Review: Revisiting the Theis solution derivation to enhance understanding and application

The Theis solution is perhaps the most influential and frequently used analytical model in groundwater hydrology. Its publication in 1935 led to immediate and continued use for simulating hydraulic head drawdown, in both confined and unconfined aquifers, as a tool in aquifer parameter estimation. For educational purposes, the Theis solution and the related Jacob’s approximation often serve as the backbone for teaching pumping-well theory, including topics such as boundary conditions in aquifers, image well theory, linear superposition, and pumping-induced streamflow depletion. Clearly, a thorough understanding of the Theis solution is critical for groundwater engineers and hydrologists. However, the solution often is presented as a “black box”, neglecting the actual origins of its derivation and accompanying physical context. This can lead to misconceptions about the model and its inherent limitations. In this paper, a physically based detailed derivation of the Theis solution is presented, along with a method of calculating drawdown from a pumping well without resorting to the final Theis equation. Examples of both constant-rate pumping and variable-rate pumping are presented and compared to results using the original Theis solution. In particular, variable pumping rates are accounted for by direct numerical integration of an earlier form in the original Theis derivation, removing the need for linear superposition of solutions in time. In this way, it is hoped the paper will provide a method of calculation that ties the model user to the physical meaning of the solution, including its assumptions.