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Derive the expression for coefficient of permeability by variable head method.
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This method is suitable for grained soil. Because quantity of water collected through the soil mass is very less and cannot be measured accurately same mould is used for this test also this experimental setup is shown below:

Let,

$h_i $= initial head

$h_f $= final head

$q$ = discharge through the sample $a$ = cross sectional area of the flow through sand pipe Now, a$dh$ = -q$dt$ But, q = A*k*i a$dh$ = -Aki$dt$ a$dh$ = -A$\frac{h}{L}$.k$dt$ $\frac{A.k.dt}{aL} = -\frac{dh}{h}$ $\frac{Ak}{aL} \int_{t_1}^{t_2} dt = \int_{h_1}^{h_2} \frac{dh}{h} $ $\frac{Ak}{aL} (t_2 - t_1) = \log_e {\frac{h_1}{h_2}}$ $k = \frac{aL}{At} \log_e \frac{h_1}{h_2}$ where t =$ t_2 - t_1$ = time interval during which head reduces from $h_1$ to $h_2$ $k = \frac{2.30 * L}{At} \log_10 \frac{h_1}{h_2}$ where $k = 10^{-2} m/s to 10^{-9} m/s$

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