- d)} & b-\ell < z < b-d \\ 0 &
    \text{otherwise}\end{cases},$$ where $\ell$ and $d$ are the upper and lower boundaries of the pumping well screen, as measured from the initial top of the aquifer.

The two sources of non-linearity in the unconfined problem are: 1) the boundary condition is applied at the water table, the location of which is unknown *a priori*; 2) the boundary condition applied on the water table includes $s^2$ terms.

In order to solve this non-linear problem both Boulton and Dagan linearized it by disregarding second order components in the free-surface boundary condition and forcing the free surface to stay at its initial position, yielding $$\label{boulton_linear_water_table}
  \frac{\partial s}{\partial t}=-
  \frac{K_z}{S_y}\frac{\partial s}{\partial z} \qquad\qquad z=h_0,$$ which now has no horizontal flux component after neglecting second-order terms. Equation can be written in non-dimensional form as $$\label{dimensionless_MWT_bc}
  \frac{\partial s_D}{\partial t_D}=-
  K_D \sigma^{\ast} \frac{\partial s_D}{\partial z_D} \qquad\qquad z_D=1,$$ where $K_D=K_z/K_r$ is the dimensionless anisotropy ratio and $\sigma^{\ast}=S/S_y$ is the dimensionless storage ratio.

Both [@boulton1954a] and [@dagan1967] solutions reproduce the intermediate and late segments of the typical unconfined time-drawdown curve, but neither of them reproduces the early segment of the curve. Both are solutions to the Laplace equation, and 