Ensemble Modeling of Hydrologic and Hydraulic Processes at One Shot: Application to Kinematic Open-Channel Flow under Uncertain Channel Properties and Uncertain Lateral Flow Conditions by the Stochastic Method of Characteristics

Abstract

A stochastic kinematic wave model for open channel flow is developed under uncertain channel properties and uncertain lateral flow conditions. Applying a known methodology, the Fokker-Planck equation (FPE) of the kinematic open-channel flow process under uncertain channel properties and uncertain lateral flow conditions is derived using the method of characteristics. Because every stochastic partial differential equation has a one-to-one relationship with a nonlocal Lagrangian-Eulerian Fokker-Planck equation (LEFPE), the LEFPE for the governing equation of any hydrologic or hydraulic process can be developed as the physically based stochastic model of the particular process. To quantify the ensemble behavior of a process, LEFPE provides a quantitative description of the time-space evolution of the probability density function of the state variables of the process at one shot. The nonlocal LEFPE reduces to the classical local FPE, which is more convenient to solve, under certain assumptions. The developed methodology is applied to two test problems under varying channel and lateral flow conditions, and the results are validated by Monte Carlo simulations. The numerical applications show that the developed FPE can express the ensemble behavior of the kinematic wave process under uncertain channel properties and uncertain lateral flow conditions adequately.