This problem asks for the effective hydraulic conductivity of a soil mass composed of two horizontal layers with different hydraulic conductivities, subjected to Darcian flow under the same hydraulic gradient.
Determining Flow Condition
The soil mass has two horizontal layers stacked one above the other. The problem states that there is the same hydraulic gradient ($i$) for both layers. This condition implies that the flow is occurring parallel to the layers. In such a scenario, the layers are effectively acting in parallel with respect to the flow path.
Calculating Effective Conductivity for Parallel Flow
For flow in parallel layers, the effective hydraulic conductivity ($k_{eff}$) is the arithmetic mean of the individual hydraulic conductivities, weighted by their respective cross-sectional areas perpendicular to the flow. Since the layers have equal thickness and equal width, their cross-sectional areas are equal.
Let $k_1$ and $k_2$ be the hydraulic conductivities of the two layers.
Given:
- $k_1 = 5 \times 10^{-2}$ cm/s
- $k_2 = 3 \times 10^{-2}$ cm/s
The formula for effective hydraulic conductivity in parallel flow, assuming equal areas, is:
$k_{eff} = \frac{k_1 + k_2}{2}$Executing the Calculation
Substitute the given values into the formula:
$k_{eff} = \frac{(5 \times 10^{-2} \text{ cm/s}) + (3 \times 10^{-2} \text{ cm/s})}{2}$ $k_{eff} = \frac{(5+3) \times 10^{-2} \text{ cm/s}}{2}$ $k_{eff} = \frac{8 \times 10^{-2} \text{ cm/s}}{2}$ $k_{eff} = 4 \times 10^{-2} \text{ cm/s}$Converting to decimal form:
$k_{eff} = 0.04 \text{ cm/s}$The result is rounded to two decimal places as required.
Important Questions from Hydrogeology (Darcy Flow/Flow Rate)
- 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]
- 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)
