$15\mathrm{.}8.($a$)$ For a confined aquifer, the Thiem equation can be used to express the drawdown surrounding a well in terms of well pumping rate, $Q_w,$ transmissivity, $T,$ and radius of influence, $R.$ Explain why this equation does not accurately account for the drawdown at the well. Show that by selecting the radius of influence as

$R=r_w$exp$(\frac{2T{s}_w}{Q_w})$

the Thiem equation can be made to accurately account for the drawdown at the well. What are the limitations of the Thiem equation and why does it have these limitations$?$ Under what circumstances would you use the Thiem equation? $($b$)$ In deriving the Thiem equation, it is assumed that Darcy's law is applicable in the aquifer immediately surrounding the well intake. If a well has a pumping rate of $30\frac{L}{s},$ diameter of $10cm,$ and a $1-m$ long intake, estimate the range of hydraulic conductivities for which the Thiem equation could be applied without violating Darcy's law.