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Assume a 0.1 m long cylindrical capillary tube of uniform diameter of 0.1 m, which is filled with saturated sand. The tube is placed horizontally, and flow is taking place from right to left.

If the flow rate across the capillary is 0.005 m\(^3\) h\(^{-1}\) and the pressure (potential) difference across the capillary is 0.02 m, calculate the hydraulic conductivity of the soil in the capillary.

Answer :

To calculate the hydraulic conductivity of soil in the capillary, we can use Darcy's law, which relates the flow rate and pressure difference across a porous medium. Here's how you can calculate it:

1. Convert the flow rate from m^3/h to m^3/s:

- Flow rate = 0.005 m^3/h = 0.005 m^3/h × (1/3600) h/s = 1.39 × 10^-6 m^3/s

2. Calculate the cross-sectional area of the capillary tube:

- The diameter of the capillary tube is given as 0.1 m, so the radius is 0.05 m.

- The cross-sectional area (A) of the capillary tube can be calculated using the formula A = πr^2:

- A = π × (0.05 m)^2 = 0.00785 m^2

3. Calculate the velocity (v) of water flow in the capillary:

- Velocity (v) can be calculated using the formula v = Q / A, where Q is the flow rate and A is the cross-sectional area.

- v = (1.39 × 10^-6 m^3/s) / 0.00785 m^2 = 1.77 × 10^-4 m/s

4. Calculate the hydraulic conductivity (K) using Darcy's law:

- Darcy's law states that the flow rate (Q) is equal to the hydraulic conductivity (K) multiplied by the cross-sectional area (A) and the pressure difference (Δh):

- Q = K × A × Δh

- Rearrange the equation to solve for K:

- K = Q / (A × Δh)

- Given that the flow rate (Q) is 1.39 × 10^-6 m^3/s and the pressure difference (Δh) is 0.02 m, we can calculate the hydraulic conductivity:

- K = (1.39 × 10^-6 m^3/s) / (0.00785 m^2 × 0.02 m)

- K ≈ 8.92 × 10^-5 m/s

Therefore, the hydraulic conductivity of the soil in the capillary is approximately 8.92 × 10^-5 m/s.

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