ANALYTICAL HAYAMI SOLUTION FOR THE DIFFUSIVE WAVE FLOOD ROUTING PROBLEM WITH LATERAL INFLOW

The diffusive wave equation is generally used in flood routing in rivers. The two parameters of the equation, celerity and diffusivity, are usually taken as functions of the discharge. If these two parameters can be assumed to be constant without lateral inflow, the diffusive wave equation may have an analytical solution: the Hayami model. A general analytical method, based on ‘Hayami’s hypothesis, is developed here which resolves the diffusive wave flood routing equation with lateral inflow or outflow uniformly distributed over a channel reach. Flood routing parameters are then identified using observed inflow and outflow and the Hayami model used to simulate outflow. Two examples are discussed. Firstly, the prediction of the hydrograph at a downstream section on the basis of a knowledge of the hydrograph at an upstream section and the lateral inflow. The second example concerns lateral inflow identification between an upstream and a downstream section on the basis of a knowledge of hydrographs at the upstream and downstream sections. The new general Hayami model was applied to flood routing simulation and for lateral inflow identification of the River Allier in France. The major advantages of the method relate to computer simulation, real-time forecasting and control applications in examples where numerical instabilities, in the solution of the partial differential equations must be avoided.     

Diffusive wave solutions for open channel flows with uniform and concentrated lateral inflow

The diffusive wave equation with inhomogeneous terms representing hydraulics with uniform or concentrated lateral inflow into a river is theoretically investigated in the current paper. All the solutions have been systematically expressed in a unified form in terms of response function or so called K-function. The integration of K-function obtained by using Laplace transform becomes S-function, which is examined in detail to improve the understanding of flood routing characters. The backwater effects usually resulting in the discharge reductions and water surface elevations upstream due to both the downstream boundary and lateral inflow are analyzed. With a pulse discharge in upstream boundary inflow, downstream boundary outflow and lateral inflow respectively, hydrographs of a channel are routed by using the S-functions. Moreover, the comparisons of hydrographs in infinite, semi-infinite and finite channels are pursued to exhibit the different backwater effects due to a concentrated lateral inflow for various channel types.     

Stochastic modelling of a diffusive wave for flood propagation using the random walk particle tracking method in a hypothetical city

Deterministic numerical schemes have been widely used for the solution of the diffusive wave (DW) equation, however, these schemes are computationally costly and suffer instability issues. This paper presents a stochastic random walk particle tracking (RWPT) method to solve such an equation for a dam‐break flow problem. Three different wave duration scenarios are presented for simulations of the DW for flood flows in a hypothetical city. The hypothetical city is represented by a domain of size 2,000 m by 500 m in x and y directions, respectively. The domain is divided into 25 m by 25 m cells. A dam is located at the upstream of the hypothetical city. Each scenario has a distinct propagation pattern after the dam is breached. Analysed and presented are 18 different simulations, which are composed of three different building configurations, two different bed slopes, and three different shapes of hydrographs. In this method, the flood volume is divided into a large number of particles where each particle carries a fixed amount of the flood volume. These particles undergo convective and diffusive movements, and their superposition represents propagation of the DW in the flow domain. The solution algorithm of the RWPT‐based equations is used to compute flood inundation depths in the hypothetical city. Comparison is drawn among the simulated results from three different shapes of the inflow hydrographs. The proposed stochastic method has two major advantages over traditional deterministic schemes: (a) greater efficiency, thus lesser computational costs, and (b) no instability issues.     

Assessment of river flow with significant lateral inflow through reverse routing modeling

The discharge hydrograph estimation in rivers based on reverse routing modeling and using only water level data at two gauged sections is here extended to the most general case of significant lateral flow contribution, without needing to deploy rainfall–runoff procedures. The proposed methodology solves the Saint‐Venant equations in diffusive form also involving the lateral contribution using a “head‐driven” modeling approach where lateral inflow is assumed to be function of the water level at the tributary junction. The procedure allows to assess the discharge hydrograph at ends of a selected river reach with significant lateral inflow, starting from the stage recorded there and without needing rainfall data. Specifically, the MAST 1D hydraulic model is applied to solve the diffusive wave equation using the observed stage hydrograph at the upstream section as upstream boundary condition. The other required data are (a) the observed stage hydrograph at the downstream section, as benchmark for the parameter calibration, and (b) the bathymetry of the river reach, from the upstream section to a short distance after the downstream gauged section. The method is validated with different flood events observed in two river reaches with a significant intermediate basin, where reliable rating curves were available, selected along the Tiber River, in central Italy, and the Alzette River, in Luxembourg. Very good performance indices are found for the computed discharge hydrographs at both the channel ends and along the tributaries. The mean Nash‐Sutcliffe value (NS_q) at the channel ends of two rivers is found equal to 0.99 and 0.86 for the upstream and downstream sites, respectively. The procedure is also validated on a longer stretch of the Tiber River including three tributaries for which appreciable results are obtained in terms of NS_q for the computed discharge hydrographs at both the channel ends for three investigated flood events.     

A fuzzy dynamic flood routing model for natural channels

A fuzzy dynamic flood routing model (FDFRM) for natural channels is presented, wherein the flood wave can be approximated to a monoclinal wave. This study is based on modification of an earlier published work by the same authors, where the nature of the wave was of gravity type. Momentum equation of the dynamic wave model is replaced by a fuzzy rule based model, while retaining the continuity equation in its complete form. Hence, the FDFRM gets rid of the assumptions associated with the momentum equation. Also, it overcomes the necessity of calculating friction slope (S_f) in flood routing and hence the associated uncertainties are eliminated. The fuzzy rule based model is developed on an equation for wave velocity, which is obtained in terms of discontinuities in the gradient of flow parameters. The channel reach is divided into a number of approximately uniform sub‐reaches. Training set required for development of the fuzzy rule based model for each sub‐reach is obtained from discharge‐area relationship at its mean section. For highly heterogeneous sub‐reaches, optimized fuzzy rule based models are obtained by means of a neuro‐fuzzy algorithm. For demonstration, the FDFRM is applied to flood routing problems in a fictitious channel with single uniform reach, in a fictitious channel with two uniform sub‐reaches and also in a natural channel with a number of approximately uniform sub‐reaches. It is observed that in cases of the fictitious channels, the FDFRM outputs match well with those of an implicit numerical model (INM), which solves the dynamic wave equations using an implicit numerical scheme. For the natural channel, the FDFRM outputs are comparable to those of the HEC‐RAS model. Copyright © 2009 John Wiley & Sons, Ltd.     

Flood routing based on diffusion wave equation using mixing cell method

A one-dimensional non-linear diffusion wave equation is derived from the Saint Venant equations with neglect of the inertia terms. This non-linear equation has no general analytical solution. Numerical schemes are therefore employed to discretize the space and time axes and convert the differential equation to difference form. In this study, the mixing cell method is used to convert the diffusion wave equation to difference form, in which the difference term can be eliminated by selecting an optimal space step size Δx when time step size Δt is given. When the time step size Δt→0, the space step size Δx=Q/(2S₀BC]_k) where Q is discharge, S₀ is bed slope, B is channel width and C_k is kinematic wave celerity, which is the same as the characteristic length proposed by Kalinin and Milyukov. The results of application to two cases show that the mixing cell and linear channel flow routing methods produce hydrographs that are in agreement with the observed flood hydrographs. © 1997 John Wiley & Sons, Ltd.     

A fuzzy dynamic wave routing model

In this paper a fuzzy dynamic wave routing model (FDWRM) for unsteady flow simulation in open channels is presented. The continuity equation of the dynamic wave routing model is preserved in its original form while the momentum equation is replaced by a fuzzy rule based model which is developed on the principle that during unsteady flow the disturbances in the form of discontinuities in the gradient of the physical parameters will propagate along the characteristics with a velocity equal to that of velocity of the shallow water wave. The model gets rid off the assumptions associated with the momentum equation by replacing it with the fuzzy rule based model. It overcomes the necessity of calculating friction slope (S_f) in flow routing and hence the associated uncertainties are eliminated. The robustness of the fuzzy rule based model enables the FDWRM to march the solution even in regions where the aforementioned assumptions are violated. Also the model can be used for flow routing in curved channels. When the model is applied to hypothetical flood routing problems in a river it is observed that the results are comparable to those of an implicit numerical model (INM) which solves the dynamic wave equations using an implicit numerical scheme. The model is also applied to a real case of flow routing in a field canal. The results match well with the measured data and the model performs better than the INM. Copyright © 2007 John Wiley & Sons, Ltd.     

MÉTHODE D'ÉTUDE DES NAPPES LIBRES DANS LES AQUIFÈRES ANISOTROPES

Abstract

The necessary and sufficient conditions for non-zero phase shift and non-zero attenuation in linear flood routing can be derived from the continuity equation alone and are found to depend on the existence of an imaginary part in the expression for frequency or in the expression for wave number. It is shown that in linear flood routing the phase lag between flow rate and area of flow is directly related to the attenuation per unit wave length. The effects of using various forms of the momentum equation, in addition to the continuity equation, are exemplified by deriving analytical expressions in terms of the frequency, both for attenuation per unit channel length and for phase shift, for the kinematic wave, the general diffusion analogy, and the complete St. Venant equation.     

A comparison of hillslope drainage area estimation methods using high-resolution DEMs with implications for topographic studies of gullies

Topographic models provide a useful tool for understanding gully occurrence in the landscape but require reliable estimates of gully head drainage areas. Modern high-resolution topography data (collected using structure from motion photogrammetry or light detection and ranging) is increasingly used for topographic studies of gullies, but little work has been done to assess the variability of gully head drainage area estimates using different methods. This study evaluated alternative approaches to using high-resolution digital elevation models (DEMs) so that gully topographic models can be more readily applied to any area with suitably high-resolution data. Specifically, we investigated the impact of single- or multiple-direction flow routing algorithms, DEM hydrologic-enforcement procedures and spatial resolution on gully head drainage area estimation. We tested these methods on a 40 km² site centred on Weany Creek, a low-relief semi-arid landscape draining towards the Great Barrier Reef, Australia. Using a subroutine to separate gully heads into those with divergent or convergent flow patterns upslope, we found that divergent flow conditions occurred at half of 484 studied gullies. Drainage areas estimated by different flow routing algorithms were more variable in these divergent cases than for convergent cases. This variation caused a significant difference between topographic threshold parameters (slope b and intercept k) derived from single- or multiple-direction flow routing algorithms, respectively. Different methods of hydrologic enforcement (filling or breaching) also affected threshold analysis, resulting in estimates of the exponent b being ~188% higher if the DEM was filled than if breached. The testing of the methods to date indicates that a finer resolution (≤2 m) DEM and a multiple-direction flow routing algorithm achieve the most realistic drainage area estimates in low-relief landscapes. For Weany Creek we estimated threshold parameters k = 0.033 and b = 0.189, indicating that it is highly susceptible to gully erosion.     

Non-hyperbolic moveout inversion of wide-azimuth P-wave data for orthorhombic media

The azimuthally varying non‐hyperbolic moveout of P‐waves in orthorhombic media can provide valuable information for characterization of fractured reservoirs and seismic processing. Here, we present a technique to invert long‐spread, wide‐azimuth P‐wave data for the orientation of the vertical symmetry planes and five key moveout parameters: the symmetry‐plane NMO velocities, V⁽¹⁾_nmo and V⁽²⁾_nmo , and the anellipticity parameters, η⁽¹⁾, η⁽²⁾ and η⁽³⁾ . The inversion algorithm is based on a coherence operator that computes the semblance for the full range of offsets and azimuths using a generalized version of the Alkhalifah–Tsvankin non‐hyperbolic moveout equation. The moveout equation provides a close approximation to the reflection traveltimes in layered anisotropic media with a uniform orientation of the vertical symmetry planes. Numerical tests on noise‐contaminated data for a single orthorhombic layer show that the best‐constrained parameters are the azimuth ? of one of the symmetry planes and the velocities V⁽¹⁾_nmo and V⁽²⁾_nmo , while the resolution in η⁽¹⁾ and η⁽²⁾ is somewhat compromised by the trade‐off between the quadratic and quartic moveout terms. The largest uncertainty is observed in the parameter η⁽³⁾ , which influences only long‐spread moveout in off‐symmetry directions. For stratified orthorhombic models with depth‐dependent symmetry‐plane azimuths, the moveout equation has to be modified by allowing the orientation of the effective NMO ellipse to differ from the principal azimuthal direction of the effective quartic moveout term. The algorithm was successfully tested on wide‐azimuth P‐wave reflections recorded at the Weyburn Field in Canada. Taking azimuthal anisotropy into account increased the semblance values for most long‐offset reflection events in the overburden, which indicates that fracturing is not limited to the reservoir level. The inverted symmetry‐plane directions are close to the azimuths of the off‐trend fracture sets determined from borehole data and shear‐wave splitting analysis. The effective moveout parameters estimated by our algorithm provide input for P‐wave time imaging and geometrical‐spreading correction in layered orthorhombic media.     

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

Impulse 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.     

WRF-Hydro模型与新安江模型在陈河流域的应用对比

本文利用全球陆面数据同化系统与降雨观测数据,以陕西半湿润区陈河流域为研究对象,驱动WRF-Hydro模型,研究该模型的表现和适用性,并在结构、参数、输入输出和模拟结果方面与新安江模型对比.考虑到次表面层与实际包气带的区别,引入土层厚度乘子ZSOILFAC对前者进行等比缩放,发现其与新安江模型反推包气带的厚度有较好的一致性.研究表明:在陈河流域中WRF-Hydro计算步长须在建议值的基础上缩小; WRF-Hydro模型善于模拟洪水细节,新安江模型表现好且稳定;前者的径流深和洪峰合格率平于或略低于后者;在两个指标均合格的洪水中,前者平均均方根误差比后者小21.5%,但对于其他洪水,前者平均均方根误差比后者大56.2%; WRF-Hydro在洪水起涨时刻模拟较好,表现出其在中小流域应用的潜力.     

基于连续方程的河道水流双变量耦合演算模型

针对现有的河道水流洪水演算模型只能模拟单一变量(流量或水位)的问题,以水流连续方程和河段蓄水量的两种不同表达形式(蓄水量等于平均过水断面面积与河段长乘积,蓄水量等于河段平均流量与传播时间的乘积)为基础,对马斯京根模型进行了通用性改进,提出了双变量耦合通用演算模型.选取了四大水系(包括内陆河流和入海河流)的16个河段汛期洪水资料进行模型检验,模型验证考虑了地理范围、不同的河段特征和水力特征、洪水量级等因素,全面地检验了模型结构的合理性和模拟实际洪水的有效性.将双变量耦合通用演算模型与传统的马斯京根法进行了效果比较,结果表明双变量耦合通用演算模型的模拟精度高于马斯京根法,模拟效果比马斯京根法稳定一些,而且具有较好的通用性.     

Simulating the hydraulic characteristics of the lower Yellow River by the finite‐volume technique

The finite‐volume technique is used to solve the two‐dimensional shallow‐water equations on unstructured mesh consisting of quadrilateral elements. In this paper the algorithm of the finite‐volume method is discussed in detail and particular attention is paid to accurately representing the complex irregular computational domain. The lower Yellow River reach from Huayuankou to Jiahetan is a typical meandering river. The generation of the computational mesh, which is used to simulate the flood, is affected by the distribution of water works in the river channel. The spatial information about the two Yellow River levee, the protecting dykes, and those roads that are obviously higher than the ground, need to be used to generate the computational mesh. As a result these dykes and roads locate the element interfaces of the computational mesh. In the model the finite‐volume method is used to solve the shallow‐wave equations, and the Osher scheme of the empirical function is used to calculate the flux through the interface between the neighbouring elements. The finite‐volume method has the advantage of using computational domain with complex geometry, and the Osher scheme is a method based on characteristic theory and is a monotone upwind numerical scheme with high resolution. The flood event with peak discharge of 15 300 m³/s, occurring in the period from 30 July to 10 August 1982, is simulated. The estimated result indicates that the simulation method is good for routing the flood in a region with complex geometry. Copyright © 2002 John Wiley & Sons, Ltd.     

Simulation of summer storms of differing recurrence intervals in a semiarid environment using a kinematic routing scheme

A kinematic flood routing procedure has been devised for a small dendritic headwater gully network on the Western slope of Colorado. the program is spatially-distributed, incorporating lateral inflows from 103 field sites on the network for which channel geometry variables are known. This model, in which a lateral inflow algorithm for the sideslopes between each channel site is convoluted into a Freeze-type (1978) numerical scheme, is fully developed in this paper. Although the field basis of the lateral inflow algorithm has been tested elsewhere (Faulkner, 1990), sensitivity tests were needed for the roughness and hillslope velocity estimates used in the routing procedure. After these successful tests, a suitably precalibrated run of the model was compared with a field-monitored runoff event on the watershed, and results again were encouraging. However, peak attentuation downstream was more pronounced in reality than on the simulation, so the model was also modified by inclusion of allowances for transmission loss. the tendency that the model had displayed for peak size attenuation downstream was considerably enhanced. Using the model, the geomorphic role of the flashfloods which affect the watershed in the summer months is briefly considered by applying the model to existing records of local summer storm rainfall events as a basis for event simulation. These simulations show that downstream attenuation of the flood wave on concave networks in steep semiarid terrain was likely to be a common occurrence, possibly resulting in down-net deposition and differences in geomorphic behaviour between upstream and downstream sites. the discussion is finally broadened to consider the relative importance of ‘common’ as compared to ‘freak’ watershed events in maintaining these differences.     

Streamflow routing in the indian arid zone

A lumped model for streamflow routing in arid ephemeral channels has been developed. The governing equations for movement of flood waves subjected to transmission losses are simplified through a time averaging process to develop an ordinary differential equation describing transmission losses as a function of distance, inflow, channel width, time parameters of flow and effective hydraulic conductivity. The resulting equation has an analytical solution and simulates runoff volume and peak discharge rates for individual storm events. The outflow hydrograph is fairly well approximated with a triangular approximation. The model is simplified and constructed to require a minimum of observed data for calibration. It can also be used for ungauged basins in arid regions through parameterization.     

Estimation of reservoir flood control operation risks with considering inflow forecasting errors

A method for quantifying inflow forecasting errors and their impact on reservoir flood control operations is proposed. This approach requires the identification of the probability distributions and uncertainty transfer scheme for the inflow forecasting errors. Accordingly, the probability distributions of the errors are inferred through deducing the relationship between its standard deviation and the forecasting accuracy quantified by the Nash–Sutcliffe efficiency coefficient. The traditional deterministic flood routing process is treated as a diffusion stochastic process. The diffusion coefficient is related to the forecasting accuracy, through which the forecasting errors are indirectly related to the sources of reservoir operation risks. The associated risks are derived by solving the stochastic differential equation of reservoir flood routing via the forward Euler method. The Geheyan reservoir in China is selected as a case study. The hydrological forecasting model for this basin is established and verified. The flood control operation risks in the forecast-based pre-release operation mode for different forecasting accuracies are estimated by the proposed approach. Application results show that the proposed method can provide a useful tool for reservoir operation risk estimation and management.     

A direct three-parameter Muskingum procedure incorporating lateral inflow

ABSTRACT

An efficient method of analysis is presented which yields directly estimates of the coefficients in the Muskingum flood routing equation:

O_j+1 = c₁I_j + c₂I_j+1 + c₃O_j

that are best in the least squares sense. A simple conceptual model of lateral inflow is proposed and leads to an extended version of the Muskingum method that has a lateral inflow parameter in addition to the standard storage and weighting parameters, K and x. Applications to a number of flood events are given, and proposals for further studies suggested.     

Reservoir attenuation of floods from ungauged basins

Abstract

In a typical reservoir routing problem, the givens are the inflow hydrograph and reservoir characteristic functions. Flood attenuation investigations can be easily accomplished using a hydrological or hydraulic routing of the inflow hydrograph to obtain the reservoir outflow hydrograph, unless the inflow hydrograph is unavailable. Although attempts for runoff simulation have been made in ungauged basins, there is only a limited degree of success in special cases. Those approaches are, in general, not suitable for basins with a reservoir. The objective of this study is to propose a procedure for flood attenuation estimation in ungauged reservoir basins. In this study, a kinematic-wave based geomorphic IUH model was adopted. The reservoir inflow hydrograph was generated through convolution integration using the rainfall excess and basin geomorphic information. Consequently, a fourth-order Runge-Kutta method was used to route the inflow hydrograph to obtain the reservoir outflow hydrograph without the aid of recorded flow data. Flood attenuation was estimated through the analysis of the inflow and outflow hydrographs of the reservoir. An ungauged reservoir basin in southern Taiwan is presented as an example to show the applicability of the proposed analytical procedure. The analytical results provide valuable information for downstream flood control work for different return periods.     

Urban fluvial flood modelling using a two‐dimensional diffusion‐wave treatment,part 1: mesh resolution effects

High‐resolution data obtained from airborne remote sensing is increasing opportunities for representation of small‐scale structural elements (e.g. walls, buildings) in complex floodplain systems using two‐dimensional (2D) models of flood inundation. At the same time, 2D inundation models have been developed and shown to provide good predictions of flood inundation extent, with respect to both full solution of the depth‐averaged Navier–Stokes equations and simplified diffusion‐wave models. However, these models have yet to be applied extensively to urban areas. This paper applies a 2D raster‐based diffusion‐wave model to determine patterns of fluvial flood inundation in urban areas using high‐resolution topographic data and explores the effects of spatial resolution upon estimated inundation extent and flow routing process. Model response shows that even relatively small changes in model resolution have considerable effects on the predicted inundation extent and the timing of flood inundation. Timing sensitivity would be expected, given the relatively poor representation of inertial processes in a diffusion‐wave model. Sensitivity to inundation extent is more surprising, but is associated with: (1) the smoothing effect of mesh coarsening upon input topographical data; (2) poorer representation of both cell blockage and surface routing processes as the mesh is coarsened, where the flow routing is especially complex; and (3) the effects of (1) and (2) upon water levels and velocities, which in turn determine which parts of the floodplain the flow can actually travel to. It is shown that the combined effects of wetting and roughness parameters can compensate in part for a coarser mesh resolution. However, the coarser the resolution, the poorer the ability to control the inundation process, as these parameters not only affect the speed, but also the direction of wetting. Thus, high‐resolution data will need to be coupled to a more sophisticated representation of the inundation process in order to obtain effective predictions of flood inundation extent. This is explored in a companion paper. Copyright © 2005 John Wiley & Sons, Ltd.