Assume the water is flowing into a well in a horizontal radial direction consider an elementary cylinder of soil having radius r thickness $dr$ and height h.

Let,

q = he rate of discharge

$dh$ = fall in water level

q = kiA

i $\cong$ $\frac{dh}{dr}$ (Dupit's assumption)

A = 2$\pi$rh

q = k. $\frac{dh}{dr}$ 2$\pi$rh

Re-arranging and integrating

$\int_{r_1}^{r_2} \frac{dr}{r} = \frac{2\pi k}{d} \int_{h_1}^{h_2} h .dh$

$k = \frac{2.303 .d \log_{10} \frac{r_2}{r_1}} {\pi (h_2^2 - h_1^2)}$