chapter 16: morphodynamics of bedrock-alluvial transitions

1

1D SEDIMENT TRANSPORT MORPHODYNAMICSwith applications to

RIVERS AND TURBIDITY CURRENTS© Gary Parker November, 2004

CHAPTER 16:MORPHODYNAMICS OF BEDROCK-ALLUVIAL TRANSITIONS

An alluvial river has a bed that is completely covered with sediment that the river can move freely during flood flow.

A bedrock river has patches of bed that are not covered by alluvium, where bedrock is exposed. In some bedrock rivers the bed is almost completely bare of sediment. This is, however, not the usual case. In most cases of interest there is a mixture of patches covered by alluvium and patches where bedrock is exposed.

A bedrock river in Kentucky (tributary of Wilson Creek) with

a partial alluvial covering. Image courtesy A. Parola.

2



THE CONCEPT OF TRANSPORT CAPACITY

Big Box Creek, USA, a bedrock river with a stepped profile.

Image courtesy E. Wohl.

Equilibrium bedrock streams transport alluvium under below-capacity conditions, whereas alluvial streams transport sediment under at-capacity conditions. These concepts can be explained as follows.

If the sediment supply of an alluvial river is increased, the bed can be expected to aggrade toward a new, steeper slope capable of carrying the extra sediment.

A bedrock stream, on the other hand, may experience no aggradation when sediment supply is increased. Instead, the stream responds by reducing the fraction of the bed covered by bedrock and increasing the fraction covered by alluvium. Only when the bed is completely covered with alluvium can the river respond to increased sediment supply by aggrading.

3



QUANTIFICATION OF TRANSPORT CAPACITYThe concept of a mobile-bed equilibrium state was outlined in Chapter 14. In the case of the Chezy resistance relation and the sample sediment transport relation introduced in that chapter, the governing equations of this equilibrium state take the following forms:

3/12wf

gSqCH

tn

c

3/23/12wf

tt RDS

gqCDRgDq

Now let grain size D, sediment submerged specific gravity R, resistance coefficient Cf, critical Shields number c* and the parameters g, t and nt be given. The relations specify two equations in the following four parameters: depth H, bed slope S, water discharge per unit width qw and volume total bed material sediment discharge per unit width qt.

Consider a stream with given values of water discharge per unit width qw and bed slope S. The capacity transport qt is that computed from the above equation.

4



BELOW-CAPACITY CONDITIONSNow suppose that for given values of qw and S, the actual sediment supply qts is less that the value qt associated with mobile-bed equilibrium, i.e.

tn

c

3/23/12wf

tt RDS

gqCDRgDq

An alluvial stream would degrade to a lower slope S that would allow the above equation to be satisfied with qts. A bedrock stream, however, cannot degrade. So in the event that for given values of qw and S the sediment supply rate qts is less than the equilibrium mobile-bed value qt, the river responds by exposing bedrock on its bed instead of degrading. As qts is further reduced the river responds by increasing the fraction of the bed over which bedrock is exposed (Sklar and Dietrich, 1998). The river so adjusts itself to transport sediment at the rate qts which is below its capacity qt for the given values of qw and S. This allows a below-capacity equilibrium.

In the event that the actual sediment supply qts is greater than the capacity transport rate qt at the given slope S , the river will aggrade to a new, higher slope in consonance with qts that satisfies the above equation. There is noabove-capacity equilibrium.

5



A SAMPLE CALCULATIONThe following values are assumed in the sediment transport relation below: t = 3.97, nt = 1.5, c* = 0.0495 (Wong and Parker, submitted, modification of Meyer-Peter and Müller), R = 1.65, g = 9.81 m2/s and Cf = 0.01.

tn

c

3/23/12wf

tt RDS

gqCDRgDq

Consider a river with D = 20 mm, flood Qw = 90 m3/s and width B = 30 m. The flood value of qw = Qw/B = 3 m2/s. For any slope S, then, the capacity value of qt can be computed from the above relation.

Assume that a bedrock river is just barely completely covered with alluvium at the slope S. How will the river respond if sediment supply qts is reduced or increased?

The following two slides illustrate that the river will aggrade to a new mobile-bed equilibrium when qts > qt. When qts < qt, the river cannot degrade due to the presence of bedrock, and instead reaches a below-capacity equilibrium with exposed bedrock.

6



bedrock

bedrock

qts = qt(qw,S)thin covering of

alluvium

bedrock

qts < qt(qw,S)bedrock exposed

bedrock

qts = qt(qw,S)thin covering of

alluvium

supply decreased

supply increased

bed has aggraded to new mobile-bed

equilibrium

A SAMPLE CALCULATION contd.

7



0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

S

q ts m

2 /s

capacity transport curve with bedrock barely covered by

alluvium:qts = qt below-capacity sediment

supply: stream will expose bedrock

above-capacity sediment supply: stream will aggrade to new alluvial equilibrium

A SAMPLE CALCULATION contd.

8



ILLUSTRATION OF BELOW-CAPACITY TRANSPORT OF 7 MM GRAVEL OVER A BEDROCK BED

The video clip is from the Ph.D. research of Phairot Chatanantavet.

rte-bookbelowcaptrans.mpg: to run without relinking, download to same folder as PowerPoint presentations.

9



BEDROCK-ALLUVIAL TRANSITIONS: THE “FALL LINE”

The southeastern coastal plain of the United States is characterized by a feature called the “Fall Line.” Upstream (westward) of this line the streams are in bedrock. Downstream (eastward) of this line they are in alluvium. It is of interest to speculate how the position of the fall line might respond to changing sea level.

Image of the southeastern coastal plain of the United States from NASA

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

10



EQUILIBRIUM STATE WITH BEDROCK-ALLUVIAL TRANSITION

bedrock

alluvium

bedrock-alluvial transition

base level (bed elevation) maintained hereqts

btsw S)D,q,q(SS

A bedrock channel has constant slope Sb and carries flood discharge per unit width qw. Sediment with size D is fed in at the upstream end at rate qts. The at-capacity slope S consonant with qst, qw and D (as computed, for example, from the transport relation of Slide 5) is less than Sb. Base level is maintained at some elevation at the downstream end; this level is higher than the elevation of the bedrock basement there. An equilibrium bedrock-alluvial transition must occur. To find it, draw a straight line with slope S and intercept at the point of base level maintenance, and extend it upstream until it intersects the bedrock profile.

bS

11



DYNAMICS OF THE MIGRATION OF BEDROCK-ALLUVIAL TRANSITIONS

t0

t1

bedrocksurface

alluvium

Bedrock-alluvial transitions can migrate upstream or downstream due to the effects of e.g. changing sediment supply from upstream or changing base level downstream. The figure below shows a case where the alluvial region is (for whatever reason) aggrading, resulting in an upstream migration of the bedrock-alluvial transition.

12



CONTINUITY CONDITION AT THE BEDROCK-ALLUVIAL TRANSITION

t0

t1

bedrocksurface

alluvium

The elevation profile of the bedrock basement is denoted as base(x); it is assumed to be unchanging in time. The elevation profile of the alluvial zone is denoted as (x, t); it can change in time due to aggradation or degradation. The position of the bedrock-alluvial transition is denoted as x = sba(t). It is a function of time because the position of the transition can change in time. In order for the bedrock channel to join continuously with the alluvial channel, the following condition must hold:

baba sxbasesx)x()t,x(

or )]t(s[]t),t(s[ babaseba

Now take the derivative with respect to time of both sides of the equation. For example,

bass

ba

ssba sS

tdtds

xt]t),t(s[

dtd

babababa

where S = -/x denotes the alluvial bed slope and = dsba/dt denotes the speed of migration of the bedrock-alluvial transition.

bas

13



CONTINUITY CONDITION AT THE BEDROCK-ALLUVIAL TRANSITION contd.

t0

t1

bedrocksurface

alluvium

Taking the derivative of both sides of the relation

results in:

where Sb = -base/x = the slope of the bedrock channel. Reducing, the following cute little relation is obtained:

)]t(s[]t),t(s[ babaseba

basbbass

sSsSt baba

ba

baba

ba

ssb

sba

SS

ts

(Parker and Muto, 2003). Now since x = sba denotes a bedrock-alluvial transition, it can always be expected that the bedrock slope Sb exceeds the alluvial slope S there. So the continuity condition says simply:If the bed aggrades, the transition moves upstream; and if the bed degrades the transition moves downstream.

14



MOVING-BOUNDARY FORMULATION FOR RIVER MORPHODYNAMICS WITH A BEDROCK-ALLUVIAL TRANSITION

The downstream end of the reach is located at the constant value x = sd, where base level is maintained. The bedrock-alluvial transition is located at x = sba(t) < sd. The goal is to describe the morphodynamics of the evolution of the stream so as to obtain both the change in the alluvial profile (x,t) as a function of time and the trajectory sba(t) of the transition as a function of time.

To this end we introduce the coordinate transformation

Note that the bedrock-alluvial transition is located at , and the downstream end of the reach is located at .

Using the chain rule,

tt,)t(ss)t(sxx

bad

ba

0x 1x

xxx

txt

x,

xtx

ttt

t

15



TRANSFORMATION OF THE EXNER EQUATION TO MOVING-BOUNDARY COORDINATES

tt,)t(ss)t(sxx

bad

ba

xx

xxt

tx,

tx

xtt

tt

Evaluating the derivatives,

badbad

ba

ss1

xx,)x1(

sss

tx,0

xt,1

tt

Transforming the Exner equation of sediment continuity

to the moving-boundary coordinate system results in the formxqI

t)1( t

fp

-

xq

ssI

xsss)x1(

t)1( t

bad

f

bad

bap

16



TRANSFORMATION OF THE CONTINUITY CONDITION TO MOVING-BOUNDARY COORDINATES

)SS(xss

st

SS

ts

0x0xb

0xbad

ba

0x

ssb

sba

baba

ba

However slope is given as

xss1

xS

bad

Between these two relations,

0x0xbba tS

1s

Now from Slide 12 and the moving-boundary coordinate transformation.

17



CHARACTER OF THE MORPHODYNAMIC PROBLEM

xq

ssI

xsss)x1(

tt

bad

f

bad

ba

0x0xbba tS

1s

There is one more variable to solve than before, i.e. the speed of the moving boundary, but there is one more equation as well. Further reducing the continuity condition with Exner,

bas

0x

tf

0xba

bad0xbba x

qIx

s)ss(S

1s

or thus

)ss(SxqI

)SS

1(

1sbad0xb

0x

tf

0xb

0xba

18



DISCRETIZATON FOR NUMERICAL SOLUTION

txq

ssI

xsss)x1(

ii x

t

bad

f

xbad

baiitti

The domain from to (x = sba to x = sd) is discretized into M intervals bounded by M+1 nodes. The node i = 1 denotes the bedrock-alluvial transition and the node i = M+1 denotes the point where base level is maintained.

0x 1x

i=1 2 3

1

x

M -1 i = M+1 M+1

M

x

The sediment feed rate qtf during floods is specified at the node i = 1; no ghost node is needed in this formulation. The Exner equation discretizes to

19



DISCRETIZATION FOR NUMERICAL SOLUTIONAs noted in the previous slide, the Exner equation discretizes to

The derivatives discretize to the forms

Derivatives need not be evaluated at i = M+1 because bed elevation M+1 is held constant. The shock condition discretizes to

1ibad1i

bad1ib

1i

tf

1ib

1ibababattba xss

1S,)ss(S

xqI

)SS

1(

1s,tsss

txq

ssI

xsss)x1(

ii x

t

bad

f

xbad

baiitti

M..1i,x

qqxq i,t1i,t

i

t

M..2i,x2

1i,x

x 1i1i

12

i

20



INTRODUCTION TO RTe-bookBedrockAlluvialTrans.xls The treatment of bedrock-alluvial transitions is implemented in RTe-bookBedrockAlluvialTrans.xls.

The formulation uses the Engelund-Hansen (1967) total bed material transport relation for sand-bed streams. (Could you modify the code for gravel-bed streams?) The downstream end of the reach is located at the fixed point sd. The bedrock channel is assumed to have constant slope Sb. (Could you modify the code to let Sb vary in space?)

Initially the alluvial zone has bed slope Sinit and length sd, so the bedrock-alluvial transition is located at sba = 0. The water discharge per unit width qw and volume sediment bed material feed rate per unit width qtf are specified along with the grain size D of the bed material and the constant Chezy resistance coefficient Cz.

The flow is computed using the normal (steady, uniform) flow approximation. (Could you change it to use a backwater formulation?). The values of submerged specific gravity R and bed porosity p are set equal to 1.65 and 0.4, respectively, in “Const” statements in the code. They can be changed by modifying the relevant “Const” statements.

21



A SAMPLE CALCULATION: UPSTREAM MIGRATION OF THE BEDROCK-ALLUVIAL TRANSITION TO A NEW, STABLE POSITION

The initial bed slope of the alluvial region Sinit = 0.00009 is too low for the specified water discharge, sediment feed rate (qtf = 0.0006 m2/s) and grain size. So the bed must aggrade to a final equilibrium slope Sfinal, which is less that the bedrock slope Sb. The result is that the bedrock-alluvial transition must move upstream to a new equilibrium from its initial point at sba = 0. The results of the computation follow.

Input cellInput Information cell: do not input numbers

qw 6 m2/s Water discharge/unit width during floodsIf 0.1 Intermittency factor for floods: 0 < I 1qtf 0.0006 m2/s Volume sediment feed rate/width at upstream end during floodsD 0.5 mm Grain size of alluviumCz 15 Chezy resistance coefficientSb 0.00035 Slope of bedrock basementSinit 0.00009 Initial slope of alluvial regionsd 50000 m Position of the downstream end of the reach, = initial length of alluvial regionM 20 Number of spatial intervalsdt 0.005 years Time stepMtoprint 10000 Number of steps to printoutMprint 20 Number of printouts

5.02E+03 Bed material feed rate in metric tons/meter/year1000 Time of calculation in years

22



Evolution of Alluvial River Profile with Bedrock-Alluvial Transition

0

2

4

6

8

10

12

14

16

18

20

-60000 -40000 -20000 0 20000 40000 60000

x m

eta

m

0 year50 year100 year150 year200 year250 year300 year350 year400 year450 year500 year550 year600 year650 year700 year750 year800 year850 year900 year950 year1000 year

23



Trajectory of the bedrock-alluvial transition

-45000

-40000

-35000

-30000

-25000

-20000

-15000

-10000

-5000

0

0 200 400 600 800 1000 1200

time, years

Posi

tion

s ba

of t

he b

edro

ck-a

lluvi

al

tran

sitio

n, m

0

2

4

6

8

10

12

14

16

18

20

Elev

atio

n up

of b

edro

ck-

allu

vial

tran

sitio

n, m

sbaetaup

24



A SAMPLE CALCULATION: DOWNSTREAM MIGRATION OF THE BEDROCK-ALLUVIAL TRANSITION TO A NEW, STABLE POSITION

The initial bed slope of the alluvial region Sinit = 0.0002 is too high for the specified water discharge, sediment feed rate (qtf = 0.0001 m2/s) and grain size. So the bed must degrade to a final equilibrium slope Sfinal, which is less that the bedrock slope Sb. The result is that the bedrock-alluvial transition must move downstream to a new equilibrium from its initial point at sba = 0. The results of the computation follow.

Input cellInput Information cell: do not input numbers

qw 6 m2/s Water discharge/unit width during floodsIf 0.1 Intermittency factor for floods: 0 < I 1qtf 0.0001 m2/s Volume sediment feed rate/width at upstream end during floodsD 0.5 mm Grain size of alluviumCz 15 Chezy resistance coefficientSb 0.00035 Slope of bedrock basementSinit 0.0002 Initial slope of alluvial regionsd 50000 m Position of the downstream end of the reach, = initial length of alluvial regionM 20 Number of spatial intervalsdt 0.005 years Time stepMtoprint 2500 Number of steps to printoutMprint 20 Number of printouts

8.36E+02 Bed material feed rate in metric tons/meter/year250 Time of calculation in years

25



Evolution of Alluvial River Profile with Bedrock-Alluvial Transition

0

2

4

6

8

10

12

0 10000 20000 30000 40000 50000 60000

x m

eta

m

0 year12.5 year25 year37.5 year50 year62.5 year75 year87.5 year100 year112.5 year125 year137.5 year150 year162.5 year175 year187.5 year200 year212.5 year225 year237.5 year250 year

26



Trajectory of the bedrock-alluvial transition

0

5000

10000

15000

20000

25000

0 50 100 150 200 250 300

time, years

Posi

tion

s ba

of t

he b

edro

ck-a

lluvi

al

tran

sitio

n, m

0

2

4

6

8

10

12

Elev

atio

n up

of b

edro

ck-

allu

vial

tran

sitio

n, m

sbaetaup

27



SOME COMMENTS FOR FUTURE WORK

The simple analysis presented in this chapter invites a wide range of extensions. A few are suggested below.

• How does sea level change affect the position of the fall line on coastal plains of the southeastern United States?

• When a dam is placed on a bedrock stream, how far upstream does the alluvial-bedrock transition created by deltaic deposit behind the dam migrate upstream?

• What happens to the position of a bedrock-alluvial transition if climate changes, resulting in changes in flood sediment supply qtf, flood water discharge qw and flood intermittency If?

• How could the analysis be generalized to gravel-bed streams and sediment mixtures?

28



NOTE IN CLOSING: BEDROCK INCISIONIn the analysis described here it is assumed that the bedrock platform is fixed in time, and is not free to undergo incision. This assumption is correct on the time scale of many alluvial problems. It is erroneous, however, to assume that bedrock channels cannot incise their beds over sufficiently long time scales (e.g. Whipple et al., 2000). The ability of a river to incise through bedrock is amply illustrated by the image below. Bedrock incision is considered in more detail in Chapters 29 and 30.

Image of a Bolivian river from NASAhttps://zulu.ssc.nasa.gov/mrsid/mrsid.pl

29



REFERENCES FOR CHAPTER 16

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

Parker, G. and Muto, T., 2003, 1D numerical model of delta response to rising sea level, Proc. 3rd IAHR Symposium, River, Coastal and Estuarine Morphodynamics, Barcelona, Spain, 1-5 September.

Sklar, L., and W. E. Dietrich, 1998, River longitudinal profiles and bedrock incision models: Stream power and the influence of sediment supply, in Rivers Over Rock: Fluvial Processes in Bedrock Channels, Geophys. Monogr. Ser., vol. 107, edited by K. J. Tinkler and E. E. Wohl, pp. 237–260, AGU, Washington, D. C.

Whipple, K. X., G. S. Hancock, and R. S. Anderson, 2000, River incision into bedrock: Mechanics and relative efficacy of plucking, abrasion, and cavitation, Geol. Soc. Am. Bull., 112, 490–503.

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 16: morphodynamics of bedrock-alluvial transitions

Documents