dredge slot between the island and the main navigation channel and
b) the main channel subsequently avulses (jumps) into the dredge slot. How would
the river evolve subsequently?
dredge slot
navigation
channel
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
FILLING OF A DREDGE SLOT IN A SAND-BED RIVER contd.
before avulsion
just after avulsion
some time after
avulsion
Note how a) a delta builds into the dredge slot from upstream and b) degradation
propagates downstream of the slot.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
MORPHODYNAMICS OF DREDGE SLOT EVOLUTION
In the modeling performed here, the following form of the Exner equation of
sediment continuity is used:

qt
qb
qs
(1  p )
 - f
 - f
- f
t
x
x
x
where  = bed elevation, qt = total volume bed material load transport rate per unit
width, qb = volume bedload transport rate per unit width and qs = volume bed
material suspended load transport rate per unit width. Here qs is computed based
on the assumption of quasi-steady flow. An alternative formulation from Chapter 4
is, however,

qb
(1  p )
 - f
  f v s cb  E
t
x
where cb is the near-bed concentration of suspended sediment, E is the
dimensionless rate of entrainment of suspended sediment from the bed and vs
denotes the fall velocity of the sediment. That is, vs cb denotes the volume rate of
deposition of suspended sediment per unit area per unit time on the bed, and vsE
denotes the corresponding volume rate of entrainment of sediment into
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suspension from the bed per unit area per unit time.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
ALTERNATIVE ENTRAINMENT FORMULATION FOR EXNER
If the alternative formulation for conservation of bed sediment

qb
(1  p )
 - f
  f v s cb  E
t
x
is used, then a) the quasi-steady form of the equation of conservation of suspended
sediment, i.e.
UCH
 v s E  cb 
x
must be solved simultaneously, and in addition the above relations must be closed
by relating cb to C. For example, writing cb = roC, the following quasi-steady
evalution is obtained from the material of Chapter 10:

cb  roC , ro  Cz

 u H
 1  (1  ) /  
,  , , b    
b (1   ) /  
v
k
b
b
 s c


11H
U
k c  (  Cz ) , Cz 
e
u
vs
u
 H 
n 30  d
 kc 
This alternative formulation is not used here. The reasons for this are explained
toward the end of the chapter.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
BEDLOAD TRANSPORT AND ENTRAINMENT RELATIONS
An appropriate relation for bedload transport in sand-bed streams is that of Ashida
and Michiue (1972) introduced in Chapter 7. Where bs denotes the boundary shear
stress due to skin friction and
bs
 
RgDs50

s
the relation takes the form

qb  17 s  c
qb
, q 
RgDs50 Ds50

b


s  c ,
c  0.05
An appropriate relation for the entrainment of sand into suspension is that of Wright
and Parker (2004) introduced in Chapter 10:
AZu5
E
A 5
1
Zu
0.3
us 
bs

, Zu 
us
Re p0.6 S0f.07 ,
vs
RgD s50 Ds50
, Re p 

A  5.7x107
Note that in the disequilibrium
formulation considered here, bed
slope S has been replaced with
friction slope Sf.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
RELATION FOR SUSPENDED SEDIMENT TRANSPORT RATE
The method of Chapter 10, which is strictly for equilibrium flows, is hereby extended
to gradually varied flows. A backwater calculation generates the depth H, the
friction slope Sf and the depth due to skin friction Hs everywhere. Once these
parameters are known, the parameters u* = (gHSf)1/2, Cz = U/u*, kc =
11H/[exp(Cz)], u*s = (gHsSf)1/2 and E can be computed everywhere.
The volume transport rate per unit width of suspended sediment is thus computed
as
qs 
EuH


where
 u H
 1  (1  ) /  
 , , b    
b (1   ) /  
v
k
b
b
 s c


vs
u
 H 
n 30  d
 kc 
In the case of the Wright-Parker formulation, b = 0.05.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SUMMARY OF MORPHODYNAMIC FORMULATION

(qb  qs )
(1   p )
 - f
t
x

qb  17 RgD s50 Ds50 s  c
AZu5
E
A 5
1
Zu
0.3
, Zu 


s  c ,
c  0.05
gHsSf
vs
Re p0.6 S0f.07 , R f 
v s (R f )
RgD s50
 u H
 1  (1  ) /  
Eu H
qs 
 ,  , , b    
b (1   ) /  

v
k
b
b
 s c


vs
u
 H 
n 30  d
 kc 
In order to finish the formulation, a resistance relation that includes the effect
of bedforms is required. This relation must be adapted to mildly disequilibrium
flows and implemented in a backwater formulation.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CALCULATION OF GRADUALLY VARIED FLOW IN SAND-BED RIVERS
INCLUDING THE EFFECT OF BEDFORMS
The backwater equation presented in Chapter 5 is
dH S  Sf

dx 1  Fr 2
where H denotes depth, x denotes downstream distance, S is bed slope and Sf is
friction slope. In addition the Froude number Fr = qw/(g1/2H3/2) where qw is the water
discharge per unit width and g is the acceleration of gravity. The friction slope can
be defined as follows:
b
U2
S f  Cf

 b  gSf H
gH gH
In a sand-bed river, the boundary shear stress b and depth H can be divided into
components due to skin friction bs and Hs and form drag due to dunes bf and Hf so
b  bs  bf  (Cfs  Cff )U2
, H  Hs  Hf
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where Cfs and Cff denote resistance coefficients due to skin friction and form drag.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
GRADUALLY VARIED FLOW IN SAND-BED RIVERS WITH BEDFORMS contd.
As shown in Chapter 9, the formulation for Wright and Parker (2004) for gradually
varied flow over a bed covered with dunes reduces to the form
Cfs1/ 2 
 H 
qw
 8.32  s 
H gHsSf
 3Ds90 
1
6
for skin friction and the form
s  0.05  0.7(Fr 0.7 )0.8
for form drag. Reducing the above equation using the friction slope Sf rather than
bed slope S in order to be able to capture quasi-steady flow,

HsS f
HS f
 0.05  0.7
 RD s50
RD 50

  qw 

3/2
  g H 
0.7




0.8
If H, qw, Ds50, Ds90 and R are known, Hs and Sf can be calculated iteratively from the
above equations. Note that a stratification correction in the first equation
above (specified in Wright and Parker, 2004) has been set equal to unity for
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simplicity in the present calculation.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
LIMITS TO THE WRIGHT-PARKER FORMULATION
The ratio s of bed shear stress due to skin friction to total bed shear stress can be
defined as

s 
bs s Hs
  
b

H
Now by definition s must satisfy the condition s ≤ 1 (skin friction must not exceed
total friction). The following equation for s is obtained from the Wright-Parker
relation:
0.05  0.7( Fr 0.7 )0.8
s 

For any given value of Froude number Fr, it is found that a minimum Shields
number min* exists, below which s > 1. In order to include values * < min*, the
relation must be amended to:
 0.05  0.7( Fr 0.7 )0.8

,

s  




1
,




min
  min
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
PLOT OF s VERSUS * FOR THE CASE Fr = 0.2
10
Fr = 0.2
s
0.05  0.7( Fr 0.7 )0.8
s 

1
s = 1 for * < min* = 0.0922
0.1
0.01
0.1
1
*
10
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
ITERATIVE COMPUTATION OF min* AS A FUNCTION OF Fr
The equation for min* takes the form
min  0.05  0.7(min Fr 0.7 )0.8
or
G(min )  min  0.05  0.7(min )4 / 5Fr 14 / 25  0
This equation cannot be solved explicitly. It can, however, be solved implicitly
using, for example the Newton-Raphson technique. Let p = 1, 2, 3… be an index,
and let min,p* be an estimate of the root of the above equation. A better estimate is
min,p+1*, where

G
(

min, p )
min, p1  min, p 
G(min, p )
, G(min, p ) 
dG
dmin
min, p
4
 1  0.7 (min )1/ 5 Fr 14 / 25
5
The calculation proceeds until the relative error  between min,p* and min,p+1* drops
below some specified small tolerance  << 1, i.e.
min, p1  min, p

1 2 (min, p1  min, p )
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
IMPLEMENTATION OF ITERATIVE COMPUTATION OF min* AS A FUNCTION OF
Fr
A sample calculation for the case Fr = 0.2 is given below.

G
(

min, p )
min, p1  min, p 
G(min, p )

min

min
G( )  

4/5
min
 0.05  0.7( )
Fr
14 / 25
4
G(min, p )  1  0.7 (min )1/ 5 Fr 14 / 25
5
0
min, p1  min, p

1 2 (min, p1  min, p )
Fr
min*
p
1
2
3
4
5
1
0.13829
0.09313
0.09222
0.09222
0.2
G
G'

0.665767 0.773
0.029906 0.662 1.514
0.000578 0.634
0.39
3.27E-07 0.634
0.01
1.06E-13 0.634 6E-06
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
PLOT OF min* VERSUS Fr
0.4
0.35
0.3
0.25
min* 0.2
0.15
0.1
0.05
0
0
0.2
0.4
0.6
Fr
0.8
1
15
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Sub Find_tausmin(xFr, xtausmin)
CODE FOR COMPUTING min*
(Dim statements deleted)
xtausmin = 0.4
Found_tausmin = False
Bombed_tausmin = False
ittau = 0
Do
ittau = ittau + 1
Ft = xtausmin - 0.05 - 0.7 * (xtausmin ^ (4 / 5)) * (xFr ^ (14 / 25))
Ftp = 1 - 0.7 * (4 / 5) * (xtausmin ^ (-1 / 5)) * (xFr ^ (14 / 25))
xtausminnew = xtausmin - Ft / Ftp
er = Abs(2 * (xtausminnew - xtausmin) / (xtausminnew + xtausmin))
If er < ep Then
Found_tausmin = True
Else
If ittau > 200 Then
Bombed_tausmin = True
Else
xtausmin = xtausminnew
End If
End If
Loop Until Found_tausmin Or Bombed_tausmin
If Found_tausmin Then
xtausmin = xtausminnew
Else
Worksheets("ResultsofCalc").Cells(1, 5).Value = "Calculation of tausmin failed to converge"
End If
End Sub
16
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CALCULATION OF Hs AND Sf FROM KNOWN DEPTH H
In order to implement a backwater calculation that includes the effects of bedforms,
it is necessary to compute Hs and Sf at every point for which depth H is given. The
governing equations are:
 H 
qw
 8.32  s 
H gHsSf
 3Ds90 
1
6
0.8
0.7



0.05  0.7 HS f   qw   ,
 RD   g H3 / 2  
HsSf 

s 50  


 

RD 50 
HS f
HS f
,
 min

RD s50
RD s50

HS f
 min
RD s50
Now writing  = Hs/H, the top equation can be solved for Sf to yield
Sf  s4 / 3Snom
 Fr 
, Snom  

8.32


2
 H 


3
D
 s90 
1/ 3
17
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CALCULATION OF Hs AND Sf FROM KNOWN DEPTH H contd.
The middle equation of the previous slide can be reduced with the definition  =Hs/H
and the last equation of the previous slide to yield

s1/ 3nom


0.05  0.7s16 / 15 (nom
)4 / 5 Fr 14 / 25 , s4 / 3nom
 min



s4 / 3nom
, s4 / 3nom
 min

where

nom

HS nom
RD s50
This relation reduces to
15 / 16


 s1/ 3 nom
 0.05 


3/ 4
s  
,


(

/

)
s
nom
min

4/5
14 / 25 
F(s )  
0
 0.7(nom ) Fr



s  1 , s  ( nom
/ min )3 / 4

18
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
NEWTON-RAPHSON SCHEME FOR s
A Newton-Raphson iterative solution is implemented for s. Thus if p is an index
and s,p is the pth guess for the solution of the above equation, a better guess is
given by
s,p1  s,p 
F(s,p )
F(s,p )
where the prime denotes a derivative with respect to s. Computing the derivative,
4 / 3 
 15  1/ 3   0.05  31 / 16 1

nom


3/ 4
s
nom
s
1 
,


(

/

)
s
nom
min


F(s )   16  0.7(nom
0
)4 / 5 Fr 14 / 25 
3 0.7(nom
)4 / 5 Fr 14 / 25


1 , s  (nom
/ min )3 / 4

The solution is initiated with some guess s,1. The calculation is continued until the
relative error  is under some acceptable limit  (e.g.  = 0.001). where

s,p1  s,p
1 2 (s,p1  s,p )
19
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SAMPLE IMPLEMENTATION OF NEWTON-RAPHSON SCHEME
The following parameters are given at a point: H = 3 m, qw = 5 m2/s, Ds50 = 0.5 mm,
Ds90 = 1 mm, R = 1.65. Thus ks = 3Ds90 = 3 mm and
2




qw
Fr

  0.000136 ,   HS nom  0.496
Fr 

0
.
307
,
S

nom
nom
1/ 6 

RD s50
g H3 / 2
 8.32 H  
k  

 s 
An iterative computation of min* yields the value 0.113. Implementing the iterative
scheme for s with the first guess s = 0.99, the following result is obtained:
s
p
1
2
3
4
5
0.99
0.38019
0.33461
0.33374
0.33374
F(s)
0.506295
0.03149
0.000577
2.42E-07
4.26E-14
F'(s)

Hs
Sf
0.83026
2.97 0.000138
0.69079
0.89 1.14058 0.000495
0.66455 0.128 1.00383 0.000587
0.664 0.003 1.00122 0.000589
0.664 1E-06 1.00122 0.000589
Note that the relative error  is below 0.1% by the fourth iteration (p = 5).
20
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
NOTE OF CAUTION CONCERNING THE NEWTON-RAPHSON SCHEME
There always seems to be a first guess of s for which the Newton-Raphson scheme
converges. When nom* is only slightly greater than min* (in which case s is only
slightly less than 1), however, the right initial guess is sometimes hard to find. For
example, the scheme may bounce back and forth between two values of s without
converging, or may yield at some point a negative value of s, in which case Sf
cannot be computed from the first equation at the bottom of Slide 17.
The following technique was adopted to overcome these difficulties in the programs
presented in this chapter.
1. The initial guess for s is set equal to 0.9
2. Whenever the iterative scheme yields a negative value of s, s is reset to 1.02
and the iterative calculation recommenced.
3. Whenever the calculation does not converge, it is assumed that s is so close to 1
that it can be set equal to 1.
These issues can be completely avoided by using the bisection method rather than
the Newton-Raphson method for computing s. The bisection method,
21
however, is rather slow to converge.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
APPLICATION TO BACKWATER CALCULATIONS
The backwater equation takes the form
dH S  S f

dx 1  Fr 2
Hi-1
Let the water discharge qw, the grain sizes
Ds50 and Ds90 and the submerged specific
gravity R be specified. In addition the
upstream and downstream bed elevations i-1
and i are known, along with the downstream
depth Hi. The downstream values Hs,i and Sf,i
are computed in accordance with the
procedures of the previous three slides.
A first guess of Hi-1 is given as Hi-1,pred, where
Hi  Hi1,pred
x
or thus:

S  Sf ,i
1  Fr
2
i

i-1
[( i1   i ) x ]  Sf ,i
Hi1,pred  Hi 
1  Fri2
i1  i  Sf ,ix
1  Fr
2
i
flow
Hi
x
i
qw
, Fri 
g H3i / 2
22
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
APPLICATION TO BACKWATER CALCULATIONS contd.
Once Hi-1,pred is known, the associated
parameters Hs,i-1,pred and Sf,i-1,pred can be
computed using the Newton-Raphson
formulation outlined in previous slides.
Having obtained these values, a predictorcorrector scheme is used to evaluate Hi-1:
Hi-1
flow
i-1
Hi  Hi1 1  [(i1  i ) x ]  Sf ,i [(i1  i ) x ]  Sf ,i1,pred 
 


2
x
2 
1  Fri
1  Fri-21,pred

or thus:
Hi
x
, Fri1,pred 
i
qw
g Hi3/12,pred
1  i1  i  Sf ,ix i1  i  Sf ,i1,pred x 
Hi1  Hi  


2
2 
1  Fri
1  Fri-21,pred

Once Hi-1 is known the Newton-Raphson scheme can be used again to
compute Hs,i-1 and Sf,i-1.
23
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CALCULATION OF NORMAL DEPTH FROM GIVEN VALUES OF
qw, D, R and S
The calculations for the morphodynamic response to a dredge slot begin with a
computation of the normal flow conditions prevailing in the absence of the dredge
slot. It is assumed that the water discharge per unit width qw, bed slope S, sediment
grain size D and sediment submerged specific gravity R are given parameters. The
parameters H and Hs associated with normal flow are to be computed.
The governing equations are the same as the first two of Slide 17, except that Sf =
S at normal flow.
 H 
qw
 8.32  s 
H gHsS
 3Ds90 
1
6
0.8
0.7



0.05  0.7 HS   qw   ,
HsS 
 RD s50   g H3 / 2  



RD 50 
HS
HS
,
 min

RD s50
RD s50

HS
 min
RD s50
24
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CALCULATION OF NORMAL DEPTH contd.
Again introducing the notation s = Hs/H, the first equation of the previous slide
reduces to:


qw

s (H)  
1
6
 H gHS 8.32  H 
 3D 

 s90 








3/2
The second equation of the previous slide then reduces with the above equation to

 RD 50

Hs (H)   S



14 / 25
4/5







HS
q
RD

w
50
min


0.05  0.7
, H

 RD   g H3 / 2 
S


s 50 





RD 50 min
H, H 
S
25
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CALCULATION OF NORMAL DEPTH contd.
The second equation of the previous slide can also be rewritten as
14 / 25
4/5








RD
HS
q
RD

50
w
50
min
H (H) 


0.05  0.7
, H

s
 RD   g H3 / 2 

S
S


s 50 



FN (H)  
0



RD 50 min
Hs (H)  H , H 

S

The corresponding Newton-Raphson scheme is
Hp1  Hp 
FN (Hp )

FN (Hp )
where the term FN’ is given on the following slide. The initial guess for H can be
based on the depth for normal flow that would prevail in the absence of bedforms
(form drag only):
 (3D90 ) q 
H

2

gS
r


1/ 3
2
w
3 / 10
,
r  8.32
26
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CALCULATION OF NORMAL DEPTH contd.
The scheme thus becomes
Hp1  Hp 
FN (Hp )

FN (Hp )
14 / 25
4/5






1
RD
HS
q
RD

50
w
50
min

s (H)  Hs (H) 
 
0.7
,
H

3/2


25H S
S
 RD s50   g H 
FN (H)  

RD 50 min
s (H)  Hs (H)  1 , H 

S

3/2




5
qw



s (H)  
s (H) , s (H)  
1 
6
2H
 H gHS 8.32  H  
 3D  

 s90  

For a given value Hp, the parameter min* is computed using the iterative
method of Slide 13.
27
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CALCULATION OF NORMAL DEPTH AND BACKWATER CURVE:
INTRODUCTION TO RTe-bookBackwaterWrightParker.xls
All three iterative schemes (i.e. for min*; Sf and Hs; and normal depth Hn). are
implemented in this workbook. The user specifies a flow discharge Qw, a channel
width B, a median grain size D50, a grain size D90 such that 90% of the bed material
is finer, a sediment submerged specific gravity R and a (constant) bed slope S.
Clicking the button “Click to compute normal depth” allows for computation of the
normal depth Hn.
Downstream bed elevation is set equal to 0, so that at normal conditions the
downstream water surface elevation d = Hn. The user may then specify a value of
d that differs from Hn (as long as the corresponding downstream Froude number is
less than unity), and compute the resulting backwater curve by clicking the button
“Click to compute backwater curve”.
The program generates a plot of bed and water surface elevations  and  versus
streamwise distance, as well as a plot of depth H and depth due to skin friction Hs
versus streamwise distance.
28
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SAMPLE CALCULATION WITH RTe-bookBackwaterWrightParker.xls: INPUT
Input cell
Output cell: do not write in these cells
(Qww)
(B)
(D50)
(D90)
(Rr)
(S)
Qw
B
D50
D90
R
S
1800
275
0.35
1
1.65
0.0001
3
m /s
m
mm
mm
(Hnorm) Hn
(xidnorm) dn
6.7491 m
d
Frd
L
M
20 m
(xid)
6.7491 m
0.0234
200000 m
50
Flow discharge
Channel width
Median grain size (sand)
Grain size such that 90% is finer (sand)
Submerged specific gravity of sediment
Bed slope
First input the above 6 parameters and
Click to compute normal
click to compute the normal depth.
depth
Downstream water surface elevation (must be >= dn)
Choose d so that the downstream Froude number Frd < 1
Reach length
Number of spatial nodes
Then input the above three parameters
Click to compute
(with d such that Frd < 1) and click
backwater curve
to compute the backwater curve.
29
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SAMPLE CALCULATION WITH RTe-bookBackwaterWrightParker.xls: OUTPUT
Bed elevation  (m) and water
surface elevation  (m)
30
25
20
eta (m)
xi (m)
15
10
5
0
0
50000
100000
150000
Distance x in downstream direction (m)
200000
30
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SAMPLE CALCULATION WITH RTe-bookBackwaterWrightParker.xls: OUTPUT
contd.
Depth H and depth due to skin friction Hs versus
downstream distance x
25
H (m), Hs(m)
20
15
H (m)
Hs (m)
10
5
0
0
50000
100000
150000
Downstream distance x (m)
200000
250000
31
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CALCULATION OF DREDGE SLOT EVOLUTION: INTRODUCTION TO
RTe-bookDredgeSlotBW.xls
The Excel workbook RTe-bookDredgeSlotBW.xls implements the formulation given
in the previous slides for the case of filling of a dredge slot. The code used in RTebookBackwaterWrightParker.xls is also used in RTe-bookDredgeSlotBW.xls.
The code first computes the equilibrium normal flow values of depth H, depth due to
skin friction Hs and volume bed load and bed material suspended load transport
rates per unit width qb and qs for given values of flood water discharge Qw, flood
intermittency If, channel width B, bed sediment sizes D50 and D90 (both assumed
constant), sediment submerged specific gravity R and (constant) bed slope S.
A dredge slot is then excavated at time t = 0. The hole has depth Hslot, width B and
length (rd - ru)L, where L is reach length, ruL is the upstream end of the dredge slot.
and rdL is the downstream end of the dredge slot. Once the slot is excavated, it is
allowed to fill without further excavation. Specification of the bed porosity p, the
number of spatial intervals M, the time step t, the number of steps to printout
Mtoprint, the number of printout after the one corresponding to the initial bed Mprint
and the upwinding coefficient au completes the input.
32
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SAMPLE CALCULATION OF DREDGE SLOT EVOLUTION: INPUT
This calculation for a very short time illustrates the state of the bed just after
excavation of the dredge slot.
Input parameters for computation of normal flow before installation of dredge slot
3
Qw
1650 m /s
Flow discharge
If
0.3
Flood intermittency
The colored boxes:
B
275 m
Channel width
indicate the parameters you must specify.
D50
0.35 mm
Median grain size (sand)
The rest are computed for you.
D90
1 mm
Grain size such that 90% is finer (sand)
R
1.65
Submerged specific gravity of sediment
S
0.0001
Bed slope
Input parameters for computation of morphodynamics after dredge slot installation at t = 0
The downstream water surface elevation is held at the value corresponding to normal flow throughout the ca
L
Hslot
ru
rd
p
M
x
t
Mtoprint
Mprint
au
10000
10
0.3
0.45
0.4
50
200
0.01
1
6
1
0.06
m
m
m
year
years
Reach length
Depth of dredge slot
Fraction of reach length defining upstrean end of dredge slot
Fraction of reach length defining downstream end of dredge slot
Bed Porosity
Number of intervals
Spatial step length
Click to run
Time step
Number of time steps to printout
Number of printouts
33
Upwinding coefficient: au = 1 corresponds to full upwinding (0.5 < au <= 1)
Duration of calculation
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SAMPLE CALCULATION OF DREDGE SLOT EVOLUTION: RESULT
The result indicates the backwater set up by the dredge slot.
Bed evolution (+ Water Surface at End of Run)
10
8
Elevation in m
6
bed 0 yr
bed 0.01 yr
bed 0.02 yr
bed 0.03 yr
bed 0.04 yr
bed 0.05 yr
bed 0.06 yr
ws 0.06 yr
4
2
0
-2
-4
-6
-8
-10
0
2000
4000
6000
Distance in m
8000
10000
34
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE DREDGE SLOT 3 YEARS LATER
Changing Mtoprint from 1 to 50 allows for a calculation 3 years into the future. The
bed degrades both upstream and downstream of the slot as it fills.
Bed evolution (+ Water Surface at End of Run)
10
8
Elevation in m
6
bed 0 yr
bed 0.5 yr
bed 1 yr
bed 1.5 yr
bed 2 yr
bed 2.5 yr
bed 3 yr
ws 3 yr
4
2
0
-2
-4
-6
-8
-10
0
2000
4000
6000
Distance in m
8000
10000
35
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE DREDGE SLOT 12 YEARS LATER
12 years later (Mtoprint from 1 to 200 the slot is nearly filled. Degradation upstream
of the slot is 0.3 ~ 0.4 m, and downstream of the slot it is on the order of meters.
Bed evolution (+ Water Surface at End of Run)
10
8
Elevation in m
6
bed 0 yr
bed 2 yr
bed 4 yr
bed 6 yr
bed 8 yr
bed 10 yr
bed 12 yr
ws 12 yr
4
2
0
-2
-4
-6
-8
-10
0
2000
4000
6000
Distance in m
8000
10000
36
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE DREDGE SLOT 24 YEARS LATER
The slot is filled and the degradation it caused is healing.
Bed evolution (+ Water Surface at End of Run)
10
8
Elevation in m
6
bed 0 yr
bed 4 yr
bed 8 yr
bed 12 yr
bed 16 yr
bed 20 yr
bed 24 yr
ws 24 yr
4
2
0
-2
-4
-6
-8
-10
0
2000
4000
6000
Distance in m
8000
10000
37
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE DREDGE SLOT 48 YEARS LATER
The slot and its effects have been obliterated; normal equilibrium has been restored.
Bed evolution (+ Water Surface at End of Run)
10
8
Elevation in m
6
bed 0 yr
bed 8 yr
bed 16 yr
bed 24 yr
bed 32 yr
bed 40 yr
bed 48 yr
ws 48 yr
4
2
0
-2
-4
-6
-8
-10
0
2000
4000
6000
Distance in m
8000
10000
38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
A SPECIAL CALCULATION: DREDGE SLOT PLUS BACKWATER
Consider the case of a very long reach:
Input parameters for computation of normal flow before installation of dredge slot
3
Qw
1650 m /s
Flow discharge
If
0.3
Flood intermittency
The colored boxes:
B
275 m
Channel width
indicate the parameters you must specify.
D50
0.35 mm
Median grain size (sand)
The rest are computed for you.
D90
1 mm
Grain size such that 90% is finer (sand)
R
1.65
Submerged specific gravity of sediment
S
0.0001
Bed slope
Input parameters for computation of morphodynamics after dredge slot installation at t = 0
The downstream water surface elevation is held at the value corresponding to normal flow throughout the
L
Hslot
ru
rd
p
M
x
t
Mtoprint
Mprint
au
200000
10
0.3
0.45
0.4
200
1000
0.1
400
6
1
240
m
m
m
year
years
Reach length
Depth of dredge slot
Fraction of reach length defining upstrean end of dredge slot
Fraction of reach length defining downstream end of dredge slot
Bed Porosity
Number of intervals
Spatial step length
Click to run
Time step
Number of time steps to printout
Number of printouts
Upwinding coefficient: au = 1 corresponds to full upwinding (0.5 < au <= 1)39
Duration of calculation
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
DREDGE SLOT PLUS BACKWATER: MODIFICATION TO CODE
Some minor modifications to the code allow the specification of a depth of 20 m at the
downstream end, insuring substantial backwater in addition to the dredge slot:
Sub Set_Initial_Bed_and_Time()
'sets initial bed, including dredge slot
Dim i As Integer: Dim iup As Integer: Dim idn As Integer
For i = 1 To M + 1
x(i) = dx * (i - 1)
eta(i) = S * L - S * dx * (i - 1)
Next i
time = 0
'xi(M + 1) = xid
xid = 20 'debug: use this statement and the statement below to specify
'backwater at downstream end
xi(M + 1) = xid 'debug
40
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
DREDGE SLOT PLUS BACKWATER: 240 YEARS LATER
Filling in the dredge slot retards filling in the backwater zone downstream (which
might be due to a dam).
Bed evolution (+ Water Surface at End of Run)
30
Elevation in m
25
bed 0 yr
bed 40 yr
bed 80 yr
bed 120 yr
bed 160 yr
bed 200 yr
bed 240 yr
ws 240 yr
20
15
10
5
0
0
50000
100000
Distance in m
150000
200000
41
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
DREDGE SLOT PLUS BACKWATER: 960 YEARS LATER
The dredge slot is filled and the backwater zone downstream is filling.
Bed evolution (+ Water Surface at End of Run)
30
Elevation in m
25
bed 0 yr
bed 160 yr
bed 320 yr
bed 480 yr
bed 640 yr
bed 800 yr
bed 960 yr
ws 960 yr
20
15
10
5
0
0
50000
100000
Distance in m
150000
200000
42
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
TRAPPING OF WASH LOAD
A sufficiently deep dredge slot can capture wash load (e.g. material finer than 62.5
m) as well as bed material load. As long as the dredge slot is sufficiently deep to
prevent re-entrainment of wash load, the rate at which wash load fills the slot can be
computed by means of a simple settling model.
Let Cwi = concentration of wash load in the ith grain size range, and vswi =
characteristic settling velocity for that range. Wash load has a nearly constant
concentration profile in the vertical, so that ro  1. Neglecting re-entrainment, the
equation of conservation of suspended wash load becomes
UHC wi
Cwi
 qw
 v swi Cwi
x
x
 v swi

 Cwi  Cuwi exp
(x  Lsu )
 qw

where Cuwi denotes the value of Cwi at the upstream end of the slot and Lsu denotes
the streamwise position of the upstream end of the slot. Including wash load, Exner
becomes

(qb  qs )
(1  p )
 - f
  f  v swi Cwi
t
x
As the slot fills, however, wash load is resuspended and carried out of the hole well
43
before bed material load.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
WHY WAS THE ENTRAINMENT FORMULATION OF EXNER NOT USED?
That is, why was not Exner implemented in terms of the relations

qb
(1  p )
 - f
  f v s cb  E
t
x
UCH
C
 qw
 v s E  roC
x
x
for bed material load as given in Slide 5?
The reason has to do with the relatively short relaxation distance for suspended sand.
To see this, consider the following case: Qw = 300 m3/s, If = 1, B = 60 m, D50 = 0.3
mm, D90 = 0.8 mm, R = 1.65 and S = 0.0002. The associated fall velocity for the
sediment is 0.0390 m/s (at 20 C). From worksheet “ResultsofCalc” of workbook
RTe-bookDredgeSlotBW.xls, it is found that the depth-averaged volume concentration
C is 5.97x10-5. The same calculation yields a value of E of 0.00705 m/s (must add a
line to the code to have this printed out), and thus the quasi-equilibrium value of ro =
E/C of 11.82.
44
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
RELAXATION DISTANCE FOR SUSPENDED SEDIMENT PROFILE
Now suppose that at time t = 0 this normal flow prevails everywhere, except that at x
= 0 the sediment is free of suspended sediment. The flow is free to pick up sediment
downstream of x = 0. As described in the pickup problem of Chapter 10 (and
simplified here with a depth-averaged formulation), the equation to be solved for the
spatial development of the profile of suspended sediment is
qw
dC
 v s (E  roC) , C x 0  0
dx
The solution to this equation is
 ro v s
E
C  1  exp 
ro 
 qw
H
u
c
x0
rigid bed

x 

erodible bed
45
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
RELAXATION DISTANCE FOR SUSPENDED SEDIMENT PROFILE contd.
Further setting qw = Qw/B = 5 m2/s, ro = 11.82, vs = 0.0390 m/s and E = 0.00705, the
following evaluation is obtained for C(x);
0.00007
0.00006
0.00005
C
0.00004
0.00003
0.00002
0.00001
0
0
10
20
30
40
50
x
Note that C is close to its equilibrium value of 5.97x10-5 by the time x = 30 m.
46
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
RELAXATION DISTANCE FOR SUSPENDED SEDIMENT PROFILE contd.
It is seen from the solution

 ro v s
C  roE1  exp 
 qw


x 

that the characteristic relaxation distance Lsr for adjustment of the suspended
sediment profile is
qw
Lsr 
ro v s
In the present case, Lsr is found to be 10.8 m.
Whenever Ls is shorter than the spatial step length x used in the calculation, it is
appropriate to assume that the suspended sediment profile is everywhere nearly
adapted to the flow conditions, allowing the use of the formulation

qt
qb
qs
(1  p )
 - f
 - f
- f
t
x
x
x
in place of the more complicated entrainment formulation. This is true for the cases
47
considered in this chapter.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
MASS CONSERVATION OF A MORPHODYNAMIC FORMULATION
An accurate morphodynamic formulation satisfies mass conservation. That is, the
total inflow of sediment mass into a reach must equal the total storage of sediment
mass within the reach minus the total outflow of sediment mass from the reach.
Consider a reach of length L. During floods, the mass inflow rate of bed material load
is B(R+1)qt(0, t) and the mass outflow rate is B(R+1)qt(L, t) where qt denotes the
volume bed material load per unit width. The Exner equation of sediment
conservation is

qt
qb
qs
(1  p )
 - f
 - f
- f
t
x
x
x
Integrating this equation from x = 0 to x = L yields the result
 L
(1  p )  dx  -  f qt (0, t)  qt (L, t)
t 0
48
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
MASS CONSERVATION OF A MORPHODYNAMIC FORMULATION contd.
In the present formulation

qt
(1  p )
 - f
t
x
is discretized to the form
qt,i
(1  p )i  - f
t
x
where I = i,new - I and
[au (qtf  qt,i )  (1  au )(qt,i  qt,i1)] / x , i  1

qt,i 
 [au (qt,i1  qt,i )  (1  au )(qt,i  qt,i1)] / x , 1  i  M  1
x 
(qt,i1  qt,i ) / x , i  M  1

In the above relation qtf is the sediment feed rate at x = 0 and au is an upwinding
coefficient.
49
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
MASS CONSERVATION OF A MORPHODYNAMIC FORMULATION contd.
Summing the discretized relation from i = 1 to i = M+1 and rearranging yields the
result
M1
M1
i1
i1
(1  p )x  i  - f  qt,i   f [uqtf  (1  u )qt,1  qt,M1]t
Thus as long as the volume inflow rate of bed material load per unit width is
interpreted as auqtf + (1 - au)qt,1 the method is mass-conserving: the volume storage
of sediment in the reach per unit width in one time step = the volume input of
sediment per unit width – the volume output of sediment per unit width, both over one
time step.
The method is specifically mass-conserving in terms of qtf if pure upwinding (au = 1) is
employed. (This method is numerically less accurate than partial upwinding,
however).
Mass conservation is tested numerically on worksheet “ResultsMassBalance” of
workbook RTe-booKDredgeSlotBW.xls.
50
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REFERENCES FOR CHAPTER 21
Ashida, K. and M. Michiue, 1972, Study on hydraulic resistance and bedload transport rate in
alluvial streams, Transactions, Japan Society of Civil Engineering, 206: 59-69 (in Japanese).
Wright, S. and G. Parker, 2004, Flow resistance and suspended load in sand-bed
rivers: simplified stratification model, Journal of Hydraulic Engineering, 130(8), 796-805.
51