RESPONSE OF A SAND-BED RIVER TO A DREDGE SLOT

1
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.s
sc.nasa.gov/mrsid/mrsid.pl
2
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
3
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.
4
MORPHODYNAMICS OF DREDGE SLOT EVOLUTION
In the modeling performed here, the following
form of the Exner equation of sediment continuity
is used 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, where 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
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 suspension
from the bed per unit area per unit time.
5
ALTERNATIVE ENTRAINMENT FORMULATION FOR EXNER
If the alternative formulation for conservation
of bed sediment is used, then a) the
quasi-steady form of the equation of conservation
of suspended sediment, i.e. must be solved
simultaneously, and in addition the above
relations must be closed by relating to C.
For example, writing roC, the following
quasi-steady evalution is obtained from the
material of Chapter 10 This alternative
formulation is not used here. The reasons for
this are explained toward the end of the chapter.
6
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 the relation takes the form An
appropriate relation for the entrainment of sand
into suspension is that of Wright and Parker
(2004) introduced in Chapter 10
Note that in the disequilibrium formulation
considered here, bed slope S has been replaced
with friction slope Sf.
7
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),
us (gHsSf)1/2 and E can be computed
everywhere. The volume transport rate per unit
width of suspended sediment is thus computed
as where In the case of the
Wright-Parker formulation, ?b 0.05.
8
SUMMARY OF MORPHODYNAMIC FORMULATION
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.
9
CALCULATION OF GRADUALLY VARIED FLOW IN SAND-BED
RIVERS INCLUDING THE EFFECT OF BEDFORMS
The backwater equation presented in Chapter 5 is
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 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 where Cfs
and Cff denote resistance coefficients due to
skin friction and form drag.
10
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 for skin friction and the form 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, 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 simplicity in the present calculation.
11
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 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 For any given value of Froude number
Fr, it is found that a minimum Shields number
?min exists, below which ?s gt 1. In order to
include values ? lt ?min, the relation must be
amended to
12
PLOT OF ?s VERSUS ? FOR THE CASE Fr 0.2
13
ITERATIVE COMPUTATION OF ?min AS A FUNCTION OF Fr
The equation for ?min takes the form or 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,p1, where The calculation
proceeds until the relative error ? between
?min,p and ?min,p1 drops below some specified
small tolerance ? ltlt 1, i.e.
14
IMPLEMENTATION OF ITERATIVE COMPUTATION OF ?min
AS A FUNCTION OF Fr
A sample calculation for the case Fr 0.2 is
given below.
15
PLOT OF ?min VERSUS Fr
16
Sub Find_tausmin(xFr, xtausmin) (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 lt ep
Then Found_tausmin True
Else If ittau gt 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
CODE FOR COMPUTING ?min
17
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
Now writing ? Hs/H, the top equation can be
solved for Sf to yield
18
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
where
This relation reduces to
19
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
where the prime denotes a derivative with respect
to ?s. Computing the derivative,
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
20
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
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
Note that the relative error ? is below 0.1 by
the fourth iteration (p 5).
21
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.
• The initial guess for ?s is set equal to 0.9
• Whenever the iterative scheme yields a negative
  value of ?s, ?s is reset to 1.02 and the
  iterative calculation recommenced.
• 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,
• however, is rather slow to converge.

22
APPLICATION TO BACKWATER CALCULATIONS
The backwater equation takes the form
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
or thus
23
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 predictor-corrector scheme is
used to evaluate Hi-1
or thus Once Hi-1 is known the Newton-Raphson
scheme can be used again to compute Hs,i-1 and
Sf,i-1.
24
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.
25
CALCULATION OF NORMAL DEPTH contd.
Again introducing the notation ?s Hs/H, the
first equation of the previous slide reduces
to The second equation of the previous
slide then reduces with the above equation to
26
CALCULATION OF NORMAL DEPTH contd.
The second equation of the previous slide can
also be rewritten as The corresponding
Newton-Raphson scheme is 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)
27
CALCULATION OF NORMAL DEPTH contd.
The scheme thus becomes
For a given value Hp, the parameter ?min is
computed using the iterative method of Slide 13.
28
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.
29
SAMPLE CALCULATION WITH RTe-bookBackwaterWrightPar
ker.xls INPUT
30
SAMPLE CALCULATION WITH RTe-bookBackwaterWrightPar
ker.xls OUTPUT
31
SAMPLE CALCULATION WITH RTe-bookBackwaterWrightPar
ker.xls OUTPUT contd.
32
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 RTe-bookBackwaterWrightParker.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.
33
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.
34
SAMPLE CALCULATION OF DREDGE SLOT EVOLUTION
RESULT
The result indicates the backwater set up by the
dredge slot.
35
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.
36
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.
37
THE DREDGE SLOT 24 YEARS LATER
The slot is filled and the degradation it caused
is healing.
38
THE DREDGE SLOT 48 YEARS LATER
The slot and its effects have been obliterated
normal equilibrium has been restored.
39
A SPECIAL CALCULATION DREDGE SLOT PLUS BACKWATER
Consider the case of a very long reach
40
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
41
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).
42
DREDGE SLOT PLUS BACKWATER 960 YEARS LATER
The dredge slot is filled and the backwater zone
downstream is filling.
43
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 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 As
the slot fills, however, wash load is resuspended
and carried out of the hole well before bed
material load.
44
WHY WAS THE ENTRAINMENT FORMULATION OF EXNER NOT
USED?
That is, why was not Exner implemented in terms
of the relations 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.
45
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 The solution to this equation is
46
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)
Note that C is close to its
equilibrium value of 5.97x10-5 by the time x 30
m.
47
RELAXATION DISTANCE FOR SUSPENDED SEDIMENT
PROFILE contd.
It is seen from the solution that the
characteristic relaxation distance Lsr for
adjustment of the suspended sediment profile
is 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 in place of the more complicated
entrainment formulation. This is true for the
cases considered in this chapter.
48
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(R1)qt(0, t) and the mass outflow rate is
B(R1)qt(L, t) where qt denotes the volume bed
material load per unit width. The Exner equation
of sediment conservation is Integrating this
equation from x 0 to x L yields the result
49
MASS CONSERVATION OF A MORPHODYNAMIC FORMULATION
contd.
In the present formulation is discretized to
the form where ??I ?i,new - ?I and In
the above relation qtf is the sediment feed rate
at x 0 and au is an upwinding coefficient.
50
MASS CONSERVATION OF A MORPHODYNAMIC FORMULATION
contd.
Summing the discretized relation from i 1 to i
M1 and rearranging yields the result 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.
51
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.