$\beta$ provides additional adjustment of the shape of the intermediate portion of the time-drawdown curve (beyond adjustments possible with $K_D$ and $\sigma^\ast$ alone), leading to improved estimates of $S_y$. When $\beta=0$ simplifies to .

Hybrid Water Table Boundary Condition
-------------------------------------

The solution of [@neuman1972; @neuman1974] was accepted by many hydrologists “as the preferred model ostensibly because it appears to make the fewest simplifying assumptions” [@moenchetal2001]. Despite acceptance, [@nwankwor1984] and [@moench1995] pointed out that significant difference might exist between measured and model-predicted drawdowns, especially at locations near the water table, leading to significantly underestimated $S_y$ using Neuman’s models (e.g., see Figure \[fig:capecod\]). @moench1995 attributed the inability of Neuman’s models to give reasonable estimates of $S_y$ and capture this observed behavior near the water table due to the later’s disregard of “gradual drainage”. In an attempt to resolve this problem, [@moench1995] replaced the instantaneous moving water table boundary condition used by Neuman with one containing a @boulton1954b delayed yield convolution integral, $$\label{moench_hybrid}
  \int _0^t\frac{\partial s}{\partial \tau}
  \sum _{m=1}^M \alpha _m e^{-\alpha _m (t-\tau )}\;\mathrm{d}\tau =-
  \frac{K_z}{S_y}\frac{\partial s}{\partial z}$$ This hybrid boundary condition ($M=1$ in @moench1995) included the convolution source term @boulton1954b [@boulton1963] and @streltsova1972a [@streltsova1972b] used in their depth-averaged governing flow equations. 