Welcome to lecture number 14 of advanced geotechnical

engineering course in the previous lecture we have introduced ourselves two methods for

measuring the permeability we said that there are two types of methods in the laboratory

one is a constant head test and falling head test and we also discussed about the achieve

difference to achieve differences between these two test methods in this lecture which

is permeability and seepage part three we will try to discuss about the factors affecting

permeability and then we will introduce our ourselves to different types of the flows

and the mathematics which is connected with these page phenomenon. So this part of the lecture is permeability

and see page part three so as shown in this slide the permeability and drainage characteristics

of soils are shown the coefficient of permeability K which is actually mentioned in meter per

second, if you look into this the one which is actually then there in the yellow color

wearing it actually has in a good drainage characteristics when it comes to this pink

color this particular range has the poor drainage characteristics and the color which is in

orange here that is beyond 10 – 8m/s which is actually has the practically impervious

drainage characteristics. So this particular these term abilities are

possible for soils which are a clean gravel or clean science and clean sand with gravel

mixtures the poor drainage characteristics or the soils which are actually having permeability

in the range of 10 – 6 to 10 – 8 m/s this is possible for very fine science organic

and in organic cells mixtures of sand silt and clay till stratified clay deposits so

for this type of soils it is possible that the permeability can be in the range of 10

– 6 to 10 – 8 m/s for certain type of soils like homogeneous clay below the join

of weathering. These soils can actually possess the permeability

in the range of 10 – 8 to 10 –m/s we have different ranges of permeability ‘s and there

is a unique property for the soil and the soils which can have the granular soils mostly

have good permeability or very high permeability fine-grained soils and how low permeability. The factors affecting the permeability if

you wanted to look into 8 the coefficient of permeability is a measure of the ease with

which water flows to the permeability materials, so this is a you know put forward by Kozeny

Carman and this in he has proposed an equation they have proposed an equation which is v

=1 / CS and S suffix T2 and multiplied by γw / μ into eq / 1 + e into I so this is

nothing but v=Ki the component K that is coefficient of permeability is indicated here

by 1/ CS Ss T2 γw / μ into eq / 1 + e. So this is according to Kozeny Carman this

is basically valid for coarse grain soils and the taylor1 948 he has also proposed the

equation reflecting the influence of the permeate and soil characteristics on the k and this

is using the this is deduced by using the poisonous law which is given like this v=c

into d2 the d is nothing but the particle size γw/ μ γw is nothing but the unit weight

of the permeate μ is nothing but the dynamic viscosity of the permeate eq / 1 + e into

I so both this equation assumed that interconnect voids are visualized as a number of capillary

tubes through which the water can flow. So we have two sets of equations one is proposed

by Kozeny Carman basically is valid for coarse-grained soils the other one is Taylor 1948 which is

deduced based on the poisonous law which is given mice v=CD2 γw/ μ into 1 + e into

I. Now if you look into the Kozeny Carman equation

the V which is nothing but defined as discharge velocity and the Cs is defined as a shape

factor for granular soils typically CS=2.5 SS is nothing but the surface area per unit

volume of solids, so the surface area of the unit volume of the soil suppose if you see

here it is in the denominator and the factor T which is nothing but the touristy factor

which is defined as ratio of the tortillas length that is nothing but a path taken by

the water flowing through the soil along the voids.

That means that this particular length which is indicated here the wavy pattern is nothing

but the tortillas path the length L is nothing but the length of the sample through which

the flow is occurring, so the tortillas T is nothing but ratio of the tortillas length

that is L1 to L so for granular soils the tortillas t factor is 1.414, so we are here

one parameter which is defined which is called as the absolute permeability or intrinsic

permeability which is going to be constant for typical soil skeleton.

So same value will be there for a particular soil and particular state so the permeability

coefficient of permeability is now connected with capital K and the γw/ μ so capital

K is equal to capital K is nothing but now 1 / CS and SST2 x q / 1 + e, so the units

for absolute permeability are intrinsic permeability are generally given in Darcy’s or a centimeter

square or meter square, so the units of absolute permeability which is also called as intrinsic

permeability and which is formed to be you know function of the soil skeleton and it

possesses the same value for a particular soil.

And one Darcy is equal to 0.98 7 into 10 – 8, so here we have said that coefficient of permeability

is a function of the number of parameters like 1 least specific surface area and Tata

Steel factor and void ratio and the shape factor for basically for granular soils, so

the factors affecting the permeability can be summarized like this. So based on the previous discussions by my

equations proposed by Kozeny Carman or Taylor 1948 shape and size of soil particles that

is shape of the soil particle whether it is angular or whether it is having if plate shape

the particle and the size of the soil particles that means that larger the soil particle are

smaller the soil particle and void ratio, so k increases with increase in the void ratio

and degree of saturation also like k increases with increase in the degree of saturation

for a partially saturated soils the permeability will be is a be less because of the partial

saturation. The composition of soil particles it also

depends upon the mineralogy type of the mineral present in the soils so soil structure and

viscosity of the Permeant at density and Centration of the perimeter, so list of the factors affecting

the permeability are the shape and size of the soil particles void ratio degree of saturation

composition of soil particles soil structure and viscosity of the permeant and density

and concentration of the permeant and also like the compactive effort you know with the

with the different compact your and different molding water contents there will be change

in the permeability. So here then the factors affecting the permeability

what we discussed is that effect of void ratio and basically here on the left hand side the

permeability which is which is permeability fact that is permeability factors like different

eq / 1 + e e2 / 1 + e and 2 are given and in the y axis and the per mobility in mm per

second which is actually given on the x axis, so for branded soils k is proportional to

approximately e3 / 1+ x degree of saturation cube, so this is approximately valid for this

relationship is approximate for s that is degree of saturation less than 100%.

On the right hand side there is a plot which is actually shown for void ratio on the y-axis

and log logarithmic of the clay logarithmic of k on the x axis, so it can be seen here

this is for a saturated natural clay soil saturated natural clay soils, so this factor

CK which is nothing but the permeability change factor and K0 is the permeability in the at

void ratio E0 K0 is the permeability at initial void ratio E0, so once the pressure is applied

or when the load is applied the primitive void ratio decreases in the process what will

happen the permeability changes. So the permeability change factor is defined

as de / d log K that is the difference between K2 and K1 permeability at different void ratios,

so this is approximately 1 / 321 / 2 e0 for a which is approximated as CK as 1 / 2 to

1 /3times the initial void ratio so k by knowing this permeability change factor we can determine

permeability and in different stages and this is K is equal to K0 into 10 place to – e0

/ K where e is the void ratio at a particular time and e0 is the initial void ratio and

the CK is the permeability change factor which is approximated as 1 / 3 to 1 / 2 times the

e0 and K0 is the initial permeability at initial void ratio. So in this plot the relationship between the

permeability change index or permeability change factor CK and e0 for all places tested

or shown and this is after Terrance at all 1983, so here it can be seen that the permeability

change index CK Is approximated as 1/ 2 to 1 / 2 to 1/ 3 e0 naught and there are upper

bound and lower bound values which are actually shown this is based on the clays for all the

types of place tested and reported by Devon’s at all 1983. Now the next factor is that effect of grain

size the permeability of the grain size depends mainly on the cross sectional area of the

pore channels, so we knew that when you have what vary the pore size which is nothing but

the D is proportional to the effective particle size let us say, so in that case we can approximate

D=d 10 the pore size is approximated as 20% of the d10 that means that the smaller

the particle size the finer is the pore channel since the average diameter of the portion

in a soil at a given process T increasing proportional to the average grain says the

permeability the granular soils might be expected to increase as a square of the some characteristic

grain size so the permeability of grander soils. Might be expected to increase as a square

of the some characteristic grain size generally it is considered as d-10 but the recent studies

indicate that d5 that is soil which actually has 5% particles passing. So the traditional the empirical formula for

estimating the permeability was given by Hazen and the Hazen empirical formula for predicting

K basically valid for clean sands is actually given here and which is actually valid for

soil which is having less than 5% fines, so k=CD 102 so what has been done is that number

of sandy type of soils having less than 5% fines were taken and the constant head permeability

tests were conducted and the correlation actually has been plotted and which actually indicates

that the K in meter per second can be obtained with the constant C.

Which is housing having a value of 10 – 2 and d10 that is effective particle size in

millimeter once we have this C=10 – 2 and d10 in millimeters the permeability can

be obtained in meter per second so for a by knowing the effective particle size at the

first end you know to in order to estimate the permeability this particular relationship

can be used, so C basically is a constant in this case it is equal into 10 – 2 which

includes the effect of the shape of the pore channels in the direction of the flow and

the total volume of pores. So C is a constant which includes the shape

of the pore channels in the direction of the flow and total volume of pores so d10 is selected

because the smaller particles control the size of the pore channels, so in this case

the Hazen’s actually has considered the d-10 because the smaller particles control

the size of the pore channels. This particular correlation is presented by

Kenney at all 1984 and this is with d5 that is on the x axis which is represented on the

log scale and the hydraulic conductivity K which is actually represented on the y axis

and the or sands which are actually having relative density 80% it at 80% dual-density

more or less greater than 18% into density was considered and particle size of the sands

which are actually used the soil which is used in the power of the various specimens

it ranges from 0.04 to25.4mm and the quotient of uniformity is in the range of 1.04 to 12

and it is said that the K the absolute permeability in mm2.

Is given as 0.05 – 1 into D52 well D5 is in mm and this particular relationship was proposed

by Kenney 1984 and the effective grain size D5 would be better choice compared to D10

according to the you know data correlated and presented by Kenney at all in 1984 the

effective grain size was D5 was reported as a better choice compared to detail and the

factors affecting the permeability further ones to discuss the effect of the degree of

saturation, so we have said that with increasing degree of saturation the permeability increases

so K is actually proportional to degree of saturation. At low saturation there will be reduction

the flow channels available for the flow because the part of the voids is actually occupied

by the air, so with increase in saturation degree of saturation the coefficient of permeability

of the soil increases, so here a measured data which is presented by after Das 1987

where degree of saturation is plotted on the x-axis and permeability is plotted on the

y-axis for a typical sign where it can show that with increase in the degree of saturation

there is an increase in the permeability Further there is a important aspect which

is required is that the soil fabric or soil structure or the arrangement of the soil particles

within the given soil mass the permanent permeability of the soil deposit is significantly affected

by it is in place soil structure his loose granular soil would have higher void ratio

than a dense soil and therefore would permit greater flow, so loose granular soil would

higher water I show then a denser soil so and then there would be a so it would permit

greater flow similarly when you have what a fine-grained soil with the flocculated structure

will have a higher permeability than the dispersed structure.

So if you look into the two types of extreme to soil one is closed and soil where can have

a looser granular structure or same coarse grained soils can have a dense granular structure

a loose granular structure can have higher permeability than the density internal structure

similarly a fine grained soil with a flocculent structure or flocculent arrangement will have

higher permeability with than the dispersive structure. So here a fine grained soil with the flocculent

structure will have higher permeability than the soil with a dispersed structure that is

what we said even it does similar void ratios a clay with undisturbed flocculated structure

will possess large wide openings then the same clay having a dispersed structure, so

the path which is actually this is with the dispersed structure where if you can see and

the permeability in this direction is found to be less and when you have a flow which

is actually taking place in this direction because of certain available higher hydraulic

gradient. The permeability will be on the on higher

side in this direction along the you know the flow which is actually taking place along

the platelet particles. In the flow through clusters of the particles

in clay soils so flow mainly controlled by the voids between the flocks and the flock

sizes is a function of the particle size shape and environment in which these flocks have

been formed and therefore marine a lytic place the permeability which is actually Kh and

kV will be equivalent to 1 1 to1.5 and the quotient of permeability of a soil with flocculent

structure will be isotropic in nature in the sense that the flow the number of flow channels

available to flow in any direction will be equal identical for a flocculent structure.

Whereas in case of a dispersed structure the flow along the shape of the dispersal this

the panel laid particles will be higher compared to the their perpendicular direction because

of the increase in taught city for the flow, so for the Marine electric plays the Kh that

is coefficient of permeability in the horizontal direction and the question of permeability

in the vertical direction the ratio can be equal to 1 to1.5 depending upon the type of

the normal in which they got deposited. Similarly when you have got the compactive

effort with an increase in compactive effort the permeability decreases, so for example

here on they-axis there is a γd which is plotted here which is also shown here and

the water content on the x-axis, so as we have seen for a typical clay initially this

is this particular portion is the optimum moisture content and this side is the wet

side of optimum and this side is the dry side of optimum, so at this point the density is

actually maximum so lower void ratio will be there and here the density is less than

the higher word ratio is possible. And so as the water content is increased you

can notice that the soil fabric changes from more or less from the flocculent structure

to a dispersed structure, so the particles undergo in the process of the compaction the

particles undergo rotation by about 900 in the sense that what will happen is that the

particles finally whence they reach of the wet side of optimum they start getting arranged

parallel to each other, so in the process you know what we can say is that than the

wet side of optimum the predominant soil structure in case of when you compact the soils is the

disparity in nature. There are same soil with the higher low lower

water content but same density can actually have a flocculent structure at the maximum

brightly dry unit weight and water up to water contained the soil structure is neither flocculent

nor dispersed it actually has got the blend of both flocculent and dispersed structures,

so here with the increase in the you know compactor for this is the lower compactor

foot and this is the standard proctor compactor foot say and this is the modified proctor

compact our effect and with increase in compact effect there will be a decrease in the permeability.

Because with increase in compactive effort there is an increase in the density and then

decrease in the void ratio that means that the permeability decreases. So here the variation of the k with water

content and γd is actually discussed here the importance of the fabric is brought out

here so in this particular slide which is actually shown here this is the compaction

curve and this is the 100% saturation line at 0 air voids line and this is the line of

Optimum’s that is with increase in compact effort the compaction curves the maximum Peaks

will be the occurring here and on the plot below what you see is the logarithmic K versus

water content. So what we notice that initially the permeability

will be high and once it reaches to the optimum water content the permeability takes a dip

and that is decreases and further there is an increase in the permeability, but towards

the wet side of optimum you can see here up to certain extent here there is a possibility

that the permeability is actually decreasing towards the wet side of the optimum, so the

dry side of optimum the dense aggregates with larger voids will be there because of this

also we discussed the different fabric or fluctuated structure will be there.

Because of that the high permeability is resulted in when they when we consider the Wet side

of optimum the more uniform distribution of particles with small voids hence the low permeability

can result and especially this is attributed to the dispersant fabric which is prevalent

on the Wet side of optimum at same γd that is the dry unit weight you can notice that

the permeability of the wet play is actually less than the permeability of the dry clay,

so that is a reason why for certain type of applications it is advised to compact the

clay. Especially for constructing clay barriers

it is advised to compact the clay towards the Wet side of optimum because the permeability

will be les because there it is not the strength of the soil which is important the soil which

is actually having you know the target permeability is important. So here which is actually given like again

with a minimum K for a given compact effort for example here the same plot which is actually

shown here, but I would like to draw your attention to the two curves which is actually

shown here this is the curve A and this curve B, so curve A well if you notice here K minimum

is actually occurring at WM is equal to at the Wopt that is the optimum water content

that is at maximum γd the minimum void ratio, so k in moves actually occurring at the WM

is equal to Wopt in case of curve B that is here which is actually shown here curve B

which is here. Where came in who is actually occurring at

wm greater than wopt so the fabric is actually more important than decreasing so this indicates

that the particle arrangement is actually more important than the decrease in the γd,

so in the field always use wm greater than or equal to wr to get low permeability especially

for clay various that is what we actually have discussed in the previous slide also,

so in this particular slide variation of K with water content come into d and real test

to data. Is actually reported by Daniel and Benson

1980 is presented here on the vertical axis what you see is the permeability is given

in K cm/s and the molding water content is actually given on the x axis and this is basically

a silty clay with the liquid limit 37% and plastic limit 23% hence the plasticity index

is about 14%, so these are the three types of compactive efforts are actually represented

here or considered one is the low compaction low Proctor compaction that means that in

this case the energy compactive energy is less compared to the standard.

Proctor medium is nothing but the standard Proctor and the high effort is nothing but

the modified Proctor, so you can see that the effect of the compactive effort on the

optimum where we can see that different up with the increase in the compact effort there

is a decrease in the optimum water content and the second issue is that the typical distinct

variation of the permeability with molding water content, so within increasing molding

water content there is a decrease in the permeability and we see that beyond optimum for all the

different all the types of compact efforts irrespective the compact effects you can see

that it increased which actually happens beyond the outdoor, so this is for a higher effort

we can see that the permeability decreases and here you can see that.

So beyond the optimum content beyond the optimum content but anyway when we come to the Wet

side of optimum you know as we discussed in the previous slide we have to note that the

length the type of the arrangement the particular arrangement place a key role then the density

which is achieved. Further connecting to our discussion in affecting

the permeability effect of soil type the volume of the water that can flow through a soil

mass is related both to the size of the void work mix then the number of the total number

of whites, so we if you note down the K coefficient of permeability of the coarse grained soil

is always greater than K of the fine-grained soils even though if you look into the void

ratios are frequently greater for the fine grained soils see fine grained soils can actually

have very high void ratios. So if you say that K increases with increase

in void ratio which this argument is not really true when it comes to this particular you

know factor so the K of the coarse-grained soils is greater than K of fine-grained soils

in fact the k of the coefficient of permeability of the sandy soil is about million times than

that of the you know1 million times of the clay soil.

So the K of coarse-grained soils is a function of the particle size gradation and particle

shape roughness and wide ratio of the medium if you consider the coefficient of permeability

of the coarse grained soils when you distort the factors, it is function of the particle

size gradation particle shape roughness and the void ratio of the soil medium and a K

of the fine-grained soils which is a function of the type of the clay mineral and ads or

ions and where the particle surface forces actually predominated.

So when you have this one so the if you look if you look into this and if you consider

the application of fluid mechanics to that then we will be able to understand why you

know the K of course students oils is greater than K of fine-grained soils. So in this particular slide what is actually

shown is a typical flow which is actually happens through a coarse grained soil having

you know let us assume that, if you have got a grain here and if you have got a grain here

and because of the presence of roughness the velocity with which the water is actually

flowing through the wines is actually decreases in the sense that along the boundary walls

the because of the frictional drag the velocity drops to 0 and but at the mid distance from

the that is D / 2D is the diameter of the pore at D / 2from the edge of the wall.

It can be seen that the velocity is actually maximum here, so the typical velocity distribution

if you assume by using the flow water through two parallel plates and two parallel plates

are actually considered as the edges of the you know the soil particles and the flow through

the pore channel in a sandy soil is represented like this, when we actually consider you know

clay soil we actually have the adsorbed water that is the adsorbed water which is actually

there and then there is a possibility that because of the flow which is actually taking

place this adsorbed water. Layer and then because of the frictional effect

the velocity here also drop down to 0, but at the center there will be maximum but when

you consider the magnitude of this and magnitude of this particular v-max in the sandy soil

v-max in the clay soil there is a marginal difference will be there similarly when you

have got say depressor double layer with decrease with the adsorbent layer then there can be

possible that you know more water flow can take place but however if you look into this

velocity distribution though it is a loggers, but this is actually several magnitudes less

than the flow through. The pore channel in sandy soil so the phenomena

of the higher permeability of the course grade soil can be explained using the concept of

the water flow through the conduit. So because of this the particular code student

soil will actually have higher permeability. But when it comes to fine grained soil we

actually have said that one is that adds or but you know when the water is actually flowing

through the adsorbed water layer there is a decrease in the velocity distribution whatever

it is nothing, but the type of the mineral which is actually present for example a for

a fine-grained soils when white spaces are they are very small all lines of lower physically

close to the wall of the conduit and therefore only low velocity occurs in place basically

the flow is already occurs in small channels and is further hampered because of these some

of the water whites is held or adsorbed. To the clay particles reducing the flow area

further and restricting the flow, so because of this particular explanation with the whatever

we have discussed so far we can say that the clay of the coefficient of the permeability

of the clay soil is much less than the coefficient of permeability of sandy soil. Now further one of the other factors which

we have discussed is that effect of the permeate like if you have got the permeant which is

actually given as you know K is proportional to unit weight of the permeant and the viscosity

remained variation of the γw that is the unit weight of the water tremendous temperature

is name visible, but variation of μ with the temperature is not negligible, so higher

the you know dynamic viscosity of the permeant will be the permeability so with increase

in you know viscosity of the pore fluid the permeability of the soil can be decreased.

So variation of the μ with the temperature is not negligible, but if you are able to

increase let us say that the pore fluid is actually replaced with another pore fluid

having a higher μ the coefficient of the permeability can be brought down and further

we also discussed from the Kozeny Carman equation effect of the specific surface area, so here

if indicates that higher the specific surface area lower would be the permeability that

means that higher will be the solution surface area means for example when you take your

light in light and multiple light the alight mineral the multiple which have actually very

high specific surface area. Compared to the can light mineral particles

so that means that with an increase in the specific surface area the permeability of

the soil decreases and also exhibits the this is attributed to the more adsorption. The classification of the soils according

to their coefficient of permeability if you look into it can be given as degree of, so

the soil can be classified based on the different values of the permeability when you say that

permeability value in meter per second if it is greater than 10 – 3 we say that the

soil actually has got high permeability and when the permeability is in the range of 10

– 3 to 10 – 5 m/s we can say that the soil is actually having medium permeability

and low which is in between 10 – 5 to 10 – 7 m/s and very low that is between 10 – 7

to 10 – 10 m/ a second. And the soil is said to be are classified

based on the permeability as practically impervious if the permeability is less than 10 – 9

m/s. So we as of now we discussed for the homogeneous

soils but we may not actually get the homogeneous soil deposits frequently, so the effect of

the you know coefficient of permeability of the statuette the soils are the stratified

soils, so in this particular case a layer 1 layer 2 layer 3 layer 4 the water can actually

flow through parallel to the layers or water can actually flow up to downwards that is

In the vertical direction that means that the in a given soil when you have got status

so the water can flow in our general direction as well as vertical direction or upward direction

because of some artesian conditions. So in that case how to determine the equivalent

permeability which we require to understand in general the natural soil deposits are stratified

and if the stratification is continuous the effective coefficient of permeability in the

horizontal vertical directions can be readily calculated. So in this particular discussion if you simplify

by using our for determining the this particular condition of flow occurring parallel to the

layers that means that if you have got say H1 H2 H3 2 H and n number of layers in a method

or horizontally the flow in the horizontal direction that is parallel to the layers when

it is actually happening let us assume that when we are the left hand side limb and the

difference in head between these two is say HL which is nothing but the head loss between

this point and this point. And so the input is nothing but the water

which is Q so these soils can actually have permeability is k1 k2 k3 k4 2 so on 2 Kn,

so the equivalent permeability in the horizontal direction is that K equivalent in the horizontal

direction are KH and the total thickness of the soil layer is nothing but H 1 + H2 + H3

so on – hm so here the condition is that q1=q out with that what will happen is that

the flow gets divided into you know depending upon the pyramid to the soil which is apportioned

as q=q1 + q2 + q3 so on to qn. So the condition here is that the head loss

it actually occurs over you know a length of the sample and the discharge q=q1 + q2

+ q3 so on to qn and then q which is actually comes out. So with the based on that discussion for the

flow in the horizontal direction parallel to the layers for horizontal flow the head

drop HL or the same flow path length L will be the same for each layer, so because as

the head law at last which actually occurs for a length of the sample yell though it

is actually having the different type of the soil layers and the hydraulic gradient which

actually gets dissipated in layer 1layer 2 layer 3 I1 is equal to I2 is equal to I3 is

equal to in, so the flow rate through a layered block of soil of breadth B.

B is the unit perpendicular to the plane of the figure which we considered, so with that

we can say the ki a which is nothing but KH that is the equivalent permeability in the

horizontal direction ie which is nothing, but the hydraulic gradient and which is the

thickness of the soil strata and B is the bread the perpendicular to the plane of the

figure which we considered, so K is nothing but KH is similarly for layer 1layer 2 layer

3, if you write layer 1 we can write it as K1 I1 be H1 similarly for layer 2 K2 IB H2.

So when I computing the flow in the horizontal direction as q=q1 + q2 + q3 is on to qn

I can write now by simplification KH is equal to K1 H 1+ K2 H2 + K3 H2 and so on to K and

Hn / H1 + H2 + H3 so on to Hm so this is summation which is given as KH=m=1 to n KHm into

HM divided by Hm, m is equal to 1 to N, so KH is nothing but the equivalent coefficient

of permeability in the horizontal direction so for equivalent for determining the equivalent

permeability when you have got started fertilize when the flow is actually occurring through

the parallel to the layers we can determine the permeability.

And permeability has KH=K1 H1 +K2 H2 + K3 H 3 so on to K and Hn / H1 + H2 + H3 so on

– Hm. Similarly when you consider the stratified

soils and permeability particularly the flow in the vertical direction that means that

when the flow is actually happens perpendicular to the layers, so in this case here because

of the higher head here the water actually takes platter flows upwards like this, but

we have got different layers of thicknesses like K1 having thickness of H1 layer having

H2 having permeability K2 layer having thickness H 3 and having permeability K3 so on 2 layer

having thickness H on to having permeability with km.

So but here what is actually happening is that the it is the head which is a you know

gets apportioned is your I is equal to you know I1 + I2 + I3 but what actually happens

is that the q which is actually entering in the soil strata stratified soil and coming

out to be equal that is q is equal to q1=q3=qn with this condition now we can write

down. For the flow in the vertical direction that

is perpendicular to the layers for a vertical flow the flow rate q through area a of each

layer is same, so the head drop across a series of layers that is we can say that the head

drop which is nothing but head loss in layer 1and then the total head loss is equal to

had lost in layer 1 plus head loss in there – so on – head loss in the layer n, so we

can write now with I is equal to H by that is the that is ΔH / L where L is nothing

but the thickness of the layer when you put into that I can write as HL as IH that is

in terms of H is the total thickness of the stratified layers.

And I 1 is the head hydraulic gradient occurred in the layer1 and H1is the thickness of the

layer 1I do is the hydraulic gradient occurred in the layer two and hit you is the thickness

of the layer two similarly I 383 and so on- I NHL by substituting v=KI that is nothing

but I is equal to V/ K so in the case on the left hand side we can write as v / kv into

H=V / K1 into H1 +V / K2 into H2 + V / K3 into H 3 so on to V / Kn into H n so this

particular expression when you further simplify. From the flow in vertical direction that is

perpendicular to layers we can write it as kv=H1 + H2+ H3 so on to HN / H1 /K1 + H2

/ K2 +H3 / K3 so onto this is HN / Kn so the K vertical permeability is the thing but M

is equal to 1 to n Hm /m=1 ∑ 2 N2 Hm by k vertical permeability of the particular

layers, so the equivalent permeability of equivalent coefficient of permeability in

the vertical direction so this can be used for both vertical flows flow occurring perpendicular

to the soil status in the vertical direction the KV can be given as H 1 + H2 + H3 and so

on Hn to H1 / K1 + H2 / K3 so on to Hn / Kn. So the main points about the stratified soils

we should understand is that in general for stratified soils what we have seen is that

kh is not equal to kv, so when you look when we say that the horizontal permeability is

not equivalent to vertical permeability then we say that the soil is anisotropic in nature.

In case where a soil deposits firm abilities are not the same in all direction then we

say that the properties are anisotropy if the properties are the same in all the directions

then it is called isotropic. For example, when you are actually constructing

an embankment with a material obtained from borough area and when you are achieving the

you know the identical permeability because of the compaction then we can say that the

permeability is isotropic in nature. But particularly when we are actually constructing earthen

dams with different types of soils or when we are considering the flow occurring in soil

status then they are generally in a entropic in nature, so for stratified soils we always

we say that the kh is always greater than kv.

The reason which is actually attributed to if you look into it the, if you consider from

the you know coefficient of earth pressure at rest if you look into if you recall that

one and which is nothing but k=kh /k=σh / σv when k is equal to say 0.5 σh=0.5times

σv that means that for some saturated soils kh is actually less than kv σh is less than

σv and for that kh will be greater than kv, so more voids are more spaces are available

in the horizontal plane under consideration that means that each, which the flow can takes

place in along the horizontal direction is relatively higher compared to the vertical

direction. And because of this particular you have the

number of voids which are actually available for the water to flow through in the horizontal

direction is they are higher compared to you know in the vertical direction and because

predominantly because of you know low horizontal stresses. But however in case of some more

consolidated soils where the locking of the stresses takes place this particular you know

deliberation is not valid. So for stratified soils basically normally

consolidated in nature there where σh is less than σv and the permeability is mostly

that is kh is actually greater than kv. So this is an example problem based on the

study flow parallel to the soil layers, here in this particular problem there is an impermeable

layer at the bottom most and then top surface of the impermeable layer is actually given

as the datum or considered as a datum and coarse sand which actually has got permeability

of 2×10-4m/s and it is having a thickness of 3 meters about that there is a four meter

medium sign and a six meter course and K=10-4m/s and medium sand actually has got K=0.5×10-4m/s.

Now at point A that is the height above this thing is about 10 meters and the total length

is about 100 meters and the head loss is actually occurring from say A to B so we need to determine

the equivalent permeability, the solution is actually as follows. And they also assume that there is an impermeable

layer at the at the top so the flow actually takes place parallel to the layers which are

actually shown three layers are the coarse sand layer a medium sand layer and coarse

sand layer below. The solution for this problem works out like this total head at A which

is nothing but the 13m+ 10m that means that here the depending up on the location the

thickness is that 6 + 4 this is 13 meters so the total head at A which is given as 13+10,

23 meters and the total height at B that is pressure head+elevation head which actually

works out to be 17.5 +1.5 that is at B this is a above 1.5 meters, so because of that

so this is 3 +4,7+3, 10 +3, 13 +10, so 23 is the head here and total head at B is about

19 meters so the difference of these two which is nothing but the head loss between point

A and point B which is actually shown in the figure which is here point A and point B the

head loss is actually is about 4 meters or a length of 100 meters between A and B.

So hydraulic gradient is nothing but 4/100 there is nothing but 0.04 using now in determining

the equivalent permeability in the horizontal direction when the as the flow is occurring

parallel to the layers it can be given as k1H1+k2H 2+k3H 3/H1+H2+H 3 so with that we

can say that kh=1.077×10-4m/s and once we get this one the total flow can be estimated

as which is nothing but kHi and H which is nothing but the summation of H1+H2+H3 or summation

of the flow which is actually taking place in layer 1+layer 2+layer 3 that is Q1+ Q2+Q3

so here Q=Q1+Q2+Q3. Now let us consider one more example problem

in determining the permeability wherein in this particular arrangement which is actually

shown calculate the volume of the water discharged in 20 minute the cross sectional area of the

soil is 400mm2 and the this ordinate which is actually here is 225mm and this horizontal

distance is 150mm and this distance above this where the inflow and the suppress flow

is actually discharged takes place the water flow takes place here so this hike is 375mm

and above this horizontal line this height is about 150 mm, so the water flow that is

discharged they expressed from this here. So the permeability of the soil which is actually

placed here is having 4mm/s. So the solution for this problem works out like this. We need to estimate amount of flow which actually

takes place in 20 minutes so by converting 20 minutes into seconds 20×60 1200 seconds

and area of the cross section which is perpendicular to the you know the flow direction which is

nothing but the area which is given as 4,000mm2 which is can be converted into 4,000×10-6

m2 and the pyramid to the soil is in m/s it is 4×10-3m/s now here the length of the sample

is the thing but the square of you know the vertical ordinate and horizontal ordinate

and with that we can actually get as 0.375m. So the ∆h/L the hydraulic gradient is nothing

but by considering 225 + 375- 150 we will be able to get this as ∆h/375 that is the

length of the soil sample, so ∆h/=1.2 / using Q=Ak(∆h/L)t so we need to estimate the amount

of water which actually flows for 20 minutes duration, so that is given as A which is nothing

but 4000×10-6m2 and the permeability which is actually given as 4×10-3m,/s and I draw

the gradient is 1.2 so with that it works out to be 23.04×10-3m3 which is nothing but

about 23.04 liters. So in this particular lecture on CPS the permeability and the CDEEP

part three we discussed it about the factors influencing the permeability and we actually

have solved some couple of problems. In the next lecture we will try to discuss

about the flow CPS theories and then some relevant discussions pertaining to CPS theory.