Calculating Seepage Velocity in Soil

This question asks us to determine the seepage velocity of soil given its void ratio and discharge velocity. To do this, we need to understand the relationship between these parameters and the porosity of the soil.

Understanding Key Soil Hydraulic Velocities

• Discharge Velocity ($\text{v}$): This is the average velocity of water flowing through the total cross-sectional area of the soil, including both the solid particles and the void spaces. It's essentially the flow rate divided by the total area.
• Seepage Velocity ($\text{v}_\text{s}$): This is the actual average velocity of water as it moves only through the interconnected void spaces (pores) within the soil. Since water only flows through the pores, the seepage velocity is always greater than or equal to the discharge velocity.
• Porosity ($\text{n}$): This is the ratio of the volume of voids to the total volume of the soil. It represents the fraction of the soil volume that is available for water flow.
• Void Ratio ($\text{e}$): This is the ratio of the volume of voids to the volume of solid particles in the soil. It is another way to express the amount of void space in the soil.

Relationship Between Parameters

The discharge velocity ($\text{v}$) and the seepage velocity ($\text{v}_\text{s}$) are related through the porosity ($\text{n}$) by the following equation:

\(\text{v} = \text{n} \times \text{v}_\text{s}\)

Therefore, the seepage velocity can be calculated as:

\(\text{v}_\text{s} = \frac{\text{v}}{\text{n}}\)

Porosity ($\text{n}$) is related to void ratio ($\text{e}$) by the formula:

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

Step-by-Step Calculation

We are given the following values:

• Void ratio, \(\text{e} = 0.5\)
• Discharge velocity, \(\text{v} = 6 \times 10^{-7} \text{ m/s}\)

First, we calculate the porosity (\(\text{n}\)) using the given void ratio:

\(\text{n} = \frac{\text{e}}{1+\text{e}} = \frac{0.5}{1+0.5} = \frac{0.5}{1.5}\)

To simplify the fraction:

\(\text{n} = \frac{0.5 \times 2}{1.5 \times 2} = \frac{1}{3}\)

So, the porosity is \(\frac{1}{3}\).

Next, we calculate the seepage velocity (\(\text{v}_\text{s}\)) using the discharge velocity and the calculated porosity:

\(\text{v}_\text{s} = \frac{\text{v}}{\text{n}}\)

Substitute the given values:

\(\text{v}_\text{s} = \frac{6 \times 10^{-7} \text{ m/s}}{\frac{1}{3}}\)

Dividing by a fraction is the same as multiplying by its reciprocal:

\(\text{v}_\text{s} = (6 \times 10^{-7}) \times 3 \text{ m/s}\)

\(\text{v}_\text{s} = 18 \times 10^{-7} \text{ m/s}\)

The calculated seepage velocity is \(18 \times 10^{-7}\) m/s.

Summary of Calculation

Conclusion on Seepage Velocity

Based on the given void ratio of 0.5 and a discharge velocity of \(6 \times 10^{-7}\) m/s, the calculated seepage velocity is \(18 \times 10^{-7}\) m/s.

Revision Table: Soil Properties and Velocities

Additional Information: Darcy's Law and Soil Flow

The flow of water through soil is often described by Darcy's Law, especially for laminar flow conditions. Darcy's Law relates the discharge velocity to the hydraulic gradient and the permeability of the soil.

\(\text{v} = \text{k} \times \text{i}\)

Where:

• \(\text{k}\) is the hydraulic conductivity (or coefficient of permeability) of the soil (units of velocity, e.g., m/s).
• \(\text{i}\) is the hydraulic gradient, which is the head loss per unit length of flow path (dimensionless).

While Darcy's Law gives the discharge velocity, the seepage velocity represents the actual speed of water particles moving through the pore channels. The tortuosity of the flow path (the winding nature of the pores) is one reason why the seepage velocity is greater than the discharge velocity.

Understanding the difference between discharge velocity and seepage velocity is crucial in geotechnical engineering and hydrogeology for analyzing groundwater flow, pollutant transport, and consolidation settlement.