to use for estimating $K$ from pumping tests with multiple observation wells [@simmons2008]. Equation (commonly called the Thiem equation) was tested in the 1930’s both in the field (@wenzel1932recent performed a 48-hour pumping test with 80 observation wells in Grand Island, Nebraska) and in the laboratory (@wyckoff1932dupuitflow developed a 15-degree unconfined wedge sand tank to simulate converging flow). Both found the steady-state solution lacking in ability to consistently estimate aquifer parameters. @wenzel1942 developed several complex averaging approaches (e.g., the “limiting” and “gradient” formulas) to attempt to consistently estimate $K$ using steady-state confined equations for a finite system from transient unconfined data. @muskat1932partialpen considered partial-penetration effects in steady-state flow to wells, discussing the nature of errors associated with assumption of uniform flux across the well screen in a partially penetrating well. Muskat’s textbook on porous media flow [@muskat1937book] summarized much of what was known in hydrology and petroleum reservoir engineering around the time of the next major advance in well test solutions by Theis.

Confined Transient Flow
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[@theis1935] utilized the analogy between transient groundwater flow and heat conduction to develop an analytical solution for confined transient flow to a pumping well (see Figure \[fig:diagram\]). He initially applied his solution to 