A fully penetrating well of diameter $30\text{ cm}$ is pumped at a constant steady discharge from a confined aquifer of thickness $15\text{ m}$ with hydraulic conductivity $K = 20\text{ m/day}$. Two observation wells located at radial distances of $r_1 = 10\text{ m}$ and $r_2 = 50\text{ m}$ record drawdowns of $s_1 = 3.0\text{ m}$ and $s_2 = 1.0\text{ m}$, respectively. Taking $\pi \approx 3.1416$, the steady discharge of the well is ________ $\text{m}^3\text{/day}$ (round off to the nearest integer).
Answer Key:
2343
Step-by-Step Solution & Explanation
For a steady-state pumping well in a confined aquifer, the Thiem equation is:
$Q = \frac{2 \pi K b (h_2 - h_1)}{\ln(r_2 / r_1)} = \frac{2 \pi K b (s_1 - s_2)}{\ln(r_2 / r_1)}$
Given:
- $K = 20\text{ m/day}$
- $b = 15\text{ m}$
- $s_1 - s_2 = 3.0 - 1.0 = 2.0\text{ m}$
- $r_1 = 10\text{ m}, r_2 = 50\text{ m} \implies \ln(50/10) = \ln(5) \approx 1.6094$
Substitute values:
$Q = \frac{2 \times 3.1416 \times 20 \times 15 \times 2.0}{1.6094} = \frac{7539.84}{1.6094} \approx 2343.3\text{ m}^3\text{/day}$
Rounded to the nearest integer: $2343\text{ m}^3\text{/day}$.