Moving Boundaries in Earthscapes Damien T. Kawakami, V.R. Voller, C. Paola, G. Parker, J. B. Swenson NSF-STC .

Moving Boundaries in Earthscapes

Damien T. Kawakami, V.R. Voller, C. Paola, G. Parker, J. B. Swenson

NSF-STC

www.nced.umn.edu

1km

Examples of Sediment Fans

How does sediment-basement interfaceevolve

Badwater Deathvalley

An Ocean Basin

How does shoreline respond to changes in sea level and sediment flux

Sediment mass balance gives

Sediment transported and deposited over fan surfaceby fluvial processes

xxt

From a momentum balance anddrag law it can be shown thatthe diffusion coefficient is a function of a drag coefficientand the bed shear stress

when flow is channelized = cont.

when flow is “sheet flow”

diffusion will be non-linear Conic shaped Fan A first order approx. analysis indicates 1/r r radial distance from source

Convex shape

An Ocean Basin

How does shoreline respond to changes in sea level and sediment flux

A large Scale Experiment by Paola and Parker at SAFL has addressed this problem

“Jurassic Tank”

~1m

Computer controlled subsidence

xxt

How does shore line move in response to sea-level changes

Swenson et al can be posed as a generalized Stefan Problem

Base level

Measured and Numerical results ( calculated from 1st principles)

Numerical Solution1-D finite difference deforming grid

The Desert Fan Problem -- A 2D Problem

xxt )t,s(,0x s

A Stefan problem with zero Latent Heat

A two-dimensional version (experiment)

• Water tight basin filled with sand– First layer: gravel to allow

easy drainage– Second layer: F110 sand

with a slope ~5º.• Water and sand poured in

corner plate

• Sand type: Sil-Co-Sil at ~45 mm

• Water feed rate: ~460 cm3/min

• Sediment feed rate: ~37 cm3/min

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

.05 grid sizeDetermine height at fillPosition of toe

• Pictures taken every half hour– Toe front recorded

• Peak height measure every half hour

• Grid of squares 10cm x 10cm

Experimental Measurements

Observations (1)• Topography

– Conic rather than convex– Slope nearly linear across position and time – bell-curve shaped toe

Observations (2)

• Three regions of flow– Sheet flow– Large channel flow– Small channel flow

• Continual bifurcation governed by shear stress

Numerical results

Constant diffusion model @ t=360min

= 4.7

as a function of radius @ t=360min

=170/r, where

r=[(iDx)2+(jDy)2]1/2

111213141

0

20

heig

ht [c

m]

distance [x5cm]

111

2131

41

0

20

heig

ht [c

m]

distance [x5cm]

Diffusion chosen to match toe position

constant nu comparison

0

10

20

30

40

50

60

70

80

0 50 100 150 200

x-position

y-p

ositi

on

60 min [numerical]

60 min [experimental]

150 min [numerical]

150 min [experimental]

360 min [numerical]

360 min [experimental]

Toe Position constant model

Nu(r) comparison

0

10

20

30

40

50

60

70

80

0 50 100 150 200

x-position

y-po

sitio

n

60 min [numerical]

60 min [experimental]

150 min [numerical]

150 min [experimental]

360 min [numerical]

360 min [experimental]

Toe Position r model

peak heigh vs. time

0

2

4

6

8

10

12

14

16

18

0 30 60 90 120 150 180 210 240 270 300 330 360

time [min]

heig

ht [c

m]

measured

constant nu

peak height vs. time

0

2

4

6

8

10

12

14

16

18

0 30 60 90 120 150 180 210 240 270 300 330 360

time [min]

heig

ht [c

m]

measured

nu(r)

111

2131

41

0

20

heig

ht [c

m]

distance [x5cm]111213141

0

20

heig

ht [c

m]

distance [x5cm]

Constantr-model

Non-Linear Diffusion model shows promise

Moving Boundaries in Earthscapes

A number of moving boundary problems in sedimentary geology have beenidentified.

It has been shown that these problems can be posed as Generalized Stefan problems

Fixed grid and deforming grid schemes have been shown to produce results inReasonable agreement with experiments

Improvements in model are needed

Utilize full range of moving boundary numerical technologies to arrive at a suite of methods with geological application

Use large scale general purpose solution packages