chapter 21 response of a sand-bed river to a dredge slot

1

1D SEDIMENT TRANSPORT MORPHODYNAMICSwith applications to

RIVERS AND TURBIDITY CURRENTS© Gary Parker November, 2004

CHAPTER 21RESPONSE OF A SAND-BED RIVER TO A DREDGE SLOT

Grey Cloud Island

Mississippi River near Grey Cloud Island south of St. Paul, Minnesota

Image from NASAhttps://zulu.ssc.nasa.gov/mrsid/mrsid.pl

Lock and Dam no. 2

2



FILLING OF A DREDGE SLOT IN A SAND-BED RIVERThe 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?

navigation channel

dredge slot

3



FILLING OF A DREDGE SLOT IN A SAND-BED RIVER contd.

Note how a) a delta builds into the dredge slot from upstream and b) degradation propagates downstream of the slot.

before avulsion

just after avulsion

some time after avulsion

4



MORPHODYNAMICS OF DREDGE SLOT EVOLUTIONIn 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 intosuspension from the bed per unit area per unit time.

xq

xq

xq

t)1( s

fb

ft

fp

---

Ecvxq

t)1( bsf

bfp

-

bc

bc

5



ALTERNATIVE ENTRAINMENT FORMULATION FOR EXNERIf 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.

Ecvxq

t)1( bsf

bfp

-

bc bc

bcE

sv

xUCH

1

c

uv

bbb

csoob

b

s

dkH30n

/)1(/)1(,

kH,

vu,Czr,Crc

uUCz,

eH11k )Cz(c

6



BEDLOAD TRANSPORT AND ENTRAINMENT RELATIONSAn 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:

05.0,17q ccscsb

50s50s

bb

50s

bss DRgD

qq,RgD

50s50sp

bss

707.0f

6.0p

s

su

5u

5u

DRgD,u

10x7.5A,SvuZ,

Z3.0

A1

AZ

Re

ReENote that in the disequilibrium formulation considered here, bed slope S has been replaced with friction slope Sf.

7



RELATION FOR SUSPENDED SEDIMENT TRANSPORT RATEThe 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

where

In the case of the Wright-Parker formulation, b = 0.05.

1

c

uv

bbb

cs b

s

dkH30n

/)1(/)1(,

kH,

vu

HEuqs

8



SUMMARY OF MORPHODYNAMIC FORMULATION

1

c

uv

bbb

css

b

s

dkH30n

/)1(/)1(,

kH,

vu,HEuq

x)qq(

t)1( sb

fp

-

05.0,DRgD17q ccscs50s50sb

50s

sf

07.0f

6.0p

f

fsu

5u

5u

RgDv,S

)SgH

Z,Z

3.0A1

AZ

RRe

R(vEs

In order to finish the formulation, a resistance relation that includes the effectof bedforms is required. This relation must be adapted to mildly disequilibriumflows 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

2f

1SS

dxdH

Fr

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.

fs2

fffsbfbsb HHH,U)CC(

HgSgHgH

UCS fbb

2

ff

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 equationabove (specified in Wright and Parker, 2004) has been set equal to unity for simplicity in the present calculation.

61

90s

s

fs

w2/1fs D3

H32.8SgHH

qC

8.07.0

2/3w

50s

f

50

fs

Hgq

RDHS7.005.0

RDSH

8.07.0s )(7.005.0 Fr

11



LIMITS TO THE WRIGHT-PARKER FORMULATIONThe 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 > 1. In order to include values * < min*, the relation must be amended to:

HHss

b

bss

8.07.0

s)(7.005.0 Fr

min

min

8.07.0

s

,1

,)(7.005.0 Fr

12



PLOT OF s VERSUS * FOR THE CASE Fr = 0.2

0.1

1

10

0.01 0.1 1 10

*

s

Fr = 0.2

s = 1 for * < min* = 0.0922

8.07.0

s)(7.005.0 Fr

13



ITERATIVE COMPUTATION OF min* AS A FUNCTION OF FrThe 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,p+1*, where

The calculation proceeds until the relative error between min,p* and min,p+1* drops below some specified small tolerance << 1, i.e.

8.07.0minmin )(7.005.0 Fr

0)(7.005.0)(G 25/145/4minminmin Fr

25/145/1min

minpmin,

pmin,

pmin,pmin,1pmin, )(

547.01

ddG)(G,

)(G)(G

pmin,

Fr

)(21 pmin,1pmin,

pmin,1pmin,

14



IMPLEMENTATION OF ITERATIVE COMPUTATION OF min* AS A FUNCTION OF Fr

A sample calculation for the case Fr = 0.2 is given below.

0)(7.005.0)(G 25/145/4minminmin Fr 25/145/1

minpmin, )(547.01)(G Fr

)(G)(G

pmin,

pmin,pmin,1pmin,

Fr 0.2

p min* G G'

1 1 0.665767 0.7732 0.13829 0.029906 0.662 1.5143 0.09313 0.000578 0.634 0.394 0.09222 3.27E-07 0.634 0.015 0.09222 1.06E-13 0.634 6E-06

)(21 pmin,1pmin,

pmin,1pmin,

15



PLOT OF min* VERSUS Fr

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.2 0.4 0.6 0.8 1

Fr

min*

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 < 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

CODE FOR COMPUTING min*

17



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:

61

90s

s

fs

w

D3H32.8

SgHHq

min50s

f

50s

f

min50s

f

8.07.0

2/3w

50s

f

50

fs

RDHS,

RDHS

RDHS,

Hgq

RDHS7.005.0

RDSH

Now writing = Hs/H, the top equation can be solved for Sf to yield

3/1

90s

2

nomnom3/4

sf D3H

8.32S,SS

Fr

CALCULATION OF Hs AND Sf FROM KNOWN DEPTH H

18



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 to50s

nomnom RD

HS

CALCULATION OF Hs AND Sf FROM KNOWN DEPTH H contd.

minnom3/4

snom3/4

s

minnom3/4

s25/145/4

nom15/16

snom

3/1s ,

,)(7.005.0 Fr

0

)/(,1

)/(,)(7.0

05.0)(F

4/3minnomss

4/3minnoms

16/15

25/145/4nom

nom3/1

ss

s

Fr

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

)(F)(F

p,s

p,sp,s1p,s

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

)(21 p,s1p,s

p,s1p,s

0

)/(,1

)/(,)(7.03

1)(7.0

05.016151)(F

4/3minnoms

4/3minnoms25/145/4

nom

nom3/4

s

16/31

25/145/4nom

nom3/1

s

s

FrFr

20



SAMPLE IMPLEMENTATION OF NEWTON-RAPHSON SCHEMEThe 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

Note that the relative error is below 0.1% by the fourth iteration (p = 5).

496.0RDHS,000136.0

kH32.8

S,307.0Hg

q

50s

nomnom

2

6/1

s

nom2/3w

FrFr

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:

p s F(s) F'(s) Hs Sf

1 0.99 0.506295 0.83026 2.97 0.0001382 0.38019 0.03149 0.69079 0.89 1.14058 0.0004953 0.33461 0.000577 0.66455 0.128 1.00383 0.0005874 0.33374 2.42E-07 0.664 0.003 1.00122 0.0005895 0.33374 4.26E-14 0.664 1E-06 1.00122 0.000589

21



NOTE OF CAUTION CONCERNING THE NEWTON-RAPHSON SCHEMEThere 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 0.92. 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,however, is rather slow to converge.

22



APPLICATION TO BACKWATER CALCULATIONS

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.

The backwater equation takes the form

2f

1SS

dxdH

Fr

A first guess of Hi-1 is given as Hi-1,pred, where

2/3i

wi

i,fi1ii,fpred,1ii

Hgq,

1S]x)[(

1SS

xHH

FrFrFr 2

i2i

Hi

Hi-1

i-1

i

flow

x

or thus:2iFr

1xS

HH i,fi1iipred,1i

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:

2/3pred,1i

wpred,1i2

pred1,-i

pred,1i,fi1ii,fi1i1ii

Hgq,

1S]x)[(

1S]x)[(

21

xHH

Fr

FrFr2i

Hi

Hi-1

i-1

i

flow

x

or thus:

Once Hi-1 is known the Newton-Raphson scheme can be used again to compute Hs,i-1 and Sf,i-1.

2pred1,-i

pred,1i,fi1ii,fi1ii1i 1

xS1

xS21HH

FrFr2i

24



CALCULATION OF NORMAL DEPTH FROM GIVEN VALUES OFqw, 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.

61

90s

s

s

w

D3H32.8

SgHHq

min50s50s

min50s

8.07.0

2/3w

50s

50

s

RDHS,

RDHS

RDHS,

Hgq

RDHS7.005.0

RDSH

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

2/3

61

90s

ws

D3H32.8gHSH

q)H(

SRDH,H

SRDH,

Hgq

RDHS7.005.0

SRD

)H(H

min50

min50

25/14

2/3w

5/4

50s

50

s

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):

0

SRDH,H)H(H

SRDH,

Hgq

RDHS7.005.0

SRD)H(H

)H(F

min50s

min50

25/14

2/3w

5/4

50s

50s

N

)H(F

)H(FHH

pN

pNp1p

32.8,gS

q)D3(H r

10/3

2r

2w

3/190

27



CALCULATION OF NORMAL DEPTH contd.

SRDH,1)H(H)H(

SRDH,

Hgq

RDHS7.0

SRD

H251)H(H)H(

)H(Fmin50

ss

min50

25/14

2/3w

5/4

50s

50ss

N

2/3

61

90s

wsss

D3H32.8gHSH

q)H(,)H(H25)H(

The scheme thus becomes

)H(F

)H(FHH

pN

pNp1p

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-bookBackwaterWrightParker.xls: INPUT

Input cellOutput cell: do not write in these cells

(Qww) Qw 1800 m3/s Flow discharge(B) B 275 m Channel width(D50) D50 0.35 mm Median grain size (sand)(D90) D90 1 mm Grain size such that 90% is finer (sand)(Rr) R 1.65 Submerged specific gravity of sediment(S) S 0.0001 Bed slope

First input the above 6 parameters and (Hnorm) Hn 6.7491 m click to compute the normal depth.(xidnorm)

dn 6.7491 m

(xid) d 20 m Downstream water surface elevation (must be >= dn)

Frd 0.0234 Choose d so that the downstream Froude number Frd < 1L 200000 m Reach lengthM 50 Number of spatial nodes

Then input the above three parameters(with d such that Frd < 1) and clickto compute the backwater curve.

Click to compute normal depth

Click to compute backwater curve

30



SAMPLE CALCULATION WITH RTe-bookBackwaterWrightParker.xls: OUTPUT

0

5

10

15

20

25

30

0 50000 100000 150000 200000

Distance x in downstream direction (m)

Bed

ele

vatio

n

(m) a

nd w

ater

su

rfac

e el

evat

ion

(m)

eta (m)xi (m)

31



SAMPLE CALCULATION WITH RTe-bookBackwaterWrightParker.xls: OUTPUT contd.

Depth H and depth due to skin friction Hs versus downstream distance x

0

5

10

15

20

25

0 50000 100000 150000 200000 250000

Downstream distance x (m)

H (m

), H

s(m)

H (m)Hs (m)

32



CALCULATION OF DREDGE SLOT EVOLUTION: INTRODUCTION TORTe-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: INPUTThis 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 slotQw 1650 m3/s Flow dischargeIf 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 sedimentS 0.0001 Bed slope

Input parameters for computation of morphodynamics after dredge slot installation at t = 0The downstream water surface elevation is held at the value corresponding to normal flow throughout the calculation

L 10000 m Reach lengthHslot 10 m Depth of dredge slotru 0.3 Fraction of reach length defining upstrean end of dredge slotrd 0.45 Fraction of reach length defining downstream end of dredge slot

p 0.4 Bed PorosityM 50 Number of intervalsx 200 m Spatial step lengtht 0.01 year Time stepMtoprint 1 Number of time steps to printoutMprint 6 Number of printoutsau 1 Upwinding coefficient: au = 1 corresponds to full upwinding (0.5 < au <= 1)

0.06 years Duration of calculation

Click to run

34



SAMPLE CALCULATION OF DREDGE SLOT EVOLUTION: RESULTThe result indicates the backwater set up by the dredge slot.

Bed evolution (+ Water Surface at End of Run)

-10

-8

-6

-4

-2

0

2

4

6

8

10

0 2000 4000 6000 8000 10000

Distance in m

Elev

atio

n in

m

bed 0 yrbed 0.01 yrbed 0.02 yrbed 0.03 yrbed 0.04 yrbed 0.05 yrbed 0.06 yrws 0.06 yr

35



THE DREDGE SLOT 3 YEARS LATERChanging 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.


-10

-8

-6

-4

-2

0

2

4

6

8

10

0 2000 4000 6000 8000 10000

Distance in m

Elev

atio

n in

m

bed 0 yrbed 0.5 yrbed 1 yrbed 1.5 yrbed 2 yrbed 2.5 yrbed 3 yrws 3 yr

36



THE DREDGE SLOT 12 YEARS LATER12 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.


-10

-8

-6

-4

-2

0

2

4

6

8

10

0 2000 4000 6000 8000 10000

Distance in m

Elev

atio

n in

m

bed 0 yrbed 2 yrbed 4 yrbed 6 yrbed 8 yrbed 10 yrbed 12 yrws 12 yr

37



THE DREDGE SLOT 24 YEARS LATERThe slot is filled and the degradation it caused is healing.


-10

-8

-6

-4

-2

0

2

4

6

8

10

0 2000 4000 6000 8000 10000

Distance in m

Elev

atio

n in

m


38



THE DREDGE SLOT 48 YEARS LATERThe slot and its effects have been obliterated; normal equilibrium has been restored.


-10

-8

-6

-4

-2

0

2

4

6

8

10

0 2000 4000 6000 8000 10000

Distance in m

Elev

atio

n in

m


39



A SPECIAL CALCULATION: DREDGE SLOT PLUS BACKWATERConsider the case of a very long reach:

Input parameters for computation of normal flow before installation of dredge slotQw 1650 m3/s Flow dischargeIf 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 sedimentS 0.0001 Bed slope

Input parameters for computation of morphodynamics after dredge slot installation at t = 0The downstream water surface elevation is held at the value corresponding to normal flow throughout the calculation

L 200000 m Reach lengthHslot 10 m Depth of dredge slotru 0.3 Fraction of reach length defining upstrean end of dredge slotrd 0.45 Fraction of reach length defining downstream end of dredge slot

p 0.4 Bed PorosityM 200 Number of intervalsx 1000 m Spatial step lengtht 0.1 year Time stepMtoprint 400 Number of time steps to printoutMprint 6 Number of printoutsau 1 Upwinding coefficient: au = 1 corresponds to full upwinding (0.5 < au <= 1)

240 years Duration of calculation

Click to run

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).


0

5

10

15

20

25

30

0 50000 100000 150000 200000

Distance in m

Elev

atio

n in

m


42



DREDGE SLOT PLUS BACKWATER: 960 YEARS LATER The dredge slot is filled and the backwater zone downstream is filling.


0

5

10

15

20

25

30

0 50000 100000 150000 200000

Distance in m

Elev

atio

n in

m


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.

)Lx(qvexpCCCv

xCq

xUHC

suw

swiuwiwiwiswi

wiw

wi

wiswifsb

fp Cvx

)qq(t

)1(

-

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.

Ecvxq

t)1( bsf

bfp

-

CvxCq

xUCH

sw orE

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

0C,)CrE(vdxdCq

0xosw

H

rigid bed erodible bed

cu

x

qvrexp1

rEC

w

so

o

0x

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.

0

0.00001

0.00002

0.00003

0.00004

0.00005

0.00006

0.00007

0 10 20 30 40 50

x

C

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.

x

qvrexp1ErCw

soo

so

wsr vr

qL

xq

xq

xq

t)1( s

fb

ft

fp

---

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(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

Integrating this equation from x = 0 to x = L yields the result

xq

xq

xq

t)1( s

fb

ft

fp

---

)t,L(q)t,0(qdxt

)1( ttf

L

0p

-

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.

txq

)1( it,fip

-

xq

t)1( t

fp

-

1Mi,x/)qq(1Mi1,x/)]qq)(a1()qq(a[

1i,x/)]qq)(a1()qq(a[

xq

i,t1i,t

1i,ti,tui,t1i,tu

1i,ti,tui,ttfuit,

50



MASS CONSERVATION OF A MORPHODYNAMIC FORMULATION contd. Summing the discretized relation from i = 1 to i = M+1 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.

t]qq)1(q[qx)1( 1M,t1,tutfuf

1M

1iit,f

1M

1iip

-

51



REFERENCES FOR CHAPTER 21Ashida, 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.

chapter 21 response of a sand-bed river to a dredge slot

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