Geotechnical Engineering

DHARM

SEEPAGE AND FLOW NETS 173

Vertical component

of flow

Z

Y

X

dz

dy

dx

(x, y, z)

Fig. 6.10 Flow through an element of soil

By Darcy’s law,

qz = k · i · A,

where A is the area of the bottom face and qz is the flow into the bottom face.

= kz F−

HG

I

KJ

∂

∂

h

z

dx · dy,

where kz is the permeability of the soil in the Z-direction at the point (x, y, z) and h is the total
head.

Flow out of the top of the element is given by:

qz + ∆qz = k k

z

dz

h

z

h

z

F z+ z dz

HG

I

KJ

−−

F

HG

I

KJ

∂

∂

∂

∂

∂

∂

,.

2

2.^ dx^ dy

Net flow into the element from vertical flow:

∆qz = inflow – outflow

= kz −

F

HG

I

KJ

∂

∂

h

z

dxdy – k k

z

dz h

z

h

z

F z+ z dz

HG

I

KJ −

−

F

HG

I

KJ

∂

∂

∂

∂

∂

∂

..

2

2 dx dy

∴ ∆qz = k h

z

kh

z

k

z

dz

h

z z

..∂ zz

∂

∂∂

∂

∂

∂

∂

∂

2

22

2

++ 2

F

HG

I

KJ

dx dy dz

Assuming the permeability to be constant at all points in a given direction, (that is, the
soil is homogeneous),
∂
∂

k

z

z = 0

∴ ∆qz = k h

z z

∂

∂

2

2

F

HG

I

KJ

dx dy dz

Similarly, the net inflow in the X-direction is:

∆qx = k h

x x

.∂

∂

2

2

F

HG

I

KJ

dx dy dz

For two-dimensional flow, qy = 0

∴ ∆q = ∆qx + ∆qz = k h

x

k h

xzz

..∂

∂

∂

∂

2

2

2

+ 2

F

HG

I

KJ

dx dy dz