Analytical Solutions for Sequentially Coupled Multi-Species Reactive Transport Problems

Abstract

Multi-species reactive transport equations coupled through sorption and sequential first-order reactions are commonly used to model sites contaminated with radioactive wastes, chlorinated solvents and nitrogenous species. Although researchers have been attempting to solve various forms of this reactive transport problem for over fifty years, a general closed-form analytical solution to this problem is not available in the published literature. In the first part of this research work, a closed-form analytical solution to this problem is deduced involving a generic spatially-varying initial condition. Two distinct solutions are derived for Dirichlet and Cauchy boundary conditions each with Bateman-type source terms. The proposed solution procedure employs a combination of Laplace and linear transform methods to uncouple and solve the system of partial differential equations. The final solution is organized and presented in a general format that represents the solutions to both the boundary conditions. In addition, the mathematical concepts for deriving the solution within a generic framework that can be used for solving similar transport problems are also presented. In the second part of this research work, the computational techniques for implementing the new solutions are discussed. These techniques are then adopted to develop a general computer code which is used to verify the solutions. In addition, several special-case solutions for simpler transport problems involving zero initial condition, identical retardation factors, zero advection, zero dispersion and steady-state condition are also derived. Where ever possible, these special-case solutions are compared against previously published analytical solutions to establish the validity of the new solution. The performance of the new solution is tested against other published analytical and semi-analytical solutions using a set of example problems. Finally, an investigation into extending the general solution to multiple dimensions using the approximate Domenico solution is also presented.