Permeability
Learning Objectives (1 of 2)
• Recognize that the permeability of soil is due to the existence of
interconnected voids where water can flow, causing seepage.
• Apply Bernoulli’s equation to the flow of water through permeable soil
mediums.
• Identify Darcy’s law as a simple equation used to calculate the discharge
velocity of water through saturated soils.
• Discuss the many factors of hydraulic conductivity, such as fluid viscosity,
pore-size distribution, grain-size distribution, void ratio, roughness of
mineral particles, and degree of soil saturation.
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Learning Objectives (2 of 2)
• Assess hydraulic conductivity using the constant head test and the falling-head
test.
• Evaluate the relationships for hydraulic conductivity for granular soils.
• Evaluate the relationships for hydraulic conductivity for cohesive soils.
• Interpret the directional variation of permeability.
• Devise the equivalent hydraulic conductivity in stratified soil.
• Discuss experimental verification of equivalent hydraulic conductivity.
• Employ a permeability test in the field by pumping from wells.
• Identify the hydraulic conductivity of compacted clayey soils.
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Introduction
• Soils are permeable due to interconnected voids through which water can
flow from points of high energy to low energy.
• Study of the flow of water through permeable soil is important in soil
mechanics. It is necessary for:
• Estimating the quantity of underground seepage under various hydraulic
conditions
• Investigating problems involving the pumping of water for underground
construction
• Making stability analyses of earth dams and earth-retaining structures subject to
seepage forces
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Bernoulli’s Equation (1 of 3)
• The total head at a point in water under motion is related by Bernoulli’s
equation:
u v2
h= + +Z
w 2g
Where:
h = total head
u = pressure
v = velocity
g = acceleration due to gravity
γw = unit weight of water
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Bernoulli’s Equation (2 of 3)
From Figure 7.1, we can deduce:
The head loss between two points is
equal to:
h = hA − hB
The hydraulic gradient is the
nondimensionalized form of the head
loss:
h
i=
L
Figure 7.1 Pressure, elevation, and total heads for flow of
water through soil
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Bernoulli’s Equation (3 of 3)
• The three zones of the fluid flow
are the laminar flow,
transition, and turbulent flow
zones.
• Most flows through soil are
laminar, and velocity is linearly
related to the hydraulic gradient:
vi
Figure 7.2 Nature of variation of v with hydraulic gradient, i
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Darcy’s Law (1 of 3)
• The discharge velocity, v, is the quantity of water flowing in unit time
through a unit gross cross-sectional area of soil at right angles to the flow
direction.
• Darcy’s law states that:
v = ki
k is the hydraulic conductivity, and is also called the coefficient of
permeability
• Darcy’s law is valid for a wide range of soils.
• This equation was based primarily on Darcy’s observations about the flow of
water through clean sands.
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Darcy’s Law (2 of 3)
• The actual velocity of water (the seepage velocity, vs) is greater than the
discharge velocity, v.
• The seepage velocity, vs, may be related to the discharge velocity through the
void ratio, e, or the porosity, n, by Eq (7.10):
1+ e v
vs = v =
e n
Figure 7.3 Derivation of Eqs. (7.7) and (7.8)
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Darcy’s Law (3 of 3)
• Hansbo (1960) found a variation of
discharge velocity with hydraulic
gradient while studying four
undisturbed natural clays.
• For very low discharge velocities,
the relationship between v and i is
non-linear as can be observed in
Figure 7.4.
Figure 7.4 Variation of discharge velocity with hydraulic
gradient in clay
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Hydraulic Conductivity (1 of 3)
• The hydraulic conductivity depends on many factors, including fluid
viscosity, pore-size distribution, grain-size distribution, void ratio, and
soil saturation.
• The hydraulic conductivity may be related to the unit weight of water, γw, and
dynamic viscosity of water, η:
w
k= K
K is the absolute permeability of the soil
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Hydraulic Conductivity (2 of 3)
Table 7.1 Typical Values of Hydraulic Conductivity of Saturated Soils
k
Soil type cm/s
Clean gravel 100−1.0
Coarse sand 1.0−0.01
Fine sand 0.01−0.001
Silty clay 0.001−0.00001
Clay <0.000001
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Hydraulic Conductivity (3 of 3)
• Hydraulic conductivity depends on the viscosity of water, which varies with
temperature.
• The value of hydraulic conductivity at 20°C may be related to the hydraulic
conductivity at the test temperature T:
T C
k20C = kT C
20C
• Table 7.2 in the text gives the value of ηT° C/η20° C for 15°C ≤ T ≤ 30°C.
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Laboratory Determination of Hydraulic
Conductivity
• Two standard laboratory tests are used to determine the
hydraulic conductivity of soil:
• The constant-head test
• The falling-head test
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Constant-head Test (1 of 2)
• Once a constant flow rate is
established, the hydraulic
conductivity is calculated based on
how much water is collected after a
set duration of time (t) by:
QL
k=
Aht
• Q is the volume of water collected, A
is the cross-sectional area, and L is
the length of the specimen
Figure 7.5 Constant-head permeability test
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Constant-head Test (2 of 2)
Figure 7.6 A constant-head permeability
test in progress (Courtesy of Khaled
Sobhan, Florida Atlantic University, Boca
Raton, Florida)
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Falling-head Test (1 of 2)
• Water from a standpipe flows
through the soil so that the final
head difference at time t = t2 is h2.
• The rate of flow of the water
through the specimen at any time
can be given by:
dh
q = −a
dt
where q is the flow rate and a is the
cross-sectional area of the pipe
Figure 7.7 Falling-head permeability test
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Falling-head Test (2 of 2)
• By rearranging and integrating the previous equation, the hydraulic gradient,
k, can be calculated by:
aL h1
k = 2.303 log10
At h2
A is the cross-sectional area of the soil specimen
L is the length of the specimen
h1 is the head difference at t = 0
h2 is the head difference at t = t2
• In most cases, laboratory tests for determination of hydraulic conductivity are
conducted with no effective overburden pressure.
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Relationships for Hydraulic Conductivity–Granular Soil (1 of 3)
• For fairly uniform sand, Hazen proposed:
cm
k = cD 2
10
sec
c is a constant between 1.0 and 1.5
D10 is the effective size, in mm
• This equation is based primarily on Hazen’s (1930) observation of loose,
clean, filter sands.
• A small quantity of silts and clays, when present in a sandy soil, may change
the hydraulic conductivity substantially.
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Relationships for Hydraulic Conductivity–Granular Soil (2 of 3)
• The equation based on the Kozeny-Carman equation gives fairly good
results in estimating the hydraulic conductivity of sandy soil
1 w e3
k=
CS S S2T 2 1 + e
where Cs = shape factor, which is a function of the shape of flow channels
Ss = specific surface area per unit volume of particles
T = tortuosity of flow channels
γw = unit weight of water
η = viscosity of permeant
e = void ratio
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Relationships for Hydraulic Conductivity–Granular Soil (3 of 3)
• On the basis of laboratory
experiments, the US Department
of Navy (1986) provided an
empirical correlation between k
and D10 for granular soils.
• This correlation is valid for
uniformity coefficients varying
between 2 and 12.
Figure 7.11 Hydraulic conductivity of granular soils
(Redrawn from US Department of Navy, 1986)
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Relationships for Hydraulic Conductivity–Cohesive Soil (1 of 3)
• Taylor (1948) proposed a linear relationship for cohesive soils as:
e0 − e
log ( k ) = log ( k0 ) −
Ck
k0 is the in situ hydraulic conductivity at a void ratio e0
k is the hydraulic conductivity at a void ratio e
Ck is the hydraulic conductivity change index and may be taken to be
about 0.5e0
• This equation is good for e0 less than about 2.5.
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Relationships for Hydraulic Conductivity–Cohesive Soil (2 of 3)
• Mesri and Olson (1971) suggested the use of a linear relationship between
log k and log e in the form
log k = A log e + B
Figure 7.14 Variation of hydraulic conductivity of sodium
clay minerals (Based on Mesri and Olson, 1971)
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Relationships for Hydraulic Conductivity–Cohesive Soil (3 of 3)
• Samarasinghe (1982) conducted laboratory tests on New Liskeard clays and
proposed that, for normally consolidated clays:
en
k = C
1+ e
where C and n are constants to be determined experimentally
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Directional Variation of Permeability
• Most soils are not isotropic with
respect to permeability.
• In Figure 7.16, there is a soil layer
through which water flows at an
angle α from the vertical.
• kV and kH correspond to the
hydraulic conductivity in the vertical Figure 7.16 Directional variation of permeability
and horizontal directions,
respectively.
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Equivalent Hydraulic Conductivity in Stratified Soil (1 of 4)
• For soils where the hydraulic conductivity varies from layer to
layer, an equivalent hydraulic conductivity may be calculated from
the hydraulic conductivities of the layers.
• This value is different for horizontal and vertical flows.
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Equivalent Hydraulic Conductivity in Stratified Soil (2 of 4)
• For horizontal flow, the total flow is the sum of the flow through each layer.
• Applying this knowledge, we get an expression:
k H ( eq ) =
1
H
(
k H1 H1 + k H 2 H 2 + + kHn H n )
H is the total height of the soil cross section
Hn is the height of the ith layer of soil
Figure 7.17 Equivalent hydraulic conductivity
determination—horizontal flow in stratified soil
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Equivalent Hydraulic Conductivity in Stratified Soil (3 of 4)
• For flow in the vertical direction, the total losses will be the sum of the
losses through each layer Accordingly:
H
kV ( eq ) =
H1 H 2 Hn
+ + +
kV 1 kV 2 kV n
Figure 7.18 Equivalent hydraulic conductivity determination—vertical
flow in stratified soil
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Equivalent Hydraulic Conductivity in Stratified Soil (4 of 4)
• Varved soil is a rhythmically
layered sediment of coarse and
fine minerals.
• It is a good example of naturally
deposited layered soil.
• Figure 7.19 shows the layer
variation seen in New Liskeard,
Canada, varved soil.
Figure 7.19 Variation of moisture content and grain-size distribution in New
Liskeard varved soil. (Source: Based on Laboratory Investigation of
Permeability Ratio of New Liskeard Varved Clay,” by H. T. Chan and T. C.
Kenney, 1973, Canadian Geotechnical Journal, 10(3), pp. 453–472.)
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Experimental Verification of Equivalent Hydraulic Conductivity
• Sridharan and Prakash (2002)
showed that if the thickness of soil
layers (H) are not equal, then
kV −exit
H
kV ( eq ) = f exit
kV −inlet
H
inlet
Figure 7.21 Effect of kV −exit kV −inlet on kV(eq) for a two-layered system (Based on Sridharan and Prakash, 2002)
H exit H inlet
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Permeability Test in the Field by Pumping from Wells
• The average hydraulic conductivity • The expression for the rate of flow of
for a soil deposit may be determined groundwater into the well, which is
using pumping tests from wells. equal to the rate of discharge from
pumping can be written as
• Water is pumped from a test well
with a perforated casing at a
constant rate and a steady state is r1
established when the water level in 2.303q log10
the test and observation wells k= r2
becomes constant. ( h12 − h22 )
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Pumping from a Confined Aquifer
• For a confined aquifer, the
hydraulic conductivity can be
calculated as:
r1
q log10
cm r2
k =
s 2.727 H ( h1 − h2 )
H is the thickness of the aquifer Figure 7.26 Pumping test from a well penetrating the full depth
in a confined aquifer
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Hydraulic Conductivity of Compacted Clayey Soils (1 of 2)
• It was shown in Chapter 6 that when a clay is compacted at a lower
moisture content, it possesses a flocculent structure.
• At optimum moisture content of compaction, the clay particles have a lower
degree of flocculation.
• A further increase in moisture content at compaction provides a greater
degree of particle orientation, but the dry weight decreases.
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Hydraulic Conductivity of Compacted Clayey Soils (2 of 2)
• Observations can be made from Figure 7.27
1. For similar compaction effort and molding
moisture content, the magnitude of k decreases
with the decrease in clod size.
2. For a given compaction effort, the hydraulic
conductivity decreases with the increase in
molding moisture content, reaching a minimum
value at about the optimum moisture content
(that is, approximately where the soil has a higher
unit weight with the clay particles having a lower
degree of flocculation). Beyond the optimum
moisture content, the hydraulic conductivity
increases slightly. Figure 7.27 Tests on a clay soil: (a) Standard and modified Proctor
compaction curves; (b) variation of k with molding moisture content
(Source: Benson, Daniel, “Influence of Clods on Hydraulic Conductivity of
Compacted Clay,” Journal of Geotechnical Engineering, 116(8), 1990, pp.
1231–1248. With permission from ASCE.)
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Summary (1 of 2)
• Darcy’s law can be expressed as
k i
=
discharge velocity hydraulic conductivity hydraulic gradient
• Seepage velocity of water through the void spaces can be given as:
discharge velocity
S =
porosity of soil
• Hydraulic conductivity is a function of viscosity (and hence temperature) of
water.
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Summary (2 of 2)
• Constant-head and falling-head types of tests are conducted to determine the
hydraulic conductivity of soils in the laboratory.
• There are several empirical correlations for hydraulic conductivity in granular
and cohesive soil.
• For layered soil, depending on the direction of flow, an equivalent hydraulic
conductivity relation can be developed to estimate the quantity of flow.
• Hydraulic conductivity in the field can be determined by pumping from wells.
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