Fly River, Papua New Guinea

1

1D SEDIMENT TRANSPORT MORPHODYNAMICSwith applications to

RIVERS AND TURBIDITY CURRENTS

Fly River, Papua New Guinea

CHAPTER 27:MORPHODYNAMICS OF GRAVEL-SAND TRANSITIONS

Strickland River

This box expanded in next slide

Image from NASA website:https://zulu.ssc.nasa.gov/mrsid/mrsid.pl

flow

2



Fly River

THE BOX IS EXPANDED IN THE NEXT SLIDE TO SHOW A GRAVEL-SAND TRANSITION

Ok Tedi


flow

flow

3




GRAVEL-SAND TRANSITION ON THE OK TEDI, PAPUA NEW GUINEA

Wandering gravel-bed river

Meandering sand-bed river

Gravel-sand transition

flow

flow

4


RIVERS AND TURBIDITY CURRENTSGRAVEL-SAND TRANSITION ON THE BENI RIVER, BOLIVIA

Andes mountains: zone of high tectonic uplift

Foredeep: zone of tectonic susidence


flow

flow

Image courtesy R. Aalto: see Aalto (2002)

5



Beni River Granulometry Profile

1

10

100

1000

10000

100000

8350000 8400000 8450000 8500000 8550000 8600000 8650000 8700000 8750000 8800000 8850000UTM Latitude (zone 19S)

Poin

t Bar

Sub

surf

ace

D50

(um

)

Bed D50

Bed D50 (Guyot)

Bar D50

Gravel-Sand Transition

Gradual fining downstream of Forebulge

GRAVEL-SAND TRANSITION ON THE BENI RIVER, BOLIVIA contd.

flow


Note the discontinuity in grain size at the gravel-sand transition.

6



Beni River DGPS Survey

100

120

140

160

180

200

220

8350000 8400000 8450000 8500000 8550000 8600000 8650000 8700000 8750000 8800000 8850000UTM Latitude (zone 19S)

Cor

rect

ed W

ater

Ele

vatio

n (m

)

sub-Andean range front

gravel-sand transition

Forebulge

5m Cachuela

wedge-top basin?

Beni Foreland BasinSecondary Basin

Madidi River

GRAVEL-SAND TRANSITION ON THE BENI RIVER, BOLIVIA contd.

flow


Note the discontinuity in slope at the gravel-sand transition.

7



GRAVEL-SAND TRANSITION:KINU RIVER, JAPAN

Both the gravel-bed and sand-bed reaches have upward-concave profiles, and show downstream fining.

Note the sharp breaks in slope and grain size!

Sambrook-Smith and Ferguson (1995) have documented many relatively sharp gravel-sand transitions in rivers around the world.

Long profile showing downstream fining and

gravel-sand transition in the Kinu River, Japan (Yatsu,

1955)

8



SHARP GRAVEL-SAND TRANSITIONS ARE LIKELY ASSOCIATED WITH A RELATIVE PAUCITY OF MATERIAL IN THE RANGE 2-8 MM IN MANY RIVERS

0

5

10

15

20

25

30

35

40

Grain size range in mm

Num

ber o

f rea

ches

AlbertaJapan

Sand-bed Gravel-bedTransitional

0

5

10

15

20

25

30

35

40

Grain size range in mm

Num

ber o

f rea

ches

AlbertaJapan

Sand-bed Gravel-bedTransitional

This paucity was illustrated in Chapters 2 and 3. It is common, but by no means universal.

From Chapter 2 From Chapter 3

9



THE SIMPLEST WAY TO MODEL LONG PROFILES WITH GRAVEL-SAND TRANSITIONS IS TO CONSIDER A TWO-GRAIN SYSTEM

sgs

gravel sand

L


hg h

s

The bed material of the gravel-bed reach is characterized with a single size Dg. The bed material of the sand-bed reach is characterized with a single size Ds. The position of the gravel-sand transition is x = sgs. It is assumed that the sand is transported through the gravel-bed reach as wash load.

L = reach lengthhg = elevation of gravel bedhs = elevation of sand bed

10



SIMPLIFICATIONS OF THE PRESENT MODEL

The model of this chapter focuses on gravel-sand transitions in subsiding basins, and in rivers-floodplain complexes subject to sea-level rise. The following simplifications are introduced.

• The gravel is characterized with a single grain size Dg, and the sand is characterized with a single grain size Ds. Grain size mixtures of gravel and sand are not considered.• The total length of the gravel-bed reach plus the sand-bed reach = the constant

value L. The position of the gravel-sand transition x = sgs(t) may change in time.

• No allowance is made for delta progradation.• Abrasion of gravel to sand is neglected.• It is assumed that there are no significant tributaries along the entire reach from x

= 0 to x = L, so that water discharge during floods is constant downstream.• Each reach (gravel-bed and sand-bed) is assumed to have a constant width.

None of these assumptions would be overly difficult to relax.

11


RIVERS AND TURBIDITY CURRENTSPARAMETERS AND EXNER EQUATIONS

x = downchannel spatial coordinate [L]t = time [L]hg, hs = bed elevation on gravel-bed, sand-bed reach [L]qg, qs = total volume gravel load, sand load per unit width [L2/T]pg, ps = bed porosity of gravel-bed, sand-bed reach [1]Ifg, Ifs = flood intermittency on gravel-bed, sand bed reach [1]g, s = channel sinuosity on gravel-bed, sand-bed reach [1]sg, ms = volume fraction sand deposited per unit gravel, volume fraction mud deposited

per unit sand in channel-floodplain complex [1]rBg, rBs = ratio of channel width Bc to depositional width Bd (basin or floodplain width) in

gravel-bed, sand-bed reach (Bd,grav/Bc,grav or Bd,sand/Bc,sand) = subsidence rate [L/T]Based on the formulation of Chapter 25, the conservation relations for gravel and sand on the gravel-bed reach are

The conservation relation for sand on the sand-bed reach isxq

r)1()1(I

tg

Bgpg

gsgfgg

h

xq

xq g

sgs

xq

r)1()1(I

ts

Bsps

smsfss

h

12



CONTINUITY CONDITION AT THE GRAVEL-SAND TRANSITION

Let ssg(t) denote the position of the gravel-sand transition, and Sggs and Ssgs denote the gravel bed slope and sand bed slope, respectively, at the gravel-sand transition, so that

gsgs sx

ssgs

sx

gggs x

S,x

S

h

h

sgs

gravel sand

L


hg h

s

Sggs Ssgs

13



In analogy to the treatment of bedrock-alluvial transitions in Chapter 16, bed elevation continuity at the gravel-sand transition is expressed in the following form:

Taking the derivate of both sides of the above equation with respect to t and rearranging with the definitions of Sggs and Ssgs of the previous slide, it is found that

CONTINUITY CONDITION AT THE GRAVEL-SAND TRANSITION contd.

)t(sxs)t(sxggsgs

)t,x()t,x(

hh

sgsggs

sx

s

sx

g

sg SS

tts gsgs

h

h

where = dssg/dt denotes the migration speed of the gravel-sand transition.

Since gravel is harder to move than sand, it can be expected that Sggs > Ssgs. Now suppose that near the gravel-sand transition the sand-bed reach is aggrading faster than the gravel-bed reach, i.e. hs/t > hg/t. According to the above equation, then, and the gravel-sand transition migrates upstream.

sgs

0ssg Sggs Ssgs

14



THE LOCATION OF THE GRAVEL-SAND TRANSITION CAN STABILIZE!Consider a subsiding system that has reached a steady state, as described in Chapter 26:

In such a case the continuity condition yields the result

i.e. an arrested gravel-sand transition (Parker and Cui, 1998; Cui and Parker, 1998). If such a steady-state position exists, the system will naturally evolve toward it.

Sea level rise at a constant rate can also lead to an arrested gravel front when the following condition is satisfied:

0SS

tts

sgsggs

sx

s

sx

g

sggsgs

h

h

xq

r)1()1(I

tg

Bgpg

gsgfgg

h

xq

r)1()1(I

ts

Bsps

smsfss

h

dsx

s

sx

g

gsgstt

h

h

d

15



MAXIMUM REACH LENGTH FOR STEADY STATE SYSTEM

In the case of a steady-state system entering a basin subsiding at constant rate with constant base level, the governing equations for the gravel-bed reach reduce to

and the governing equation for the sand-bed reach reduces to

The corresponding forms for the case of a constant rate of base level (sea level) rise in the absence of subsidence are

These forms are closely allied to the steady-state forms developed in Chapter 26.

dxdq

dxdq,

)1(Ir)1(

dxdq g

sgs

gsgfg

Bgpgg

dsmsfs

Bspss

gsg

sd

gsgfg

Bgpgg

)1(Ir)1(

dxdq

dxdq

dxdq,

)1(Ir)1(

dxdq

smsfs

Bspss

)1(Ir)1(

dxdq

Gravel-bed reach

Sand-bed reach

16



MAXIMUM REACH LENGTH FOR STEADY STATE SYSTEM contd. In general, then, the steady-state equations can be written as

where vv = for the case of constant subsidence without base level rise and vv = for the case of base level rise at a constant rate without subsidence.

Over the gravel-bed reach, the top two equations integrate to

where qg,feed and qs,feed denote the feed rates of sand andgravel (volume feed rate per unit width) at x = 0.

vsmsfs

Bspss

gsg

sv

gsgfg

Bgpgg

v)1(Ir)1(

dxdq

dxdq

dxdq,v

)1(Ir)1(

dxdq

Gravel-bed reach

Sand-bed reach

d

xv)1(Ir)1(

q)qq(qq

xv)1(Ir)1(

qq

vgsgfg

Bgpgsgfeed,sfeed,gsgfeed,ss

vgsgfg

Bgpgfeed,gg

Gravel and sand fill the accommodation space of the gravel-bed reach created by subsidence or sea level rise.

17



MAXIMUM REACH LENGTH FOR STEADY STATE SYSTEM contd. The gravel transport rate drops to zero (qg = 0) at the steady-state position of the gravel-sand transition x = ssg,ss given by the relation

The sand transport rate qs at the point where the gravel runs out is

Note that in order for sand to be available for transport beyond x = Lgrav,max the following condition must be satisfied:

The relation for the sand-bed reach (second equation of previous slide) then integrate to give

feed,gvBgpg

gsgfgss,gs q

vr)1()1(I

s

feed,gsgfeed,sss,sgvgsgfg

Bgpgsgfeed,ssxs qqsv

)1(Ir)1(

qqss,sg

feed,gsgfeed,s qq

)sx(v)1(Ir)1(

qqq ss,sgvsmsfs

Bspsfeed,gsgfeed,ss

18



MAXIMUM REACH LENGTH FOR STEADY STATE SYSTEM contd.

The sand transport rate drops to zero (qs = 0) at x = Lmax, given by the relation

or thus

If the reach length is longer than Lmax it is not possible to reach a steady state which maintains a specified base level at the downstream end of the reach. This is because there is not enough sediment (gravel and sand) available to fill the accomodation space created by subsidence or sea level rise. The result is the formation of an embayment (drowned river valley) at the downstream end.

feed,gsgfeed,svBsps

smsfsss,gsmax qq

vr)1()1(IsL

feed,gsgfeed,s

Bsps

smsfsfeed,g

Bgpg

gsgfg

vmax qq

r)1()1(Iq

r)1()1(I

v1L

19



gsgs sx

g

Bgpg

gsgfg

sx

s

Bsps

smsfs

sgsggsgs x

qr)1()1(I

xq

r)1()1(I

)SS(1s

REDUCTION OF THE CONTINUITY CONDITION TO A RELATION FOR THEMIGRATION SPEED OF THE GRAVEL-SAND TRANSITION

Returning to the non-steady-state problem, the continuity condition

reduces with the forms for Exner of Slide 11, i.e.

to yield the following equation for the migration speed of the gravel-sand transition:

sgsggs

sx

s

sx

g

sg SS

tts gsgs

h

h

xq

r)1()1(I

tg

Bgpg

gsgfgg

h

xq

r)1()1(I

ts

Bsps

smsfss

h

20



The gravel-sand transition is free to move about in time. It thus constitutes a moving boundary problem. Moving boundary analysis was developed in the context of a migrating bedrock-alluvial transition in Chapter 16. Here it is adapted for the case of a gravel-sand transition.

Moving boundary coordinates for the gravel-bed and sand-bed reaches can be defined as:

Note that on the gravel-bed reach, and on the sand-bed reach.

The Exner equation for gravel conservation on the gravel-bed reach of the previous slide transforms to:

TRANSFORMATION TO MOVING BOUNDARY COORDINATES

tt,)t(sL)t(sx

x;tt,)t(s

xx sgs

gssg

gsg

1x0 g 1x0 s

g

g

Bggspg

gsgfg

g

g

gs

ggs

g

g

xq

rs)1()1(I

xsxs

t

h

h

21



The Exner equation for the conservation of sand on the gravel-bed reach, given in Slide 11, transforms to:

The Exner equation for the conservation of sand on the sand-bed reach, given in Slide 15, transforms to:

The continuity condition of Slide 12 describing the migration speed of the gravel-sand transition transforms to:

s

s

Bsgsps

smsfs

s

s

gs

sgs

s

s

xq

r)sL)(1()1(I

x)sL()x1(s

t

h

h

TRANSFORMATION TO MOVING BOUNDARY COORDINATES contd.

sg

gsg

g

s

xq

xq

1xg

g

gsBgpg

gsgfg

0xs

s

gsBsps

smsfs

sgsggsgs

gsxq

sr)1()1(I

xq

)sL(r)1()1(I

)SS(1s

22



The spatial discretization involves MG gravel-bed intervals followed by MS sand-bed intervals, bounded by MG + MS + 1 nodes. The dimensionless spatial steps for the gravel-bed and sand-bed reaches are given as

The node i = MG + MS + 1 defines the downstream end of the reach, i.e. x = L. The node i = MG + 1 defines the gravel-sand transition, i.e. x = sgs. Gravel and sand are fed in at a ghost node one step upstream of node i = 1.

MS1x,

MG1x sg

SPATIAL DISCRETIZATION

L

i=12

3MG MG+MS+1


MG+MSMG+1MG+2ghost

gravel-bed reach sand-bed reach

gx sx

23



A backwater formulation is used to compute the flow (which is assumed to be barely confined to the channel). The friction coefficients on the gravel-bed and sand-bed reaches, denoted correspondingly as Cfg and Cfs, are assumed to be specified constants. In accordance with Chapter 5, then, the backwater formulation for the gravel-bed reach is

where Hgrav denotes flow depth on the gravel-bed reach, qw denotes the water discharge per unit width (during floods) and Sg denotes bed slope on the gravel-bed reach, and the corresponding formulation for the sand-bed reach is

where Ss denotes the slope and Hsand denotes the flow depth on the sand-bed reach.

CALCULATION OF FLOW

xS,

gHq1

gHqCS

xH g

grav

3grav

2w

3grav

2w

fgggrav

h

xS,

gHq1

gHqCS

xH s

g

3sand

2w

3sand

2w

fsssand

h

24



Transforming the relations of the previous page to moving boundary coordinates results in the forms

CALCULATION OF FLOW contd.

g

g

gsg

3grav

2w

3grav

2w

fgg

g

grav

gs xs1S,

gHq1

gHqCS

xH

s1

h

s

s

gss

3sand

2w

3sand

2w

fss

s

sand

gs xsL1S,

gHq1

gHqCS

xH

sL1

h

25



The boundary condition on the backwater formulation is specified at x = L, where downstream water surface elevation d is specified. Here may be a specified constant do , or it may change in time at some constant rate . Thus in general

or

In addition, a continuity condition must be satisfied at the gravel-sand transition;


ddo1xsandss

)H( h

d

1xsddo1xsandss

H

h

0xsand1xgravsg

HH

26



At any given time, the backwater curve above the bed at that time can then be solved numerically by implementing the formulation of Chapter 20 adapted to the present problem. That is, for the sand-bed reach


1MSMG,sandddo1MSMG,sandH h

sp,sandback,s1i,sandback,s1i,sandi,sand

s1i,sandback,s1i,sandp,sand

x)H(F)H(F21HH

x)H(F21HH

1MGtoMSMGi

sgs

1i,si,ss

gs3sand

2w

3sand

2w

ssandback,s

x)sL(S

)sL(gH

q1gH

qS)H(F

hh

27



The corresponding formulation for the gravel-bed reach is


1MG,sand1MG,grav HH

1toMGi

gp,gravback,g1i,gravback,g1i,gravi,grav

g1i,gravback,g1i,gravp,grav

x)H(F)H(F21HH

x)H(F21HH

gsg

1i,gi,gg

sg3grav

2w

3grav

2w

ggravback,g

xsS

sgH

q1gH

qS)H(F

hh

28



The submerged specific gravity R = s/ - 1 is assumed to be the same for the gravel as it is for the sand. Recall from Chapters 5 and 20 that boundary shear stress b is given as

where H denotes flow depth, and that the Shields number * is given as

where D is an appropriate grain size. Let U = qw/H. The Shields number sand,i* at the ith node of the sand-bed reach is thus given as

and the corresponding value grav,i* for the ith node of the gravel-bed reach is given as

CALCULATION OF SHIELDS NUMBERS

1MSMGto1MGi 2i,sands

2wfs

i,sand HRgDqC

2

2w

f2

fb HqCUC

2

2w

f

2f

RgDHqC

RgDUC

2i,gravg

2wfg

i,grav HRgDqC

1MGto1i

29



In the present implementation the gravel transport on the gravel-bed reach is calculated using the Parker (1979) approximation of the Einstein (1950) relation introduced in Chapter 7; where qg denotes the volume gravel transport per unit width and the subscript “i” denotes the ith node,

The sand transport on the sand-bed reach is calculated using the Engelund-Hansen (1967) formulation introduced in Chapter 12; where qs denotes the volume sand transport per unit width and the subscript “i” denotes the ith node,

CALCULATION OF SEDIMENT TRANSPORT

5.2i,sand

fsssi,s C

05.0DRgDq

5.4

i,grav

5.1i,gravggi,g

03.012.11DRgDq

1MSMGto1MGi

1MGto1i

30



The implementation of Exner on the gravel-bed reach is as follows: where qg,feed denotes the volume feed rate per unit width of gravel at x = 0,

CALCULATION OF BED EVOLUTION OF GRAVEL-BED REACH

MG..1i,txq

r)1(s)1(I

ttxs

xs

g

i,g

Bgpggs

gsgfg

g

i,g

gs

i,ggsti,gtti,g

h

hh

MG..2i,xqq

1i,xqq

xq

g

1i,gi,g

g

feed,gi,g

g

i,g

hh

hh

h

MG..2i,x2

1i,x

xg

1i,g1i,g

g

i,g1i,g

g

i,g

31



The model is designed so that washload (e.g. sand for a gravel-bed stream) can be captured as the gravel-bed channel aggrades over its depositional width. This results in a downstream decrease in qs over the gravel-bed reach, even though sand is traveling as wash load. The decrease is calculated by discretizing the following relation from Slide 21:

so yielding

where qs,feed denotes the volume feed rate per unit width of sand at x = 0.

CALCULATION OF CAPTURE OF SAND IN THE GRAVEL-BED REACH

xq

xq g

sgs

MG..2i,)qq(q

1i,qq

i,g1i,gsg1i,s

feed,si,s

32



The implementation of Exner on the sand-bed reach is as follows:

CALCULATION OF BED EVOLUTION OF SAND-BED REACH

1MSMG..1MGi

,txq

r)1)(sL()1(I

ttx)sL(

)x1(s

s

i,s

Bspsgs

gmsfs

s

i,s

gs

i,sgsti,stti,s

h

hh

1MSMG..2MGi,xqq

1MGi,xqq

xq

s

1i,si,s

s

MG,si,s

s

i,s

MSMG

1MSMGi,x

..2MGi,x2

1MGi,x

x

s

1i,si,s

s

1i,s1i,s

s

i,s1i,s

s

i,s

hh

hh

hh

h

33



The migration speed of the gravel-sand transition is given as:

This relation translates to the following moving-boundary form:

where

The new position of the gravel-sand transition is thus given as

CALCULATION OF MIGRATION OF GRAVEL-SAND TRANSITION

0xs

g

sgBgpg

gsgfg

1xg

s

gsBsps

smsfs

sgsggsgs

sgxq

)sL(r)1()1(I

xq

sr)1()1(I

)SS(1s

sgs

1MG,g2MG,gsgs

ggs

MG,g1MG,gggs x)sL(

S,xs

S

hh

hh

g

MG,g1MG,g

gsBgpg

gsgfg

s

MG,s1MG,s

gsBsps

smsfs

sgsggsgs x

qqsr)1(

)1(Ix

qq)sL(r)1(

)1(I)SS(

1s

tsss gstgsttgs

34



The analysis of the previous slides is implemented in the workbook RTe-bookGravelSandTransition.xls. The code utilizes a large number of input parameters in worksheet “InData”, as enumerated below and on the next slideQbf bankfull discharge: same for gravel- and sand-bed reach [L3/T] Ifg flood intermittency for gravel-bed reach [1]Ifs flood intermittency for sand-bed reach [1]Qgrav,feed volume feed rate of gravel at x = 0 (qg,feed = Qgrav,feed/Bc,grav) [L3/T]Qsand,feed volume feed rate of sand at x = 0 (qs,feed = Qsand,feed/Bc,sand) [L3/T]Bc,grav bankfull width of gravel-bed stream [L]Bc,sand bankfull width of sand-bed stream [L]Bd,grav depositional width of gravel-bed reach (rBg = Bd,grav/Bc,grav) [L]Bd,sand depositional width of sand-bed reach (rBs = Bd,sand/Bc,sand) [L]g sinuosity of gravel-bed reach [1]s sinuosity of sand-bed reach [1]sg volume fraction of sand deposited per unit gravel in gravel-bed reach [1]ms volume fraction of mud deposited per unit sand in sand-bed reach [1]Dg characteristic size of gravel [L]Ds characteristic size of sand [L]

INTRODUCTION TO RTe-bookGravelSandTransition.xls, A CALCULATOR FOR THE EVOLUTION OF THE LONG PROFILE OF A RIVER WITH A GRAVEL-SAND

TRANSITION THAT IS FREE TO MIGRATE

35



More input parameters specified in worksheet “InData” of RTe-bookAgDegNormalGravMixSubPW.xls are defined below. Czg Chezy resistance coefficient of gravel-bed reach (Cfg = Czg

-2) [1]Czs Chezy resistance coefficient of sand-bed reach (Cfs = Czs

-2) [1]L Reach length [L]sgsI Initial value of distance sgs to gravel-sand transition [L]SgI Initial slope of gravel-bed reach [1]SsI Initial slope of sand-bed reach [1] Subsidence rate [L/T]do Initial value of sea level elevation [L]

rate of sea level rise [L/T]Yearstart Year in which sea level rise starts [T]Yearstop Year in which sea level rise stops [T]t Time step [T]MG Number of gravel intervalsMS Number of sand intervalsMtoprint Number of time steps to printoutMprint Number of printoutsThe following parameters are specified in worksheet “AuxiliaryData”: porosity of deposit on gravel-bed reach pg, porosity of deposit on sand-bed reach ps and sediment submerged specific gravity R (assumed to be the same for sand and gravel).

INTRODUCTION TO RTe-bookGravelSandTransition.xls contd.

d

36


RIVERS AND TURBIDITY CURRENTSNOTES AND CAVEATS

1. The code locates the gravel-sand transition at a point determined by the continuity condition. At this point the gravel transport rate is only a small fraction of the feed value, but it is not precisely zero. In rivers, the small residual gravel load at gravel-sand transitions is either buried or consists of grains that easily break down to sand. In the code, the residual gravel load at the gravel-sand transition is added to the sand load.

2. In the case of sea level rise at constant rate , rise can be commenced and halted at specified times Yearstart and Yearstop in worksheet “InData”.

3. The reach length L should be chosen to be less than the maximum value Lmax, in order to ensure that there is enough sediment supply to fill the accomodation space created by subsidence or sea level rise. Guidance in this regard is provided in Cell C41 of worksheet “InData”.

4. The initial downstream bed elevation is taken to be zero. As a result, the initial downstream water surface elevation do also equals the initial downstream depth. In order to ensure subcritical flow (and thus keep the calculation from crashing), do must be exceed the critical flow depth Hc = [(Qbf/Bc,sand)2/g]-1/3. Guidance is provided in Cell C44 of worksheet “InData”.

5. Depending on the input values, there may be no steady-state solution allowing a gravel-sand transition to equilibrate at a position between 0 and L. For example, if = 0 and = 0, the only steady-state solution is one for which the sand is all driven into the sea. In such cases, the code will fail. (It would be an easy job to modify the code to handle such cases, but it has not been done). The code can be run, however, to a time at which the gravel-sand

transition is nearly driven out of the domain of interest. Examples appear in succeeding slides.

d

d

37



EVOLUTION OF RIVER PROFILES WITH MIGRATING GRAVEL-SAND

TRANSITIONS: CASE OF SEA LEVEL RISE(WITH VANISHING SUBSIDENCE)

A parametric study is presented with rates of sea level rise varying from 0 to 14 mm/year. Input data for a base case ( = 6 mm/year) are given to the left and below.

d

d

Qbf 750 m3/s Bankfull water discharge at floodIfg 0.05 Flood intermittency, gravel-bed reachIfs 0.05 Flood intermittency, sand-bed reachQgrav,feed 0.075 m3/s Feed rate of gravelQsand,feed 0.15 m3/s Feed rate of sand

3.14E-01 Mt/a Annual gravel load supply to reach6.27E-01 Mt/a Annual sand load supply to reach

Bc,grav 90 m Gravel-bed channel widthBc,sand 90 m Sand-bed channel widthBd,grav 4000 m Depositional or floodplain width, gravel-bed reachBd,sand 4000 m Depositional or floodplain width, sand-bed reach

g 1.5 Sinuosity of gravel-bed reach

s 2 Sinuosity of sand-bed reach

sg 0.5 Fraction of sand deposited per unit gravel in depositional zone of gravel-bed reach

ms 1 Fraction of mud deposited per unit sand in depositional zone of sand-bed reachDg 30 mm Grain size of gravelDs 0.25 mm Grain size of sandCzg 15 Dimensionless Chezy resistance coefficient gravel-bed reach; = (C fg)

-1/2

Czs 25 Dimensionless Chezy resistance coefficient sand-bed reach; (C fs)-1/2

L 30000 m Reach lengthsgsI 15000 m Initial position of gravel-sand transition (must be < L)SgI 0.0015 Initial slope of gravel-bed reachSsI 0.00015 Initial slope of sand-bed reach 0 mm/year Subsidence rate

do 5 m Initial water surface base level6 mm/year Rate of base level rise

Yearstart 0 year Year of start of sea level riseYearstop 12000 year Year of stop of sea level riset 0.2 years Time stepMG 50 No. of fluvial gravel intervalsMS 50 No. of fluvial sand intervalsMtoprint 5000 No. of steps until a printout of results is madeMprint 6 No. of printouts after the initial one

6000 Calculation time in years

d

pg 0.4 Bed porosity, gravel-bed reach

ps 0.4 Bed porosity, sand-bed reach

R 1.65 Submerged specific gravity of sediment

38


RIVERS AND TURBIDITY CURRENTSd/dt = 0 mm/yearSEA LEVEL RISE OF 6 MM/YEAR FOR 6000 YEARS

Bed Elevation Profiles (h), Final Water Surface Elevation Profile ()

-10

0

10

20

30

40

50

60

70

0 5000 10000 15000 20000 25000 30000 35000

x (m)

h , fi

nal

(m)

0 yr1000 yr2000 yr3000 yr4000 yr5000 yr6000 yrfinal w.s.


39


RIVERS AND TURBIDITY CURRENTSd/dt = 0 mm/yearSEA LEVEL RISE OF 6 MM/YEAR FOR 6000 YEARS


-10

0

10

20

30

40

50

60

70

0 5000 10000 15000 20000 25000 30000 35000

x (m)

h , fi

nal

(m)


Position of gravel-sand transition migrates downstream and stabilizes as river aggrades.

40



Bed Slope Profiles ("G" = gravel, "S" = sand in legend)

0

0.0005

0.001

0.0015

0.002

0.0025

-5000 0 5000 10000 15000 20000 25000 30000 35000

x (m)

S

G 0 yrS 0 yrG 1000 yrS 1000 yrG 2000 yrS 2000 yrG 3000 yrS 3000 yrG 4000 yrS 4000 yrG 5000 yrS 5000 yrG 6000 yrS 6000 yr

d/dt = 0 mm/yearSEA LEVEL RISE OF 6 MM/YEAR FOR 6000 YEARS

The high slope near the gravel-sand transition is an artifact of the calculation and should be ignored: see next slide.

Slope break at gravel-sand transition at steady state

Gravel-bed

Sand-bed

41




0

0.0005

0.001

0.0015

0.002

0.0025

-5000 0 5000 10000 15000 20000 25000 30000 35000

x (m)

S


d/dt = 0 mm/yearREASON FOR THE SPURIOUSLY HIGH GRAVEL-BED SLOPE NEAR THE

GRAVEL-SAND TRANSITIONIn a backwater formulation, the actual continuity condition is not the one given in Slide 13 in terms of bed elevation but rather one expressed in terms of water surface elevation:

Since = h + H, this leads to the form

)t(sxs)t(sxggsgs

)t,x()t,x(

h

h

gsgs

gsgs

sx

ssgs

sx

gggs

sx

ss

sx

gg

sg

xHS

xH

S

t)H(

t)H(

s

The extra terms would likely remove the spurious slope, but would otherwise not change the analysis much.

42



Gravel and Sand Loads ("G" = gravel, "S" = sand in legend)

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

0.0018

-5000 0 5000 10000 15000 20000 25000 30000 35000

x (m)

q g, q

s m2 /s


d/dt = 0 mm/yearSEA LEVEL RISE OF 6 MM/YEAR FOR 6000 YEARS

Gravel load drops nearly to zero at steady-state gravel-sand transition

Sand load does not drop to zero even at steady state

Gravel

Sand

43



Position of Gravel-Sand Transition

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

0 1000 2000 3000 4000 5000 6000 7000

time (years)

s gs (

m)

sgs

SEA LEVEL RISE OF 6 MM/YEAR FOR 6000 YEARS

The gravel-sand transition migrates downstream nearly to its steady state position within 2000 years.

44





0

10

20

30

40

50

60

70

80

90

100

0 5000 10000 15000 20000 25000 30000 35000

x (m)

h , fi

nal

(m)


The model eventually fails shortly after 2160 years as the gravel-sand transition migrates downstream out of the domain. This is to be expected for a vanishing sea level rise.

45





0

10

20

30

40

50

60

70

80

90

100

0 5000 10000 15000 20000 25000 30000 35000

x (m)

h , fi

nal

(m)


Again the gravel-sand transition migrates downstream out of the domain, this time shortly after 3780 years. The rate of sea level rise is still not sufficient to stabilize the gravel-sand transition within the domain.

46





0

10

20

30

40

50

60

70

80

90

100

0 5000 10000 15000 20000 25000 30000 35000

x (m)

h , fi

nal

(m)


Gravel-sand transition location does not stabilize by 6000 years, but neither does it migrate downstream out of the domain.

47





0

10

20

30

40

50

60

70

80

90

100

0 5000 10000 15000 20000 25000 30000 35000

x (m)

h , fi

nal

(m)


Gravel-sand transition migrates downstream and starts to stabilize by 6000 years.

48





0

10

20

30

40

50

60

70

80

90

100

0 5000 10000 15000 20000 25000 30000 35000

x (m)

h , fi

nal

(m)


Gravel-sand transition migrates downstream modestly and stabilizes by 6000 years.

49





0

10

20

30

40

50

60

70

80

90

100

0 5000 10000 15000 20000 25000 30000 35000

x (m)

h , fi

nal

(m)


Gravel-sand transition migrates slightly upstream and stabilizes by 6000 years.

50





0

10

20

30

40

50

60

70

80

90

100

0 5000 10000 15000 20000 25000 30000 35000

x (m)

h , fi

nal

(m)


Gravel-sand transition migrates supstantially upstream and nearly stabilizes by 6000 years.

51





0

10

20

30

40

50

60

70

80

90

100

0 5000 10000 15000 20000 25000 30000 35000

x (m)

h , fi

nal

(m)


Gravel-sand transition migrates even farther upstream and nearly stabilizes by 6000 years.

52





0

10

20

30

40

50

60

70

80

90

100

0 5000 10000 15000 20000 25000 30000 35000

x (m)

h , fi

nal

(m)


Rate of sea level rise is so large that there is insufficient sediment to fill the accommodation space. As a result, an embayment forms.

53




-10

0

10

20

30

40

50

60

70

80

0 5000 10000 15000 20000 25000 30000 35000

x (m)

h , fi

nal

(m)


SEA LEVEL RISE OF 0 MM/YEAR UP TO YEAR 1000, 12 MM/YEAR FROM YEAR 1000 TO 4000, 0 MM/YEAR FROM YEAR 4000 TO 6000

Transition progrades out again

Transition progrades outTransition moves upstream

In this run Yearstart = 1000 years and Yearstop = 4000 years.

54




0

10

20

30

40

50

60

70

80

90

100

0 5000 10000 15000 20000 25000 30000 35000

x (m)

h , fi

nal

(m)


Transition progrades out again

Transition progrades out

Transition moves upstream

SEA LEVEL RISE OF 4 MM/YEAR UP TO YEAR 1000, 12 MM/YEAR FROM YEAR 1000 TO 4000, 4 MM/YEAR FROM YEAR 4000 TO 6000

Note: this calculation required changing 5 lines of code.

55



EVOLUTION OF RIVER PROFILES WITH MIGRATING GRAVEL-SAND

TRANSITIONS: CASE OF SUBSIDENCE(WITH VANISHING SEA LEVEL RISE)

A reach with a length of 180,000 m and a subsidence rate of 2 mm/year is considered. Two cases are considered: an initial position sgsI of the gravel-sand transition of 15,000 m, and one with an initial position of 60,000 m.

Qbf 750 m3/s Bankfull water discharge at floodIfg 0.05 Flood intermittency, gravel-bed reachIfs 0.05 Flood intermittency, sand-bed reachQgrav,feed 0.05 m3/s Feed rate of gravelQsand,feed 0.15 m3/s Feed rate of sand

2.09E-01 Mt/a Annual gravel load supply to reach6.27E-01 Mt/a Annual sand load supply to reach

Bc,grav 90 m Gravel-bed channel widthBc,sand 90 m Sand-bed channel widthBd,grav 4000 m Depositional or floodplain width, gravel-bed reachBd,sand 4000 m Depositional or floodplain width, sand-bed reach

g 1.5 Sinuosity of gravel-bed reach

s 2 Sinuosity of sand-bed reach

sg 0.5 Fraction of sand deposited per unit gravel in depositional zone of gravel-bed reach

ms 1 Fraction of mud deposited per unit sand in depositional zone of sand-bed reachDg 30 mm Grain size of gravelDs 0.25 mm Grain size of sandCzg 15 Dimensionless Chezy resistance coefficient gravel-bed reach; = (Cfg)

-1/2

Czs 25 Dimensionless Chezy resistance coefficient sand-bed reach; (C fs)-1/2

L 180000 m Reach lengthsgsI 15000 m Initial position of gravel-sand transition (must be < L)SgI 0.001 Initial slope of gravel-bed reachSsI 0.00015 Initial slope of sand-bed reach 2 mm/year Subsidence rate

do 5 m Initial water surface base level0 mm/year Rate of base level rise

Yearstart 0 year Year of start of sea level riseYearstop 12000 year Year of stop of sea level riset 0.2 years Time stepMG 150 No. of fluvial gravel intervalsMS 50 No. of fluvial sand intervalsMtoprint 10000 No. of steps until a printout of results is madeMprint 6 No. of printouts after the initial one

12000 Calculation time in years

d

pg 0.4 Bed porosity, gravel-bed reach

ps 0.4 Bed porosity, sand-bed reach

R 1.65 Submerged specific gravity of sediment

56



INITIAL POSITION OF THE GRAVEL-SAND TRANSITION sgsI is 15,000 m


-10

0

10

20

30

40

50

60

0 50000 100000 150000 200000

x (m)

h , fi

nal

(m)


Gravel-sand transition migrates downstream and stabilizes by 12,000 years

Subsidence rate = 2 mm/year

57





0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

-50000 0 50000 100000 150000 200000

x (m)

S


Gravel-bed

Sand-bed


58





0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

0.0018

-50000 0 50000 100000 150000 200000

x (m)

q g, q

s m2 /s


Gravel

Sand


59





0

50000

100000

150000

0 2000 4000 6000 8000 10000 12000 14000

time (years)

s gs (

m)

sgs

Gravel-sand transition migrates downstream and stabilizes by 12,000 years


60





-10

0

10

20

30

40

50

60

70

80

90

0 50000 100000 150000 200000

x (m)

h , fi

nal

(m)


Gravel-sand transition migrates upstream, but has not quite stabilized by 48,000 years


61





0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

-50000 0 50000 100000 150000 200000

x (m)

S


Gravel-bed

Sand-bed


62





0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

0.0018

-50000 0 50000 100000 150000 200000

x (m)

q g, q

s m2 /s


Gravel

Sand


63





0

50000

100000

150000

0 10000 20000 30000 40000 50000 60000

time (years)

s gs (

m)

sgs

Gravel-sand transition migrates upstream, but has not quite stabilized by 48,000 years


64



FURTHER COMMENTS

1. The model presented in this chapter allows neither downstream fining nor abrasion of gravel. Parker (1991a,b) provides a formulation of abrasion in the context of downstream fining of gravel-bed rivers, and Parker and Cui (1998) and Cui and Parker (1998) incorporate this formulation in a treatment of gravel-sand transitions.

2. Gravel-sand transitions have also been treated by Paola et al. (1992). In their treatment the location of the gravel-sand transition is determined by the point where the gravel runs out (gravel transport drops to zero).

3. Recent modeling work by Ferguson (2003) merits review by the interested reader.

65


RIVERS AND TURBIDITY CURRENTSREFERENCES FOR CHAPTER 27

Aalto, R., 2002, Geomorphic form and Process of Sediment Flux within an Active Orogen: Denudation of the Bolivian Andes and Sediment Conveyance across the Beni Foreland, PhD thesis, University of Washington, USA, 365 p.

Cui, Y. and Parker, G., 1998, The arrested gravel front: stable gravel-sand transitions in rivers. Part 2: General numerical solution, Journal of Hydraulic Research, 36(2): 159-182.

Einstein, H. A., 1950, The Bed-load Function for Sediment Transportation in Open Channel Flows, Technical Bulletin 1026, U.S. Dept. of the Army, Soil Conservation Service.

Engelund, F. and E. Hansen, 1967, A Monograph on Sediment Transport in Alluvial Streams, Technisk Vorlag, Copenhagen, Denmark.

Ferguson, R. I., 2003, Emergence of abrupt gravel-sand transitions along rivers through sorting processes, Geology 31, 159-162.

Paola, C., Heller, P. L., and Angevine, C. L., 1992, The large-scale dynamics of grain-size variation in alluvial basins, 1: Theory, Basin Research, 4, 73-90.

Parker, G., 1979, Hydraulic geometry of active gravel rivers, Journal of Hydraulic Engineering, 105(9), 1185‑1201.Parker, G., 1991a, Selective sorting and abrasion of river gravel: theory, Journal of Hydraulic

Engineering, 117(2), 131-149.Parker, G., 1991b, Selective sorting and abrasion of river gravel: applications, Journal of

Hydraulic Engineering, 117(2), 150-171.Parker, G., and Y. Cui, 1998, The arrested gravel front: stable gravel-sand transitions in rivers.

Part 1: Simplified analytical solution, Journal of Hydraulic Research, 36(1): 75-100.

66


RIVERS AND TURBIDITY CURRENTSREFERENCES FOR CHAPTER 27 contd.

Sambrook Smith, G. H. and Ferguson, R., 1995, The gravel-sand transition along river channels, Journal of Sedimentary Research, A65(2): 423-430.

Shaw, J. and R. Kellerhals, 1982, The Composition of Recent Alluvial Gravels in Alberta River Beds, Bulletin 41, Alberta Research Council, Edmonton, Alberta, Canada.

Yatsu, E., 1955, On the longitudinal profile of the graded river, Transactions, American Geophysical Union, 36: 655-663.

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Fly River, Papua New Guinea

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