DHARM

COMPRESSIBILITY AND CONSOLIDATION OF SOILS 227

This is written as:

∂

∂

=

∂

∂

u

t

c

u

v z

.

2

2 ...(Eq. 7.19)

where cv =

k

γwv.m

cv is known as the “Coefficient of consolidation”. u represents the hydrostatic excess pressure

at a depth z from the drainage face at time t from the start of the process of consolidation.

The coefficient of consolidation may also be written in terms of the coefficient of

compressibility:

cv =

k

m

ke

γγwv avw

=

()1+ 0

...(Eq. 7.20)

Equation 7.19 is the basic differential equation of consolidation according to Terzaghi’s
theory of one-dimensional consolidation. The coefficient of consolidation combines the effect of
permeability and compressibility characteristics on volume change during consolidation. Its
units can be shown to be mm^2 /s or L^2 T–1.

The initial hydrostatic excess pressure, ui, is equal to the increment of pressure ∆σ, and
is the same throughout the depth of the sample, immediately on application of the pressure,
and is shown by the heavy line in Fig. 7.21 (b). The horizontal portion of the heavy line indi-
cates the fact that, at the drainage face, the hydrostatic excess pressure instantly reduces to
zero, theoretically speaking. Further, the hydrostatic excess pressure would get fully dissi-
pated throughout the depth of the sample only after the lapse of infinite time*, as indicated by
the heavy vertical line on the left of the figure. At any other instant of time, the hydrostatic
excess pressure will be maximum at the farthest point in the depth from the drainage faces,
that is, at the middle and it is zero at the top and bottom. The distribution of the hydrostatic
excess pressure with depth is sinusoidal at other instants of time, as shown by dotted lines.
These curves are called “Isochrones”.
Aliter
With reference to Fig. 7.21, the
hydraulic gradient at depth iz

h

z

u

(^1) w z
U^1
VW=
∂
∂
= ∂
∂γ
.
Hydraulic gradient at depthi
zdz
u
z
u
z
dz
w
2
2
2
1
()+.
UV
W

∂
∂

• ∂
  ∂
  F
  HG
  I
  γ KJ
  Rate of inflow per unit area = Velocity at depth z = k.i 1 , by Darcy’s law.
  Rate of outflow per unit area = velocity at (z + dz) = k.i 2
  Water lost per unit time = k(i 2 – i 1 ) =
  ku
  z
  dz
  γw
  ..
  ∂
  ∂
  2
  2

• As the process of consolidation is in progress, the hydrostatic excess pressure causing flow
  decreases, which, in turn, slows down the rate of flow. This, again, reduces the rate of dissipation of pore
  water pressure, and so on. This results in an asymptotic relation between time and excess pore pres-
  sure. Therefore, mathematically speaking, it takes infinite time for 100% consolidation. Fortunately, it
  takes finite time for 99% or even 99.9% consolidation. This is good enough from the point of view of
  engineering accuracy.