Mod-02 Lec-14 Permeability and Seepage – 3

Mod-02 Lec-14 Permeability and Seepage – 3


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.

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