Ensemble modeling of stochastic unsteady open-channel flow in terms of its time–space evolutionary probability distribution – Part 1: theoretical development

Abstract

Abstract. The Saint-Venant equations are commonly used as the governing equations to solve for modeling the spatially varied unsteady flow in open channels. The presence of uncertainties in the channel or flow parameters renders these equations stochastic, thus requiring their solution in a stochastic framework in order to quantify the ensemble behavior and the variability of the process. While the Monte Carlo approach can be used for such a solution, its computational expense and its large number of simulations act to its disadvantage. This study proposes, explains, and derives a new methodology for solving the stochastic Saint-Venant equations in only one shot, without the need for a large number of simulations. The proposed methodology is derived by developing the nonlocal Lagrangian-Eulerian FokkerPlanck equation of the characteristic form of the stochastic Saint-Venant equations for an open-channel flow process, with an uncertain roughness coefficient. A numerical method for its solution is subsequently devised. The application and validation of this methodology are provided in a companion paper, in which the statistical results computed by the proposed methodology are compared against the results obtained by the Monte Carlo approach.

Ensemble modeling of the two-dimensional stochastic confined groundwater flow through the evolution of the hydraulic head's probability density function

“…However, this assumption could break down depending on the behavior of the 𝜂 functions. For 𝜂 functions with similar periodicity and close frequencies, the crosscovariance terms could become more comparable in magnitude to the autocovariance values, contrary to the previous simplification (Dib and Kavvas, 2018)…”

Ensemble modeling of stochastic unsteady open-channel flow in terms of its time–space evolutionary probability distribution – Part 2: numerical application

“…This study applied the proposed FPE methodology derived in the companion paper by Dib and Kavvas (2018) to a stochastic unsteady open-channel flow problem, with an uncertain roughness coefficient. The equations used to describe the open-channel flow problem were the Saint-Venant equations, transformed into their characteristic form by the method of characteristics.…”

“…From this probabilistic solution, one can then obtain the ensemble mean and ensemble variance of the process as they evolve in time and space. This new methodology, which is proposed, explained, and derived in the companion paper by Dib and Kavvas (2018), makes use of the ensemble averaging technique developed in Kavvas (2003) to obtain a Fokker-Planck Equation (FPE) that specifically describes an unsteady openchannel flow process. Some other hydrologic processes have been successfully simulated by following a similar procedure, which involved applying the ensemble averaging technique of Kavvas (2003) to their corresponding governing equations and obtaining the FPEs specific to their case.…”

“…Therefore, in the wake of the preceding discussions, the first objective of this study is to use the FPE methodology derived in the companion paper for the unsteady open-channel flow process (Dib and Kavvas, 2018) and to apply it to a representative stochastic unsteady open-channel flow problem in order to solve for the probability density of the state variables of the flow process and to provide a quantitative de-scription of the expected behavior and variability of the system in one single simulation. The second objective is to evaluate the performance of the proposed methodology and to validate its results by comparing the statistical properties of the flow variables computed by the FPE methodology against those calculated by the MC approach.…”

“…The result is shown in Eq. (5), while its detailed derivation can be found in the companion paper by Dib and Kavvas (2018).…”