Geotechnical Engineering

DHARM

SEEPAGE AND FLOW NETS 175

But kx. ∂h/∂x = vx and kz. ∂h/∂z = vz, by Darcy’s law.

∴ ∂

∂

∂

∂

v

x

v

z

xz+ = 0 ...(Eq. 6.5)

This is called the ‘Equation of Continuity’ in two-dimensions and can be got by setting
∆q = 0 (or net inflow is zero) during the derivation of Eq. 6.2.

The flow net which consists of two sets of curves – a series of flow lines and of equipotential
lines–is obtained merely as a solution to the Laplace’s equation – Eq. 6.4. The fact that the
basic equation of steady flow in isotropic soil satisfies Laplace’s equation, suggests that, the
flow lines and equipotential lines intersect at right-angles to form an orthogonal net – the ‘flow
net’. In other words, the flow net as drawn in the preceding sections is a theoretically sound
solution to the flow problems.
The ‘velocity potential’ is defined as a scalar function of space and time such that its
derivative with respect to any direction gives the velocity in that direction.

Thus, if the velocity potential, φ is defined as kh, φ being a function of x and z,

Similarly,

∂φ

∂

∂

∂

∂φ

∂

∂

∂

x

k

h

x

v

z

k

h

z

v

x

y

==

==

U

V

|

W

|

|

.

. ...(Eq. 6.6)

In view of Eq. 6.4 for an isotropic soil and in view of the definition of the velocity poten-
tial, we have:

∂φ

∂

∂φ

∂

2

2

2
xz+^2 = 0 ...(Eq. 6.7)
This is to say the head as well as the velocity potential satisfy the Laplace’s equation in
two-dimensions.

The equipotential lines are contours of equal or potential. The direction of seepage is
always at right angles to the equipotential lines.

The ‘stream function’ is defined as a scalar function of space and time such that the
partial derivative of this function with respect to any direction gives the component of velocity
in a direction inclined at + 90° (clockwise) to the original direction.

If the stream function is designated as ψ(x, z),

and

∂ψ

∂

∂ψ

∂

z

v

x

v

x

z

=

=−

U

V

|

W

|

| ...(Eq. 6.8)

by definition.

By Eqs. 6.6 and 6.8, we have:

and

∂φ ∂ ∂ψ ∂

∂φ ∂ ∂ψ ∂

//

//

xz

zx

=

=−

U
V
W ...(Eq. 6.9)
These equations are known as Cauchy-Riemann equations. Substituting the relevant
values in terms of ψ in the continuity equation (Eq. 6.5) and Laplace’s equation (Eq. 6.7), we
can show easily that the stream function ψ(x, z) satisfies both these equations just as φ(x, z)
does.