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# Craig's Soil Mechanics 7th Edition

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The values of k for different types of soil are typically within the ranges shown in

Table 2.1. For sands, Hazen showed that the approximate value of k is given by

k ¼ 10�2D210 ðm=sÞ ð2:3Þ

where D10 is the effective size in mm.
On the microscopic scale the water seeping through a soil follows a very tortuous

path between the solid particles but macroscopically the flow path (in one dimension)
can be considered as a smooth line. The average velocity at which the water flows
through the soil pores is obtained by dividing the volume of water flowing per unit

Permeability 31

time by the average area of voids (Av) on a cross-section normal to the macroscopic
direction of flow: this velocity is called the seepage velocity (v0). Thus

v0 ¼ q
Av

The porosity of a soil is defined in terms of volume:

n ¼ Vv
V

However, on average, the porosity can also be expressed as

n ¼ Av
A

Hence

v0 ¼ q
nA

¼ v
n

or

v0 ¼ ki
n

ð2:4Þ

Determination of coefficient of permeability

Laboratory methods

The coefficient of permeability for coarse soils can be determined by means of the
constant-head permeability test (Figure 2.1(a)). The soil specimen, at the appropriate
density, is contained in a Perspex cylinder of cross-sectional area A: the specimen rests
on a coarse filter or a wire mesh. A steady vertical flow of water, under a constant total
head, is maintained through the soil and the volume of water flowing per unit time (q)

Table 2.1 Coefficient of permeability (m/s) (BS 8004: 1986)

Clean
gravels

Clean sands
and sand – gravel
mixtures

Very fine sands,
silts and clay-silt
laminate

Unfissured clays and
clay-silts (>20%
clay)

1 10–1 10–2 10–3 10–4 10–5 10–6 10–7 10–8 10–9 10–10

Desiccated and fissured clays

32 Seepage

is measured. Tappings from the side of the cylinder enable the hydraulic gradient (h/l )
to be measured. Then from Darcy’s law:

k ¼ ql
Ah

A series of tests should be run, each at a different rate of flow. Prior to running the test
a vacuum is applied to the specimen to ensure that the degree of saturation under flow
will be close to 100%. If a high degree of saturation is to be maintained the water used
in the test should be de-aired.
For fine soils the falling-head test (Figure 2.1(b)) should be used. In the case of

fine soils, undisturbed specimens are normally tested and the containing cylinder in
the test may be the sampling tube itself. The length of the specimen is l and the
cross-sectional area A. A coarse filter is placed at each end of the specimen and
a standpipe of internal area a is connected to the top of the cylinder. The water
drains into a reservoir of constant level. The standpipe is filled with water and a
measurement is made of the time (t1) for the water level (relative to the water level
in the reservoir) to fall from h0 to h1. At any intermediate time t the water level
in the standpipe is given by h and its rate of change by �dh/dt. At time t the

Figure 2.1 Laboratory permeability tests: (a) constant head and (b) falling head.

Permeability 33

difference in total head between the top and bottom of the specimen is h. Then,
applying Darcy’s law:

�a dh
dt

¼ Ak h
l

; � a
Z h1
h0

dh

h
¼ Ak

l

Z t1
0

dt

; k ¼ al
At1

ln
h0

h1

¼ 2:3 al
At1

log
h0

h1

Again, precautions must be taken to ensure that the degree of saturation remains close
to 100%. A series of tests should be run using different values of h0 and h1 and/or
standpipes of different diameters.
The coefficient of permeability of fine soils can also be determined indirectly from

the results of consolidation tests (see Chapter 7).
The reliability of laboratory methods depends on the extent to which the test

specimens are representative of the soil mass as a whole. More reliable results can
generally be obtained by the in-situ methods described below.

Well pumping test

This method is most suitable for use in homogeneous coarse soil strata. The procedure
involves continuous pumping at a constant rate from a well, normally at least 300mm in
diameter, which penetrates to the bottom of the stratum under test. A screen or filter is
placed in the bottom of the well to prevent ingress of soil particles. Perforated casing is
normally required to support the sides of the well. Steady seepage is established, radially
towards the well, resulting in the water table being drawn down to form a ‘cone of
depression’. Water levels are observed in a number of boreholes spaced on radial lines at
various distances from the well. An unconfined stratum of uniform thickness with a
(relatively) impermeable lower boundary is shown in Figure 2.2(a), the water table being
below the upper surface of the stratum.A confined layer between two impermeable strata is
shown inFigure 2.2(b), the originalwater tablebeingwithin theoverlying stratum.Frequent
recordings are made of the water levels in the boreholes, usually by means of an electrical
dipper. The test enables the average coefficient of permeability of the soil mass below the
cone of depression to be determined. Full details of the test procedure are given in BS 6316.
Analysis is based on the assumption that the hydraulic gradient at any distance r from

the centre of the well is constant with depth and is equal to the slope of the water table, i.e.

ir ¼ dh
dr

where h is the height of the water table at radius r. This is known as the Dupuit
assumption and is reasonably accurate except at points close to the well.
In the case of an unconfined stratum (Figure 2.2(a)), consider two boreholes located

on a radial line at distances r1 and r2 from the centre of the well, the respective water

34 Seepage

levels relative to the bottom of the stratum being h1 and h2. At distance r from the well
the area through which seepage takes place is 2
rh, where r and h are variables. Then
applying Darcy’s law:

q ¼ 2
rhk dh
dr

; q
Z r2
r1

dr

r
¼ 2
k

Z h2
h1

h dh

; q ln
r2

r1

� �
¼
kðh22 � h21Þ

; k ¼ 2:3q logðr2=r1Þ

ðh22 � h21Þ

For a confined stratum of thickness H (Figure 2.2(b)) the area through which
seepage takes place is 2
rH, where r is variable and H is constant. Then

q ¼ 2
rHk dh
dr

; q
Z r2
r1

dr

r
¼ 2
Hk

Z h2
h1

dh

r1

r2

r1

r2

q

q

Well

Observation
boreholes

h1
h2

h2h1

H

W.T.

W.T.

(a)

(b)

Figure 2.2 Well pumping tests: (a) unconfined stratum and (b) confined stratum.

Permeability 35

; q ln
r2

r1

� �
¼ 2
Hkðh2 � h1Þ

; k ¼ 2:3q logðr2=r1Þ
2
Hðh2 � h1Þ

Borehole tests

The general principle is that water is either introduced into or pumped out of a
borehole which terminates within the stratum in question, the procedures being
referred to as inflow and outflow tests, respectively. A hydraulic gradient is thus
established, causing seepage either into or out of the soil mass surrounding the bore-
hole and the rate of flow is measured. In a constant-head test the water level is
maintained throughout at a given level (Figure 2.3(a)). In a variable-head test the
water level is allowed to fall or rise from its initial position and the time taken for
the level to change between two values is recorded (Figure 2.3(b)). The tests indicate
the permeability of the soil within a radius of only 1–2m from the centre of the
borehole. Careful boring is essential to avoid disturbance in the soil structure.
A problem in such tests is that clogging of the soil face at the bottom of the borehole

tends to occur due to the deposition of sediment from the water. To alleviate the

q

hc

h

h1
h2

W.T.

W.T.

W.T.

(a)

(e) (d)

(b) (c)

Seepage

B
A

i = hAB

Figure 2.3 Borehole tests.

36 Seepage

problem the borehole may be extended below the bottom of the casing, as shown in
Figure 2.3(c),```