If the velocity of flow of water through the soil is 0.01 cm/s and the piezometer readings at the end points A and B of soil sample are 10 cm and 5 cm, respectively. The distance between the points A and B is 20 cm. Find the coefficient of permeability of the soil sample.
- A4 cm/s
- B0.04 cm/sCorrect
- C0.25 cm/s
- D0.025 cm/s
Solution
Concept
Darcy's law: flow velocity equals permeability times hydraulic gradient (v = k i).
Formula
v = k i, i = Δh / L, k = v / i
Step-by-step solution
1. Head loss Δh = 10 - 5 = 5 cm. 2. Length L = 20 cm. 3. Hydraulic gradient i = Δh / L = 5 / 20 = 0.25. 4. Darcy's law: v = k i, so k = v / i. 5. k = 0.01 / 0.25 = 0.04 cm/s.
Explanation
Water flows from A to B because of a head difference of 5 cm over 20 cm, which gives a gradient of 0.25. By Darcy's law, k = v / i = 0.01 / 0.25 = 0.04 cm/s.
Why the other options are wrong
A: 4 cm/s comes from a wrong division (0.01 × 400 type error). C: 0.25 is the hydraulic gradient, not k. D: 0.025 cm/s comes from multiplying v by 2.5 or using the wrong gradient.
Shortcut
k = v × L / Δh = 0.01 × 20 / 5 = 0.04 cm/s.
Common mistake
Taking the hydraulic gradient itself as k, or using 10/20 instead of the head difference (10 - 5)/20.
Correct answer: B) 0.04 cm/s