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GATE 2025 CE (CE1) – Question 59

Geotechnical Engineering · Permeability and seepage: flow nets, uplift pressure, piping, capillarity · 2 marks · Numerical answer

Two soils of permeabilities $k_1$ and $k_2$ are placed in a horizontal flow apparatus, as shown in the figure. For Soil 1, $L_1 = 50$ cm, and $k_1 = 0.055$ cm/s; for Soil 2, $L_2 = 30$ cm, and $k_2 = 0.035$ cm/s. The cross sectional area of the horizontal pipe is 100 cm$^2$, and the head difference ($\Delta h$) is 150 cm. The discharge (in cm$^3$/s) through the soils is ___________ (*rounded off to 2 decimal places*).

A horizontal pipe connecting two standpipes, with the water level 150 cm higher on the left. Soil 1 of length $L_1$ and Soil 2 of length $L_2$ sit one after the other in the pipe, and the flow is from left to right through Soil 1 and then Soil 2.

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Correct answer: 8.45 to 8.53

Explanation

The soils are in series along the flow, so the same discharge passes through both. The equivalent permeability is $k_{eq} = \frac{L_1 + L_2}{\frac{L_1}{k_1} + \frac{L_2}{k_2}} = \frac{80}{\frac{50}{0.055} + \frac{30}{0.035}} = \frac{80}{1766.2} = 0.04530$ cm/s. The hydraulic gradient is $i = \frac{150}{80} = 1.875$, so $Q = k_{eq}iA = 0.04530 \times 1.875 \times 100 = 8.49$ cm$^3$/s.