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ID: past-exam-of-the-topics-in-fluid-mechanics-course-of-the-university-of-oxford/2023/2/a/iv/solution

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Past exam of the topics in fluid mechanics course of the University of Oxford / 2023 / 2 / a / iv / Solution by

Codex  0 2026-10-25

x = L \tilde{x}

,

h = H \tilde{h}

, and

t = T \tilde{t}

. The Boussinesq equation for an unconfined aquifer becomes

\frac{ϕ H}{T} \tilde{h}_{\tilde{t}} = \frac{K H ^{2}}{L ^{2}} (\tilde{h} \tilde{h}_{\tilde{x}})_{\tilde{x}} + R .

Balance rainfall recharge with lateral drainage, and then with storage:

H = L \frac{R}{K}, T = \frac{ϕ H}{R} = \frac{ϕ L}{K R} .

Dropping tildes, the dimensionless equation is

h_{t} = (h h_{x})_{x} + 1.

The horizontal Darcy velocity, vertical Darcy velocity, and discharge scales are respectively

K R

,

R

, and

R L

. Thus the vertical thickness scale is set by rainfall balanced against hydraulic conductivity over the finite drainage length $L$ , rather than by porosity. The latter determines the storage time

T

. The Dupuit approximation requires

H / L = R / K ≪ 1

for these scales. The initially dry initial condition is

h (x, 0) = 0

.

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