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