Constant head test, k at porosity 39%

Given

$D = 7.5$ cm, $L = 18$ cm, head loss $h = 24.7$ cm, $Q = 626$ ml in $t = 60$ s, porosity $n = 0.44$.

$$A = \frac{\pi}{4}(7.5)^2 = 44.18\ \text{cm}^2, \qquad q = \frac{626}{60} = 10.43\ \text{cm}^3/\text{s}, \qquad i = \frac{24.7}{18} = 1.372$$

Permeability

$$k = \frac{q}{A\,i} = \frac{10.43}{44.18 \times 1.372} = 0.172\ \text{cm/s}$$

Flow (discharge) velocity

$$v = \frac{q}{A} = \frac{10.43}{44.18} = 0.236\ \text{cm/s} \quad (= ki)$$

Seepage velocity

$$v_s = \frac{v}{n} = \frac{0.236}{0.44} = 0.537\ \text{cm/s}$$

Permeability at $n = 0.39$

For the same soil, $k \propto \dfrac{e^3}{1+e}$ (Kozeny–Carman type relation).

$$e_1 = \frac{0.44}{0.56} = 0.786, \qquad e_2 = \frac{0.39}{0.61} = 0.639$$ $$\frac{k_2}{k_1} = \frac{e_2^3/(1+e_2)}{e_1^3/(1+e_1)} = \frac{0.1594}{0.2716} = 0.587$$ $$k_2 = 0.172 \times 0.587 = 0.101\ \text{cm/s}$$

Answer: $k = 0.172$ cm/s; $v = 0.236$ cm/s; $v_s = 0.537$ cm/s; $k$ at $n = 0.39$ is $0.101$ cm/s.