granular avalanche modeling methodology working group 2 october 2006
TRANSCRIPT
Granular Avalanche Modeling
Methodology Working Group
2 October 2006
Atenquique, Mexico 1955
Atenquique, Mexico 1955
Volcan Colima, Mexico
San Bernardino Mountain: Waterman Canyon
Guinsaugon. Phillipines, 02/16/06
Heavy rain sent a torrent of earth, mud and rocks down on the village of Guinsaugon. Phillipines, 02/16/06 A relief official says 1,800 people are feared dead.
Model Topography and Equations(2D)
ground
geophysical 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:
sbb
tb
sst
s
et
t
FF:0),(Fat
0FF:0),(Fat
vx
vx
Elevation data from public and private DEMs - different sources and different formal resolutions.
z is the direction normal to the hillside
Model System-Basic Equations
The equations for a continuum incompressible medium are:
semi-empirical relationship between the stress tensor T and u are derived from Coulomb theory
Boundary conditions for stress:
bbb
r
rbbbbbbb
sss
t
t
nTnu
unTnnnTxF
nTxF
tan:0),(at
0:0),(at
gTuuu
u
000 ρρρ
0
t
Model System-Depth Average Theory
Depth average
the continuity equation:
where
),(),,(),,( yxbtyxstyxh
0)()(
y
vh
x
vh
t
h yx
s
b
yy
s
b
xx dzvvhdzvvh ,,
s
b
s
b
s
b
dzh
dzh
dzh
uTu ρ1
,1
,1
Basic model
• System of 3 PDEs for (h, hvx hvy) in space and time (x,y,t)
randomness includes topography
)0),((,, datainitial
)(
sinsgntan1
)5.(
0
int/int/
int2
22
22
txvvh
y
hghk
y
vhvg
vv
vhg
y
vhv
x
hgkhv
t
hv
y
hv
x
hv
t
h
yx
bedbed
zap
xbedx
xz
yx
xx
xyzapxx
yx
Also need to provide initial mass M, location of this mass, initial velocity.
Abstracting
• Y = F(X,θ) + εmodel
• Uncertain parameters
θ = (φint, φbed, M, x0, v0, θrest)
• And would like to include uncertainty in topography
An Aside on M
It is those rare very large flows that cause enormous damage and loss of life.
An Aside on friction
TITAN2D (choose grid ↔ speed)TITAN2D (choose grid ↔ speed) Use adaptive meshing for computational efficiency
Large scale computations to produce realistic simulations of mass flows
Integrated with GIS to obtain terrain data (massaging required)
Need to manipulate DEM grids to computational grids
Integrated with multi-scale visualization tools
Runs efficiently on a range of computers – laptops to large clusters
Code is GRID enabled for remote access through a portal http://grid.ccr.buffalo.edu
All software (source code) freely available for use at http://www.gmfg.buffalo.edu
Other Computer Models (fast)
• Flow3D – similar integration of terrain. Code simulates a frictional block sliding downhill under gravity (can’t run up over an embankment)
• LaharZ – combines terrain data with statistical estimate (based on historical data) and potential energy of the mass, to estimate the volumetric flow from one terrain block into the next
‘Field’ Data
• Table top experiments, reasonably controlled. But scaling up doesn’t work!
• In the field, geologists can measure flow depth (i.e., the “h” after flow stops) at selected sites on the deposit field [take core samples]
• In some instances they can estimate flow speed at locations, by examining run-up near bends
• Both are highly prone to error• Rare event: geologists on site during a flow!
Table Top Experiment
Comparison to experiment
Comparison to experiment
Effect of different initial volumes
Left – block and Ash flow on Colima, V =1.5 x 105 m3
Right – same flow -- V = 8 x105 m3
Real Topography (Little Tahoma, WA)
Tahoma peak (deposit area extent)
Tahoma peak, Mount Rainier (debris avalanche, 1963)
The 2005 Vazcún Valley Lahar
• 12 February 2005. • Vazcún Valley, north-east
flank of Volcán Tungurahua, Ecuador
• Small ash-rich lahar • Volume: 50,000m3
(calculated from field observations) to 70,000m3 (calculated).
• Velocity: 7m/s and 3m/s Photo: Defensa Civil
Flow Thickness
• At El Salado Baths
• As measured 5 months later, flow depth was 3.00 m.
• Two-phase code gives max flow depths of 3.5m at this section.
• Model flow depth is within 50cm of agreement.
Our Halting Early Attempts I
• Generate a response surface
fp = ∑ α θp + e(α, θ)
• Vary M, v0
• Latin hypercube for initial set of runs• Next θ by maximizing variance point, until little
further change in variance. Then up the order of the polynomial.
• “Truth” surface – 10,000 runs on reasonable spatial grid, cross product grid on θs
Hazard map
Conditioned on
a large event
occurring!!
Probability that flow thickness will exceed 1 m.
Our Halting Early Attempts II
INPUT UNCERTAINTY PROPAGATION
• Model inputs are often uncertain – range data and distributions may be estimated– need to propagate this to range and distributions on desired outputs
• Polynomial Chaos (PC)– Assume that the field variables are functions of a random variable(s)
– Expand in terms of “orthogonal polynomials” and use orthogonality to obtain the coefficients – Fourier series like process. Provably accurate methodology.
– Complicated for highly nonlinear problems
n
jjj
n
iii yykk
11
)()(),()( kydt
dy
Quantifying Uncertainty -- Approach
• Polynomial Chaos (PC): approximate pdf with sum of finite
number of orthogonal polynomials i
U(ø) =P
i=1N Ui i(ø)
= h 2m(ø)i1 hf (U(ø;t) m(ø)i
• Multiply by m and integrate to use orthogonality
• Coupling among all the equations for the coefficients
Wiener 34 Xiu and Karniadakis’02
< @t@
PUi i; m >= @t
@Um
),( )(Uft
U
Example
),(),(
)(
mijijmii
jjii
ukdt
du
kktuu
kudt
du
Polynomial Chaos Quadrature
– Instead of Galerkin projection, integrate by quadrature weights
– Leads to a method that has the simplicity of MC sampling and cost of PC
– Can directly compute all moment integrals– Degrades for large number of random
variables
NISP / Polynomial Chaos Quadrature
G(øq) = U(øq; t0) +R
t0
t1f (U(øq; t))dt
hU(t1)Ni = hGNi =P
qwqG(øq)N
Pqwq
2m(øq)
Pqwqf (U(øq;t) m(øq)
@t@Um(t) =
Replace integration with quadrature and interchangeorder of time and stochastic dimension integration → ”smart” sampling