Question

In a hydraulic conductivity study, a 15 cm thick water-saturated soil layer was allowed a 15 cm water layer above it vertically, and the water was allowed to pass through the soil for 45 minutes. A total of 540 cc of water was collected. The cross-sectional area of the soil layer was 30 cm². Calculate the discharge rate of flow of water through the soil.

• A. 6.0 cm/h
• B. 12.0 cm/h
• C. 18.0 cm/h
• D. 24 cm/h

Hint:

To calculate water flux $v$ (velocity) directly, divide the total volume by the product of area and time: $v = \frac{\text{Volume}}{\text{Area} \times \text{Time}}$. This is independent of Darcy's gradient.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The rate of water flow through saturated soil is governed by Darcy's Law.
The question asks for the discharge rate of flow, which can be interpreted as the discharge velocity or flux ($v = \frac{Q}{A}$).
Key Formula or Approach:
The discharge velocity (flux) is: \[ v = \frac{V}{A \cdot t} \] Where: - $V$ is the volume of water collected (in $\text{cm}^3$ or $\text{cc}$).
- $A$ is the cross-sectional area of the soil (in $\text{cm}^2$).
- $t$ is the time of flow (in hours, $\text{h}$).

Step 2: Detailed Explanation:
Given values:
- Volume of water collected ($V$) = $540\text{ cc} = 540\text{ cm}^3$
- Cross-sectional area ($A$) = $30\text{ cm}^2$
- Time ($t$) = $45\text{ minutes} = \frac{45}{60}\text{ hours} = 0.75\text{ hours}$
Calculate the discharge velocity (flux): \[ v = \frac{540}{30 \times 0.75} \] \[ v = \frac{540}{22.5} = 24\text{ cm/h} \]

Step 3: Alternative Interpretation (Hydraulic Conductivity, K):
If the question intended to find the Hydraulic Conductivity ($K$) using Darcy's Law: \[ Q = K \cdot A \cdot \frac{\Delta h}{L} \implies v = K \cdot \frac{\Delta h}{L} \] Where: - Soil thickness ($L$) = $15\text{ cm}$
- Water head above soil = $15\text{ cm} \implies$ Total hydraulic head difference ($\Delta h$) = $15 + 15 = 30\text{ cm}$
- Hydraulic gradient ($i$) = $\frac{\Delta h}{L} = \frac{30}{15} = 2$
Substitute the values: \[ 24 = K \times 2 \implies K = 12\text{ cm/h} \] Since the question asks for the "discharge rate of flow" in units of $\text{cm/h}$, the flux value is 24 cm/h.

Step 5: Final Answer:
The discharge rate of flow is 24 cm/h, corresponding to Option (D).