The logarithm of time fitting method is based on characteristic of U and log₁₀T_v in intersection of tangent at point of inflection and asymptote of lower portion is at ordinate of 100% U.

The coefficient of volume change m_v is defined as change in volume of a soil per unit of initial volume due to a given increase in unit pressure.
? \(m_v= \frac{-?e}{1+e_o} \frac{1}{???}. \)

Since quantity of water q is equal to velocity of flow time area of cross-section through which it flows,
? q=v*A. The area of element is dxdy.
? The quantity of water entering element is vdxdy.

Soils, by nature, are non-homogeneous media, and anisotropic in general. This means that ir properties vary within relatively short distances in vertical and horizontal directions. For simplicity in solving equations soil is considered to be homogenous.

Out of many methods available, two important methods are:

• square root of time fitting method
• logarithm of time fitting method.

In order to facilitate drainage of pore water that is between soil particles, porous stones are kept at top and bottom of soil specimen in consolidation test.

The secondary consolidation is represented by equation of,
\(?e = -? log_{10}?\frac{t_2}{t_1}\)
Where ? = coefficient representing rate of secondary compression
t₁ and t₂ = total elapsed time since load was applied to soil.

The consolidation of soil takes a lot of time, so in order to test it, specimen is allowed to consolidate under a number of increments of vertical pressures of 10, 20, 50, 100, 200, 400, 800 and 1000 kPa and each pressure is maintained constant until compression virtually ceases. This generally takes 24 hours or even more to complete.

Given, double drainage for both specimen and field,
t₁=20 min
d₁=24 mm/2 = 2.4 cm/2 = 1.2 cm
d₂ =4/2 m 200cm
Since, for same degree of consolidation of 50%, T_v is same,
\(t?\frac{d^2}{C_r}\)
Therefore, \(\frac{t_2}{t_1} =(\frac{d_2}{d_1})^2\)
So, \(t_2=t_1 (\frac{d_2}{d_1} )^2=20(\frac{200}{1.2})^2=365 \,days.\)

Since coefficient of permeability is given by equation,
k=C_v m_v ?_w ??????(1)
The coefficient of volume change is given by,
\(m_v=\frac{a_v}{1+e_0},\) refore substituting in (1),
\(k=\frac{C_v a_v ?_w}{1+e_o} \).