1. First IMPACT Workshop Wallingford, UK, 16-17 May 2002
A state of the art review on mathematical modelling of flood propagation
General overview
References to original/classics and to recent work > present
Author biases
Apologies for omissions
First IMPACT Workshop
Wallingford, UK,
16-17 May 2002
F. Alcrudo
University of Zaragoza
Spain

2. Overview The modelling process
Mathematical models of flood propagation
Solution of the Model equations
Validation

3. 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
Validation makes no statement about the prediction capabilities -> The mother of all uncertainties

4. Discretization errors
The modelling process
REALITY
Analisis
Computer Simulation
& Validation
Data uncertainties
Conceptual
errors & uncertainties
Discretization errors
COMPUTER MODEL
MATHEMATICAL MODEL
Numerics &
Implementation

5. The flow characteristics
3-D
time dependent
incompresible
free surface
fixed bed
(no erosion – deposition)
turbulent (very high Re)

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

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

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

9. Simplified mathematical models
Shallow Water Equations (SWE)
Depth integrated NS
Mass and momentum conservation in horizontal plane
Pseudo compressibility h u v

10. Inertial & Pressure fluxes
Convective Momentum transport
Hydrostatic pressure distribution

11. Diffusive fluxes Benqué et al. (1982) Fluid viscosity Turbulence
Velocity dispersion (non-uniformity)
Benqué et al. (1982)

12. Sources Bed slope Bed friction (empirical)
Infiltration / Aportation (Singh et al Fiedler et al. 2000)

13. 1-D SWE models

14. 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)

15. 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)

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

17. Flat pond model of Rio Verde area (Estrela 1999)

18. 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
Validation makes no statement about the prediction capabilities -> The mother of all uncertainties

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

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

21. Finite elements Variational formulation Practical applications
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

22. Mesh configurations Structured Unstructured Quad-Tree
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

23. 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 , 2001)
Practical applications (Borthwick et al. 2001)

24. Numerical schemes Space – Time discretization Time integration
Space discretizations +
Time integration of resulting ODE
Time integration
Explicit usu 2-step, Runge-Kutta (Subject to CFL constraints)
Implicit (not frequent)

25. 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)

26. 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

27. 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)

28. 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

29. Validation Model accuracy Accuracy loss:
Differences between model output & real life
Determined with respect to experimental data
Accuracy loss:
Uncertainty Due to lack of knowledge
Errors Recognizable defficiencies
Validation makes no statement about the prediction capabilities -> The mother of all uncertainties

30. 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)
Validation makes no statement about the prediction capabilities -> The mother of all uncertainties

31. Validation against data from laboratory experiments
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
Causon et al., Wang et al., Borthwick et al. etc
Validation against data from laboratory experiments
CADAM work, Tseng et al. 2000, Sakarya & Toykay 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 ?
Validation makes no statement about the prediction capabilities -> The mother of all uncertainties

32. 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
Validation makes no statement about the prediction capabilities -> The mother of all uncertainties