In order to determine the average permeability of a bed of sand 12.5 m thick overlying an impermeable stratum, a well was sunk through the sand and a pumping test carried out. After some time the discharge was 850kg/min and the drawdowns in observation wells 15.2m and 30.4m from the pump were 1.625 m and 1.360 m respectively. If the original water table was at a depth of 1.95 m below ground level, find the permeability of the sand (in m/s) and an approximate value for the effective grain size.
To determine the permeability of the sand and an approximate value for the effective grain size, we can use Darcy's Law and the equation for the coefficient of permeability.
Darcy's Law relates the flow rate (Q) through a porous medium to the cross-sectional area (A) perpendicular to the flow, the hydraulic gradient (i), and the coefficient of permeability (k):
Q = k * A * i
In this case, we have the discharge (Q) of 850kg/min, which we need to convert to a volumetric flow rate in a unit that is compatible with the other parameters. Let's convert it to m^3/s:
850 kg/min = (850 kg/min) * (1 min/60 s) * (1 m^3/1000 kg) = 0.1417 m^3/s
We also have the hydraulic gradient (i) which is the difference in head (h) between points 15.2m and 30.4m from the pump and divided by the distance (L) between those points:
i = (h_2 - h_1) / L
= (1.360 m - 1.625 m) / (30.4 m - 15.2 m)
= -0.265 / 15.2
= -0.01743421
Next, we can calculate the cross-sectional area (A) based on the thickness (12.5m) of the sand bed and the distance (r) between the observation well and the pump:
A = π * r^2
For the observation well at 15.2m, the radius (r1) is 15.2m since it is at a distance of 15.2m from the pump. Similarly, for the observation well at 30.4m, the radius (r2) is 30.4m.
A1 = π * (15.2m)^2
= 723.79 m^2
A2 = π * (30.4m)^2
= 2904.32 m^2
Now, we can rewrite Darcy's Law using the known values:
Q = k * A1 * i
0.1417 m^3/s = k * 723.79 m^2 * -0.01743421
Solving for k:
k = (0.1417 m^3/s) / (723.79 m^2 * -0.01743421)
≈ -0.000138 m/s
Note: The negative sign is due to the negative hydraulic gradient. The permeability is always considered positive, indicating the direction of flow.
The permeability of the sand is approximately 0.000138 m/s.
To find the approximate value for the effective grain size, we can use the Hazen's empirical formula for sand:
k = (d^2/10) * (n^2 - 1)
Where:
k = permeability (in m/s)
d = effective grain size (in mm)
n = uniformity coefficient
The uniformity coefficient (n) is a measure of the uniformity in the particle size distribution. For sands, n usually ranges between 1.0 and 10.0. We can assume a value of 2.5 for this calculation.
Substituting the values and rearranging the equation, we get:
d^2 = (10 * k) / (n^2 - 1)
= (10 * 0.000138 m/s) / (2.5^2 - 1)
≈ 7.3968e-6 m^2/s
Taking the square root, we can find the approximate value for the effective grain size (d):
d ≈ √(7.3968e-6 m^2/s)
≈ 0.00272 m
Therefore, the approximate value for the effective grain size is approximately 0.00272 meters or 2.72 mm.