unconfined flow, assuming instantaneous drainage due to water table movement. The analytical solution was based on a Green’s function heat conduction solution in an infinite axis-symmetric slab due to an instantaneous line heat source or sink [@carslaw1921]. With the aid of mathematician Clarence Lubin, Theis extended the heat conduction solution to a continuous source, motivated to better explain the results of pumping tests like the 1931 test in Grand Island. [@theis1935] gave an expression for drawdown due to pumping a well at rate $Q$ in a homogeneous, isotropic confined aquifer of infinite radial extent as an exponential integral $$\label{theis}
  s(r,t)=\frac{Q}{4\pi T}\int_{r^2 /(4 \alpha_s t)}^{\infty}\frac{e^{-u}}{u}
  \;\mathrm{d}u,$$ where $s=h_0(r)-h(t,r)$ is drawdown, $h_0$ is pre-test hydraulic head, $T=Kb$ is transmissivity, and $S=S_s b$ is storativity. Equation is a solution to the diffusion equation, with zero-drawdown inital and far-field conditions, $$s(r,t=0) = s(r\rightarrow \infty,t) = 0.$$ The pumping well was approximated by a line sink (zero radius), and the source term assigned there was based upon , $${\label{boulton_sink_well}}
  \lim_{r \rightarrow 0} r\frac{\partial s}{\partial r}=-\frac{Q}{2 \pi T}.$$

![Unconfined well test diagram[]{data-label="fig:diagram"}](well-diagram.eps){width="60.00000%"}

Although the transient governing equation was known through analogy with heat conduction, the transient storage mechanism (analogous to specific heat capacity) was not completely 