Introducing the novel method of equation-oriented modeling for pollutant transport in different types of river networks

Rivers are vital for ecosystems, water resources, and sustaining life on Earth. Modeling pollutant transport in river networks is essential for reducing environmental contamination but classical numerical methods struggle with the complexities of these networks. This research introduces a novel Equation Oriented Modeling approach using open-source mathematical software to enhance flexibility, transparency, ease of modification and integration of multiple physics in complex river networks. The EOM approach formulates the problem as a coupled system of partial differential equations. In this approach, the boundary conditions at internal nodes are defined to ensure mass conservation and concentration continuity, while also accounting for the effect of dispersion in the transport of pollution to other branches which often overlooked in classical methods. Once the boundary and initial conditions for the network are established, the system is solved using method of lines. unlike classical method, EOM using the power of mathematical software to discretize the equations to reduce coding, especially in loop networks, provides the option to define equation terms to apply more complexity to modeling and integrate multiphysics in modeling, and consider the effect of dispersion in mixing and rejection of pollution at the nodes and also facilitates coupling with other tools for post-processing. To verify this method, two networks of tree-type and loop-tree type were utilized. According to the results, the error parameters such as RMSE and MAE are close to zero and the R² parameter was between 98 and 100. However, considering the results of the error parameters and graphs, especially in the case 2 test at times 0.33, 0.5, 0.58 and 0.67, it can be stated that whenever the dispersion mechanism becomes more pronounced due to the increase in the dispersion coefficient and in addition to the time difference in the pollution reaching the node, the difference between the results of the classical method and EOM becomes apparent, so that at time 0.58 the values of R2, RMSE, MAE reached 0.87, 0.0542 and 0.0296, respectively. In addition, the results of the EOM model had appropriate stability due to the severe changes in the flow parameters in the case 2 test and convergence was achieved.