Inverse Flood Routing Using Simplified Flow Equations

Abstract

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.

Simulation of the Entire Process of an Interbasin Water Transfer Project for Flow Routing

“…The results showed that the second method had a better fitting effect. Similarly, G ąsiorowski et al [21] verified the rationality and accuracy of the two equations in the open-channel flow inversion model by performing a reverse integral inverse solution in the x direction of the linear motion wave equation and the linear Muskingum equation. At the same time, Badfar et al [22] established a flow inversion model based on the Muskingum model.…”

Explicit Scheme for a Hydrological Channel Routing: Mathematical Model and Practical Application

“…Open channel routing is a procedure aimed at determining the timing and magnitude of flow at specific points along a channel branch, utilizing a measured or estimated inflow hydrograph [1][2][3]. The routing is carried out using lumped (hydrological) or distributed (hydraulic) methods [4].…”

A Physically‐Based Reverse‐Stage Routing Model Considering Lateral Flow for Establishing Normal Rating Curves at Ungauged Upstream River Sections

“…In this study, we have broadly classified the available reverse routing algorithms into two classes: (a) methods those solve the de Saint Venant's equations or its simplified forms, such as the linear kinematic wave model, the linear Muskingum type model linked to the kinematic wave model (Koussis et al., 2012), and the linear Advection‐Diffusion (AD) model expressed by convolution (Gąsiorowski & Szymkiewicz, 2022) and using numerical methods (Bruen & Dooge, 2007; Dooge & Bruen, 2005; Eli et al., 1974; Spada et al., 2017; Szymkiewicz, 1996) and (b) methods those solve the parametric governing equations with an additional assumption of knowing the shape or the probability density function (PDF) of the unknown inflow hydrograph a priori (Badfar et al., 2021; D’Oria & Tanda, 2012; D’Oria et al., 2014; Kaya et al., 2017; Tayfur & Moramarco, 2022; Zucco et al., 2015). These assumptions make the second approach more parametric than the first category and may not serve well in field conditions, when the shape and the PDF can change significantly from one flood event to another.…”

“…For reverse routing, Szymkiewicz (1993Szymkiewicz ( , 1996 attempted to generalize the implicit finite difference scheme by limiting the spatial (Ψ) and temporal (θ) weighting parameters, wherein θ and Ψ were responsible for stable solutions in the forward and reverse routing studies, respectively. Recently, Gąsiorowski and Szymkiewicz (2022) noted that a set of θ and Ψ will result in an exact upstream solution if the corresponding diffusion coefficient (D) is the same as that resulted from the exact upstream flow hydrograph. However, for the field application, the D at the inlet section of a reach cannot be estimated without knowing the inflow a priori.…”