$(20$ points$)$ The initial piezometric head in a confined aquifer, which has a thickness of $11\mathrm{.}6$ m $,$ is $85\mathrm{.}7$ m above sea level. A well with a radius of $0\mathrm{.}5$ m pumps at a constant rate of $1,240\frac{m^3}{d}ay.$ After the cone of depression has achieved equilibrium $($i$.$e$.,$ steady state$),$ the piezometric heads at two observation wells $40$ m $($well $1)$ and $95$ m $($well $2)$ from the pumping well are measured as $78\mathrm{.}9$ m and $83\mathrm{.}4$ m above sea level, respectively.

$($a$)(5$ points$)$ determine the aquifer transmissivity.

$($b$)(5$ points$)$ determine the radius of influence of the pumping.

$($c$)(5$ points$)$ determine the total well losses in and around the pumping well if the observed head in the pumping well during equilibrium is $46\mathrm{.}6$ m $.$

$($d$)(5$ points$)$ A tracer was injected into well $2.$ We find it takes $50$ days for the injected dye to travel to the well $1.$ Estimate the aquifer porosity $($n$).$