Soil Seepage Velocity Calculation with Void Ratio

The question asks us to determine the seepage velocity of a soil, given its discharge velocity and void ratio. This is a fundamental concept in soil mechanics related to groundwater flow.

Let's first understand the terms:

• Discharge Velocity (v): This is the average velocity of water flowing through the gross cross-sectional area of the soil sample (soil solids + voids). It's based on Darcy's Law.
• Seepage Velocity ($v_s$): This is the actual average velocity of water flowing only through the void spaces (pores) in the soil. Since water only flows through the voids, the seepage velocity is always greater than the discharge velocity.
• Void Ratio (e): This is the ratio of the volume of voids to the volume of soil solids. It's a measure of how much empty space is in the soil.
• Porosity (n): This is the ratio of the volume of voids to the total volume of the soil (volume of voids + volume of solids). It represents the percentage or fraction of the total volume occupied by voids.

Relationship between Velocities and Soil Properties

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

\begin{equation*} v_s = \frac{v}{n} \end{equation*}

where $n$ is the porosity of the soil.

The relationship between porosity ($n$) and void ratio ($e$) is given by:

\begin{equation*} n = \frac{e}{1+e} \end{equation*}

Calculating Seepage Velocity

We are given:

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

First, we need to calculate the porosity ($n$) using the given void ratio ($e$):

\begin{equation*} n = \frac{e}{1+e} = \frac{0.5}{1+0.5} = \frac{0.5}{1.5} \end{equation*}

To simplify the fraction $\frac{0.5}{1.5}$, we can multiply the numerator and denominator by 10:

\begin{equation*} n = \frac{0.5 \times 10}{1.5 \times 10} = \frac{5}{15} = \frac{1}{3} \end{equation*}

So, the porosity $n = \frac{1}{3}$.

Now, we can calculate the seepage velocity ($v_s$) using the discharge velocity ($v$) and the porosity ($n$):

\begin{equation*} v_s = \frac{v}{n} = \frac{6 \times 10^{-7} \text{ m/s}}{1/3} \end{equation*}

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

\begin{equation*} v_s = 6 \times 10^{-7} \times 3 \text{ m/s} \end{equation*}

\begin{equation*} v_s = 18 \times 10^{-7} \text{ m/s} \end{equation*}

Thus, the seepage velocity of the soil is $18 \times 10^{-7}$ m/s.