For an unconfined (water-table) tube well discharging Q with drawdown s through a screen (strainer) of length L and well radius r_w, with radius of influence R, which expression gives the required screen length in terms of aquifer hydraulic conductivity K?

Introduction / Context:
Design of unconfined tube wells requires selecting an adequate screen length to safely pass the target discharge at the allowable drawdown. This uses steady radial-flow relations (Thiem equation) adapted to water-table conditions and distributed inflow along the screen length.

Given Data / Assumptions:

• Unconfined aquifer with hydraulic conductivity K.
• Well radius r_w and radius of influence R (R >> r_w).
• Steady discharge Q with drawdown s at the well under design pumping.
• Uniform inflow along screen length L (first-approximation design).

Concept / Approach:
For unconfined flow, the Thiem relation connects discharge to head difference and geometry. Spreading the inflow over the screen length gives an average entrance flux. Rearrangement provides the screen length L necessary to limit entrance flux so that the overall drawdown is s for discharge Q. The commonly used expression is:
L = (Q / (2 * pi * K * s)) * ln(R / r_w)
with natural logarithm ln().

Step-by-Step Solution:

Verification / Alternative check:
Dimensional check: Q has L^3/T; denominator 2 * pi * K * s is L/T; multiplying by the dimensionless ln term yields L (length), as required.

Why Other Options Are Wrong:

• Options with inverted fraction or missing 2 * pi lead to incorrect scaling.
• Replacing ln(R / r_w) with (R / r_w) is incorrect; radial flow solution yields the logarithmic term.

Common Pitfalls:

• Using confined-aquifer formulas directly without adapting to water-table conditions.
• Forgetting the logarithmic dependence on R / r_w.

Final Answer:
L = (Q / (2 * pi * K * s)) * ln(R / r_w)