analytically derived solutions.

Early Well Test Solutions
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Dupuit’s Steady-State Finite-Domain Solutions
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[@dupuit1857] considered steady-state radial flow to a well pumping at constant volumetric flowrate $Q$ \[L$^3$/T\] in a horizontal homogeneous confined aquifer of thickness $b$ \[L\]. He used Darcy’s law [@darcy1856] to express the velocity of groundwater flow $u$ \[L/T\] in terms of radial hydraulic head gradient $\left(\partial
  h/\partial r\right)$ as $$\label{darcys-law}
  u=K\frac{\partial h}{\partial r},$$ where $K=kg/\nu$ is hydraulic conductivity \[L/T\], $k$ is formation permeability \[L$^2$\], $g$ is the gravitational constant \[L/T$^2$\], $\nu$ is fluid kinematic viscosity \[L$^2$/T\], $h=\psi+z$ is hydraulic head \[L\], $\psi$ is gage pressure head \[L\], and $z$ is elevation above an arbitrary datum \[L\]. Darcy derived a form equivalent to for one-dimensional flow through sand-packed pipes. Dupuit was the first to apply to converging flow by combining it with mass conservation $Q=\left(2\pi rb \right)u$ across a cylindrical shell concentric with the well, leading to $$\label{dupuit_1}
  Q=K\left( 2\pi rb\right)\frac{\partial h}{\partial r}.$$ Integrating between two radial distances $r_1$ and $r_2$ from the pumping well, Dupuit evaluated the confined steady-state head difference between the two points as $$\label{dupuit_confined}
  h(r_{2})-h(r_{1})=\frac{Q}{2\pi Kb}\log\left( \frac{r_2}{r_1}\right).$$ This is the solution for flow to a well at the center of a circular island, 