PERMEABILITY
AND
CAPILLARITY
Pavement
Original ground surface
Frozen soil
GWT
Capillary rise and frost action
Ice lenses
fed by
capill
ary
water
from
water
table.
CAPILLARITY
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Movement of soil moisture through small pores.
Pores serve as capillary tubes and soil moisture rises above GWT.
Water held in this manner is in a state of suction or negative pressure.
Height of capillary rise depends on soil type, the finer the voids the greater the
height.
Capillary water is continuously connected to the GWT
It rises up against the force of gravity due to capillary action.
It can be removed by drainage only.
Water in the capillary fringe (zone) held by surface tension forces, cannot be
drained by any drainage system, because capillary flow does not obey the law
of gravity.
If the GWT is lowered, the whole capillary fringe can be lowered.
Capillary water can also be removed by evaporation.
In cohesive soils its decreases cohesion and stability and the soil is
transferred into a plastic state.
In sandy soils it adds to the stability provided the soil is laterally confined.
Height of capillary rise varies inversely with size of pores, which is a function
of the particle size and density of soil.
Up to this height above the water table the soil is sufficiently close to full
saturation.
The capillary height is determined by capillarimeter.
h = capillary rise.
r = radius of tube.
Downward gravity force =
wt. of the liquid column
in the tube
= w(r2h)---- 1
T = surface tension force per
unit length
 = Contact angle
Upward surface tension
force = T  2r cos  -- 2
Fig: Capillary rise in a tube above
the free water surface.
Water will stop rising, when the upward force will be balanced by the downward
force.
w (r2 h) = 2 T r cos 
2T cos 
h
w  r
(3)
Equation shows that h increases as r decreases.
It may be noted that for pure water in contact with clean glass the value of angle
 = 0. The equation is then simplified as:
2T
h
w r
(4)
The value of T at room temperature is 0.064 N/m or 73.0 dynes/cm. In applying
the development of capillary rise in tubes to capillary rise in soil these values
of T are sufficiently accurate for many practical problems.
Equation for capillary rise can be expressed as:
0.31
hc 
cm
d
d = Effective capillary diameter = 1/5 D10
(5)
Terzaghi and Peck (1948) equation for capillary rise
C
hC max
eD10
(6)
Where, hc max. is in millimeter, C is a constant depending upon the shape of the
grains and the surface impurities (varying from 10.0 to 50.0 mm2) and D10 is
the effective size expressed in millimeters.
Approximate capillary height for different soil types.
SOIL TYPE
APPROX. CAPILLARY HEIGHT (cm)
Fine gravel
2-10
Coarse sand
10-15
Fine sand
30-100
Silt
100-1000
Clay
1000-3000
Colloids
>3000
CAPILLARY MOVEMENT IN SOIL
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Soil moisture moves continuously, even through unsaturated soils.
Direction of movement depends on the relative potential.
Moisture moves from higher potential towards lower potential.
The demand of a moist soil for additional moisture above GWT and the
gravitational pull provide the principal potentials, which influence moisture
movement.
Demand or capillary attraction for water is exerted in all directions.
Therefore capillary water may also move horizontally in soil depending upon
the relative potential.
GWT is a free water surface at which the pressure is atmospheric.
For unsaturated soil, the capillary potential is always negative while in a
saturated soil, it is zero.
Water therefore rises above the GWT.
When the capillary potential is balanced with the gravitational potential,
capillary moisture will be in static equilibrium, and no flow will occur.
Field moisture seldom reaches a state of equilibrium, because of relatively
slow rate of capillary movement and the continuously changing weather
conditions.
During dry season upper soil is drier with low capillary potential than lower
soil and upward movement occurs. After rain fall, downward movement due
to combined effect of gravity and capillarity takes place.
IMPORTANCE OF CAPILLARITY IN CIVIL ENGINEERING
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Moisture increases due to capillary rise.
An increase of moisture always reduces the strength of soil (especially fine
grained soil).
Study of capillarity is important for the following projects.
Pavement Sub-grade
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Sub-grade performance is much influenced by capillary water.
Pavement provides an impervious cover on the sub-grade.
Due to pavement, rain water is kept out of sub-grade soil, and
evaporation and transpiration are prevented.
Pavement also reduces the range and frequency of changes of
temperature in the underlying soil.
These conditions help the establishment of equilibrium with the ground
water table.
As a result moisture tends to accumulate in a pavement sub-grade after
the pavement is laid.
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Such accumulation occurs slowly and requires 3-5 years.
Consequently he soil loses its shear strength.
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If the study indicates the possibility of increase of capillary moisture to
alarming level, techniques to avoid the accumulation of capillary moisture
(within the zone of higher stresses below the pavement) are adopted.
Techniques include,
1. Provision of impermeable membrane
2. Drainage layer
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at a design level (depth) to stop the rise of capillary moisture above that level.
Excavations
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Additional bonding force within particles is developed due to negative capillary
potential.
This bonding force is known as apparent cohesion.
Apparent cohesion permits short term deep excavation in fine grained soil
without support.
Apparent cohesion however disappears with loss of capillary potential and the
excavation may collapse.
Shallow trenches in moist sand without support are also possible due to negative
capillary potential, while in dry and saturated sand trenches without support
can not may made.
Fig: Different zones of capillary water, capillary potential and capillary movement.
Capillary Siphoning
Fig: Capillary siphoning in dams above the impervious core
To prevent the loss water, the crest of the impervious core should be kept
sufficiently high.
FACTORS AFFECTING CAPILLARITY OR CAPILLARY POTENTIAL
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Capillary potential depends on surface tension of water and the radii of
curvature of the air-water surfaces of the tiny wedges of water between the
soil particles.
The amount of soil water to affect the radii of curvature depends on the
following,
1. Particle size,
2. Density,
3. Temperature,
4. Degree of wetting in terms of angle of contact,
5. Percentage of dissolved salts in the soil water.
1. Effect of Particle Size
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For same moisture content in a fine and coarse soil, the fine soil due to more
surface area will have more points of contacts between the soil particles.
At each point of contact, less water will be collected and the radii of curvature
and the corresponding surface tension will be greater. A lower capillary
potential and a greater attraction for moisture is observed in fine grained soils
as compared to coarse-grained soils.
2. Effect of Soil Density
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If two particles of moist soil are packed closer, the meniscus curvature will be
reduced, lowering the surface tension and increasing the capillary potential,
resulting in lesser attraction for water.
When a relatively dry soil mass is sufficiently compressed it become saturated,
although the moisture content remain unchanged.
The capillary potential, which initially was negative in loose state, increases to
the maximum value of zero on densification. Actually the process of
densification gradually reduces the curvature of air-water interfaces; and,
finally, no curvature remains at saturation.
Hence the capillary potential increases with increase of density.
3. Effect of Temperature
The surface tension varies inversely with temperature. A decrease in
temperature increases surface tension and hence the capillary potential is
reduced.
4. Effect of Angle of Contact
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The angle of contact between the menisci and the soil particles depends
on the mineral composition of soil.
An increase in the angle of contact will show low degree of wetting and
hence tend to decrease the curvature of the menisci and thereby increase
the capillary potential of soil at that particular water content.
A soil with an angle of contact greater than zero will have less attraction
for water than a soil in which the particles are completely wetted.
5. Effect of Dissolved Salts
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An increase in percentage of dissolved salts in soil water increase its
surface tension, and reduces the capillary potential, and hence indicate
more attraction for moisture. However the effect of dissolved salts is very
small.
Fig. Influence of state of packing of soil on curvature of air-water interface.
(a) Closely-packed soil; (b) Loosely-packed soil.
Fig. Influence of wettability of soil grains on curvature of air-water interface.
(a) Low wettability; (b) High wettability.
Fig. Effect of particle size upon curvature of air-water interface.
(a) Coarse-grained soil; (b) Fine-grained soil.
PERMEABILITY
Permeability means the ease or difficulty with which water or any other liquid
flow through soil.
The knowledge of permeability of soil is important for the following:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Evaluating the amount of seepage through or beneath dams and levees and into
water wells.
Evaluating uplift or seepage pressure beneath hydraulic structures for stability
analyses.
Providing control of seepage velocities, so that, fine particles are not eroded
from the soil mass.
Rate of settlement (consolidation) studies where volume changes occur as water
is expelled from the voids.
Controlling seepage from sanitary landfills and hazardous liquid waste dumps.
Evaluating the yield from wells as a source of water supply.
Designing the highways sub-drainage system.
Designing sub-drainage for water logging and salinity control.
Ground water lowering (Dewatering).
Investigation of contaminated lands.
Design of landfill sites.
Soil Hydrologic Cycle
Source: Vepraskas, M.J, et. Al. “ Wetland Soils”, 2001.
Flow
direction can
change
DARCY’S LAW OF FLOW THROUGH SOIL
According to Darcy’s law, the velocity of flow through soil is directly proportional
to the hydraulic gradient.
Fig: Loss of head, due to flow of water through soil.
h = difference of head.
L = length of soil between two points along the flow path
Where piezometers are installed.
V = velocity of flow through soil = Q/A
where:
Q is the discharge & A is the gross cross-sectional area of
i = hydraulic gradient = h/L
According to Darcy’s law.
Vi
V = ki
the soil
Where K is the constant of proportionality and is known as coefficient of
permeability.
Assumption of Darcy’s Law:
1. The continuity of the flow condition in the soil mass must be satisfied with
no velocity changes taking place during the flow.
2. The flow must be with voids saturated through out the flow and no
compressible air present in the voids.
3. The flow must be in a steady state. i.e., the velocity of flow must be constant
at any particular section with respect to time.
4. The flow must be laminar.
FECTORS EFFECTING PERMEABILITY
1. Soil grain size
Permeability depends mainly on the size of voids, which in turn depends on the
size, shape and state of packing of the soil particles.
Permeability appears to be proportional to the square of the effective grain
size.
K  (D10)2
For sandy soil, A.Hazen developed the following empirical equation.
K = C (D10)2
cm/sec.
Where,
D10, is the effective grain size in centimeters.
K, is coefficient of permeability in centimeters per second.
C, is a constant, according to Hazen, it varies from about 40 to 150 expressed as
1/cm.sec.
Table: Values of coefficient C for different grades of sand.
C
40-80
80-120
120-150
Sand
Very fine, well graded or with appreciable fines
Medium coarse, poorly graded; clean, coarse but
Well graded very coarse, very poorly graded,
gravelly, clean.
Generally for sandy soil K = 100(D10)2
(C.G.S. units)
2. Properties of the liquid
Permeability varies with density and viscosity of fluid flowing through the soil
a. Density of the fluid:
K 
Where,
 = Density of fluid.
b. Viscosity of the fluid:
K 1/
Where:
 = Absolute viscosity of fluid.
Since viscosity changes with temperature following equation may be used to
find ‘K1’ for any temp.
K 1

K1 
Where:
K is co-efficient of permeability at standard temperature commonly 20 oC.
K1 is the co-efficient of permeability at any test temperature.
 is viscosity at standard temperature i.e., 20oC.
1 is viscosity at test temperature.
(
3. Void ratio
a. For cohesive soil – K  e2
b. For non cohesive soils – K  e3/1+e
Cassagrande equation for fine or medium clean sand is:
K = 0.85 1.4e2 K0.85
Where:
K = permeability at any void ratio
K0.85 = permeability at a void ratio of 0.85
4. Soil Structure
Stratified soil usually has greater permeability in the horizontal direction than in
the vertical direction.
Rock/Sediment with high porosity
and low permeability?
5. Degree of Saturation
Increase in the degree of saturation increases permeability.
KS
Where, ‘S’ is degree of saturation.
6. Entrapped air within the soil
Entrapped air/gases reduce the degree of saturation and permeability.
Entrapped air/gases may be due to,
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Chemical decomposition of soil.
Disintegration of rock and animal remains.
Dissolved air.
According to D.W. Taylor the following simple Eq. relates ‘K’ to a number of
factors which influence permeability.
 w e3
K  Ds
C
 1 e
2
Where:
DS = Effective, grain size
w = Density of permeant (water/fluid)
 = Viscosity of permeant (water/fluid)
e = Void ratio
C = constant which depends on shape and arrangement of pores.
SUPERFICIAL AND SEEPAGE VELOCITY
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Water moves through the soil pores.
Area of flow is actually, the cumulative pore area, which is difficult to
determine.
Gross cross-sectional area of the soil cylinder is generally used for
common permeability/ seepage calculations.
Velocity in this case is known as superficial velocity.
Using the area of voids, velocity is known as seepage velocity.
However for practical problems term velocity of flow is used which is
based on total cross-sectional area of soil.
Q = A  V (where V is superficial velocity & A is the total crosssectional area of soil)
Q = Av  Vs (where Vs = seepage velocity & Av = area of voids).
Porosity = n = Vv/V
For a unit thickness, n = Av/A
or
Av = n  A
Putting the value of Av in Eq. for Q.,
Q = n x A x Vs
or
n x A x Vs = A V
V = n x Vs.
Since n is always less than 1 therefore V < Vs.
MEASUREMENT OF PERMEABILITY
Laboratory Methods
• Constant head permeameter.
• Variable head permeameter.
First is suitable for relatively coarse grained (sandy) soils, while the second is
recommended for fine grained (silty/clayey) soils.
Field Method
• Pumping-out test/discharge well test.
• Pumping-in test/recharge well test.
CONSTANT HEAD PERMEAMETER
Fig: Constant head Permeameter.
A = Cross-sectional area of soil sample
L = Length of sample
h = drop in head between the two piezometers
l = distance between peizometers
Vol. = Volume collected in time T.
T = time of test
According to DARCY’S law,
Vi
V=Ki
h
Since i 
l
h
V K
l
Multiplying ‘A’ on both sides
h
AV  KA
l
h
Q  KA
l
Vol .
Since Q 
Time
Vol .
h
 KA
T
l
Vol .  l
K
AhT
In highly impervious soil, quantity of flow is small and accurate measurement of
its value is not possible. Therefore the constant head permeameter is mainly
applicable to relatively previous soil such as sands and gravel.
VARIABLE HEAD PERMEAMETER
It is used to find the permeability
of relatively less permeable soil
(fine grained soil, silt and clay).
it is also known as falling head
permeameter.
Let a = cross-sectional area of the stand pipe.
A = cross-sectional area of soil sample.
L = Length of sample.
h1= Initial head at time t1.
h2= Final head at time t2.
dh = the drop in head in time dt
Velocity of fall of water level in the stand-pipe
V = -dh/dt (-ve sign indicates a fall of head).
dh
V 
dt
Multiplying by ‘a’ on both sides
Vxa=-a
dh
dt
Q=-a
dh
dt
Q is the discharge
KAi = -a 
dh
dt
h
dh
KA    a 
L
dt
KA
dh
dt  
aL
h
Integrating the above equation.
KA
aL
t2

t1
dt 
h2

h1
1
 dh
h
KA
(t 2  t1 )  (ln h2  ln h1 )
aL
aL
h1
KA
h1
K  2.303
log10
 t  ln 
or
At
h2
aL
h2
FIELD DETERMINATION OF PERMEABILITY
i- Discharge Well (Pumping-out Test)
If water is pumped out of the test well, it is called discharge well test. The
method is extensively used by water supply Engineers.
In foundation engineering the important problem is the draw down of the
ground water table, which is necessary to get the dry foundation pit to
start the construction work.
ii- Recharge Well (Pumping-in Test)
When water is pumped into the well from an out side source, it is called a
recharge well test. All the procedure is similar to discharge well method,
except an inverted cone of depression is developed.
THEORY OF ORDINARY PERFECT WELLS (DUPUIT THIEM’S
THEORY)
For the derivation of an analytical equation for the discharge Q of the well or
the permeability of soil K, following assumptions are made.
1. The soil is homogeneous, uniform and porous medium of infinite areal extent.
2. The well takes the ground water from the entire thickness of the permeable
water bearing stratum.
3. There exists an unconfined, uniform, steady, laminar and radial ground
water flow to the cylindrical well from a concentric boundary.
4. For small inclination of the free surface of the ground water gravity flow
system, the streamlines can be taken as horizontal.
5. The horizontal velocity is independent of depth.
6. The hydraulic gradient is equal to the slope of the tangent at any point on the
depression curve, of the free ground water table.
7. The coefficient of permeability K of the soil is constant at all times and at all
places.
8. The well is being pumped continuously at a uniform rate until the flow of
water to the well is stabilized.
DERIVATION OF EQUATION FOR THE CO-EFFICIENT OF
PERMEABILITYITY ‘K’
a- Unconfined Aquifer:
ro = Radius of the pumping well
H = Thickness of water bearing stratum
R = Radius of influence i.e. where the draw down S is zero
x-x = Original static ground water table
Smax = Maximum draw down which is at the pumping well.
Upon pumping, the water table in the well lowers by an amount of S termed as
draw down. The dewatered zone in the soil ABCDA takes the form of a
funnel or cone known as cone of depression.
The rate of flow, Q, towards the well from the surrounding soil at stabilized flow
is expressed by means of Darcy’s law as follows
Q = VA = Kai
V = Velocity of flow
A = Area of flow
dy
i
dx
hydraulic gradient
Fig. Pumping test in an
unconfined aquifer to
find permeability
Area of flow =A= 2 x y
dy
QK
2 xy
dx
When
x = ro, y = h
x = R ,y = H
Integrating for the above limits.
H
Q
h ydy  2K
R
dx
r x
o
( 2.14)
1
Q
R
2
2
(H  h ) 
loge
2
2K
ro
(2.16)
Q
R
H h 
loge
K
ro
2
2
R
Q loge
ro
K
 (H 2  h2 )
x = r1
x = r2
y = h1
y = h2
Integrating for the above limits.
(2.17)
(2.18)
h2
Q
r2
dx
h ydy  2  K r x
1
1
1
Q
r2
(h2  h1 ) 
loge
2
2 K
r1
r2
Q loge
r1
K
 (h2 2  h12 )
EQUATION FOR THE CONE OF DEPRESSION
The curve of depressed water table Cone of depression) can be drawn, using an
equation having x and y variables.
x = ro,
y=h
x = x,
y=y
y
Q
x
dx
h ydy  2  K ho x
x
Q loge
ro
2
2
y h 
 K
Q
x
y h 
loge
( which is the generaleq. of depressionline)
K
ro
2
2
Q
x
y h 
loge
K
ro
2
Similarly the equation of depression line can be written in another form i.e. when:
x = x,
x = R,
y=y
y=H
H
R
Q
dx
y yd y  2 K x x
1
Q
R
(H 2  y 2 ) 
loge
2
2 K
x
Q
R
y H 
loge
K
x
2
Putting different values of x in the above Equation, the corresponding values of y
are calculated, then with the set of x and y co-ordinates, curve of depression
line can be drawn provided K is already known.
If the value of drawn down S at any point x, from the center of well (along the
curve of depression) is required, the following basic equation is used.
Draw down S = H – y
Q
R
SH  H 
loge
K
x
2
For pumping water out of the well (lowering of ground water table) use minus
sign and for recharge of water into the well use plus sign before the square
root.
Maximum draw down at the pumping well can be determined by putting x = rO
S max  H  y  H  H 2 
Q
R
loge
K
ro
YIELD OF WELL
Yield of a well means, the maximum discharge capacity of the well installed in
any water bearing strata. The yield depends on the permeability of water
bearing strata. Yield can be determined by any of the following equations.
Q
 K (H 2  h2 )

 K ( H  h)(H  h)
R
R
loge
ro
ro
 K ( H  h)(S max )  K ( H  H  S max)(S max )
Q

R
R
loge
loge
ro
ro
 K (2 H  S max )(S max )
Q
R
loge
ro
loge
Specific Yield
The specific yield of the well, q, is defined as it’s yield per unit length (1m) of
draw down in the well.
RADIUS OF INFLUENCE
The radius of influence of the depression cone, R, is to be estimated from
experience or it is to be determined from observation in several bore holes
(observation wells) made at different distances from the test well. It can also
be determined from any of the above equation, which contain the term R.
According to Sichardt, for stabilized flow, R is given by an empirical equation
as follows.
R  3000S K
(in metersunits)
Where:
S = Maximum draw down in meters
K = Coefficient of permeability of soil in m/sec.
Kozeny gave an expression for the calculation of the radius of influence, R, in
terms of time, t, during which yield from the well of Q (m3/sec) has been
attained.
12t
R
n
QK

(in m etersunits)
(26)
Where:
n = Porosity of soil in decimal fractions.
K = coefficient of permeability of soil (m/sec).
The radius of influence increases with the fourth root of discharge, Q, and
permeability K.
b- Confined Aquifer:
In the case of confined aquifer the
pump is drawing water from the
layer of thickness D.
The area of flow = A = 2xD
dy
QK
2xD
dx
Separating the variables.
dx 2KD

dy
x
Q
Integrating between the limits.
1
2KD
r x dx  Q h dy
1
1
r2
h2
 r2  2KD
h2  h1 
ln   
Q
 r1 
r2
Q ln ( )
r1
K
2D (h2  h1 )
Fig. Pumping test in a confined aquifer to find permeability.