CHAPTER 14: 1D AGGRADATION AND DEGRADATION OF RIVERS: NORMAL FLOW ASSUMPTION

1

1D SEDIMENT TRANSPORT MORPHODYNAMICSwith applications to

RIVERS AND TURBIDITY CURRENTS© Gary Parker November, 2004

CHAPTER 14:1D AGGRADATION AND DEGRADATION OF RIVERS: NORMAL FLOW

ASSUMPTION

The disposal of large amounts of waste sediment from the Ok Tedi Copper Mine, Papua New Guinea, has caused significant aggradation, or bed level rise, in the Ok Tedi (“Ok” means “river”) and Fly Rivers.

Aggradation in gravel-bed reaches of Ok Tedi

The mine

5 m aggradation at bridge

Aggradation where the Ok Ma joins the

Ok Tedi

2



CHANNEL AGGRADATION AND FLOODPLAIN DEPOSITION OF OK TEDI AT GRAVEL-SAND TRANSITION

River slope drops by an order of magnitude in the transition zone from braided gravel-bed to meandering sand-bed stream, leading to massive deposition of sand.

Sediment depositing on the floodplain has destroyed the forest.

Sand is dredged from the river to ameliorate the deposition.

3



RESPONSE OF A RIVER TO SUDDEN VERTICAL FAULTING CAUSED BY AN EARTHQUAKE

View in November, 1999, shortly after the earthquake caused a sharp 3 m elevation drop at a

fault.

View in May, 2000 after aggradation and degradation

have smoothed out the elevation drop.The above images of the Deresuyu River, Turkey, are

courtesy of Patrick Lawrence and François Métivier (Lawrence, 2003)

4



RESPONSE OF A RIVER TO SUDDEN VERTICAL FAULTING CAUSED BY AN EARTHQUAKE contd.

Upstream degradation (bed level lowering) and downstream aggradation (bed level increase) are realized as the river responds

to the knickpoint created by the earthquake (Lawrence, 2003)

Inferred initial profile immediately after faulting

in November, 1999

Profile in May, 2001

5



BACKGROUND AND ASSUMPTIONS

Here “base level” loosely means a controlling elevation at the downstream end of the reach of interest. It means water surface elevation if the river flows into a lake or the ocean, or a downstream bed elevation controlled by e.g. tectonic uplift or subsidence at a point where the river is not flowing into standing water.

Change in channel bed level (aggradation or degradation) can occur in response to:• increase or decrease in upstream sediment supply;• change in hydrologic regime (water diversion or climate change);• change in river slope (e.g. channel straightening, as outlined in Chapter 2);• increased or decreased sediment supply from tributaries;• sudden inputs of sediment from debris flows or landslides;• faulting due to earthquakes or other tectonic effects such as tilting along the reach,

and;• changing base level at the downstream end of the reach of interest.

Base level of this reach of the Eau Claire river, Wisconsin, USA is controlled by a reservoir, Lake Altoona

6



THE EQUILIBRIUM STATE contd.

Rivers are different in many ways from laboratory flumes. It nevertheless helps to conceptualize rivers in terms of a long, straight, wide, rectangular flume with high sidewalls (no floodplain), constant width and a bed covered with alluvium. Such a “river” has a simple mobile-bed equilibrium (graded) state at which flow depth H, bed slope S, water discharge per unit width qw and bed material load per unit width qt remain constant in time t and in the streamwise direction x. A recirculating flume (with both water and sediment recirculated) at equilibrium is illustrated below.

pump

water

sediment

7



Friction relations:

where kc is a composite bed roughness which may include the effect of bedforms (if present).

THE EQUILIBRIUM STATE contd.The hydraulics of the equilibrium state are those of normal flow. Here the case of a plane bed (no bedforms) is considered as an example. The bed consists of uniform material with size D. The governing equations are (Chapter 5):

UHqw Water conservation: gHSb

2fb UC )StricklerManning(

kHCor)Chezy(constC

6/1

cr

2/1ff

Momentum conservation:

Generic transport relation of the form of Meyer-Peter and Müller for total bed material load: where t and nt are dimensionless constants:

tn

cb

tt

RgDDRgDq

8



THE EQUILIBRIUM STATE contd.In the case of the Chezy resistance relation, the equations governing thenormal state reduce to:

3/12wf

gSqCH

tn

c

3/23/12wf

tt RDS

gqCDRgDq

In the case of the Manning-Stickler resistance relation, the equationsgoverning the normal state reduce with to:

10/3

2r

2w

3/1c

gSqkH

tn

c

10/710/3

2r

2w

3/1c

tt RDS

gqkDRgDq

Let D, kc and R be given. In either case above, there are two equations for four parameters at equilibrium; water discharge per unit width qw, volume sediment discharge per unit width qt, bed slope S and flow depth H. If any two of the set (qw, qt, S and H) are specified, the other two can be computed. In a sediment-feed flume, qw and qt are set, and equilibrium S and H can be computed from either of the above pair. In a recirculating flume, qw andH are set (total water mass in flume is conserved), and qt and S can be computed.

9



THE EQUILIBRIUM STATE contd.The basic nature of the arguments of the previous slide do not change if a) total bed material transport is divided into bedload and suspended load components, each with its own predictor, b) bed shear stress is divided into skin friction and form drag, each with its own predictor, and c) transport/entrainment relations for uniform material are replaced with relations for sediment mixtures. Each new variable is accompanied by one new constraint (governing equation). For example, consider the case of gravel transport in the absence of bedforms. Using the gravel bedload transport relation of Powell et al. (2001) as an example, and setting kc = nkDs90 (no form drag), the problem reduces with the relations of Chapters 5 and 7 to

10/3

2r

2w

3/190s

3/1k

gSqDnH

4.5

i

10/710/3

2r

2w

3/190s

3/1k

74.0

50s

i20/9

2r

2w

3/190s

3/1k

20/2120/1

ibi

RDS

gqDn

DD03.0

1qDnRSgFq

Recalling that qbT = qbi and bedload fractions pi =qbi/qbT, if any two of the set (H, S, qw, qT) and either the bed surface fractions Fi or the bedload fractions pi are specified, the equilibrium values of the other parameters can be computed from the above equations. For example, if S, qw and Fi (from which Ds50 can be computed according to the relations of Chapter 2) arespecified, H, qbT and pi can be computed directly from the above relations.

10



SIMPLIFICATIONSThe concepts of aggradation and degradation are best illustrated by using simplified

relations for hydraulic resistance and sediment transport. Here the following simplifications are made in addition to the assumptions of constant width and the absence of a floodplain:

1. The case of a Manning-Strickler formulation with constant composite roughness kc is considered;

2. Bed material is taken to be uniform with size D;3. The Exner equation of sediment conservation is based on a computation of total

bed material load, which is computed via the generic equation

where s 1 is a constant to convert total boundary shear stress to that due to skin friction (if necessary). For example, to recover the corrected version of Meyer-Peter and Müller (1948) relation of Wong and Parker (submitted) for gravel transport, set t = 3.97 , nt = 1.5, c* = 0.0495 and s = 1. For the bed material load relation of Engelund and Hansen (1967) for sand transport, which uses total boundary shear stress, not that due to skin friction,

t = 0.05/Cf, nt = 2.5, c* = 0 and s = 1.

tn

cbs

tt

RgDDRgDq

11



SIMPLIFICATIONS contd.4. The full flood hydrograph or flow duration curve of discharge variation is

replaced by a flood intermittency factor If, so that the river is assumed to be at low flow (and not transporting significant amounts of sediment) for time fraction 1 – If, and is in flood at constant discharge Q, and thus constant discharge per unit width qw = Q/B for time fraction If (Paola et al., 1992). The implied hydrograph takes the conceptual form below:

In the long term, then, the relation between actual time t and time that the river has been in flood tf is given as

Let the value of the total bed material load at flood flow qt be computed in m2/s. Then the total mean annual sediment load Gt in million tons per year is given as

t

Qlow flow

flood

tIt ff

)10x1/(tBIqG 6aftst

12



SIMPLIFICATIONS contd.There are many reasonable ways to compute the intermittency factor If. One

reasonable way to do so is to:a) compute the volume bed material transport rate Qtbf at bankfull flow;b) use the full flow duration curve to compute the mean annual volume bed

material transport rate Qtanav as

where qt,k denotes the value of qt in the kth discharge range, and pk denotes the fraction of time the flow is in this range, and

c) Compute the flood value of qt and If as

In this way If denotes the fraction of time per year that continuousbankfull flow would yield the annual sediment yield.

t

Qlow flow

flood

kk,tavtan pQQ

tbf

avtanf

bf

tbft

QQI

BQq

13



SIMPLIFICATIONS contd.

Generalization to the case of varying width is also rather straightforward, and is implemented in future chapters of this e-book. Including the floodplain, however, is more difficult, especially in the case of meandering rivers. This is because when the floodplain is inundated and the floodplain depth is substantial, the thread of high velocity may no longer completely follow the river channel. The simplest reasonable assumption is that the bed material load at above-bankfull flows is equal to that at bankfull flow.

It is important to realize that none of these simplifications are necessary. Oncemethods for computing aggradation and degradation are developed using the above simplifications, however, the analysis easily generalizes to cases with mixed grain sizes, a distinction between bedload and suspended bed material load, a computation of both form drag and skin friction, and computations using the full flow hydrograph or flow duration curve. The Minnesota River, USA

near Le Sueur during the flood of record in 1965

channel

14



AGGRADATION AND DEGRADATION AS TRANSIENT RESPONSES TO IMPOSED DISEQIUILBRIUM CONDITIONS

Aggradation or degradation of a river reach can be considered to be a response to disequilibrium conditions, by which the river tries to reach a new equilibrium. For example, if a river reach has attained an equilibrium with a given sediment supply from upstream, and that sediment supply is suddenly increased at t = 0, the river can be expected to aggrade toward a new equilibrium.

h

antecedent equilibrium bed profile established with load qta

final equilibrium bed profile in balance with load qt > qta

transient aggradational profile

sediment supply increases from qta

to qt at t = 0

15



NORMAL FLOW FORMULATION OF MORPHODYNAMICS: GOVERNING EQUATIONS

In this chapter the flow is calculated by approximating it with the normal flow formulation, even if the profile itself is in disequilibrium. The approximation is of loose validity in most cases of interest, and becomes more rigorously valid with increasing Froude number. Gradually varied flow is considered in Chapter 20. Using the Exner formulation of Chapter 2 and the Manning-Strickler formulation for flow resistance, the morphodynamic problem has the following character:

xqI

t)1( t

fp

h

-

xS,

RDS

gqkDRgDq

tn

c

10/710/3

2r

2w

3/1c

stt h

In the above relations t denotes real time (as opposed to flood time) and the intermittency factor If accounts for the fact that the river is only occasionally in flood (and thus morphologically active).

16



THE NORMAL FLOW MORPHODYNAMIC FORMULATION AS A NONLINEAR DIFFUSION PROBLEM

The previous formulation can be rewritten as:

where d is a kinematic “diffusivity” of sediment (dimensions of L2/T) given by the relation

h

h

x)S(

xt d

tn

c

10/710/3

2r

2w

3/1c

stp

fd RD

Sgqk

S)1(DRgDI

The top equation is a diffusion equation. In the bottom equation, it is seen that d is dependent on S = - h/x, so that the diffusion formulation is nonlinear.

The problem is second-order in x and first order in t, so that one initialcondition and two boundary conditions are required for solution.

17



INITIAL AND BOUNDARY CONDITIONS

The initial condition is that of a specified bed profile;

The simplest example of this is a profile with specified initial downstream elevation hId at x = L and constant initial slope SI;

The upstream boundary condition can be specified in terms of given sediment supply, or feed rate qtf, which may vary in time;

The simplest case is that of a constant value of sediment feed.

The downstream boundary condition can be one of prescribed base level in terms of bed elevation;

Again the simplest case is a constant value, e.g. hd = 0.

)x()t,x( I0thh

)xL(S)t,x( IId0thh

)t(q)t,x(q tf0xt

)t()t,x( dLxhh

The reach over which morphodynamic evolution is to be described must have a finite length L. Here it extends from x = 0 to x = L.

18



NOTES ON THE DOWNSTREAM BOUNDARY CONDITION

Alluvial Kaiya River, Papua New Guinea, and downstream bedrock exposureBedrock makes a

good downstrea

m b.c.

In principle the best place to locate the downstream boundary condition is at a bedrock exposure, as illustrated below. In most alluvial streams, however, such points may not be available. Three alternatives are possible:

a) Set the boundary condition at a point so far downstream that no effect of e.g. changed sediment feed rate is felt during the time span of interest;

b) Set the boundary condition where the river joins a much larger river; orc) Set the boundary condition at a point of known water surface elevation, such as

a lake (see Chapter 20 and the use of the gradually varied flow model).

19



DISCRETIZATION FOR NUMERICAL SOLUTION

hh

hh

hh

1Mi,x

M..2i,x2

1i,x

S

1MM

1i1i

21

i

Bed slope can be computed by the relations to the right. Once the slope Si is computed the sediment transport rate qt,i can be computed at every node. At the ghost node, qt,g = qtf.

The morphodynamic problem is nonlinear and requires a numerical solution. This may be done by dividing the domain from x = 0 to x = L into M subreaches bounded by M + 1 nodes. The step length x is then given as L/M. Sediment is fed in at an extra “ghost” node one step upstream of the first node.

MLx 1M..1i,x)1i(x i Feed sediment here!

L

x

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

20



M..1i,tIx

q1

1f

i,t

ptitti

hh

xqq

)a1(xqq

axq i,t1i,t

u1i,ti,t

ui,t

DISCRETIZATION OF THE EXNER EQUATION

and au is an upwinding coefficient. In a pure upwinding scheme, au = 1. In a central difference scheme, au = 0.5. A central difference scheme generally works well when the normal flow formulation is used.

At the ghost node, qt,g = qtf. In computing qt,i/x at i = 1, the node at i – 1 (= 0) is the ghost node. At node M+1, the Exner equation is not implemented because bed elevation is specified as hM+1 = hd.

Let t denote the time step. Then the Exner equation discretizes to

where

21



INTRODUCTION TO RTe-bookAgDegNormal.xls

The basic program in Visual Basic for Applications is contained in Module 1, and is run from worksheet “Calculator”.

The program is designed to compute a) an ambient mobile-bed equilibrium, and b) the response of a reach to changed sediment input rate at the upstream end of the reach starting from t = 0.

The first set of required input includes: flood discharge Q, intermittency If, channel (bankfull) width B, grain size D, bed porosity p, composite roughness height kc and ambient bed slope S (before increase in sediment supply). Composite roughness height kc should be equal to ks = nkD, where nk is in the range 2 – 4, in the absence of bedforms. When bedforms are expected kc should be estimated at bankfull flow using the techniques of Chapter 9 and 10 (compute Cz from hydraulic resistance formulation; kc = (11 H)/exp(Cz)).

Various parameters of the ambient flow, including the ambient annual bed material transport rate Gt in tons per year, are then computed directly on worksheet “Calculator”.

22



INTRODUCTION TO RTe-bookAgDegNormal.xls contd.The next required input is the annual average bed material feed rate Gtf imposed after t > 0. If this is the same as the ambient rate Gt then nothing should happen; if Gtf > Gt then the bed should aggrade, and if Gtf < Gt then it should degrade.

The final set of input includes the reach length L, the number of intervals M into which the reach is divided (so that x = L/M), the time step t, the upwinding coefficient au (use 0.5 for a central difference scheme), and two parameters controlling output, the number of time steps to printout Ntoprint and the number of printouts (in addition to the initial ambient state) Nprint.

The downstream bed elevation hd is automatically set equal to zero in the program.

Auxiliary parameters, including r (coefficient in Manning-Strickler), t and nt (coefficient and exponent in load relation), c* (critical Shields stress), s (fraction of boundary shear stress that is skin friction) and R (sediment submerged specific gravity) are specified in the worksheet “Auxiliary Parameters”.

23



INTRODUCTION TO RTe-bookAgDegNormal.xls contd.

The parameter s estimating the fraction of boundary shear stress that is skin friction, should either be set equal to 1 or estimated using the techniques of Chapter 9.

In any given case it will be necessary to play with the parameters M (which sets x) and t in order to obtain good results. For any given x, it is appropriate to find the largest value of t that does not lead to numerical instability.

The program is executed by clicking the button “Do a Calculation” from the worksheet “Calculator”. Output for bed elevation is given in terms of numbers in worksheet “ResultsofCalc” and in terms of plots in worksheet “PlottheData”

The formulation is given in more detail in the worksheet “Formulation”, which is also available as a stand-alone document, Rte-bookAgDegNormalFormul.doc.

24



Sub Main() Clear_Old_Output Get_Auxiliary_Data Get_Data Compute_Ambient_and_Final_Equilibria Set_Initial_Bed_and_time Send_Output j = 0 For j = 1 To Nprint For w = 1 To Ntoprint Find_Slope_and_Load Find_New_eta Next w More_Output Next jEnd Sub

MODULE 1 Sub MainThis is the master subroutine that controls the Visual Basic program.

25



Sub Set_Initial_Bed_and_time() For i = 1 To N + 1 x(i) = dx * (i - 1) eta(i) = Sa * L - Sa * dx * (i - 1) Next i time = 0 End Sub

MODULE 1 Sub Set_Initial_Bed_and_time

This subroutine sets the initial ambient bed profile.

26



Sub Find_Slope_and_Load() Dim i As Integer Dim taux As Double: Dim qstarx As Double: Dim Hx As Double Sl(1) = (eta(1) - eta(2)) / dx Sl(M + 1) = (eta(M) - eta(M + 1)) / dx For i = 2 To M Sl(i) = (eta(i - 1) - eta(i + 1)) / (2 * dx) Next i For i = 1 To M + 1 Hx = ((Qf ^ 2) * (kc ^ (1 / 3)) / (alr ^ 2) / (B ^ 2) / g / Sl(i)) ^ (3 / 10) taux = Hx * Sl(i) / Rr / D If fis * taux <= tausc Then qstarx = 0 Else qstarx = alt * (fis * taux - tausc) ^ nt End If qt(i) = ((Rr * g * D) ^ 0.5) * D * qstarx Next i End Sub

MODULE 1 Sub Find_Slope_and_Load

This subroutine computes the load at every node.

27



Sub Find_New_eta() Dim i As Integer Dim qtback As Double: Dim qtit As Double: Dim qtfrnt As Double: Dim qtdif As Double For i = 1 To M If i = 1 Then qtback = qqtf Else qtback = qt(i - 1) End If qtit = qt(i) qtfrnt = qt(i + 1) qtdif = au * (qtback - qtit) + (1 - au) * (qtit - qtfrnt) eta(i) = eta(i) + dt / (1 - lamp) / dx * qtdif * Inter Next i time = time + dt End Sub

MODULE 1 Sub Find_New_eta

This subroutine implements the Exner equation to find the bed one time step later.

28



A SAMPLE COMPUTATIONCalculation of River Bed Elevation Variation with Normal Flow Assumption

Calculation of ambient river conditions (before imposed change)Assumed parameters

(Qf) Q 70 m^3/s Flood discharge(Inter) If 0.03 Intermittency The colored boxes:(B) B 25 m Channel Width indicate the parameters you must specify.(D) D 30 mm Grain Size The rest are computed for you.(lamp)

p 0.35 Bed Porosity(kc) kc 75 mm Roughness Height If bedforms are absent, set kc = ks, where ks = nk D and nk is an order-one factor (e.g. 3).

(S) S 0.008 Ambient Bed Slope Otherwise set kc = an appropriate value including the effects of bedforms.

Computed parameters at ambient conditionsH 0.875553 m Flow depth (at flood)* 0.141503 Shields number (at flood)q* 0.232414 Einstein number (at flood)qt 0.004859 m^2/s Volume sediment transport rate per unit width (at flood)Gt 3.05E+05 tons/a Ambient annual sediment transport rate in tons per annum (averaged over entire year)

Calculation of ultimate conditions imposed by a modified rate of sediment input

Gtf 7.00E+05 tons/a Imposed annual sediment transport rate fed in from upstream (which must all be carried during floods)qtf 0.011161 m^2/s Upstream imposed volume sediment transport rate per unit width (at flood)

ult 0.211523 Ultimate equilibrium Shields number (at flood)

Sult 0.014207 Ultimate slope to which the bed must aggrade Click the button to perform a calculationHult 0.736984 m Ultimate flow depth (at flood)

Calculation of time evolution toward this ultimate state

L 10000 m length of reach Ntoprint 200 Number of time steps to printoutqt,g 0.011161 m^2/s sediment feed rate (during floods) at ghost node Nprint 5 Number of printoutsx 1.67E+02 m spatial step M 60 Intervalst 0.01 year time step

u 0.5 Here 1 = full upwind, 0.5 = central differenceDuration of calculation 10 years

The ambient sediment transport rate is 305,000 tons/year. At time t = 0 this is increased to 700,000 tons per year. The bed must aggrade in response.

29



RESULTS OF SAMPLE COMPUTATION

Bed evolution

0

20

40

60

80

100

120

140

160

0 2000 4000 6000 8000 10000

Distance in m

Elev

atio

n in

m

0 yr2 yr4 yr6 yr8 yr10 yrUltimate

30



INTERPRETATIONThe long profile of a river is a plot of bed elevation h versus down-channel distance x. The long profile of a river is called upward concave if slope S = -h/x is decreasing in the streamwise direction; otherwise it is called upward convex. That is, a long profile is upward concave if

0xx

S2

2

h

x

h

upward-concave

upward-convex

Aggrading reaches often show transient upward concave profiles. This is because the deposition of sediment causes the sediment load to decrease in the downstream direction. The decreased load can be carried with a decreased Shields number *, and thus according to the normal-flow formulation of the present chapter, a decreased slope:

RDS

gqk 10/710/3

2r

2w

3/1c

31



INTERPRETATION contd.The transient long profile of Slide 29 is upward concave because the river is aggrading toward a new mobile-bed equilibrium with a higher slope. Once the new equilibrium is reached, the river will have a constant slope (vanishing concavity). This process is outlined in the next slide (Slide 32), in which all the input parameters are the same as in Slide 28 except Ntoprint, which is varied so that the duration of calculation ranges from 1 year (far from final equilibrium) to 250 years (final equilibrium essentially reached).

Slide 33 shows a case where the profile degrades to a new mobile-bed equilibrium. During the transient process of degradation the long profile of the bed is downward concave, or upward convex. This is because the erosion which drives degradation causes the load, and thus the slope to increase in the downstream direction. The input conditions for Slide 32 are the same as that of Slide 28, except that the sediment feed rate Gtf is dropped to 70,000 tons per year. This value is well below the ambient value of 305,000 tons per year (see Slide 28), forcing degradation and transient downward concavity. In addition, Ntoprint is varied so that the duration of calculation varies from 1 year to 250 years.

It will be seen in Chapter 25 that factors such as subsidence or sea level risecan drive equilibrium long profiles which are upward concave.

32



Bed evolution

0

20

40

60

80

100

120

140

160

0 2000 4000 6000 8000 10000

Distance in m

Elev

atio

n in

m

0 yr0.2 yr0.4 yr0.6 yr0.8 yr1 yrUltimate

Bed evolution

0

20

40

60

80

100

120

140

160

0 2000 4000 6000 8000 10000

Distance in mEl

evat

ion

in m


Bed evolution

0

20

40

60

80

100

120

140

160

0 2000 4000 6000 8000 10000

Distance in m

Elev

atio

n in

m


Bed evolution

0

20

40

60

80

100

120

140

160

0 2000 4000 6000 8000 10000

Distance in m

Elev

atio

n in

m


AGGRADATION TO A NEW MOBILE-BED EQUILIBRIUM

33



DEGRADATION TO A NEW MOBILE-BED EQUILIBRIUM

Bed evolution

0

10

20

30

40

50

60

70

80

90

0 2000 4000 6000 8000 10000

Distance in m

Elev

atio

n in

m


Bed evolution

0

10

20

30

40

50

60

70

80

90

0 2000 4000 6000 8000 10000

Distance in m

Elev

atio

n in

m


Bed evolution

0

10

20

30

40

50

60

70

80

90

0 2000 4000 6000 8000 10000

Distance in m

Elev

atio

n in

m

0 yr0.2 yr0.4 yr0.6 yr0.8 yr1 yrUltimate

Bed evolution

0

10

20

30

40

50

60

70

80

90

0 2000 4000 6000 8000 10000

Distance in mEl

evat

ion

in m


34



ADJUSTING THE NUMBER M OF SPATIAL INTERVALS AND THE TIME STEP t

The calculation becomes unstable, and the program crashes if the time step t is too long. The above example resulted in a crash when t was increased from the value of 0.01 years in Slide 29 to 0.05 years. The larger the value M of spatial intervals is, the smaller is the maximum value of t to avoid numericalinstability. Acceptable values of M and t can be found by trial and error.

35



AN EXTENSION:RESPONSE OF AN ALLUVIAL RIVER TO VERTICAL FAULTING DUE TO AN

EARTHQUAKE

h

The code in RTe-bookAgDegNormal.xls represents a plain vanilla version of a formulation that is easily extended to a variety of other cases. The spreadsheet RTe-bookAgDegNormalFault.xls contains an extension of the formulation for sudden vertical faulting of the bed. The bed downstream of the point x = rfL (0 < rf < 1) is suddenly faulted downward by an amount hf at time tf. The eventual smearing out of the long profile is then computed.

36



RESULTS OF SAMPLE CALCULATION WITH FAULTING

Bed evolution

-6

-4

-2

0

2

4

6

8

10

12

0 2000 4000 6000 8000 10000

Distance in m

Elev

atio

n in

m 0 yr0.05 yr0.1 yr0.15 yr0.2 yr0.25 yr

37



RESULTS OF SAMPLE CALCULATION WITH FAULTING contd.In time the fault is erased by degradation upstream and aggradation downstream, and a new mobile-bed equilibrium is reached.

Bed evolution

-6

-4

-2

0

2

4

6

8

10

12

0 2000 4000 6000 8000 10000

Distance in m

Elev

atio

n in

m 0 yr0.001 yr0.002 yr0.003 yr0.004 yr0.005 yr

Bed evolution

-6

-4

-2

0

2

4

6

8

10

12

0 2000 4000 6000 8000 10000

Distance in m

Elev

atio

n in

m 0 yr0.025 yr0.05 yr0.075 yr0.1 yr0.125 yr

Bed evolution

-6

-4

-2

0

2

4

6

8

10

12

0 2000 4000 6000 8000 10000

Distance in m

Elev

atio

n in

m 0 yr0.5 yr1 yr1.5 yr2 yr2.5 yr

Bed evolution

-6

-4

-2

0

2

4

6

8

10

12

0 2000 4000 6000 8000 10000

Distance in m

Elev

atio

n in

m 0 yr5 yr10 yr15 yr20 yr25 yr

38



REFERENCES FOR CHAPTER 14

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

Lawrence, P., 2003, Bank Erosion and Sediment Transport in a Microscale Straight River, Ph.D. thesis, University of Paris 7 – Denis Diderot, 167 p.

Meyer-Peter, E. and Müller, R., 1948, Formulas for Bed-Load Transport, Proceedings, 2nd Congress, International Association of Hydraulic Research, Stockholm: 39-64.

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

Powell, D. M., Reid, I. and Laronne, J. B., 2001, Evolution of bedload grain-size distribution with increasing flow strength and the effect of flow duration on the caliber of bedload sediment yield in ephemeral gravel-bed rivers, Water Resources Research, 37(5), 1463-1474.

Wong, M. and Parker, G., submitted, The bedload transport relation of Meyer-Peter and Müller overpredicts by a factor of two, Journal of Hydraulic Engineering, downloadable at http://cee.uiuc.edu/people/parkerg/preprints.htm .

http://cee.uiuc.edu/people/parkerg/preprints.htm

CHAPTER 14: 1D AGGRADATION AND DEGRADATION OF RIVERS: NORMAL FLOW ASSUMPTION

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