therefore disregard confined aquifer storage, causing pressure pulses to propagate instantaneously through the saturated zone. Both solutions exhibit an instantaneous step-like increase in drawdown when pumping starts.

Delayed Yield Unconfined Response
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[@boulton1954b] extended Theis’ transient confined theory to include the effect of delayed yield due to movement of the water table in unconfined aquifers. Boulton’s proposed solutions ([-@boulton1954b; -@boulton1963]) reproduce all three segments of the unconfined time-drawdown curve. In his formulation of delayed yield, he assumed as the water table falls water is released from storage (through drainage) gradually, rather than instantaneously as in the free-surface solutions of [@boulton1954a] and [@dagan1967]. This approach yielded an integro-differential flow equation in terms of vertically averaged drawdown $s^\ast$ as $$\label{boulton_solution}
  \frac{\partial ^2 s^\ast}{\partial r^2}+
  \frac{1}{r}\frac{\partial s^\ast}{\partial r}=
  \left[\frac{S}{T}\frac{\partial s^\ast}{\partial t} \right]+
  \left\lbrace\alpha S_y \int _0^t \frac{\partial s^\ast}{\partial \tau}
  e^{-\alpha (t-\tau )}\;\mathrm{d}\tau \right \rbrace$$ which Boulton linearized by treating $T$ as constant. The term in square brackets is instantaneous confined storage, the term in braces is a convolution integral representing storage released gradually since pumping began, due to water table decline. [@boulton1963] showed the time when delayed yield effects become negligible is approximately equal to $\frac{1}{\alpha}$, leading to the term “delay index”. [@prickett1965] 