Unconfined aquifer pumping: ratio of discharges for two drawdowns A well fully penetrating an unconfined aquifer has initial saturated thickness 100 m. Under equilibrium (steady) Dupuit conditions and the same radius of influence, compute the ratio of discharges when the drawdown at the well is 20 m versus 40 m.

Introduction / Context:
For unconfined aquifers, the steady pumping discharge from a fully penetrating well depends on the difference of squares of the piezometric head (measured above an impervious base) at two radii. This is a standard application of the Dupuit–Thiem equation and illustrates how discharge grows nonlinearly with drawdown because the saturated thickness reduces as the water table falls.

Given Data / Assumptions:

• Unconfined aquifer with initial saturated thickness b = 100 m.
• Steady (equilibrium) flow; homogeneous, isotropic aquifer.
• Same radius of influence for both cases; well fully penetrating.
• Drawdowns at the well: s1 = 20 m, s2 = 40 m.

Concept / Approach:

Dupuit–Thiem for unconfined conditions gives (ignoring constants common to both cases): Q ∝ (H_R^2 − h_w^2), where H_R is head at the radius of influence (approximately the original water table height), and h_w is head at the well. With initial thickness b, take H_R = b. For a drawdown s, the head at the well is h_w = b − s, so the driving term becomes b^2 − (b − s)^2.

Step-by-Step Solution:

Verification / Alternative check:

Using the equivalent expression b^2 − (b − s)^2 = 2 b s − s^2 yields 210020 − 20^2 = 3600 and 210040 − 40^2 = 6400, confirming the same ratio 9/16.

Why Other Options Are Wrong:

• 3/4, 1/2, 4/9, 5/4: These ignore the squared-thickness dependence in unconfined flow; only 9/16 matches the correct difference-of-squares relationship.

Common Pitfalls:

• Treating the problem as confined flow (where Q is proportional to drawdown, not to head-squared difference).
• Using b − s instead of b^2 − (b − s)^2 in the numerator of the discharge relation.

Final Answer:

9/16.