can be viewed as an approximation to the zero-order linearized free-surface boundary condition of [@boulton1954a] and [@dagan1967]. Streltsova considered the initial condition $\xi
(r,t=0)=b$ and used the same boundary condition at the pumping well and the outer boundary $(r\rightarrow \infty )$ used by [@theis1935] and [@boulton1963]. Equation has the solution [@streltsova1972b] $$\label{strelstsova_sol}
  \frac{\partial \xi}{\partial t} =
  -\alpha_T \int _0^t e^{-\alpha_T (t-\tau)}
  \frac{\partial s^\ast}{\partial \tau}\;\mathrm{d}\tau$$ where $\alpha_T = K_z/(S_yb_z)$. Substituting into produces solution of [@boulton1954b; @boulton1963]; the two solutions are equivalent. Boulton’s delayed yield theory (like that of Streltsova) does not account for flow in unsaturated zone but instead treats water table as material boundary moving vertically downward under influence of gravity. [@streltsova1973] used field data collected by [@meyer1962] to demonstrate unsaturated flow had virtually no impact on the observed delayed process. Although Streltsova’s solution related Boulton’s delay index to physical aquifer properties, it was later found to be a function of $r$ [@neuman1975; @herrera1978]. The delayed yield solutions of Boulton and Streltsova do not account for vertical flow within the unconfined aquifer through simplifying assumptions; they cannot be extended to account for partially penetrating pumping and observation wells.

Prickett’s pumping test in the vicinity of Lawrenceville, Illinois [@prickett1965] showed that 