7 - Water in Soil mechanics geotechnical engineering

Geotechnical Engineering A
4
Water in Soil

Contents
1. Groundwater
2. Permeability
3. Coefficient of permeability:
i. Coefficient of permeability published data
ii. Coefficient of permeability laboratory tests
iii. Coefficient of permeability in-situ tests

1. Groundwater
• Groundwater in soil may be one of two types, occurring in two
distinct zones separated by the water table (or phreatic surface).
The zone below the water table is known as gravitational water (or phreatic), which:
1. Is subject to gravitational forces.
2. Saturates the pore spaces in the soil below the water table.
3. Has an internal pore pressure greater than atmospheric pressure.
4. Tends to flow laterally.
The zone above the water table is known as
vadose water, which is the unsaturated
zone and comprises:
1. Slow moving water, percolating downwards
to join the phreatic water below the water
table.
2. Capillary water held above the water table
by surface tension forces (with internal pore
pressure less than atmospheric).

Normal and Perched water table
• The water table is the level at which the pore water pressure is equal to the
atmospheric pressure.
• In general, the water table corresponds to the free water surface; such as
may be found at a river or lake or in an excavation,
– where groundwater lies above isolated bodies of soil, perched water tables may
occur.

Confined groundwater
• Artesian conditions are said to exist where the water table ties above ground level.
• Sub-artesian conditions occur when the water table lies between the surface and the aquifer.
High Permeability
Low Permeability
} Aquifer
Artesian conditions
(water table above ground)
Sub-artesian conditions
(water table below ground)
Where a stratum of reasonably high permeability, is confined above and below strata
of low permeability = an aquifer - there may be no obvious groundwater table.
However, the water level in standpipes or wells would indicate the level of the water
table. The porewater pressure in a confined aquifer depends on the condition at the
place where the layer is unconfined. If a rise in the water table occurred in this area, it
is likely that the artesian pore water pressure will also rise

Capillary water
• Capillary water is held above the water table by surface tension
Capillary Tube Detail at Surface
hc can be established theoretically
from an understanding of the
vertical forces at the surface of the
liquid
Capillary rise
hc
For a soil an approximation, based on the voids ratio, e, and the
effective size of the soil particles, D10, may be established from: 10
30
.
hc
e D


• While the value of hc
represents the maximum capillary rise, the soil will only be
saturated with capillary moisture up to the capillary saturation level hcs. The
approximate relationship between capillary rise and capillary saturation level is
given on the graph below.
Effective size, D10, (mm) (log scale)
Capillary
Rise
(mm)
(log
Scale)
0 0.002 0.006 0.02 0.06 0.2 0.6 2 6
104
103
102
101
0
Saturation level hcs
Capillary rise hc
Approximate relationship between capillary rise and soil type
CLAY
SILT
SAND
GRAVEL

Partially saturated
with percolating water only
Groundwater zones
Saturated with phreatic water
Partially saturated with
capillary water
Saturated with capillary water
Free water
surface
closed face
open face
Excavation
Water table

2. Permeability
• Permeability may be defined as the rate at which water under pressure can
flow through the interconnected voids (or pore spaces) within a soil.
• The flow properties of water through soil was first investigated
by H. Darcy in 1856.
– He showed that under steady flow conditions through beds of sand of varying
thickness and under various pressures, the rate of flow, q, was always proportional
to the hydraulic gradient, i, (the fall in hydraulic head per unit thickness of sand).
– This principal is known as Darcy’s Law, which is given below. This equation has been
found to be valid for most types of fluid flow through soils.
q=Aki
where: q = rate of flow (m3
/sec)
A = area of flow (m2
)
k = coefficient of permeability (m/sec)
i = hydraulic gradient

h1-h2=h
• The hydraulic gradient, i, is the ratio of the difference in total head on either side of a soil
layer, to the thickness of the layer measured in the direction of the flow. This can be
illustrated using the examples given below.
Hydraulic gradient
Horizontal
flow through
a tube of soil.
X
If a standpipe or
piezometer tube is
inserted in the soil at
any point X, the water
will rise in the tube up
to a level (h) which
indicates the static
pressure of water at X.
The level of water in
the pipe is called the
piezometric level and
the pressure of the
water is the height of
the column of water in
the pipe above point x
The difference in
piezometric level
between P & Q is
the hydraulic head
between these two
points, which may
be denoted as h1-h2.
P Q
h1
h2
L
The hydraulic gradient, i, is the ratio of the difference
in head to the distance, L, between P & Q
Pressure = gh
1 2
h h h
i
L L
 
  

Inclined flow under a hydraulic gradient
• If the flow of water is not horizontal, as illustrated below, the difference in level between points S and T
must be taken into account. This is achieved by measuring the heights of the piezometric levels h3
and h4
above a common datum, which gives the total head at each point. The length, L, is the distance between
points S and T measured along the direction of flow. The calculation of the hydraulic gradient is similar to
that given above:
3 4
h h h
i
L L
 
  
S
T
h3
h4
Datum Level
L

Coefficient of Permeability, k
• The coefficient of permeability, k, may be
defined as the mean discharge velocity of water
flow under the action of a unit hydraulic gradient
and is usually expressed in m/s.
• The coefficient of permeability may be assessed
from:
• Published data,
• Laboratory or in-situ permeability tests or
Derived from other empirical (based on
observation and experiment) relationships.
These methods of assessment are considered in more detail
below.

3.1 Coefficient of permeability – published data
CLAYS GRAVELS
SILTS SANDS

3.2 Coefficient of permeability – laboratory tests
• There are two basic methods of determining the coefficient of permeability of a soil in the laboratory:
1. The Constant Head (used for soils of high permeability e.g. gravels & sands k >10-4
m/s)
2. The Falling Head (used for soils of low permeability e.g silts & clays)
1. The Constant Head Permeability (BS1377: Part 5: 1990)
• This piece of apparatus effectively reproduces Darcy’s equation in the laboratory.
• Water from a constant head supply flows through the soil sample of cross sectional area A,
until steady flow conditions prevail and the two manometer readings h1
and h2
remain
constant.
• When steady flow conditions are attained the head loss, h1
- h2 is measured.
• If the length of the sample, L, is known it is then possible to calculate the hydraulic gradient,
i, from the expression:
1 2
h h
i
L



h1 h2
L
Steady flow conditions
h
1 2
h h
i
L


• The rate of flow of water through the sample may be calculated by measuring the
quantity of water, Q, collected in a measuring cylinder in a measured time period. The
rate of flow, q, may then be calculated from the expression:
Q
q
t

Q
t

• Given that the cross sectional area of the sample is known, the coefficient of
permeability may be calculated from Darcy’s equation rearranged as follows.
q = Aki (Darcy’s equation)
q
k
Ai

 
Q
q rateof flowof water
t

 
1 2
h h h
i hydraulic gradient
L L
 
 
q
k
Ai

.
. .
Q L
k
t A h


or
Q
k
t Ai

NB Care is required to ensure that the units used when calculating values of k are appropriate.

Example 1: during a constant head permeability test the following data was recorded
for a sample having a diameter of 75mm. Determine the average value of the
coefficient k.
Flow quantity for 3 minutes (ml) 1509 1442 1257 1106 974
Difference in manometer levels (mm) 81 72 59 52 37
Distance in manometer levels (mm) =120mm
Length of flow path =120mm
Cross-sectional area of sample =
2
2
75
4418
4
A mm
 
 
Q = Q(ml) x103
mm3
t = 3 x 60secs = 180sec
3
10 120
4418. . 180
Q
k
h
 
 
 
Average k = (2.81+3.02+3.22+3.21+3.97)/ 5
.
. .
Q L
k
t A h


1
1509
0.151
81
k  2
1442
0.151 3.02 /
72
k mm s
  3 4 5
3.22, 3.21, 3.97
k k k
  
= 3.25x10
= 3.25x10-3
-3
m/s
m/s
= 3.25mm/s
2.81 /
mm s

0.151 ( / )
Q
mm s
h


• Steady flow conditions are impossible to attain through soils of low permeability and the
rate of flow would be too low to record using the constant head method of test.
• Consequently, the falling head test was developed specifically for low permeability soils.
• The falling head permeability test utilises a similar sample to that described above, but in
this case a vertical standpipe leads directly to the top of the sample.
• The bottom of the sample is immersed in a dish in which the water level is kept constant.
2. The Falling Head Permeability
• At any instant in time, the difference in height
between the water level in the standpipe and
that in the dish is effectively the head loss
over the sample length L.
• The flow rate is related to the drop in height
of the water in the standpipe and the cross
sectional area of the standpipe.
• However, the solution is complicated by the
fact that the head, hence the flow rate, are
constantly changing with time.
• The formula for the calculation of the
coefficient of permeability, k, is derived using
integral calculus, and is:
h1
h2
L
flow rate

 
 
1 2
2 1
ln /
aL h h
k
A t t
 
 


where:
a = cross sectional area of standpipe
A = cross sectional area of sample
L = length of sample
h1,h2= heights of water in the standpipe
measured from the water level in the
dish, at times t1 and t2 respectively
ln = logarithm to base e
or
 
 
10 1 2
2 1
2.3 log /
aL h h
k
A t t
 
 


log
If a = cross-sectional area of standpipe
A = cross-sectional area of sample
L = length of sample
Quantity of water flowing in time dt = Q = -a.dh
Also from Darcy’s Law, Q = k.A.i.dt
in which i = h/L
Then Q= -a.dh
. . .
k A h dt
L

Rearranging and integrating:
2 2
1 1
h t
h t
dh kA
dt
h aL
 
 
 
2
2 1
1
log
h kA
t t
h aL
  

Example 2: during a falling head permeability test the following data was recorded for a
sample having a diameter of 100 mm and a length of 150 mm. Determine the average value
of the coefficient k (standpipe diameter 9.00 mm).
Initial standpipe level (mm) h1 1200 900 750
Final standpipe level (mm) h2 900 750 500
Time onterval (s) t2 - t1 65 41 95
Cross-sectional area of sample =
Average k =5.32x10-3
mm/s = 5.32x10-6
m/s
Cross-sectional area of standpipe =
 
 
1 2
2 1
ln /
aL h h
k
A t t
 
 


 
 
1 2
2 1
63.6 150 ln /
7854
h h
k
t t
 
  
 
 
 
 
1 2
2 1
1.215 ln /
h h
k
t t
 
 
 

 
1
1.215 ln 1200/900
65
k
 
 
  =5.38x10-3
mm/s
 
2
1.215 ln 900/ 750
41
k
 
 
  =5.40x10-3
mm/s
 
1.215 ln 750/500
3
95
k
 
 
  =5.19x10-3
mm/s

3.3 Coefficient of permeability
in-situ test
• Laboratory measurement of permeability may be unreliable
because, among other reasons, it is difficult to obtain
undisturbed samples of granular soils. In addition, the small
sample used may not be representative of the soil mass
(macrostructure).
• There are three basic types of in-situ test, which are:
i. Constant/Falling head tests,
ii. Packer tests,
iii. Pumping tests.
– These methods are briefly discussed below.

• These tests are undertaken within boreholes sunk using any of the various drilling
techniques.
• It should be noted that these tests are only applicable to soil below the water table and
are not normally undertaken in rock.
• During the site work it is necessary to take the following measurements:
i) Constant head and falling head permeability in-situ tests
Falling Head In-situ
Falling Head In-situ
Constant Head In-situ
Constant Head In-situ
Water table
d
Hc
c
q
k
fdH

f is a shape factor which is depends on the conditions at
the bottom of the casing (cases A to F)
Constant head
Water table
d
Ho
 
ln /
.
o t
H H
A
k
fd t

Flow Rate (q)
Ht
to
t1
Borehole
Ground Level
Borehole
Ground Level
Time t
ln
(H
o
/H
t
)
fdk
gradient
A

SAND
CLAY

Ref. Soil Mechanics Principles and Practice by GE Barnes

d
Water table
Borehole
ii) Packer in-situ permeability tests
Flow Rate (q)
L
Double Packer
Double Packer
(used in completed borehole)
Flow Rate (q)
Single Packer
Single Packer
(used during drilling)
Packer (rubber
bag is inflated
against the sides
of the
boreholes)
Applied head (Hp)
Gravity head
(Hg)
q
k
Hfd

Total head (H) = Hp+Hg

SAND
CLAY
r1
r2
r3
z2
h2
z3
h3
H
GW
h0
Fully Penetrating well in an unconfined aquifer; overlying impervious layer, with observation wells
iii) Pumping in-situ permeability tests
Constant Head Pumping Test
   
3
2
2
1
2 2 2 2
2 1 3 2
log
log e
e
r
r
q
q
r
r
k
h h h h
 
 
 
   
   
 
 
 
2 2 2
1 2 3
1 2 3
2
z z z
h
z z z
 

 
An approximate value of k can be obtained
from the above expression even if the well is
not fully penetrating but the depth H must be
known in order to calculate values of h from
draw down z. h can be found approximately if
there are 3 observation wells spaced such that
r/r2 = r2/r1
z1
h1
Pumping water
out at a rate q

CLAY
CLAY
SAND
D
h1 h2 h3
r1
r2
r3
Fully Penetrating well in an confined aquifer; between impervious layers, with observation wells
   
3
2
2
1
2 1 3 2
log
log
2 2
e
e
r
r
q
q
r
r
k
D h h D h h
 
 
 
   
   
 
 
Strictly, only two observation wells are needed but the
third well acts as a check. The formula for k is true only
for confined flow: the piezometric head in the main
well must be above the top of the aquifer
Pumping water
out at a rate q

CLAY
CLAY
SAND
ho
h1
h1
h2
L1
L2
Fully Penetrating Trench; in an unconfined aquifer with a line source
   
1 2
2 2 2 2
1 2
2 2
o o
qL qL
k
h h h h
 
 
These expressions assume flow into one side of the
trench only
   
2 2 2 2
1 2
1 2
2 2
o o
k h h k h h
or q
L L
 
 

Example 3: A pumping test was carried out to determine the permeability of a sand layer in an unconfined aquifer with a well arrangement as
shown below. At steady-state pumping rate of 58.7 m3
/h, the draw downs in the observation wells were respectively 2.91 m and 0.88 m.
Calculate the coefficient of permeability k.
SAND
CLAY
15 m
35 m
16.2 m
GW
2.91 m
58.7 m3
/h
1.85 m
3
58.7
/
60 60
Flowrate q m s


= 0.0163 m3
/s
h1= 16.2 - 1.85 - 2.91 = 11.44 m
h2= 16.2 - 1.85 - 0.88 = 13.47 m
h1
h2
= 11.44 m
= 13.47 m
 
2
1
2 2
2 1
loge
r
q
r
k
h h

 
 
 


 
 
2 2
35
0.0163log
15
13.47 11.44
e
k

 

= 8.70x10
= 8.70x10-5
-5
m/s
m/s
0.88m
1.85 m

CLAY
CLAY
SAND
D = 7.6 m
15 m
32 m
5.7 m
2.18 m
1.62 m
0.47 m
Example 4: A permeability pumping test was carried out in a confined aquifer.the arrangement of wells and relevant dimensions is shown
below. The draw downs indicated were observed at a steady-state pumping rate of 15.6 m3
/h. Calculate the coefficient of permeability k.
 
2
1
2 1
log
2
e
r
q
r
k
D h h

 
 
 


3
15.6
/
60 60
Flowrate q m s


= 4.33x10-3
m3
/s
h1= 5.7+7.6-2.18-1.62 = 9.50m
h2= 5.7+7.6-2.18-0.47 = 10.65m
= 6.0x10
= 6.0x10-5
-5
m/s
m/s
 
 
3 32
4.33 10 log
15
2 7.6 10.65 9.50
e
k




  
h1 h2
15.6m3
/h

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