Venant Equations Research Articles

Development of a diffusive wave flood routing model for an ungauged basin: a case study in Kulsi River Basin, India

Flood is the most devastating and frequent disaster in North-East India, resulting in loss of human life and damage of properties. Its deleterious effects can be minimized by appropriate modeling, analysis, and management methods. Such modeling and analyzing techniques are hindered in flood prediction in an ungauged basin due to the lack of hydro-meteorological data. The main objective of this work is to develop a numerical approach for flood routing in an ungauged basin using the rainfall-runoff model and the diffusive wave flood routing model (DWFRM). GIS approach is used to extract the geographical information of the study area. The SCS-CN rainfall-runoff model is employed to obtain the inflow, and lateral inflow hydrographs of the ungauged sub-basins and DWFRM is employed to anticipate the flood hydrograph at the outlet of the ungauged basin. A new form of diffusive wave momentum equation of the Saint–Venant equation is derived for cross-section having a parabolic and rectangular channel. The modeling approach is employed to the Kulsi River Basin, India hypothetically considered ungauged basin, and the results obtained are compared with the observed data at the outlet of the basin. The performance of the model is validated by considering four parameters RMSE, peak flow error, peak time error, and total volume error. The results reveal that the proposed model diffusive wave flood routing model (DWFRM) is efficient to predict the flood hydrograph at the outlet of an ungauged basin.

Observing Rivers With Varying Spatial Scales.

The National Aeronautics and Space Administration/Centre national d’études spatiales Surface Water and Ocean Topography (SWOT) mission will estimate global river discharge using remote sensing. Synoptic remote sensing data extend in situ point measurements but, at any given point, are generally less accurate. We address two questions: (1)What are the scales at which river dynamics can be observed, given spatial sampling and measurement noise characteristics? (2) Is there an equation whose variables are the averaged hydraulic quantities obtained by remote sensing and which describes the dynamics of spatially averaged rivers? We use calibrated hydraulic models to examine the power spectra of the different terms in the momentum equation and conclude that the measurement of river slope sets the scale at which rivers can be observed. We introduce the reach‐averaged Saint Venant equations that involve only observable hydraulic variations and which parametrize within‐reach variability with a variability index that multiplies the friction coefficient and leads to an increased “effective” friction coefficient. An exact expression is derived for the increase in the effective friction coefficient, and we propose an approximation that requires only estimates of the hydraulic parameter variances. We validate the results using a large set of hydraulic models and find that the approximated variability index is most faithful when the river parameters obey lognormal statistics. The effective friction coefficient, which can vary from a few percent to more than 50% of the point friction coefficient, is proportional to the riverbed elevation variance and inversely proportional to the depth. This has significant implications for estimating discharge from SWOT data.

A new one-dimensional numerical model for unsteady hydraulics of sediments in rivers

This paper presents the model UMHYSER-1D (Unsteady Model for the HYdraulics of SEdiments in Rivers 1-D), a one-dimensional hydromorphodynamic model capable of representing water surface profiles in a single river or a multiriver network, with different flow regimes considering cohesive or non-cohesive sediment transport. It has both steady and unsteady flow and sediment modules. For steady gradually varied flows, UMHYSER-1D uses the standard step method to solve the energy equation and the “NewC” scheme for the de St Venant equations. For sediment transport, UMHYSER-1D uses two methods: for long-term simulation, the unsteady terms of the sediment transport continuity equation are ignored, and a non-equilibrium sediment transport method is used. For short-term simulation, the convection–diffusion equation, with a source term arising from sediment erosion/deposition is solved using the fractional step method. The equation without the source term is solved with an implicit finite-volume method, then the equation with source term is solved. Internal boundary conditions, such as time-stage tables, rating curves, weirs, bridges, and gates are simulated. Incorporated is the active layer concept, which allows selective erosion, enabling the simulation of bed armoring. Non-cohesive sediment transport equations and cohesive sediment physical processes are applied to calculate the sediment deposition and erosion. Finally, UMHYSER-1D empirically accounts for bed geometry adjustments by using a relationship between erosion width and flow rate, an angle of repose condition for bank stability and three minimization theories. The presented validation and application cases show UMHYSER-1D’s capabilities and predicts its promising role in solving complex, real engineering cases.

Refinement of the channel response system by considering time‐varying parameters for flood prediction

AbstractImpulse response functions derived from different types of flood wave equations (simplified shallow water equations) are continuously developed to conduct the linear channel routing (LCR), which is based on the linearized Saint Venant equation and has been widely applied to avoid any possibility of numerical instability. The impulse response function proposed by Dooge, Napiórkowski, and Strupczewski (1987) and derived from the dynamic wave equation with complete force terms has been acknowledged as a classic work to establish a good physical interpretation for the LCR model; however, the flexibility of altering the shape of impulse response still needs to be improved. Based on the concept of this work, this study intends to introduce the time‐varying parameters in the model, so the values of parameters can be adjusted according to the inflow condition, flood stage, and the cross‐sectional shape. Moreover, an integrated routing procedure is proposed to formulate the impulse response function for lateral‐flow inputs and then to connect multiple inputs from subwatersheds or alongside the main channel with the impulse response function of each channel segment to reflect the spatial variation of hydraulic characteristics among different segments. In the discussion of this article, the impulse response function is analysed to show its sensitivity to hydraulic variables with spatial and temporary variations. Flood‐event simulations of a studied watershed are also provided to verify the applicability of the proposed channel routing system.

Identification of Parameters Influencing the Accuracy of the Solution of the Nonlinear Muskingum Equation

Two nonlinear versions of the Muskingum equation are considered. The difference between both equations relates to the exponent parameter. In the first version, commonly used in hydrology, this parameter is considered as free, while in the second version, it takes a value resulting from the kinematic wave theory. Consequently, the first version of the equation is dimensionally inconsistent, whereas the proposed second one is consistent. It is shown that the difference between the results provided by both versions of the nonlinear Muskingum equation depends on the longitudinal bed slope of a channel. For an increasing slope, when a propagating wave becomes more kinematic, the value of the exponent considered as the free parameter tends to its value resulting from the kinematic wave theory. Consequently, when the character of an open channel flow tends to a kinematic one, the dimensionally inconsistent version of the nonlinear Muskingum equation becomes a consistent one. The results of the numerical analysis suggest that apart from the parameters involved in the Muskingum equation, usually considered as free, the parameters of the numerical method of the solution (the number of reservoirs and the time step) should be considered also as free parameters. This conclusion results from the fundamental property of the Muskingum equation, relating to the numerical roots of the wave attenuation process. All numerical examples and tests relate to the solutions of the system of Saint Venant equations, considered as the benchmark.

Multi-Objective Model Predictive Control for Real-Time Operation of a Multi-Reservoir System

This paper presents an extended Model Predictive Control scheme called Multi-objective Model Predictive Control (MOMPC) for real-time operation of a multi-reservoir system. The MOMPC approach incorporates the non-dominated sorting genetic algorithm II (NSGA-II), multi-criteria decision making (MCDM) and the receding horizon principle to solve a multi-objective reservoir operation problem in real time. In this study, a water system is simulated using the De Saint Venant equations and the structure flow equations. For solving multi-objective optimization, NSGA-II is used to find the Pareto-optimal solutions for the conflicting objectives and a control decision is made based on multiple criteria. Application is made to an existing reservoir system in the Sittaung river basin in Myanmar, where the optimal operation is required to compromise the three operational objectives. The control objectives are to minimize the storage deviations in the reservoirs, to minimize flood risks at a downstream vulnerable place and to maximize hydropower generation. After finding a set of candidate solutions, a couple of decision rules are used to access the overall performance of the system. In addition, the effect of the different decision-making methods is discussed. The results show that the MOMPC approach is applicable to support the decision-makers in real-time operation of a multi-reservoir system.

Experimental Analysis and Numerical Simulation of Hydraulic Jump

Saint Venant equations are numerically solved to simulate the formation of hydraulic jump in a rectangular channel having a small bed slope. The MacCormack’s scheme is used for the solution by applying specified initial and boundary conditions until a steady state flow is reached. The location of the hydraulic jump is determined as a part of these computations. The artificial viscosity technique should be used in the computations to dampen the superior oscillations near the steep gradient of the simulated hydraulic jump. Twenty laboratory experiments were carried out for verification of the numerical model. The upstream Froude number for these experiments ranged from 2.17 to 7.0 in three different bed slopes 0, 0.02174, 0.0475. The simulated hydraulic jump profiles using the MacCormack’s scheme shows a good agreement with the experimental data. An empirical equation was developed to determine the location of hydraulic jump using regression analysis based on simulated data. Software based on Computational Fluid Dynamics (CFD) was also used to simulate two of these experiments. The results obtained from CFD analysis matched fairly with the experimental results.

Undular and broken surges in dam-break flows: a review of wave breaking strategies in a Boussinesq-type framework

The water waves resulting from the collapse of a dam are important unsteady free surface flows in civil and environmental engineering. Considering the basic case of ideal dam break waves in a horizontal and rectangular channel the wave patterns observed experimentally depends on the initial depths downstream (hd) and upstream (ho) of the dam. For r = hd/ho above the transition domain 0.4–0.55, the surge travelling downstream is undular, a feature described by the dispersive Serre–Green–Naghdi (SGN) equations. In contrast, for r below this transition domain, the surge is broken and it is well described by the weak solution of the Saint–Venant equations, called Shallow Water Equations (SWE). Hybrid models combining SGN–SWE equations are thus used in practice, typically implementing wave breaking modules resorting to several criteria to define the onset of breaking, frequently involving case-dependent calibration of parameters. In this work, a new set of higher-order depth-averaged non-hydrostatic equations is presented. The equations consist in the SGN equations plus additional higher-order contributions originating from the variation with elevation of the velocity profile, modeled here with a Picard iteration of the potential flow equations. It is demonstrated that the higher-order terms confer wave breaking ability to the model without using any empirical parameter, such while, for r > 0.4–0.55, the model results are essentially identical to the SGN equations but, for r < 0.4–0.55, wave breaking is automatically accounted for, thereby producing broken waves as part of the solution. The transition from undular to broken surges predicted by the high-order equations is gradual and in good agreement with experimental observations. Using the solution of the new higher-order equations it was further developed a new wave breaking index based on the acceleration at the free surface to its use in hybrid SGN–SWE models.

High temporal resolution rainfall–runoff modeling using long-short-term-memory (LSTM) networks

Accurate and efficient models for rainfall runoff (RR) simulations are crucial for flood risk management. Most rainfall models in use today are process-driven; i.e. they solve either simplified empirical formulas or some variation of the St. Venant (shallow water) equations. With the development of machine-learning techniques, we may now be able to emulate rainfall models using, for example, neural networks. In this study, a data-driven RR model using a sequence-to-sequence Long-short-Term-Memory (LSTM) network was constructed. The model was tested for a watershed in Houston, TX, known for severe flood events. The LSTM network's capability in learning long-term dependencies between the input and output of the network allowed modeling RR with high resolution in time (15 minutes). Using 10-years precipitation from 153 rainfall gages and river channel discharge data (more than 5.3 million data points), and by designing several numerical tests the developed model performance in predicting river discharge was tested. The model results were also compared with the output of a process-driven model Gridded Surface Subsurface Hydrologic Analysis (GSSHA). Moreover, physical consistency of the LSTM model was explored. The model results showed that the LSTM model was able to efficiently predict discharge and achieve good model performance. When compared to GSSHA, the data-driven model was more efficient and robust in terms of prediction and calibration. Interestingly, the performance of the LSTM model improved (test Nash-Sutcliffe model efficiency from 0.666 to 0.942) when a selected subset of rainfall gages based on the model performance, were used as input instead of all rainfall gages.

Resistance Formulations in Shallow Overland Flow Along a Hillslope Covered With Patchy Vegetation

AbstractDescribing the effects of surface roughness on flow resistance remains a first‐order challenge for modeling shallow overland flow using the Saint Venant equations (SVE). This challenge has resulted in a proliferation of roughness schemes relating the properties of a uniform rough surface to bulk velocity and resistance, making selection of an appropriate roughness scheme daunting—especially on heterogeneous surfaces. For hydrological predictions, a roughness scheme is appropriate if the resulting SVE solutions predict outcomes such as water balance partitioning between runoff and infiltration, the hillslope hydrograph, and the discharge velocity with uncertainties that are less than or comparable to uncertainty in measurements. To assess the sensitivity of hydrological predictions to the choice of a roughness scheme, multiple schemes are first calibrated to each other under equilibrium flow conditions while imposing the kinematic wave approximation at the outlet. This approach yields analytical relations between the parameters of different roughness schemes when applied to the same hillslope and discharge. The approach is used to provide a parameterization of five commonly used roughness schemes for a common surface, and the predictions of the schemes for multiple patchily vegetated hillslopes are compared. The results suggest that, once calibrated, there is minimal prediction sensitivity to the choice of roughness scheme across a wide range of rainfall conditions. Operationally, the results demonstrate that parameterizing a roughness schemes is of higher significance for predicting hydrological patterns than the precise formulation employed.

Numerical evaluation of the general flow hydraulics and estimation of the river plain by solving the Saint–Venant equation

Continuity and momentum equations govern the movement of unsteady flows in open channels, known as Saint–Venant equations. Numerical models of flood routing are classified according to the simplifications made in these equations. Among the flood routing methods, one can refer to dynamic wave and kinematic models and the Att-Kin and Muskingum hydrological models. The kinematic-wave model solves the Saint–Venant equations approximately by considering the gravity and the frictional force terms. This method assumes that the slopes of the energy gradient and the bed slope are equal. The full dynamic wave model uses all the terms in the Saint–Venant equation to simulate unsteady flows. Besides, given the historical records of the past floods and based on the flow continuity equation, the Att-Kin and Muskingum hydrological models deal with flood routing operations. This study describes the principles of Saint–Venant equations and numerical models of dynamic and kinematic-waves, as well as Att-Kin and Muskingum models, the results of which are compared. For this purpose, the Mehrian River range was used. The maximum flood discharge, the maximum area of the river flood plain, and the maximum flood flow width were estimated by presenting a new mathematical relation. The results showed that the MIKE 11 dynamic wave model yields better results than other models by solving partial differential equations. Moreover, among the hydrological methods, the mathematical flood routing and zoning indicate the closeness of Muskingum hydrological results using the least-squares model and the least error with the numerical values of the software for the observed flood.

2D numerical simulation of urban dam break and its effect to building using lax scheme with numerical filter

Dam-break is one of the disasters that can occurred due to earthquake. The earthquake vibration may damage the dam construction and therefore causing a dam-break flow. The flow can cause severe destruction to the downstream urban area. Dam-break modeling offers a way to analyze its effect of buildings. In this study, a 2D model for analyzing dam break flow is developed based on the Saint Venant equations and solved using Lax Scheme. The initial condition of the modelling is a rectangular channel with obstacles at both sides of the channel and two similar columns in the middle of the channel. These obstacles are considered as buildings in an urban area. Numerical filter is used to increase the stability of the simulation. The developed model is able to perform well in simulating a case of urban dam-break based on a previous experiment. In addition, the applied numerical filter is able to handle shock, therefore maintaining the stability of the model while reducing the simulation time without the needs to use a higher order numerical scheme.

Well-Balanced Discontinuous Galerkin Method for Shallow Water Equations with Constant Subtraction Techniques on Unstructured Meshes

The classical Saint–Venant shallow water equations on complex geometries have wide applications in many areas including coastal engineering and atmospheric modeling. The main numerical challenge in simulating Saint–Venant equations is to maintain the high order of accuracy and well-balanced property simultaneously. In this paper, we propose a high-order accurate and well-balanced discontinuous Galerkin (DG) method on two dimensional (2D) unstructured meshes for the Saint–Venant shallow water equations. The technique used to maintain well-balanced property is called constant subtraction and proposed in Yang et al. (J Sci Comput 63:678–698, 2015). Hierarchical reconstruction limiter with a remainder correction technique is introduced to control numerical oscillations. Numerical examples with smooth and discontinuous solutions are provided to demonstrate the performance of our proposed DG methods.

Historical Changes in Lower Columbia River and Estuary Floods: A Numerical Study

AbstractOver the past 150 years, the Lower Columbia River Estuary controlling depth has approximately doubled, the majority of historical wetlands and floodplain have been reclaimed, numerous infrastructure projects have altered and confined flow pathways, and significant natural and anthropogenic changes to the discharge hydrograph have occurred. To investigate the effect of these changes on tides, river slope, and flood water levels, we construct and validate numerical models that simulate flow over late nineteenth‐century and present‐day bathymetry. The models are validated using archival (1853–1877) and modern tide measurements throughout the Lower Columbia River Estuary and river stage measurements from the tidal river (1876–present). Historical flood plain roughness and levee heights are validated iteratively by requiring simulations to match the observed roll off in the river stage rating curve during floods. Measurements and model results show that environmental change has amplified tidal constituents, with peak change about 60 km from the coast. By contrast, increased depth has reduced river slope for low and moderate river discharge. For rarely observed extreme floods of 30 × 103m3/s, simulated modern water levels exceed historical in Portland (OR). These observations highlight competing hydrodynamic effects, which are investigated by scaling the St. Venant equations for a simulated 25 × 103‐m3/s flood: While larger modern depth reduces frictional effects and decreases surface slope, reduced floodplain access confines modern flow into channels, increasing velocity, bed stress, and water levels. However, the highly frictional historical floodplain conveyed little flow, limiting the effect of floodplain to storage effects; hence, most simulated historical floods exceed modern levels.

Dynamic flood wave routing in prismatic channels with hydraulic and hydrologic methods

AbstractTime-dependent, unsteady flow has been studied in prismatic open channels with symmetric trapezoidal and triangular cross sections and small bottom slope. The St Venant equations without lateral inflow have been discretized in explicit as well as in implicit form and solved numerically, for unsteady, subcritical flow. The inflow hydrograph used can be applied for different flood events by adjusting its parameters accordingly. The results presented are derived from the explicit schemes Lax-Diffusive, MacCormack, Lambda as well as the implicit Preissmann scheme, and are compared to those from the Muskingum-Cunge method and the widely used commercial software HEC-RAS. The peak flow computed by the Lax-Diffusive scheme was reduced at the downstream end of the channel and the arrival time of the peak increased if compared to the other methods. The Muskingun-Cunge method forecasted the shortest peak flow arrival time at the downstream end cross section. Mass conservation computed from inflow and outflow hydrographs has been confirmed, since the maximum error did not exceed 2.60%. All codes were implemented in house using Matlab®.

A novel well-balanced scheme for spatial and temporal bed evolution in rapidly varying flow

This paper presents a novel method of preserving well-balanced conditions for dam-break flows on irregular beds. The research on preservation of well-balanced conditions commonly deals only with spatial variation of beds. It is challenging to design a well-balanced scheme that can model both spatial and temporal variation of beds particularly on domains discretized with unstructured meshes. This challenge arises due to the non-trivial nature of the governing equations modeling hydro-morpho dynamics. This study proposes a new and simple well-balanced method applicable for both spatial and temporal bed variations without requiring additional treatment and complicated discretization of slope source terms. The work relies on a cell-centered Godunov-type finite volume method for solving coupled Saint Venant and sediment continuity equations. The two most important issues, i.e., the ‘lake at rest condition’ and ‘water depth positivity’ of shallow-water flows over irregular beds, are tested and verified in the proposed numerical scheme. The authors employed Riemann solvers to compute fluxes through the control volume surfaces. The proposed scheme is evaluated for various benchmark cases of dam-break flows on rigid and erodible beds. Comparisons between the results of the proposed numerical scheme, analytical solution, experimental data, and previous numerical observations show significant improvement for some cases and a good agreement in the rest.

Analytical Approach of Long Waves Dynamics in an Estuary (Case Study in Karang Mumus River Estuary)

A mathematical model is developed for describing a propagating long wave in an estuary. Wave dynamics in an estuary will employ the Saint Venant equation in the form of a nonlinear partial differential equation (PDE), which consists of equations of mass conservation and momentum conservation. Waves propagate in an estuary usually generated by tides entering a gentle slope channel. The analytical approach is used to solve this non-linear PDE approach using a perturbation method. This paper considers only a zero order. A cross-differentiation between conservation of mass and conservation of momentum equations result in the Bessel differential equation. The solution of zero order gives decreasing amplitude of the long wave to the end of the estuary.

Flood risk assessment in Quzhou City (China) using a coupled hydrodynamic model and fuzzy comprehensive evaluation (FCE)

To evaluate urban flood risk while considering spatial and temporal characteristics, this paper establishes an assessment method based on a coupled hydrodynamic model and a fuzzy comprehensive evaluation method. An assessment system is developed using hydrologic, social and economic factors. In addition, hydrology factors are obtained by the coupled hydrodynamic model. The hydrodynamic model is solved using the three-cascade method of the Saint–Venant equations for a river system, and the surface runoff is solved using a modified Roe-type approximate Riemann solver. The hydrology results are used as hydrologic factors and directly reflect the inundation of the subsequent assessment. Afterward, an analytic hierarchy process is used to calculate the weights of the assessment indices. Then, the urban flood risk is analyzed using the fuzzy comprehensive evaluation method. Five different rainfall events in Quzhou City are simulated and evaluated. The numerically computed results compare well with the statistical data and indicate an extreme inundation depth of more than 3 m. In addition, the assessment risk of Quzhou City is at level IV (high). This study shows that combining the assessment method with the coupled hydrodynamic model efficiently reflects the urban flood risk by considering flood, social and economic data.

Contributions to simulation of the non-permanent movement in sewerage collectors

The water movement of the sewer collectors is of a non-permanent type because of the variation in flow over time. Non-permanent movement in collectors is mathematically modelled by the Saint Venant equations. To solve the equation system, the characteristic method was used, provided that both equations are quasi-linear in the unknown h and Q. Non-dynamic contouring conditions are given by the hydraulic feature of the collectors / tanks located on the collector. Dynamic contour conditions are rendered by the collector-specific equations (river level effluent hydrograph, manoeuvring parameters of the boiler, manoeuvring characteristics at pumping stations, etc.). For the mathematical model implementation, a computational program was developed in the Matlab programming environment (Variation Gradual Collector Program). The program was used to simulate the non-permanent movement in the main sewerage collector “Cacaina” from the sewerage network of the city of Iasi. The simulation of the non-permanent movement was done in two collector evacuation scenarios (gravity and pumping). The analysis of the results obtained for scenario 2 highlights the differential operation of the eight collector sections separated by the side feeders. Simulation of the movement has highlighted the exceeding of the minimum and maximum values of the parameters imposed by the operating rules.

Emulation of the Saint Venant Equations Enables Rapid and Accurate Predictions of Infiltration and Overland Flow Velocity on Spatially Heterogeneous Surfaces

AbstractThe interleaving of impermeable and permeable surfaces along a runoff flow path controls the hillslope hydrograph, the spatial pattern of infiltration, and the distribution of flow velocities in landscapes dominated by overland flow. Predictions of the relationship between the pattern of (im)permeable surfaces and hydrological outcomes tend to fall into two categories: (i) generalized metrics of landscape pattern, often referred to as connectivity metrics, and (ii) direct simulation of specific hillslopes. Unfortunately, the success of using connectivity metrics for prediction is mixed, while direct simulation approaches are computationally expensive and hard to generalize. Here we present a new approach for prediction based on emulation of a coupled Saint Venant equation‐Richards equation model with random forest regression. The emulation model predicts infiltration and peak flow velocities for every location on a hillslope with an arbitrary spatial pattern of impermeable and permeable surfaces but fixed soil, slope, and storm properties. It provides excellent fidelity to the physically based model predictions and is generalizable to novel spatial patterns. The spatial pattern features that explain most of the hydrological variability are not stable across different soils, slopes, and storms, potentially explaining some of the difficulties associated with direct use of spatial metrics for predicting landscape function. Although the current emulator relies on strong assumptions, including smooth topography, binary permeability fields, and only a small collection of soils, slope, and storm scenarios, it offers a promising way forward for applications in dryland and urban settings and in supporting the development of potential connectivity indices.