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 Incoming links: Separable aquifer drawdown

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 332 / 1 / a / iii / Solution Created 2026-09-28 Updated 2026-10-03

After rainfall stops, the late finite-domain solution is the separable aquifer drawdown

h (x, t) \sim \frac{n L ^{2}}{K t} F (X), X = \frac{x}{L} .

The dimensionless profile is determined by

(F F^{'})^{'} + F = 0, F (0) = 0, F^{'} (1) = 0.

If

C_{d} = lim_{X ↓ 0} F F^{'} > 0

, then

q (0, t) \sim - \frac{n ^{2} L ^{3}}{K t ^{2}} C_{d},

and

F (X) \sim (2 C_{d} X)^{1/2}, h (x, t) \sim \frac{n L ^{2}}{K t} (\frac{2 C _{d} x}{L})^{1/2} .

Thus aquifer thickness decays as $t^{- 1}$ and river discharge as $t^{- 2}$ .

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