Problem $3$:

Given that the ground water table is depressed due to the pumping rate $Q$ of $0\mathrm{.}35f\frac{t^3}{sec}$ at the

well of $2ft$ diameter in a $90-ft$ deep unconfined horizontal layer of medium fine sand as

shown in Fig. $3.$ Two field observations from the boreholes: $($a$)$ the water table at a radial

distance $r_2$ of ${160}^{\text{'}}$ from the axis of the well is ${20}^{\text{'}}$ below the ground surface and $($b$)$ the water

table is lowered to ${36}^{\text{'}}$ below the ground surface at a radial distance $r_l$ of $20\text{'}$ from the well's

axis.

$($a$)$ Determine the permeability constant $k$ in Darcy's law using the relevant case of Dupuit's

$1$D seepage model$/$equation$,$

$($b$)$ Taking $r_2$ as $r_{\text{f}\text{a}\text{r}}$ whose total head as $h_{\text{f}\text{a}\text{r}}$ with the elevation datum at the base of the

stratum, determine the equation for the depressed top profile of the water table $($equivalent to

the total head $h(r)$ as a function of the radial distance $r$ from the well axis from $1\text{'}$ to $160\text{'}.$

$($c$)$ Use Excel or any graphics$/$plotting software to plot it from $r=1\text{'}$ to $160\text{'}$