A semi-analytical solution for simulating contaminant transport subject to chain-decay reactions

We present a set of new, semi-analytical solutions to simulate three-dimensional contaminant transport subject to first-order chain-decay reactions. The aquifer is assumed to be areally semi-infinite, but finite in thickness. The analytical solution can treat the transformation of contaminants into daughter products, leading to decay chains consisting of multiple contaminant species and various reaction pathways. The solution in its current form is capable of accounting for up to seven species and four decay levels. The complex pathways are represented by means of first-order decay and production terms, while branching ratios account for decay stoichiometry. Besides advection, dispersion, bio-chemical or radioactive decay and daughter product formation, the model also accounts for sorption of contaminants on the aquifer solid phase with each species having a different retardation factor. First-type contaminant boundary conditions are utilized at the source (x=0 m) and can be either constant-in-time for each species, or the concentration can be allowed to undergo first-order decay. The solutions are obtained by exponential Fourier, Fourier cosine and Laplace transforms. Limiting forms of the solutions can be obtained in closed form, but we evaluate the general solutions by numerically inverting the analytical solutions in exponential Fourier and Laplace transform spaces. Various cases are generated and the solutions are verified against the HydroGeoSphere numerical model.