Semi-analytical solution of single-well push–pull test under transient flow conditions

The single-well push–pull (SWPP) test has been extensively used in parameter estimation models. Numerous analytical solutions to the problem are available, all of which consider solute transport in a non-uniform flow field from steady radial flow created by the SWPP test. However, the non-uniform flow velocity is based on the Thiem equation and varies in a spatial manner, rather than a temporal manner. No analytical solution has been previously described for the solute transport equation of SWPP under fully transient flow. In this study, the generalized integral transform technique (GITT) was used to develop a new semi-analytical solution for solute transport in the SWPP test under fully transient flow. Four phases of the SWPP test were included: injection, chaser, rest, and extraction. With the proposed solution, the differences between a transient flow SWPP model solution and a piecewise steady-state flow SWPP model solution were non-negligible; such differences increased with decreases in the dimensionless parameter related to aquifer flow properties, which is proportional to transmissivity, aquifer thickness, and porosity, but inversely proportional to the storage coefficient and pumping rate. Additionally, long tails of BTCs are characteristic of type curves under the transient flow model. The long tails of BTCs under transient flow could result in overestimation of dispersivity if the SWPP model uses piecewise steady-state flow, rather than transient flow, for parameter estimation. The proposed semi-analytical solution provides a useful tool for curve-fitting during parameter estimation when the effects of transient flow are significant. The proposed solution can also serve as a performance comparison for testing numerical solutions to the SWPP test.