River quality management: Integrating uncertainty, failure probability, and assimilation capacity

Managing river water quality is challenging due to uncertainties in hydraulic and hydrologic parameters. This study integrates the symmetric exponential function (SEF) approach for solving the advection-dispersion equation with the Monte Carlo method in MATLAB. This combination allows us to explore the river's assimilation capacity and the failure probability ($P_f$) of maintaining desired water quality standards. Here, $P_f$ represents the likelihood of pollutant concentrations exceeding acceptable limits under varying river conditions. A key contribution of this study is the introduction of a novel equation, developed using the Genetic Programming (GP) soft computing tool, to calculate assimilation capacity considering the failure probability of water quality provision. This equation provides a valuable tool for risk assessment in water resource management by quantifying pollutant assimilation dynamics. Its robustness is validated through high Coefficient of Determination (R²) and Overall Index (OI) values near 1, along with low Root Mean Square Error (RMSE) and Mean Absolute Error (MAE). The study identifies critical river characteristics, such as flow velocity and pollutant load, significantly influencing the reliability index ($\beta$). By outlining how adjustments in these parameters can achieve a target reliability index ($\beta =4.526$), our study offers a practical approach to safeguarding river ecosystems. For example, increasing flow velocity by 76 % can shift the river from a safe state ($P_f=3\times {10}^{-5}$) to a hazardous state ($P_f=1$), while a 44 % decrease in velocity allows for 57 % more pollutant assimilation. These findings highlight the importance of flow control as a cost-effective strategy for mitigating high pollutant concentrations and ensuring sustainable water quality management. By integrating numerical approaches with reliability sampling methods and soft computing techniques, this study enhances understanding of river system dynamics and supports informed decision-making for protecting water resources.