A 45 cm diametre well penetrates an unconfined aquifer of 30 m thick. Under the steady pumping rate for a long time, the drawdown's at two observation wells 10 m and 20 m from the pumping well are 5 m and 3.5 m respectively. What will be the discharge (cumec), if the permeability of the aquifer is given as 20 m/day?
Try it first: pick an option, then read the explanation.
Detailed Explanation
This is a full field pumping-test problem for an UNCONFINED aquifer — water table conditions where the aquifer's own upper boundary is the free water surface being drawn down by pumping, rather than a confined layer under pressure.
Given: aquifer thickness H = 30 m; observation well radii r1 = 10 m, r2 = 20 m; drawdowns S1 = 5 m, S2 = 3.5 m; permeability K = 20 m/day.
Step 1 — Water depths at each observation well: h1 = H − S1 = 30 − 5 = 25 m; h2 = H − S2 = 30 − 3.5 = 26.5 m
Step 2 — Convert K to m/s: K = 20 m/day = 20/(24×60×60) = 20/86400 m/s
Step 3 — Discharge formula for unconfined aquifer: Q = πK(h2² − h1²) / [2.303·log₁₀(r2/r1)]
Q = π × (20/86400) × (26.5² − 25²) / ln(2) = π × (20/86400) × (702.25 − 625) / 0.693 ≈ 0.08 m³/s
Why "0.08" Is Correct
Substituting all given (and unit-converted) values into the unconfined-aquifer discharge formula gives Q ≈ 0.08 m³/s (cumecs).
Why the Other Options Are Wrong
- 0.05: too low — likely results from a units slip somewhere in the K conversion (m/day to m/s) or in the h1²/h2² terms.
- 0.067: close to, but not matching, the correct calculation — likely a rounding or intermediate-step arithmetic difference.
- 1.12: off by more than an order of magnitude — almost certainly from forgetting to convert K from m/day to m/s before substituting into the formula.
Key Point
Always convert given permeability values to consistent units (here, m/day → m/s) BEFORE substituting into the discharge formula — this is the single most common source of large (order-of-magnitude) errors in field pumping-test problems.
Key Concepts for Students
- For an unconfined aquifer, water depth (h) is measured from the aquifer's base up to the drawn-down water table at each observation point — different from a confined aquifer's approach (see Q21).
- The formula's (h2² − h1²) term, rather than a simple linear difference, reflects the specific mathematics of radial flow toward a well in an unconfined (water-table) aquifer.
- See Q21 for the parallel confined-aquifer version of this same well-pumping problem type, using a linear (not squared) head difference.
- See the Going Deeper section for why unconfined and confined aquifers need genuinely different discharge formulas.
Going Deeper: Unconfined vs. Confined Aquifer Well Formulas
Radial flow toward a pumped well behaves differently depending on whether the aquifer is unconfined (a water-table aquifer, where the upper boundary is the free water surface itself) or confined (an aquifer sandwiched between impermeable layers, under pressure).
| Aquifer Type | Discharge Formula | Key Difference |
|---|---|---|
| Unconfined (water table) | Q = πK(h2² − h1²) / [2.303·log10(r2/r1)] | Uses squared water depths (h2², h1²), since the flow cross-section itself changes with depth as the water table lowers |
| Confined | Q = 2πKD·Sw / ln(R/Rw) | Uses a simple linear drawdown (Sw) and the FIXED aquifer thickness D, since the flow cross-section (bounded by impermeable layers) doesn't change with drawdown |
The physical reason for this difference: in an unconfined aquifer, the flow cross-sectional area at any radius depends on the LOCAL water table height (which changes with distance from the well), leading to the squared-head form. In a confined aquifer, the flow cross-section is fixed by the aquifer's constant thickness (D) regardless of drawdown, so only a simple linear drawdown term (Sw) is needed. See Q21 for a fully worked confined-aquifer example using this second formula.