The drawdown or change in hydraulic head, $s$ (a measure of the water pressure above some geodetic datum) a distance $r$ from a well at time $t$ from which water is being pumped at a constant rate, $Q$, can be modelled using the Theis equation, $$ s(r, t) = H_0 - H(r,t) = \frac{Q}{4\pi T}W(u), \quad \mathrm{where}\quad u = \frac{r^2S}{4Tt}. $$ Here $H_0$ is the hydraulic head in the absence of the well, $S$ is the aquifer storage coefficient (volume of water released per unit decrease in $H$ per unit area) and $T$ is the transmissivity (a measure of how much water is transported horizontally per unit time). The Well Function, $W(u)$ is simply the exponential integral, $E_1(u)$.

For a well being pumped at $Q = 1000\;\mathrm{m^3\,day^{-1}}$ from an aquifer described by the parameters $H_0 = 20\;\mathrm{m}$, $S = 0.0003$, $T = 1000\;\mathrm{m^2\,day^{-1}}$, determine the height of the hydraulic head as a function of $r$ after $t = 1\;\mathrm{day}$ of pumping.

Compare your answer with the approximate version of the Theis equation known as the Jacob equation, in which the well function is taken to be appoximately $W(u) \approx -\gamma - \ln u$ where $\gamma=0.577215664\cdots$ is the Euler-Mascheroni constant.