A novel efficient and robust treatment of the friction source term in 2D shallow water inundation models

Barbero G., Costabile P., Costanzo C., Ferraro D., Petaccia G.

• Use of 2D-SWEs model to provide evidence on the variability of the lag time in small basins. • Lag time estimation on the basis of characteristic times influencing the hydrodynamic response. • Satisfying results in relation to the observed lag times. • Competitive predictions in respect to literature formulas. The use of integrated flood models represents an approach of growing interest in the literature, being the hydrological and hydrodynamic flood processes described entirely within the two-dimensional (2D) hydrodynamic unsteady flow equations. Due to its ability in simulating complex spatial–temporal dependence of both the hydrodynamic and hydrologic responses of a catchment to rainfall events, this kind of approach paves the way to the development of novel lines of research in catchment hydrology. With particular reference to the lag time estimation, the paper focuses on three interrelated issues: 1) the use of the 2D-SWEs model to provide evidence on the variability of the lag time in small basins, 2) the description of the hydrologic response of small catchments based on characteristic times influencing the hydrodynamic response to rainfall events and 3) the representativeness of hydrodynamic-based rainfall-runoff scenarios in the description of the hydrologic response observed in real events. To that purpose, synthetic rainfall-runoff scenarios for different return periods are firstly generated to analyze the hydrologic response time in three ungauged basins. Then regressive formulas, based on ad hoc variables representing characteristic times, are introduced to interpret the variability shown by the computed lag time. Finally, the physical soundness of the proposed approach is tested against observed rainfall-runoff data in four additional basins. Despite all the simplifications introduced in the proposed approach, the regressive formulas provided a reasonably good agreement with the estimations of the lag time of the observed events, showing errors of the order of 30%. The comparison with the predictions of empirical literature formulas further confirms the potential of the proposed approach. This study represents the basis for future research on basin response to rainfall events through the use of a 2D integrated hydrodynamic-hydrologic modeling.

Martínez-Aranda S., Murillo J., Morales-Hernández M., García-Navarro P.

• Novel explicit upwind approaches are proposed for 2D basal resistance discretization. • The classical explicit upwind discretization generates mesh-alignment in the flow. • The implicit centered method for quadratic basal resistance fails in steady states. • The proposed explicit upwind approaches ensures mesh independence in realistic flows. In the context of two-dimensional models for complex geophysical surface flows such as debris flows, muddy slurries, oil spills over land, hyperconcentrated floods, lava flows, etc, depth-averaged rheological models relate the shear stress state within the fluid column to the depth-averaged local flow features. Despite it is the most influencing term on the mobility of complex shallow flows, the numerical treatment of the resistance contribution to the flow momentum is still a challenging topic, especially when dealing with 2D large-scale applications. In this work, two novel strategies for the explicit upwind discretization of generalized non-Newtonian resistance terms in two-dimensional numerical models are proposed, called integral and differential approaches. These new strategies are applicable to generalized rheological formulations in any type of mesh topology. Results from benchmark tests running in orthogonal, triangle structured and triangle unstructured meshes demonstrate that both approaches represent an improvement for the explicit upwind integration of the 2D resistance force compared with previous procedures. It is shown that the alignment of the flow with the mesh main-axis, which has been previously attributed to faults of 2D FV numerical methods and insufficient mesh refinements, is directly related to the loss of the rotational invariance of the integrated resistance force. This is caused by the erroneous procedure for including the 2D resistance term into the local flux balance at the cell edges. Furthermore, a novel implicit centered method for the integration of the 2D resistance force has also been derived for the quadratic frictional non-linear resistance formulation. Despite the implicit procedure fails to converge to steady uniform flow states, the differential explicit upwind and the implicit centered methods show similar level of accuracy, robustness and computational efficiency for transient 2D frictional visco-plastic flows.

Cea L., Vila G., García-Alén G., Puertas J., Pena L.

Zhao J., Liang Q.

• New variable reconstruction schemes are derived in the context of a Godunov-type finite volume method for simulation of shallow overland flows. • Key numerical challenges in the simulation of overland flows are discussed in detail and addressed by the new simulation schemes. • The new variable reconstruction and source term discretisation schemes are implemented to develop first- and second-order hydrodynamic models. • The performance of the new numerical schemes is demonstrated through application to theoretical and real-world test cases. Due to the recent advances in computing and data acquisition technologies, numerical models solving the 2D shallow water equations (SWEs) are now widely used in predicting overland flow and surface water flooding process in natural catchments. In catchment-scale flood modelling, in addition to the calculation of convective terms that are usually achieved through solving local Riemann problems to capture transient flow dynamics, correct discretisation of source terms is also essential to handle complex domain topography and ensure accurate and stable numerical solutions. The external forces induced by gravity and bed friction create the key source terms that drive the change of momentum in the SWEs and affect the stability and accuracy of the adopted numerical scheme, especially for applications involving very shallow water depth (e.g. wet and dry fronts, overland flows). Herein, new schemes are introduced to reconstruct the face values of flow variables and a fully implicit algorithm is improved to discretise the stiff friction terms, in the context of developing a Godunov-type finite volume hydrodynamic model for accurate and stable simulation of overland flow and surface water flooding involving very shallow water depth. Five analytical test cases are considered to validate the resulting models and the numerical results are compared with those produed by two alternative numerical schemes. The performance of the models are further demonstrated by reproducing a flood event in the 2500 km 2 Eden Catchment, UK.

Xing Y., Chen H., Liang Q., Ma X.

Flood modelling can provide useful information to support flood risk assessment and management. The accuracy of flood simulation results is highly dependent on the quality of input data. In particular, digital elevation models (DEMs) may directly influence the performance of flood predictions and improper representation of complex urban features including buildings and bridges may lead to incorrect prediction of flooding paths and extents, and consequently miscalculate flood risk. In this work, a geographic information system (GIS)-based correction method is proposed to make modifications in high-resolution DEMs by adding building complexes and removing unphysical representations of bridges for a more realistic description of flood paths in considering the flow connectivity in intensely urbanized areas and with the objective of obtaining more accurate flood simulation results. The proposed DEM correction method is applied to support large-scale urban flood modelling in Fuzhou City, China, using an established hydrodynamic flood model known as High-Performance Integrated hydrodynamic Modelling System (HiPIMS). Comparisons are made to the simulation results with and without the DEM improvements using the proposed correction method. The results demonstrate that correct representation of the artificial structures in the urban DEM can significantly improve the flood simulation results.

Ion S., Marinescu D., Cruceanu S.

Hillslope hydrology is a very important part of research based on watershed hydrology. In this study, we focus on water flow over a soil surface with vegetation in a hydrographic basin. We introduce a partial-differential-equation model based on the general principles of fluid mechanics where the unknowns are the depth and velocity of water. The effect of vegetation on the dynamics of water is explained in terms of porosity (a quantity that is related to the density of vegetation) that is a function defined over the hydrological basin. Using a Finite Volume scheme for discretization in space, we introduce an ordinary-differential-equation system that constitutes the base of the discrete model that we are working with. We discuss and investigate several properties of this model that have a physical relevance. Finally, we perform different quantitative validation tests by comparing numerical results with exact solutions or with laboratory-measured data. We also consider some qualitative validation tests by numerically simulating the flow on a theoretical vegetated soil and on a real hydrographic basin.

Bulteau S., Badsi M., Berthon C., Bessemoulin-Chatard M.

The aim of this paper is to prove the preservation of the diffusive limit by a numerical scheme for the shallow-water equations with a generalized Manning friction source term. This asymptotic behavior coincides with the long time and stiff friction limit. The adopted discretization was initially developed to preserve all the steady states of the model under concern. In this work, a relevant improvement is performed in order to preserve also the diffusive limit of the problem and to exactly capture the moving and non-moving steady solutions. In addition, a second-order time and space extension is detailed. Involving suitable linearizations, the obtained second-order scheme exactly preserves the steady states and the diffusive behavior. Several numerical experiments illustrate the relevance of the designed schemes.

Cozzolino L., Varra G., Cimorelli L., Pianese D., Della Morte R.

Friction decoupling, i.e. the computation of friction vector components making separate use of the corresponding velocity components, is common in staggered grid models of the SWE simplifications (Zero-Inertia and Local Inertia Approximation), due to the programming simplicity and to the consequent calculations speed-up. In the present paper, the effect of friction decoupling has been studied from the theoretical and numerical point of view. First, it has been found that friction vector decoupling causes the reduction of the computed friction force and the rotation of the friction force vector. Second, it has been demonstrated that decoupled-friction models lack of rotational invariance, i.e. model results depend on the alignment of the reference framework. These theoretical results have been confirmed by means of numerical experiments. On this basis, it is evident that the decoupling of the friction vector causes a major loss of credibility of the corresponding mathematical and numerical models. Despite the modest speed-up of decoupled-friction computations, classic coupled-friction models should be preferred in every case.

Ponce V.M., Simons D.B., Li R.

The applicability of the kinematic and diffusion models of open channel flow is assessed by comparing the propagation characteristics of sinusoidal perturbations to the steady uniform flow for the kinematic, diffusion, and dynamic models (the dynamic model is that based on the complete Saint Venant equations). The comparison allows the determination of inequality criteria that need to be satisfied if the kinematic or diffusion models are to simulate the physical phenomena within a prescribed accuracy. It is shown that bed slope and wave period (akin to wave duration in waves of shape other than sinusoidal) are the important physical characteristics in determining the applicability of the approximate models. Larger bed slopes or long wave periods, or both, will satisfy the inequality criteria. In practice, larger bed slopes are those of overland flow, and long wave periods are those corresponding to slow-rising flood waves.

Dong J.

A second-order surface reconstruction (SR) method for the shallow water equations with a discontinuous bottom topography and a Manning friction source term is presented. We redefine the water surface level at the cell interface by using the minimum difference between the bottom level and the original water surface level. The reconstructed water surface level is used to define the intermediate bottom level and the intermediate water height at the cell interface. We propose an explicit-implicit method to address the friction source term. The new second-order SR scheme together with the explicit-implicit method can preserve a special steady-state solution of the system and can maintain the positivity of the water depth. We also extend the new scheme to two-dimensional shallow water flows. To demonstrate the robustness and effectiveness of the new scheme, we use several classical numerical experiments for the shallow water flows over a complex bottom topography.

Cozzolino L., Cimorelli L., Della Morte R., Pugliano G., Piscopo V., Pianese D.

Attention of the researchers has increased towards a simplification of the complete Shallow water Equations called the Local Inertia Approximation (LInA), which is obtained by neglecting the advection term in the momentum conservation equation. In the present paper it is demonstrated that a shock is always developed at moving wetting-drying frontiers, and this justifies the study of the Riemann problem on even and uneven beds. In particular, the general exact solution for the Riemann problem on horizontal frictionless bed is given, together with the exact solution of the non-breaking wave propagating on horizontal bed with friction, while some example solution is given for the Riemann problem on discontinuous bed. From this analysis, it follows that drying of the wet bed is forbidden in the LInA model, and that there are initial conditions for which the Riemann problem has no solution on smoothly varying bed. In addition, propagation of the flood on discontinuous sloping bed is impossible if the bed drops height have the same order of magnitude of the moving-frontier shock height. Finally, it is found that the conservation of the mechanical energy is violated. It is evident that all these findings pose a severe limit to the application of the model. The numerical analysis has proven that LInA numerical models may produce numerical solutions, which are unreliable because of mere algorithmic nature, also in the case that the LInA mathematical solutions do not exist. The applicability limits of the LInA model are discouragingly severe, even if the bed elevation varies continuously. More important, the non-existence of the LInA solution in the case of discontinuous topography and the non-existence of receding fronts radically question the viability of the LInA model in realistic cases. It is evident that classic SWE models should be preferred in the majority of the practical applications.

Sanders B.F., Schubert J.E.

Simulation of flood inundation at metric resolution is important for making hazard information useful to a wide range of end-users involved in flood risk management, and addressing the alarming increase in flood losses that have been observed over recent decades. However, high data volumes and computational demands make this challenging over large spatial extents comparable to the metropolitan areas of major cities where flood impacts are concentrated, especially for time-sensitive applications such as forecasting and repetitive simulation for uncertainty assessment. Additionally, several factors present difficulties for numerical solvers including combinations of steep and flat topography that promote transcritical flows, the need to resolve flow in relatively narrow features such as drainage channels and roadways in urban areas which channel flood water during extreme events, and the need to depict compound hazards resulting from the interaction of pluvial, fluvial and coastal flooding. A new flood inundation model is presented here to address these challenges. The Parallel Raster Inundation Model ( PRIMo ) solves the shallow-water equations on an upscaled grid that is far coarser than the underlying raster digital topographic model (DTM), and uses a subgrid modeling approach so that the solution benefits from DTM-scale topographic data. Additionally, an approximate Riemann solver is applied in an innovative way to integrate fluxes between cells, as needed to update the solution by the finite volume method, which makes the method applicable to subcritical, supercritical and transcritical flows. PRIMo is implemented using a two-dimensional domain decomposition approach to Single Process Multiple Data (SPMD) parallel computing, and overlapping communications and computations are implemented to yield ideal parallel scaling for well-balanced test cases. With both a subgrid model and ideal parallel scaling, the model can scale to meet the demands of any application. Several benchmarks are presented to demonstrate predictive skill and the potential for timely, whole-city, metric-resolution flooding simulations. Limitations of the methods and opportunities for improvements are also presented.

Hou J., Wang T., Li P., Li Z., Zhang X., Zhao J., Hinkelmann R.

Aiming at resolving the numerical problems caused by the improper friction source term treatment when simulating overland flow using 2D shallow water flow models, a proposed implicit method computing the friction source term is developed in this work. The method is able to not only accurately evaluate the tricky thin overland flow by considering the flow velocity varying in a single time step, but also eliminates the redundant iterations for regular implicit approach through converting the implicit scheme to an explicit equation. It is therefore an improved one in terms of accuracy and efficiency comparing to the existing methods. Furthermore, it is an independent part and can be straightforwardly and universally incorporated into any FVM based shallow water flow model. Such performances are validated against three test cases involving theoretical and practical overland flow problems. The results prove the proposed method treating friction source term could simulate overland flow problems in a relatively more accurate and efficient way, and therefore can effectively help extend the applicability of the shallow water models reliably computing both the open channel and overland flows.

Xia X., Liang Q.

Discretisation of the friction terms to ensure numerical stability and accuracy remains to be challenging for the development of robust numerical schemes to solve the shallow water equations (SWEs), particularly for applications involving very shallow flows (e.g. overland flows and wet/dry fronts) over complex domain topography. The key challenge is to ensure relaxation of the flow towards an equilibrium state characterised by the balance between friction and gravity in a computationally efficient way. To overcome this numerical challenge, this paper proposes a novel approach for discretising the friction source terms in the SWEs in the context of an explicit finite volume method. The overall numerical scheme adopts the HLLC Riemann solver and surface reconstruction method (SRM) to explicitly discretise the flux and bed slope source terms. Whilst a fully implicit scheme is used to handle the friction source terms, solution to the implicit formulation is analytically derived to explicitly update the flow variables. Compared with the existing approaches, the proposed scheme effectively resolves the issue associated with stiff relaxation without necessity to use an iteration method and it supports efficient simulation using time steps controlled only by the Courant–Friedrichs–Levy (CFL) condition. The current friction term discretisation scheme is not coupled with flux and bed slope calculation and therefore may be readily implemented in any other explicit finite volume SWE models. After being successfully validated against two benchmark tests with analytical solutions, the resulting new SWE model is applied to reproduce a rainfall-flooding event in the Upper Lee catchment in the UK.

Gu S., Zheng X., Ren L., Xie H., Huang Y., Wei J., Shao S.

Varra G., Cozzolino L., Della Morte R., Soares-Frazão S.

Climate change and urbanization, among various factors, are expected to exacerbate the risk of flood disasters in urban areas. This prompts the construction of appropriate modeling tools capable of addressing full-scale urban floods for hazard and risk assessment. In this view, sub-grid porosity models based on the classic shallow water equations (SWE) appear to be a promising approach for full-scale applications in urban environments with reduced computational cost with respect to classic SWE models on high-resolution grids. The present work focuses on the recently proposed two-dimensional binary single porosity (BSP) model, which is a porosity flooding model written in differential form and based on the use of a binary indicator function to locate obstacles and buildings. Several applications (synthetic, experimental, and real-world cases) show that (i) the BSP results tend to the classic SWE solution for sufficiently refined mesh and that (ii) the BSP model can be successfully applied to realistic conditions with complicated terrain and obstacle distribution on coarser grids. Clearly, the adoption of medium/coarse grids makes the BSP model inherently less accurate than the classic SWE model on high-resolution grids, but the corresponding reduction of computational cost makes the use of the BSP model promising in full-scale urban flood applications when (i) multiple simulations are needed to perform stochastic or scenario analysis, (ii) no detailed information of local flow characteristics is required, and/or (iii) for complementing classic SWE models in a nesting cascade.