(Up to third decimal place)
Calculating Groundwater Flow Velocity
This solution details the calculation of groundwater flow velocity using the provided parameters and Darcy's Law.
Understanding the Parameters
The problem provides the following key information for an isotropic aquifer:
- Hydraulic Conductivity (K): 10 m/day
- Drop in Hydraulic Head (Δh): 4 m
- Distance over which head drops (ΔL): 750 m
The goal is to determine the groundwater flow velocity (v) in m/day.
Applying Darcy's Law
Darcy's Law describes the flow of fluid through a porous medium. The average linear velocity (or seepage velocity) v can be calculated using the formula:
$v = K \times i$
Where:
- v is the groundwater flow velocity
- K is the hydraulic conductivity
- i is the hydraulic gradient
The hydraulic gradient (i) is defined as the change in hydraulic head divided by the distance:
$i = \frac{\Delta h}{\Delta L}$
Substituting the expression for i into Darcy's Law gives:
$v = K \times \left( \frac{\Delta h}{\Delta L} \right)$
Step-by-Step Calculation
-
Calculate the Hydraulic Gradient (i):
Using the given values for head drop and distance:
$i = \frac{4 \text{ m}}{750 \text{ m}}$
-
Calculate the Groundwater Flow Velocity (v):
Now, substitute the hydraulic conductivity (K) and the calculated gradient (i) into the velocity formula:
$v = 10 \text{ m/day} \times \frac{4}{750}$
$v = \frac{40}{750} \text{ m/day}$
$v = \frac{4}{75} \text{ m/day}$
Converting the fraction to a decimal:
$v \approx 0.05333... \text{ m/day}$
-
Round the Result:
Rounding the velocity to the third decimal place gives:
$v \approx 0.053 \text{ m/day}$
Final Result Validation
The calculated groundwater flow velocity is approximately 0.053 m/day. This value falls within the range of 0.05 to 0.055 m/day, confirming the result.
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]
