2D Shallow Water Equations Research Articles

Simulation of the Full‐Process Dynamics of Floating Vehicles Driven by Flash Floods

AbstractFlash flooding has become more prominent under climate change, threatening people's life and property. Post‐event investigations of recent events emphasize the role of floating debris, such as vehicles, in exacerbating damage. Few modeling methods and tools have been developed to simulate the full‐process dynamics of floating debris driven by large‐scale flood waves in real world. In this work, a fully coupled model is developed for simulating the full‐process interactive movements of vehicles driven by flash flood hydrodynamics, from entrainment, transport to deposition. The proposed coupled modeling system consists of a finite volume shock‐capturing hydrodynamic model solving the 2D shallow water equations and a 3D discrete element method (DEM) model. The proposed two‐way coupling approach estimates the hydrostatic and hydrodynamic forces acting on solid objects using the water depth and velocity predicted by the hydrodynamic model; the resulting counter forces on the fluid flow are then considered by adding extra source terms in the hydrodynamic model. A multi‐sphere method is further embedded in the DEM model to better represent vehicle shapes. New calculation modules are further implemented to represent the vehicle entrainment, contact and stopping motions. The coupled model is applied to reproduce a flash flood event hit Boscastle in the UK in 2004. Over 100 vehicles were moved and carried downstream by the highly transient flood flow. The model well predicts the hydrodynamics, interactive transport process and the final locations of vehicles. The proposed coupled model provides a new tool for simulating large‐scale flash flooding processes, including debris dynamics.

Derivation and numerical resolution of 2D shallow water equations for multi-regime flows of Herschel–Bulkley fluids

This paper presents mathematical modelling and simulation of thin free-surface flows of viscoplastic fluids with a Herschel–Bulkley rheology over complex topographies with basal perturbations. Using the asymptotic expansion method, depth-averaged models (lubrication and shallow water type models) are derived for 3D (three-dimensional) multi-regime flows on non-flat inclined topographies with varying basal slipperiness. Starting from the Navier–Stokes equations, two flow regimes corresponding to different balances between shear and pressure forces are presented. Flow models corresponding to these regimes are calculated as perturbations of the zeroth-order solutions. The classical reference models in the literature are recovered by considering their respective cases on a flat-inclined surface. In the second regime case, a pressure term is non-negligible. Mathematically, it leads to a corrective term to the classical regime equations. Flow solutions of the two regimes are compared; the difference appears in particular in the vicinity of sharp changes of slopes. Nonetheless, both regime models are compared with experiments and are found to be in good agreement. Furthermore, numerical examples are shown to illustrate the robustness of the present shallow water models to simulate viscoplastic flows in 3D and over an inclined topography with local perturbations in basal elevation and basal slipperiness. The derived models are adequate for direct (engineering and geophysical) applications to real-world flow problems presenting Herschel–Bulkley rheology like lava and mud flows.

Tsunami inundation and flow velocity within a partially sheltered structural array

ABSTRACT This work examines tsunami run-up processes within a building array partially sheltered by a high-elevation, finite length, seawall. Work was performed using numerical modeling of 2D shallow water equations with varying seawall length, wave height, time scale, and structure size. Inundation depth and flow velocity were output at many locations of interest throughout the structural array. The most important influence on results when compared to the no-wall case was the sheltering or clearance angle as defined from the end of the wall. Behind the wall, inundation depth and cross-shore velocity declined significantly as sheltering angles increased (more sheltered). Alongshore inundation velocity (parallel to the seawall) increased strongly with increasing sheltering angle. A longer wave time scale resulted in significant increase in inundation depth and velocity especially at locations sheltered by the wall. Larger structure size or blockage ratio also led to increased inundation depth and flow velocity. However, varying wave height gave little change in inundation and velocity patterns. These results will serve to provide a better understanding of the wave inundation within a coastal urban region in a tsunami event when the seawall is breached, as well as the influence of coastal structure size and spacing on the tsunami inundation.

Implementation of a GPU-enhanced multiclass soil erosion model based on the 2D shallow water equations in the software Iber

We present the implementation of a new fully distributed multiclass soil erosion module. The model is based on a 2D finite volume solver (Iber+) for the 2D shallow water equations that computes the overland flow water depths and velocities. From these, the model evaluates the transport of sediment particles due to bed load and suspended load, including rainfall-driven and runoff-driven erosion processes, and using well-established physically-based formulations. The evolution of the mass of sediment particles in the soil layer is computed from a mass conservation equation for each sediment class. The solver is implemented using High Performance Computing techniques that take advantage of the computational capabilities of standard Graphical Processing Units, achieving speed-ups of two orders of magnitude relative to a sequential implementation on the CPU. We show the application and validation of the model at different spatial scales, ranging from laboratory experiments to meso-scale catchments.

Development of a Semi-Analytical Dynamic Force Model

A moving water mass generates force which is exerted on its moving path. Cyclone generated storm surge or earthquake generated tsunami are specific examples of moving water mass generated force along the coasts. In addition to human lives, these moving water masses cause severe damages to the coastal infrastructure due to tremendous force exerted on these structures. To assess the damage on these infrastructures, an essential parameter is the resultant force exerted on these structures. To evaluate the damages, there is hardly any quantitative method available to compute this force. In this paper we have developed a semi-analytical model, named as Dynamic Force Model (DFM), by using Variational Iteration Method to compute this force. We have derived the governing equation on the basis of Saint-Venant equations which are basically 1D shallow water equations derived from the Navier-Stokes equations. DFM is verified, calibrated, validated, and applied in Bangladesh coastal zone to compute dynamic thrust force due to tropical cyclone SIDR.

Physics-informed neural networks for solving flow problems modeled by the 2D Shallow Water Equations without labeled data

This paper investigates the application of physics-informed neural networks (PINNs) to solve free-surface flow problems governed by the 2D shallow water equations (SWEs). Two types of PINNs are developed and analyzed: a physics-informed fully connected neural network (PIFCN) and a physics-informed convolutional neural network (PICN). The PINNs eliminate the need for labeled data for training by employing the SWEs, initial and boundary conditions as components of the loss function to be minimized. Results from a set of idealized and real-world tests showed that the prediction accuracy and computation time (i.e., training time) of both PINNs may be less affected by the resolution of the domain discretization when compared against solutions by a Finite Volume (FV) model. Overall, the PICN shows a better trade-off between computational speed and accuracy than the PIFCN. Also, our results for the idealized problems indicated that PINNs can provide more than 5 times higher prediction accuracy than the FV model, while the FV simulation with coarse resolution (e.g., 10 m) can provide sub-centimeter accurate (RMSE) solutions at least one order of magnitude faster than the PINNs. Results from a river flood simulation showed that PINNs delivered better speed-accuracy trade-off than the FV model in terms of predicting the water depth, while FV models outperformed the PINNs for predictions of total flow discharge.

Application of 1D model for overland flow simulations on 2D complex domains

High computational demand limits the applications of two-dimensional (2D) shallow water equation models for high resolution overland flow simulations, while one-dimensional (1D) models can achieve higher computing efficiency. This study applied a 1D hydrodynamic model to surrogate 2D models for overland flow simulations on complex 2D domains. With one dimension reduced, the surrogate model simulation would have acceptable accuracy with much higher computing efficiency. The surrogating is fulfilled through mimicking 2D models in mesh generations, so that the 1D channel network is generated in such a way that it geometrically covers the whole domain without overlapping and intersection, and hydrologically follows the steepest slopes. Several benchmark cases on 2D domains in both laboratory and field scales with complex geometry, where no 1D models have been ever applied, are used to compare the 1D and 2D model simulations. The comparisons demonstrate that the 1D model does have potentials to efficiently simulate overland flow on 2D complex domains with accuracy comparable to 2D models.

Iber-PEST: Automatic calibration in fully distributed hydrological models based on the 2D shallow water equations

Calibrating physically based hydrological models manually is time-consuming and challenging. Automatic calibration tools have become prevalent in these models; however, its effective use requires not only a deep knowledge of the calibration procedure itself, but also understanding the structure of the model input and output files. In this study, we introduce Iber-PEST, a novel framework combining the parameter estimation and uncertainty analysis package, PEST, with Iber, a 2D model based on the shallow water equations. We demonstrate its capabilities by successfully calibrating eight storm events in northwestern Spain's basins, achieving promising results (mean NSE equal to 0.84). Furthermore, by applying the iterative ensemble smoother included in PEST, we demonstrate the feasibility to use the calibration data generated by Iber-PEST with alternative PEST algorithms not directly integrated in the Iber-PEST framework. This work successfully addresses the significant implementation barrier associated with the automatic calibration of Iber hydrological models with PEST.

On the physical vacuum free boundary problem of the 1D shallow water equations coupled with the Poisson equation

This paper is concerned with the vacuum free boundary problem of the 1D shallow water equations coupled with the Poisson equation. We establish the local-in-time well-posedness of classical solutions to this system, and the solutions possess higher-order regularity all the way to the vacuum free boundary, though the density degenerates near the vacuum boundary. To deal with the force term generated by the Poisson equation, we make use of the structure of the momentum equation formulated in a fixed domain by the Lagrangian coordinates. The proof is built on some higher-order weighted energy functionals and weighted embeddings corresponding to the degeneracy near the initial vacuum boundary.

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

The technological advancements of the last few decades have fostered the use of two-dimensional (2D) Shallow water Equations (SWE) flood simulations not only in academic research but also in practical real-world applications and territorial planning. The evaluation of flow resistance due to friction is crucial as it plays a relevant role in a variety of flow conditions. However, problems arise in dam-break and overland flow applications involving very small water depth because the time scale connected to the friction term may be much smaller than the hydrodynamic time scale. To cope with this issue, known as source term stiffness, an implicit treatment of the friction term is frequently adopted in Finite Volume (FV) numerical schemes, but there are cases (null roughness coefficient or null flow velocity) where a division by zero may occur during calculations, leading to a crash of the algorithm. Herein, we propose a reformulation of the implicit friction term that is general, more stable and computationally efficient (with a speed-up of approximately 3 times for the routine running the friction computation) than currently available approaches. The novel implicit method allows to manage special conditions with null roughness or null discharge, without resorting to ad-hoc thresholds, and can be easily implemented in existing schemes with pointwise friction treatment. The novel friction approach is implemented in a purposely simple FV numerical scheme (hydrostatic reconstruction to account for the bed slope terms, first-order accuracy in time and space) - to better focus on the friction term treatment - and it is validated against analytical, synthetic, laboratory and real-world case studies, showing promising capabilities.

A new 2D ESPH bedload sediment transport model for rapidly varied flows over mobile beds

A 2D Eulerian meshless bedload sediment transport model is developed using smoothed particle hydrodynamics (SPH) to simulate rapidly varied flows over mobile beds. In the developed model, we adopt a weakly coupled numerical approach to explicitly solve the governing equations, including 2D shallow water equations for fluid flow motion and the Exner equation for bed sediment movement at the same time step. A defined virtual bedload velocity in a weakly coupled approach is involved in the calculation of time step sizes. However, nonphysical virtual bedload velocities, which often occur in cases with nearly flat beds, require extremely small time step sizes. A formulation of the lower and upper bounds of information propagation speeds for an HLLC approximate Riemann solver is thus proposed to remedy the problem. Four case studies involving dam break flows in prismatic and erodible channels, knickpoint migration, dam erosion due to overtopping flow and dam-break flow in an erodible channel with an abrupt expansion are adopted to validate the developed model. In addition, to compare the performances of different particle interaction configurations under Cartesian uniform particle arrangements, four and eight interacting particles are obtained by changing the smoothing length. Against the measured results, the case of eight interacting particles shows more accurate predictions of erosion peaks along the measured cross-sections because the lateral flows are significant. The good agreement between the simulated and measured results shows that the developed 2D Eulerian SPH bedload sediment transport model with eight interacting particles is well suited for simulating complicated flow-induced bed erosion.

A well-balanced, positive, entropy-stable, and multi-dimensional-aware finite volume scheme for 2D shallow-water equations with unstructured grids

In this article, we present a multi-dimensional-aware Eulerian Riemann Solver (RS) and its associated Finite Volume (FV) scheme for the 2D Shallow-Water (SW) equations. This RS, appropriately derived from its associated Lagrangian version, presents the specific feature of coupling all cells in the vicinity of the current one. Consequently, this solver is no longer a 1D RS across one edge. Contrarily, it encounters for genuine multidimensional effects and for the presence of the source term of the SW equations. The associated first order FV numerical scheme ensures well-balancing for lake at rest steady states, positivity preservation and entropy stability properties. Moreover, a second-order accurate extension is proposed based on Runge-Kutta time discretization and piecewise linear limited reconstructions, that preserve the well-balanced character of the first order scheme. We present several 2D tests assessing the good behaviors of the obtained numerical scheme on unstructured mesh. The numerical scheme seems insensitive to spurious numerical instabilities such as the carbuncle effect.

FloodGNN-GRU: a spatio-temporal graph neural network for flood prediction

Abstract Classical approaches for flood prediction apply numerical methods for the solution of partial differential equations that capture the physics of inundation processes (e.g., the 2D Shallow Water equations). However, traditional inundation models are still unable to satisfy the requirements of many relevant applications, including early-warning systems, high-resolution (or large spatial domain) simulations, and robust inference over distributions of inputs (e.g., rainfall events). Machine learning (ML) approaches are a promising alternative to physics-based models due to their ability to efficiently capture correlations between relevant inputs and outputs in a data-driven fashion. In particular, once trained, ML models can be tested/deployed much more efficiently than classical approaches. Yet, few ML-based solutions for spatio-temporal flood prediction have been developed, and their reliability/accuracy is poorly understood. In this paper, we propose FloodGNN-GRU, a spatio-temporal flood prediction model that combines a graph neural network (GNN) and a gated recurrent unit (GRU) architecture. Compared to existing approaches, FloodGNN-GRU (i) employs a graph-based model (GNN); (ii) operates on both spatial and temporal dimensions; and (iii) processes the water flow velocities as vector features, instead of scalar features. We evaluate FloodGNN-GRU using a LISFLOOD-FP simulation of Hurricane Harvey (2017) in Houston, Texas. Our results, based on several metrics, show that FloodGNN-GRU outperforms several data-driven alternatives in terms of accuracy. Moreover, our approach can be trained 100x faster and tested 1000x faster than the time required to run a comparable simulation. These findings illustrate the potential of ML-based methods to efficiently emulate physics-based inundation models, especially for short-term predictions.

Estimación del gasto máximo en cauces que producen inundaciones mediante simulación numérica, información geoespacial y algoritmos evolutivos

Floods produce enormous human and material losses every year. Evaluating their extent and severity, and especially simulating possible future scenarios can improve the response, mitigation and prevention of the effects of this phenomenon. This paper presents a methodology to reproduce the extent of floods produced by channel overflows and recorded by satellite images, identifying the maximum discharge that produced it by means of the numerical solution of the 2D shallow water equations and Differential Evolution. The objective is to minimize the difference between the extent of the flooded area recorded in the satellite images, and that obtained in the simulation by adjusting the maximum value of the flow curve used at the entrance of the channel in the solution domain. The proposal is applied to data and images corresponding to an area located south of the city of Villahermosa, in the Mexican state of Tabasco, which is an area susceptible to flooding by the overflow of the Río de la Sierra. Our proposal shows that it is possible to have more accurate information on the extent and height of water in the flooded area than that shown by the satellite, which can be used as information for prevention and mitigation plans for the adverse effects of flooding.

Uncertainty analysis of simplified 1D and 2D shallow water equations via the Karhunen–Loéve expansion and Monte Carlo simulations

Uncertainty analysis of simplified 1D and 2D shallow water equations via the Karhunen–Loéve expansion and Monte Carlo simulations

Coping with geometric discontinuities in porosity-based shallow water models

The use of classic two-dimensional (2D) shallow water equations (SWE) for flooding simulation in complex urban environments is computationally expensive, due to the need of refined meshes for the representation of obstacles and building. Aiming to reduce the computational burden, a class of sub-grid SWE models, where small-scale building features are preserved on relatively coarse meshes by means of macroscale porosity parameters, has been recently introduced in the literature. Among the other porosity-based models, the single porosity (SP) model is relevant because the corresponding one-dimensional (1D) Riemann problem is the building block for the construction of many porosity-based numerical schemes. Like the Riemann problem connected to mathematical models such as the SWE with variable bed elevation and the 1D Euler equations in contracting pipes, the SP Riemann problem may exhibit multiple solutions for certain initial conditions. This ambiguity can be solved by restoring the microscale information of the 2D SWE model that is lost at the SP macroscale. In the present paper, we disambiguate the solutions' multiplicity by systematically comparing the solution of the SP Riemann problem at local porosity discontinuities with the corresponding 2D SWE numerical solutions in contracting channels. An additional result of this comparison is that the SP Riemann problem should incorporate an adequate amount of head loss when strongly supercritical flows past sudden porosity reductions occur. An approximate Riemann solver, able to pick the physically congruent solution among the alternatives and equipped with the required head loss amount, shows promising results when implemented in a 1D single porosity finite volume scheme.

Numerical three-dimensional modeling of earthen dam piping failure

A physically-based numerical three-dimensional earthen dam piping failure model is developed for homogeneous and zoned soil dams. This model is an erosion model, coupled with force/moment equilibrium analyses. Orifice flow and two-dimensional (2D) shallow water equations (SWE) are solved to simulate dam break flows at different breaching stages. Erosion rates of different soils with different construction compaction efforts are calculated using corresponding erosion formulae. The dam's real shape, soil properties, and surrounding area are programmed. Large outer 2D-SWE grids are used to control upstream and downstream hydraulic conditions and control the boundary conditions of orifice flow, and inner 2D-SWE flow is used to scour soil and perform force/moment equilibrium analyses. This model is validated using the European Commission IMPACT (Investigation of Extreme Flood Processes and Uncertainty) Test #5 in Norway, Teton Dam failure in Idaho, USA, and Quail Creek Dike failure in Utah, USA. All calculated peak outflows are within 10% errors of observed values. Simulation results show that, for a V-shaped dam like Teton Dam, a piping breach location at the abutment tends to result in a smaller peak breach outflow than the piping breach location at the dam's center; and if Teton Dam had broken from its center for internal erosion, a peak outflow of 117 851 m3/s, which is 81% larger than the peak outflow of 65 120 m3/s released from its right abutment, would have been released from Teton Dam. A lower piping inlet elevation tends to cause a faster/earlier piping breach than a higher piping inlet elevation.

Building Treatments in Numerical Modeling of Dam Break Induced Flow in Urban Area

Dam break is one of the most dangerous disasters, which cause a large mass of water from the reservoir is suddenly released to the downstream. Although dam break events are still rare, the occurrences can cause severe damage to the downstream area, especially if the downstream is an urban area. Urban flood due to dam break has a different characteristic compared to the flood in nature. Modeling the flood is good way to learn the urban flood characteristic. Therefore, further studies are needed to develop an urban dam break model. Numerical modeling is one of the most effective methods to simulate the flood. In numerical modeling, buildings can be modeled with 3 approaches, as a wall boundary, as an area with high manning value, and as a contour. This paper discusses about two-dimensional flow of a dam break which flows through a sloping channel and single building, particularly comparing 3 approaches to modeled the building. The case is based on laboratory experiment of previous study. The model was developed with 2D Shallow Water Equations (SWE) which are based on the conservation of mass and momentum derived from Navier Stokes equation. The equations then discretized using Mac-Cormack numerical scheme. Numerical filter is used to reduce the numerical instability or oscillation in the model. The numerical model results are then compared to the experimental result from previous study. The accuracy of the numerical model was checked using Root Mean Square Error (RMSE) to determine the best methods to modeled the building. The results indicate that modeling the building as a wall boundary have the best results with lowest RMSE value and also shown the closest comparison of water level compared to the experimental results. By conducting this study, it is hoped to assist the urban dam break modeling in the future.

Comparative analysis of numerical solutions of 2D unsteady dambreak waves using FVM and SPH method

Abstract This work presents a comparison of two-dimensional numerical solutions of unsteady free surface flow. This is a simulation of the dam-break wave with different configurations using based-mesh finite volume method and meshless smoothed particle hydrodynamics (SPH). Two well-known approaches, widely used in the computational fluid dynamics (CFD). These techniques have proven their robustness in the numerical treatment of such conservation laws. The main goal is to check the ability of the SPH method and the first order finite volume HLLC solver to reproduce the numerical solutions of the 2D shallow water equations. Based on many benchmark tests, one investigates the effect of the topographic variation along the x and y directions on behavior of the numerical solutions namely at the wet-dry front. The comparison between the simulated results, the analytical solutions and the experimental measurements shows a good correlation, although the finite volume approach remains more advantageous in terms of accuracy and the CPU time.

Simulation of shallow landslides induced by typhoon in July 2018 in Gogoshima, Ehime Prefecture, Japan based on a simple model

Every year, many typhoons make landfall in Japan causing disasters on mountain slopes across the country. Rainfall induced landslides in Japan mostly occur at shallow depths. Developing simple models of rainfall-induced landslides provides a practical alternative to finite element methods in large-scale landslide modelling and prediction. Therefore, authors previously developed a simple method to calculate the advance of the wetting front into fine sand slopes subjected rainfall. However, during intense rainfall, the excess rainfall that cannot be absorbed into the soil will generate runoff on the slope surface, which has not been included in the previous model. Thus, this study integrated the model with a surface water modeling module to simulate the disasters more comprehensively. The surface water is controlled by 2D shallow water equations and the digital elevation model depression removal process. The combined model is able to provide the maps of the wetting front, surface water depth, and factor of safety of slope during rainstorms. The newly integrated model was used to simulate rainfall-induced disasters in the Gogoshima Island during the July 2018 typhoon. The simulation results disclosed that both moving of the wetting front and surface water flows contributed to shallow landslides along catchments in this island.