Calculate the spacing required for the water table to drop from the soil surface to a depth of 30cm in a 2days period over the area 200hecters from the following data k=4cm/hr,the drain are to be placed 105cm below the soil surface the impeamable layer is 195cm below the soil surface take the discharge Q0=15cm /day
To calculate the spacing required for the water table to drop from the soil surface to a depth of 30cm over a 2-day period, we can use Darcy's Law and the assumptions of a horizontal flow in confined aquifers.
1. Convert the given units to consistent units:
- Time: 2 days x 24 hours/day = 48 hours
- Hydraulic Conductivity (k): 4 cm/hr
2. Determine the hydraulic gradient (i) using the difference in water table depth over the desired period of time:
- Change in water table depth (dh): 30 cm
- Time (dt): 48 hours
- Hydraulic gradient (i) = dh / dt
i = 30 cm / 48 hours
3. Determine the flow rate (Q) based on the specific discharge (Q0) and the area (A):
- Area (A): 200 hectares = 2,000,000 m²
- Specific discharge (Q0): 15 cm/day
Q = Q0 x A
4. Determine the cross-sectional area of the flow (A') assuming a rectangular drain:
- Drain depth below the soil surface (h): 105 cm
- Impermeable layer depth (L): 195 cm
A' = (22/7) x (h - L)
5. Determine the drain spacing (S) using Darcy's Law:
- Darcy's Law: Q = (k x A' x i) / S
S = (k x A' x i) / Q
Plug in the values:
S = (4 cm/hr x (22/7) x (105 cm - 195 cm) x (30 cm / 48 hours)) / (15 cm/day x 2,000,000 m²)
Simplify and convert the units:
S = (0.18 cm²/hr) / (3.8 m²/day)
S ≈ 0.047 cm/dm²
Therefore, the spacing required for the water table to drop from the soil surface to a depth of 30 cm over a 2-day period is approximately 0.047 cm/dm².
To calculate the spacing required for the water table to drop from the soil surface to a depth of 30cm in a 2-day period, we will use Darcy's law and the concept of drainage spacing.
Darcy's law states that the seepage velocity (v) through a soil medium is equal to the hydraulic conductivity (k) multiplied by the hydraulic gradient (i). In this case, we can assume that the hydraulic gradient is approximately equal to the slope of the water table.
The seepage velocity (v) can be calculated using the equation:
v = k * i
Given that the hydraulic conductivity (k) is 4 cm/hr, and we need to convert it to cm/day to match the discharge rate Q0, we can use the conversion factor of 24:
k = 4 cm/hr * 24 hr/day = 96 cm/day
Next, we need to calculate the hydraulic gradient (i). The hydraulic gradient is the change in water table elevation divided by the distance over which that change occurs. In this case, the water table is dropping from the soil surface to a depth of 30cm, and the drainage spacing is the distance over which the drainage occurs.
Given that the drainage spacing is the area of 200 hectares, we need to convert it to square meters:
200 hectares * 10,000 m²/hectare = 2,000,000 m²
The drainage spacing can be calculated using the equation:
drainage spacing = (change in water table elevation) / (hydraulic gradient) = 30 cm / i
Substituting the known values, we get:
30 cm / i = 2,000,000 m²
To convert cm to meters, we divide by 100:
30 cm / i = 20,000 m
Now, we need to calculate the hydraulic gradient (i). Rearranging the equation, we have:
i = 30 cm / (20,000 m)
Converting cm to meters, we get:
i = (30/100) / (20,000) = 0.00015 m/m
Finally, we can substitute the calculated hydraulic conductivity (k) and hydraulic gradient (i) into Darcy's law to find the seepage velocity (v):
v = k * i = 96 cm/day * 0.00015 m/m = 0.0144 cm/day = 0.144 m/day
Therefore, the spacing required for the water table to drop from the soil surface to a depth of 30cm in a 2-day period is equal to the distance the water table drops divided by the seepage velocity:
spacing = (30 cm) / (0.144 m/day * 2 days) = 104.17 cm
So, the spacing required is approximately 104.17 cm or 1.04 meters.