A semi-analytical solution for one-dimensional pollutant transport equation in different types of river networks

This study proposes a semi-analytical solution to a one-dimensional advection-dispersion equation considering reaction and source terms to predict and describe pollutant transport in various river networks using the Laplace transform approach. Moreover, the proposed semi-analytical solution is expanded for variable coefficients of the transport equation by increasing the number of middle nodes and associated intervals. The presented method solves the mass transport equation by considering advection and dispersion phenomena for one branch of the river network. The final mass balance equation is thus obtained in different types of river networks for each node. Regarding the connection matrix, the matrices of the characteristics of flow, pollutant, and network geometry are first determined as the model inputs. Then, by considering the diffusion and final mass conservation (Laplace transform of mass balance equation) equations for each node in the arbitrary river network, the mass balance and diffusion matrices are calculated in terms of the Laplace variable (s). Thus, a complex system of nonlinear algebraic equations is produced in terms of s, which can be solved using inverse Laplace algorithms to determine the concentration value at every node. The present study introduces four practical scenarios of pollutant transport in river networks in order to evaluate the proposed semi-analytical solution; the first is a branch-type river network with constant coefficients. The second case is a loop-type river network with variable coefficients. The third case simultaneously considers a point pollutant source and a distributed pollutant source in a branchtype river network. The fourth case is a branch-type river network under the dispersion-dominant phenomenon with a point source. Results show that the proposed semi-analytical solution is adequately capable of modeling pollutant transport and indicating its critical features in complex states of various river networks. Moreover, exploring the literature shows that no analytical solution has been introduced for river networks before undertaking the present study.