Shallow-water equations
Flood water is shallow compared with its horizontal extent, so the depth h(x,y,t) and the depth-averaged velocity (u,v) obey ∂h/∂t + ∂(hu)/∂x + ∂(hv)/∂y = 0 and ∂(hu)/∂t + ∂(hu² + g h²/2)/∂x + ∂(huv)/∂y = -g h ∂zb/∂x - g n² u |u| / h1/3 (and the same for y). zb is the ground level, n the Manning roughness (0.035 in the channel, 0.028 on streets, 0.045 on lawns, 0.05 on fields). The model integrates these on 17,920 cells of 20 m with a finite-volume scheme: HLL fluxes (with the contact wave of HLLC so that the flow along a bank is not smeared), the hydrostatic reconstruction of Audusse et al. that keeps a lake at rest exactly at rest on a rough bed and handles the advancing and retreating water edge, and implicit friction. The time step follows the CFL condition, about 1 s.
Hydrograph, return period, probability
The inflow is a flood wave Q(t) = Qb + (Qp - Qb) (t/tp e1 - t/tp)2.5. How rare a peak is comes from a Gumbel (extreme-value type I) model of annual maxima, Q(T) = u - β ln(-ln(1 - 1/T)) with u = 900 m³/s and β = 420 m³/s (a made-up river). A return period T means a chance 1/T per year; the chance of at least one such flood in n years is P = 1 - (1 - 1/T)n, which the checks compare with random draws.
Levees, weirs and confinement
A levee is a raised ridge in the ground. When the river is above the crest, the flow over it behaves like a broad-crested weir, Q = C L H3/2 with C of 1.4 to 1.7 (m1/2/s), L the length and H the head above the crest. The model shows the measured overflow beside this formula. Levees also have a side effect: by closing off the flood plain they force the water into a narrower section, so the stage in the channel rises (for the 1-in-100-year flood here the stage is 1.5 m above the bank top in the open valley and 3.9 m between the levees). That is why the required crest is computed for the leveed section and not for the open valley, and why a levee that protects one place can raise the flood levels elsewhere.
Manning estimate
For steady uniform flow each strip of the section carries q = h5/3 √S / n per metre. Summing the strips between the levees gives a rating curve Q(stage). The 2-D model gives a stage 0.2 to 0.25 m higher: the fast deep channel drags the slow water on the banks along (momentum exchange the strip method ignores), the river bends, and the 20 m cells cannot follow the bank shape exactly. The simple estimate is therefore a lower bound, and the freeboard is there for such differences.
Pumps, basin and damage
A pump at the foot of the levee lifts water from the ditch to the river; its delivery falls with the lift H, Q = Qrated √(1 - (H/H0)²) with a shut-off head H0 = 12 m, so a high river both fills the ditch and weakens the pumps. The retention basin (about 1.5 million m³) is filled through the three-bay inlet gate when the river is high and emptied afterwards through a flap-gated culvert. It trims the peak of a short sharp flood but is small against a long flood. Damage per block comes from a depth-damage curve (3 % at 0.1 m, 22 % at 0.6 m, 56 % at 2 m, 90 % at 6 m) weighted by the value of the block type. These numbers are generic.
What is simplified
The river and city are fictional. Fluvial flooding only: no rain on the city, no groundwater or seepage under the levee, no levee breach or erosion (a breach is deliberately not simulated), no sewers. Buildings are solid obstacles of 60 m blocks (water cannot enter them or the courtyards); the streets are 20 m wide cells. There is no sediment or debris; the brown colour of the water and the mud left on the ground are only visual. Bridges are modelled as clear spans with the deck above the water. The grid is coarse (20 m): depths within a cell are averages, narrow channels and kerbs are not resolved, and the outflow boundary assumes uniform flow. Pump, culvert and gate flows use simple formulas. Costs are arbitrary units. Treat everything as a teaching model.
Educational only. In a real flood follow the instructions of the civil-protection authorities, do not enter flood water and move to higher ground when told to. No real river, city or organisation is represented.