Permeability Calculation Using Darcy's Law
This solution details the calculation of the coefficient of permeability ($k$) for a sand column using data from a constant-head permeability test and applying Darcy's law.
Input Parameters
| Parameter | Symbol | Value | Unit |
| Column Height | $L$ | 40 | cm |
| Cross-sectional Area | $A$ | 25 | cm² |
| Head Loss | $h$ | 50 | cm |
| Volume of Water | $V$ | 300 | cm³ |
| Time Duration | $t$ | 120 | sec (2 minutes) |
Darcy's Law Application
Darcy's law describes the flow of fluid through a porous medium. The formula used to calculate the coefficient of permeability ($k$) in a constant-head test is:
$ k = \frac{V \cdot L}{A \cdot h \cdot t} $
Where:
- $k$ = Coefficient of permeability (cm/sec)
- $V$ = Volume of water collected (cm³)
- $L$ = Length of the soil specimen (cm)
- $A$ = Cross-sectional area of the specimen (cm²)
- $h$ = Head loss across the specimen (cm)
- $t$ = Time duration (sec)
Calculation Steps
Substitute the given values into Darcy's law formula:
$ k = \frac{300 \text{ cm³} \cdot 40 \text{ cm}}{25 \text{ cm²} \cdot 50 \text{ cm} \cdot 120 \text{ sec}} $
Calculate the numerator and denominator:
$ \text{Numerator} = 300 \times 40 = 12000 \text{ cm⁴} $
$ \text{Denominator} = 25 \times 50 \times 120 = 150000 \text{ cm³} \cdot \text{sec} $
Compute the value of $k$:
$ k = \frac{12000}{150000} \text{ cm/sec} $
$ k = 0.08 \text{ cm/sec} $
Final Result
The calculated coefficient of permeability ($k$) is 0.08 cm/sec.
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]
- The hydraulic conductivity (K) of an isotropic aquifer is 10 m/day. If the hydraulic head within the aquifer drops 4 m over a distance of 750 m, the groundwater flow velocity within the aquifer is _________ m/day.
(Up to third decimal place)
