Void Ratio and Soil Flow

This question asks us to determine the seepage velocity through a soil given its void ratio and the discharge velocity. These are important concepts in soil mechanics related to the flow of water through soil pores.

Relationship Between Velocities and Void Ratio

In soil mechanics, there are two main velocities we consider for water flow:

• Discharge Velocity (\(v\)): This is the average velocity of water flow through the gross cross-sectional area of the soil sample. It's often the velocity calculated directly using Darcy's law.
• Seepage Velocity (\(v_s\)): This is the actual velocity of water flow through the interconnected pore spaces (voids) in the soil. Since water only flows through the voids, the seepage velocity is always greater than the discharge velocity.

The relationship between discharge velocity (\(v\)) and seepage velocity (\(v_s\)) is given by:

\(v_s = \frac{v}{n}\)

where \(n\) is the porosity of the soil.

Calculating Porosity from Void Ratio

Porosity (\(n\)) is defined as the ratio of the volume of voids (\(V_v\)) to the total volume (\(V\)) of the soil. Void ratio (\(e\)) is defined as the ratio of the volume of voids (\(V_v\)) to the volume of solids (\(V_s\)). The relationship between porosity and void ratio is:

\(n = \frac{e}{1+e}\)

We are given the void ratio \(e = 0.25\). We can calculate the porosity \(n\):

\(n = \frac{0.25}{1+0.25} = \frac{0.25}{1.25}\)

\(n = \frac{25/100}{125/100} = \frac{25}{125} = \frac{1}{5} = 0.2\)

So, the porosity of the soil is \(0.2\).

Calculating Seepage Velocity

Now that we have the porosity (\(n = 0.2\)) and the discharge velocity (\(v = 3 \times 10^{-4}\) mm/s), we can calculate the seepage velocity (\(v_s\)) using the formula \(v_s = \frac{v}{n}\):

\(v_s = \frac{3 \times 10^{-4} \text{ mm/s}}{0.2}\)

\(v_s = \frac{3}{0.2} \times 10^{-4} \text{ mm/s}\)

\(v_s = \frac{30}{2} \times 10^{-4} \text{ mm/s}\)

\(v_s = 15 \times 10^{-4} \text{ mm/s}\)

The seepage velocity through the soil is \(15 \times 10^{-4}\) mm/s.

Summary of Calculation

• Given void ratio, \(e = 0.25\)
• Given discharge velocity, \(v = 3 \times 10^{-4}\) mm/s
• Calculate porosity, \(n = \frac{e}{1+e} = \frac{0.25}{1.25} = 0.2\)
• Calculate seepage velocity, \(v_s = \frac{v}{n} = \frac{3 \times 10^{-4}}{0.2} = 15 \times 10^{-4}\) mm/s

The calculated seepage velocity matches one of the given options.