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V_e = \pi r_e^2 h, \quad dV = 2\pi r \, h dr 

p_r = \frac{1}{V_e} \int p(r) dV = \frac{2}{r_e^2} \int p(r) \, r \, dr

p(r) = p_i + \frac{q_t}{2 \pi \sigma} \, \ln \frac{r}{r_e} = p_{wf} + \frac{q_t}{2 \pi \sigma} \, \ln \frac{r}{r_w} , \quad r_{wf} < r \leq r_e

p_r =  \frac{2}{r_e^2} \int \bigg[ p_i -+ \frac{q_t}{2\pi \sigma} \ln \frac{r}{r_we} \bigg] \, r \, dr = p_i - \frac{q_t}{4\pi \sigma}