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





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.





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.





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.





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





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.





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





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.





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





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.





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.





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





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





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,





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





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.





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





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





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





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’





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,





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.





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,





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,





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).





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

ANJUMAN COLLEGE OF ENGINEERING & …...UNIT – III 3. Permeability: Darcy’s law & its validity,...

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