Theis equation revisited: From infinite to finite aquifer and from fully to partially penetrating well

The Theis equation has seen widespread use in the fields of hydrogeology, groundwater flow analysis, and pumping test analysis. Originally derived from the principles of heat conduction with an assumption of an infinite line heat source, this equation has been widely applied in hydrogeology. However, in reality, wells have finite lengths, and the implications of assuming an infinite length have not been thoroughly explored. This study aims to derive and validate an analytical solution for flow towards wells in a confined aquifer while considering both a finite well length and the presence of aquifer boundaries, using a point-source approach. The resulting solution can also account for anisotropy in the porous media and applies to both fully and partially penetrating wells. To assess its accuracy, the study compares its results with those obtained from the Theis equation and a numerical model, which serves as a benchmark. This study considers two scenarios: one with a relatively large screen length compared to the aquifer thickness, and the other with a relatively small screen length in comparison to the aquifer thickness. The study's findings indicate that the Theis equation accurately represents outcomes only when the screen length-to-aquifer thickness ratio is high. However, the Theis equation falls short when this ratio is small, which is often the case in the majority of pumping wells. In such situations, the Theis equation tends to significantly underestimate or overestimate the actual drawdown in the aquifer, with the extent of error depending on the proximity to the well screen.