Extending models of granular avalanche flows

Extending models of granular avalanche flows

GEOPHYSICAL GRANULAR & PARTICLE-LADEN FLOWS

Newton Institute @ Bristol 28 October 2003

Bruce PitmanThe University at Buffalo

and if you see my reflection in the snow covered hills

well the landslide will bring it down

the landslide will bring it down

M. Fleetwood

Supported by NSF

Interdisciplinary team:Camil Nichita (Math)

Abani Patra, Kesh Kesavadas, Eliot Winer,

Andy Bauer (MAE)

Mike Sheridan, Marcus Bursik (Geology)

Chris Renschler (Geography)

and a cast of students – Long Le (Math)

Casita disaster, Nicaragua

2D - depth averaged equations, dry flow:

• two parameters – internal and basal friction

Model System – Dry Flow

y topographspecified and

)0,(,, data initial andboundary ith together w

momentum-yfor equation similar

sinsgntan1

)5.(

0

int2

22

22

txvvh

y

hghk

y

vhvg

vv

vhg

y

vhv

x

hgkhv

t

hv

y

hv

x

hv

t

h

yx

zap

xbedx

xz

yx

xsx

xyzapxx

yx

TITAN 2D

Simulation environment, currently for dry flow only

Integrate GRASS GI data for topographical map

High order numerical solver, adaptive mesh, parallel computing

Extension to include erosion (Bursik)

Little Tahoma Peak, 1963 avalanche

•several avalanches, total of 107 m3 of broken lava blocks and other debris

•6.8 km horizontal and 1.8 km vertical run

•estimate pile run-up on terminal moraine gives reasonable comparison with mapped flow; we miss the run-up on Goat Island Mt.

Little Tahoma Peak, 1963 avalanche

Tahoma peak (deposit area extent)

Tahoma peak, Mount Rainier (debris avalanche, 1963)

Debris Flows

Mass flows containing fluid ubiquitous and important

Iverson (’97) 1D Mixture model; Iverson and Denlinger 2D mixture model and simulations

How to model fluid/pore pressure motion?

2-Fluid Approach

Model equations used in engineering literature

Continuum balance laws of mass and momentum for interpenetrating solids and fluid

Drag terms transfer momentum

2-Fluid Approach

gvu

Tuuu

gvu

TTvvv

u

v

ft

fst

t

t

)-(1)()-(1

)-(1))(-(1

)()-(1

)(

0)-(1)-(1

0

f

f

s

s

f f

s s

2-Fluid Approach

•Decide constitutive relations for solid and fluid stresses (frictional solids, Newtonian fluid)•Phenomenological volume-fraction dependent function in drag

•Depth average – introduceserrors that we will examine (and live with)

m)1/(

Free boundary and basal surface

ground

flowing mass

),( yxbz

),,( tyxsz

bsh

Upper free surface

Fs(x,t) = s(x,y,t) – z = 0,

Basal material surface

Fb(x,t) = b(x,y) – z = 0

Kinematic BC:

0FF:0),(Fat

0FF:0),(Fat

bbt

b

sst

s

t

t

vx

vx

bbb

r

rbbbbbbb

sss

t

t

nTnu

unTnnnTxF

nTxF

tan:0),(at

0:0),(at

Scales

Characteristic length scales (mm to km)

e.g for Mount St. Helens (mudflow –1985) Runout distance 31,000 mDescent height 2,150 mFlow length(L) 100-2,000Flow thickness(H) 1-10 mMean diameter of sediment material 10-3-10 m

Scale: ε═H/L – several terms small and are dropped

(data from Iverson 1995, Iverson & Denlinger 2001)

Model System-Depth Average Theory 2D to 1D

Depth average

solids conservation:

where

)(),(),( xbtxstxh

0)(

x

v

tx

s

b

xx dzvvh

s

b

s

b

s

b

dzvh

dzTh

dzvh

ρ1

,1

,1

Model System – 1D

33

33

33

33

))(1(/])1(/[

'

)(

/

sijsij

zfxsfm

T

ff

zsfzsz

zfz

TTstresssolidassume

gTvuv

ionapproximatlikeArcyDsotermsinertiafluiddrop

ITTpressureaisstressfluidsassume

ggT

gT

Model System – 1D

xxsf

zsfsxt

xt

xt

ghhg

ghbhTvhvh

momentumsolidsaverageddepth

vhh

continuitysolidsaverageddepth

uhvhh

yieldsaveragingdepthibleincompressisflowweightedvolume

/

])[/1()(

0)(

0))1((

113133112

Model System – 1D

int

2

bed2

int2

cos

1])}tan1(cos1{1[2

)(

apk

tcoefficienpressureearthinvolvetscoefficien

sourcemomentumasentersxbtopography

Errors in modeling

Special Solutions

Special Solutions

Special Solutions

hφ constant (lower curve)

h evolves in time (upper

curve)

Special Solutions

Special Solutions

constant velocities u,v

hφ faster

h slower

Time Evolution

Mixed hyperbolic-parabolic system

zf

zfxsfm

T

xt

hgT

gTvuv

uhvhh

hforequationcontinuitytheConsider

~

))(1(/])1(/[

0))1((

:

Time Evolution

Time Evolution

On inclined plane, volume fraction changes smallspecial solution

Interaction with ‘topography’ induces variation in φ

Modeling questions

Evolution equation for fluid velocity?

Efficient methods for computing 2D system including realistic topography

dzgvu

TTuuu

xfs

fzfxthb

])()/(

)()([ 3111

Comments on model•Continuum model

In situ, there is a distribution of particle sizes. Models are operating at the edge where the discreteness of solids particles cannot be ignored

•Depth averaged velocity

Are recirculation and basal slip velocity important?

•There is no simple scaling arguments from tabletop experiments to real debris avalanches (No Re, Ba, Sa)