1d morphodynamics of mountain rivers: uniform sediment

1

National Center for Earth-surface DynamicsShort Course

Morphology, Morphodynamics and Ecology of Mountain Rivers

December 11-12, 20051D MORPHODYNAMICS OF MOUNTAIN RIVERS: UNIFORM SEDIMENT

Morphodynamics is the study of the variation of morphology in response to net erosion or deposition of sediment. The images below illustrate a reach of the Mad River, California which has undergone bed lowering (degradation) in response to gravel mining.

2



December 11-12, 2005AGGRADATION AND DEGRADATIONA river reach aggrades (bed elevation increases) when it is supplied more sediment than it exports.

A river reach degrades (bed elevation decreases) when it exports more sediment than it is supplied.

Degraded reach of the Uria River, Venezuela after the Vargas disaster, 1999. Cour. J. Lopez.

The Ok Tedi in Papua New Guinea had aggraded some 5 m near this bridge in response to mine disposal.

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December 11-12, 2005RESPONSE 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)

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December 11-12, 2005RESPONSE 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

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December 11-12, 2005BEDLOAD TRANSPORTThe morphodynamics of mountain rivers is controlled by the differential transport of gravel moving as bedload. Bedload particles slide, roll, or saltate just above the bed, as opposed to suspended particles, which can be wafted high in the water column by turbulence. Bedload transport of uniform 7-mm gravel in a flume is illustrated below. The video clip is from the experiments of Miguel Wong.

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December 11-12, 2005BELOW-CAPACITY BEDLOAD TRANSPORTThe video clip is from the experiments of Phairot Chatanantavet.

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December 11-12, 2005EXNER EQUATION FOR THE CONSERVATION OF BED SEDIMENT

In 1D morphodynamics bed elevation variation is considered in the absence of width variation, and local bed features such as bars are not specifically modeled. 1D morphodynamics describes the time variation of the longitudinal profile of river bed elevation in response to net sediment deposition or erosion.

The first step in characterizing 1D morphodynamics is the derivation of the Exner (19??) equation of bed sediment conservation. The parameters defined below are used in the derivation.

qb = volume bedload transport rate per unit width [L2T-1]s = sediment density [ML3/T]gb = sqb = mass bedload transport rate per unit width [ML-1T-1] = bed elevation [L]p = porosity of sediment in bed deposit [1]

(volume fraction of bed sample that is holes rather than sediment: 0.25 ~ 0.55 for noncohesive material)

x = streamwise coordinate [L]t = time [T]B = channel width [L]

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December 11-12, 2005CONSERVATION OF BED SEDIMENT

1qq1gg1x)1(t xxbxbsxxbxbps

or thus

x

q

t)1( b

p

-

This corresponds to the original form derived by Exner.

bed sediment + pores

water

x

1

x

x +x

qb

qb

/t(gravel mass in control volume of bed) = mass gravel inflow rate – mass gravel outflow rate

The control volume has a unit width normal to x

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December 11-12, 2005BACKGROUND AND ASSUMPTIONS FOR 1D MORPHODYNAMICS

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);• 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

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December 11-12, 2005THE 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

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December 11-12, 2005

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 (see lecture on hydraulics):

UHqw Water conservation: gHSb

2fb UC )StricklerManning(

k

HCor)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:

t

t

n

cb

tbn

ct RgDDRgD

qor)(q

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December 11-12, 2005THE EQUILIBRIUM STATE contd.In the case of the Chezy resistance relation, the equations governing thenormal state reduce to (Slide 26 of the lecture on hydraulics):

3/12wf

gS

qCH

tn

c

3/23/12wf

tb RD

S

g

qCDRgDq

In the case of the Manning-Stickler resistance relation with an exponent of 1/6, the equations governing the normal state reduce to (Slide 30 of the lecture on hydraulics, with kc ks and g r):

10/3

2r

2w

3/1s

gS

qkH

tn

c

10/710/3

2r

2w

3/1s

tb RD

S

g

qkDRgDq

Let D, ks 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, qb, S and H) are specified, the other two can be computed. In a sediment-feed flume, qw and qb are set, and equilibrium S and H can be computed from either of the above pair. In a recirculating flume, qw and H are set (total water mass in flume is conserved), and qb and S can be computed.

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December 11-12, 2005SIMPLIFICATIONSThe 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 roughness ks is considered;

2. Bed material is taken to be uniform with size D;3. Only the portion of boundary shear stress due to skin friction is available to

transport sediment;4. The Exner equation of sediment conservation is based on a computation of

bedload, 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 (2003) gravel transport, set t = 3.97 , nt = 1.5, c* = 0.0495 and s = 1. A setting s = 0.75 implies that 75%of the total resistance is skin friction and 25% is form drag.

tn

cbs

tb

RgDDRgD

q

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December 11-12, 2005SIMPLIFICATIONS contd.5. 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 bed material load at flood flow qb 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 6afbst

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December 11-12, 2005AN ISSUE OF NOTATION

A numerical method and program for computing the 1D morphodynamics of rivers using the normal flow approximation is introduced in the succeeding slides. The code was originally written for a generic river (gravel-bed or sand-bed), for which the total volume bed material load per unit width is denoted as qt. Here this code is applied to mountain rivers, so wherever qt appears, the user of this lecture material should make the transformation

bt qq

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December 11-12, 2005AGGRADATION 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.

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

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December 11-12, 2005NORMAL 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. It is particularly justifiable in the case of mountain rivers, as shown in the lecture on hydraulics Using the Exner formulation for sediment conservation and the Manning-Strickler formulation for flow resistance, the morphodynamic problem has the following character:

x

qI

t)1( t

fp

-

xS,

RD

S

g

qkDRgDq

tn

c

10/710/3

2r

2w

3/1s

stt

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).

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December 11-12, 2005THE 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

x)S(

xt d

tn

c

10/710/3

2r

2w

3/1s

stp

fd RD

S

g

qk

S)1(

DRgDI

The top equation is a (nonlinear) diffusion equation. In the bottom equation, it is seen that d is dependent on S = - /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.

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December 11-12, 2005INITIAL 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 Id 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. d = 0.

)x()t,x( I0t

)xL(S)t,x( IId0t

)t(q)t,x(q tf0xt

)t()t,x( dLx

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.

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December 11-12, 2005NOTES ON THE DOWNSTREAM BOUNDARY CONDITION

Alluvial Kaiya River, Papua New Guinea, and downstream bedrock exposure

Bedrock 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.

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December 11-12, 2005DISCRETIZATION FOR NUMERICAL SOLUTION

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.

M

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

L

x

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

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M..1i,tIx

q

1

1f

i,t

ptitti

x

qq)a1(

x

qqa

x

q i,t1i,tu

1i,ti,tu

i,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 M+1 = d.

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

where

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December 11-12, 2005INTRODUCTION 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). Here composite roughness height is meant to include the effect of bedforms. For mountain streams in the absence of form drag, it is appropriate to set kc equal to ks = nkD, where nk is in the range 3 – 4. When form drag is present it is appropriate to increase nk to somewhat larger values (~ 5 or 6).

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”.

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December 11-12, 2005INTRODUCTION 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 d 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”.

25



December 11-12, 2005INTRODUCTION 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.

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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 Main

This is the master subroutine that controls the Visual Basic program.

27




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.

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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.

29




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.

30



December 11-12, 2005A 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 calculation

Hult 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 difference

Duration 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.

31



December 11-12, 2005RESULTS OF SAMPLE COMPUTATION

Bed evolution

0

20

40

60

80

100

120

140

160

0 2000 4000 6000 8000 10000

Distance in m

Ele

vati

on

in m

0 yr2 yr4 yr6 yr8 yr10 yrUltimate

32



December 11-12, 2005INTERPRETATION

The long profile of a river is a plot of bed elevation versus down-channel distance x. The long profile of a river is called upward concave if slope S = -/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

x

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:

RD

S

g

qk 10/710/3

2r

2w

3/1c

33



December 11-12, 2005INTERPRETATION contd.The transient long profile of Slide 30 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 33), in which all the input parameters are the same as in Slide 29 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 34 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 34 are the same as those of Slide 29, 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 29), forcing degradation and transient downward concavity. In addition, Ntoprint is varied so that the duration of calculation varies from 1 year to 250 years.

Factors such as subsidence or base level rise can drive equilibrium long profiles which are upward concave.

34




Bed evolution

0

20

40

60

80

100

120

140

160

0 2000 4000 6000 8000 10000

Distance in m

Ele

vat

ion

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 mE

leva

tio

n in

m


Bed evolution

0

20

40

60

80

100

120

140

160

0 2000 4000 6000 8000 10000

Distance in m

Ele

vati

on

in m


Bed evolution

0

20

40

60

80

100

120

140

160

0 2000 4000 6000 8000 10000

Distance in m

Ele

vati

on

in m


AGGRADATION TO A NEW MOBILE-BED EQUILIBRIUM

35



December 11-12, 2005DEGRADATION 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

Ele

va

tio

n in

m


Bed evolution

0

10

20

30

40

50

60

70

80

90

0 2000 4000 6000 8000 10000

Distance in m

Ele

vati

on

in m


Bed evolution

0

10

20

30

40

50

60

70

80

90

0 2000 4000 6000 8000 10000

Distance in m

Ele

vati

on

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 m

Ele

vati

on

in m


36



December 11-12, 2005ADJUSTING 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.

37



December 11-12, 2005AN EXTENSION:RESPONSE OF AN ALLUVIAL RIVER TO VERTICAL FAULTING DUE TO AN

EARTHQUAKE

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 f at time tf. The eventual smearing out of the long profile is then computed.

38



December 11-12, 2005 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

Ele

vati

on

in m 0 yr

0.05 yr0.1 yr0.15 yr0.2 yr0.25 yr

39



December 11-12, 2005 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

Ele

vati

on

in m 0 yr

0.001 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

Ele

vat

ion

in m 0 yr

0.025 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

Ele

vat

ion

in m 0 yr

0.5 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

Ele

vat

ion

in m 0 yr

5 yr10 yr15 yr20 yr25 yr

40



December 11-12, 2005REFERENCES

Exner, F. M., 1920, Zur Physik der Dunen, Sitzber. Akad. Wiss Wien, Part IIa, Bd. 129 (in German).

Exner, F. M., 1925, Uber die Wechselwirkung zwischen Wasser und Geschiebe in Flussen, Sitzber. Akad. Wiss Wien, Part IIa, Bd. 134 (in German).

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.

Wong, M., 2003, Does the bedload equation of Meyer-Peter and Müller fit its own data?, Proceedings, 30th Congress, International Association of Hydraulic Research, Thessaloniki, J.F.K. Competition Volume: 73-80.

For more information see Gary Parker’s e-book:1D Morphodynamics of Rivers and Turbidity Currents http://cee.uiuc.edu/people/parkerg/morphodynamics_e-book.htm

41



December 11-12, 20051D CONSERVATION OF BED SEDIMENT FOR SIZE MIXTURES, BEDLOAD ONLY

1qq1xzdf)1(t xxbixbis0 ips

fi'(z', x, t) = fractions at elevation z' in ith grain size range above datum in bed [1]. Note that over all N grain size ranges:

qbi(x, t) = volume bedload transport rate of sediment in the ith grain size range [L2/T]

x

qzdf

t)1( bi

0 ip

Or thus:

1fN

1ii

x 1

z'

qbi qbi

42



December 11-12, 2005ACTIVE LAYER CONCEPT

The active, exchange or surface layer approximation (Hirano, 1972):

Sediment grains in active layer extending from - La < z’ < have a constant, finite probability per unit time of being entrained into bedload.Sediment grains below the active layer have zero probability of entrainment.

x

z'

qbi qbi La

43




aiaIiL i

L

0 i0 i LFt

)L(t

fzdft

zdft

zdft a

a

where the interfacial exchange fractions fIi defined as

REDUCTION OF SEDIMENT CONSERVATION RELATION USING THE ACTIVE LAYER CONCEPT

ai

aii Lz,)z,x(f

zL,)t,x(F)t,z,x(f

Thus

aLiIi ff

Fractions Fi in the active layer have no vertical structure.Fractions fi in the substrate do not vary in time.

describe how sediment is exchanged between the active, or surface layer and the substrate as the bed aggrades or degrades.

44




aiaIiL i

L

0 i0 i LFt

)L(t

fzdft

zdft

zdft a

a

REDUCTION OF SEDIMENT CONSERVATION RELATION USING THE ACTIVE LAYER CONCEPT contd.

Between

x

qzdf

t)1( bi

0 ip

and

it is found that

x

qLF

t)L(

tf)1( bi

aiaIip

(Parker, 1991).

45



December 11-12, 2005REDUCTION contd.The total bedload transport rate summed over all grain sizes qbT and the fraction pbi of bedload in the ith grain size range can be defined as

bT

bibi

N

1ibibT q

qp,qq

The conservation relation can thus also be written as

Summing over all grain sizes, the following equation describing the evolution of bed elevation is obtained:

x

pqLF

t)L(

tf)1( bibT

aiaIip

x

q

t)1( bT

p

Between the above two relations, the following equation describing the evolution of the grain size distribution of the active layer is obtained:

x

qf

x

pq

t

LfF

t

FL)1( bT

IibibTa

Iiii

ap

46



December 11-12, 2005EXCHANGE FRACTIONS

0t

,p)1(F

0t

,ff

bii

Lzi

Iia

where 0 1 (Hoey and Ferguson, 1994; Toro-Escobar et al., 1996). In the above relations Fi, pbi and fi denote fractions in the surface layer, bedload and substrate, respectively.

That is:The substrate is mined as the bed degrades.A mixture of surface and bedload material is transferred to the substrate as the

bed aggrades, making stratigraphy.Stratigraphy (vertical variation of the grain size distribution of the substrate)

needs to be stored in memory as bed aggrades in order to compute subsequent degradation.

47




Rivers often sort their sediment. An example is downstream fining: many rivers show a tendency for sediment to become finer in the downstream direction.

Long profiles showing downstream fining and gravel-sand

transition in the Kinu River, Japan (Yatsu,

1955)

elevation

bed slope

median bed material grain size

WHY THE CONCERN WITH SEDIMENT MIXTURES?

48



December 11-12, 2005WHY THE CONCERN WITH SEDIMENT MIXTURES ? contd.

upstream

downstream

Downstream fining can also be studied in the laboratory by forcing aggradation of heterogeneous sediment in a flume.

Downstream fining of a gravel-sand mixture at St. Anthony Falls

Laboratory, University of Minnesota (Toro-Escobar et al.,

2000)

Many other examples of sediment sorting also motivate the study of the transport, erosion and deposition of sediment mixtures.

49



December 11-12, 2005REFERENCES FOR CHAPTER 4

Hirano, M., 1971, On riverbed variation with armoring, Proceedings, Japan Society of Civil Engineering, 195: 55-65 (in Japanese).

Hoey, T. B., and R. I. Ferguson, 1994, Numerical simulation of downstream fining by selective transport in gravel bed rivers: Model development and illustration, Water Resources Research, 30, 2251-2260.

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., 1991, Selective sorting and abrasion of river gravel. I: Theory, Journal of Hydraulic Engineering, 117(2): 131-149.

Toro-Escobar, C. M., G. Parker and C. Paola, 1996, Transfer function for the deposition of poorly sorted gravel in response to streambed aggradation, Journal of Hydraulic Research, 34(1): 35-53.

Toro-Escobar, C. M., C. Paola, G. Parker, P. R. Wilcock, and J. B. Southard, 2000, Experiments on downstream fining of gravel. II: Wide and sandy runs, Journal of Hydraulic Engineering, 126(3): 198-208.

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

1d morphodynamics of mountain rivers: uniform sediment

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