Question

For a liquid, the permeability of a sandy soil having a void ratio of 0.60 was determined as 0.14 cm/s. Considering the same liquid and by using Taylor's equation, the permeability (in cm/s) of this soil corresponding to the void ratio of 0.80 is (rounded off to two decimal places).

Hint:

Use Taylor's relation k is proportional to e^3/(1+e) to compare permeability at two void ratios.

Answer 1

Correct Answer: 0.29

Step 1: Recall Taylor's equation relating permeability to void ratio.
For a given soil and permeant liquid, Taylor's relation (based on the Kozeny-Carman model) states that permeability varies with void ratio as
\[ k \propto \frac{e^3}{1+e} \]
So for the same soil and liquid tested at two different void ratios \(e_1\) and \(e_2\), the ratio of permeabilities is
\[ \frac{k_2}{k_1} = \frac{e_2^3/(1+e_2)}{e_1^3/(1+e_1)} \]

Step 2: Substitute the given values.
Here \(e_1 = 0.60\), \(k_1 = 0.14\) cm/s, and \(e_2 = 0.80\).
\[ \frac{e_1^3}{1+e_1} = \frac{(0.60)^3}{1.60} = \frac{0.216}{1.60} = 0.1350 \]
\[ \frac{e_2^3}{1+e_2} = \frac{(0.80)^3}{1.80} = \frac{0.512}{1.80} = 0.2844 \]

Step 3: Find the permeability ratio and solve for \(k_2\).
\[ \frac{k_2}{k_1} = \frac{0.2844}{0.1350} = 2.107 \]
\[ k_2 = 0.14 \times 2.107 = 0.295 \ \text{cm/s} \]

Step 4: Interpret the result.
An increase in void ratio from 0.60 to 0.80 means the soil is looser, with larger and better connected pore channels, so the permeability rises even though the soil grains and the permeant liquid stay the same.

Final Answer:
The permeability of the soil at a void ratio of 0.80 works out to about 0.29 cm/s.
\[ \boxed{k_2 \approx 0.29 \ \text{cm/s}} \]