ANJUMAN COLLEGE OF ENGINEERING & …...UNIT – III 3. Permeability: Darcy’s law & its validity,...
Transcript of ANJUMAN COLLEGE OF ENGINEERING & …...UNIT – III 3. Permeability: Darcy’s law & its validity,...
ANJUMAN COLLEGE OF ENGINEERING & TECHNOLOGY
MANGALWARI BAZAAR ROAD, SADAR, NAGPUR - 440001.
DEPARTMENT OF CIVIL ENGINEERING
Prof. Rashmi G. Bade, Department of Civil Engineering, Geotechnical Engineering – I 1
Geotechnical Engineering – I
B.E. FOURTH SEMESTER
ANJUMAN COLLEGE OF ENGINEERING & TECHNOLOGY
MANGALWARI BAZAAR ROAD, SADAR, NAGPUR - 440001.
DEPARTMENT OF CIVIL ENGINEERING
Prof. Rashmi G. Bade, Department of Civil Engineering, Geotechnical Engineering – I 2
UNIT – III 3. Permeability: Darcy’s law & its validity, Discharge & seepage velocity, factors affecting
permeability, Determination of coefficients of permeability by Laboratory and field methods,
permeability of stratified soil.
4. Seepage: Seepage pressure, quick sand condition, characteristics & uses of flownets, Preliminary
problems of discharge estimation in homogeneous soils, Effective, Neutral and total stresses in soil
mass.
ANJUMAN COLLEGE OF ENGINEERING & TECHNOLOGY
MANGALWARI BAZAAR ROAD, SADAR, NAGPUR - 440001.
DEPARTMENT OF CIVIL ENGINEERING
Prof. Rashmi G. Bade, Department of Civil Engineering, Geotechnical Engineering – I 3
3.1 INTRODUCTION
Permeability is defined as the property of a porous material which permits the passage or seepage of
water through its interconnecting voids. A material having continuous voids is called permeable.
Gravels are highly permeable while stiff clay is the least permeable and hence such clay may be
termed as impermeable for all practical purpose.
The flow of water through soils may either be a laminar flow or a turbulent flow. In laminar flow ,
each fluid particle travels along a definite path which never crosses the path of any other particle. In
turbulent flow, the paths are irregular and twisting, crossing and recrossing at random. In most of the
practical flow problems in soil mechanics, the flow is laminar.
The study of seepage of water through soil’s is important for the following engineering problems:
1. Determination of rate of settlement of a saturated compressible soil layer.
2. Calculation of seepage through the body of earth dams, and stability of slopes.
3. Calculation of uplift pressure under hydraulic structure and their safety against piping
4. Ground water flow towards wells ad drainage of soils.
3.2 DARCY’S LAW & ITS VALIDITY
The law of flow of water through soil was studied by Darcy’s (1856)who demonstrated
experimentally that for laminar flow condition in a saturated soil, the rate of flow or the discharge
per unit time is proportional to the hydraulic gradient.
q= k i A ---(1)
v=q/A=k i
Where q=discharge per unit time
A = total cross –sectional area of soil mass, perpendicular to the direction of flow.
I = hyraulic gradient
K = Darcy’s coefficient of permeability
V = velocity of flow, or average discharge velocity
Fig. 1. Flow of Water through soil.
ANJUMAN COLLEGE OF ENGINEERING & TECHNOLOGY
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DEPARTMENT OF CIVIL ENGINEERING
Prof. Rashmi G. Bade, Department of Civil Engineering, Geotechnical Engineering – I 4
If a soil sample of length L and cross – sectional area A, is subjected to differential head of water, h1
– h2, the hydraulic gradient I will be equal to
and, we have,
----------- (i)
From fig.1 when hydraulic gradient is unity, k is equal to v. Thus, the coefficient of
permeability, or simply permeability, is defined as the average velocity of flow that will occur
through the total cross – sectional area of soil under unit hydraulic gradient. The dimensions of the
coefficient of permeability k are the same as these of velocity. It is usually expressed as cm/sec or
m/day or feet/day.
Typical value of coefficient of permeability
Soil type Coefficient of permeability
in mm/sec Drainage properties
Clean gravel 101 to10
2 Very good
Coarse and medium sands 10 -2
to 101
Good
Fine sand, loose silt 10-4
to 10-2
Fair
Dense silt, clayey soil 10 -5
to 10-4
Poor
Silty clay, clay 10 -8 to 10 -5 Very poor
Validity of Darcy’s law
Darcy’s law is valid if the flow through soil is laminar. The flow of water through soil depends upon
the dimension of interstices, which in turn, depends upon the particle size. In fine-grained soil, the
dimensions of the interstices are very small and the flow is necessarily laminar. In coarse-grained
soil, the flow is also generally laminar. However, in coarse grained soils, such as coarse gravel, the
flow may be turbulent.
For flow of water through pipes, the flow is laminar when the Reynolds number is less than 2000.
For flow through soil, it has been found that the flow is laminar if the Reynolds number is less than
unity. For flow through soil, the characteristic length in the Reynolds numbers is taken as the average
particle diameter (D).
Thus,
Where ρ=is the mass density and η is the coefficient of viscosity.
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Prof. Rashmi G. Bade, Department of Civil Engineering, Geotechnical Engineering – I 5
It can be shown that the maximum diameter of the particle for the flow to be laminar is about
0.50mm.
The value of the critical Reynolds number is unity is, however, conservative. It has been
demonstrated that flow remain laminar even upto the Reynolds number of 75.it has been observed
that Darcy’s law is valid for flow in clays, silts and fine sands. In coarse sands, gravels and boulders.
The flow may be turbulent and Darcy’s law may not be applicable. It is difficult to predict the exact
range of the validity of Darcy’s law. The best method to ascertain the range is to conduct
experiments and determine the actual relationship between the velocity ν and the hydraulic gradient.
For Darcy’s law to be valid, this relationship should be approximately linear.
For flow through coarse sands, gravels and boulders, the actual relationship between the velocity
and the hydraulic gradient is non-linear. The following equation for the velocity when turbulent.
Where n=exponent , with a value of 0.65In extremely fine grained soils, such as colloidal clay, the
interstices are very small. The velocity therefore very small. In such soils also, the Darcy law is not
valid
3.3 DISCHARGE & SEEPAGE VELOCITY
Darcy’s law represents the statistical macroscopic equivalent of the Navier- Stokes equations of
motion for the viscous flow of ground water. The velocity of flow ν is the rate of discharge of water
per unit of total cross-sectional area A of soil. This total area of cross-sectional is composed of the
area of solids As and the area of voids Aν . Since the flow takes through the voids, the actual or true
velocity of flow will be more than the discharge velocity. This actual velocity is called the seepage
velocity ν, and is defined as the rate of percolating water per unit cross-sectional area of voids
perpendicular to the direction of flow.
From the definitions of the discharge velocity and seepage velocity, we have
but
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Prof. Rashmi G. Bade, Department of Civil Engineering, Geotechnical Engineering – I 6
the seepage velocity νs is also proportional to the hydraulic gradient:
κν =coefficient of percolation
where from Darcy’s law
3.4 FACTORS AFFECTING PERMEABILITY OF SOILS
The following factors affect the permeability of soils:-
1. Grain size.
2. Properties of the pore fluid.
3. Void ratio of the soil.
4. Structural arrangement of the soil particles.
5. Entrapped air and foreign – matter.
6. Adsorbed water in clayed soils.
1. Effect of size and shape of particles:-
Permeability varies approximately as the square of the grain size. Since soils consist of many
different – sized grains, some specific grain size has to be used for comparison. Allen – Hazen
(1892), based on his experiments on filter sands of particle size between 0.1 and 3 mm, found
that the permeability can be expressed as
K = CD102
Where,
K = coefficient of permeability (cm/sec)
D10 = effective diameter (cm)
C = constant, approximate equal to 100 when D10 is expressed in centimeter.
Attempts have been made to correlate the permeability with the specific surface of the soil particles.
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2. Effect of properties of pore fluid:-
Permeability is directly proportional to the unit weight of water and inversely proportional to its
viscosity. Though the unit weight of water does not change much with the change in temperature,
there is great variation in viscosity with temperature. Hence, when other factors remain constant,
the effect of the property of water on the values of permeability can be expressed as
It is equal to convert the permeability results to a standard temperature (27
0 C) for
comparison purposes, by expression:
Where,
K27 = permeability at 270C
27 = viscosity at 270C
k = permeability at test temperature
viscosity at test temperature.
3. Effect of the voids ratio:-
The effect of voids ratio on the values of permeability can be expressed as
Laboratory experiments have shown that the factor C changes very little with the changes in the
voids ratio of un – stratified sand samples. However, for clays, it varies appreciably. Thus, for
coarse grained soils, the equation reduces to
Based on another concept of mean hydraulic radius for the soils, the following relationship is
obtained:
Fig.2. Variation of k with e
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It has been found that a semi – logarithmic plot of voids ratio versus permeability is approximately a
straight line for both coarse grained as well as fine grained soils.
4. Effect of Structural arrangement of particles and stratification:-
The structural arrangement of the particles may vary, at the same voids ratio, depending upon the
method of deposition or compacting the soil mass. The structure may be entirely different for a
disturbed sample as compared to an undisturbed sample which may possess stratification. The
effect of structural disturbance on permeability is much pronounced in fine – grained soils. When
flow through natural soil deposits is under consideration, permeability should be determined on
undisturbed soil as its natural structural arrangement.
5. Effect of degree of saturation and other foreign matter:-
The permeability is greatly reduced if air is entrapped in the voids thus reducing its degrees of
saturation. The dissolved air in the pore fluid (water) may get liberated, thus changing the
permeability. Organic foreign matter also has the tendency to move towards critical flow
channels and choke them up, thus decreasing the permeability.
6. Effect of adsorbed water:-
The adsorbed water surrounding the fine – soil particles is not free to move, and reduces the
effective pore space available for the passage of water. According to a crude approximation after
Casagrande, 0.1 may be taken as the voids ratio occupied by adsorbed water, and the
permeability may be roughly assumed to be proportional to the square of he net voids ratio of (e
– 0.1).
3.5 DETERMINATION OF COEFFICIENTS OF PERMEABILITY BY LABORATORY
AND FIELD METHODS
The coefficient of permeability of a soil can be determined using the following methods:
a) Laboratory Methods:-the coefficient of permeability of a soil sample can be determined by the
following methods:
i) Constant head permeability test.
ii) Variable head permeability test. OR Falling head permeability test.
b) Field Methods:-The coefficient of permeability of a soil deposit in-situ conditions can be
determined by the following field methods.
i) Pumping- out tests
ii) Pumping –in tests
The pumping –out test influence a large area the pumping well and give an overall value of the
coefficient of permeability of the soil deposit. The pumping –in test influence a small area around the
hole and therefore gives a value of the coefficient of permeability of the soil surrounding the hole.
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c) Indirect Methods:-The coefficient of permeability of the soil can also be determined indirectly
from the soil parameters by
i) Computation from the particle size or its specific surface.
ii) Computation from the consolidation test data.
The first methods is used if the particle size is known. The second is used when the coefficient of
volume change has been determined from the consolidation test on the soil.
d) Capillarity –Permeability test:- The coefficient of permeability of an unsaturated soil can be
determined by the capillarity –permeability test .
I] CONSTANT HEAD PERMEABILITY TEST
The coefficient of permeability of a relatively more permeable soil can be determined in a
laboratory by the constant-head permeability test. The test is conducted in an instrument known as
constant-head permeameter. It consist of a metallic mould,100mm internal diameter, 127.3mm
effective height and 1000ml capacity according to IS:2720(PartXVII). The mould is fitted with a
drainage base plate with a recess for porous stone. The mould is fitted with a drainage cap having an
inlet vale and an air release valve. The drainage base and cap have fittings for clamping to the mould.
Fig. shows a schematic sketch .The soil sample is placed inside the mould between two porous discs.
The porous discs should be at least ten times more permeable than the soil. The porous discs should
be de aired before. The water tubes should also be deaired .The sample can also be prepared in the
permeameter by pouring the soil into it and tamping it to obtain the required density. The base is
provided with a dummy plate, 12mm thick and 108 mm in diameter, which is used when the sample
is compacted in the mould. It is essential that the sample is fully saturated. This is done by one of the
following three methods.
i) By pouring the soil in the permeameter filled with water and thus depositing the soil under water.
ii) By allowing water to flow upward from the base to the top after the soil has been placed in the
mould. This is done by attaching the constant- head reservoir to the drainage base. The upward
flow is maintained for sufficient time till all the air has been expelled out.
iii) By applying a vacuum pressure of about 700mm of mercury through the drainage cap for about
15 minutes after closing the drainage valve. Then the sol is saturated by allowing deaired water
to enter from the drainage base. The air-release valve is kept open during saturation process.
After the soil sample has been saturated , the constant-head reservoir is connected to the drainage
cap. Water is allowed to flow out from the drainage base for some time till a steady-state is
established. The water level in the constant-head chamber in which the mould is placed is kept
constant. The chamber is filled to the brim at the start of the experiment. The water which enter
the chamber after flowing through the sample spills over the chamber and is collected in a
graduated jar for a convenient period. The head causing flow (h) is equal to the difference in
water levels between the constant- head reservoir and the constant-head chamber.
If the cross-sectional area of the specimen is A, the discharge is given by
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Where
L= length of specimen,
h= head causing flow.
The discharge q is equal to the volume of water collected divided by time.
The finer particles of the soil specimen have a tendency to migrate towards the ends faces when
water flows through it. This results in the formation of a filter skin at the ends. The coefficient of
permeability of these end portion is quite different from that of the middle portion. For more accurate
results it would be preferable to measure the loss of head h' over a length L' in the middle to
determine the hydraulic gradient (i) ..
Thus i= h'/L'
The temperature of the permeating water should be preferably somewhat higher than that of the
soil sample. This will prevent release of the air during the test. It also helps in removing the
entrapped air in the pores of the soil. As the water cools, it has a tendency to absorb air.
To reduce the chances of formation of large voids at points where the particles of the soil touch the
permeameter walls, the diameter of the permeameter is kept at least 15 to 20 times the particles size.
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To increase the rate of flow for the soils of low permeability, a gas under pressure is applied to the
surface of water in the constant-head reservoir. The total head causing flow in that case increase to
(h+p/γw), where p is pressure applied.
The bulk density of the soil in the mould may be determined from the mass of the soil in the
mould and its volume. The bulk density should be equal to that in the field. The undisturbed sample
can also be used instead of the compacted sample. For accurate result, the specimen should have the
same structure as in natural condition.
The constant head permeability test is suitable for clean sand and gravel with κ > 10-2
mm/sec.
II] FALLING HEAD PERMEABILITY TEST
For relatively less permeable soils, the quantity of water collected in the graduated jar of the
constant-head permeability test is very small and cannot be measured accurately. For such soils, the
variable –head permeability test is used. The permeameter mould is the same as that used in the
constant-head permeability test. A vertical , graduated stand pipe of known diameter id fitted to the
top of permeameter . The sample is placed between two porous discs. The whole assembly is placed
in a constant head chamber filled with water to the brim at the start of the test. Fig shows a schematic
sketch. The porous discs and water tubes should be de-aired before the sample is available, the same
can be used; otherwise the soil is taken in the mould and compacted to the required density.
The valve at the drainage base is closed and a vacuum pressure is applied slowly through the
drainage cap to remove air from the soil. The vacuum pressure is increased to 700mm of mercury
and maintained for about 15 minutes. The sample is saturated by allowing deaired water to flow
upward from the drainage base when under vacuum. When the soil is saturated, both the top and
bottom outlets are closed. The standpipe is filled with water to the required height.
The test is started by allowing the water in the stand pipe to flow through the sample to the
constant-head chamber from which it overflows and spills out. As the water flows through the soil,
the water level in the standpipe falls. The time required for the water level to fall from a known
initial head (h1) to a known final head (h2) is determined. The head is measured with reference to the
level of water in the constant-head chamber.
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Lets us consider the instant when the head is h. For the infinitesimal small time dt, the head falls
by dh. Let the discharge through the sample be q. From continuity of flow,
Where a is cross-sectional area of the standpipe
Or
Or
Or
integrating
or
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where t = (t2-t1), the time interval during which the head reduces from h1 to h2.
The rate of fall of water level in the stand pipe and the rate of flow can be adjusted by changing the
area of the cross-sectional of the standpipe. The smaller diameter pipes are required for less pervious
soils.
The coefficient of permeability is reported at 27°C as per IS :2720(part XVII). The void ratio of
the soil is also generally determined.
The variable head permeameter is suitable for very fine sand and silt with κ = 10-2
to 10-5
mm/sec.
III] FIELD METHOD OF PERMEABILITY TEST
Pumping –Out test
The laboratory methods for the determination of the coefficient of permeability, as discussed before ,
do not give correct result. The samples used are generally disturbed and do not represents the true in-
situ structure. For more accurate, representative values, the field tests are conducted. The field test
may be in the form of pumping out test where n the water is pumped into the drilled holes,.
For large engineering project, it is usual practice to measure the permeability of soils by pumping-
out tests. The methods are extremely useful for a homogeneous, coarse grained deposits for which it
is difficult to obtain undisturbed samples. In a pumping out test, the soil deposit over a large area is
influenced and therefore the result represent an overall coefficient of permeability of a large mass of
soil. However, the test are very costly and can be justified only for large projects.
Ground water occurs in pervious soil deposits known as aquifers. The aquifers are reservoirs of
ground water that can be easily drained or pumped out. An aquiclude is a soil deposit which is
impervious. If an aquifer does not have an aquiclude at its top and the water table is in the acquifer
itself. It is called an unconfined aquifer. If the acquifer is confined between two aquicludes, one at its
top and the other at its bottom, it is known as confined aquifer. The coefficient of permeability of the
soil can be found using the equations developed below separately for unconfined aquifer and
confined aquifer.
(a) Unconfined Aquifer:- In an unconfined aquifer, a tube well is drilled as shown in fig. The well
reachs the underlying impervious stratum. The tube used for the well is perforated so that water
can enter the well. The tube is surrounded by a screen called strainer to check the flow of soil
particles into the well. Water is pumped out of the tube well till a steady state is reached. At that
stage, the discharge becomes
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Constant and the water level in the well does not change. The water table, which was originally
horizontal before the pumping was started, is depressed near the well. The water table near the well
forms an inverted cone, known as the cone of depression. The maximum depression of the water
table is known as the drawdown(d).
The expression for the coefficient of permeability can be derived making the following assumption,
known as Dupuit’s assumption.
(1) The flow is laminar and Darcy’s law is valid.
(2) The soil mass is isotropic and homogenous.
(3) The well penetrates the entire thickness of aquifer.
(4) The flow is steady.
(5) The coefficient of permeability remains constant throughout.
(6) The flow towards the well is radial and horizontal.
(7) Natural ground water regime remains constant.
(8) The slope of the hydraulic gradient line is small, ans can be taken as the tangent of the angle
in place of the sine of angle i.e
i=dz/dr (1)
Lets us consider the flow through a cylindrical surface of height z at a radial distance of r from the
centre of the well. From Darcy’s law,
Substituting the value of i from eq.(1) and taking A equal to 2πr z,
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Integrating
Or
or
Near the test well, there is rapid drop in head and the slope of the hydraulic gradient is steep, and
assumption (8) is not satisfied .the observation wells 1 and 2 should be drilled at considerable
distance from the well for accurate measurement
b) Confined aquifer:- fig shows the confined aquifer of thickness band lying between the two
aquicludes.the piezometric surface is above the top of the aquifer,the water pressure is indicated
piezometric surface(P.S).
thus the piezometric surface is the water table equal equivalent for a confined aquifer.
Intially,the piezometer surface is horizontal.when the pumping is started from the well,it is
depressed and a cone of depression is formed.The expression for the co-efficient of permeability can
be derived by making the assumption as in the case of unconfined aquifer.let us consider the
discharge through a cylindrical surface at a ridal distance r from the centre and the height z.from
Darcy’s law
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Integrating
Where z1= height of water level in observation well(1) at a radial distance of r1 and
z2= height of water level in observation well() at a radial distance of r2 .
As in the case of an unconfined aquifer , an approximate value of k can be determined if the radius of
influence R is known or estimated . in this case
3.6 PERMEABILITY OF STRATIFIED SOIL DEPOSITS
In nature, soil mass may consist of several layers deposited one above the other. Their bedding
planes may be horizontal inclined or vertical. Each layer, assumed to be homogeneous and isotropic,
has its own value of coefficient of permeability. The average permeability of the whole deposit will
depend upon the direction of flow with relation to the direction of the bedding planes. We shall
consider both the cases of flow:-
i) Parallel to the bedding planes
ii) Perpendicular to the bedding planes.
(A) Flow parallel to the planes of stratification:-
Lets us consider a deposit consisting of two horizontal layer od soil of thickness H1 and H2 as shown
in fig
For flow parallel to plane of stratification,the loss of head(h) over the length L is same for both the
layer.Therefore the hydraulic gradient(i) for each layer is equal to the hydraulic gradient of the entire
deposit. The system is analogus to the two resistance in parallel in an electrical circuit,wherein the
potential drop is the same in both the resistance.
From the continuity equation, the total discharge (q) per unit width is equal to the sum of discharges
in theindividual layer, i.e.
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Let (kh)1 and (kh)2 be the permeability of the layer 1 and 2 respectively,parallel to the plane of
stratification and (kh) be the overall permeability of that direction.from eq 1 using darcy’s law
If there are n number of layer instead of two
(B) Flow normal to the plane of stratification:-
Lets us consider a deposit consisting of two layer of thickness H1 and H2 in which the flow
occurs normal to the plane of stratification as shown
Let (kv)1 and (kv)2 be the coefficient of permeability of the layers 1 and 2 in the direction
perpendicular to the plane of stratification and kv be the average coefficient of permeability of
the entire deposit in that direction. In this case, the discharge per unit width is the same for each
layer and is equal to the discharge in the entire deposit. The case is analogous to the resistances
in series in an electrical circuit , wherein the current is the same for all resistance.
Therefore
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Using Darcy’s law, considering unit area perpendicular to flow,
Where iv = overall hydraulic gradient ,(iv)1 = hydraulic gradient in layer 1,
(iv)2 = hydraulic gradient in layer 2.
As the total loss of head (h) over the entire deposit is equal to the sum of the loss of heads in the
individual layers,
Writing in terms of hydraulic gradient (i) and the distance of flow, remembering
In general when there are n such layer
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Evan (1962) proved that for isotropic (kv = kh) and homogeneous layers, the average permeability of
the entire deposit parallel to the plane of stratification is always greater than that normal to this plane.
For illustration, let us consider a deposit consisting of two layers of thickness 1 m and 2 m, having
the coefficient of permeability of 1 x 10-2
cm/sec and 1 x 10-4
cm/sec respectively.
Thus
It may be noted that the average permeability parallel to the plane of stratification depends mainly on
the permeability of the most permeable layer and its value is close to the permeability of that layer.
on the other hand, the average permeability normal to the plane of stratification is close to that for
the most impermeable layer. In other words , the average flow parallel to the plane of stratification is
governed by the most permeable layer and that perpendicular to the plane of stratification by the least
permeable layer.
4.1 SEEPAGE PRESSURE
By virtue of the viscous friction exerted on water flowing through soil pores, an energy
transfer is affected between the water and the soil. The force corresponding to this energy transfer is
called the seepage force or seepage pressure. Thus, seepage pressure exerted by water on the soil
through which it percolates. It is this seepage pressure that is responsible for the phenomenon known
as quick sand and is of vital importance in the stability analysis of earth structures subjected to the
action of seepage.
4.2 QUICK SAND CONDITION
When flow takes place in an upward direction, the seepage pressure also acts in the upward
direction and the effective pressure is reduced. If the seepage pressure becomes equal to the pressure
due to submerged weight of the soil, the effective pressure is reduced to zero. In such a case, a
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cohesionless soil loses all its shear strength, and the soil particles have a tendency to move up in the
direction of flow. This phenomenon of lifting of soil particles is called quick condition, boiling
condition or quick sand. Thus, during the quick condition,
σ’ = z γ’ – ps = 0
or ps = z γ’
or izγw = zγ’
The hydraulic gradient at such a critical state is called the critical hydraulic gradient. For
loose deposits of sand or silt, if voids ratio ‘e’ is taken as 0.67 and G as 2.67, the critical hydraulic
gradient works out to be unity. It should be noted that quick sand is not a type of sand but a flow
condition occurring within a cohesionless soil when its effective pressure is reduced to zero due to
upward flow of water.
Fig.1 shows the setup to demonstrate the phenomenon of quick sand. Water flows in an
upward direction through a saturated soil sample of thickness ‘z’ under a hydraulic head ‘h’. This
head can be increased or decreased by moving the supply tank in the upward or downward direction.
When soil particles are in the state of critical equilibrium, the total upward force at the bottom of the
soil becomes equal to the total weight of all the materials above the surface considered.
Equating the upward and downward forces at the level a-a, we have,
(h+z)γw.A = z γsat A
h γw = z (γsat - γw ) – zγ’
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4.3 TWO DIMENSIONAL FLOW: LAPLACE EQUATION
The quantity of water flowing through a saturated soil mass , as well as the distribution of
water pressure can be estimated by the theory of flow of fluids through porous medium. While
computing these quantities with the help of theoretical analysis that follow, the following
assumptions are made:
1) The saturated porous medium is incompressible. The size of the pore spaces does not change
with time, regardless of water pressure.
2) The seeping water flows under a hydraulic gradient which is due only to gravity head loss, or
Darcy’s law for flow through porous medium is valid.
3) There is no change in the degree of saturation in the zone of soil through which water seeps
and the quantity of water flowing into any element of volume is equal to the quantity which
flows out in the same length of time.
4) The hydraulic boundary conditions at entry and exit are known.
5) Water is incompressible.
Let us consider an element of soil of size dx by dz through which flow is taking place. The third
dimension along y-axix is large. For convenience, it is taken as unity. Let the velocity at the inlet and
outlet faces be vx and in x-direction and vz and in z-direction.
Fig.1 Two dimensional flow.
As the flow is steady and the soil is incompressible, the discharge entering the element is
equal to that leaving the element.
Thus,
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This is a continuity equation for two-dimensional flow.
Let ‘h’ be the total head at any point. The horizontal and vertical components of the hydraulic
gradient are, respectively,
The minus indicates that the head decreases in the direction of flow.
From Darcy’s law,
Substituting in equation (i),
or
As the soil is isotropic, kx = kz. Therefore,
Equation (ii) is the Laplace equation in terms of head ‘h’.
Sometimes, the Laplace equation is represented in terms of velocity potential Ф, given by
Ф = - k h
Therefore,
and
substituting vx and vz in above equation, we get,
Equation (iii) is the Laplace equation in terms of velocity potential.
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4.4 APPLICATIONS OF FLOW NET
i) Determination of seepage:- The portion between two successive flow lines is known as at
flow channel. The portion between two successive equipotential lines and successive flow lines
is known as field such as that shown in fig.2.
Fig.2
Let ‘b’ and ‘l’ be the width and length of the field,
Δh = head drop through the field
Δq = discharge passing through the flow channel.
H = total hydraulic head causing flow.
= difference between upstream and downstream heads.
Then, from Darcy’s law of flow through soils:
If Nd = total number of potential drops in the complete flow net then,
Hence,
The total discharge through the complete flow net is given by
Where Nf = total number of flow channels in the net.
The field is square; hence b = i
Thus,
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This is the required expression for the discharge passing through a flow nwt and is valid for isotropic
soils in which kx = ky = k.
ii) Determination of hydrostatic pressure:- The hydrostatic pressure at any point within the soil mass
is given by
u = hw γw
where u = hydrostatic pressure
hw = piezometric head
The hydrostatic pressure in terms of piezometric head hw is calculated from the following
relation
hw = h – Z
where h= hydraulic potential at the point under consideration
Z = position head of the point above datum, considered positive upwards.
iii) Determination of Seepage pressure:- The hydraulic potential ‘h’ at any point located ‘n’
potential drops, each of value Δh is given by
h = H – nΔh
The seepage pressure at any point equals the hydraulic potential or the nbalance hydraulic
head multiplied by the unit weight of water and hence, is given by
ps = h γw – (H - nΔh) γw. This pressure acts in the direction of flow.
iv) Determination of exit gradient:- The exit gradient is the hydraulic gradient at the downstream
end of the flow line where the percolating water leaves the soil mass and emerges into the free
water at the downstream. The exit gradient can be calculated from the following expression, in
which Δh represents the potential drop and ‘l’ the average length of last field in the flow net at
exit end:
4.5 SEEPAGE THROUGH ANISOTROPIC SOIL
The coefficient of permeability of stratified soil deposits parallel to the plane of stratification is
generally greater than that normal to this plane. Such soils are anisotropic in permeability. Let us
take the axes x –x and z – z parallel and perpendicular to the plane of stratification, respectively.
Therefore kx > kz. Form Darcy’s law,
Substituting the values of vx and vz in the continuity equation, we get,
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As equation (1) is not a Laplace equation, the principles of flow net construction, are not
applicable to anisotropic soils.
Equation (10 can however be converted to Laplace’s equation by transformation. Let the ‘x’
coordinate be transformed to the new coordinate xt by the transformation.
……….(3)
Transformation of coordinates.
Equation (1) can be written as
Equation (2) is the Laplace equation in xt and z. Therefore, the principles of flow net construction
can be used for anisotropic soils after transformation.
The cross-section of the soil mass whose flow net is required is redrawn keeping the z – scale
unchanged but reducing the x – scale by the ratio . The flow net is constructed for the
transformed section by usual methods. (fig. b). The flow net for the actual section is obtained by
transferring back the flow net to the natural section by increasing the x – scale in the ratio .
Obviously, the flow net for the natural section does not have the flow lines and the equipotential
lines orthogonal to each other. (fig. a).
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4.6 FLOW NET FOR ANISOTROPIC SOILS.
The discharge through an anisotropic soil mass can be obtained from an equation,
where k’ is the modified coefficient of permeability as determined below.
Discharge through a flow channel on the transformed scale per unit width is given by
Since the discharge is the same in both the channels,
or
Using Eq. (3),
or
Hence the discharge is given by
4.7 EFFECTIVE, NEUTRAL, TOTAL PRESSURE
At any plane in a soil mass, the total stress or unit pressure ‘σ’ is the total load per unit area. This
may be due to;
a) Self weight of soil
b) Over burden on the soil.
The total pressure consists of two distinct components: Intergranular pressure or effective
pressure and the neutral pressure or pore pressure.
Effective pressure σ’ is the pressure transmitted form particle to particle of the soil mass.
Such a pressure is also termed as intergrannular pressure.
Effective pressure is effective in decreasing the void ratio of soil mass and in mobilizing its
shear strength.
The neutral pressure or pore pressure transmitted through the pore fluid.
This pressure is equal to water load per unit area, above the plane. It does not have any
measurable influence on the void ratio or shearing strength. Therefore this pressure is called as
Neutral pressure ‘u’.
Since the total vertical pressure at any plane is equal to the sum of the effective pressure and
pore pressure.
σ = σ’ + u