1 1d sediment transport morphodynamics with applications to rivers and turbidity currents © gary...

1

1D SEDIMENT TRANSPORT MORPHODYNAMICSwith applications to

RIVERS AND TURBIDITY CURRENTS© Gary Parker November, 2004

CHAPTER 26:RIVERS FLOWING INTO SUBSIDING BASINS: UPWARD CONCAVITY OF LONG

PROFILE AND DOWNSTREAM FINING

As noted in Chapter 25, basin subsidence can drive an upward-concave long profile of a river. An upward-concave long profile in turn implies a bed slope that decreases in the downstream direction. When sediment mixtures are transported over a bed with a slope that declines downstream, the coarser material can be expected to be preferentially deposited upstream. The result is a pattern of downstream fining (e.g. Paola et al., 1992, Cui et al., 1996; Cui and Parker, 1997).

Rivers entering a (subsiding) graben in

eastern Taiwan.Image from NASA

website:https://zulu.ssc.nasa.gov/mrsid/mrsid.pl

2



CHARACTERIZATION OF PROFILE CONCAVITY AND DOWNSTREAM FINING

)D(n sg2s

x

h

As noted in Chapter 25, an upward-concave long profile is characterized by a bed slope S = - h/x that declines downstream. Downstream fining can in turn be characterized in terms of a surface geometric geometric mean size Dsg [or corresponding arithmetic mean ] or surface median size Ds50 [or corresponding arithmetic median ] that declines in the downstream direction.

)D(n sg2s )D(n 50s250s

3



SUBSIDENCE AND PROFILE CONCAVITY: UNIFORM SEDIMENT TRANSPORTED AT CONSTANT FLOW IN A FLUME-LIKE SETTING

Before considering the case downstream fining associated with the transport of sediment mixtures in subsiding basins, it is of value to consider how a river carrying uniform sediment responds to a subsiding basin.

In a simple first analysis, the flow is considered to be constant (If = 1) and the river is taken to be flume-like, i.e. infinitely high vertical walls confining the flow to a straight channel. From Chapter 4, the appropriate form of the Exner equation of sediment conservation is

whereh = bed elevation [L]x = spatial coordinate [L]t = time [T]qt = total volume bed material load per unit width [L2/T]p = bed porosity [1] = subsidence rate [L/T]

x

q

t)1( t

p

h

-

4



STEADY STATE UPWARD-CONCAVE LONG PROFILE IN THE PRESENCE OF SUBSIDENCE

Let the subsidence rate be constant in space and time, and the upstream total volume feed rate per unit width of bed material qtf be constant in time. If the profile is allowed to evolve for a sufficient amount of time, it will achieve a steady state

for which the governing equation isx

q

t)1( t

p

h

-

)1(xd

qdp

t

This equation defines a perfect balance between the creation of accomodation space to store sediment by subsidence, and filling of this accomodation space by sediment deposition.

x

h Dt

deposit over time Dt

5



STEADY STATE UPWARD-CONCAVE LONG PROFILE IN THE PRESENCE OF SUBSIDENCE contd.

The equation

subject to the boundary condition

integrates to yield

)1(xd

qdp

t

x

h Dt

deposit over time Dtx)1(qq ptft

tf0xt qq

Note that the sediment transport rate declines linearly downstream as it is consumed in filling the accomodation space created by subsidence.

6



STEADY STATE UPWARD-CONCAVE LONG PROFILE IN THE PRESENCE OF SUBSIDENCE contd.

The sediment transport rate qt drops to zero where x = Lmax, where

The downstream variation in sediment transport rate can then be written as

)1(

qL

p

tfmax

maxtf

t

L

x1

q

q

Here Lmax denotes the maximum length of basin that the sediment supply can fill. At this length the sediment transport rate out of the basin drops precisely to zero, i.e.

0qmaxLxt

qt/qtf

x

1

0Lmax

7



SOLUTION FOR STEADY STATE PROFILE WITH SUBSIDENCE: UNIFORM MATERIAL

In Chapter 14 the following generic sediment transport relation for uniform sediment was introduced:

In the above relation g denotes the acceleration of gravity, D denotes grain size, R denotes the submerged specific gravity of the sediment ( = s/ - 1), * denotes the Shields number, c* denotes a critical Shields number at the threshold of motion, s denotes the fraction of the bed shear stress that is skin friction, t is a coefficient and nt is an exponent. In Chapter 14 this relation was further reduced with the normal flow assumption and the Manning-Strickler resistance relation to the form

where S = - h/x = bed slope and kc = composite bed roughness height.

tn

c

10/710/3

2r

2w

3/1c

stt RD

S

g

qkDRgDq

tn

cstt

DRgD

q

8



SOLUTION FOR STEADY STATE PROFILE WITH SUBSIDENCE: UNIFORM MATERIAL contd.

Reducing the two relations below

yields the following analytical solution for the steady state slope profile:

Assuming h = 0 at x = Lmax, the steady state bed elevation profile is given as

maxtft

n

c

10/710/3

2r

2w

3/1c

stt L

x1qq,

RD

S

g

qkDRgDq

t

7/3

2w

3/1c

2r

7/10

s

7/10

c

n/1

t

maxtf

qk

gRD

DRgD

Lx

1q

S

t

hhh

max

max

L

xLxSdx0andS

dx

d

9



SAMPLE IMPLEMENTATION FOR UNIFORM MATERIAL

t 4

nt 1.5

c* 0.0495

s 1

R 1.65

D 20 mm

kc 60 mm

qw 3 m2/s

qtf 0.004 m2/s

p 0.4

4000 mm/yr

m52600)1(

qL

p

tfmax

That is, if the subsidence rate is equal to 4000 mm/year (4 m/year), the maximum basin length at which the sediment transport rate runs to zero is 52.6 km.

It should be noted that a subsidence rate of 4 m/year is at least three orders of magnitude too high, whether the processes involved be either tectonic or associated with the compaction of sediment under its own weight (consolidation). The issue is resolved in succeeding slides. This notwithstanding, the results of the sample implementation are given in the next slide.

10



SAMPLE IMPLEMENTATION FOR UNIFORM MATERIAL contd.

Steady State Profiles: Bed Elevation, Bed Slope, Load

0

50

100

150

200

250

300

350

0 10000 20000 30000 40000 50000 60000

x (m)

h (m

)

0

0.002

0.004

0.006

0.008

0.01

0.012

qt (

m2/s

), S

etaqtS

Load declines linearly downstream, slope declines downstream, bed elevation profile is upward concave.

= 4000 mm/year

11



The Exner equation in question, i.e. the one below,

requires adaptation in order to allow application to real rivers.

The first adaptation is the inclusion of a constant flood intermittency If, as first introduced in Chapter 14, in recognition of the fact that most of the time rivers are not morphologically active. Thus the Exner equation is modified to

GENERALIZATION OF THE EXNER EQUATION FOR UNIFORM SEDIMENT FROM A FLUME-LIKE SETTING TO A RIVER

x

q

t)1( t

p

h

-

In point of fact the analysis could be modified to include entire hydrographs, as was done in Chapter 19.

x

qI

t)1( t

fp

h

-

t

Qlow flow

flood

12



The second adaptation recognizes the fact that in an aggrading river sediment deposits not only in the channel itself, but also in a much wider belt (e.g. the floodplain or basin width, due to overbank deposition, channel migration and avulsion. Here channel width is denoted as Bc (which can be taken to be synonymous with bankfull width) and effective depositional width is denoted as Bd. Both of these are taken as constant here for simplicity. Otherwise the necessary adaptation is based on a generalization of that given in Chapter 25. The Exner equation now takes the form

GENERALIZATION OF THE EXNER EQUATION FOR UNIFORM SEDIMENT FROM A FLUME-LIKE SETTING TO A RIVER contd.

x

qBI

tB)1( t

cfdp

h

-

Bd

Bc

The above cross-section shows channel bodies resulting from migration and avulsion across a depositional surface as the river aggrades.

13



The third adaptation recognizes the fact that recognizes that much of the sediment that deposits over the depositional width can be expected to be effective “washload” in terms of the channel, i.e. sand in the case of a gravel-bed stream or mud in the case of a sand-bed stream. Here it is assumed that for each unit of bed material load deposited units of wash load are deposited. The adaptation is that given in Chapter 25. The Exner equation now takes the form


x

qB)1(I

tB)1( t

cfdp

h

-

Architecture of fill across depositional width: view looking downstream.

Overbank deposits:mixture of bed materialload and wash load

Channel deposits: bed material load

14



The fourth adaptation recognizes the fact that channels may be sinuous. Here it is assumed that the channel has sinuosity , but that the depositional surface across which it wanders is rectangular. The appropriate modification of the Exner equation of sediment continuity is given in Chapter 25. The result is given below: note that x remains a down-channel coordinate.

The above equation may be rewritten as


x

qB)1(I

tB)1( t

cfdp

h

-

Bd

Bc

x

q

r)1(

)1(I

tt

Bp

f

h

-

where

c

dB B

Br

15



The steady-state profile of a river with constant flow in a flume-like setting was studied in Slide 5 with the following form of the Exner equation:

The generalizations for a real river encompassed in the Exner equation of the previous slide result in the following form for a steady state profile:

In real rivers the effect of subsidence tends to be greatly amplified as compared with a flume-like setting with constant flow. Consider, for example the case rB = 60, If = 0.025, = 1 and = 1.5. The amplification factor is given as

REVISITATION OF THE CASE OF STEADY STATE PROFILE WITH SUBSIDENCE:RIVER VERSUS FLUME-LIKE SETTING

)1()1(I

r

xd

qdp

f

Bt

)1(xd

qdp

t

800)1(I

r

f

B

so that the downstream rate of loss of sediment to fill the hole created by subsidence is 800 times greater than for the case of constant flow in a flume-like setting.

16



In the example of the previous slide.the amplification factor is given as

The amplification is due to the fact that rB = Bd/Bc > 1 and If < 1 in most natural cases of interest. That is a) at any given time the river deposits its sediment only in or near the river itself, whereas subsidence is assumed to be occurring at average rate over the entire depositional width, and b) the river is morphologically active only a fraction of the time, whereas the basin is assumed to be subsiding at average rate all the time. The parameters > 0 and > 1 reduce, not amplify the effective subsidence rate, but their effect us usually not nearly so profound as rB and If.

THE REASONS FOR THE AMPLIFICATION OF SUBSIDENCE IN A THE CASE OF A RIVER AS COMPARED TO A FLUME-LIKE SETTING WITH CONSTANT FLOW

800)1(I

r

f

B

t

Qlow flow

floodBd

Bc

17



The solution for the steady-state profile of Slides 6 is correspondingly modified to yield the forms

The solution of Slide 9 for the profiles of bed slope and elevation remain unmodified:

REVISITATION OF THE STEADY-STATE PROFILE contd.

It was noted in the previous slide that the values rB = 60, If = 0.025, = 1 and = 1.5 lead to an amplification of subsidence by a factor of 800. As a result, the above values combined with a subsidence rate of only 5 mm/year leads to exactly the same profiles as the case of a flume-like setting with constant flow and a subsidence rate of 4000 mm/year.

,L

x1

q

q

maxtf

t

)1(

q

r

)1(IL

p

tf

B

fmax

,qk

gRD

DRgD

Lx

1q

S7/3

2w

3/1c

2r

7/10

s

7/10

c

n/1

t

maxtf

t

h maxL

xSdx

18



SAMPLE IMPLEMENTATION FOR UNIFORM MATERIAL: GENERALIZATION TO RIVER

t 4

nt 1.5

c* 0.0495

s 1

R 1.65

D 20 mm

kc 60 mm

qw 3 m2/s

qtf 0.004 m2/s

p 0.4

5 mm/yr

rB 60

If 0.025

1

1.5

Now a subsidence rate of only 5 mm/year produces a maximum depositional length that is precisely equal to the value associated with a subsidence rate of 4000 mm/year in a flume-like setting with constant flow.

The resulting profiles for bed material load qt, bed slope S and bed elevation h given in the next slide are exactly the same as those given in Slide 10.

m200,56)1(

q

r

)1(IL

p

tf

B

fmax

19



GENERALIZATION TO RIVER contd.

Steady State Profiles: Bed Elevation, Bed Slope, Load

0

50

100

150

200

250

300

350

0 10000 20000 30000 40000 50000 60000

x (m)

h (m

)

0

0.002

0.004

0.006

0.008

0.01

0.012

qt (

m2/s

), S

etaqtS

Load declines linearly downstream, slope declines downstream, bed elevation profile is upward concave.

= 5 mm/year(but rB = 60, If = 0.025, = 1.5, = 1)

20



APPROACH TO STEADY STATE FOR UNIFORM MATERIAL

The approach to steady state can be studied by solving the full form of the Exner equation of Slide 14, i.e.

with the load equation of Slide 8,

x

q

r)1(

)1(I

tt

Bp

f

h

-

tn

c

10/710/3

2r

2w

3/1c

stt RD

S

g

qkDRgDq

and the boundary conditions

where L Lmax (so that there is always sufficient sediment to fill the basin). The initial condition is set in terms of an initial bed slope SI.

0,qqLxtf0xt h

x

h Dt

deposit over time Dt downstream point of fixed

elevation

qtf

x = L < Lmaxx = 0

21



APPROACH TO STEADY STATE FOR UNIFORM MATERIAL contd.

The formulation is almost identical to that of Chapter 14. The spatial grid is defined in terms of M intervals bounded by M + 1 nodes (+ a ghost node):

The Exner equation discretizes to

where

and au is an upwinding coefficient that can be set equal to 0.5 here.

M

Lx D 1M..1i,x)1i(x i DFeed sediment here!

L

Dx

i=1 2 3 M -1 i = M+1ghost M

M..1i,tIx

q

r)1(

)1(If

i,t

Bp

ftitti D

DD

hhD

x

qq)a1(

x

qqa

x

q i,t1i,tu

1i,ti,tu

i,t

D

D

DD

22



INTRODUCTION TO RTe-bookAgDegNormalSub.xls , A CALCULATOR FOR THE APPROACH TO EQUILIBRIUM IN A RIVER CARRYING UNIFORM

MATERIAL AND FLOWING INTO A SUBSIDING BASIN

The program RTe-bookAgDegNormalSub.xls is a descendant of RTe-bookAgDegNormal.xls. Three relatively minor changes have been implemented as follows.

a) The data input worksheet “Calculator” has been modified to reflect the purpose of the code, and so as to include the following parameters: subsidence rate , ratio of depositional width to channel width rB, ratio of wash load deposited per unit bed material load and channel sinuosity .

b) The code has been modified so as to include subsidence in the calculation of mass balance.

c) The output has been modified to show the time evolution of not only the profile of bed elevation h, but also the profiles of bed slope S and the ratio qt/qtf, where qtf denotes the volume feed rate of bed material load per unit width.

23



Input parametersqw qw 3 m2/s Water discharge per unit width during flood

If Inter 0.025 Flood intermittencyD D 20 mm Grain size

p lamp 0.4 Bed porosity

kc kc 60 mm Roughness height

SI SI 0.005 Initial bed slope

qtf qqtf 0.004 m2/s Volume sediment feed rate per unit width subrate 5 mm/yr Subsidence raterB rB 60 Ratio of depositional width to channel width omega 1.5 Channel sinousity lamda 1 Unit wash load deposited in channel-floodplain per unit bed material loadLmax 52596 m Maximum reach lengthL L 50000 m Length of reach (must be less than maximum reach length)

M 50 IntervalsNtoprint 1000 Number of time steps to printoutNprint 5 Number of printouts

Dt dt 1 year time stepau au 0.5 Here 1 = full upwind, 0.5 = central differenceDx dx 1000 m Spatial step length

5000 years Duration of calculation

Auxiliary Input Parameters

r alr 8.1

t alt 4

nt nt 1.5

c* tausc 0.0495

s fis 1R Rr 1.65

The following input parameters were used to compute a case for a gravel-bed stream using RTe-bookAgDegNormalSub.xls. Calculations were performed for durations of 5000 and 25,000 years. Note L = 50,000 m is slightly less than Lmax.

CALCULATIONS FOR A GRAVEL-BED RIVER USING RTe-bookAgDegNormalSub.xls

24




Bed elevation evolution

0

50

100

150

200

250

300

350

0 10000 20000 30000 40000 50000

Distance in m

Ele

vati

on

in m 0 yr

1000 yr2000 yr3000 yr4000 yr5000 yr

By 5000 years; steady state not yet achieved.

25



Bed slope evolution

0.001

0.01

0.1

0 10000 20000 30000 40000 50000

Distance in m

Bed

slo

pe

0 yr1000 yr2000 yr3000 yr4000 yr5000 yr



26



Bed material load evolution

0

0.5

1

1.5

2

0 10000 20000 30000 40000 50000

Distance in m

qt/q

tf

0 yr1000 yr2000 yr3000 yr4000 yr5000 yr



27




0

50

100

150

200

250

300

350

0 10000 20000 30000 40000 50000

Distance in m

Ele

vati

on

in m 0 yr

5000 yr10000 yr15000 yr20000 yr25000 yr

By 25,000 years; steady state achieved


upward-concave elevation profile

In this and the next two slides Ntoprint = 4000 rather than 1000 of Slide 23

28



Bed slope evolution

0.001

0.01

0.1

0 10000 20000 30000 40000 50000

Distance in m

Bed

slo

pe

0 yr5000 yr10000 yr15000 yr20000 yr25000 yr



bed slope declines downstream

29




0

0.5

1

1.5

2

0 10000 20000 30000 40000 50000

Distance in m

qt/q

tf

0 yr5000 yr10000 yr15000 yr20000 yr25000 yr



bed material load decreases linearly downstream

30



NOTE ON USE OF RTe-bookAgDegNormalSub.xls


-60

-10

40

90

140

190

240

0 10000 20000 30000 40000 50000

Distance in m

Ele

vati

on

in m 0 yr

4000 yr8000 yr12000 yr16000 yr20000 yr

The calculation below is based on exactly the same input parameters as used for Slide 27, with the exception that the initial slope SI was reduced from 0.005 to 0.002. If the initial slope SI is not sufficiently high, subsidence will overwhelm sediment deposition at the downstream end of the reach in the early stages of the calculation, and it is not possible to reach steady state while maintaining constant bed elevation at x = L. The problem is physical, not numerical. If one wishes to attain the correct final steady state using the lower initial slope, the problem can be fixed by changing the boundary condition at x = L to one of vanishing sediment transport whenever h < 0 at x = L. This is done in Chapter 33.

Insufficient sediment to fill basin at downstream

end during evolution toward steady state

31



The following input parameters were used to compute a case for a sand-bed stream using RTe-bookAgDegNormalSub.xls. Calculations were performed for a duration of 5000 years. Note L = 50,000 m is slightly less than Lmax.

Input parametersqw qw 4 m2/s Water discharge per unit width during flood

If Inter 0.1 Flood intermittencyD D 0.5 mm Grain size

p lamp 0.4 Bed porosity

kc kc 20 mm Roughness height

SI SI 0.0005 Initial bed slope

qtf qqtf 0.0003 m2/s Volume sediment feed rate per unit width subrate 2 mm/yr Subsidence raterB rB 60 Ratio of depositional width to channel width omega 2 Channel sinousity lamda 1 Unit wash load deposited in channel-floodplain per unit bed material loadLmax 52596 m Maximum reach lengthL L 50000 m Length of reach (must be less than maximum reach length)

M 50 IntervalsNtoprint 100 Number of time steps to printoutNprint 5 Number of printouts

Dt dt 10 year time stepau au 0.5 Here 1 = full upwind, 0.5 = central differenceDx dx 1000 m Spatial step length

5000 years Duration of calculation

Auxiliary Input Parameters

r alr 8.1

t alt 4

nt nt 1.5

c* tausc 0.0495

s fis 0.6R Rr 1.65

CALCULATIONS FOR A SAND-BED RIVER USING RTe-bookAgDegNormalSub.xls

32




0

5

10

15

20

25

30

35

0 10000 20000 30000 40000 50000

Distance in m

Ele

vati

on

in m 0 yr

1000 yr2000 yr3000 yr4000 yr5000 yr


upward-concave elevation profile

33



Bed slope evolution

0.0001

0.001

0.01

0 10000 20000 30000 40000 50000

Distance in m

Bed

slo

pe

0 yr1000 yr2000 yr3000 yr4000 yr5000 yr


bed slope declines downstream

34




0

0.5

1

1.5

2

0 10000 20000 30000 40000 50000

Distance in m

qt/q

tf

0 yr1000 yr2000 yr3000 yr4000 yr5000 yr


bed material load decreases linearly downstream

35



EXNER EQUATION FOR MIXTURES INCLUDING SUBSIDENCE

The analysis for uniform sediment can be generalized to sediment mixtures, so as to allow study of downstream fining as well as profile concavity. The appropriate form for sediment conservation of mixtures in a river under conditions of subsidence was derived in Chapter 4:

In the above equation:x denotes the streamwise coordinate [L];t denotes time [T];h denotes bed elevation [L];Fi denotes the fraction of sediment in the ith grain size range (which is characterized by size Di) in the active (surface) layer [1];La denotes the thickness of the active (surface) layer [L];qtT denotes the total volume bed material transport rate per unit width [L2/T];pti denotes the fraction of material in the ith size range in the bed material load [1];fIi denotes the interfacial fraction in the ith grain size range exchanged between surface and substrate as the bed aggrades or degrades [1]; denotes the subsidence rate [L/T]; andp denotes the porosity of the bed deposit [1].

x

pqLF

t)L(

tf)1( titT

aiaIip

h

36



ADAPTATION OF THE EXNER EQUATION FOR SEDIMENT MIXTURES

The Exner equation of the previous slide can be adapted to realistic rivers flowing into subsiding basins by virtue of the same steps as pursued in Slides 11 – 14 for uniform material. The result is the form

where rB, If, and have the same meanings as before. Summing over all grain sizes, the relation describing the evolution of the bed is found to be:

Between the two equations above, the equation for evolution of the surface size distribution is found to be:

xpq

r)1(I

LFt

)L(t

f)1( titT

B

faiaIip

h

xq

r)1(I

t)1( tT

B

fp

h

x

qf

x

pq

r

)1(I

t

LfF)1(

t

FL)1( tT

IititT

B

faIiip

iap

37



CASE OF A STEADY STATE LONG PROFILE MAINTAINED BY A PERFECT BALANCE BETWEEN SUBSIDENCE AND DEPOSITION

As outlined for the case of uniform sediment, basin subsidence creates a “hole” into which sediment can be deposited. Sediment can fill this “hole.” The long profile of the river becomes invariant in time when the deposition rate precisely balances the rate of creation of space created by subsidence. In the case of mixtures, the relevant forms of sediment conservation reduce to:

x

h Dt


x

q

r

)1(I

t)1( tT

B

fp

h

x

qf

x

pq

r

)1(I

t

LfF)1(

t

FL)1( tT

IititT

B

faIiip

iap

38



CASE OF A STEADY STATE LONG PROFILE MAINTAINED BY A PERFECT BALANCE BETWEEN SUBSIDENCE AND DEPOSITION contd.

The Exner equations of the previous slide thus simplify to the following forms:

x

h Dt


)1(I

r)1(

dxdq

f

BptT

The first of these equations integrates to give exactly the same forms as for uniform material:

)pf()1(I

r)1(

dx

dpq tiIi

f

BptitT

where qtTf denotes the feed rate of total bed material load.

maxtTf

tT

L

x1

q

q

)1(

q

r

)1(IL

p

tTf

B

fmax

39



CASE OF A STEADY STATE LONG PROFILE MAINTAINED BY A PERFECT BALANCE BETWEEN SUBSIDENCE AND DEPOSITION contd.

The equation

combined with

and appropriate relations for flow resistance and sediment transport of mixtures, can be solved to determine the steady-state downstream variation in bed slope S and surface layer fractions Fi and thus surface geometric mean and median sizes Dsg and Ds50).

In the case of sediment mixtures an analytical solution is not possible; implementation involves an iterative numerical scheme. Here an alternative scheme is pursued in terms of solution of the time-varying problem that naturally relaxes to steady state.

)pf()1(I

r)1(

dxdp

qtiIi

f

BptitT

maxtTf

tT

L

x1

q

q

40



GRAVEL-BED RIVERS:EVOLUTION OF THE PROFILE TOWARD STEADY STATE

The case of gravel-bed rivers is considered. The treatment here is an extension of that of Chapter 17. For this case the bed material load is gravel and the wash load is usually mostly sand. The relations governing the evolution toward steady state are those given in Slide 36, but with qtT →qbT and pti → pbi, where the subscript “b” denotes bedload (Chapters 4 and 7);

xq

r)1(I

t)1( bT

B

fp

h

x

qf

x

pq

r

)1(I

t

LfF)1(

t

FL)1( bT

IibitT

B

faIiip

iap

Bedload transport for gravel mixtures is computed using one of the methodologies introduced in Chapter 6 and implemented in Chapter 17;

where u* denotes bed shear velocity and fW is specified in terms of the bedload transport relations of Parker (1990) or Wilcock and Crowe (2003) (Chapter 6).

)(fRg

uFq iW

3

ibi

i

2

i RgD

u

41



GRAVEL-BED RIVERS:EVOLUTION OF THE PROFILE TOWARD STEADY STATE contd.

Flow resistance is computed using the Manning-Strickler formulation introduced in Chapter 5 and implemented in Chapter 17; where U denotes depth-averaged flow velocity,

The flow is calculated using the normal flow assumption. As in Chapter 17, the shear velocity is computed as

The volume bedload transport rate per unit width qbi is then computed from the surface fractions Fi, shear velocity u*, the grain sizes Di and an appropriate bedload transport relation for mixtures as outlined in Chapter 7.

6/1

sr k

H

u

U

90sks Dnk

20/720/7

20/3

2r

2w

3/1s Sg

qku

42




The interfacial exchange fractions fIi are computed from the relation

of Chapter 4. In the above relation is a user-specified constant between 0 and 1 and fi denotes the fractions in the substrate.

The upstream boundary conditions at x = 0 consist of a specified total volume bedload feed rate per unit width qbTf specified bedload feed fractions pfi. The downstream boundary condition at x = L consists of a fixed downstream bed elevation hd (e.g. hd = 0). Thus

Here L must be less than Lmax, where from Slide 38

h

h

h

0t

,p)1(F

0t

,ff

bii

Lzi

Iia

dLxbfi0xbibTf0xbT ,pp,qq hh

)1(

q

r

)1(IL

p

bTf

B

fmax

43




The initial conditions consist of specified, constant initial bed slope SI and specified initial fractions in the surface layer FIi. These initial surface fractions are taken to be the same for every node.

In addition to the above parameters, it is necessary to specify the fractions in the substrate. These fractions are assumed to be the same at every node, and independent of vertical position within the bed.

In a complete description, the vertical structure of the substrate would be updated whenever the river degraded into an existing deposit, and then aggraded subsequently. This step is not implemented here.

44



NUMERICAL IMPLEMENTATION FOR MIXTURES

The formulation is identical to that of Chapter 17 except for the fact that subsidence and the parameters rB, If, and are included. As outlined in Chapter 17, M + 1 nodes bound M intervals. Sediment is fed in at a ghost node. The Exner formulations are implemented as

where k is an index denoting node, and ranging from k = 1 to M + 1 (in addition to a ghost node where sediment is fed), and i is an index denoting grain size range.

tx

q

r)1(

)1(It

k

bT

Bp

fkttk D

DhhD

tx

qf

x

pq

r)1(L

)1(It

t

LfF

L

1FF

k

bTk,Ii

k

bibT

Bpk,a

fk,ak,Iik,i

k,ak,ittk,i D

D

D

45



INTRODUCTION TO RTe-bookAgDegNormalGravMixSubPW.xls , A CALCULATOR FOR THE APPROACH TO EQUILIBRIUM IN A GRAVEL RIVER CARRYING A

MIXTURE OF SIZES AND FLOWING INTO A SUBSIDING BASIN

The code in RTe-bookAgDegNormalGravMixSubPW.xls is a modestly modified version of RTe-bookAgDegNormalGravMixPW.xls, in which the modification allow for a) the inclusion of subsidence at rate and b) characterization of the river and subsiding basin in terms of the parameters If, rB, and . The code allows implementation of either the bedload relation of Parker (1990) or the one of Wilcock and Crowe (2003).

The output consists of numerical data and plots for a) profiles of bed elevation, b) profiles of bed slope, c) profiles of surface geometric mean size, d) profiles of the ratio of total bedload transport rate.

Calculations are presented here using: Case A) the relation of Wilcock and Crowe (2003) and widely distributed sediment feed; Case B) the relation of Wilcock and Crowe (2003) and a uniform feed sediment with the same geometric mean size as that of Case A; and Case C) the relation of Parker (1990) and the same feed distribution as in Case A.

46



CALCULATIONS FOR USING RTe-bookAgDegNormalGravMixSubPW.xls WITH THE GRAVEL TRANSPORT RELATION OF WILCOCK AND CROWE (2003)

Dd,i mm Feed Initial Surface Substrate

256 100 100 100128 97 97 97

64 90 90 9032 65 65 6516 45 45 45

8 30 30 304 20 20 202 18 18 181 15 15 15

0.5 5 5 50.25 0 0 0

0.125 0 0 0

Input parametersqw qw 6 water discharge/width, m^2/sqbTf qbTf 4.00E-03 gravel input rate, m^2/sIf Inter 0.03 Flood intermittencyh

d etad 0 base level, m subrate 5 subsidence rate, mm/yearrB rB 60 ratio of depositional width to channel width lamda 1 ratio of wash load deposited per unit bed material load deposited Sinu 1.5 channel sinuosityLmax Lmax 52596 Maximum reach lengthL L 50000 Length of reach (must be less than maximum reach length)SI SI 5.00E-03 initial bed slopeDt dt 292.2 time step, days

M 25 no. of intervalsNtoprint 20000 no. of steps until a printout of results is madeNprint 6 no. of printouts after the initial one

96000 years calculation time

Auxiliary Parameters (all dimensionless)nk nk 2 Factor by which surface Ds90 is multiplied to obtain roughness height ks

na nactive 2 Factor by which surface Ds90 is multiplied to obtain active layer thickness La

r ar 8.1 Coefficient in Manning-Strickler resistance relationR Rr 1.65 Submerged specific gravity of gravel

p lps 0.4 Bed porosity, gravelau au 0.75 Upwinding coefficient for load spatial derivatives in Exner equation (value > 0.5 suggested) atrans 0.5 Coefficient for material transferred to substrate as bed aggrades

Percents finer

Case A: Sediment Mixture

47




Downstream Variation in Bed Elevation

0

50

100

150

200

250

300

0 10000 20000 30000 40000 50000

Distance m

Ele

vati

on

m

0 yr16000 yr32000 yr48000 yr64000 yr80000 yr96000 yr

Steady-state upward-concave profile nearly achieved


48




Downstream Variation in Bed Slope

0.0001

0.001

0.01

0 10000 20000 30000 40000 50000

Distance m

Slo

pe

0 yr16000 yr32000 yr48000 yr64000 yr80000 yr96000 yr

Slope declines strongly downstream


49




Downstream Variation in Surface Geometric Mean Size

10

100

0 10000 20000 30000 40000 50000

Distance m

Su

rfac

e G

eom

etr

ic M

ean

Siz

e m

m

0 yr16000 yr32000 yr48000 yr64000 yr80000 yr96000 yr

Surface geometric mean grain size declines downstream


50




Downstream Variation of qbT/qbTf, where qbT = Bedload Transport Rate and qbTf = Upstream Bedload

Feed Rate

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 10000 20000 30000 40000 50000

Distance m

qb

T/q

bT

f

0 yr16000 yr32000 yr48000 yr64000 yr80000 yr96000 yr

Nearly linear decline in total bedload transport rate by end of run.


51




Plot of fractions finer in a) substrate (Ffs), b) sediment feed (pfeed) c) final surface at upstream node (Ffup) and d) final

surface at downstream node (Ffdwn)

0

10

20

30

40

50

60

70

80

90

100

0.1 1 10 100 1000

D mm

Pe

rce

nt

fin

er

FfspfeedFfupFfdwn

Final surface size distribution at downstream end is noticeably finer than that that at upstream end


52




Case B: Uniform SedimentInput parametersqw qw 6 water discharge/width, m^2/sqbTf qbTf 4.00E-03 gravel input rate, m^2/sIf Inter 0.03 Flood intermittencyh




Auxiliary Parameters (all dimensionless)nk nk 2 Factor by which surface Ds90 is multiplied to obtain roughness height ks





256 100 100 100128 100 100 100

64 100 100 10032 100 100 100

13.2 100 100 10012 0 0 0

4 0 0 02 0 0 01 0 0 0

0.5 0 0 00.25 0 0 0

0.125 0 0 0

The sediment is uniform with a size of 12.6 mm, i.e. the geometric mean size of the input distributions of Case A

53





0

50

100

150

200

250

300

0 10000 20000 30000 40000 50000

Distance m

Ele

vat

ion

m

0 yr16000 yr32000 yr48000 yr64000 yr80000 yr96000 yr

Case B: Uniform Sediment

Steady-state profile is somewhat less concave than Case A

54





0.001

0.01

0 10000 20000 30000 40000 50000

Distance m

Slo

pe

0 yr16000 yr32000 yr48000 yr64000 yr80000 yr96000 yr


Downstream slope decline is not as strong as Case A

55





10

100

0 10000 20000 30000 40000 50000

Distance m

Su

rfa

ce G

eom

etri

c M

ean

Siz

e m

m

0 yr16000 yr32000 yr48000 yr64000 yr80000 yr96000 yr


Since sediment size is uniform it cannot decline downstream

56





Feed Rate

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 10000 20000 30000 40000 50000

Distance m

qb

T/q

bT

f

0 yr16000 yr32000 yr48000 yr64000 yr80000 yr96000 yr


Steady-state downstream load variation is the same as Case A

57



CALCULATIONS FOR USING RTe-bookAgDegNormalGravMixSubPW.xls WITH THE GRAVEL TRANSPORT RELATION OF PARKER (1990)

Input parametersqw qw 6 water discharge/width, m^2/sqbTf qbTf 4.00E-03 gravel input rate, m^2/sIf Inter 0.03 Flood intermittencyh





256 100 100 100128 91.463415 91.4634 91.46341

64 60.97561 60.9756 60.9756132 36.585366 36.5854 36.5853716 18.292683 18.2927 18.29268

8 6.097561 6.09756 6.0975614 3.6585366 3.65854 3.6585372 0 0 01 0 0 0

0.5 0 0 00.25 0 0 0

0.125 0 0 0Auxiliary Parameters (all dimensionless)nk nk 2 Factor by which surface Ds90 is multiplied to obtain roughness height ks




The percents finer are based on the values used for the implementation of Wilcock-Crowe, but they have been renormalized to exclude the sand.

Case C: Sediment Mixtures

58





0

50

100

150

200

250

300

0 10000 20000 30000 40000 50000

Distance m

Ele

vati

on

m

0 yr16000 yr32000 yr48000 yr64000 yr80000 yr96000 yr


Profile is upward concave, but concavity is muted compared to Case A

59





0.001

0.01

0 10000 20000 30000 40000 50000

Distance m

Slo

pe

0 yr16000 yr32000 yr48000 yr64000 yr80000 yr96000 yr


Slope is steeper and downstream slope variation is muted compared to Case A

60





10

100

0 10000 20000 30000 40000 50000

Distance m

Su

rfac

e G

eom

etri

c M

ean

Siz

e m

m

0 yr16000 yr32000 yr48000 yr64000 yr80000 yr96000 yr

Downstream decline in geometric mean size of surface is muted compared to Case A


61





Feed Rate

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 10000 20000 30000 40000 50000

Distance m

qb

T/q

bT

f

0 yr16000 yr32000 yr48000 yr64000 yr80000 yr96000 yr

Steady-state downstream load variation is the same as Case A


62




Plot of fractions finer in a) substrate (Ffs), b) sediment feed (pfeed) c) final surface at upstream node (Ffup) and d) final

surface at downstream node (Ffdwn)

0

10

20

30

40

50

60

70

80

90

100

0.1 1 10 100 1000

D mm

Pe

rce

nt

fin

er

FfspfeedFfupFfdwn


Difference in upstream and downstream grain size distributions is less than Case A

63



DOWNSTREAM FINING IN SAND-BED STREAMS FLOWING INTO SUBSIDING BASINS

Long reaches of sand-bed rivers also usually show profile upward concavity and downstream fining. As discussed in Chapter 25, basin subsidence is only one of several factors that can drive these tendencies.

For example, the Mississippi River downstream of Cairo, Illinois USA shows both upward profile concavity and downstream fining (in terms of the surface median size Ds50) (from Wright and Parker, in press a).

Distance downstream from Cairo Il (km)

Bed

ele

vatio

n (m

)

Ds5

0 (

mm

)

64



DOWNSTREAM FINING IN SAND-BED STREAMS FLOWING INTO SUBSIDING BASINS contd.

Wright and Parker (in press a, in press b) have studied upward concavity and downstream fining in large, low-slope sand-bed rivers subject to a) delta progradation, b) sea level rise and c) subsidence.

Their model uses the formulation of Wright and Parker (2004) for flow resistance in sand-bed streams, the bedload formulation for mixtures of Ashida and Michiue (1972), the suspension model for mixtures of Wright and Parker (2004). Otherwise the overall structure of the model is similar to the structure presented here for gravel-bed streams.

Mississippi River near New Orleans, USA

65



REFERENCES FOR CHAPTER 26

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).

Cui, Y., Parker, G. and Paola, C., 1996, Numerical simulation of aggradation and downstream fining, Journal of Hydraulic Research, 34(2), 185-214.

Cui, Y. and Parker, G., 1997, A quasi-normal simulation of aggradation and downstream fining with shock fitting, International Journal of Sediment Research, 12(2): 68-82.

Paola, C., P. L. Heller and C. L. Angevine, 1992, The large-scale dynamics of grain-size variation in alluvial basins, I: Theory, Basin Research, 4, 73-90.

Parker, G., 1990, Surface-based bedload transport relation for gravel rivers, Journal of Hydraulic Research, 28(4), 417-436.Wilcock, P. R., and Crowe, J. C., 2003, Surface-based transport model for mixed-size sediment,

Journal of Hydraulic Engineering, 129(2), 120-128.Wright, S. and Parker, G., 2004, Flow resistance and suspended load in sand-bed rivers:

simplified stratification model, Journal of Hydraulic Engineering, 130(8), 796-805.Wright, S. and Parker, G., in press, Modeling downstream fining in sand-bed rivers I:

Formulation, Journal of Hydraulic Research (preprint available at http://cee.uiuc.edu/people/parkerg/preprints.htm ).

Wright, S. and Parker, G., in press, Modeling downstream fining in sand-bed rivers II: Application, Journal of Hydraulic Research (preprint available at http://cee.uiuc.edu/people/parkerg/preprints.htm ).

1 1d sediment transport morphodynamics with applications to rivers and turbidity currents © gary...

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