unsaturated flow equation , @kroszynskidagan1975 linearized by adopting the [@gardner1958] exponential model for the relative hydraulic conductivity, $k(\psi )=e^{\kappa_a (\psi -\psi_a)}$, where $\kappa_a$ is the sorptive number \[1/L\] (related to pore size). They adopted the same exponential form for the moisture capacity model, $\theta (\psi
)=e^{\kappa_k (\psi -\psi_k)}$, where $\psi_k$ is the pressure at which $k(\psi)=1$, $\kappa_a=\kappa_k$, and $\psi_a=\psi_k$, leading to the simplified form $C(\psi)=S_y\kappa_a e^{\kappa_a (\psi
  -\psi_a)}$. In the limit as $\kappa_k=\kappa_a \rightarrow \infty$ their solution reduces to that of [@dagan1967]. The relationship between pressure head and water content is a step function. [@kroszynskidagan1975] took unsaturated flow above the water table into account but ignored the effects of confined aquifer storage, leading to early-time step-change behavior similar to @boulton1954a and @dagan1967.

Increasingly Realistic Saturated-Unsaturated Well Test Models
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[@mathiasbutler2006] combined the confined aquifer flow equation with a one-dimensional linearized version of for a vadose zone of finite thickness. Their water table was treated as a fixed boundary with known flow conditions, decoupling the unsaturated and saturated solutions at the water table. Although they only considered a one-dimensional unsaturated zone, they included the additional flexibility provided by different exponents $(\kappa_a \neq \kappa_k)$. [@mathiasbutler2006] did not consider a partially penetrating well, 