Kinematic Wave Equation Research Articles

Random fractional kinematic wave equations of overland flow: The HPM solutions and applications

This paper presents new findings from analyses of a random fractional kinematic wave equation (rfKWE) for overland flow. The rfKWE is featured with orders of temporal and spatial fractional derivatives and with the roughness parameter, the effective rainfall intensity and infiltration rate as random variables. The new solutions are derived with the aid of a numerical method named the homotopy perturbation method (HPM) and approximate solutions are presented for different situations. The solutions are evaluated with data from overland flow flumes with simulated rainfall in the laboratory. The results suggest that on an infiltrating surface the temporal nonlocality of overland flow represented by the temporal order of fractional derivatives diminishes over time while the spatial nonlocality manifested by the spatial order of fractional derivatives continue if there is overland flow. It shows that the widely used unit discharge-height relationship is a special case of the solution of the rfKWE. Procedures are demonstrated for determining the fractional roughness coefficient, nf, the order of spatial fractional derivative, ρ, and the steady-state infiltration rate during the overland flow, As. The analyses of the data show that the mean spatial order of fractional derivative is ρ=1.25, the mean flow pattern parameter m=1.50, and the mean fractional roughness coefficient is nf=0.002 which is smaller than the conventional roughness coefficient, n=0.108. With these average values of the parameters and their standard deviations, simulations were performed to demonstrate the use of the methods, which is also a comparison of the classic KWE and rfKWE models.

Macropore flow in relation to the geometry and topology of soil macropore networks: Re-visiting the kinematic wave equation

The rapid flow of water through soil macropores significantly affects the partitioning of precipitation between surface runoff and infiltration and also the rate of solute transport in soil, both of which have an impact on the risk of contamination of surface water and groundwater. The kinematic wave equation is often employed as a model of gravity-driven water flow through soil macropores. The exponent in this simple model influences the pore water velocity attained in the macropores at any given input rate and is usually estimated by inverse modelling against measured flow rates or water contents. In theory, the exponent in the kinematic wave equation should depend on the geometry and topology of the conducting macropore networks, although these relationships have not so far been investigated. In this study, we related metrics of soil structure derived from X-ray images to values of the kinematic exponent estimated from drainage experiments on twenty-two columns sampled at three different field sites under two contrasting land uses and at three different depths.We found that smaller values of the exponent in the kinematic wave equation, which would equate to more rapid flow of water through soil macropores, were found in plough pan and subsoil columns of smaller macroporosity, for which biopores comprised a significant fraction. The macroporosity in these columns was more vertically oriented and poorly inter-connected, though still continuous across the sample. In contrast, topsoil columns from both arable land and grassland had better connected, denser and more isotropically-distributed macropore networks and larger values of the kinematic exponent. Our results suggest that for predictive modelling at large scales, it may be feasible to estimate the kinematic exponent using class pedotransfer functions based on pedological information such as land use and horizon type.

Element Free Galerkin (EFG) Formulation in Solving Kinematic Wave Equation for Hydrological Modelling

Meshfree methods such as Element Free Galerkin Method (EFG) are the cutting edge of numerical methods as it does not require user to perform meshing which often is time consuming and tedious. The method being "meshfree" for the user can theoretically generate more accurate models which follow more complex boundary conditions such as natural rivers etc. The time saving and potential of higher accuracy in modelling makes EFG formulation an interesting research topic. EFG formulation is used to solve kinematic wave equations which is detailed in this paper. Galerkin weighted residual method using EFG’s shape function parameters is used to discretize the partial differential equation (PDE) of kinematic wave equations. To discretize the equations, a forward difference scheme is used for temporal discretization and Picard direct substitution method is employed to solve the nonlinear system at each time step. The formulations are validated by making a comparison between well-established numerical techniques against actual recorded datasets. After which it is found that EFG method agrees with conventional established techniques, thus validating the formulations as useful. To further investigate the usefulness of this formulation, an optimal performance study was also conducted to determine the most optimal shape function parameters. The optimal shape function values improved the performance to where it was found that established methods such as FDM were surpassed and on par with FEM. This shows that EFG is a good alternative for hydrological modelling.

Coupling water, solute, and sediment transport into a new computationally efficient hydrologic model

While surface–subsurface interactions strongly impact water quality and quantity in ecosystems, only relatively little attention has been given to solute mass transfers at this boundary. The KINEROS2 (K2) model solves the 1D kinematic wave equation for water flow and the advection–dispersion equation for non-absorbing solute (or sediment) transport at the surface, whereas the HYDRUS-1D (H1D) model solves the Richards equation for flow and the advection–dispersion equation for solute transport in the subsurface. We previously coupled water flow in the surface and subsurface environments using the K2 and H1D models (H1D-K2). Herein, an innovative framework was developed to additionally couple solute transport at the surface–subsurface boundary in the simple, accurate, and computationally efficient H1D-K2 model. To test the H1D-K2 model’s accuracy and robustness, simulation results are compared to analytical solutions, experimental data, and results from the two-dimensional HYDRUS-2D (H2D) code. The relative differences in water and solute balance components between the H2D and H1D-K2 models are less than 1% for homogeneous cases and 4% for heterogeneous cases. The simulated hydrograph and chemograph are almost identical for the two models. Furthermore, simulated Bromide (Br) concentration in runoff is in good agreement with observations. The proposed model also accurately reproduces soil erosion under different rainfall patterns and intensities.

Analytical Solution of Kinematic Wave Equation for Overland Flow due to Storms Moving at a Velocity Lower than Flow Velocity

Analytical Solution of Kinematic Wave Equation for Overland Flow due to Storms Moving at a Velocity Lower than Flow Velocity

Forecasting water discharges of the river Marmarik (Armenia) according to the kinematic wave equation

The article provides an overview of methods for short-term forecasting of water flow in mountain rivers based on physical-statistical and regression equations, conceptual approaches. Requirements for forecast models for the Armenian rivers are noted, which correspond to the kinematic wave model, but so far there are no forecast results for the Armenian rivers using this model. The purpose of the study was to test a mathematical model of a kinematic wave for a short-term forecast of water discharges on the mountainous Armenian river Marmarik. The model of a kinematic wave can be obtained after simplifying
 the model of channel runoff in a general form, i.e., after excluding from consideration deformations, local constant and variable backwaters from narrowing of channels, confluence of rivers, surge phenomena, and also after excluding the influence of the lower boundary condition, which is responsible for the characteristic of the diffusion spreading of the flood wave.
 A kinematic wave model was tested on the Marmarik River, which is an important water body in Armenia. We used series of observations of urgent water discharges during the spring flood from April 1 to June 30 for 2015—2018 at the top of the river Marmarik — Hankavan settlement, lower alignment — Aghavnadzor settlement and at the alignment on the main tributary of the river Gomraget — Meghradzor village. It was found that in the dynamic parametrization of the model, the anthropogenic impact should be introduced as an additional parameter. The obtained values of water discharges when making verification forecasts in the aggregate characterize the kinematic wave model as an effective approach according to
 the criteria for assessing the quality of forecast methods of the Hydrometeorological Center and Nash—Sutcliffe. The kinematic wave model is easy to use with a small amount of initial information without meteorological data, which will facilitate its implementation for practical applications.

Application of Leak 2D to Describe Preferential Water Flow in a Soil Containing Artificial Macropores

Leak 2D is a new two-dimensional dual permeability mathematical model for the simulation of the preferential flow in the vadose zone. In this model, water flow in the soil matrix domain is described by the two-dimensional h-based Richards’ equation. Water flow in the fracture domain is estimated using the kinematic wave approach. Richards’ equation is solved by a combination of the alternating direction implicit (A.D.I.) method and the Douglas and Jones predictor−corrector method. The kinematic wave equation is solved explicitly. In the present paper, Leak 2D is calibrated and validated with data obtained in a Hele–Shaw apparatus filled with sand. Preferential flow is achieved by inserting four artificial macropores of various sizes into the soil. Six irrigations of various intensities and durations were used for the calibration and validation process. The water content at various depths was recorded by five sensors that were inserted into the soil. A comparison of the simulated water content with the measured profiles shows that Leak 2D can sufficiently describe preferential flow into the unsaturated zone of the soil, even under extreme irrigation conditions.

A New Two-dimensional Dual-permeability Model of Preferential Water Flow in the Vadose Zone

LEAK2D (L2D) is a new, two-dimensional, dual-permeability model for the simulation of preferential water flow in the vadose zone, allowing for the continuous exchange of water between the matrix and the fracture domain. It is based on the two-dimensional Richards equation for the simulation of flow in the matrix domain and on the kinematic wave equation for the simulation of flow in the fracture domain. The Richards equation is solved by a combination of the Alternating Direction Implicit method and the Douglas-Jones predictor-corrector method. This combination leads to a very efficient, stable, and time-consuming method. A variable time step is used by which any instability of the numerical solution is avoided. The water transfer from the fracture to the matrix domain is estimated as a first-order approximation of the water diffusion equation. The model was used to satisfactorily simulate preferential flow under an extreme rainfall/irrigation event. The exchange of water between the two domains depends on parameters which have physical meaning; however, their exact values are difficult to be determined or measured. Based on the most common values of these parameters found in the literature, a sensitivity analysis was performed to define their effect on the output of the model.

Impact of Slope Orientation on Inlet Spacing: Gutter Flow Analyses

A roadway’s capacity to drain itself is of utmost importance for the safety and comfort of its users. Standing water and any amount of channelized flow on roadways create nuisances to the users, and the extent of encroachment into the lanes and the water-film thickness over the lanes are crucial for motorists with relatively high speed. Guidelines cover a wide range of subjects from size and type of inlets, which capture the channelized flow for conveyance into enclosed drains, to the decision for slope orientation, but the guidelines seem to lack in checking the depth of channelized flow. HEC-22 (the urban drainage design manual of US Department of Transportation) endorses limiting the flow depths to curb height (as if the concern is no longer the roadway users) and fixes the criterion for the inlet spacing (restricted to 90 to 150 m) to maximum allowable flow spreads. This study analyzed the maximum allowable inlet spacing via setting three criteria: fixed maximums to flow depth, spread for the channel flow, and to over-lane water-film thickness. The impact of slope orientation on inlet spacing is tested along with some other factors for roadways of two types (local and highway). The results were graphed for various uniform slope orientations under a wide range of rainfall intensities for the determined inlet spacing values. This was performed by combining a kinematic wave equation solution to dismiss the conditions that lead to hydroplaning depths when using the Rational Method and Manning’s equation to obtain water depths and inlet spacings for an inlet of full capture capacity. It is found that the allowable spacing values do not constitute any major restrictions in highway setting (3 m shoulder) in terms of recommended spacing. In the local setting, however, with a maximum spread of 1.8 m, maximum allowable inlet spacing becomes a limitation in many orientations, and slope optimization under such conditions becomes crucial at times when providing the same spacing for two orientations.

A computationally efficient hydrologic modeling framework to simulate surface-subsurface hydrological processes at the hillslope scale

Considering surface and subsurface interactions is imperative to predict water movement, water quantity and quality in the environment. However, substantial execution time and over-parameterization presently limit the applicability of integrated hydrologic models at larger scales. Herein, a new efficient coupling routine for one-dimensional (1D) surface–subsurface modeling was developed by externally coupling two widely used open-source codes. KINEROS2 (K2) solves the 1D kinematic wave equation for overland flow, and HYDRUS-1D (H1D) solves the Richards equation for subsurface flow. A novel approach, combined with water balance and boundary condition switching, is used to account for surface ponding and water exchange between the two model domains. A weighting factor related to the surface water depth is used to assign the time series of exchange fluxes in one H1D profile to multiple surface nodes in the K2 calculation. This novel approach enables us to represent the entire subsurface below each overland-flow plane by one vertical soil column with different soil layering. When soil properties vary horizontally, multiple soil columns can represent spatial heterogeneities. The performance of the coupled H1D-K2 model is examined by comparing simulation results with the HYDRUS-2D (H2D) model for six benchmark problems. The solution's robustness, stability, and accuracy are assessed for a wide variety of cases, including multiple rainstorms, different slopes, heterogeneous subsurface, and different bottom boundary conditions resulting in 41 cases. The simulated hydrographs, surface water levels, and water balance components are all in good agreement. Relative mass balance errors are very small, typically not exceeding 4.0% for the coupled H1D-K2 model. Compared with the H2D model, the coupled model achieves a factor of 1.3–10 speedup by applying one H1D soil profile to an entire overland flow plane rather than to individual surface nodes.

SELECTING THE TIME OF CONCENTRATION CONSIDERING GROUNDWATER DEPTH AND RUNOFF COEFFICIENT IN THE ARGENTINIAN PAMPA REGION

The time of concentration (Tc) is one of the most sensitive parameters for calculating the maximum runoff in a basin and plays a key role in the determination of design flows. In this work, the consistency of the equations that make up the analytical methods used to estimate the Tc is evaluated through the contrast of observed events from the application of the graphical method in a plain basin located in the Pampa region, Argentina. The methodology used considers the complexity of the runoff generation process and of the water dynamics of the region taking into account the analysis of the groundwater depth and the runoff coefficient. This study shows that the equations available may generate Tc predictions with errors by up to 95% in plain basins. The results indicated that Ventura’s and Pasini’s equations are suitable to determine Tc under water deficit, while Izzard’s and Kinematic Wave equations are appropriate for saturated conditions of the system.

Hydrological Considerations in Designing Roadways: Avoiding Hydroplaning

Safety on highways is essential, but high water levels on lanes risk that. Drainage structures are mostly concerned with water spread at sideways and ignore the build-up flow on the surface. This study addresses whether optimizing cross slopes prevents hydroplaning. Water depths obtained using kinematic wave equation were tested against several studies for verification. Wide range of rainfall intensities and cross slopes was tested. Findings revealed that cross slope optimization for grades up to 10% prevents hydroplaning for intensities below 250mm/hr with widths up to 15m. The study shows cross slope optimization must be considered simultaneously with inlet design work.

Inverse Flood Routing Using Simplified Flow Equations

The paper considers the problem of inverse flood routing in reservoir operation strategy. The aim of the work is to investigate the possibility of determining the hydrograph at the upstream end based on the hydrograph required at the downstream end using simplified open channel flow models. To accomplish this, the linear kinematic wave equation, the diffusive wave equation and the linear Muskingum equation are considered. To achieve the hydrograph at the upstream end, an inverse solution of the afore mentioned equations with backward integration in the x direction is carried out. The numerical solution of the kinematic wave equation and the Muskingum equation bases on the finite difference scheme. It is shown that both these equations are able to provide satisfying results because of their exceptional properties related to numerical diffusion. In the paper, an alternative approach to solve the inverse routing using the diffusive wave model is also presented. To this end, it is described by a convolution which involves the instantaneous unit hydrograph (IUH) corresponding to the linear diffusive wave equation. Consequently, instead of a solution of partial or ordinary differential equations, the integral equation with Laguerre polynomials, used for the expansion of the upstream hydrograph, is solved. It was shown that the convolution approach is more reliable comparing to the inverse solution of the simplified models in the form of differential equations.

Improvement of soil moisture and groundwater level estimations using a scale‐consistent river parameterization for the coupled ParFlow-CLM hydrological model: A case study of the Upper Rhine Basin

• Scaling of Manning coefficient and permeability in hydrological models were proposed. • We investigate the impact of the scaling approach on results of ParFlow-CLM model. • The validity of the results is examined through an innovative application of FORM. • The average bias in soil moisture was decreased from 0. 17 mm 3 /mm 3 to 0. 1 mm 3 /mm 3. • The accuracy of simulations is more than 95 and 92 percent for Autumn and Summer. Accurate implementation of river interactions with subsurface water is critical in large-scale hydrologic models with a constant horizontal grid resolution when models apply kinematic wave approximation for both hillslope and river channel flow. The size of rivers can vary greatly in the model domain, and the implemented grid resolution is too coarse to accurately account for river interactions. Consequently, the flow velocity is underestimated when the width of the rivers is much narrower than the selected grid size. This leads to inaccuracy and uncertainties in calculations of water quantities. In addition, the rate of exfiltration and infiltration between the river and the subsurface may be overestimated as the modeled area of water exchange between rivers and subsurface is larger than reality. Therefore, the present study tests the approximation of subscale channel flow by a scaled roughness coefficient in the kinematic wave equation. For this purpose, a relationship between grid cell size and river width is used to correct flow velocity, which follows a simplified modification of the Manning-Strickler equation. The rate of exfiltration and infiltration between the subsurface and river is also corrected across riverbeds by a scaled saturated hydraulic conductivity based on the grid resolution even though the grid size is relatively large. The scaling methodology is implemented in a hydrological model coupling ParFlow (PARallel FLOW) v3.5 and the Community Land Model (CLM) v4.5. The model is applied over the Upper Rhine Basin (between France and Germany) for a time period from 2012 to 2014 and at a spatial resolution of 0.055° (∼6 km). The validity of the results is examined with satellite and in situ data through an innovative application of the First Order Reliability Method (FORM). The scaling approach shows that soil moisture estimates have improved, particularly in the summer and autumn seasons when cross-validated with independent soil moisture observations provided by the Climate Change Initiative (CCI). The results underline the use of a simple scaling procedure of the Manning coefficient and saturated hydraulic conductivity to account for the real infiltration/exfiltration rate in large-scale hydrological models with constant horizontal grid resolution. The scaling procedure also shows overall improvements in groundwater level estimation, particularly where the groundwater level is shallow (less than 5 m from the surface). By using the scaling approach, the average bias in soil moisture for the study domain was decreased from 0.17 mm 3 /mm 3 to 0.1 mm 3 /mm 3 . The FORM results show that the probability of a substantial divergence between the ParFlow-CLM-S soil moisture results and the CCI-SM observation, which is defined as more than 0.25% of the CCI-SM observation value, is less than 0.05, 0.11, 0.15, and 0.08 for autumn, winter, spring, and summer, respectively.

Kinematic reverse flood routing in natural rivers using stage data

In many developing countries, due to economic constraints, a single station on a river reach is often equipped to record flow variables. On the other hand, hydrographs at the upstream sections may also be needed for especially assessing flooded areas. The upstream flow hydrograph prediction is called the reverse flood routing. There are some reverse flood routing pocedures requiring sophisticated methods together with substantial data requirements. This study proposes a new reverse flood routing procedure, based upon the simple kinematic wave (KW) equation, requiring only easily measurable downstream stage data. The KW equation is first averaged along a channel length at a fixed time, t, assuming that channel width is spatially constant, and then the spatially averaged equation is averaged in time, Δt. The temporally averaged terms are approximated as the arithmetical mean of the corresponding terms evaluated at time t and t + Δt. The Chezy roughness equation is employed for flow velocity, and the upstream flow stage hydrograph is assumed be described by a two parameter gamma distribution (Pearson Type III). The spatially averaged mean flow depth and lateral flow are related to the downstream flow stage. The resulting routing equation is thus obtained as a function of only downstream flow stage, meaning that the method mainly requires measurements of downstream flow stage data besides the mean values of channel length, channel width, roughness coefficient and bed slope. The optimal values of the parameters of reverse flood routing are obtained using the genetic algorithm. The calibration of the model is accomplished by using the measured downstream hydrographs. The validation is performed by comparing the model-generated upstream hydrographs against the measured upstream hydrographs. The proposed model is applied to generate upstream hydrographs at four different river reaches of Tiber River, located in central Italy. The length of river reaches varied from 20 to 65 km. Several upstream hydrographs at different stations on this river are generated using the developed method and compared with the observed hydrographs. The method predicts the time to peak with less than 5% error and peak rates with less than 10% error in the short river reaches of 20 km and 31 km. It also predicts the time to peak and peak rate in other two brances of 45 km and 65 km with less than 15% error. The method satisfactorily generates upstream hydrographs, with an overall mean absolute error (MAE) of 42 m3/s.

Variable storage behavior controlled by rainfall intensity and profile structure upon saturation excess overland flow generation

The effects of rainfall intensity and terrain structure on hillslope storage behavior upon saturation excess overland flow (SOF) generation are not yet fully understood. Analytical solution techniques that provide direct insights into the hillslope storage behavior were adopted in this study. The Threshold of StOrage (TSO) at critical rainfall intensity N0 (the minimum rainfall intensity to generate SOF) and the storage upon SOF generation (SO) at rainfall intensity N (>N0) were derived using analytical solutions of the kinematic wave equation over unit-width shallow sloping aquifers. The derived analytical solutions match the results of the numerical models of the Boussinesq equation for 25 idealized hillslopes under various rainfall intensities. It was found that as N = N0, TSO is determined by hillslope structures, i.e., TSO on the convex hillslopes is always greater than that on concave hillslopes with the same gradient, and TSO on convex and concave hillslopes gradually approaches as the gradient increases. As N > N0, hillslope storage dynamics is characterised by variable storage behavior as rainfall intensity increasing, i.e., SO increases with N. An inflection point of rainfall intensity N1, below which the curvature of the bedrock profile has a limited effect on SO, was found. Although N1 was greater on steeper hillslope, the corresponding values of SO1 on hillslopes under different gradients are close, which means that the effects of rainfall intensity and profile structure on SO is balanced at N = N1. This study helps to deepen our understanding of the quantitative role of rainfall intensity and profile structure in the process of runoff generation, and has the potential to improve the free storage capacity parameter in conceptual models.

Impact of the Finite Element Mesh Structure on the Solution Accuracy of a Two-Dimensional Kinematic Wave Equation

This paper presents the influence of the finite element mesh structure on the accuracy of the numerical solution of a two-dimensional linear kinematic wave equation. This equation was solved using a two-level scheme for time integration and a modified finite element method with triangular elements for space discretization. The accuracy analysis of the applied scheme was performed using a modified equation method for three different uniform triangular meshes with the same resolution, but with a different structure. The modified equation approach based on the Taylor series truncation allowed the numerical diffusivity and dispersivity tensors to be derived, which are directly associated with numerical errors. The derived tensors depend on parameters such as the space and time interval, flow velocity, and weighting coefficients. A detailed analysis carried out for the particular values of these parameters enabled an assessment of the numerical errors that may be generated in the solution for the assumed mesh structure. The theoretical analysis was confirmed by using numerical simulations carried out for an arbitrary domain and auxiliary conditions. According to the obtained results, it appears that it is possible to improve the accuracy of the numerical solution by choosing the proper mesh structure and numerical parameters for the applied algorithm.

Water Field Distribution Characteristics under Slope Runoff and Seepage Coupled Effect Based on the Finite Element Method

The unsaturated seepage field coupled with heavy rainfall-induced surface flow mainly accounts for the slope instability. If the slope contains macropores, the coupled model and solution process significantly differ from the traditional one (without macropores). Most of the studies on the variation of the water field under the coupled effect of runoff and seepage on the slope did not consider the macropore structure. In this paper, two coupled Richards equations were used to describe the MF (Macropore Flow), and along with the kinematic wave equation, they were applied to establish a coupled model of SR (Slope Runoff) and MF. The numerical solving of the coupled model was realized by the COMSOL PDE finite element method, and an innovative laboratory test was conducted to verify the numerical results. The effects of different factors (i.e., rainfall intensity, rainfall duration, saturated conductivity, and slope roughness coefficient) on water content and ponding depth with and without macropores were compared and analyzed. The results show that infiltration is more likely to happen in MF than UF (Unsaturated Flow, without macropore). The depths of the saturation zone and the wetting front of MF are obviously greater than those of UF. When SR occurs, rainfall duration has the most significant influence on infiltration. When macropores are considered, the ponding depth is smaller at the beginning of rainfall, while the effects are not obvious in the later period. Rain intensity and roughness coefficient have significant influences on the ponding depth. Therefore, macropores should not be ignored in the analysis of the slope seepage field.

Anomalous overland flow on hillslopes: A fractional kinematic wave model, its solutions and verification with data from laboratory observations

This paper analyses a fractional kinematic wave equation (fKWE) for overland flow and evaluates its solutions for applications using data from overland flow flumes with simulated rainfall in the laboratory. Solutions of fKWE presented have been derived for large time or when the Laplace transform variable s→0, which is one of the most important situations for overland flow. The solutions include expressions for the depth, velocity, and unit discharge, of overland flow. Fitting the approximate solution for the depth of overland flow to the data yields the values of the order of space-fractional derivatives, ρ, which is around ρ=1.5. It is found that ρ increases with the slope gradient and the rainfall intensity. The approximate order of ρ=1.5, which is about the average of order 1 for the advection equation and 2 for the diffusion equation, implies that diffusive mechanisms manifest in the overland flow. The findings mean that either an fKWE is used or a diffusion term is needed to account for dynamic forces in overland flow as the fKWE captures more physical mechanisms.

Уравнения кинематической волны для размываемых русел

The mathematical model of kinematic wave, that is widely used in hydrological calculations, is generalized to compute processes in deformable channels. Self-similar solutions to the kinematic wave equations, namely, the discontinuous wave of increase and the “simple” wave of decrease are generalized. A numerical method is proposed for solving the kinematic wave equations for deformable channels. The comparison of calculation results with self-similar solutions revealed a good agreement.