Question

A well fully penetrates a \(30\) m thick confined aquifer. After a long period of pumping at a constant rate of \(0.06\) m\(^3\)/s, drawdowns of \(3.6\) m and \(2.0\) m are found at distances \(60\) m and \(120\) m away from the well, respectively. The hydraulic conductivity of the aquifer is \(n \times 10^{-4}\) m/s. The value of \(n\) is ________ (Rounded off to two decimal places)

Hint:

Apply the Thiem equation for a confined aquifer using the two drawdown observations.

Answer 1

Correct Answer: 1.38

Step 1: Use the log-cycle form of the Thiem equation.
Instead of natural logs, write the confined-aquifer Thiem relation with base-$10$ logs: \[ K = \frac{2.303\,Q\,\log_{10}(r_2/r_1)}{2\pi b(s_1-s_2)} \] This is the same relation as the natural-log form, just carried through base-$10$ logs.

Step 2: Plug in the numbers.
$Q=0.06$ m$^3$/s, $b=30$ m, $s_1-s_2=3.6-2.0=1.6$ m, and $\log_{10}(120/60)=\log_{10}(2)=0.3010$.

Step 3: Substitute and simplify.
\[ K = \frac{2.303\times0.06\times0.3010}{2\pi\times30\times1.6} = \frac{0.04159}{301.59} \]
Step 4: Evaluate.
\[ K \approx 1.38\times10^{-4}\ \text{m/s} \]
Final Answer:
Both log forms of Thiem's equation agree on a conductivity near $1.38\times10^{-4}$ m/s. \[ \boxed{n \approx 1.38} \]