pumping well, it is convenient to use $L_C = b$ as a characteristic length, $T_C = Sb^2/T$ as a characteristic time, and $H_C = Q/(4 \pi T)$ as a characteristic head. The dimensionless diffusion equation is $$\label{diffusion}
  \frac{\partial ^2 s_D}{\partial r_D^2}+\frac{1}{r_D}\frac{\partial
    s_D}{\partial r_D}=\frac{\partial s_D}{\partial t_D},$$ where $r_D=r/L_C$, $s_D=s/H_c$, and $t_D=t/T_C$ are scaled by characteristic quantities.

The [@theis1935] solution was developed for field application to estimate aquifer hydraulic properties, but it saw limited use because it was difficult to compute the exponential integral for arbitrary inputs. [@wenzel1942] proposed a type-curve method that enabled graphical application of the [@theis1935] solution to field data. [@cooperjacob1946] suggested for large values of $t_D$ ($t_D
\geq 25$), the infinite integral in the [@theis1935] solution can be approximated as $$\label{JacobCooper}
s_D(t_D,r_D) = \int_{r^2/(4\alpha_st)}^{\infty} \frac{e^{-u}}{u} \; \mathrm{d}u \approx
\log_e \left(\frac{4 Tt}{r^2S}\right) - \gamma$$ where $\gamma \approx 0.57722$ is the Euler-Mascheroni constant. This leads to Jacob and Cooper’s straight-line simplification $$\Delta s \approx 2.3 \frac{Q}{4 \pi T}$$ where $\Delta s$ is the drawdown over one log-cycle (base 10) of time. The intercept of the straight-line approximation is related to $S$ through This approximation made estimating hydraulic parameters much simpler at large $t_D$. @hantush1961 later extended Theis’ confined solution for 