Polynomial-based approximate solutions to the Boussinesq equation near a well

•Constructed approximate analytical solutions to the Boussinesq equation that models the injection from a single well in an unconfined aquifer.•Obtained approximate solutions that contain two parts: one that models the behavior near the well and the other that models the behavior in the whole solution domain.•Removed the singular term from the solution, by modifying the equation, since decoupling for the nonlinear problems is not possible in general.•Showed how to use higher moments of the solution in order to increase the degree of approximating polynomials that model the solution away from the wellbore.