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. 1 - 9
• 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 is the primary means for rivers to receive sewage, and the permitted loadings of pollutants, based on the pollutant mixing zone, are critical parameters in discharge management. The adjoint equation method has demonstrated considerable benefits in addressing inverse problems in hydraulics. However, optimization objectives based on the discrepancies between predicted and observed concentrations cannot be straightforwardly employed for determining the permissible loadings, thus restricting 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 tailored to lateral effluents utilizing the depth-averaged pollution transport equations for lateral effluents. Given the narrow and elongated nature of the pollutant mixing zone in lateral discharges, a new formula for the error source term is introduced, targeting the length of the pollutant mixing zone as the primary objective. The adjustment value for lateral effluents is calculated by solving the adjoint equations and employing the BFGS optimization algorithm, iteratively determining the permitted pollutant loadings from lateral discharges. The simulation of the forward problem lays the groundwork for the inverse problem. This research focuses on an outlet from a sewage treatment facility in the upper reaches of the Yangtze River to assess the hydrodynamic and water quality model. The findings indicated that the water quality model can accurately simulate the pollutant mixing zone, with the prediction error for the permanganate index (CODMn) maintained at 16.7%, satisfying the precision requirements of water quality simulations in practical engineering. Following the accuracy verification in the forward problem, an experiment is conducted to assess the efficacy of the proposed inversion method. The inversion outcomes revealed that, after 18 iterations, the computational precision for the length of the pollutant mixing zone is maintained below 0.01 m despite experiencing two fluctuations in the convergence process due to inherent limitations of the BFGS method. In practical engineering applications, the demand for precision in 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, offering valuable technical support for managing effluents in natural rivers.