v.r. voller+, j. b. swenson*, w. kim+ and c. paola+

National Center for Earth-surface Dynamicsan NSF Science and Technology Center

www.nced.umn.edu

V.R. Voller+, J. B. Swenson*, W. Kim+ and C. Paola+

+ National Center for Earth-surface Dynamics University of Minnesota, Minneapolis*Dept. Geological Sciences and Large Lake Observatory, University of Minnesota-Duluth


www.nced.umn.edu

Ganges-Brahmaputra Delta “growth” of sediment delta into oceanGrain Growth in Metal Solidification

From W.J. Boettinger

m

10km

Commonality between solidification and ocean basin formation

Geometry and Heat transfer Models of Shoreline movements

1As always “-- the material presented should be approached with an open mind, studied carefully, and critically considered.”

Cobb County Geogia


www.nced.umn.edu

Fans Toes Shoreline

Two Problems of Interest


www.nced.umn.edu

1km

Examples of Sediment FansMoving Boundary

How does sediment-basement interfaceevolve

Badwater Deathvalley


www.nced.umn.edu

sediment

h(x,t)

x = u(t)

0q

bed-rock

ocean

x

shoreline

x = s(t)

land surface


www.nced.umn.edu

An Ocean Basin

Melting vs. Shoreline movement


www.nced.umn.edu

pressurizedwater reservoir

to water supply

solenoidvalve

stainless steelcone

to gravel recycling

transport surface

gravel basement

rubber membrane

experimental deposit

Experimental validation of shoreline boundary condition

~3m


www.nced.umn.edu

Experimental validation of shoreline boundary condition

dt

dS)su(

21

dx)x(dt

dZ)su()t,s(q

dtds

S)su( f2u

s

blsff

eXperimental EarthScape facility (XES)

Flux balance at shoreline

Flux base subsidence slope

Calculated frontvelocity from

exp. measurment of RHS

measured


www.nced.umn.edu

Base level

Measured and Numerical results ( calculated from 1st principles)

1-D finite difference deforming grid vs. experiment

xxt+Shoreline balance


www.nced.umn.edu

Limit Conditions: A Fixed Slope Ocean

q=1

s(t)

similarity solution

q

2

)(erfe

)(erf1,t2s

22/1

21

21

2

21

21

0

5

10

15

20

25

0 100 200 300

Time

sh

ore

line

0if),x(LH

2

2

xt

H

Enthalpy Sol.

A Melting Problem driven by a fixed flux with SPACE DEPENDENT

Latent Heat L = s

dt

dss

x)t(sx0,

xt s2

2

s Depth at toe


www.nced.umn.edu

h(x,y,t)

q

bed-rock

ocean

y

shoreline

x = s(t)

land surface

(x,y,t)

A 2-D Front -Limit of Cliff face Shorefront But Account of Subsidence and relative ocean level

0hif),t,y,x(LhH

)h(t

H

Enthalpy Sol.

xy]/H,1[MINfrac

Solve on fixed gridin plan view

Track Boundary by calculating in each cell


www.nced.umn.edu

0

5

10

15

20

25

30

0 100 200 300 400 500

time

shor

elie

n po

sitio

n

0

5

10

15

20

25

30

0 500 1000 1500 2000

time

shor

elin

e po

sitio

nnumerical

steady state

s(t)

s(t)

Hinged subsidenceq

2

)(erfe

)(erf1,t2s

22/1

21

21

2

21

21


www.nced.umn.edu

A 2-D problem Sediment input into an oceanwith an evolving trench driven By hinged subsidence

First look at case whereOcean is at constant depthNO TRENCH

Then Look at case with Trench


www.nced.umn.edu


www.nced.umn.edu

With Trench


www.nced.umn.edu


www.nced.umn.edu

No Trench Trench

Plan view movement of fronts


www.nced.umn.edu

s(t) s(t)

R

0

5

10

15

20

25

30

35

40

45

50

0 20 40 60 80 100

shoreline

sea-level

geometric – modelof shoreline movementwith changing sea level

q=1

Assumption of rapid fluvial transport allow for a geometric balance

22

221

2

22

RBQ2RR

u

0RQ

Ruu2

2

)uR(

2

uRuQ

NOTE: REVERSE of shoreline!

u(t)

t

0

t

0

dtqQ

dtrR


www.nced.umn.edu

0

0.5

1

1.5

2

0 5 10 15 20 25 30

0

0.5

1

1.5

2

0 5 10 15 20 25 30

0

0.5

1

1.5

2

0 5 10 15 20 25 30

0

0.5

1

1.5

2

0 5 10 15 20 25 30

sediment

Movement of sediment plug behind a dam Dam reservoir profileWith sediment plug downstream of dam

At time t = 0 water levelin reservoir dropped at a Constant rate

assume cliff faceno flow in or out

Describe movement ofSediment by

2

2

xt

H

)t(LH

Water depth


www.nced.umn.edu

Experiments by Chris Bromley, University of Nottingham

Ekwha dam Oregon


www.nced.umn.edu


www.nced.umn.edu

0

0.5

1

1.5

2

0 5 10 15 20 25 30

0

0.5

1

1.5

2

0 5 10 15 20 25 30

0

0.5

1

1.5

2

0 5 10 15 20 25 30

0

0.5

1

1.5

2

0 5 10 15 20 25 30

sediment

0

2

4

6

8

10

12

14

0 200 400 600 800 1000 1200 1400 1600

0.001

0.00250.005

Movement of toeGoes as t2

Movement of sediment plug behind a dam drawdown rate


www.nced.umn.edu

WHY Build a model

Stratigraphy and Shoreline

-30-20-10

0102030405060

0 10 20 30

Models can predict stratigraphy“sand pockets” = OIL

The Po

Shoreline position is signature of channels


www.nced.umn.edu

Shoreline Tracking Model has beenValidated (Experiments)

And a numerical method based on HeatTransfer concepts has been Verified.

0hif),t,y,x(LhH

)h(t

H

Enthalpy Sol.

Will allow for a first cut simulationof how sea-level and subsidence Could effect the motion of shorelines

Can be used to model short time systemsRelated to dam removal

0

0.5

1

1.5

2

0 5 10 15 20 25 30

Space and time dependent latent heat

Other Systems of interest


www.nced.umn.edu

e.g. the Dessert Sediment Fan

1km

How does sediment-basement interfaceevolve

Badwater Deathvalley


www.nced.umn.edu

An experiment

• Water tight basin -First layer: gravel to allow easy drainage-Second layer: F110 sand with a slope ~4º.

• Water and sand poured in corner plate

• Sand type: Sil-Co-Sil at ~45 mm• Water feed rate:

~460 cm3/min• Sediment feed rate: ~37cm3/min


www.nced.umn.edu

The Desert Fan Problem

xxt )t,s(,0x s

A Stefan problem with zero Latent Heat


www.nced.umn.edu

The Numerical Method-Explicit, Fixed Grid, Up wind Finite Difference VOF like scheme

Flux out of toe elements =0Until Sediment height >Downstream basement

fill point

P

)qq(t

out2PnewP in

E

The Toe Treatment

EPq

Square grid placed onbasement

At end of each time stepRedistribution scheme is requiredTo ensure that no “downstream” covered areas are higher

r

Determine height at fill : Position of toe

.05 grid size


www.nced.umn.edu

y – (x,t) = 0

0y)t,x(,0xW

),y,x(Qxxxxt

),x(,0 s n

On toe0

0.10.20.30.40.50.60.7

00.511.5

x-location (m)

y-location (m

)

r

k

0

0.05

0.1

0.15

0 100 200 300

time (min)

feed

hig

ht

(m)

height at input

fan with time

v.r. voller+, j. b. swenson*, w. kim+ and c. paola+

Documents