Aquifer Area Calculation using Darcy's Law
This problem involves calculating the cross-sectional area ('A') of a confined aquifer perpendicular to groundwater flow, utilizing Darcy's Law.
Darcy's Law Formula Application
Darcy's Law describes the rate of flow ($Q$) through a porous medium. The fundamental equation is:
$ Q = k \cdot A \cdot i $
Where:
- $Q$ represents the total daily flow rate ($m^3$/day).
- $k$ is the coefficient of permeability (m/day).
- $A$ is the cross-sectional area perpendicular to flow ($m^2$).
- $i$ is the hydraulic gradient (dimensionless).
Calculating Aquifer Area
To find the unknown area 'A', we rearrange Darcy's Law formula:
$ A = \frac{Q}{k \cdot i} $
Step-by-Step Calculation
The problem provides the following values:
- $Q = 250$ m$^3$/day
- $k = 2$ m/day
- $i = 0.005$
Substitute these known values into the rearranged formula:
$ A = \frac{250 \text{ m}^3/\text{day}}{(2 \text{ m/day}) \times (0.005)} $
First, calculate the product of permeability and gradient in the denominator:
$ k \times i = 2 \times 0.005 = 0.01 $
Now, perform the division to find the area:
$ A = \frac{250}{0.01} $
$ A = 25000 \text{ m}^2 $
Final Result
The calculated cross-sectional area ('A') of the confined aquifer is 25000 m$^2$.
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]
