Soil MechanicUniversity of Babylon – Civil Eng. Dept.
Chapter five
5-1 Introduction
Water can flow through the interconnected soil pore. Water in the pore is under pressure,
the pressure is measured relative to the atmospheric pressure.
Water table ( W.T) : is the level at which the water pressure is atmospheric , also called
phreatic surface.
Notes :
Below water table the soil is assumed to be fully saturated
- The level of water table changes according to climate conditions and also construction
operation can change it.
- Artesian conditions can exist if an inclined layer of high permeability is confined locally
by an overlying layer if low permeability.
• Water in soil is
1- Either static : the hydrostatic pressure depend on the depth below the water table.
2- Or seepage through the soil under hydraulic gradient . her Bernoulli's equation applies
to the pore water.
The problems of fluid flow in soil are
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1- Rate of flow of fluid through an earth dams ( eg. Determination of rate of leakage through
an earth dam).
2- Problems involving compression (eg. Determination of rate of settlement of foundation ).
3- Problems involving strength (eg. The evaluation of factor of safety of a given soil under a
given loading ).
5-2-1 Permeability Definition-
• The property of soils that allows water to pass through them at some
rate.
• This property is a product of the granular nature of the soil, although it
can be affected by other factors (such as water bonding in clays)
• Different soils have different perm abilities, understanding of which is
critical to the use of the soil as a foundation or structural element
• Soil and rock are porous materials
• Fluid flow takes place through interconnected void spaces between
particles and not through the particles themselves
• No soil or rock material is strictly “impermeable”
The study of flow of water through porous media is necessary for-
Estimation Seepage Loss
3- Evaluation Quicksand Conditions
4- Dewatering System Design
Straight path in soil problems
but is a winding path in actual Flow path on soils
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5-Drainage System Design
5-2-1-1 Hydraulic gradient :
According to Bernoulli's equation, the total head of a point under motion can be given by:
Where
• In reality, an energy balance of the soil as it flows through the ground
• Kinetic Component can usually be ignored then
ht=
+ Z = hp + he
and this is defined as the piezometer head therefore, in soil engineering
problems the total head = piezometer head = pressure head + elevation head
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The loss of head between A & B, can be given by
h can be expressed in non-dimensional form as hydraulic gradient
i= h
where L =length of flow over which loss of head h
5-3 capillary in soils
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Water can rise and remain above the line of atmospheric pressure ( phreatic line ) in a
very fine pore due to attraction ( surface tension ) between adjacent molecules in the surface . at
phreatic line u = 0
The water is drawn upwards by this force until, at height h, the weight of water in the column is
in equilibrium with the magnitude of the surface tension.
Then if atmospheric pressure = 0, equating vertical forces at the surface :
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W= FT cos
Ts is the surface tension
Giving capillary pore pressure u= 2
cos or putting u = where unit weight of water
γwπr2h = Ts2πrcos
hc ∝ 1
Hence :
- height of water rise depend on size of soil voids.
-All point above W.T is negative charge (-ve since h is a against atmospheric pressure )
-The soil almost saturated in the capillary zone
- the water pressure is negative ( less than the atmospheric pressure in capillary zone)
- the negative pressure of water held above the water table results in attractive force between
particles. This phenomena is called soil suction. If is a function of the pore size and water
content.
5-3 Permeability :
Methods that are in common use for determining the coefficient of permeability k can be
classified under laboratory and field methods.
Laboratory methods: 1. Constant head permeability method
2. Falling head permeability method
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Field methods: 1. Pumping tests 2. Bore hole tests
Indirect Method:
Empirical correlations
The various types of apparatus which are used in soil laboratories for determining the
permeability of
soils are called permeameters. The apparatus used for the constant head permeability test is
called a
constant head permeameter and the one used for the falling head test is a falling
headpermeameter.
The soil samples used in laboratory methods are either undisturbed or disturbed. Empirical
correlations have been developed relating grain size and void ratio to hydraulic conductivity .
In one dimensional, water flows through a fully saturated soil in accordance with Darcy's law
q = VA
V
q is a volume of water flowing per unit time
A is a cross sectional area of flow path and
V is seepage velocity
i= is hydraulic gradient
k= coefficient pf permeability (L/t)
k is depend permeability on the average pore size which is related to the followings:
1- Particle size distribution
In general
The smaller soil particle, the smaller is the average size of pore and lower is the coefficient of
permeability. For given soil, the coefficient of permeability is a function of the void ratio.
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5-3-1 Permeability in stratified soil
Hydraulic conductivity of a disturbed sample may be different from that of the undisturbed
sample
even though the void ratio is the same. This may be due to a change in the structure or due to the
stratification of the undisturbed soil or a combination of both of these factors. In nature we may
find
fine grained soils having either flocculated or dispersed structures. Two fine-grained soils at the
same void ratio, one dispersed and the other flocculated, will exhibit different permeabilities.
Soils may be stratified by the deposition of different materials in layers which possess
different permeability characteristics. In such stratified soils engineers desire to have the average
permeability either in the horizontal or vertical directions. The average permeability can be
computed if the permeabilities of each layer are determined in the laboratory. The procedure is as
follows:
k , k2, ..., kn = hydraulic conductivities of individual strata of soil either in the vertical or
horizontal direction.
Z = z1+z2+z3+……Zn
kh = average hydraulic conductivity parallel to the bedding planes (usually horizontal).
kv - average hydraulic conductivity perpendicular to the bedding planes (usually vertical).
k=
of the water
characteristics of the soil skeleton only
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- The velocity by which water flow through soil is varying from point A to point B in
magnitude and direction.
- The average velocity at which water flows through soil is called seepage velocity (ν).
Qin = Qout VA = νs x Av .and
.n=
νs=

where νs= velocity of water through the soil between point A and B
5-3-2 Pressure Head
Pore pressure needed to compute, effective stress can be obtaine from the pressure head.
Pore pressure = pressure head x γw
Or u = hp x γw
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Pressure head can be either
i- Measured or
a- Monometer or stand pipe (m).
b- Boueden gage ( kN/m2 or kPa.)
5-3-3 Effective stress in soil with fluid flow
1- Now flow
u= γw ( H+Z)
2- With flow
a- Dawnward flow
b- Upward flow
• At Downward flow inlet is higher than the storage tank and the flow from up to dawn
as shown in figure below
σ = γw x H + γsat. x H0
u= γw (H + H0 - h)
σ= (γsat.-γw )H0 + γwh
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• Upward flow in this case the flow direction from dawn to up of soil sample as
shawn in figure below.
u= γw (H + H0 + h)
σ= (γsat.-γw )H - γwh
Seepage force : Is applied by moving water to the soil skelton and acts in the flow direction.
Hence:
i- The total boundary force OR
ii- Buoyancy force + seepage force
Seepage force per unit volume (J)
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J=
Notes :
1- Seepage force exerted by the flowing water act as on external force on the soil
Skelton.
2- J is a constant value for a given soil mass
Ex: for the figure below find the J if h= 0.6 m and L = 0.6 m
J = iγw
J=
=
a- Total weight + Boundary water force
F = Zγw A + LAγt – (h +Z+L)γw A
= LA (γt –γw ) – hAγw
F = LAγb – hAγw
F = σ A
• Quick condition ( Boiling )
S = shear strength , can be determine using Colomb equation as below:
* Aσ u* A
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S = C+ σ tanφ
C = cohesion of soil ( fine grained soil property )
φ = internal friction angle ( coarse grained soil property )
• The boiling occure in sandy soils therefore, for cohesionless soils ( coarse grain ) C = 0 then
S = σ tanφ then
then σ = 0
= critical hydraulic gradient , this hydraulic gradient accrue at quick condition
= γb
1+e γw
The safety factor (Fs) is the factor refer to the state of the soil condition , if the soil without
boiling this is safe state while at boiling case the soil under failure condition and the soil at
critical condition when the shear strength of the soil is higher than shear stress which try to fail
the soil.
Fs = ic
ie where( ic) is the critical hydraulic gradient and (ie) is the exist hydraulic gradient
Note : if γb not given can assume it (γb) = 1 and become ic ≈ 1
5-3-4 Permeability in stratified soil ( equivalent permeability )
the coefficient of permeability of a given layer of soil may vary with the direction of flow.
Therefore, the equivalent permeability coefficient must be determine for both direction
a- X- direction flow
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1- Assume a cross – section of unit length passing through H layer and perpendicular to
the direction of flow.
When
V = average discharge velocity
q= v1*1*h1+v2*1*h2+v3 *h3*1+v4*1*h4
+… vn*1*hn
Vav. = kheq. *ieq
kheq. *ieq * H = kh1*h1*i1+ kh2*h2*i2+ kh3*h3*i3+ kh4*h4*i4+……, khn*hn*in
kheq.= kh1∗h1+ kh2∗h2+ kh3∗h3+ kh4∗h4+…,khn∗hn
H
OR
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In this case the velocity of flow through all the layers is the same. However, the total head loss (h) is
equal to the summation of the head loss in each layer.
5-3-5 Types of flow
1- Dawn-ward fluid flow
2- Upward fluid flow
4- Horizontal fluid flow
5- Flow through inclined layer
for first three types the flow is vertical flow while number four deals with the horizontal flow
in soil. Also the fifth items deal with the inclined soil therefore the flow will be inclined flow. In
next section will be discuss each type of flow individually.
Direction of flow
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5-4 Determination of the permeability coefficient
A- Laboratory methods
1- Constant head permeability test method ( for coarse grain soil )
A steady flow under constant head is maintained
through the soil and volume of water per unit time
(q) is measure
vacuum is applied to specimen before the test to
bring (S close to 100%) if a high degree of
saturation it to be maintained during the test, de-
aired water should be used.
Example : In a laboratory, a constant-head permeability test
was conducted on a brown sand
with a trace of mica. For the constant-head permeameter (Figure above), the following data were
obtained:
1. Quantity of water discharged during the test = 25cm3 .
2. Length of specimen between manometer outlets = 11.43 cm .
3. Time required for given quantity of water to be discharged = 65.0 s.
4- Head (difference between manometer levels) = 5.5 cm .
5. Temperature of water = 20 0C.
6. Diameter of specimen 10.16 cm.
Required
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2- Falling head
As shown in Figure below, the velocity of fall in the burette is
given by = −dh
with the minus sign used to indicate a falling
(and therefore decreasing)
head. The flow of water into the specimen is therefore =
− dh
, and the
flow through and out of the specimen is, from = h
.
Therefore
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Or
Where
A is cross –sectional area of specimen
k= permeability coefficient of soil test.
h1 = initial height of water before the test.
h2 = finial height of water after the test.
h the difference in total head between the top and bottom at time (t).
L = length of specimen.
The coefficient of permeability as determined by both methods is the value for the particular
water temperature at which the test was conducted. This value is ordinarily corrected to that for
20°C by multiplying the computed value by the ratio of the viscosity of water at the test
temperature to the viscosity of water at 20°C.
Permeability determined in a laboratory may not be truly indicative of the in situ permeability.
There are several reasons for this in addition to the fact that the soil in the permeameter does not
exactly duplicate the structure of the soil in situ, particularly that of nonhomogeneous soils and
granular materials. For one thing, the flow of water in the permeameter is downward, whereas
flow in the soil in situ may be more nearly horizontal or in a direction between horizontal and
vertical. Indeed, the permeability of a natural soil in the horizontal direction can be considerably
greater
than that in its vertical direction. For another thing, naturally occurring strata in the in situ soil
will not be duplicated in the permeameter. Also, the relatively smooth walls of the permeameter
afford different boundary conditions from those of the in situ soil.
Finally, the hydraulic head in the permeameter may differ from the field gradient. Another
concern with the permeability test is any effect from entrapped air in the water and test
specimen. To avoid this, the water to be used in the test should be de-aired by boiling distilled
water and keeping it covered and non-agitated until used.
Example
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B- In situ tests method 1- Well pumping test
- This method most suitable for use in homogenous coarse soil strata.
- This method involves continuous pumping at constant rate from a well ( 300mm diam. At
test ) which penetrate to the bottom of the stratum under test.
- A screen of filter is placed in the bottom of the well to prevent ingress of soil particles.
Perforated casing is used to support the sides of the well.
- Radially steady –state seepage towards the well. The water table will be drawn down to
form a cone of depression.
- Water levels are observed in number of bore –holes spaced on radial lines at varies
distances from the pumping well.
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- Analysis is made on the assumption that the hydraulic gradient at distance ( r ) from the
center of well.
2- Equal to the slop of water table
icr = dh

where
h is a height of water table at distance ( r) " this is Dupuit assumption "
• In pumping test method there are two formula can used to determine the permeability
coefficient. The type of formula used depend on the soil stratum condition as confined
stratum or unconfined stratum.
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Example: A pumping test was performed in a well penetrating an unconfined aquifer to evaluate
the coefficient of permeability of the soil in the aquifer. When equilibrium flow was reached, the
following data were obtained:
1. Equilibrium discharge of water from the well = 200 gl/min.
2. Water levels (h1 and h2) is 15ft and 18 ft at distances from the well (r1 and r2) of 60 and 180 ft,
respectively.
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2- Pumping test method with confined stratum
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Example A pumping test was performed in a well penetrating a confined aquifer to evaluate the
coefficient of permeability of the soil in the aquifer. When equilibrium flow was reached, the
following data were obtained:
1. Equilibrium discharge of water from the well = 200 gl/min.
2. Water levels (h1 and h2) is 15ft and 18 ft at distances from the well (r1 and r2) of 60 and 180 ft,
respectively.
Required
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3- Empirical Relations for Hydraulic Conductivity
Several empirical equations for estimating hydraulic conductivity have been
proposed over the years.
Granular Soil
For fairly uniform sand (that is, a small uniformity coefficient), Hazen (1930)
proposed an empirical relationship for hydraulic conductivity in the form:
k= CD2 10
D10 = the effective size (mm)
The above equation is based primarily on Hazen’s observations 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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Examples
1- The permeability coefficient of a sand at a void ratio of 0.55 is 0.1 ft/min. estimate its
permeability coefficient at avoid ratio of 0.7. Use Casagrande empirical relationship
2- for normally consolidated clay soil, the following are given:
Estimate the permeability coefficient of clay at
void ratio of 1.2 .Use Samarasingh et. al. relation.
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3- A pumping test from Gravity well in a permeable layer underlain by an impervious
stratum was made. When steady state was reached, the following observations were made
q = 100 gpm; h1 = 20 ft; h2 = 15 ft; r1= 150 ft; and r2 = 50 ft. Determine the
permeability coefficient of the permeable layer.
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4- A permeable soil layer is underlain by an impervious layer, as shown in Figure below.
With k = 5.3x10-5 m/sec for the permeable layer, calculate the rate of seepage through it in
m3/hr/m width if H= 3 m and α= 8°.
Solution
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Problems
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5-For a falling-head permeability test, the following are given: length of specimen = 380
mm; area of specimen = 6.5 cm2; k= 0.175 cm/min. What should be the area of the standpipe
for the head to drop from 650 cm to 300 cm in8 min?
6- For a falling-head permeability test, the following are given:
• Length of soil specimen= 700 mm
• Area of the soil specimen= 20 cm2
• Area of the standpipe= 1.05 cm2
• Head difference at time t = 0 is 800 mm
• Head difference at time t= 8 min is 500 mm
a. Determine the absolute permeability of the soil.
b. What is the head difference at time = 6 min?
Assume that the test was conducted at 20= oC, and at 20= oC, γw= 9.789 kN/m3 and
η= 1.005x103 Ns/m2.
7- A sand layer of the cross-sectional area shown in Fig. below has been determined to exist for
a 800 m length of the levee. The hydraulic conductivity of the sand layer is 2.8 m/day.
Determine the quantity of water which flows into the ditch in m3/min.
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8- A permeable soil layer is underlain by an impervious layer, as shown in Figure7.28. With k=
5.2 x 104 cm/sec for the permeable layer, calculate the rate of seepage through it in m3/hr/m
length if H= 3.8 m and α= 8o.
9-The hydraulic conductivity of a sand at a void ratio of 0.5 is 0.022 cm/sec. Estimate its
hydraulic conductivity at a void ratio of 0.7. 1- For a sand, the following are given: porosity, n = 0.31 and k = 0.2 ft/min.
Determine k when n = 0.4. 2- The maximum dry density determined in the laboratory for a quartz sand is 1800 kg/m3.
In the field, if the relative compaction is 80%, determine the hydraulic conductivity of the sand
in the field-compaction condition (given that D10 = 0.15 mm, Cu= 2.2 and Gs = 2.66). use
shahada et.al formula .
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