Question

Constant head permeability tests were performed on two soil specimens, S1 and S2. The ratio of height of the two $(L_{S1}:L_{S2})$ is 1.5, the ratio of the diameter of specimens $(D_{S1}:D_{S2})$ is 0.5, and the ratio of the constant head $(h_{S1}:h_{S2})$ applied on the specimens is 2.0. If the discharge from both the specimens is equal, the ratio of the permeability of the soil specimens $(k_{S1}:k_{S2})$ is

• A. 3

Answer 1

The correct option is A 3
For constant head permeability test,

$k=\frac{QL}{ah}$
Q $\to$ discharge
L $\to$ length of specimen
a $\to$ area of cross-section of specimen
h $\to$ constant head

Given,

$\frac{L_{s_1}}{L_{s_2}}=\frac{3}{2}$

$\frac{D_{s_1}}{D_{s_2}}=0.5$

$\Rightarrow \frac{a_{s_1}}{a_{s_2}}={\left(\frac{D_{s_1}}{D_{s_2}}\right)}^2=(0.5{)}^2$

$\frac{h_{s_1}}{h_{s_2}}=2$

$Q_{s_1}=Q_{s_2}$

$\therefore \frac{K_{s1}}{K_{s2}}=\frac{\left(\frac{Q_{s_1}{L}_{s_1}}{a_{s_1}{h}_{s_1}}\right)}{\left(\frac{Q_{s_2}{L}_{s_2}}{a_{s_2}{h}_{s_2}}\right)}$

$=\frac{Q_{s_1}}{Q_{s_2}}\times \frac{L_{s_1}}{L_{s_2}}\times \frac{h_{s_2}}{h_{s_1}}\times \frac{a_{s_2}}{a_{s_1}}$

$=1\times \frac{3}{2}\times \frac{1}{2}\times \frac{1}{(0.5{)}^2}$

$=\frac{3}{2^2}\times 2^2=3$