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 1998 -> present Author biases Apologies for omissions First IMPACT Workshop Wallingford, UK, 16-17 May 2002 F. Alcrudo University of Zaragoza Spain

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

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

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

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

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

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

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 2001... River flows Casulli & Stelling (Q-hydrostatic) 1998 Sinha et al. 1998, Ye &McCorquodale 1998...

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

Inertial & Pressure fluxes Convective Momentum transport Hydrostatic pressure distribution

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

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

1-D SWE models

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)

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)

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)

Flat pond model of Rio Verde area (Estrela 1999)

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

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)

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

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

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

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)

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)

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)

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 ...

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)

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

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

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

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

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