A state of the art review on mathematical modelling of flood propagation

1
A state of the art review on mathematical
modelling offlood propagation

• First IMPACT Workshop
• Wallingford, UK,
• 16-17 May 2002

F. Alcrudo University of Zaragoza Spain
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Overview

• The modelling process
• Mathematical models of flood propagation
• Solution of the Model equations
• Validation

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The modelling process

• Understanding of flow characteristics
• Formulation of mathematical laws
• Numerical methods
• Programming
• Validation of model by comparison of results
  against real life data
• Prediction Ability to FOREtell not to PASTtell

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The modelling process
REALITY
Analisis
Computer Simulation Validation
Data uncertainties
Conceptual errors uncertainties
Discretization errors
MATHEMATICAL MODEL
COMPUTER MODEL
Numerics Implementation
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The flow characteristics

• 3-D
• time dependent
• incompresible
• free surface
• fixed bed
• (no erosion deposition)
• turbulent (very high Re)

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Mathematical models

• 3-D Navier-Stokes (DNS)
• Chimerical
• 3-D RANS
• Turbulence models ?
• Still too complex
• Euler (inviscid)
• Simpler, requires much less resolution
• Could be an option soon

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Mathematical models

• Tracking of the free surface
• VOF method (Hirt Nichols 1981)
• MAC method (Welch et al. 1966)
• Moving mesh methods

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NS, RANS Euler

• 2-D dam break and overturning waves
• Zwart et al. 1999
• Mohapatra et al. 1999
• Stansby et al. (Potential) 1998
• Stelling Busnelli 2001...
• River flows
• Casulli Stelling (Q-hydrostatic) 1998
• Sinha et al. 1998, Ye McCorquodale 1998...

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Simplified mathematical models

• Shallow Water Equations (SWE)
• Depth integrated NS
• Mass and momentum conservation in horizontal
  plane
• Pseudo compressibility

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• Inertial Pressure fluxes
• Convective Momentum transport
• Hydrostatic pressure distribution

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• Diffusive fluxes
• Fluid viscosity
• Turbulence
• Velocity dispersion (non-uniformity)

Benqué et al. (1982)
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• Sources
• Bed slope
• Bed friction (empirical)
• Infiltration / Aportation (Singh et al. 1998
  Fiedler et al. 2000)

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• 1-D SWE models

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Issues in SWE models

• Corrections for non-hydrostatic pressure,
  non-zero vertical movement
• Boussinesq aproximation (Soares 2002)
• Stansby and Zhou 1998 (in NS-2D-V)
• Flow over vertical steps (Zhou et al. 2001)
• (Exact solutions Alcrudo Benkhaldoun 2001)
• Corrections for non-uniform horizontal velocity ?
• (Dispersion effects)

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Issues in SWE models (cont.)

• Turbulence modelling in 2D-H
• Nadaoka Yagi (1998) river flow
• Gutting Hutter (1998) lake circulation (K-e)
• Gelb Gleeson (2001) atmospheric SWE model
• Bottom friction
• Non-uniform unsteady friction laws ?
• Distributed friction coefficients (Aronica et al.
  1998)
• Bottom induced horizontal shear generation
  (Nadaoka Yagi 1998)

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Simplified models

• Kinematic diffusive models
• Arónica et al. (1998)
• Horrit and Bates (2001)
• Flat Pond models
• Tous dam break inundation (Estrela 1999)

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Flat pond model of Rio Verde area (Estrela 1999)
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Solution of the model equations(Restricted to
SWE models)

• Discretization strategies
• Mesh configurations
• Numerical schemes
• Space-Time discretizations
• Front propagation
• Source term integration
• Wetting and drying

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Discretization strategies

• Finite differences
• Decaying use (less flexible)
• Usually structured grids
• Scheme development/testing (Liska Wendroff
  1999, Glaister 2000 ...)
• Practical appications (Bento-Franco 1996,
  Heinrich et al. 2000, Aureli et al. 2000)

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• Finite volumes
• Both structured unstructured grids
• Cell-centered or cell-vertex
• Extremely flexible intuitive
• Many practical applications (CADAM 1998-1999,
  Brufau et al. 2000, Soares et al. 1999, Zoppou
  1999)
• Most popular

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• Finite elements
• Variational formulation
• Conceptually more complex
• More difficult front capture operator (Ribeiro et
  al. 2001, Hauke 1998)
• Practical applications
• Hervouet 2000, Hervouet Petitjean 1999
• Supercritical / subcritical, tidal flows, Heniche
  et al. 2000

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Mesh configurations

• Structured
• Cartesian / Boundary fitted (mappings)
• Less flexible / Easy interpolation
• Unstructured
• Flexible but Indexing / Bookkeeping overheads
• More elaborated Interpolation (Sleigh 1998,
  Hubbard 1999)
• Easy refining (Sleigh 1998, Soares 1999) and
  adaptation (Benkhaldoun 1994, Ivanenko et al.
  2000)
• Quad-Tree

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Mesh configurations

• Quad-Tree
• Cartesian with grid refining/adaptation
• Hierarchical structure / Interpolation operators
• Needs bookkeeping
• Usually specific boundary treatments (Cartesian
  cut-cell approach Causon et al. 2000, 2001)
• Practical applications (Borthwick et al. 2001)

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Numerical schemes

• Space Time discretization
• Space discretizations
• Time integration of resulting ODE
• Time integration
• Explicit usu 2-step, Runge-Kutta (Subject to CFL
  constraints)
• Implicit (not frequent)

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• Front propagation
• Shock capturing or through methods
• Approximate Riemann solvers (Most popular Roe,
  WAF second)
• Higher order interpolations limiters (either
  flux or variables), TVD, ENO
• Mostly in FV FD but progressively incorporated
  into FE (Sheu Fhang 2001)
• Plenty of methods (or publications)

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• Multidimensional upwind
• Wave recognition schemes (opposed to classical
  dimensional splitting)
• Consistent Higher resolution of wave patterns
• Usually in unstructured (cell vertex) grids
  (mostly triangles)
• Considerably more expensive
• Hubbard Baines 1998, Brufau Garcianavarro
  2000 ...

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• Source term integration (bed slope)
• Flow is source term dominated in most practical
  applications
• Flux discretization must be compatible with
  source term
• Source term upwinding (Bermudez Vazquez 1994)
• Pressure splitting (Nujic 1995)
• Flux lateralisation (Capart et al. 1996, Soares
  2002)
• Surface gradient method (Zhou et al. 2001)
• Discontinuous bed topography (Zhou et al. 2002)

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• Wetting-drying
• Intrinsic to flood propagation scenarios
• Instabilities due to coupling with friction
  formulae and to sloping bottom (Soares 2002)
• Threshold technique (CADAM 1998), simple, widely
  used but no more than a trick
• Fictitious negative depth (Soares 2002)
• Boundary treatment at interface (Bento-Franco
  1996, Sleigh 1998), modification of bottom
  function (Brufau 2000)
• Bottom function modification, ALE (Quecedo and
  Pastor (2002) in Taylor Galerkin FE

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Validation

• Model accuracy
• Differences between model output real life
• Determined with respect to experimental data
• Accuracy loss
• Uncertainty Due to lack of knowledge
• Errors Recognizable defficiencies

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• Main losses of accuracy in flood propagation
  models
• Errors in the math description (SWE or worse)
• Uncertainties in data (topography, friction
  levels, initial flood characteristics)
• Additional errors
• Inaccurate solution of model equations (grid
  refining)

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• Much validation work of numerical methods against
  analytical /other numerical solutions
• Chippada et al., Hu et al., Aral et al. 1998
• Holdhal et al., Liska Wendroff , Zoppou
  Roberts etc ... 1999
• Causon et al., Wang et al., Borthwick et al. etc
  ... 2001
• Validation against data from laboratory
  experiments
• CADAM work, Tseng et al. 2000, Sakarya Toykay
  2000 etc ...
• Validation against true real flooding data
• CADAM 1999, Hervouet Petitjean (1999), Hervouet
  (2000), Horritt (2000), Heinrich et al. (2001),
  Haider (2001)
• Sensitiviy analysis (usually friction)
• Urban flooding ?

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Conlusions

• Present feasible mathematical descriptions of
  flood propagation are known to be erroneous but
  ...
• Better mathematical models are still far ahead
• The level of accuracy of present models has not
  yet been thoroughly assessed
• There are enough methods at hand to solve the
  mathematical models (most are good enough)
• Exhaustive validation programs against real data
  are needed