Hydrogeological setup of a hypothetical alluvial area (where contact X-Y between two sands is vertical) is given in the schematic section. Hydraulic heads are indicated as $h_1$, $h_2$ and hydraulic conductivities as $K_1$ and $K_2$. The hydraulic head at the contact (X-Y) is ___________ m. (round off to 2 decimal places).
To find the hydraulic head at the contact (X-Y) between two sands in the alluvial area, we'll use Darcy's Law and the principle of continuity. Given:
- $h_1=50m$, $h_2=30m$
- $K_1=20m/day$, $L_1=3km=3000m$
- $K_2=30m/day$, $L_2=4km=4000m$
Flow across the contact must be equal due to continuity:
Flow in Sand-I: $Q_1=K_1 \cdot \left(\frac{h_1-h_x}{L_1}\right) \cdot A$
Flow in Sand-II: $Q_2=K_2 \cdot \left(\frac{h_x-h_2}{L_2}\right) \cdot A$
Setting $Q_1$ equal to $Q_2$ and simplifying (as $A$ cancels out):
$K_1 \cdot \left(\frac{h_1-h_x}{L_1}\right) = K_2 \cdot \left(\frac{h_x-h_2}{L_2}\right)$
Substituting the values:
$20 \cdot \left(\frac{50-h_x}{3000}\right) = 30 \cdot \left(\frac{h_x-30}{4000}\right)$
Simplifying further:
$20 \cdot (50-h_x) \cdot 4000 = 30 \cdot (h_x-30) \cdot 3000$
$80000 \cdot (50-h_x) = 90000 \cdot (h_x-30)$
Expanding both sides:
$4000000 - 80000h_x = 90000h_x - 2700000$
Combining like terms:
$6700000 = 170000h_x$
Solving for $h_x$:
$h_x = \frac{6700000}{170000} = 39.41m$
The hydraulic head at contact X-Y is 39.41 m, which falls within the range 39 to 39.8.
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]
