A novel method for analytically solving multi-species advective–dispersive transport equations sequentially coupled with first-order decay reactions

Abstract

Analytical solutions for coupled multi-species solute transport problems are difficult to derive and relatively few in subsurface hydrology. Decomposition strategy such as linear transform format or matrix diagonalization method which decomposes the set of coupled advective–dispersive transport equations into a system of independent differential equations have been widely used to derive the analytical solution for coupled multi-species solute transport problem. These decomposition techniques are generally limited to derive the analytical solution for an infinite or a semi-infinite domain. In this study, we present a novel method for analytically solving multi-species advective–dispersive transport equations sequentially coupled by first-order decay reactions. The method first performs Laplace transform with respect to time and the generalized integral transform technique with respect to the spatial coordinate to convert the set of partial differential equations into a system of algebraic equations. Subsequently, the system of algebraic equations is solved using simple algebraic manipulation, thus the concentrations in the transformed domain for each species can be independently obtained. Ultimately, the concentrations in the original domain for all species are obtained by successive application of Laplace and the corresponding generalized integral transform inversions. A coupled four-species transport problem in a finite domain is used to demonstrate the robustness of the proposed method for deriving the analytical solutions associated with sequentially coupled multi-species solute transport problem. The developed analytical solution is tested by comparing their results against those generated with the corresponding numerical solutions. Results show perfect agreements between the analytical and numerical solutions. Moreover, the developed analytical solution is compared with the analytical solutions for a semi-infinite domain available in literature to illustrate the impacts of the exit boundary conditions on coupled multi-species transport. It is observed that significant discrepancies exist between two solutions for small Peclet numbers, whereas two solutions deviate negligibly each other for medium Peclet numbers.