Section 4: Soil and Water Conservation Engineering
Hydrology
Subtopic: Runoff estimation
Question Statement:
Groundwater Hydrology: Groundwater occurrence; ground water movement; Darcy's Law, steady and unsteady flow in confined and unconfined aquifers, groundwater exploration techniques; overview of groundwater recharge estimation and artificial recharge techniques.
A 20 cm diameter well fully penetrates a confined aquifer of thickness 20 m. After long pumping at 1500 L/min, steady drawdowns in wells at 30 m and 50 m are 2.0 m and 1.5 m respectively. Assuming π = 3.14, the transmissivity of the aquifer (m² s⁻¹) is
Options:
(A) 0.244
(B) 14.676
(C) 352.224
(D) 880.560
Official Answer Key:
MARKS TO ALL
Step-by-Step Solution & Explanation
Using the Thiem (steady-state confined-aquifer) equation for two observation wells:
T = Q·ln(r2/r1) / [2π(s1−s2)]
Q = 1500 L/min = 0.025 m³/s; r1 = 30 m, r2 = 50 m; s1 = 2.0 m, s2 = 1.5 m.
T = 0.025 × ln(50/30) / (2×3.14×0.5) = 0.025×0.5108/3.14 = 0.01277/3.14 = 0.00407 m²/s.
This value (0.00407 m²/s) does not match any of the four listed options when the units are taken literally as "m² s⁻¹" as printed in the question. However, converting the SAME computed transmissivity to other common time bases reproduces each option almost exactly:
• per minute: 0.00407 × 60 = 0.244 m²/min → option (A)
• per hour: 0.00407 × 3600 = 14.65 m²/h → option (B)
• per day: 0.00407 × 86,400 = 351.6 ≈ 352.2 m²/day → option (C)
This confirms the question statement's unit label ("m² s⁻¹") is inconsistent with the intended (and listed) answer options, which are actually expressed in m²/min, m²/h, and m²/day respectively. Because of this unit inconsistency, GATE authorities awarded marks to all candidates for this question — consistent with the official key. The physically computed transmissivity is T ≈ 0.00407 m²/s = 352 m²/day.
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