Inversion Method for Permitting Loadings of Pollutant from Lateral Effluents Based on Adjoint Equations

• Published in: 工程科学与技术 Vol. 57; pp. 176 - 184
• Main Authors: SHI Xiaoyan, ZHANG Hong, TAO Chunhua, LU Lingjiang, WAN Xin, LIU Zhaowei
• Format: Journal Article
• Language: English
• Published: Editorial Department of Journal of Sichuan University (Engineering Science Edition) 01.07.2025
• Subjects: adjoint equation; BFGS method; inverse problems; lateral effluents in rivers; permitting loadings of pollutant; water quality simulation
• ISSN: 2096-3246

Lateral discharge serves as the primary pathway through which rivers receive sewage, and the permitted pollutant loadings, determined based on the pollutant mixing zone, represent critical parameters in discharge management. The adjoint equation method demonstrates substantial benefits in solving inverse problems in hydraulics. However, optimization objectives that rely on discrepancies between predicted and observed concentrations cannot be directly applied to determine the permissible loadings, limiting the application of the adjoint equation method to this issue. This study applies the adjoint equation method to derive both the control equation and boundary conditions specifically suited to lateral effluents utilizing the depth-averaged pollution transport equations for lateral discharges. Considering the narrow and elongated characteristics of the pollutant mixing zone in lateral discharges, a new formula for the error source term is introduced, with the length of the pollutant mixing zone defined as the primary objective. The adjustment value for lateral effluents is calculated by solving the adjoint equations and employing the BFGS optimization algorithm, which iteratively determines the permitted pollutant loadings from lateral discharges. The simulation of the forward problem establishes the foundation for solving the inverse problem. This research focuses on an outlet from a sewage treatment facility located in the upper reaches of the Yangtze River to evaluate the hydrodynamic and water quality model. The findings indicated that the water quality model accurately simulates the pollutant mixing zone, with the prediction error for the permanganate index (CODMn) maintained at 16.7%, meeting the precision requirements of water quality simulations in practical engineering. Following the accuracy verification in the forward problem, an experiment is conducted to evaluate the performance of the proposed inversion method. The inversion outcomes revealed that, after 18 iterations, the computational precision for the length of the pollutant mixing zone remains below 0.01 m despite two fluctuations during the convergence process due to inherent limitations of the BFGS method. In practical engineering applications, the required precision for controlling the mixing zone length is comparatively modest and is achieved within six iterations, reducing the error to 1 m. These results highlight the method's high computational accuracy and rapid convergence rate, providing valuable technical support for managing effluents in natural rivers.