Geotechnical Engineering

DHARM

174 GEOTECHNICAL ENGINEERING

∆q may be obtained in a different manner as follows:

The volume of water in the element is:

Vw =

Se

e

.

()1+

. dx dy dz

∆q = rate of change of water in the element with time:

=

∂

∂

V

t

w = (∂/∂t) Se

e

dx dy dz

.

()

.

1 +

L

N

M

O

Q

P

dx dy dz

()1+e

is the volume of solids, which is constant.

∴∆q =

dx dy dz

()1+e

(∂/∂t) (S. e)

Equating the two expressions for ∆q, we have:

k h

x

k h

xzz

..∂

∂

∂

∂

2

2

2

+ 2

F

HG

I

KJ

dx dy dz =

dx dy dz

()1+e

. (∂/∂t) (S. e)

or kx

∂

∂

2

2

h

x

+ kz.

∂

∂

2

2

h

z

=

1

() 1

.

+

F +

HG

I

e KJ

e S

t

S e

t

∂

∂

∂

∂

...(Eq. 6.2)

This is the basic equation for two-dimensional laminar flow through soil.
The following are the possible situations:
(i) Both e and S are constant.
(ii)e varies, S remaining constant.
(iii)S varies, e remaining constant.
(iv) Both e and S vary.
Situation (i) represents steady flow which has been treated in Chapter 5 and this chapter.
Situation (ii) represents ‘Consolidation’ or ‘Expansion’, depending upon whether e decreases
or increases, and is treated in Chapter 7. Situation (iii) represents ‘drainage’ at constant volume
or ‘imbibition’, depending upon whether S decreases or increases. Situation (iv) includes
problems of compression and expansion. Situations (iii) and (iv) are complex flow conditions
for which satisfactory solutions have yet to be found. (Strictly speaking, Eq. 6.2 is applicable
only for small strains).

For situation (i), Eq. 6.2 reduces to:

kx. ∂

∂

2

2

h

x

+ kz. ∂

∂

2

2

h

z

= 0 ...(Eq. 6.3)

If the permeability is the same in all directions, (that is, the soil is isotropic),

∂

∂

∂

∂

2

2

2

2

h

x

h

z

+ = 0 ...(Eq. 6.4)

This is nothing but the Laplace’s equation in two-dimensions. In words, this equation
means that the change of gradient in the X-direction plus that in the Z-direction is zero.

From Eq. 6.3,

(∂/∂x) k

h

x x

.

∂

∂

F

HG

I

KJ + (∂/∂z) k

h

z z

.

∂

∂

F

HG

I

KJ = 0