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)
