How Does Urban Form Impact Flooding?

A mathematical equation relating the depth and velocity with which water flows along a sloping surface or pipe was first developed in the 18th century by the French hydrologist Antoine Chézy (1718-98). The equation met an important need to estimate the flow of water in rivers and canals, and to design pipes for water supply and drainage. A century later, an alternative equation was developed by French Engineer Philippe Gauckler (1826–1905) and Irish engineer Robert Manning (1816-1897), and that equation – known today as the Manning equation – is widely used to estimate how deep water flows considering the volumetric flow rate, slope and surface roughness resisting flow.
The Manning equation remains in use today for many practical applications, from the sizing of pipes for water systems, to the estimation of river stages during floods. However, when floods spread out laterally and encounter complex obstructions such as woody debris and structures built upon flood plains, the Manning equation is no longer applicable. Obstructions to flow introduce form drag—like the force felt on your hand if you stretch it out of a car window at speed – which adds to the surface resistance approximated by the Manning equation.
Today there is grave concern about flooding in areas where people and assets are concentrated—urban areas—because those areas are most vulnerable to life losses and damages. And unfortunately, the Manning equation is not a reliable predictor within urban areas, as flow resistance is controlled by form drag. Efforts to understand and estimate flood hazards facing urban areas are left dependent upon advanced fluids mechanics models that are data intensive and computational demanding, which limits capacity to examine urban flood risks across the major cities of the world and to understand why types of urban growth patterns would work best from a flood resilience perspective.
Balaian and colleagues have now developed a relatively simple mathematical equation capable of estimating urban flood depths and velocities with the same fidelity that the Manning equation estimates depths and velocities in channels. By running thousands of computer model simulations over different configurations of city blocks and building arrangement, Balaian et al. have derived new mathematical equation that predicts flood depth and velocity given a volumetric flow rate (e.g., from intense rainfall) and two parameters descriptive of the urban form of the city: the two-dimensional porosity of the city representative of the space available for flood water to move, and the chord length distance representative of how far water can move in the downhill direction before encountering an obstruction. In essence, the equation predicts deeper flooding with denser cities and shallower flooding with longer chord lengths which enable the acceleration of flow. The equation is especially valuable for practical applications because its two parameters are readily deduced from globally available data describing ground slopes and building footprints on the land surface.
What does the new equation reveal about the susceptibility of cities to flooding globally based on urban form alone, i.e., independent of the severity of rainfall? Cities with high densities and longer urban chord lengths promote both relatively deep flooding and fast moving floods and include Tokyo, Lagos, Jakarta, Karachi, and Sao Paulo. In the U.S., Chicago, Miami and Houston emerge as a cities susceptible to deep flooding due to building density and mild slopes, and San Francisco emerges as a city vulnerable to shallow but fast flooding due to steep slopes. On the other hand, Seattle emerges as a city with relatively low flood hazard based on its intermediate building density and slopes.
Validation of our new equation is challenging because extreme flood events occur infrequently, and urban flood depths are not well documented. However, based on limited data on flood damages in major cities during heavy rainfall events, Balaian and colleagues found a correlation between the hazard severity predicted by the new urban flooding equation and the losses incurred during the flood. More research is needed to fully explore the descriptive power of the newly derived equation and its utility for risk management, including urban planning.