A novel analytical power series solution for solute transport in a radially convergent flow field

The concentration breakthrough curves at a pumping well for solute transport in a radially converge at flow field are governed by an advective-dispersive second order partial differential equation with a radial distance-dependent velocity and dispersion coefficient. The Laplace transform is generally first employed to eliminate the temporal derivative to solve the partial differential equation. The Laplace transformed equations are then converted to the standard form of the special Airy function through successive applications of variable change. This study presents the solution of the Laplace-transformed equation without using the special Airy function. A direct power series method and a power series method with variable changes to eliminate the advection term that usually results in numerical errors for large Peclet numbers are applied to obtain an analytical solution in the Laplace domain. The obtained solutions are compared to other Airy function-formed solutions to examine the method's robustness and accuracy. Analytical results indicate that the Laplace transform power series method with variable change can effectively and accurately handle the radial advection-dispersion equation of high Peclet numbers, whereas the direct power series method can only evaluate the solution for medium Peclet numbers. The novel power series technique with variable change is valuable for future quantitative hydrogeological issues with variable dependent differential equation and can be extended to higher dimensional problems.