Kinematic Wave Model Research Articles

Physics-informed neural network for solution of forward and inverse kinematic wave problems

The one-dimensional kinematic wave (KW) model (KWM) is widely used in surface water and water quality hydrology as well as for modeling the movement of traffic on long highways. The KW model involves a first-order nonlinear partial differential equation which is solved numerically for realistic geometric representation, input, and initial and boundary conditions. This paper developed a method to solve the partial differential equation using a neural network structure, called the physics-informed neural network (PINN). The PINN was applied to solve the one-dimensional nonlinear KWM for forward and inverse problems. To enhance the accuracy and convergence and to suppress the spurious oscillation at the steep front, a technique with redistributed collocation points as an improved algorithm of PINN was employed. To deal with the fractional order of the nonlinear advection term, a stop-gradient technique was proposed. For validation, the KWM for steady and unsteady overland flow, and open channel flow without lateral inflow was analyzed for the forward problem. For the inverse problem, the unknown Manning coefficient was correctly computed, demonstrating the capability of the proposed algorithm for the identification of KWM parameters.

A new kinematic wave model that describes lateral subsurface flow and percolation in hillslopes

Kinematic wave (KW) models are commonly used for flood forecasting in large river basins, but have limitations in their ability to simulate percolation effects. Here, we suggest an extended KW model wherein the vertical pressure head is under non-hydrostatic conditions. The proposed model uses the Brooks Corey and Mualem models to estimate the water retention curve and unsaturated hydraulic conductivity and adopts the pressure head gradient as a new variable to calculate runoff based on changing water content distributions over time due to percolation. The increase in the pressure head gradient with rainfall infiltration is determined from the relationship between water storage and pressure head distribution, while the decrease in pressure head gradient due to percolation is modeled as exponential decay with time. A comparison of the proposed KW model and a numerical solution of the Richards equation shows that the proposed KW model can reproduce the storage effect caused by percolation, delaying runoff and reducing peak discharge. The agreement between the proposed KW model and the Richards equation is high for soils with high water retention. Conversely, the agreement becomes lower for soil with low water retention or deep soil layer thickness, where the complexity of water content distribution requires further refinement. The proposed model can easily be incorporated into distributed models while low computational cost with keeping the important percolation effects.

Flood Routing in Rivers by Numerical Modeling Preissmann Scheme, Saint-venant Equation and Muskingum’s Method Based on Kinematic Wave Model

Flood Routing in Rivers by Numerical Modeling Preissmann Scheme, Saint-venant Equation and Muskingum’s Method Based on Kinematic Wave Model

Parameter Estimation Using the Inverse Problem Method for Simulating Lateral Inflow and Runoff Depth in a small catchment of Amazon.

The inverse problem method can be applied to determine the properties of hydrological phenomena and estimate the parameters, which cannot be measured directly. This type of inverse focus can facilitate the implementation of the kinematic wave model (direct model-DM), to fill gaps for lateral inflow rate and runoff depth in watersheds. Thus, the goal of the study was the application of the inverse problem method (IP). The lateral inflow rate was generally obtained as a Fourier transform to represent any watersheds. The study was developed using a small catchment in the Amazon where intense rainfall events occur, producing runoff and sediments, which affect rural populations. Lateral inflow rate and runoff depth were derived using precipitation data and parameters estimated through the KINEROS2 (K2)/direct model (DM) model and the ensuing solution methods with MCMC (Markov chains Monte Carlo)/Fourier transform. The developed method was applied to four rainfall-runoff events, leading to a good fit between the observed and predicted data (Nash-Sutcliffe coefficients between 0.76 and 0.85 and RMSE values between 1.80 mm and 6.72 mm).

Three-dimensional X-ray imaging of macropore flow

Macropores are known to be important pathways for the rapid transport of water, solutes and colloids in soil. Nevertheless, we still know very little about how the topology and geometry of macropore networks govern water flow configurations and velocities in natural soil. In this study, we aimed at gaining more insight into macropore flow by using X-ray tomography to quantify air–water distributions in the macropore networks of undisturbed topsoil and subsoil columns of a clay soil at varying steady-state flow rates. We observed that while large fractions of the macropore network remained air-filled, the air phase only became entrapped when the irrigation rate was very close to the saturated hydraulic conductivity of the soil. The data enabled us to parameterize a kinematic wave model for water flow following the approach proposed in Jarvis et al. (2017a). Follow-up experiments would be required to evaluate whether these kinematic wave parameters derived by X-ray imaging match with those obtained from outflow measurements. We found that quantitative X-ray imaging of macropore flow through soils still remains a challenging task. We recommend that future experiments are conducted on smaller soil samples to improve image resolution and minimize experimental time spans as well as X-ray image noise and illumination bias. Such experiments could also include 3-D tracer imaging to identify the imaged macropore networks transporting most of the water (i.e. the backbone) at varying steady irrigation rates.

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.

Temporal changes in lahar sediment run‐off characteristics and run‐off coefficients in the Arimura River basin of Sakurajima volcano, Japan

AbstractThe rapid flowing nature of lahars can cause serious hazards in the proximal to distal areas of volcanoes. The accumulation of ash can form an impermeable layer on the ground and prevent the infiltration of precipitation. The surface water thus formed erodes the ash layer and causes the surface permeability to rise again. Over time, this accumulation and erosion sequence changes the lahar risk. Large numbers of lahars are observed each year around Sakurajima, which is known as one of the most active volcanoes in Japan. In this study, the temporal change of lahar sediment run‐off properties and run‐off coefficients have been investigated between 2015 and 2020 in the Arimura River basin of Sakurajima volcano, Japan. X Band Multi‐Parameter (XMP) radars are used to gather precipitation data in the basin. The sediment data have been obtained by Load, Vibration, Pressure (LVP) sensors, and a kinematic wave model is adopted to calculate surface run‐off. Lahars are categorized as regular and irregular considering the correspondence of measured and calculated waveforms of the hydrographs based on the kinematic wave model. Analyses reveal that run‐off coefficient changes depend on the volcanic activity and decreases after an irregular lahar compared with a previous regular lahar. LVP sensor analysis results show that sediment concentrations of the lahars change in time. It is found that irregular and regular lahars are distinct in terms of velocity and sediment concentrations. It is estimated that irregular lahars carry larger material than regular lahars based on the vibration data. This study underlines that the properties of lahars change over time under the influence of ash fall thickness, and these changes can be detected in detail with our kinematic wave model and LVP sensors.

An analytical model for simulating the rainfall-interception-infiltration-runoff process with non-uniform rainfall

The rainfall runoff process is a critical factor in determining the transport of surface materials. Simulating the surface runoff process is fundamental to accurately characterize soil erosion and soil nutrient loss. This research aims to develop a comprehensive simulation model for rainfall-interception-infiltration-runoff under vegetation cover conditions. The model integrates three key components: a vegetation interception model, Philip's infiltration model, and a kinematic wave model. By combining these models, an analytical solution is derived to simulate slope runoff considering vegetation interception and infiltration during non-constant rainfall events. To validate the reliability of the analytical solution, a numerical solution was obtained using the Pressimann Box scheme method and compared with the analytical results. The comparison confirms the accuracy and robustness of the analytical solution (R2 = 0.984, RMSE = 0.0049 cm/min, NS = 0.969). Moreover, this study investigates the influence of two significant parameters, Intm and k, on the production flow process. The analysis reveals that both parameters exert a significant impact on the timing of production initiation and the magnitude of runoff. Specifically, Intm exhibits a positive correlation with runoff intensity, while k displays a negative correlation. This research introduces a novel simulation method that enhances our understanding and modeling of rainfall production and convergence under complex slope conditions. The proposed model provides valuable insights into rainfall-runoff dynamics, particularly in scenarios characterized by varying rainfall patterns and vegetation cover. Overall, this study contributes to advancing the field of hydrological modeling and offers a practical approach for quantifying soil erosion and nutrient loss under different environmental conditions.

Optimal Deployment of Sensors along Freeway Corridors for Traffic Accident Detection

Sensor deployment is important to the detection performance of traffic accidents on freeways. However, the temporally and spatially varying effects of accident risks are seldom considered in sensor deployment, which can induce a decrease in accident detection accuracy and detection timeliness. To address this problem, this paper proposes an optimal deployment method of sensors according to the temporal-spatial effect of accidents on traffic flow. Considering the uncertainty of traffic accidents, the temporal-spatial distribution of accident risk is estimated according to historical accident data. Then the kinematic wave model and Van Aerder model are introduced to analyze the temporal-spatial effect of accidents on traffic flow. To obtain the optimal locations of sensors, the sensor deployment problem is converted into a coverage problem on accident risk in time and space, and an optimal deployment model of sensors is built by considering the temporal-spatial effect of accident risk, the coverage rate on accident risk, the detection timeliness of the accident, and deployment cost. The particle swarm optimization algorithm is introduced to solve it. A case study is carried out to validate the proposed method using consistency, sensitivity, reliability, and comparison experiments. The results show that the proposed method is effective and reliable under different traffic volumes, and it outperforms the uniform spacing method in traffic accident detection.

Self-organized criticality of traffic flow: Implications for congestion management technologies

Self-organized criticality (SOC) is a celebrated paradigm from the 90’s for understanding dynamical systems naturally driven to its critical point, where the power-law dynamics taking place make predictions practically impossible, such as in stock prices, earthquakes, pandemics and many other problems in science related to phase transitions. Shortly thereafter, it was realized that traffic flow might be in the SOC category, implying that conventional traffic management strategies seeking to maximize the local flows can become detrimental. This paper shows that the Kinematic Wave model with triangular fundamental diagram, and many other related traffic models, indeed exhibit SOC, thanks in part to the fractal nature of traffic exposed here on the one hand, and our need to get to our destinations as soon as possible, on the other hand.Important implications for congestion management of traffic near the critical region are discussed, such as: (i) Jam sizes obey a power-law distribution with exponent 1/2, implying that both its mean and variance become ill-defined and therefore impossible to estimate. (ii) Traffic in the critical region is chaotic in the sense that predictions becomes extremely sensitive to initial conditions. (iii) However, aggregate measures of performance such as delays and average speeds are not heavy tailed, and can be characterized exactly by different scalings of the Airy distribution, (iv) Traffic state time–space “heat maps” are self-affine fractals where the basic unit is a triangle, in the shape of the fundamental diagram, containing 3 traffic states: voids, capacity and jams. This fractal nature of traffic flow calls for analysis methods currently not used in our field.

Developing a 1D kinematic wave model for simulating the downstream flow of Tra Khuc river

Developing a 1D kinematic wave model for simulating the downstream flow of Tra Khuc river

An eco-driving algorithm based on vehicle to infrastructure (V2I) communications for signalized intersections

Efficient operations of traffic signals are of critical importance in urban areas, as signalized intersections prevent the smooth flow of traffic and cause delays. This paper devises an eco-driving algorithm based on connected vehicle technologies, with basic kinematic wave and car-following models. The objectives of the proposed algorithm are to increase the throughput of signal intersections and decrease fuel consumption. Specifically, we focus on a signalized intersection under mixed traffic flows with connected and autonomous vehicles (AVs) and human-driven vehicles (HVs). Through the proposed algorithm, the vehicle speeds at the intersection (i.e., the intersection control speed) and signal timings can be adjusted in response to the real-time traffic conditions. According to the signal timing and the speed at the intersection, the algorithm estimates the time points of each vehicle entering the intersection. An advisory speed limit approach is formulated for each AV, making the vehicle enter the intersection at the allocated timing with the control speed. An onboard alert is set for each HV to stop or pass through. The algorithm is evaluated under various market penetration rates of AVs, different congestion levels, and with signal actuation. The results indicate that the eco-driving algorithm can increase the throughput and average travel speed at signalized intersections in addition to gaining fuel savings.

Efficient evaluation of stochastic traffic flow models using Gaussian process approximation

This paper studies a Gaussian process approximation for a class of stochastic traffic flow models. It can be used to efficiently and accurately evaluate the joint (in the spatial and temporal sense) distribution of vehicle-density distributions in road traffic networks of arbitrary topology. The Gaussian approximation follows, via a scaling-limit argument, from a Markovian model that is consistent with discrete-space kinematic wave models. We describe in detail how this formal result can be converted into a computational procedure. The performance of our approach is demonstrated through a series of experiments that feature various realistic scenarios. Moreover, we discuss the computational complexity of our approach by assessing how computation times depend on the network size. We also argue that the (debatable) assumption that the vehicles’ headways are exponentially distributed does not negatively impact the accuracy of our approximation.

Numerical Examples of Non-Dissipative Discontinuous Kinematic Waves in Open Channels

The kinematic wave model under the assumption of balanced gravity and friction forces has been applied in open channel hydraulics and surface hydrology. There persists a severe misunderstanding that a discontinuity of a kinematic wave occurs due to a discontinuity of input and then dissipates. This study clarifies that a discontinuity can develop without dissipation under the smoothness of all input. The theory of first-order quasilinear partial differential equations shows that Cauchy problems for the kinematic wave model have unique measurable and bounded solutions, which are possibly discontinuous. Numerical examples are presented to visualize the fundamental properties of discontinuous kinematic waves.

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.

Experimental Dispersion identification using a fitted state-space model

An identification method that can estimate the dispersion relation of waveguides experimentally using an efficient and accurate procedure is presented. The method fits a linear state-space model before resorting to a kinematic wave model in the frequency region near the pre-identified natural frequencies. The eigenvectors, or mode-shapes, are computed at the sensor locations, and based on the reduced-Bloch mode expansion method, the propagation modes are fitted to match the identified vibration mode-shape at these frequencies. Classical methods to identify the dispersion relation from measured data can be computationally expensive and time-consuming, with limited accuracy in cases of multimode propagation. Results show that the fitted dynamic model expands the frequency range of obtained dispersion curves and enhances the speed of computation and accuracy. The method is derived and verified for both lumped and distributed systems, using numerical finite element simulation. Experimental verification is carried out on two acoustical waveguides. The first is a circular ring-shaped array of coupled Helmholtz resonators, modeled using a lumped parameter model. The second is an air-filled acoustic wave-tube that is modeled as a distributed acoustic-elastic coupled waveguide. The method's strengths and weaknesses are discussed from the experimentally obtained dispersion curves, and its main feature, the ability to fit the dispersion model of weak modes, is highlighted.

Variable scale effects on hillslope soil erosion during rainfall-runoff processes

The variation of soil erosion across scales remains a controversial issue. A theoretical framework, coupling the normalized Green-Ampt equation for infiltration, 1D kinematic wave model for overland flow, and WEPP erosion modeling approaches for soil erosion, was used to explain and quantify the direct effect of scale on the soil erosion process. The results show that the relation between interrill erosion and slope length accords with a power-law decreasing trend, while the relation of rill erosion versus slope length shows a power-law increasing trend. Moreover, the power-law scaling of interrill erosion becomes more prominent with an increase of rainfall duration and intensity but is nearly independent of the interrill erodibility factor. The rainfall duration, soil type, soil critical shear stress, sediment transport coefficient, and rill erodibility factor substantially influence the magnitude of scale effect on rill erosion, while rainfall intensity, slope gradient, and Manning's roughness have minor effects. It suggests that the scale effect on soil erosion is variable and dynamic for different erosion types, rainfall and soil characteristics, which can explain why the controversial scaling relationships of soil erosion exist in previous studies. The physically based soil erosion model WEPP is compared to the empirical models USLE and RUSLE at various scales. Only when rill erosion plays a dominant role in soil loss do all these models exhibit consistent scale effect on erosion. These results provide insights into the essence of scaling in the hillslope soil erosion process, which are expected to benefit the development of upscaling techniques for the large-scale hydrological, geomorphological, and ecological processes.

Stable local dynamics for day-to-day departure time choice

Existing dynamical systems for day-to-day departure time choice are either unstable, or stable but assuming drivers to possess complete information and make decisions on both arrival and departure times. In this paper, we present a new dynamical system with local shifting of departure times, such that a driver only defers or advances his/her departure time to a time interval later or earlier with lower costs. We establish the asymmetrical upper bounds of the deferral and advance coefficients for the discrete model to be well-defined. We then derive the continuous version as a kinematic wave model and present some examples of symmetrical deferral and advance coefficients. We demonstrate that the stationary state of the dynamical system is the same as the user equilibrium, and the user equilibrium is proved with Lyapunov's second method to be stable for the symmetrical deferral and advance coefficients. With numerical examples, we verify the analytical results and examine the model's sensitivity to different factors with different combinations of heuristic asymmetrical coefficients and theoretically stable symmetrical coefficients. Both analytical and numerical results confirm that the new dynamical system is asymptotically stable in a stability region. This study provides some guidelines on how to derive new day-to-day dynamical system models of departure time user equilibrium. Such a dynamical system can potentially be applied to solve the general dynamic traffic assignment problem in the future.

Adaptive network traffic control with an integrated model-based and data-driven approach and a decentralised solution method

This paper presents an adaptive traffic controller for stochastic road networks with an integrated model-based and data-driven solution framework. The model-based optimisation component operates based upon an underlying kinematic wave model driven by stochastic demand within a prediction horizon. The data-driven optimisation component operates based upon an approximate dynamic programming (ADP) formulation which approximates the state-control interactions over future stages with a parametric approximator. The approximator reduces the computational complexity of the adaptive control problem by parameterising the state and decision space. The parametric approximator is to be iteratively updated with online feeding of realisations of traffic states via a temporal difference (TD) learning process. Our results reveal that incorporation of the model-based component facilitates the training of the ADP-based state approximator, and hence improve the overall performance of the control system. We further develop a decentralised solution approach in which individual intersections are allowed to derive their own control policies in an asynchronous manner. The data-driven ADP approximator would serve as a central agent coordinating the control policies derived at individual intersections in the network. This is shown to be able to improve and stabilise the performance of the overall control system even under congested conditions. This is a significant progress in adaptive control system design with use of decentralised optimisation techniques. The present study contributes to the adaptive network traffic control with uncertainties through use of advanced modelling and optimisation methods.

Reconstruction of the MPD system parameters at the gas kicks into well with the measured surface data in situ by the mathematical modeling

One of the main problems of managed pressure drilling (MPD) is detection and monitoring of gas kicks from the formation to the annulus. The model is proposed to estimate the parameters of the gas kick and gas-liquid flow in a well during MPD process based on measurements performed only at the surface. A gas-liquid flow is described by the kinematic wave model. The ability of gaseous hydrocarbon to dissolve in mud, when drilling on an oil based mud (OBM), is taken into account. The hydrodynamic parameters of the flow are estimated by numerically solving the equations of fluid motion jointly with measured input data: the flowrate of the mud at inlet, the flowrate of gas-liquid mixture at outlet, and the surface back pressure (SBP). The control parameters are the change in mass gas and void fraction over the well depth and on time, as well as BHP. To verify the proposed model, the calculation of gas-liquid flow parameters is conducted on base of the detailed measured data of field test during drilling of 16,000-ft deep well using a water based mud (WBM). The pattern of the gas kick into the well from the formation is reconstructed, the change in void fraction during gas rising was traced until it was kicked to the surface, and the total mass of gas entering the well is determined. The error is estimated by comparing deviations of the measured and calculated bottomhole pressure (BHP) value.