Solute transport through radial advection–dispersion is a common phenomenon in many applications, such as aquifer decontamination, heat exchange for geothermal exploration, and tracer testing. Numerous analytical solutions to the problem of tracer testing in a radially divergent flow field are available for fitting breakthrough curves. However, all of these analytical solutions assume non-uniform flow velocity based on Thiem's solution, which varies only spatially, not temporally. Although an injection well with a constant injection rate in a tracer test causes both spatial and temporal variability of the radial flow field, previous analytical studies have employed a steady-state groundwater flow solution, i.e. Thiem's solution, to derive analytical solutions for breakthrough curves or spatial concentration distribution curves. To the best of our knowledge, no analytical solution to the radial advection–dispersion equation under spatially and temporally variable flow conditions driven by injection at a constant rate is currently available. Here, we propose a novel semi-analytical solution for describing solute transport in spatially and temporally variable flow induced by an injection well with a constant injection rate. We find that, above certain threshold values of dimensionless parameters related to aquifer flow properties, solute transport parameters, and the injection rate, the differences between previous analytical strategies our proposed semi-analytical solution are negligible. These threshold values tend to increase more sharply with an instantaneous injection source than a continuous injection source. In other words, it is more difficult for an instantaneous injection source to reach the condition where the difference between our approach and previous ones is negligible than for a continuous injection source type. Our findings clarify when existing analytical solutions can be reasonably used for parameter estimation through fitting of the breakthrough curve to field data and when the proposed semi-analytical solution is a better alternative.