Granular Avalanche Modeling

1. Granular Avalanche Modeling Methodology Working Group 2 October 2006

4. Atenquique, Mexico 1955

5. Atenquique, Mexico 1955

6. Volcan Colima, Mexico

7. San Bernardino Mountain: Waterman Canyon

8. 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.

9. geophysical mass ground Model Topography and Equations(2D) 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: z is the direction normal to the hillside Elevation data from public and private DEMs - different sources and different formal resolutions.

10. 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:

11. Model System-Depth Average Theory Depth average the continuity equation: where

12. Basic model • System of 3 PDEs for (h, hvx hvy) in space and time (x,y,t) Also need to provide initial mass M, location of this mass, initial velocity.

13. Abstracting • Y = F(X,θ) + εmodel • Uncertain parameters θ = (φint, φbed, M, x0, v0, θrest) • And would like to include uncertainty in topography

14. An Aside on M It is those rare very large flows that cause enormous damage and loss of life.

15. An Aside on friction

16. 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

17. 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

18. ‘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!

19. Table Top Experiment

20. Comparison to experiment

21. Comparison to experiment

22. 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

23. Real Topography (Little Tahoma, WA) Tahoma peak, Mount Rainier (debris avalanche, 1963) Tahoma peak (deposit area extent)

24. 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

25. 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. Flow Thickness

26. Our Halting Early Attempts I • Generate a response surface fp = ∑ αθp + 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

27. Hazard map • Conditioned on • a large event • occurring!! Probability that flow thickness will exceed 1 m.

28. Our Halting Early Attempts II

29. 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) x • Expand in terms of “orthogonal polynomials” y and use orthogonality to obtain the coefficients – Fourier series like process. Provably accurate methodology. • Complicated for highly nonlinear problems

30. Quantifying Uncertainty -- Approach Wiener 34 Xiu and Karniadakis’02 • Polynomial Chaos (PC): approximate pdf with sum of finite number of orthogonal polynomials yi • Multiply by ym and integrate to use orthogonality • Coupling among all the equations for the coefficients

31. Example

32. 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

33. NISP / Polynomial Chaos Quadrature Replace integration with quadrature and interchange order of time and stochastic dimension integration → ”smart” sampling