lecture 12 morphodynamics of 1d submarine/sublacustrine fans

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LECTURE 12 MORPHODYNAMICS OF 1D SUBMARINE/SUBLACUSTRINE

FANS

CEE 598, GEOL 593TURBIDITY CURRENTS: MORPHODYNAMICS AND DEPOSITS

As the Colorado River flows into Lake Mead, USA, it forms a delta.

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STRUCTURE OF A DELTAIC DEPOSIT

Rivers flowing into lakes and reservoirs deposit their sediment in deltas. In a delta, the coarser material (e.g. sand and coarser)material is fluvially emplaced in a topset deposit and emplaced by avalanching in a foreset deposit. Finer material (e.g. mud) is preferentially emplaced in deep water beyond the toe of the foreset. A major mechanism for this deep-water emplacement consists of turbidity currents associated with plunging.

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PLUNGING IN LAKE MEAD

Image from USBR

Logjam near plunge point

plunge line

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DEPOSITION WITHIN LAKE MEAD, COLORADO RIVER, USA

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SOME DETAIL OF THE DEPOSIT

Sand deposits in the topset and foreset.

Mud deposits in the bottomset.

Two moving boundaries: topset-foreset break and foreset-bottomset break.

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MOST DELTAS SPREAD OUT LATERALLY TO MAKE FAN-DELTAS

Delta of the Selenga River as it flows into Lake Baikal, Russia

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WHEN LAKE MEAD BEGAN TO FILL, HOWEVER, THE DELTA WAS CONFINED TO A NARROW CANYON FOR

MANY YEARS

So the somewhat abstract case of a 1D delta prograding into a zone of constant width is not entirely without field analogs.

Besides, if we can understand 1D delta, the treatment can then be generalized to 2D deltas.

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HERE WE SIMPLIFY THE PROBLEM TO A 1D SUBAQUEOUS FAN

feed point

initial bed

later bed

turbidity current

x

The upstream feed point of the 1D subaqueous fan is set at x = 0. The vertical position of the feed point is, however, allowed to adjust with the evolution of the bed.

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FURTHER SIMPLIFICATIONS

feed point

initial bed

later bed

turbidity current

x

1. The flow is assumed to be continuous.2. Only a single grain size is considered.3. The flow is assumed to be supercritical everywhere, so that the

“backwater” equations for the flow can be integrated downstream from the feed point.

4. Erosion is neglected in a first formulation, so that the turbidity current is treated as purely depositional.

All of these assumptions can be relaxed in a more elaborate model.

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THE GOVERNING EQUATIONS

Here we use a 3-equation formulation for the flow. Since the flow is assumed to be dilute and continuous, the relations:

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f

w

s s o

UH U H 1 CHRg RgCHS C U

t x 2 xH UH

e Ut xCH UCH

v (E r C)t x

sw f o

sw f o

s s so s 3

s

v1 1S e (2 ) C rdH 2 2 U

dx 1v1 1

S e (1 ) C rH dU 2 2 UU dx 1

dq v RgqHr , q UCH ,

q dx U U

Ri Ri Ri

Ri

Ri Ri Ri

Ri

Ri

can be simplified to:

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PARAMETERS AND BOUNDARY CONDITIONS

In the model presented here, ro, Cf, vs, R (and of course g, because we just might not be working on Earth) must be specified. We are assuming supercritical flow, i.e. Ri < 1, so that the governing equations

are integrated in the downstream direction. Upstream boundary conditions for C, U and H must be specified at x = 0. Here we specifiy the upstream water discharge per unit width qwo = UoHo, the suspended sediment discharge per unit width qso = UoHoCo and the upstream Richardson number Rio = (RgCoHo/Uo

2) = (Rgqso/Uo3). Thus

sw f o

sw f o

s s so s 3

s

v1 1S e (2 ) C rdH 2 2 U

dx 1v1 1

S e (1 ) C rH dU 2 2 UU dx 1

dq v RgqHr , q UCH ,

q dx U U

Ri Ri Ri

Ri

Ri Ri Ri

Ri

Ri

1/ 3

so so woo o o

wo o o

q Rgq qC , U , H

q U

Ri

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BED EVOLUTION

The Exner equation of bed sediment continuity takes the form

That is, here we neglect both sediment entrainment and bedload transport:

An initial condition must be specified for the bed profile:

To simplify things here, we assume that the initial bed has a constant slope SbI and an initial bed elevation of 0 at x = 0.

bp s o s

q(1 ) v (r C E )

t x

-

p s o(1 ) v r Ct

ox 0(x,t) (x)

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FLOW OF THE CALCULATION

sw f o

sw f o

s so

s

v1 1S e (2 ) C rdH 2 2 U

dx 1v1 1

S e (1 ) C rH dU 2 2 UU dx 1

dq vHr

q dx U

Ri Ri Ri

Ri

Ri Ri Ri

Ri

At any given time t, solve the equations below over the existing bed from x = 0 to x = L, where L is some domain length.

ox 0

ox 0

s sox 0

H H

U U

q q

Use the results of this solution to find the bed some time t + t later:

p s o

s ot t t tp

(1 ) v r Ct

1v r C t

(1 )

s3

Rgq

URi

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SPATIAL DISCRETIZATION

i=1 2 3

L

x

M -1 i = M+1 M

feed point

initial bed

later bed

turbidity current

x

The problem is solved over a domain extending from x = 0 to x = L, where L is a specified parameter. This domain is discretized to M intervals of length x bounded by M + 1 points:

i

Lx (i 1) x , i 1..M 1 , x

M

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SOLUTION FOR THE FLOW

The simplest method you can use to solve for the flow is the Euler Step Method. (Yes, you can use a more accurate method if you know it). We rewrite the equations of the flow as

H s

U s

sq s

dHf (q ,H,U,S)

dxdU

f (q ,H,U,S)dxdq

f (q ,H,U,S)dx

s s s sw f o3 3 3

H ss

3

Rgq Rgq v Rgq1 1S e 2 C r

U 2 U 2 U Uf (q ,H,U,S)

Rgq1

U

where for example

The equations discretize to

where for example Hi denotes the value of H at xi. For any given bed profile (which specifies S), the equations can be solved stepwise downstream from i = 1 to i = M + 1.

i 1 i H s,i i i i 1 o

i 1 i U s,i i i i 1 o

s,i 1 s,i q s,i i i i s,1 so

H H f (q ,H ,U ,S ) x , H H

U U f (q ,H ,U ,S ) x , U U

q q f (q ,H ,U ,S ) x , q q

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Solve for the new bed elevations from Exner:

and obtain the new bed slopes Si as

And with these new bed slopes, it is possible to solve for the flow over the new bed!

SOLUTION FOR THE BED EVOLUTION

i i s o it t t tp

1v r C t

(1 )

1Mi,x

M..2i,x2

1i,x

S

1MM

1i1i

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i

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WELCOME TO THE EXCEL WORKBOOK WITH IMBEDDED CODE IN VISUAL BASIC FOR APPLICAITIONS:

Rte-book1DSubaqueousFan.xls

1D Model for Subaqueous Fans Created by Purely Depositional Turbidity Currents

Input parametersqflowo 6 input flow discharge/width, m^2/sInter 1 intermittencyCzs 30 dimensionless Chezy resistance coefficient, subaqueousro 2 ratio of near-bed to layer-averaged concentrationD 0.02 grain size of mud (or fine sand) in mmR 1.65 submerged specific gravity of mud (or fine sand) Input celllp 0.6 bed porosity, mud (or fine sand) Output cell showing useful informationqso 1.00E-02 mud (or fine sand) input rate, m^2/setabI 0 initial elevation of bottom of the foreset, mSbI 0.05 initial subaqueous bed slopeL 5000 length of domain, mnu 1.00E-06 kinematic viscosity of water, m^2/sRio 0.3 Upstream value of Richardson number (must be less than 1)dt 0.182625 time step, daysN 80 no. of nodes each zone (excluding ghost node)Ntoprint 200 no. of steps until a printout of results is made Uo 0.814099 m/s upstream flow velocityNprint 6 no. of printouts after the initial one Ho 7.3701108 m upstream layer thickness

0.6 Calculation time in years Co 0.0016667 upstream suspended sediment conc.

Click to Run Code

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If the flow were morphodynamically active for only fraction of time (i.e. a constant flow that is maitained for only a few days per year), the Exner equation must be modified to

The code allows for this possibility through the parameter = Inter.

The coefficient of bed friction Cf is related to Czs via the relation

The fall velocity vs is computed from the Dietrich (1982) relation introduced earlier. The water entrainment relation used is that of Parker et al. (1987):

NOTES ON THE PARAMETERS

bp s o s

q(1 ) v (r C E )

t x

-

2f zsC C

w 2.4

0.075e

1 718

Ri

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THE CODE

The code is found in the Visual Basic Editor. From the Excel menu, selectTools,Macro,Visual Basic Editor

If the code is (macros are) not enabled when you open the Excel file, you will have to go to the Excel Menu, selectTools,Macro,Securityand set Security no higher than “medium”.

You then have to close and open Excel in order to have the code enabled when you open the file.

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SAMPLE CALCULATION: BASE CASE

Input parametersqflowo 6 input flow discharge/width, m^2/sInter 1 intermittencyCzs 30 dimensionless Chezy resistance coefficient, subaqueousro 2 ratio of near-bed to layer-averaged concentrationD 0.02 grain size of mud (or fine sand) in mmR 1.65 submerged specific gravity of mud (or fine sand)lp 0.6 bed porosity, mud (or fine sand)qso 1.00E-02 mud (or fine sand) input rate, m^2/setabI 0 initial elevation of bottom of the foreset, mSbI 0.05 initial subaqueous bed slopeL 5000 length of domain, mnu 1.00E-06 kinematic viscosity of water, m^2/sRio 0.3 Upstream value of Richardson number (must be less than 1)dt 0.182625 time step, daysN 80 no. of nodes each zone (excluding ghost node)Ntoprint 200 no. of steps until a printout of results is madeNprint 6 no. of printouts after the initial one

0.6 Calculation time in years

Click to Run Code

These are the base input parameters. Note thatD = 0.02 mm (mud)SbI = initial bed slope = 0.05 Simulation time = 0.6 years of continuous flow

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D = 0.02 mm, SbI = initial bed slope = 0.05

Turbidity Current Fan Morphology: 1D

-300

-200

-100

0

100

200

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0 1000 2000 3000 4000 5000

Distance m

Bed

Ele

vati

on

m 0 yr0.1 yr0.2 yr0.3 yr0.4 yr0.5 yr0.6 yr

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Distance m

Bed

Ele

vati

on


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0 1000 2000 3000 4000 5000

Distance m

Bed

Ele

vati

on


Bed gets too steep because the flow is purely depositional!

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D = 0.02 mm, SbI = initial bed slope = 0.05 (again)


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-100

0

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0 1000 2000 3000 4000 5000

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Bed

Ele

vati

on


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-300

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Bed

Ele

vati

on


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-300

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0

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0 1000 2000 3000 4000 5000

Distance m

Bed

Ele

vati

on


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MATERIAL TO BE ADDED: EROSIONAL CASE

NEED TO ADD. RTe-book1DSubaqueousFanWErosTRY.xls

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REFERENCES

Ashida, K. and M. Michiue, 1972, Study on hydraulic resistance and bedload transport rate in alluvial streams, Transactions, Japan Society of Civil Engineering, 206: 59-69 (in Japanese).

UNDER CONSTRUCTION

lecture 12 morphodynamics of 1d submarine/sublacustrine fans

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