# Hydrodynamic Modeling of Urban Flood Flows and Disaster Risk Reduction

## Summary and Keywords

Communities facing urban flood risk have access to powerful flood simulation software for use in disaster-risk-reduction (DRR) initiatives. However, recent research has shown that flood risk continues to escalate globally, despite an increase in the primary outcome of flood simulation: increased knowledge. Thus, a key issue with the utilization of urban flood models is not necessarily development of new knowledge about flooding, but rather the achievement of more socially robust and context-sensitive knowledge production capable of converting knowledge into action. There are early indications that this can be accomplished when an urban flood model is used as a tool to bring together local lay and scientific expertise around local priorities and perceptions, and to advance improved, target-oriented methods of flood risk communication.

The success of urban flood models as a facilitating agent for knowledge coproduction will depend on whether they are trusted by both the scientific and local expert, and to this end, whether the model constitutes an accurate approximation of flood dynamics is a key issue. This is not a sufficient condition for knowledge coproduction, but it is a necessary one. For example, trust can easily be eroded at the local level by disagreements among scientists about what constitutes an accurate approximation.

Motivated by the need for confidence in urban flood models, and the wide variety of models available to users, this article reviews progress in urban flood model development over three eras: (1) the *era of theory,* when the foundation of urban flood models was established using fluid mechanics principles and considerable attention focused on development of computational methods for solving the one- and two-dimensional equations governing flood flows; (2) the *era of data*, which took form in the 2000s, and has motivated a reexamination of urban flood model design in response to the transformation from a data-poor to a data-rich modeling environment; and (3) the *era of disaster risk reduction,* whereby modeling tools are put in the hands of communities facing flood risk and are used to codevelop flood risk knowledge and transform knowledge to action. The article aims to inform decision makers and policy makers regarding the match between model selection and decision points, to orient the engineering community to the varied decision-making and policy needs that arise in the context of DRR activities, to highlight the opportunities and pitfalls associated with alternative urban flood modeling techniques, and to frame areas for future research.

Keywords: urban flood modeling, disaster risk reduction, flood risk, risk communication, shallow-water equations

Introduction

Flood risk threatens more people than any other natural catastrophe, because over half the world’s population lives in cities, which are generally located along waterways and the sea (Sundermann et al., 2014). Additionally, an escalation of flood risk can be expected as populations continue to migrate from rural to urban areas in search of economic opportunities. The potential impacts of higher sea levels include future flooding losses up to 1 trillion USD by 2050 (Hallegatte et al., 2013), with impacts concentrated in coastal cities (Hanson et al., 2013), but almost all metropolitan areas are exposed to flood risk, and the threat from river flooding is especially high for cities in India and China (Sundermann et al., 2014).

Urban flood models are powerful tools for responding to this growing challenge. Urban flood models describe the spatial and temporal distribution of flood attributes, such as depth and velocity, and thus present the opportunity to inform all aspects of the disaster management cycle, including mitigation, preparedness, early warning, real-time assessment, response, recovery, and redevelopment. Urban flood models resolve flooding at fine spatial scales in the range of 1 to 5 m and account for the effect of buildings and roads on flooding, in addition to formal drainage infrastructure (Henonin et al., 2013). Fine-scale output makes it possible to assess localized impacts, such as roads vulnerable to high-velocity flow, buildings vulnerable to damage as well as buildings that offer safety, and impacts to components of critical infrastructure systems, such as waste water pump stations, electrical transformers, telecommunication hubs, and emergency management facilities, such as police stations, fire stations, and hospitals. Importantly, the fine scales of urban flood model predictions align with the scales at which humans perceive flooding, creating opportunities to radically transform flood risk communication and to increase the frequency with which flood risk knowledge is transformed to action. The opportunity is best appreciated when flood risk is defined as *R* = *HEV*, where *H* represents hazard, *E* represents exposure, and *V* represents vulnerability, because visualizations and maps of flooding pose opportunities to communicate information about all three elements (i.e., hazard, exposure, and vulnerability). This definition contrasts with the interpretation of risk as expected loss, *R* = *PL*, where *P* represents probability of an event and *L* represents losses from the event, a formulation that is common to engineering disciplines. It also differs from the statistical definition of flood risk as the probability of one or more exceedances. Indeed, the power of urban flood models to support disaster risk reduction (DRR) is magnified by a broad definition of risk and an understanding of the breadth of decision makers who can be informed by model data: planners, policy makers, residents, businesses, managers of infrastructures (for water, waste, transportation, power, oil, gas, and communications), public health officials, managers of natural resources (such as parks, wetlands, rivers, and coastal waterways), and emergency management and response personnel (Price & Vojinovic, 2008). Furthermore, decision-maker engagement in the modeling process is essential for making urban flood models effective tools for DRR (Lane et al., 2011; Spiekermann et al., 2015). This motivates a transformation of the role of urban flood modelers in the practice of flood risk management (Lane et al., 2011) away from that of an expert analyst and toward that of a participant in the process of knowledge production. In particular, DRR research points to a community-engaged process of flood risk knowledge coproduction that integrates models and data, expert knowledge, local knowledge, and community context (Landström et al., 2011; Spiekermann et al., 2015).

An excellent approximation of urban flooding at fine scales is achieved by solving two-dimensional (2D) de St Venant equations, which assume flow is nearly horizontal, the vertical pressure distribution is hydrostatic, and turbulence closure is satisfactorily achieved with a bottom-resistance parameterization (Toro, 2001). Additionally, with urban flooding, a large fraction of the flood flow may be conveyed by subsurface storm sewers that can be effectively modeled as a network of links between junctions (Rossman, 2015). Henonin et al. (2013) present a classification of urban flood models, including 1D (one-dimensional), 1D–1D, 1D–2D, and 2D classes, that reflects the dimensionality of sewer and surface flow models. Henonin et al. (2013) present examples of each approach and readily available software. Surface flooding models can be further classified based on whether local and convective acceleration terms and pressure gradients are included in the horizontal momentum balance, giving rise to kinematic wave models, diffusive wave models, local acceleration models, and dynamic wave models (de Almeida & Bates, 2013). Key ideas that progressed over a period of more than four decades led to the development of 2D hydrodynamic (dynamic wave) urban flood models, the formulation needed to account for both subcritical and supercritical flow and to resolve hydraulic jumps, and the formulation most commonly adopted by operational software for urban flood modeling. The purposes of this article are: (a) to summarize the main areas of research that took place over the four decades and that persist today as defining features of operational software, (b) to highlight modeling challenges that have occupied researchers, as well as novel solutions, and (c) to highlight the potential impact of urban flood models in the context of DRR, not only to help respond to the challenges posed by urban flooding, but also to orient disciplinary modeling researchers (e.g., civil engineers) toward the transdisciplinary field of DRR where urban flood models are ultimately used, yet not always found to be useful (e.g., Leskens et al., 2014). This last point is particularly important given the levels of flood risk facing the world (Kalra et al., 2013) and promising results from modeling experience (Lane et al., 2011; Price & Vojinovic, 2008).

There have been three eras of research activity: (1) the *era of theory,* when the foundation of hydrodynamic flood models was established using fluid mechanics principles and considerable attention was focused on development of computational methods for solving the 1D and 2D equations governing flood flows; (2) the *era of data*, which took form in the 2000s, and has motivated a reexamination of urban flood model design in response to the transformation from a data-poor to a data-rich modeling environment; and (3) the *era of disaster risk reduction,* whereby modeling tools are put in the hands of communities facing flood risk and are used to codevelop flood risk knowledge and transform knowledge to action.

The Era of Theory

Urban flood model formulations based on fluid mechanics principles can be traced to several application areas, including floodplain modeling, tidal modeling in deltas and estuaries, furrow irrigation modeling, overland flow modeling, and urban drainage modeling. In these applications, at the scales that were of practical interest, researchers found that fluid pressure could be accurately approximated as hydrostatic and turbulence closure could be achieved using an empirical friction loss (or drag) formulation. Hence, researchers shared a common interest in 1D and 2D de St Venant equations (or shallow-water equations) for describing flow, which are based on the assumptions of hydrostatic pressure, a well mixed water column, and nearly horizontal flow with a fully developed boundary layer, or so-called gradually varied flow (Chow, 1959; Henderson, 1966).

Numerical solution of de St Venant equations was recognized, early on, as a major challenge due to equations of the hyperbolic type and poorly understood boundary conditions (Cunge, 1975). For example, there are three unknowns when solving the 2D flow equations: depth and two components of velocity. However, the hyperbolic form of the equations dictates that, under subcritical flow, only one constraint can be applied at outflow boundaries and two constraints can be applied at inflow boundaries, while under supercritical flow, three constraints are required at inflow boundaries and none is required at outflow boundaries. Additionally, flooding applications introduce moving boundaries where the water surface intersects dry land, the so-called wet/dry interface. Hence, the most general solver requires the ability to adapt boundary conditions based on the solution itself. Research activity occurred in several different application areas within civil engineering, such as floodplain modeling, tidal delta modeling, furrow irrigation modeling, and dam-break modeling. Collectively known as computational hydraulics, these specialty fields took form in the 1970s and research into improved models continued over the following decades until the 2000s, when a major transformation of the field began to unfold with the convergence of three factors: the advent of generalizable numerical methods for solving the shallow-water equations, widespread access to high-performance computing systems, and most importantly, the proliferation of geodetic technologies and geographical information systems to measure and organize land surface features, such as topography, bathymetry, vegetation, flood defenses, and drainage infrastructure. (This signaled the transition from the era of theory to the era of data.)

In the development of floodplain models, topographic features like levees and drainage channels figured prominently into model design, motivating researchers to conceptualize floodplains as a linked network of pools and channels, as opposed to a 2D continuum (Cunge, 1975; Lorgere et al., 1964; Zanobetti et al., 1970). The spatial domain was decomposed into a set of arbitrarily shaped subdomains, or cells, where water storage was computed, the water level could be approximated as horizontal, and 1D expressions describing the exchange of water between neighboring cells were developed in accordance with the exchange mechanism, such as a channel flow, a weir flow, or a gate flow (Cunge, 1975). The Froude number of floodplain flows is typically small, so 1D exchange terms for channels were developed under the assumption that inertial terms in the momentum balance could be neglected. This resulted in an explicit equation for exchange flows based on the difference in free surface height between neighboring cells, and factors that accounted for the cross-sectional area and other factors that affected flow. The resulting system of equations was of the parabolic type, making it well suited to solution by implicit finite difference methods, which, based on numerical stability, allowed for a relatively large and practical time step without overly sacrificing accuracy, although considerable care was needed to deal with complexities like small depths and to set up a system of linear equations (matrix problem) that could be efficiently solved (Cunge, 1975).

Numerical stability reflects the ability of numerical schemes to dampen high-frequency errors introduced by approximations of spatial and temporal derivatives in the governing equations. In particular, the centered difference approximation of the first spatial derivative, which provides second-order accuracy, introduces a so-called dispersive error proportional to the third derivative of the solution variable, and thus a centered-in-space, forward-in-time explicit solution (Euler Scheme) to the shallow-water equations is unconditionally unstable because there is no diffusive mechanism to dampen dispersive errors (Hirsch, 1988). Diffusion can be introduced through either the spatial or temporal finite difference approximation, or both; this generates so-called numerical diffusion. A pseudoviscosity can also be introduced to the governing equations to dampen spurious oscillations, but this is not viewed as sound modeling (Katopodes, 1984b). A physically meaningful diffusion mechanism can also be introduced to account for horizontal turbulent shear (Rastogi & Rodi, 1978), in which case the governing equations take on a mixed hyberbolic-parabolic type, and the discretization of diffusive turbulence terms may help to dampen dispersive oscillations associated with centered difference approximations of first derivatives. However, this approach is not recommended for urban flood modeling because horizontal shear forces have a negligible impact on solutions (see Mignot et al., 2006). Poorly designed computational solution methods may also require a turbulent viscosity parameter outside of a physically meaningful range in order to stabilize an otherwise unstable numerical solution (Gray, 1982).

Upwinding has proven to be among the most systematic means of stabilizing numerical solutions of hyperbolic equations, i.e., one-sided spatial derivatives with a diffusive truncation error, but this reduces the accuracy of models to first order. Thus, striking a balance between accuracy and stability has been a major theme in the development of computational methods for de St Venant equations.

Hydraulic engineers recognized the importance of inertia in the development of early models of tidal dynamics in well-mixed deltas and estuaries (Abbott et al., 1973). However, given the dominant role of topography in these systems, the 1D network floodplain modeling approach (see Zanobetti et al., 1970) was extended for tidal flows by accounting for inertia with only the local acceleration term in the momentum equation (Abbott & Cunge, 1975), i.e., neglecting the convective acceleration. A fully dynamic 2D flow model (with convective acceleration terms) was also developed for tidal channel flows and was solved by implicit finite difference methods, although the convective acceleration terms introduced nonlinearities that demanded greater attention from a numerical method perspective (Abbott & Cunge, 1975). Implicit finite difference methods remain popular today for 2D flow modeling and are the numerical engine of numerous software packages with 2D modeling capabilities, such as SOBEK (Deltares, Delft, The Netherlands), MIKE Flood (DHI, Horsholm, DK), FloodModeler Pro (CH2M, Englewood, Colorado), TUFLOW (BMT WBM, Brisbane, Australia), and HEC-RAS (Hydraulic Engineering Center, Davis, California). While the schemes vary in many details across available software, many models take advantage of the alternating direction implicit (ADI) method because this splits the 2D problem into a sequence of 1D problems that are far more efficient to solve (Casulli, 1990; Cunge et al., 1980). Historically, implicit schemes did not perform well in applications involving supercritical flow, hydraulic jumps, and wetting and drying fronts, but there has been research to address these limitations. To cite just a couple of examples, Casulli (1990) addressed the problem of spurious oscillations in implicit finite difference schemes by treating convective acceleration terms explicitly and remaining terms implicitly are treated explicitly, Balzano (1998) reviewed numerous methods for wetting and drying, and Stelling and Duinmeijer (2003) proposed a technique to improve modeling of hydraulic jumps and flow over rapidly varying topography.

The appropriate momentum balance for describing overland flow has been extensively researched, especially within irrigation modeling and overland flow modeling communities, leading to models without inertia, kinematic wave models, and diffusive wave models in particular (Hunter et al., 2007; Katopodes & Strelkoff, 1977; Ponce, 1991; Singh, 1997). Diffusive wave models based on explicit update equations (see Bates & de Roo, 2000) gained widespread use among researchers for modeling floodplain dynamics at reach scales characterized by many multiples of the channel width (Hunter et al., 2007), but these models are not well suited to urban flooding, even for very slow moving flows where inertial terms are negligible, because the stability criterion severely penalizes mesh refinement with a time step that scales with the square of the grid size (Hunter et al., 2006). That is, urban flood models generally resolve flow at relatively fine scales comparable to the width of flow paths, e.g., the scale of roads and buildings. In this limit, hyperbolic equations solved by explicit update equations are advantageous from a stability perspective because the maximum allowable time step scales with only the first power of the grid size. Bates et al. (2010) presented a model that ignores only convective acceleration in the momentum balance (local accelerations are retained, similar to Abbott & Cunge, 1975) and is solved by an explicit update scheme, and they showed that this leads to a more accurate and efficient model for urban flooding than a diffusive-wave flood model.

The safety concern over sudden releases of water following failure of a hydraulic structure (dam, levee, weir) motivated interest in so-called dam-break modeling, which involved highly unsteady flows with hydraulic jumps, both supercritical and subcritical flow, and water advancing over dry land. Dam-break flood waves are extremely complex flows characterized by high levels of turbulence, highly three-dimensional flows near shock waves, and entrainment of sediment and debris. However, shallow-water models have proven in both the laboratory and in the field to provide a good approximation of wave height, speed, and velocity when the volume of the flood and site topography are known (Begnudelli & Sanders, 2007b; Chen, 1980; Cunge, 1970; Gallegos et al., 2009; Valiani et al., 2002; Yevjevich, 1975). Bellos and Sakkas (1987) reviewed finite difference schemes for dam-break flood modeling and focused attention on the importance of directly discretizing the conservative form of the governing equations in dam-break applications, and the benefits of using explicit update schemes. Explicit schemes update the solution on a cell-by-cell basis and are consistent with governing equations of the hyperbolic type because information moves through the domain at a finite speed. Conversely, implicit numerical schemes update the solution in all cells simultaneously (by solving a matrix problem) and are consistent with governing equations of the parabolic type because the wave speed is infinite. However, what is theoretically advantageous is not always what becomes popular for practical applications. In particular, the FLDWAV model (Fread, 1988; Fread & Lewis, 1988), developed by the U.S. National Weather Service for dam-break flood prediction, is based on an implicit finite difference scheme.

Explicit schemes are conceptually simpler and easier to program than implicit schemes, especially for nonlinear problems, but they are at best conditionally stable in accordance with the Courant, Friedrichs, Lewy criterion (Courant et al., 1928), which states that the rate of propagation of signals in the difference scheme should be at least as large as the true maximum signal speed (Lax & Wendroff, 1960). On the other hand, implicit schemes can be unconditionally stable, and in fact Abbott and Ionescu (1967) developed an unconditionally stable scheme for solving the 1D shallow-water equations, but cautioned that stability did not guarantee accuracy—guidance also provided by Fread (1973) with respect to dam-break modeling. This remains an important message for consideration in urban flood modeling today: schemes that allow for a relatively large time step on the basis of stability may appear more efficient than models constrained by a relatively small time step, but stability does not guarantee accuracy, and when a flood front moves over dry land, the only way to promote accuracy is to track the front as it moves from cell to cell each time step.

Solution of the 2D flow equations by the Galerkin finite element method generated widespread research interest in the 1970s, 1980s, and 1990s, and formed the basis of several general-purpose software packages in use today, such as RMA (King et al., 1975; King & Norton, 1978; Norton et al., 1973), ADCIRC (Luettich & Westerink, 2004), and TELEMAC (Galland et al., 1991). The practical success of the Galerkin method is somewhat of a curiosity considering the number of reported inadequacies (Gray, 1982) and the finding by Dupont (1973) that the Galerkin method does not achieve optimal accuracy for hyperbolic equations as it does for parabolic and elliptic equations. Gray (1982) reported that Galerkin models typically rely on excessive numerical damping for stability in practical applications. Katopodes (1984a, 1984b) also showed that Galerkin methods produce spurious oscillations in the presence of a hydraulic jump, and introduced a variant of the Galerkin method with upwinding to achieve highly selective dissipation that affects only high-frequency oscillations. Lynch and Gray (1979) transformed the shallow-water equations into the so-called wave-equation form, which led to the suppression of short-wavelength noise when solved by the Galerkin method and became a foundation of the ADCIRC model (Westerink et al., 1992). TELEMAC combined the method of characteristics for acceleration terms with the finite element method for the remaining terms (Galland et al., 1991).

Figure 1 illustrates solutions to the problem of a surge wave moving along a rectangular channel, including the analytical solution, a Galerkin solution with spurious oscillations, and a pseudoviscosity solution that is overly diffusive, as first shown by Katopodes (1984b). In the development of dynamic flood models, a major goal has been numerical methods capable of sharply resolving surge fronts (Figure 1—analytical solution) without spurious oscillations (Figure 1—Galerkin solution) or nonphysical smearing (Figure 1—pseudoviscosity solution).

Robust models for wetting and drying also proved challenging to implement in the context of finite element schemes (Bates, 2000; Defina, 2000). Models were developed with adaptive grids that adjusted to the flooded domain (Lynch & Gray, 1980), but these did not easily translate to practical applications with complex topography, motivating improved schemes for fixed-grid models (Bates, 2000; Defina, 2000; Defina et al., 1994). Among important innovations was the concept of a subgrid model, which evaluated water surface area within a cell as a function of water level, and fine-scale topography, an innovation that would gain considerable traction in the era of data.

In the 1990s and 2000s, Godunov-type finite volume (GFV) methods emerged, with excellent properties for resolving hydraulic jumps and mobile wet/dry fronts (Guinot, 2003; Toro, 1999). GFV schemes use a piecewise solution structure, allowing for discontinuities between cells, and the interpretation of fluxes as the result of a Riemann problem, i.e., an initial value problem consisting of two constant states separated by a discontinuity, as proposed by Godunov (1959). GFV methods were popularized in the 1980s in aerospace applications after the introduction of approximate Riemann solvers, such as the Roe solver (Roe, 1981) and the Harten, Lax, and van Leer (HLL) solver (Harten et al., 1983), which cleverly upwinded information from neighboring cells to yield cell interface fluxes that produced monotone solutions (no spurious oscillations) irrespective of whether the solution was smooth or discontinuous. Second-order accuracy was achieved without producing spurious oscillations by the introduction of slope limiters, which prevented the introduction of new maxima and minima when data were reconstructed at cell edges for input into approximate Riemann solvers (Sweby, 1984; van Leer, 1979). Application of schemes developed for the Euler equations describing compressible gas dynamics to the shallow-water equations was straightforward for test cases involving a horizontal bed and a fully wetted domain, and Glaister (1988) presented one of the earliest GFV shallow-water models for the most basic shallow-water problem: a single 1D discontinuity (dam break) on a frictionless, horizontal, and fully wetted bed. However, variable topography, wet/dry fronts, and 2D extensions posed significant modeling challenges and occupied many researchers interested in developing general-purpose 2D shallow-water solvers useful for practical applications, especially because software available to practitioners during this time was generally not very good with respect to modeling the advance and recession of flood waves over irregular topography. Since GVF schemes directly discretize the conservative form of the shallow-water equations, horizontal pressure fluxes from the sloping channel bed and boundary of the computational cell must be precisely calculated, in the limit of a still water, to sum to zero so there is no numerical acceleration of the fluid. Hence, considerable research addressed the numerical treatment of fluxes and source terms (see Hubbard & Garcia Navarro, 2000; Liang & Marche, 2009; Zhou et al., 2002). Tracking of wet/dry fronts is especially complicated in the presence of variable topography, and modelers developed specialized procedures for computational cells at the interface between dry and wet topography. GFV schemes store the solution on a cell- average basis, while ground elevation is defined at the vertices of the cells, so it is possible for a computational cell to have a combination of wet and dry vertices and for the volume of water within a cell to be a nonlinear function of the water surface height (Begnudelli & Sanders, 2006, 2007a). Hence, many researchers focused attention on specialized procedures for such cells, with the goal of achieving exact local and global mass conservation properties while preventing spurious oscillations and promoting accuracy (Bradford & Sanders, 2002; Brufau et al., 2002, 2004; Song et al., 2011). For example, Begnudelli and Sanders (2006, 2007a) used a subgrid model to relate storage within each cell to water level in accordance with the ground elevation at mesh vertices and a horizontal free surface, a method that draws parallels with earlier work in finite elements (see Defina et al., 1994). The ability of Riemann solvers to provide a robust treatment of steep fronts and hydraulic jumps has also advanced finite element modeling with so-called discontinuous Galerkin schemes (Aizinger & Dawson, 2002; Ern et al., 2008; Schwanenberg & Harms, 2004). Available flooding software that uses Godunov-type finite volume schemes includes InfoWorks (Innovyze, Bloomfield, Colorado) and TUFLOW FV (BMT WBM, Brisbane, Australia).

The era of theory produced many schemes to solve dynamic wave equations and to describe 2D floodwater spreading in urban applications, each with different strengths and weaknesses, but none strong enough to produce one undeniably superior approach. However, some general findings are as follows. First, urban flooding focuses attention on relatively fine scales, on the order of meters, where inertia is important and cannot be ignored if the goal is to predict variability in depth and velocity in both space and time. In particular, assessment of damage and safety risks to pedestrians and vehicles requires accurate velocity predictions (Martinez-Gomariz et al., 2016; Xia et al., 2015). In many situations, urban flooding promotes supercritical flow as a result of steep slopes and smooth surfaces like concrete (Mignot et al., 2006). However, when the Froude number is relatively small (Fr < 0.5) so that flooding is exclusively subcritical, the convective acceleration terms can be ignored, but local acceleration needs to be resolved in addition to bed slope, pressure gradient, and frictional effects (Bates et al., 2010; de Almeida et al., 2013; Lin et al., 2006; Yu & Lane, 2006a). Indeed, it is possible to map certain attributes of an urban flood, such as flood extent, using relatively simple models with limited physics if conditions only slowly vary in space and time, and while these models can easily predict localized depth by subtracting localized ground elevation from a regional estimate of water surface elevation, velocity prediction demands a full momentum balance. Second, history shows that many different numerical methods can be used to solve flow equations describing urban flooding with reasonable accuracy (Hunter et al., 2007), and finite difference and finite element schemes with implicit time stepping offer the potential for relatively large time steps compared with explicit schemes that are constrained by the CFL condition. However, urban flooding can occur relatively quickly and can be highly unsteady as a result of levee failures, intense rainfall, culvert blockages, and other factors resulting in wave fronts that dynamically spread across the land surface. In these cases, the CFL criterion will be applicable for accuracy purposes irrespective of whether it is required for stability. Implicit schemes are often preferred for generating steady-state solutions because transients are not of interest, but urban flooding is rarely well characterized by a steady state. Third, despite extensive research, there is no generally accepted technique to model wetting and drying fronts in a robust way in dynamic flooding models. The wetting and drying treatment can cause fluid conservation errors and generate oscillations. Solutions methods typically involve a number of logical statements that are dependent on details of the numerical method, including the type of grid, and thus the wet/dry procedure is not easily transferred from one numerical model to another. Models may appear robust based on benchmark test cases but then break down in practical applications based on some unforeseen conditions, such as cells with steep slopes exceeding unity. Even schemes that theoretically should produce monotone predictions may yield oscillations that originate from the wet/dry interface. Extensive experience with a model is therefore required to fully understand the overall stability and accuracy with respect to wetting and drying and to gauge whether model performance is adequate for meeting the modeling goals. Signs that the model is not performing well include relative mass conservation errors greater than 0.1% (GFV schemes can easily achieve relative errors less than 1 × 10^{−7}), the appearance of floodwater in areas that are hydraulically disconnected from water sources, and spurious oscillations in depth and/or velocity predictions.

The Era of Data

Areal laser scanning of topographic heights was popularized in the 2000s and led to metric resolution topographic data with centimetric vertical accuracy, a resource that would catalyze rapid progress in the field of urban flood modeling (Hunter et al., 2007; Marks & Bates, 2000) because data at this resolution and over whole-city scales were previously too expensive to be collected by ground surveying and photogrammetry (Bates, 2012; Bates et al., 1992; Hunter et al., 2007). So-called raster flood models, named after the image file format found useful for storing ground elevation data, were popularized whereby unsteady flow calculations were made on a Cartesian grid aligned with elevation data (Bates & de Roo, 2000; Bradbrook et al., 2004). The availability of field data would also create, for the first time, the ability to examine the performance of urban flood inundation models at field scale under realistic flooding conditions and systematically to assess the factors that contribute to flood model uncertainty (Bates, 2012; Hunter et al., 2007). The ability to model flooding and to assess predictions also allowed researchers to think systematically about sources of model uncertainty, including topographic heights, resistance parameters, unknown flows, and model structure, work that deepened appreciation for the importance of high-quality topographic data for practical application of urban flood models (Bates, 2012; Hunter et al., 2008), and to some extent marginalized the importance of using a particular numerical scheme (see Hunter et al., 2007) and of higher-order schemes (see Begnudelli et al., 2008). Metric resolution fine-scale urban flood modeling was also found to be extremely demanding of computational resources at the whole-city scale, which motivated considerable interest in parallel computing on distributed memory clusters (Sanders et al., 2010) and graphical processing units (GPUs) (Brodtkorb et al., 2012; Castro et al., 2011; Vacandio et al., 2014), but also motivated a desire for less computationally intensive modeling approaches (Cea & Blade, 2015).

It could be argued that progress in hydrodynamic urban flood modeling in the era of data was catalyzed by the study of extreme flooding in Nimes, France, by Mignot et al. (2006). This was one of the first field-scale studies involving the solution of the 2D de St Venant equations on a computational grid designed to resolve flow along the road network. In this case, exogenous factors caused flood flows to enter the model domain from multiple locations along the model boundary, and the configuration of the road network, combined with sloped topography, generated a transcritical flow regime with localized areas of subcritical and supercritical flow. Inflow hydrographs were developed from the application of a hydrologic model to transform rainfall rates to runoff, and the sewer capacity was subtracted to estimate street flows. Topography data were supplied as 200 ground survey transects spanning 60 streets, with each transect consisting of 11 spot heights to resolve the curb height, gutter height, and road camber. Two flood events were presented, whereby the first was used for model assessment and sensitivity analysis, and the second was used for validation. Flood observations included 99 and 28 watermarks for the first and second events, respectively, and a crude flood extent map with limited research utility was presented for the first event.

Mignot et al. (2006) modeled flow by solving the shallow-water equations with a GFV scheme capable of resolving transcritical flows and wetting and drying, using an unstructured grid of quadrilateral and triangular cells whereby mesh edges were aligned with the edge of densely developed buildings that were assumed to block flow completely. Schubert et al. (2008) would later term this the building hole method, one of several ways of representing buildings in urban flood models. For mesh vertices between available transect data, topographic heights were linearly interpolated longitudinally from neighboring transects, thus preserving a distinct road-transect shape across the model domain. The mesh resolution was metric, with roughly 5-m lengths in the along-road direction and smaller (and in some cases submetric) resolution in the cross-road direction, which varied with the street width so as to maintain a fixed number of cells spanning the width of the road. A reference flood simulation was developed using a spatially uniform Manning resistance coefficient and eddy viscosity, sensitivity analysis was completed by considering several other model scenarios (parameters, boundary conditions, and mesh design), and repeatability was examined by applying the model to a second, smaller flood event.

The base case produced depth predictions with a maximum error of 1.6 m and average absolute height error of 0.41 m. With maximum flood depths mainly less than 2 m (in a few locations, maximum flood depth exceeded 2 m), this represented roughly a 20% height error. This result was promising. Prior to this study, localized modeling of transcritical flows at the city scale had not been attempted, yet this result showed that it could be done to yield reasonable information. Sensitivity analysis was also important because it showed that many factors contributed significantly to model uncertainty with depth changes of 10 cm or more: uncertain inflow hydrographs, mesh resolution, uncertain resistance parameters, interactions with sewers, and downstream boundary conditions. Topography measurements were not explicitly considered as a source of uncertainty; instead, they were considered indirectly through mesh coarsening, which acted to flatten out road transects because fewer transect points were sampled. Use of at least three to four wetted cells across each road was recommended for extreme flooding scenarios, being large enough to model conveyance accurately. It was also reported that the mesh need not be so fine as to resolve street gutters when the goal is to analyze extreme flooding. Resistance parameters were adjusted in a spatially distributed way and were shown to redistribute the flow, for example shifting areas of supercritical and subcritical flows as well as depths. Only one factor could be disregarded for lack of sensitivity: the eddy viscosity. When the model was applied to the second flood event, the relative error was similar.

This study was important for several reasons. First, it validated years of research into improved numerical methods for the shallow-water equations: it showed the ability of GVF scheme to simulate a very complex urban flooding regime marked by complex site geometry, transcritical flows, and wetting and drying. Second, it showed that many factors contribute to the uncertainty of the model yet it provided no clear insight into how to make the model more accurate—a call for more research in focused areas, such as topographic representation, building representation, resistance parameterization, coupled sewer modeling, and road intersection modeling. Third, this study was completed without the benefit of high-resolution geographical data, such as lidar data, building footprint polygons, and road polylines. As a result, the study captured the imagination of researchers interested in applications of these data for improved urban flood modeling. Fourth, the study showcased ways of using other models in coordination with 2D urban flood models, for example hydrologic models that transform rainfall to runoff, and sewer models that work to divide runoff into overland flow and storm sewer flows.

Whereas Mignot et al. (2006) approached urban flood modeling using traditional topographic data sources, Hunter et al. (2007) approached urban flood modeling based on the availability of a high-resolution digital elevation model (DEM), which was becoming increasingly available as a result of aerial laser scanning. Modeling was based on a 2-m resolution DEM of a neighborhood near Glasgow, Scotland, created from an aerial lidar survey, with heights representative of either ground elevation or building rooftops. The flow scenario corresponded to a common situation where the formal drainage system is overwhelmed (in this case, a culvert) and surface flows discharge across the road network from the water source. Hence, modeling focused on predicting the spreading of surface flows from a point across the sloping neighborhood. Further, several predictions were compared based on different numerical models, including implicit finite difference schemes and finite volume schemes that solved the full shallow-water equations (dynamic models), and finite difference schemes that solved the diffusive wave equation (zero inertia). Each model was run on a 2-m resolution Cartesian grid to match the available topographic data, the so-called raster-based modeling approach popularized by Bates and de Roo (2000), which avoids the high levels of preprocessing (mesh generation, etc.) typically used in 2D shallow-water modeling. The study validated the importance of inertia when modeling flows along sloping roads, as the results of dynamic wave models differed markedly from diffusive wave (zero-inertia) models, and the diffusive wave model was (ironically) far more computationally demanding due to its stability criterion. The work also showed that differences among dynamic wave models were comparable to differences that could be attributed to uncertainty in the topographic data, which in many ways shifted attention in model development away from the need for improved numerical methods for the shallow-water equations and toward improved methods for systematically using available geospatial data and managing overall model uncertainty arising from numerous factors, such as uncertain hydrologic forcing, topography data, resistance parameters, sewer flows, and interactions with buildings (Bates, 2012; Dottori et al., 2013; Hunter et al., 2008).

Along the theme of data-driven urban flood model design, Schubert et al. (2008) presented a scheme to automate unstructured mesh design and parameter estimation within a geographical information system (GIS) containing several data layers, including a lidar-based DSM, lidar-based DTM, lidar-based vegetation thickness data, a land-surface classification based on aerial imagery, and building footprint polygons derived from the lidar-based DSM. Building footprint polygons were used as input into Triangle (Shewchuk, 1996), an open-source constrained Delaunay mesh generator, which allowed for precise representation of building geometry over a wide range of grid resolutions, as shown in Figure 3, and interpolation schemes were used to estimate topographic heights at mesh vertices and resistance parameters at cell centers based on the DTM and spatially distributed Manning resistance parameter fields, respectively, where the latter were derived from a combination of land surface classification and vegetation height data. The Glasgow test case considered by Hunter et al. (2007) was revisited by Schubert et al. (2008), and while model predictions compared well with other dynamic wave model results presented by Hunter et al. (2007), the study showed that computationally efficient unstructured grid models can be built in a semi-automated way using GIS.

Gallegos et al. (2009) presented one of the first fine-scale models of an urban dam-break flood, the 1963 failure of the Baldwin Hills dam in Los Angeles, California. Previous modeling studies of urban dam-break flooding had been limited to physical model studies at the laboratory scale, where inflows were carefully controlled and buildings were modeled using solid blocks (see Testa et al., 2007). In contrast to the Nimes, France, neighborhood studied by Mignot et al. (2006), buildings in the Baldwin Hills neighborhood were sparsely spaced, allowing for significant flow between them, and in fact several structures were completely washed away. Hence, Gallegos et al. (2009) accounted for buildings using a spatially distributed Manning resistance parameter, whereby low values representative of concrete were used along roads and a value of 0.3 was assigned to all land parcels with a building, a value recommended by the U.S. Army Corps of Engineers for modeling flooding in urban areas (USACE, 1981). Model development in GIS was another theme of the work, including use of parcel boundary polygons and road network polylines to guide unstructured mesh generation, and use of aerial imagery and parcel boundary polygons to create a land surface classification that guided the spatial distribution of the Manning resistance parameter. Results showed that a complex flood extent pattern characterized by fingering of water along roads was predicted, with a flood agreement metric of 79%, defined as the intersection of the predicted and measured flooded area divided by the union of the predicted and measured flooded area. Results also showed that predictions and measurements of the discharge hydrograph downstream of the flood zone compared well, with peak discharge error of only 10% and the timing of the peak off by less than 10 minutes. Aside from showcasing the benefits of integrating GIS and computational modeling from a model development perspective, this study also identified a number of model sensitivities that aligned with the findings of Mignot et al. (2006) in the study of Nimes flooding. For example, a minimum of three wetted cells across each road was recommended for accuracy purposes, spatially distributed resistance parameters were found necessarily to accurately predict both flood extent and the discharge hydrograph, metric resolution topographic data (lidar) were needed to resolve the preferred conveyance of water along roads, and a model of sewer flows was also found to be important. Gallegos et al. (2009) used a simple model for sewer flows involving point sinks at the location of curb inlets and point sources at the location of storm sewer outlets to open channels resolved by the 2D model. Inflow rates were computed using either a weir-type or orifice-type equation, depending on the local depth of flow, and were instantaneously routed to the drainage channel, in proportion to a calibrated discharge coefficient. This assumes that sewer flow rates are limited by the capacity of curb inlets, and that routing, which physically occurs in a matter of minutes, is approximated as instantaneous. Hence, Gallegos et al. (2009) avoided solving 1D network flow equations typically associated with storm sewer models like SWMM (Rossman & Huber, 2016). Obviously, a more complex sewer flow model could be used, but this was found to be a simple and convenient approximation that could be easily calibrated by available flood extent data because the method introduced only one new parameter, a city-scale weir/orifice discharge coefficient.

The Baldwin Hills test case supported two follow-up studies that deepened understanding of the predictive power of, and best practices for, urban flood models. Gallegos et al. (2012) showed that urban flood models can be extended to predict structural damage to buildings from dam-break flooding with the aid of empirical damage functions that are scaled by depth and velocity. In particular, Gallegos et al. (2012) showed that structural damage to wood-framed residential structures was best predicted by the inertial momentum flux, and that two-way coupling between the flow model and damage model was important because the failure of structures is followed by a reduction in localized flow resistance. Given the prominence of buildings in cities and their effect on flood dynamics, Schubert and Sanders (2012) presented four different methods of representing buildings in urban flood models and evaluated their performance with respect to flood extent, flood velocity, and flood hydrograph predictions. The four methods of building representation, known as building resistance (BR), building block (BB), building hole (BH), and building porosity (BP), required different mesh designs, which are shown in Figure 4. Schubert and Sanders (2012) found that all four methods were capable of accurately predicting flood extent and flood discharge hydrographs, but only a subset of the methods were successful at predicting localized velocities along roadways. Overall, BP was the best performing method, considering both accuracy and computational effort in the Baldwin Hills test case (Sanders et al., 2008).

The application of porous media concepts in urban flooding is especially promising from a research perspective because it is one of the only application areas where a heterogeneous and anisotropic porosity distribution can be deterministically characterized, i.e., using geospatial land surface data. In geological applications, by contrast, the porosity distribution must be inferred from sparsely sampled soil and fluid measurements. The concept of porosity in flood modeling was first introduced in the 1990s (Defina et al., 1994) where it impacted finite element modeling of wetting and drying (Bates, 2000; Defina, 2000), but it was the availability of high-resolution topographic data, at scales finer than practical grid resolutions for city-scale modeling, that catalyzed further research. Yu and Lane (2006b) presented a porosity-type treatment of both storage and conveyance effects for a raster-based diffusive wave model of flood inundation, which effectively involved summing over the individual raster cells and cell edges within a larger computational cell to determine the effective storage and conveyance, respectively. An attractive feature of this approach is that topographic variability and building effects on flow are both captured by the porosity distribution, and use of both cell- and edge-based porosities introduces a mechanism for handling anisotropy in the porosity distribution. The porosity model led to much greater accuracy in relatively coarse grid flood predictions in comparison to coarse grid models based on the average topographic height in each cell, and at far less computational effort than a model run at the scale of topographic data. McMillan and Brasington (2007) also evaluated the performance of a porosity treatment for a diffusive wave model and drew similar conclusions. Hydrodynamic modeling of flooding with porosity parameters first focused on depth-independent building effects, not topographic variability, and high-Froude-number flows associated with dam-break type flooding. Guinot and Soares-Frazao (2006) and Soares-Frazao et al. (2008) presented a single porosity model to account for both storage and conveyance effects that revealed the potential for significant model speedup and a promising level of accuracy compared to fine-grid models, but the model was limited to isotropic blockage distributions and the porosity parameter was not easily inferred from blockage features because of differences between storage and conveyance attributes. Two methods to accommodate anisotropic porosity distributions include the integral porosity method (Sanders et al., 2008) and the dual porosity method (Guinot, 2012). With the integral porosity method, the porosity distribution is a deterministic function of the intersection of the computational mesh with building footprints, similar to raster-based porosity models (see Yu & Lane, 2006b).

The integral porosity method of Sanders et al. (2008) has been found to be more efficient in modeling urban dam-break floods in field-scale applications than building-resolving models (Schubert & Sanders, 2012), considering both computational cost and the ability to predict localized depths, velocities, and flood extent, but it demands a higher level of model preprocessing. Based on laboratory scale dam-break modeling, the integral porosity method has also been found to perform better than single porosity models in applications involving strongly anisotropic flow obstructions (Kim et al., 2015). A key advantage of the integral porosity and dual porosity models over single porosity models is that a representative elemental volume (REV; Bear, 2013) need not exist (Guinot, 2012; Sanders et al., 2008), which is important because the scale at which REVs exist on urban floodplains can be hundreds of meters or even kilometers, far greater than scales of practical interest (Guinot, 2012). A remaining challenge in urban flood modeling with porosity arises from what are by definition mesh-dependent solutions. A fundamental tenet of computational modeling is use of grids that are sufficiently refined that truncation errors associated with grid resolution are negligible, and with the integral porosity model, the mesh cannot be refined without changing the model parameters. Kim et al. (2015) showed that errors in the integral porosity model can be divided into structural model errors associated with the classical shallow flow approximation, porosity model errors associated with upscaling, and numerical truncation errors associated with a finite resolution mesh, and they found that by using a reference solution, such as a fine-grid solution of the shallow-water equations, porosity model errors can be quantified and reduced to smaller than structural model errors through drag parameter calibration. Hence, the traditional concept of convergence is replaced with the idea that porosity model errors and numerical truncation errors should be smaller than structural model errors. Ozgen et al. (2015) present an integral porosity model that accounts for both topographic variability and building effects, which further generalizes the integral porosity model of Sanders et al. (2008).

Defina et al.’s (1994) concept of porosity also inspired improved implicit finite volume models that can account for topographic variability. In particular, Casulli (2009) showed that an unstructured grid finite volume model could be designed upon fine-resolution topographic data that parameterizes a porosity subgrid model similar to Defina et al. (1994). The subgrid model supports use of a relatively coarse, and computationally efficient, grid compared to the size of channels and storage features.

Quadtree gridding is another promising development for shallow-water modeling in conjunction with raster formatted elevation data (Liang et al., 2008). Quadtree grids are based on successively, and sometimes adaptively, quartered Cartesian grid cells to support localized refinement in a structured way, with the finest level of refinement matching the raster file resolution. On relatively coarse grid cells that contain many raster data points, a technique to use subgrid scale elevation data is needed. Stelling (2012) adopted the quadtree method of Wang et al. (2004) and the porosity model of Casulli (2009) to develop a general-purpose hydrodynamic flood model designed to run directly on raster files. Stelling (2012) also showed that subgrid scale elevation data can also be used to model flow resistance in way that would be consistent with a fine-scale model. In a practical application involving flooding of a polder area in the Netherlands, it was reported that the subgrid model enabled a 100-fold reduction in run times while achieving the same level of accuracy. One drawback with all porosity models that upscale raster data is the need for lookup tables for each computational cell and possibly each edge connecting neighboring cells, and interpolation schemes to extract information on the fly from the tables (Stelling, 2012). These features will increase the memory requirements and computational effort of a model beyond that of a 2D model of the same resolution that doesn’t use a subgrid model, but advantages of the subgrid model in terms of accuracy gained at coarse resolution should outweigh the disadvantage of additional computational effort and memory, as was found by Stelling (2012).

In summary, porosity subgrid models provide the opportunity to run hydrodynamic urban flood models at greater spatial resolution, yielding far greater efficiencies and the ability to model greater spatial extents, but also introduce more calibration parameters than classical dynamic flood models (Kim et al., 2015; Sanders et al., 2008), and introduce scale ambiguity into the interpretation of velocity predictions. Porosity subgrid models may also require lookup tables and additional interpolation, which increase memory and computational demands, respectively, although these costs should be more than outweighed by efficiencies gained through grid coarsening (Stelling, 2012).

A final and essential progression associated with the era of data, one that is important for decision support, is the coupling of 2D flooding models with other models as needed to characterize flooding dynamics at the whole-city scale in a systematic way. Coupling of 2D flooding models with 1D drainage models has received considerable attention, as reflected by the present availability of commercial software (Henonin et al., 2013). Field studies have shown that 2D models alone cannot accurately predict extreme event flood attributes (depth, velocity) without consideration of so-called dual drainage pathways (see Gallegos et al., 2009; Kim et al., 2014; Mignot et al., 2006). Coupling of 2D flooding models with hydrologic models that account for hillslope processes that transform rainfall to runoff has also been the focus of research (Bellos & Tsakiris, 2016; Kim, Ivanov, & Katopodes, 2012; Laganier, Ayral, Salze, & Sauvagnargues, 2013; Nguyen et al., 2015), as has the coupling of 2D flooding models with ocean and wave models to address coastal flood risk (Bacapoulos & Hagen, 2014; Dietrich et al., 2012). When modeling urban flooding, the zone of flooding often occurs at a much smaller scale than the region over which the flood event develops, such as the case with fluvial flooding and coastal flooding. Such exogenous causes of flooding can often be addressed with loosely coupled modeling systems where the output of a regional model, or even field measurements, can be used to force a localized flood model covering the domain where flood intensity data is needed (see Gallien et al., 2012). On the other hand, endogenous causes of flooding (i.e., pluvial flooding) typically require more tightly coupled 1D/2D modeling systems (Henonin et al., 2013; Hsu et al., 2000; Lin et al., 2006; Schmitt et al., 2004). Relatively little research has addressed the uncertainty of 1D/2D models in the context of extreme flood events with city-scale (or even neighborhood scale) field data, especially for the purpose of deepening understanding of the process representation needed to predict flood impacts over a wide range of flood severity. For example, the level of process representation needed to predict low levels of flooding, or nuisance flooding, that disrupt traffic and temporarily interrupt business activity will be different from the process representation to predict extreme flooding caused by a dam or levee failure. The formal drainage system can be expected to dominate flood conveyance in the first case, while overland flow along streets can be expected to dominate in the latter. Accurately predicting low levels of surface flooding, i.e., below curb levels, is ironically more demanding than predicting extreme flooding (Sampson et al., 2012).

The most widely used 1D modeling approach for 1D/2D drainage models is the link-node approach of SWMM (Rossman, 2015), which resolves bulk exchange between neighboring junctions and does not account for high-frequency transients, which are known cause significant storm water management challenges, such as guysering (Vasconcelos & Wright, 2005). Models with transient modeling have been developed to account for both the air and liquid phase (Sanders and Bradford, 2010; Vasconcelos et al., 2006), and Fraga et al. (2017) showed the integration of dynamic, transient-resolving 1D models with a 2D flooding model. When and where this level of process representation is appropriate for DRR in light of uncertainties about flooding is unclear, and deserves of more research.

The Era of Disaster Risk Reduction

The era of DRR is only beginning. GIS tools that enable users to simulate floods, understand impacts, design interventions, improve emergency management, and enhance disaster recovery (e.g., Kulkarni et al., 2014) are only in their infancy. Wilkinson et al. (2015) presented a cloud-based tool for information access and knowledge exchange on local flood risk; the tool is designed for access and integration of data, models, and visualization capabilities to help stakeholders with scientifically informed management at the local scale. Price and Vojinovic (2008) envisioned GIS as a focal point for DRR activity, and Mackay et al. (2015) suggested that GIS enhances opportunities for knowledge exchange between scientists, policy makers, and local communities regarding a broad range of environmental issues. As of 2017, popular ArcGIS software (Environmental Systems Research Institute, Redlands, California) is being offered as a cloud-based tool for visualizing and sharing geospatial data over many different devices, making information sharing easier than ever. For example, flood modeling can be supported not only with improved access to information, such as DEMs, aerial imagery, land surface classifications, and storm sewer data, but also with the ability to upload and share model predictions. Figure 5 shows the application of Google Earth to visualize a coastal flood zone in Newport Beach, California.

It is important not to overstate the capabilities of urban flood models and GIS in the context of DRR. * GIS provides a mechanism to share information; it is far from a panacea for flood risk management*. For example, Leskens et al. (2014) found that a 1-m resolution Sobek (Deltares, Delft, The Netherlands) flood model supported by expert users was of little use in a disaster management exercise because information about field conditions and decision-making priorities evolved faster than the model could be configured and run. Leskens et al. (2014) also cautioned that the complexity of organizational structures in disaster management may even hinder the effectiveness of future models that can be executed much faster. On the other hand, given the important role that urban flood models can play in helping communities to identify interventions and their cost-effectiveness (see Lane et al., 2011), GIS is clearly indispensable for improving access to information.

The era of DRR will offer an important opportunity to review and redesign urban flood models based on what decision makers need to manage flood risk. In the era of theory, developers of flood models focused on minimizing differences between numerical and exact solutions of the governing equations, and differences between measurements and models at the laboratory scale. Models could not be set up and validated at the whole-city scale for lack of input data and assessment data. The era of data ushered in these opportunities, and model developers focused attention on parameter uncertainties, especially arising from topographic data, and also the limited skill with which urban flood models can be validated (Bates, 2012; Dottori et al., 2013; Schumann et al., 2009). Overall, the era of data marginalized the utility of models offering high formal accuracy and drew favor upon models adapted to the complexity of urban flooding applications exemplified by highly irregular topography, dynamic wetting and drying fronts, transcritical flows, and junctions with drainage infrastructure. How the era of DRR will affect urban flood model development remains to be seen, but surely emphasis will be placed on model responsiveness, which will put a premium on model execution speed. The capacity to support innovative risk communication strategies is a another consideration, and one that may put greater emphasis on the ability to characterize flood dynamics (i.e., evolution of flooding conditions over time) and to characterize many different aspects of the flood, such as flood extent, flood depths, flood velocities, and flood durations, in order to meet a wide range of decision-making needs. Last, it is possible that new modeling conventions will emerge. For example, Lane et al. (2011) reported that a community decision-making process was successfully shaped with the aid of an urban flood model, even though traditional model calibration and validation conventions were not followed. In general, urban flood models are sought to aid in the avoidance of disasters, and typically without the data required to calibrate and validate the model. Furthermore, whereas mitigation and preparedness aspects of DRR typically rely on probabilistic floods that cannot be validated, early warning and emergency response aspects of DRR demand realistic, real-time predictions, for which confidence in predictions is critical.

In light of the great variety in social and hydrological context setting that can be expected across the flood- vulnerable cities of the world, we should expect that urban flood modeling software will evolve with the capacity to adapt to local context, offering users different levels of mathematical and numerical approximation and resolution depending on decision-support needs, and methods of communicating uncertainty (see Bevin et al., 2015). We can also anticipate that in most urban settings, there will be diverse needs for model information. For example, the needs of city planners and the needs of emergency managers are likely to be quite different based on job responsibilities, and the model needs for communicating to the public may be quite different from the model needs for supporting specialized job functions related to flooding. The future of urban flood modeling as a discipline, therefore, calls for modeling personnel who not only are knowledgeable in traditional domains, such as flood hydrology, engineering mechanics, computational methods, and computing and information systems, but also are skilled at site investigations and community engagement to deepen understanding of local context and to show leadership in the coproduction of local flood models that offer a knowledge-to-action pathway toward flood resilience.

Concluding Remarks and Outlook

Hydrodynamic urban flood models progressed from (a) the era of theory, where governing equations were analyzed and numerical methods were developed; to (b) the era of data, when city-scale modeling became possible, allowed for a systematic investigation of uncertainty, and inspired new data-driven model formulations; and to (c) the era of DRR. It is not possible to provide a comprehensive review of available software, but it is important to deepen understanding of the theory that underpins software, as well as software strengths and weakness, and to increase the impact of future model development research by framing the context within which models can be applied for DRR, because flood risk threatens more people than any other natural hazard, and climate change portends an escalation of impacts, especially from sea level rise, urbanization, and population shifts from rural to urban areas. Additionally, the practice of flood risk management is in the process of adapting to the routine use of software with 2D modeling capabilities (Lane et al., 2013), a process that will surely accelerate with freely available flood modeling software with 2D capabilities, such as HEC-RAS (Hydraulic Engineering Center, Davis, California). Hence, the potential exists for 2D modeling software to achieve unprecedented levels of impact in the context of DRR initiatives, including support for flood mitigation measures, flood preparedness, early warning systems, real-time impact assessment, emergency response, and recovery. However, a critical issue to be confronted by scientists and engineers in the application of modeling software is that *a lack of community knowledge about flood hazards and potential impacts*, *precisely the information that can be described by 2D flood modeling software, is not the key issue in DRR*; in fact, disaster risk continues to grow despite increasing knowledge about flood hazards (Spiekermann et al., 2015). The key issue, rather, is the knowledge-to-action transformation, which “involves risk interpretation and understanding, mentalities across scales, power structures, personal attitudes, world views and budget constraints” (Spiekermann et al., 2015). Furthermore, the way forward, according to Spiekermann et al. (2015), is a more socially robust and context-sensitive paradigm of DRR involving the consideration of local priorities, both perceptions and factual depiction of community needs, and improved, target-oriented methods of communication. In short, there is a need for a high level of community engagement that includes a mechanism to integrate local and expert knowledge within local social priorities and constraints. Lane et al. (2011) provided an example wherein scientific expertise and lay expertise came together in a community that faced a high flood risk but was unable to move forward with action, and, most importantly, *a 2D urban flood model served as an invaluable tool for overcoming an impasse by acting as a trusted focal point for thinking through alternative decisions*. Hence, advances in urban flood modeling are expected to have the greatest impact on DRR to the extent that they are used to facilitate the transformation of knowledge to action.

Acknowledgments

This work was made possible by a grant from the National Science Foundation (DMS-1331611), whose support is gratefully acknowledged. The author expresses thanks to J. Schubert for assistance with graphics, to R. Matthew for stimulating dialogue about the role of simulation models in disaster risk reduction, and to N. Katopodes for an introduction to computational methods for channel flows.

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