Ponencia en  IUGG Conference on Mathematical Geophysic

Córdoba, Gustavo

Pitman, Bruce

Sheridan, Mike

Universidad de Nariño

A depth-averaged, two-phase flow code for hazard mapping that satisfies both hydraulic and granular flow extremes.ByGustavo Cordoba (1,2), Mike Sheridan (1) and Bruce Pitman (1)(1) University at Buffalo (NY), Center for GeoHazard Studies(2) Universidad de Nariño (Col.).The Titan projectAims- Create realistic 2D & 3D models that simulate geophysical mass flow- Integrate and communicate information from several linked sources including:.- Simulation results.- Remote sensing data .- GIS dataCurrent applications- One phase dense granular flowsRock avalanchesBlock-and-ash flows- Two phase dense flowsDebris flows- Hazard maps & risk assessment- Probabilistic analysisFrank slideColima 1991Mathematical basisSavage-Hutter (1989) Model- Provides a framework for later, more sophisticated models. - The model is based on assumptions:Aspect ratio of the flow is smallGranular flows have Coulomb friction behaviourTop surface is stress-freeTop surface and bed surface represented as functions Fh(X,t) =0 and Fb(X) = 0bedgeophysical massz=s(x,y,t)h=s-bz=b(x,y)The equations are depth averaged:Problems with the one phase model- Several natural flows are multiphase- Savage-Hutter model is only for dry avalanches- Interstitial fluid modelled in Titan by reducing bed and internal friction anglesHowever,- Interstitial fluid can greatly alter the dynamics of the flow . - Flows can achieve very high velocities, and can travel long distances. - Water may be supplied by rainfall, melting of snow or ice or by overflow of a crater lake.Pitman and Le model- Pitman and Le, (2005) expand the applications of TITAN to debris flows with various water contents- Independent equations of motion specified for each phase, along with an equation for volume fractions- Solid-Fluid interaction modelled using a drag force termValidation: actual prediction of the2007 Mt Rapehu LaharPreviously nonrecognized bifurcationpredicted by TitanActual path Bank leveeTitan predictionProblems with Pitman model- Numerics break down at flow-front- Non-physical velocities, as conserved variables, are divided by very small values- It does not work well close to the extremes of pure dry or pure water flows. Cordoba, Pitman and Sheridan modelCharacteristics of the proposed model - No limit on the fluid phase content- Pure dry avalanches modeled as a mixture of near maximum pack concentration of solids plus air.- Solid phase modeled as a granular flow that obeys the Coulomb friction law.- Fluid phase modeled as shallow water that follows the Darcy-Weisbachapproach for friction.- Phase equations coupled through a drag coefficient and pressure.- Mathematically the equations become the Savage and Hutter (1889)approach for pure solids and to pure shallow water (e.g. see the modelof Cun-hong et al., 2006) in case of zero solids concentration ValidationFor purposes of validation and analytical testing, the one dimensional version of the equations was tested:Dam breakSymmetryExponentially decaying topography with bumpReflected wave BumpPure waterBumpBig bump Small bumpEquation system for bi-phase flows∂t s∇⋅s vi =0∂tf 1−∇⋅f 1−ui=0∂t s vi∇⋅s vi v j=∇⋅Tsf ss g∂t f 1−ui ∇⋅f 1−uiu j=∇⋅Tff ff 1−gConservation laws, assuming a fluid saturated mixture of solids and water:Mass:Momentum:Interaction forces between phasesf s=−∇T fD v−uf f=−f sD= 34Cdf 〚v−u 〛f 2 d1−Where the drag D is approached through the empirical relationship:andf=f ReTitan2phase implementation:- Full two phase equation system implemented into the Titan2D frame.- Eigen dependent time steps of the order of O(10^-4) - Presence of a fluid phase reduces the bulk bed friction in a natural way.- Currently, constant volumetric fraction of solids assumed. Titan2D and Titan2phase Titan2D Titan2phaseNext steps: • Allowing change in particle concentration in time• Validation• Corrections and debugging• Validation• Release of a beta-version • Web distribution as free software• Implementation on VHUB Equation System

A two-phase, depth-averaged model for geophysical mass flows in the TITAN code framework

http://sired.udenar.edu.co/3638/1/A%20two-phase%2C%20depth-averaged%20model%20for%20geophysical%20mass%20flows%20in%20the%20TITAN%20code%20framework.pdf

A two-phase, depth-averaged model for geophysical mass flows in the TITAN code framework

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