above the base of the aquifer) as a moving material boundary subject to the condition $h\left( r,z=\xi,t\right)=z$. He considered the water table without recharge to be comprised of a constant set of particles, leading to the kinematic boundary condition $$\label{dynamic}
  \frac{D}{Dt}\left(h - z \right) = 0$$ which is a statement of conservation of mass, for an incompressible fluid. @boulton1954a considered the Darcy velocity of the water table as $u_z=-\frac{K_z}{S_y}\frac{\partial h}{\partial z}$ and $u_r=-\frac{K_r}{S_y}\frac{\partial h}{\partial r}$, and expressed the total derivative as $$\label{material_derivative}
  \frac{D}{Dt}=\frac{\partial}{\partial t}-
  \frac{K_r}{S_y}\frac{\partial h}{\partial r}\frac{\partial}{\partial r}-
  \frac{K_z}{S_y}\frac{\partial h}{\partial z}\frac{\partial}{\partial z},$$ where $K_r$ and $K_z$ are radial and vertical hydraulic conductivity components. Using , the kinematic boundary condition in terms of drawdown is $${\label{free_surface}}
  \frac{\partial s}{\partial t}-\frac{K_r}{S_y}
  \left( \frac{\partial s}{\partial r} \right)^2-
  \frac{K_z}{S_y} \left( \frac{\partial s}{\partial z} \right)^2=-
  \frac{K_z}{S_y}\frac{\partial s}{\partial z}.$$

@boulton1954a utilized the wellbore and far-field boundary conditions of @theis1935. He also considered the aquifer rests on an impermeable flat horizontal boundary $\left. \partial
  h/\partial z \right|_{z=0}= 0$; this was also inferred by @theis1935 because of his two-dimensional radial flow assumption. [@dagan1967] extended Boulton’s water table solution to the partially penetrating case by replacing the wellbore boundary condition with $$\lim_{r \rightarrow 0} r\frac{\partial s}{\partial r}=
  \begin{cases}\frac{Q}{2\pi K (\ell 