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In a falling head permeameter test, the soil sample had a diameter of 70mm and a length of 150mm. The stand pipe had a diameter of 16mm. A stopwatch was started when the head was 500mm and read 26.2 seconds when the head was 250mm. The test was repeated for a drop in head from 250mm to 120mm and the time was 26.4 seconds. Calculate the coefficient of permeability of the soil in mm/sec.

Answer :

To calculate the coefficient of permeability (k) of the soil using a falling head permeameter test, we use the formula:

[tex]k = \frac{(a \cdot L)}{(A \cdot t)} \cdot \ln\left(\frac{h_1}{h_2}\right)[/tex]

where:

  • [tex]a[/tex] is the cross-sectional area of the stand pipe
  • [tex]L[/tex] is the length of the soil sample
  • [tex]A[/tex] is the cross-sectional area of the soil sample
  • [tex]t[/tex] is the time it takes for the head to fall from [tex]h_1[/tex] to [tex]h_2[/tex]
  • [tex]h_1[/tex] and [tex]h_2[/tex] are the initial and final heads, respectively

Calculation Steps:

  1. Find the cross-sectional areas:

    • The area of the soil sample ([tex]A[/tex]) is given by the formula [tex]A = \pi r^2[/tex], where [tex]r[/tex] is the radius.

      [tex]A = \pi \times (35)^2 = 3848.45 \text{ mm}^2[/tex]

    • The area of the standpipe ([tex]a[/tex]) is:

      [tex]a = \pi \times (8)^2 = 201.06 \text{ mm}^2[/tex]

  2. First test (from 500 mm to 250 mm):

    • Use the first drop in head value:

      [tex]h_1 = 500 \text{ mm}, \ h_2 = 250 \text{ mm}, \ t = 26.2 \text{ seconds}[/tex]

      [tex]k = \frac{(201.06 \times 150)}{(3848.45 \times 26.2)} \times \ln\left(\frac{500}{250}\right)[/tex]

    • Calculate:

      [tex]k = \frac{(30159)}{(100683.39)} \times 0.6931[/tex]

      [tex]k \approx 0.2071 \times 0.6931 = 0.1436 \text{ mm/sec}[/tex]

  3. Second test (from 250 mm to 120 mm):

    • Use the second drop in head value:

      [tex]h_1 = 250 \text{ mm}, \ h_2 = 120 \text{ mm}, \ t = 26.4 \text{ seconds}[/tex]

      [tex]k = \frac{(201.06 \times 150)}{(3848.45 \times 26.4)} \times \ln\left(\frac{250}{120}\right)[/tex]

    • Calculate:

      [tex]k = \frac{(30159)}{(101429.88)} \times 0.7307[/tex]

      [tex]k \approx 0.2973 \times 0.7307 = 0.1347 \text{ mm/sec}[/tex]

Both tests provide estimates of the permeability. Small differences might occur due to slight inaccuracies in measurements or assumptions, but both results are consistent in showing medium permeability.

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