Permeability Calculation Using Darcy's Law

The question asks for the coefficient of permeability (k) of a cylindrical soil sample using data from a constant-head permeameter test. Darcy's Law is applied here.

The formula for discharge (Q) under laminar flow through a porous medium is given by Darcy's Law:

$Q = k \cdot i \cdot A \cdot t$

Where:

• $Q$ = Volume of water passed (cm$^3$)
• $k$ = Coefficient of permeability (cm/s)
• $i$ = Hydraulic gradient
• $A$ = Cross-sectional area of the sample (cm$^2$)
• $t$ = Time duration (s)

Calculating Input Parameters

First, extract and calculate the necessary parameters from the given data:

• Diameter of the sample, $D = 10$ cm
• Radius of the sample, $r = D/2 = 10 \text{ cm} / 2 = 5$ cm
• Cross-sectional area, $A = \pi r^2$. Using $\pi = 3.14$, $A = 3.14 \times (5 \text{ cm})^2 = 3.14 \times 25 \text{ cm}^2 = 78.5 \text{ cm}^2$.
• Volume of water collected, $Q = 250$ cm$^3$.
• Time duration, $t = 5$ minutes $= 5 \times 60$ seconds $= 300$ seconds.
• Head difference, $\Delta h = 5$ cm.
• Length of the sample (distance between tapping points), $L = 15$ cm.
• Hydraulic gradient, $i = \Delta h / L = 5 \text{ cm} / 15 \text{ cm} = 1/3$.

Determining Coefficient of Permeability (k)

Rearrange Darcy's Law to solve for k:

$k = Q / (i \cdot A \cdot t)$

Substitute the calculated values into the formula:

$k = 250 \text{ cm}^3 / ((1/3) \cdot 78.5 \text{ cm}^2 \cdot 300 \text{ s})$

Calculate the denominator:

Denominator $= (1/3) \times 78.5 \times 300 = 78.5 \times 100 = 7850 \text{ cm}^3/\text{s}$

Now calculate k:

$k = 250 \text{ cm}^3 / 7850 \text{ cm}^3/\text{s}$

$k \approx 0.031847...$ cm/s

Rounding the result to 3 decimal places gives $k \approx 0.032$ cm/s.

This value falls within the specified range of 0.03 to 0.032.