Hydraulic Conductivity Calculation using Darcy's Law
The problem requires calculating the hydraulic conductivity ($K$) of a porous medium using Darcy's Law. We are given the specific discharge ($q$), the hydraulic head difference ($\Delta h$), and the flow length ($L$).
Darcy's Law Formula
Darcy's Law states that the specific discharge ($q$) is proportional to the hydraulic gradient ($\frac{\Delta h}{L}$) and the hydraulic conductivity ($K$). The formula is:
$q = K \frac{\Delta h}{L}$
Unit Conversion
First, ensure all units are consistent. The specific discharge is given in cm/day, while head difference and flow length are in meters. Convert specific discharge to m/day:
- $q = 1 \text{ cm/day} = 0.01 \text{ m/day}$
Rearranging the Formula
To find the hydraulic conductivity ($K$), rearrange Darcy's Law:
$K = q \frac{L}{\Delta h}$
Calculation
Substitute the given values into the rearranged formula:
- $q = 0.01 \text{ m/day}$
- $\Delta h = 0.5 \text{ m}$
- $L = 20 \text{ m}$
$K = (0.01 \text{ m/day}) \times \frac{20 \text{ m}}{0.5 \text{ m}}$
$K = 0.01 \times 40 \text{ m/day}$
$K = 0.4 \text{ m/day}$
Result
The calculated hydraulic conductivity is 0.4 m/day. This value falls within the specified range.
Important Questions from Hydrogeology (Darcy Flow/Flow Rate)
- A soil mass comprises two horizontal layers (of equal thickness and equal width) stacked one above the other. The hydraulic conductivities of the two layers are $5 \times 10^{-2}$ cm/s and $3 \times 10^{-2}$ cm/s. Considering Darcian flow of water and same hydraulic gradient for both the layers, the effective hydraulic conductivity of the soil mass in cm/s is ________ (rounded off to two decimal places).
- A cylindrical soil sample of 10 cm diameter is tested in a constant-head permeameter. A volume of 250 cm$^3$ of water is collected in 5 minutes when the constant-head difference between tapping points 15 cm apart is 5 cm. Considering Darcy flow, the absolute value of coefficient of permeability in cm/s is _________ ($\pi = 3.14$) [round off to 3 decimal places]
- For a productive alluvial aquifer with hydraulic conductivity = 105 m/day and hydraulic gradient = 0.01, the flow rate is _______ m/day. [round off to 2 decimal places]
- The cross section of a river channel is approximated by a trapezium. The river has an average channel width of 40 m and average depth of 3 m. If the average flow speed is 2 m/s, the discharge rate is __________ m³/s. [in integer]
- A constant-head permeability test was performed on a vertical sand column of height 40 cm and cross-sectional area of 25 cm². During the test, when the loss of head was 50 cm, the volume of water collected in 2 minutes was 300 cm³. Applying Darcy's law, the calculated coefficient of permeability of the sand column is __________ cm/sec. [round off to 2 decimal places]
